LIBRARIES “ RETURNING MATERIALS: Place in book drop to remove this checkout from your record; FINES will be charged if book is returned after the date stamped below. DECEfoB $110131 THERMAL CONTRACTION AND CRACK FORMATION IN FROZEN SOIL BY Haaaan M, AL-Mbuaaawi A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR.OF PHILOSOPHY Department of Civil and Environmental Engineering 1988 ABSTRACT THERMAL CONTRACTION AND CRACK FORMATION IN FROZEN SOIL BY Hassan M. AL-Moussawi On cooling, frozen soil will contract if it is not constrained. If elastic soil behavior is assumed, a temperature drop of only 2 or 3 deg. C will produce significant tensile stresses with no observable strains. In tension, the soil will creep along with some stress relaxation. The creep deformation includes elastic, delayed elastic, and viscous flow (permanent deformation). The same soil on.warming will expand, but the permanent deformation will remain. Subsequent cycles of cooling and warming allow the permanent deformation to accumulate followed by eventual rupture in tension at about one percent strain. During periods of decreasing winter temperatures, thermal contraction will increase tensile stresses creating the potential for crack formation, particularly in partially saturated, weaker frozen surface soils. Duplication in the laboratory of field conditions responsible for crack formation requires facilities and equipment not available to most researchers. To simplify experimental work, a series of constant strain/stress relaxation tensile tests were conducted at a constant temperature. In addition, limited data were obtained on stress increase as a function of soil cooling rates for the same saturated frozen sand. Linear thermal contraction/expansion coefficients were determined on duplicate samples so as to permit a more accurate analysis. Preliminary tests on fiber-reinforced sand suggests a technique by which crack formation may be controlled in frozen surface soils. Crack formation in frozen surface soils may occur under two conditions: rapid cooling (severe winter storm) and contraction giving tensile stresses greater than the frozen soil strength, and the accumulation of permanent strain for a number of thermally induced load cycles leading to rupture at a relatively low total strain, less than one percent. Crack susceptibility will be greatest for partially saturated surface soils with lower tensile strengths, including landfill covers and highway subgrade materials. A numerical example illustrates that cracks propagating unstably through the frozen surface soils may extend deeper than the tensile stresses to which they owe their growth. The study shows that crack prevention will require soil enhancement (reinforcement) which will increase the tensile strength, increase the strain at failure, and reduce post-peak loss of strength. In The Name of Allah The Merciful, The Compassionate "O my Lord! Advance me in knowledge" To My Wife Hanan, My Daughter Noor, And My Son Ahmd This dissertation is dedicated to my wife, Hanan Ali Alwahab. Without her encouragement, I would not have undertaken a doctoral program Without her love, understanding, and hard work, completion of this study would not have been possible. vi ACKNOWLEDGEMENTS All praise and thanks are due to Allah, Lord of the Universe, for His merciful divine direction throughout my study. The writer is deeply indebted to his major Professor, Dr. 0. E. Andersland, Professor of Civil Engineering, for his continuous encouragement and assistance, and the time which he devoted most generously to discussions and consultations. Thanks are also due other members of the writer's doctoral comittee: Dr. R. K. Wen, Professor of Civil Engineering; Dr. Larry J. Segerlind, Professor of Agricultural Engineering, Dr. T. V. Atkinson, Professor of Chemistry; and Dr. N. Alt iero, Professor of Engineering Mechanics. Thanks are also due the Ministry of Youth, Baghdad, IRAQ; for financial support throughout my study at Michigan State University, and the Division of Engineering Research and the Department of Civil and Environmental Engineering at Michigan State University for providing the instrumentation and materials which helped make this study possible. Thanks also due Mrs. Sun Yomg Ahn for typing the manuscript. vii I. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS .......................................... vii LIST OF FIGURES .......................................... xi LIST OF TABLES .......................................... xvi LIST OF SYMBOLS .. xviii mmonucnw OOOOOOOOOOOOOOOOOOOO...OOOOOOOOOOOOOOOOOOOOO. I II. Llnnmn mm OOCOCOOOOOOOOOOOIOOOOI.0.00.00.00.00...O. 6 III. 2.1 Deformation behavior of frozen soils ................ 6 1 Thermal Contraction/Expansion PrOperties ........... 6 2 Tensile and Compressive strengths .................. 8 3 Elastic Properties -- E and v .................... 14 4 Time dependent effect (creep) . 18 4 4 l Recoverable and Residual Deformation ............. 21 .2 Fatigue Damage Accumulation ...................... 23 2.2 Frozen Soil Thermal Properties 25 2 2 l Unfrozen Water Content ............................ 25 2 2 2 Thermal Conductivity .............................. 28 2.2.3 Latent Heat ....................................... 29 2 2 4 Apparent Eeat Capacity ............................ 29 2 2 5 Thermal Diffusivity ............................... 32 2.3 Thermal Contraction Cracks in Frozen surface sails .._...O...O..OOOOOOOOOOOOOOOOOOOOOOOO 32 2 3 I origin Of cm craCks O... 00...... I... 00...... O... O. 34 2.3.2 Stresses before and after Fracture ................ 35 2 3 3 ctafl Depth aw spaCing O C C I O O O O O C C O O O O O O C I O O O O O O O O 38 Pmmm“! Enmmmn won .0 O... O... O... O... O... O... I. 40 3. 1 Thermal Contraction/Expansion Behavior . . . . . . . . . . . . 40 Materials and Preparation of Beam Specimens ........ 40 3.1.1 3.1.2 Equipment 8“ Tea: Procedure. OOOOOOOOOOOOOOOCOOOOOO ‘2 3.1.3 Thermal Contraction Coefficients ................... 45 viii IV. 302 Fiber Reinforcemnt OOOOOOOOOOCOOOOOOO...00.0.00... 3 2 1 Materials and Preparation of Compression Specimens .. 3.2.2 uni-“1‘1 compression Test 0.. 00...... O... O... O... O. 3 23 Compression Test Results 3.3 Thermal Tensile and Stress Relaxation Tests ........ 3 3 1 Materials and Preparation of Tensile Specimens ..... 3 3 2 Experimental Equipment ............................ 3.3.3 Test Procedures ................................... 3 3 4 Thermal Tensile Test Results ....................... 3 3 5 Stress Relaxation Behavior ........................ monnuL commmrlons 000......OOOOOOOOOOCCOOOOOOOOOO 4.] mthemtical Fomulation O 0.. O... O... O... O... O... O. 40"] C“.I -me step Functmn OOOOOOOOOOOOOOOOOOOOOOOO 4.1.2 C‘se II - The Linear Functim O O. O... O... O... O... 0. 4.1.3 Tension Crack in an Infinite Medium ............... 4.1.4 Tension Crack at the Free Surface of a sat-infinite Hedi-um .0. O... O... O... O... O... O... O. 4.2 Thermal Contraction and Fracture .................. 4 2 1 The Elastic Problem ............................... 4 2 2 Crack Stress Intensity Factor ..................... 4.2.3 Crack Depth ....................................... 4 2 4. Crack Growth Resistance Curve (R-Curve) 4 2 5 Experimental Method to Determine the churve ....... Plum 0.00......OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO. 5.1 Field Conditions -- Fargo, North Dakota ........... .1 $011 PrOfile halyzed OOOOOIOOOOOOOOOOOOOOOIOOOOOOO .2 Surface and Ground Temperatures ................... .3 Frost Depth Penetration OOOOOOOOOOOOOOOOO 0.0.0.0... 502 merical Results O...OOOOOOOOIOOOOOOOOOOOOOOO00.... 1 Thermal Stresses and Soil Strength ................ 2 cr‘Ck Depth Prediction 00.0.00...0.00.00.00.00...I. .3 Horizontal Stress Relief and Crack Spacing ........ 4 Inelastic Effects after Soil Rupture .............. 5.3 Discussion with Design Implications ............... l Lmdfill cont, OOCOOOOOOOOOOOOOOIOOOOOOOOOOOIOOOOO .2 Highway subgrade sails .000......OOOOOOOOOOOOOOOOOO 47 47 SO 67 67 74 74 89 99 99 104 108 111 114 121 121 123 130 132 139 140 140 140 143 152 161 161 162 172 175 176 176 178 VI . SUMY AND COKLUSIONS . . . . . . 6.1 6.2 6.3 81 BLIOGRAPHY smty OOOOCOOOOOOOOOOOOO...00......0.00.00.00.00. amelusion. OCCOOOOCOOCOCOOOOOOOOOOOOOOOOOOOOIOOOOO hemwed ResearCh O...O...OOOOOOOOOOOOOOOOIOOIOO ”mu “D‘ta OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO. A. 8. C. D. Thermal Contraction Measurements Fiber Reinforcement Tests Thermal Tensile and Stress Relaxation Tests .. Temperature Data, Fargo, ND 182 182 184 187 189 194 194 196 210 215 Figure LIST OF FIGURES Page Effect of Total Moisture Content on Unconfined Compressive Strength of Frozen Fine Sand at -12 °C (metalmd’1987). 0.0....................OO ..... 0. lo Compressive Strength vs. Strain Rate (Br.88‘ndAnder81and’ 198]). 00.00.0000...OOOOOOOOOOOO 12 (a) Eon-attenuating Creep Deformation and (h) Attenuating Creep Deformation as the Sum of the Components; (c) Attenuating Deformation 7); (d) Steady-State Deformation (Y1 a) Progressive Deformation (YIII), (Gyalov, 1986). ..... 19 Recoverable, Ye , and Residual Yp, Components of Deformation as Presented on (a) The Creep Curve and (b) The Stress-Deformation Graph (Vyalov, 1986). ........................................ 22 Schematic Representation of the Formation of an Ice Hedge According to the Contraction - Crack Theory. Width of Crack Exaggerated for Illustrative Purposes (Lachenbruch, 1963). ......................... 36 Beam Mold and Template for Preparation of Contraction/Expansion Specimens. ...................... 41 Low Friction Beam Support Consisting of 9 mm Diameter wood Dowels Placed on a Styrofoam mud. ....................................O00.0.0.0... 43 The 10-Inch Hhittemore Strain Gage. ................... 44 Effect of Temperature on the Coefficient of Thermal Contraction for Frozen Sand, Ice, and Quartz. .......... 46 Thermal Contraction/Expansion Behavior of Frozen qu‘rtz sand.. ..................................0 48 Effect of Density and Temperature on Coefficient of Thermal Contraction of Ice. ........................ 49 Mdified Triuial cell. 0.0...........O....O0.00.00.... 52 Diurnaf Test sync”. .0......I...........0......I0.. 55 xi 3-10 3-11 3-12 3-13 3-14 3-15 3-16 3-17 3-18 3-19 3-20 3-21 3-22 Uniaxial Compression Test Set-0p. (8) Loading Unit and Constant Temperature Baths, (b) Data Collection Equipment. Influence of Fiber Volume Fraction on Compressive Strength of Fiber-Reinforced Frozen Sand (steel fiber gage 17). ........ Influence of Fiber Volume Fraction on Compressive Strength of Fiber-Reinforced Frozen Sand (steel fiber gage 24). ........ Influence of Fiber Volume Fraction on Compressive Strength of Fiber-Reinforced Frozen Sand (steel fiber gage 28). ........ Influence of Fiber Diameter on Compressive Strength of Fiber-Reinforced Frozen Sand (fiber length - 0.25 inch). ... Influence of Fiber Diameter on Compressive Strength of Fiber-Reinforced Frozen Sand (fiber length - 0.25 inch). ... Influence of Fiber Diameter on Compressive Strength of Fiber-Reinforced Frozen Sand (fiber length - 0.50 inch). ... Influence of Fiber Diameter and Gage Length on the Compressive Strength of Fiber-Reinforced Frozen s‘nd. ......O........... Influence of Fiber Aspect Ratio on the Unconfined Compressive Strength of Fiber Reinforced Frozen Sand. ... Aluminum.Mbld Used in Preparation of Frozen Tensile Test Samples. ......... Tensile Test Sample Cross-Section With End Plates, Screw Rods, and Connectors in Position. ........ Tensile Test Specimen End Plate with Fabrication Details. .......... Freezer Box Modification for the Thermal Contraction Tests Including Insulation Board Location and Metal Screen for Support of the Dry Ice. ............... Diagram Showing Frozen Tensile Sample in Position for Stress Relaxation Tests or Loading by Thermal Contraction. xii 56 60 61 62 63 64 65 66 68 69 70 71 73 75 3-23 3-24 3-25 3-26 3-27 3-28 3-29 3-30 3-31 3-32 3-33 3-34 3-35 3-36 3-37 3-38 Equipment for Tensile and Stress Relaxation Tests. (a) Freezer Box and Load Frame, (b) Measurement of Axial Deformation. Frozen Sample with Thermistors in Freezer Box After Mounting on Load Frame. ...... Frozen Sand Sample (No. 4) after Tensile Failure. ............................ Temperature versus Time for Sample 4, Showing “use in dT/dt. ................... DevelOpment of Tensile Stress with Time for sm1e 4. ...................... Development of Tensile Stress in Saturated Frozen Sand over a Period of Time for Three coaling Rates. ...................... Cooling Curves for Three Saturated Frozen Sand Samples. Partial Control of dT/dt Achieved by Change in Sample Diameter and Spacing Between Dry Ice and Sample. . Tensile Stress Increase in Saturated Frozen Sand with Decrease in Temperature for Three Cooling rates -- -4.3 'C/min, -3.0 'C/min, -0063 .CImine oesoosesaeseeeaeeeessee Development of Tensile Stress in a Saturated Frozen Sand with Time for a Cooling Rate 0f-00021.C/mine ................... Summary of Stress/Temperature Cooling Curves for Saturated Frozen Sand. .......... Stress and Strain vs. Time for the Tensile Test on Frozen Saturated Sand, Load Cycle 1. Cyclic Strain (2 cycles). .......... ................... Tensile Stress Relaxation in Frozen Saturated Sand for 2 Load Cycles. ' ............ Cyclic Strain (4 cycles). ........... Stress Relaxation Curves at A Constant Temperature (-15 °C) for Frozen Saturated Quartz Sand After Cyclic Loading. ... Stress Relaxation Curves for Frozen Saturated Sand at -15 °C over 4 Loading Cycles. xiii 76 77 79 81 82 84 85 86 87 88 91 92 93 94 95 96 3-39 Load Cycle Effect on Failure Strain and Rupture Stress for Saturated Frozen Quartz 88“ at -15 °C. ............................0 ..... ...... 98 4-1 Coordinate System and Position of the Thermal craCR. ......O....................O............101 4-2 Transformation Mapping of the Exterior of an Ellipse into the Exterior of A Unit Circle (Adapted from Muskhelelishvili, 1963). ...... ..... ..... 105 4-3 Case I: Step Function (Stress Distribution). .......... 113 4-4 Case 11: The Linear Function (stre's Diatribution). ..................00............. 113 4-5 Reduction of The Solution for A Step Distribution of Stress on the Walls of A Crack in a totally Infinite Medium to the Corresponding Solution for a Semi-Infinite Medium.(Lachenbruch, 1961). ............................ 115 4-6 Tangential Stress on the Free Surface (x - 0) Due to A Step Distribution of Surface Stress on the Halls of a Crack (y - 0 +, 0 (x 0. The maximum principle stress will be represented by T(x), which is directed parallel to the y axis and varies only with x. Mathematical formulation, to study the stress around the crack tip, was based on the Airy function with utilization of complex variables to solve the differential equation (Muskhelishvili, 1963). This study can be visualized by taking the crack as two parallel lines in an infinite medium, then by pulling the centerline to the outside the shape of the crack will change to an ellipse. Working with the ellipse is difficult mathematically. Hence Muskhelishvili (1963) mapped the ellipse to a 4 more convenient function, a circle. By solving the problem.for circular cracks, mapping them.back to an ellipse, and taking the limit of the minor axis of the ellipse, the problem is returned to the square one which is two parallel lines of cracks. To obtain an approximate solution for a semi-infinite medium, the superposition concept was applied to the exact solution of an infinite medium. A complete solution for crack depth, spacing, and horizontal distance for stress relief is dependent on shape of the thermal stress distribution.with respect to depth. Both a step function and a linear function for stress distribution are represented, which can be used to solve the complex problem. In the numerical example (Chapter V) a step function appears to describe the cracking phenomena in a frozen cohesionless soil. The numerical example (Chapter V) shows that after a relatively short period of time and for temperatures typical of mid-winter for Fargo, ND, thermal stresses in the saturated frozen sand are higher than tensile strengths of the same sand to a depth of 30 cm. This indicates that a crack.would form.and would proceed down into the frozen sand. With the water table at some depth below the surface, in many cases below the frost depth, the granular surface soils will be only partially saturated. Soil tensile strengths are dependent on the degree of ice saturation. A lower soil strength will allow the thermal crack to penetrate to a greater depth. For the numerical example, initial soil temperatures were in the 0 to -15 '0 range, hence residual tensile stresses would remain from cooling prior to the winter storm. The addition of these tensile stresses along with lower tensile strengths 5 due to partial ice saturation suggests that the crack would continue down through the frozen sand to the frost line at a depth of 205 cm. With formation of a crack, tensile stresses would be reduced to a horizontal distance about equal to the crack depth. Fracture theory provides estimates on the magnitude of this stress reduction over the zone of stress relief. Crack spacing will be dependent on this stress reduction and any variation in strength of surficial materials from place to place. All cracks can be visualized as initiating at zones of weakness or "flaws". Duplication in the laboratory of field conditions responsible for thermally induced loading requires facilities and equipment not available to most researchers. To simplify experimental work, a series of constant strain/stress relation tensile tests were conducted at a constant temperature. 'In addition, limited dats.were obtained on stress increase as a function of soil cooling rates for the same saturated frozen sand. Linear thermal contraction/expansion coefficients were determined on duplicate samples so as to permit a more accurate analy- sis. Preliminary tests on fiber-reinforced sand suggests a technique by which crack formation may be controlled in frozen surface soils. The thermal contraction and crack formation problem to which this study has been directed, has application to covers and liners for hazardous waste landfills, to highway subgrade soils, and to frozen surface soils for dams, dikes, and other hydraulic structures exposed to severe winter temperature changes. Conclusions from this study, given in chapter VI, involve thermal cracking, crack depth, thermal contraction coefficients, and soil property enhancement. Several recommendations for future research are provided. CHAPTER II LITERATURE REVIEW 2.1 Deformation Behavior of Frozen Soils Mechanical properties determine the behavior of frozen soils under applied tensile forces and loads. Three basic types of deformation are involved: elastic, plastic and viscous flow (creep). A temperature My“! decrease*will cause the soil to contract; a subsequent temperature increase‘will cause the soil mass to expand in volume. The response of the frozen soil will depend on soil type, ice content, degree of ice saturation, dry density of soil solids, and temperature. Rate of loading and time dependent effects will alter the frozen soil response. All these factors, which are reviewed in this section, are relevant to thermal contraction behavior and possible rupture in frozen surface soils. 2.1.1 Thermal Contraction/Expansion Properties The linear thermal coefficient of expansion is expressed by the relative increase in length with increasing temperature. Thus, a - A—L (2-1) AT Lo where a is the linear thermal coefficient of expansion, L0 is length at some reference temperature, and AL is the change in length due to s 7 temperature change AT from some reference temperature. If the material is isotrOpic, i.e., exhibits the same thermal expansion in every direction, then a - 3/3 ' (2—2) where B is the volumetric coefficient of thermal expansion. The volumetric thermal coefficient of expansion is related to the volume of a substance, which in turn is at its minimum value at absolute zero -460 'F (-273.33 'C). This volume is increased as the temperature becomes warmer. The relative increase in volume with increasing temperature is expressed by the volumetric coefficient of thermal expansion. The average volumetric thermal expansion over a temperature interval AT is expressed by: s - Av/(A'r v0) (2-3) where B is the volumetric coefficient of thermal expansion, V0 is the volume at some reference temperature, AV is change in volume due to a temperature change AT from the same reference temperature. There are many different soil types, each with its own mineral and organic composition. Other factors include crystal orientation within the minerals and texture, which are responsible for a different thermal coefficient of expansion in different directions, Jones, et al. (1968). . Based on convenience, several methods have been used to measure the thermal coefficient, including the dilatometer, optical interferometer, strain gage, and fulcrumrtype extensometer. Books and Goetz (1964) used the Whittemore strain gage on bituminous concrete for temperatures from +30'C to -30'C. This gage has advantages over the dilatometer in that it is easy to Operate and requires little operator experience in order to obtain reproducible results. The Whittemore strain gage was used in this study to determine the coefficient of linear contraction and expansion for frozen Ottawa sand. 2.1.2 Tensile and Compressive Strengths Frozen soil tensile and compressive strengths depend on several variables including dry density, degree of ice saturation, unfrozen water content, temperature, ample size, rate of loading, and strain tat/e. The study by Haynes and Karslius (1977) on frozen silt, showed that both canpression and tensile strengths depend on temperature, machine speeds (rate of loading), and unfrozen water content. They used two machine speeds, 4.23 cm/sec and 0.423 cm/sec, along with taperatures true 0 'C to -56.7 ’C to show that compressive strength increased by about one order of magnitude and the tensile strength increased by one-half an order of magnitude from the warmest to the coldest temperature. For a higher rate of loading they observed a smaller time to failure for both compression and tension tests. Kaplar (1971) showed that strength for soilsof normal unit weight with less than 1002 saturation always i9¢,¥9§99§..!i99,...1.-9‘din8 rate. v a\'~-‘c'-" ..r—p - “—- The effect of total moisture content on the unconfined compressive strength of a fine sand at -12 'C and an axial strain rate of 2.2 x 10-6 S"1 is shown in Figure (2-1). At low water contents, less than 51, the sand behavior and strength are similar to those for dry unfrozen sand. Strength increased ”gapidly, "unwilliPEFF‘" in the ice matrix, approaching a maximum strength value at full ice saturation and maximum dry density. With higher water contents, a decrease in dry density is 9 responsible for a decrease in the unconfined compressive strength. At a moisture content close to 582 the soil particles become suspended in the ice matrix with little or no particle-to—particle contact. The compressive strength is now dependent on the behavior of the ice matrix. A small decrease in strength is noted as the soil particle volume decreases from about 422 at a moisture content of 582 to zero at a moisture content of 1002 (Andersland, 1987). Haynes, et a1. (1975) showed that at a temperature of -10°C, the compressive strength for frozen silt is very sensitive to strain rate, increasing ten times over a strain rate range of 1.2 x 10.4 sec.1 to 2.9 sec-l. The tensile strength was relatively insensitive with little change indicated. Bragg and Andersland (1981) found that at strain rates above about 10.5 sec"1 the compressive strength for frozen sand was essentially independent of strain rate as shown in Figure (2-2). However, at strain rates below 10.5 sec.1 the compressive strength increased linearly with strain rate according to a power law of the form 1/n coax - A t (2-4) where n is the creep parameter, A is a temperature-dependent proof stress evaluated at a strain rate of 1 sec-l, Q is the strain rate (sec-l), and amsx is the maximum uniaxisl compressive strength. Values of A and 1/n reported by several investigators for their data are summarized in Table (2-1). 10 .Anmm_ .uoeaauomnwnuoumaoo voomucouca no uoouooo anon-«ox annoy mo uouuuu u_I~ ouomwh N .uoouooo ououmaoz Hmuoa cog cm on oh so an ow on em cm a _ _ n . _ _ A _ _ o umxmm O vomamuopo< can noomcmoou . 14 mg ooo~ ooo~ :. coon IL oooe I. count ououmuooaoa :. coon mum ouozx~.~ «use cashew F 1 88 poem «can _ r ._ .F A. _ P _ _ egn‘qasuazns SAISSGJdmOO paurguooun 11 Table 2-1: Parameter A and 1/n for Equation 2-4 for Silica Sand with Similar Grsdation and Density (Bragg and Andersland, 1981). Temperature ('C) A 1/n é (sec-I) Source -2 341.61 0.303 5 x 10'6 to 8 x 10'5 Bragg (1980) -3 _ 0.105 1.7 x 10.4 to 2 x 10"2 Sayles and Epsnchin (1966) -3.85 _ 0.10 1.7 x 10'5 to 1.7 x 10"2 Sayles (1974) -s.s 28.76 0.09 2 x 10"7 to 2 x 10"3 Baker (1978) -6 47.37 0.115 s x 10" to 8 x 10'5 Bragg (1980) -6 _ 0.073 1 x 10'7 to 1 x 10"2 Parameswaran (1980) -6.5 _ 0.092 1.7 x 10" to 2 x 10'2 Sayles and Epsnchin (1966) -10 62.31 0.119 s x 10'6 to 8 x 10‘5 Bragg (1980) -10 __ 0.071 1 x 10'7 to 1 x 10'2 Parameswsran (1980) -10 _ 0.094 1.7 x 10.4 to 2 x 10.2 Sayles and Epsnchin (1966) -15 44.31 0.079 5 x 10'5 to 8 x 10"5 Bragg (1980) -15 0.079 1 x 10"7 to 1 x 10’2 Parsmeswaran (1980) ‘7 12 .A_om_ .vomduuovc4 one mwaumv comm cannon .a> nuwoouum o>wuauuaaoo “~1~ «momma A loom v Quad cwmuum Hmwx< a ~1oe m1oe 31o_ m1o~ o1o~ ~1oz «« 4 q a —:q.~ a a . .—quqq- -.q - —-q-udd - q —-<-dd- u q 1 o.me 1 AV 1 1 u. on 1 mu 1 n 8.0 1 O - n O ” C..p. _ Pl..r. . rthph. .1 1rCE.-. _ 7:.P.. . coo p14 1: N m/un ) qazazus aarssazdmoo <3 -o ( Z 13 Bragg and Andersland (1981) gave three equations for the effect of sample size on compressive strength, failure strain and initial tangent modulus for frozen sand. They include the following: 0' I 12.06 ' 0.24 ‘9 (2-5) at - 3.07 + 0.014 0, for T - -6 'C and (2-6) é - 1.2 x 10.4 sec-1 Bi - 1.703 + 0.231 <1), for T - -6 °C and (2-7) é - 1.2 x 10“ sec-Dl where a is compressive strength (MN/m2), at is failure strain (2), <15 is sample diameter in (cm), and Bi is initial tangent modulus (GN/mz). Parameters for the above equations were based on the least squares method. It is clear from the above equations that o, a and E vary 1: linearly with sasple di'aaeter while the compressive strength decreased with increasing diameter for both a and Bi. The initial tangent t modulus increased with increasing ample diameter. Pet tensile strength of frozen silt, am’ reported by Yuanlin and Carbee (1987), was found to be a function of strain rate, temperature, and density. A simple power-law equation gives 0‘m as am - 1: @750“ (M) where t! is a reference strain rate taken as 1 sec'l, 3: is strain rate, 11 is a dimensionless parameter, and k has dimensions of stress. Both k and n varied with temperature, density, and strain rate. A summary of their values are listed in Table (2-2). 14 Table 2-2: Values of k and n in Equation 2-8 (Insulin and Carbee 1987). —_:—‘:—: 1 J Temperature Density Strain Rate k n ('c) (g/cm3) e (3") (kg/cmz) ' 1.08 - 1.12 5 x 10" - 5 x 10'7 148.6 0.142 -5 1.2 - 1.26 1 x 10'2 - 1 x 10"5 143.4 0.151 1 x 10'5 - 6 x 10"8 48.6 0.068 1.36 - 1.41 1 x 10"3 - 7 x 10’7 105.4 0.134 -2 1.20 - 1.26 1 x 10"2 - 1 x 10'6 103.2 0.185 2.1.3 Elastic Properties, E and V Linear relations between the components of stress and strain are described by Hooke's law. The modulus of elasticity E is a linear relation between stress and strain, thus E - -- (2-9) where E is modulus of elasticity (F/Lz). a is stress (F/LZ), and e is strain (L/L). Strain extension of frozen soils in the Xkdirection is accompanied by lateral strain (contraction) in both the y, and 2 directions. This is known as Poisson's effect, which leads to Poisson's ratio v. For isotropic materials v--—'—‘---— (2-10) :2: J.- I. , .6! to! Ii 5:48 the: 135C 51: 15 where 6‘ is the strain component in the erirection, 6y is the strain component in the y-direction, and 62 is the strain component is the z- direction. Young's Modulus, E, for frozen soils is many times greater than for unfrozen soils. In frozen soils E depends on soil composition, void ratio, ice content, temperature, and external pressures (Andersland and Anderson, 1978). Tsytovich (1975), introduced an equation for the modulus of elasticity as follows: n-a+80+ye’+ 663 (2-11) where a, B, y, and 6 are experimentally determined parameters. The absolute value of the negative temperature 6 of frozen soil is in degree Celsius. Tsytovich approximated equation 2-11 with a power law equation of the form E - a + Ben (2-12) where n is a non linear parameter, and 8 is a function of normal stress a. For temperatures down to -10°C, n can be assumed equal to unity, hence E - a + 89 for 9 §.- IO'C (2-13) Values for E decreased with increasing external pressures and increased with decreasing temperatures. Haynes and Karaliua (1977) showed that frozen soils gave their highest modulus for higher machine speeds and consequently lower strains. The modulus values at 50 2 strength were slightly lower than the initial tangent modulus. Also, tension tests consistently gave (Isl “- 253. 1:25; nu, . 0 I212. 3359? CC 51 fr : '5‘: 16 higher modulus values as compared to compression tests, and higher machine speeds produced higher E values for both tension and compression tests. Yuanlin and Carbee (1987) in their investigation on tensile strength of frozen silt reported on the effects of temperature, strain rate, and density. They showed that the initial tangent modulus Ei is a function of strain rate and temperature. For mediumrdense samples, they observed a relation between E1.. and the modulus E at 50! strength which can be expressed as Ei - 6.05 E + 3.26 x 104 1 (2-14) where Bi and El are in kgf/cmz. Poisson's ratio for three soils are summarized in Table (2-3). These values were checked by equation V- -1 (2-15) 2G using Young's moduli measured from compression tests, and shear moduli from tension tests on cylinderical frozen specimens (Tsytovich, 1975). It is clear from Table (2-3) that temperature has a significant influence on Poisson's ratio, and that it will decrease with a decrease in temperature (Tsytovich, 1975). Other methods of finding the modulus of Elasticity and Poisson's ratio involves the use of compressional and shear waves in the frozen soil. Using equation 2-16 values for v, E, and C may be computed (Roethlisberger, I972), 17 Table 2-3: Poisson's Ratio for Frozen Soil (Tsytovich, 1975). water T 0 v content Temperature Stress Poisson's Soil type I 'C kN/mz ratio Sand (71 pass 0.25mm, 19.0 -0.2 196 0.41 1.41 pass 0.05am) 19.0 -0.8 588 0.13 Silt (64.42 pass 0.05am, 28.0 -0.3 147 0.35 9.22 pass 0.005mm) 28.0 -0.8 196 0.18 28.7 -4.0 588 0.13 Clay (50 + Zpass 0.005mm) 50.1 -0.5 196 0.45 53.4 -1.7 392 0.35 54.8 -5.0 1176 0.26 1 v"z - 2v: V I — -—L-—-2— (2‘16) 2 v: - v where v u E I.“ ’ v u G p 1’ 0 (1w) (1-2v) 8 v’ p v is Poisson's ratio, 0 is density, E is modulus of elasticity, G is shear modulus, VP is compressional wave velocity, and V8 is shear wave velocity. 18 2.1.4 Time Dependent Effects (Creep) A material under constant load will deform with time. This phenomenon is called creep. A test carried out at constant load is therefore called a creep test, and the measured strains are called creep strains. The slope of the creep curve (strain versus time) at any point determines the creep rate. Yyalov (1986), Andersland and Anderson (1978), and Ladanyi (1972), defined creep deformation as the sum of a hypothetically instantaneous deformation, 70’ which occurs immediately after load application and a deformation which develops with time 7(t), thus, 7 - Y0 + Y(t) (2-17) The deformation 7(t) associated with the process of creep, is illustrated in Figure (2-3). In addition to instantaneous deformation, creep displays three stages: (I) non-steady state creep (segment AB), (II) steady state flow (segment BC), and (III) progressive flow. During stage I, defbrmation.develops at.a decreasing rate and reaches a minimum value. During stage II the deformation rate remains more or less constant, 7 . coast; and is sometimes called the stage of viscoplastic flow. At stage III, defbrmation.develops at an increasing rate until failure occurs (brittle or viscous). This final process is referred to as the failure stage. The duration and effect of any particular stage of creep varies with the soil type and load. Higher.loads correspond to a shorter stage II and a more rapid development of failure, stage III. Referring to Figure (2-3) the transition time from stage to stage will be different: 19 . HHH Hmaomm_ .>o~mh>v A »v nowumanomoa oawumeuwoum “av «A rv mowumauouon oumum1hvmeum Adv "Aurv nomueauouen wmmumanouu< on “monoconaoo «no no flow emu me comumauouon guano wcwumnoouu< any one aowuoauomon ooeuo mawumocouuo1noz Amy on Aev “mIN ouowmm 20 the beginning of steady-state viscoplastic flow is denoted by tea; the transition to the stage of progressive flow by tpr' and the time of failure by t Soil deformation with time can be represented by the fail' 1:88 + Y 0 II tpr + tfail t YIII as Y - YO + YI (2-18) tpr where Y0 is hypothetically the instantaneous deformation which occurs immediately on load application, t - 0, YI is the deformation deveIOping over the period 0 < t S.t' is the deformation developing over the s’ YII period tss < t < t is the progressive deformation developing pr’ and YIII over'the period tpr < t < tfail' In most cases, it is undesirable to allow soil to Operate under the conditions of stage III. Therefore is comnonly excluded from.the YIII analysis, and creep deformation is considered as the sum.of the instantaneom and visc0p1astic defonaations, thus Y I YO 4]» ‘YI + Yn (2.19) Sayles (1973), reporting on triaxial creep tests on frozen Ottawa sand, found that at low stresses the rate of strain decreased continuously'with time such that logarithmic plots of strain versus time data lead to the expression: l/n l/n é - [tn/(tn) 1 (t) (2—20) 0 where e is the rate of strain at time t greater than zero, SR is a corresponding strain rate, t is a reference time greater than zero, and R n is the slope of the straight line on the plot. 3511 ”I "W give her W1 Of” 369 21 Eckardt (1982) studied the creep behavior of frozen soil in uniaxial compression. The lower and upper boundary conditions for his tests on sand and clayey, sandy silt included: Temperature T: -40°C _<_ T 5. -5'C Time under load t3: 1 h S tB _<_ 10,000 h Uniaxial stress a]: -300 kPa i 01 _<_ 10,000 kPa Failure deformation cf: -21 3 cf 5 71 Defamation rate 6%: l x 10-6/h i é i 5 x 10-3/h He found that creep curves for both compression and tension tests can be given in the form of axial deformation, thus Ill 6:] - ol/k(T, t3) (2-21) where O is the axial creep stress, :11 is a material constant, and K is a par-ate: dependent on temperature T, and t is the loading time. B 2.1.4.1 Recoverable and Residual Deformation When a soil specimen is unloaded, as shown in Figure (2-4), some of the deformation will be of the recoverable kind. This recovery, from an initial hypothetically instantaneous deformation yo, is initiated as soon as the load is removed and proceeds until the deformation is either totally or partially gone. Total recovery is the case when the initial deformation is of the purely elastic type, YO - Ye, so that segment 0-2 of the loading curve in Figure (2-4) is the same as segment 3-4 of the unloading curve. Partial recovery takes place when the initial deformation consists of an elastic deformation (segment 0-1), and a plastic deformation (segment 1-2), 22 .Aomm_ .>o~mh>v guano nowumauomon Imuouum any Any one o>u=o oeuuo any Amv no mounemoum mo oomumauouon mo uuamnooaou .9» Hmsvmmom one .0» .ounmuo>ooom «e1w ouawfim Russo—wagons: u .059 8 1' ‘ sseius maqvmon . " wmwemodaa .5. axw oxw 3v T: L ‘ nornsmaopa 23 Consequently, the body recovers only from the elastic deformation 73. The recovery from deformation Y1 in Figure (2-4) takes place only partially with time (segrent 4-5), the deformation consisting of an elastic segment Y; (segnent 5-6), and a plastic after effect 7'; (sepent 5-7) , thus .- e p - Y: Y1 + 71 (2 23) The deformation at stage II and III, the steady-state and progressive flow, respectively, are plastic and totally irrecoverable, thus _p _p - Yu Yu ““4 YIn Y111 (2 24) In general, soil creep deformation at any time t consists of a recoverable deformation and a residual deformation Vaylov (1986). 70 _ Ye+ YP . (2—25) 2.1.4.2 Fatigue Damage Accmulation In a body subjected to cyclic loading, the process of fracture starts with microcracks nucleat ing and growing at the initial stage of the process. Then, macrocracks begin to form and propagate, which ends in fracture of the body. For a body with considerable initial defects (such as notches, cracks, and/or inclusions) the early ("latent") stage of fracture may be very short or even non-existent. The relation 24 between time for the latent stage and time for macrocrack propagation depends on the geometry of the body and on the nature of defects. The time for crack growth may include 10 to 80 percent of the service life of the specimen. If the body is sufficiently homogeneous, and there is no stress concentration, the latent period may be quite long. It is difficult to estimate damage accumulation in the latent stage. It is also difficult, as a rule, to determine the initial time for macrocrack formation. In fatigue theory, damage accumulation is usually considered with respect to the entire fracture process without distinguishing the stage of fatigue (Rachanov, 1986). In tests with uniform load cycles the cumulative effect eventually produces fatigue failure, unless the load is below the prevailing fatigue limit. When loading and unloading doesn't occur in uniform cycles but in an irregular manner, the cumulative effect of these events may also produce fatigue failure. The term "cumulative damage" refers to the fatigue effects of loading events other than uniform cycles (Fuchs and Stephens, 1980). Fatigue damage is defined as 01 - 11m" (2-26) where N is the current number of cycles, and N’ is the number of cycles to fracture. Under conditions of cyclic loading, the analysis can generally be based on the principle of a linear summation of damage. This principle, as applied to fatigue fracture, was formulated by Palmgren in 1924 and by Miner in 1945. The percent damage already done to the sample or fraction of life used at a certain stress level can be formulated by 25 211i Reversals applied at oai I (2'27) 2N . Reversals to failure at O . f1 a1 Cycle Ratio - which equals the fraction of life used at Oai' Failure (Kachanov,1986; Fuchs and Stephens, 1980) occurs when 2111i 2 - 1 (2—28) i znfi 2.2 Frozen Soil Thermal Properties Thermal properties of frozen soil play an important role in their mechanical behavior. Thermal conductivity, latent heat, apparent heat capacity, and thermal diffusivity are essential in heat flow calculations involving frozen soils. Thermal prOperties of a frozen soil are not constant, but are a function of temperature, dry density or porosity, water content, mineral composition, additives, organic factors, and direction of heat flow. 2.2.1 Unfrozen Hater Content The amount of frozen and unfrozen water in soil has a significant effect on thermal conductivity and the specific heat of frozen soil. Unfrozen.water will reduce the tensile strength of frozen soil by reducing the ice content and by creating1weak zones adjacent to the soil particles. 26 Anderson and Tice (1972) predicted the unfrozen water content wu such that "u.' f(S, 6) (2-29) where S is specific surface area.(m2/g) and 9 is the temperature in degrees below zero degrees Celsius (32°F). Anderson et al. (1973) proposed an equation for prediction of unfrozen water contents as B wu - 09 (2-30) where'wu is the unfrozen water content, a and B are parameters characteristic of each soil type (Table 2-4). and 9 is the temperature in degrees Celsius below zero. The values of a and 8 obtained by Anderson et al. (1973) was a function of specific surface areas for each soil. They found that 1n n - 0.5519 ln 8 + 0.2618 (2-31) with a correlation coefficient of 0.90 and, In (-B) - -o.2640 ln s + 0.3711 (2—32) with a correlation coefficient of 0.86. This gives the expression for w as u -0.264 1n wu - 0.2618 + 0.5519 lnS - 1.4498 In 0 (2-33) \-——‘-.—I 01’ 0.264 wh - Exp [0.2618 + 0.5519 lnS - 1.4498' ln 9] (2-34) Equations 2-30, 2-31, and 2-32 are utilized in Chapter V to compute the unfrozen water content in frozen soil. 27 Table 2-4: Experimental Values for a, 8, Specific Surface Areas S and Correlation Coefficients r (Anderson et al., 1973). Surface ‘ Soil Area 2 a B r S - m /g Basalt 6 3.45 -1.13 0.96 Rust 10 11.05 -0.80 0.96 What Lebanon.gravel < 100 u 18 3.82 -0.64 0.94 Limonite 26 8.82 -0.83 0.99 Fair banks silt 40 4.81. -0.33 0.96 Dow field silty clay 50 10.35 -0.61 0.94 Kaolinite 84 23.80 -0.36 0.90 Suffield silty clay 140 13.92 -0.31 0.98 Hawaiian clay 382 32.42 -0.24 0.93 Uyoming bentonite 800 55.99 -0.29 0.72 Umiat bentonite 800 67.55 -0.34 0.83 9'5, 5‘. N- 1161'. 1(he- 28 2.2.2 Thermal Conductivity Thermal conductivity is the amount of heat passing per unit time through a unit cross-sectional area of the soil under a unit temperature gradient applied in the direction of heat flow (Farouki 1981, 1982). Thus k - q~ (2-35) A (T2 - T1)/LO where k is thermal conductivity (W/m.°C), A is cross-sectional area (m2), AT - (I: - T1) is drop in temperature ('C), L0 is length of the element (m), and q is the amount of heat in (W). In general, thermal conductivity of soil depends on its density, water content, mineralogical composition, temperature, solid, liquid, vapor constituents, and the state of the pore water. Kersten (1949) has determined the thermal conductivity for a wide range of both frozen and unfrozen soils at different water contents. Thus for unfrozen-sandy soils 0.6243yd kn - 0.1442 [0.7 log w + 0.4] 10 (2-36) and for frozen-sand soils kf - 0.01096 (1010'8"676‘+ 0.00461 (10)0'9115Yd w (2—37) where k is thermal conductivity (W/m,°K), Yd is soil density (g/cm9), and w is water content. .ewerage values given by equation 2-36 and 2-37 will be used in Chapter V, to compute frost depth penetration, and ground temperatures. 29 2.2.3 Latent Heat Heat energy that is released or absorbed when a material undergoes a change of phase is called latent heat. It is expressed by the number of calories of heat required to effect a complete change of one gram of the material from one phase to another. Thus, for frozen soil L - pd wL' (2-38) and for partially frozen soil L - pd w(1-wu)L' (2-39) where Wu - wu/w is the ratio of unfrozen to total water content of the soil, L is latent heat of fusion (J/m3), I.’ is latent heat of water (333.7 kJ/kg), pd is dry density of the soil (kg/m3), w is the soil water content (percent by weight of solids), and wu is the unfrozen water content (percent by weight of solids). 2.2.4 Apparent Beat Capacity The unfrozen water content decreases exponentially with temperature, gradually releasing latent heat and continuously changing the heat capacity (Johansen and Frivik, 1980). To account for this change the heat capacity of frozen soil can be expressed as the am of the heat capacities of the main constituents. In frozen soils the liquid-solid phase change is gradual and continual with change in temperature, therefore the term specific heat capacity is not appropriate. Use of apparent heat capacity is more appropriate. Therefore, the apparent heat capacity can be expressed as the sum of an appropriate term for each of these factors plus a term to account for 30 the latent heat of phase change that is continually being given off or absorbed. Thus 2 1 3w Ca - C8 + C1 (w-wh) + Cuwu + L u dT (2-40) KT 5T T1 where Ca is the apparent heat capacity of frozen soil, C8 is the heat capacity of the dry soil matrix, Ci is the heat capacity of ice, Cu is the heat capacity of the unfrozen water, w is the total water content, wu is the unfrozen water content, T is temperature, and L is latent heat of the liquid-solid phase change (Hoekstra, 1969). Volumetric heat capacity is the energy required to raise the temperature of a unit volume of soil by 1°C. For unfrozen soil Cvu - Yd [ems + C w w/IOO] (2-41) for frozen soil Cvf I Yd [cms + Cmi'w/IOO] (2-42) and for partially frozen soil cvf - yd [0“. + cmi (1+Wh) w/100] (2-43) where Uh - wu/w is the ratio of unfrozen to total water content, CI" is the mass heat capacity of soil solids (0.71 J/g 'C), Cu. is the mass heat capacity of water (4.1868 J/g 'C), Cmi is the mass heat capacity of ice (2.1 J/g 7C), and Yd is dry density of the solids (g/cma). 31 Kay and Goit (1975) found that the mass heat capacity varies linearlwaith temperature from.-73°C to +27°C (200 to 300°K). Their measurements showed that C - C - mT (2-44) ms p where Cm.‘ Cp is the mass heat in cal/'Kg, m.is a proportionally constant, and T is absolute temperature in °K. They found that specific heat decreased with decreasing temperature over their range of temperature measurement. Table (2-5) shows some typical values for Cp. Table 2-5: Correlation between Mass Heat and Temperature for Representative Soil Materials (Ray and Goit, 1975). Material Equation Correlation coefficient Clay-Na bentonite cp - 8.4 x 10" r - 0.342 0.984 SandrSauble Beach cp - 5.2 x 10“ r + 0.0247 0.970 Silt-Conestogasilt 1mm 01, - 4.9 x 10" r + 0.0369 0.995 Peat-Sphagnum 41: fiber cp - 1.15 x 10'3 r - 0.0245 0.986 Ice c - 1.76 x 10'3 r + 0.0228 0.999 32 2.2.5 Thermal Diffusivity Thermal diffusivity is the ratio of thermal conductivity k to volumetric heat capacity pc. Thus. a - k/pc - k/cv (2-45) where o is thermal diffusivity (mg/s), p is density (kg/m3), k is thermal conductivity (W/m °C), C is mass heat capacity (J/kg °C). For unfrozen soil on - ku/Cvu (2-468) and for frozen soil of - kf/Cvf (2-46b) where the subscript u, and f refer to unfrozen and frozen soil, respectively. Knowledge of the thermal diffusivity of frozen soils is necessary for transient heat transfer analysis. Haynes et al. (1980), reported a thermal diffusivity for a number of soils, over a range of temperatures, and water contents as summarized in Table (2-6). 2.3 Thermal Contraction Cracks in Frozen Surface Soils Temperatures at the earth's surface fluctuate on the order of 25°C about the mean through the combined effects of changing seasons, and shorter period random and diurnal changes. More than 90 percent of this fluctuation is confined to the surficial 10 m. For Alaska, all of Canada, and the Northern tier of states of the U.S., daily mean air temperatures will drop below freezing for periods of weeks during the 33 Table 2-6: Thermal Diffusivity for three soil types (Haynes et al., 1980). Sample Water Temperature (°C) content z -50 -35 -20 -5 20 Thermal diffusivity, cmzls x 10-3 * ... * 20-30 Ottawa 0.01 3.02 2.85 2.88 2.73 2.81 sand 8.8 13.17° 11.77 12.45 __. .__ * * 3.0 3.37 3.19 3.30 3.10 3.03 Fair banks 17.0 9.49* 8.22 7.78 6.79* ___ 311: 25.0 .__ 10.05 10.02 8.04* .__ 1.8 3.14 2.91 2.76 2.66 2.33* CRREL varved 18.9 8.26° 8.21 7.44 6.22* - clay 25.5 ' 12.19° 11.24 10.33 8.76 - * Extrapolated + For example 2.88 x 10.3 34 winter season. The surface soil strata will freeze to depths of 2 m.or more. When temperatures fall in the winter, these surface layers "try" to contract but they are constrained by the tensile strength of the frozen soil. The surface layers are stretched in a sense, although no observable displacements occur. Because frozen soil is relatively weak in tension, initial fracturing commences at the ground surface and penetrates the ground to the depth needed to relieve the tensile stresses. The cracks will be distributed over the ground surface in a pattern such that tensile stresses are reduced below the frozen soil tensile strength. 2.3.1 Origin of the Cracks When the ground surface is cooled it would contract if it were not constrained. As it cannot contract, horizontal tensions are generated with mmall horizontal strains. The horizontal thermal strain is calculated as the product of the contraction coefficient and the change in temperature from its initial value. It is the stress produced by this stretching and not the strain that determines whether cracking will occur. Frozen soil materials behave elastically in response to rapid deformation. If elastic behavior is assumed in response to slower natural thermal deformations, the stress would be proportional to the drop in temperature from some reference value. Then using elastic soil constants, stresses on the order of the soil tensile strength would develop in ice-saturated soil when the temperature drops only 2 or 3°C, and open cracks would occur with a horizontal spacing no greater than the crack depth (Lachenbruch, 1962). Since this does not generally 35 happen, it is apparent that frozen soil behavior is more complex and involves time dependent effects as outlined in section 2.3.2. From the above comments on thermal stress, it is clear that both M. (“lgwfiggmperatures and rapid cooling rates favor large tensile stresses. " "“‘"”"“""" "“‘ r 9- » u ......py." MHAbM-F‘ .1..- HW‘FW‘5-s—gsmnfi-Mu '7 - W 1 “-n‘ffl' m5“! .9” Since both of these quantities will attain greater extremes at the ground surface, the greatest thermal tensions will normally develop at the ground surface. It follows that repeated contraction cracks would be initiated at the ground surface. This behavior is reported by Lachenbruch (1962) for thermal cracks in permafrost. The original cracks (Figure 2-5a) that start ice-wedges and determine their location are initiated at the ground surface. With the approach of warmer seasonal temperatures, water seepage fills the cracks and freezes. Any remains of the crack in the surface thawed layer will disappear during the summer. The process will repeat itself the following fall with the crack forming in the weaker ice formed in the crack from.the previous season. For this case, crack initiation occurs below the surface in the ice. For each subsequent season the lateral dimensions of the ice wedge will grow as more and more ice accumulates in the permafrost (Figure 2- 5c and 2-5d). In temperate regions with.warmer mean annual temperatures, the ice wedge will not form. 2.3.2 Stresses Before And After Fracture In analyzing the stress prior to fracture, the ground is assumed to be a homogeneous semi-infinite medium. The ground temperature, with reference to the mean annual temperature as zero, is used to provide a relation between horizontal stress and temperature. Information required includes: (a) ground temperature as a function of air ’1 I 36 lst Winter 1st Fall A Scale —0 17 Feet .J Layer Perma- Frost 500 TH Winter 500 TH Fall C D Figure 2-5: Schematic Representation of the Formation of an Ice Wedge According to the Contraction - Crack Theory. Width of Crack Exaggerated for Illustrative Purposes (Lachenbruch, 1963). 37 temperatures, time, and depth; (b) the thermal contraction coefficient as a function of temperature; and (c) mechanical prOperties of the frozen soil as a function of temperature. Assuming the simplest model of deformation, the elastic solution for stress gives E l-v a AT (2-47) where o is the tensile stress at a given depth, E is Young's modulus, v is Poisson's ratio, a is the coefficient of thermal contraction, and AT is the change in temperature. Prior to fracture the stresses will be the same from place to place at a given depth and time. After crack formation there will be a reduction in tensile stress in its vicinity. These cracks must be so distributed that the stress over the entire surface is reduced below the tensile strength of the frozen ground. Consider a single isolated crack. At the surface of the crack the horizontal tension normal to the strike is zero, as this is a free surface. With increasing distance from the crack at any depth, the horizontal stress will change and asymptotically approach the initial or prefracture value at large distances. Each crack has associated with it a zone of influence or stress relief. Throughout this zone the stresses are reduced well below the tensile strength and no further cracking1would occur. Beyond the zone of stress relief the stresses near the ground surface will still exceed the frozen soil tensile strength and other cracks would be expected to occur. The spacing of the cracks are related to the width of the stress relief zone for a single fracture. This question will be examined in greater detail in section 5.2.3. 38 2.3.3 Crack Depth and Spacing Up to this point it has been assumed that cracks will initiate if and when the tensile stress exceeds some critical value - the tensile strength. Lachenbruch (1961) has used fracture mechanics theory to provide a more complete picture of crack formation in frozen soils. Energy is required to lengthen a crack, first to overcome the forces of cohesion to produce new surfaces, and second, in a brittle-plastic medium, to do work of plastic deformation in the region of elevated stress near the crack tip. At the same time, when a tension crack lengthens, it relieves some of the tension that produced it, hence strain energy is released from the medium. A crack will lengthen if, and only if, by so doing it releases at least as much (strain) energy as it consumes near the crack tip. In a frozen soil under uniform tension, the amount of strain energy (G) released with the creation of one square centimeter of crack surface increases with the length of the crack and the tensile stress. As the tensile stress is increased in a brittle-plastic medium, a value of (G) is reached at which small flaws in the frozen soil start to grow and coalesce to form.minute cracks. The growing cracks release more strain energy with increase in length, but at the same time, the size of the plastic zone near the tip grows, and so does the energy consumption. When the rate of energy consumption overtakes the rate of energy release, crack growth stops. This process is called "stable cracking" because it is self arresting. As cracks continue to lengthen stably in a brittle-plastic material, the increase in plastic-zone size becomes less important, and a crack length is reached at which the rate of energy release is growing 39 faster than the rate of energy consunption with increasing crack length. At this point the crack will extend with no further increase in tensile stress, and "unstable" fast fracture begins. The stress at which small cracks attain this critical length in uniformly stressed frozen soil is called the "tensile strength". This condition is identified with a critical value of the rate of strain energy release (or consumption) per square centimeter and is denoted by Cc. If the rate of energy release (G) for a given crack is less than Gc’ propagation, if it occurs, will be slow and stable; if it is greater, excess energy is available to accelerate crack extension to cause fast fracture. At what depth does the crack stop? The crack will stop at that depth at which the rate of energy release ceases to exceed the rate of energy consuapt ion near the crack tip; or about the depth where G falls to Gc' Crack depth is a function of stress on the crack surfaces, and stress intensity factors, thus b - f (E1, 0) (2—48) where b is crack depth, KI is stress intensity factor, and O is stress on the crack surfaces. Due to non linearity of stress distribution on the crack surface, a conbination of methods (Sih 1973; Koiter, 1965; Lachenbruch, 1962) are used in evaluating the stress intensity factor KI as will be shown in Chapter IV. Crack spacing depends primarily on how the tensile stresses are reduced along the horizontal surface and away from the crack. Crack spacing is a function of horizontal distance from the crack, crack depth, and stress magnitude and distribution along the crack surface. Calculations for crack spacing will be shown in Chapter V along with a numerical example. CHAPTER III PRELIMINARY EXPERIMENTAL WM 3.1 Thermal Contraction/Exguion Behavior 3.1.1 Materials and Preparation of Beam Specimens A commercially available Silica Sand (produced by UNDMIN Corporation, Oregon, Illinois 61061, U.S.A) used for the experimental study was evaluated relative to its thermal contraction/expansion behavior in the frozen state. The sand consisted of sub-angular quartz particles with a specific gravity of 2.65. The samples were prepared so that the sand gradation was uniform in size with all.material passing the number 30 U.S. standard sieve (0.590 mm) and retained on the number 200 sieve (0.074 mm). The coefficient of uniformity (Cu - d60/d10) was approximately 1.51. A wooden mold, 12 inches (304.8 mm) long by 2.5 inches (63.5 mm) wide by 2 inches (50.8 mm) high, Figure (3-1), was used to prepare the samples. The mold was first sealed by placement of high vacuum grease along the edges, then water was poured into the mold. Sand was placed into the water to first remove all air bubbles, then the sides were tapped to give the desired sand density. After preparation.of the unfrozen sample, the next step involved drilling partial holes at each end for placement of steel gage plugs. The holes were carefully located on the top of the compacted bean, using 40 4| MHZ-1 ' s Figure 3-1: Beam Mold and Template for Preparation of Contraction/Expansion Specimens. 42 the template shown in Figure (3-1), to give two 10 inch (254 mm) gage lines centered 1 inch (25.4 mm) from each end, and 0.75 inch (19.05 mm) from the edge of the bean. Holding the template firmly in place on the beam, the gage plugs were inserted into the hole and tapped lightly into the surface of the sand. The template was then removed. The mold, sand, and steel plugs were then placed into the freezer and frozen at a temperature close to -5°C (23°F). After 24 hours the ample was removed from the mold, placed on the wooden rollers shown in Figure (3-2), and returned to the freezer. Gage length and temperature readings were taken imediately and at appropriate intervals as the frozen beam was allowed to slowly cool down to -30 °C (-22 °F). The 9 11m: diameter wood rollers provided a very low friction support during contraction and expansion. Cage and temperature readings were continued as the sample was permitted to slowly return to -5 °C (23 °F). One cycle of realings was completed in about 2.5 days. 3.1.2 Equipnent and Test Procedures Dimensional changes of the frozen sample caused by a change in temperature were measured using a Whittenore strain gage. This gage is a mechanical device employing a dial gage with no internal multiplication of the measured defomnat ion. For this investigation, a 10 inch (254 mm) gage length and a least gage reading of 0.0001 inch (2. 54 x 10"3 10" mm) gave a strain sensitivity of 0.00001 inch/inch (2.54 x 111m). The gage and accessories are shown in Figure (3-3). The frame and bar on which the gage is mounted serves as a reference between readings. 43 .ouoom amomouhum a no nooo~m maosoa woos nouoaofin as a mo wmwuofinmoo muoaosm Boom commowum so; 0 31 —"'—-'-r WT, , ‘w 0 '2 o i .21...) 'H--"—- " -—-' --‘- 1...”. "NIM ounwwm 44 .oon :Mmuum ouoaouuwnz mucH1o_ any unln onnwwm 45 A standard separate frame is used for establishing the correct gage length on the specimens. The testing technique involved placement of the standard bar and gage in the cold box adjacent to the frozen samples so that a uniform temperature was maintained throughout the investigation. Before taking a measurement, the gage was checked against the appropriate standard bar followed by a series of three readings obtained for the left and right sample gage plugs. The gage points were then gently inserted into a set of the steel gage plugs and the reading taken. This procedure was repeated three times. for each gage line, removing and replacing the gage for each consecutive reading. It was important that the gage be held vertical and that a gentle tap be given the dial gage to ensure that consistmt readings were obtained. Temperatures were monitored using two thermistors (type 100 OHM Platinum RTD .0(B85 T.C. produced by Pyromation Inc.) embedded, prior to freezing, in the sample near its mid-section. The thermistors where monitored using a Minitrend 205 data logger produced by Doric Scientific, San Diego, California. Measurement of dimensional changes was repeated at selected temperatures with the length change. of the specimen given by the difference between initial and consecutive readings. These results were then plotted as shown in Figure (3-4). 3.1.3 Therml Contract ion Coefficients Thermal contraction/expansion tests on (1) saturated sand, (2) partially saturated sand, (3) drained partially saturated sand, and (4) snow, showed that an increasing percent age of ice saturation increased 46 .wuumoo one .ooH .oomm nowoum HON mowuoouusoo Hosanna mo umowowuuooo on» no ousuoueoaoa mo uoouuu “v1n «woman A U V summonses—ma. O Oml 90.1 ON! . mdl CAI m1 A hmmu .SUfiaroo—unm v 5335+ x A Ame .sofiroxuzm V mom X A m .oz man—5mm v Somme 1 A e .oz man—8mm v ooHQ A m .oz 6253 meow omumuzumm haofiuuom magnum—D T A~.oz man—Emmvmmom moumuzuom :mwuummm— A A .02 oaaamm v meow moumuzummmv M 1. 29¢ka x x j .3de d. o If . . ...V .1 . . 2m e G G Ina}: 54A. ON on Co on on (1-3") 9_om D uonosnuog remain, go auaporyaog 47 the coefficient of thermal contraction as shown in Figures (3-4) and (3- 5). In all samples tested and the data reported by Butkovich (1957), the effect of changing the temperature from -5 °C (23 °F) to -25 °C (-13 °F), changed the coefficient by at most :4 x 10.6 °C-1. A comparison of these coefficients of thermal contraction for the frozen soil with the coefficient for quartz and for ice (7 - 0.97 g/cm3), shows that a is almost constant with temperature between -5 °C (23 °F) and -25 °C (-13 °F). The presence of ice and an increase in frozen sand density serves to increase the coefficient of thermal contraction. This change is shown in Figures (3-5) and (3-6). In general, the density of the material haswmore effect on the coefficient of thermal contraction, than the change in temperature. 3.2 Fiber Reinforcement 3.2.1 Materials and Preparation of Compression Specimens A commercially available silica sand (produced by UNIMIN corporation, Oregon, Illinois 61061, U.S.A.) was selected for the fiber reinforced compression samples. This sand consisted of sub-angular quartz particles with a specific gravity of 2.65. The sand gradation was uniform with all material passing the number 20 (0.840 mm) U.S. standard sieve and retained on the number 200 (0.074 mm) sieve. The coefficient of uniformity (Cu - dGO/dlo) was approximately 1.49. Comercially available steel wire (gage sizes of 28, 24, and 17) was used in volume fractions of 32, 62, and 92. The wire was cut to the required length of 0.50 inch or 0.25 inch as needed for samples. A sand and steel fiber volume fraction of 64 percent was selected to give a dense fiber reinforced soil mass. This volume fraction was convenient 48 . usmm Nausea nououm mo nom>onom eowenmown\nomuomuunoo Amauosh Ama\wzv Andaman he: cououm "nIn ounmmm m.~ ~.~ ~.~ o.~ I» q — q d u — Tl - m \ \ \ I. 9\ {x a”! flMmmm11 1 Mahatma- 1 t5: . elm moan MT LL 1 Hooum as «aw summed owou can: cue. coo. ..m ..m ..m ,o.l 1.1. oIT. 1 [.9 (.3 nos 0.11 v.7. v.c. we as we I . — P F r _ 0A ON «N mm an on (13°) 9_01 x D ‘nonoenuoo 1211113111, go unsungeoo .ooH uo nowuomuueoo guanoch mo unomofiuuoou no ousumuooaoa use hugncoa mo moouuu ”01m ouswwm Ann—\mzv . X1 Duncan 49 b - a q — q q a q q 4 q . q — "I 1451.1 0| |0\\‘—-‘1 wzHMbrTllaaldlellL Uoo~1 u 2 fl ehweoa omen ..F 6 1/ A 91/311 916'0 - Doculla m. m Am. .0 ad ad T. T. 0T. 3 03 .3 o 8 .7 89 .../C. C. 9 \I 9) I S m) a Nun no a 000 In I.\ In m0 m1\ Co E ( a p . h b . . _ p . . . a — - In W 1: \O O ) 9_01 xvuopoenuog '[euuaql go guerogggaog o I- ( AU 4. 5. Vi 50 for ease of compaction and is comparable to a compacted sand volume fraction typically encountered in the field. All samples were prepared in split aluminum molds. A removable extension, 0.375 inches high, was attached to the open end of each mold to aid in sanple preparation. The mold was disassembled and cleaned prior to each sample preparation. next, a thin coat of silicone grease was applied to the inner'mold surface to reduce ice adhesion at the sample/mold interface and to aid in sample removal after freezing. The amount of oven dried sand, and steel fibers required for a given volume fraction, was predetermined based on the mold volume, weight of the sand, and steel fiber. To insure a high degree of saturation, molds were first partially filled with distilled.water. Next, a mixture of sand and steel fibers was slowly poured into the mold allowing air bubbles to readily escape to the surface. The degree of ice saturation, for suples prepared in this manner, ranged from 96.1 to 99.9 percent based on an ice density at -14 '0 of 0.9148 mg/m3 (Founder, 1967). Sample compaction was achieved by tapping the mold sides and bottom sufficiently to allow the fiber reinforced sand to settle within the mold. The mold and samples were then placed in a cold box at -14 °C and allowed to freeze for at least 12 hours. Next, mold extensions were removed and the exposed sample ends were trimed with a sharpened scraper until a uniform seating area was formed for the loading platens. Samples were then removed from the mold, weighed, and enclosed in rubber membranes to prevent sublimation during storage. Sample volume was assumed equal to the mold volume for density computations . 51 3.2.2 Uniaxial Compression Test Uniaxial constant strain rate compression tests were conducted on 1.13 inch (28.70 mm) diameter cylindrical samples with a 2:1 height to diameter ratio, at a temperature of -6 °C (21.2 °F) and strain rate of approximately 1.1 x 10.4 sec-'. Sample diameters were selected so as to stay'within the capacity of the test equipment. Prior to mounting samples in the triaxial cell, a stainless steel (disk) loading platen was placed at each end of the sample. Loading platen surfaces in contact with the sample were coated with a thin layer of Teflon to reduce end effects. Two protective membranes were placed over the samples and fastened securely to the loading platens with rubber bands. The 1.13 inch diameter samples were then mounted on the base pedestal of the triaxial cell (inside a cold box). The triaxial cell (Figure 3-7) was placed over the sample, attached to the cell base, and the loading ram.was brought into contact with the t0p loading platen. The entire triaxial cell was then transferred to the low temperature bath and a mixture of ethylene glycol and water was allowed to fill the bath and circulate around the sample. A period of at least 6 hours was allowed for sample temperatures to adjust to the cold bath temperature prior to testing. After completion of a test the triaxial cell was disassembled and the sample removed. The membrane and sample were inspected for leaks and the failure mode was noted and sketched. The sample was thawed and oven dried to permit determination of oven dry weights. The degree of ice saturation was then computed based on the mold volume, weight fractions of sand and fiber, and sample weight. 52 Load Cell Loading Ram Lubricated Washer Steel Loading Cap Thermistor Rubber Membranes Rubber Bands Ethylene Glycol- Water Coola:::> Figure 3-7: Modified Triaxial Cell. pm 11581 to. m 01 53 With higher compressive strengths due to fiber reinforcement and test equipment of limited load capacity, it was necessary to use smaller diameter frozen sand specimens in the test program. The following paragraphs present a description of the equipment and test procedures used to determine the mechanical properties of the fiber reinforced fro zen sand mat erial . Equipment: A modified triuial cell was used for the constant strain rate compression tests conducted on the 1.13 inch diameter samples. The samples rested on a stainless steel platen which was supported by the triaxial cell load pedestal. A flat load cell, universal, Model FLU: 5 SP2 - 0211 (rated capacity of 5,000 pounds), with SCM-700-strain gage signal conditioner was used to monitor axial loads. Axial deformation was measured using a displacement transducer (LVDT) type GOA-121-500 SIN 3665 with 80! series, SIN 987 strain gage signal conditioner. A schematic diagram of the triaxial cell assembly is shown in Figure (3- 7). A constant temperature was maintained during testing by immersion of the test apparatus (triaxial cell) in a circulating low temperature bath of ethylene glycol and water (50-50 mixture). The temperature of the coolant fluid was maintained to i 0.1 °C using a micro-regulated refrigeration unit and circulator. A thermometer with scale divisions of 0.1 °C was used to monitor the tenperature of the bath. The temperature of the coolant imediately adjacent to the sample was determined to i 0.05 °C using a thermistor type 100 OHM platinum RTD .00385 T.C period and a Doric, Minitrend 205 data logger. A schematic 54 diagram of the test equipment and coolant circulation system is shown in Figure (3-8). The constant strain rate uniaxial compression tests were conducted using a Wykeham-Farrance (Model T57) variable speed testing machine with a 10,000 pound load capacity. This test machine had a 30 speed gear box with the capability for displacement rates ranging from 0.075 to 0.000048 inches per minute. Results indicated that the cross-head displacement rate increased slightly during testing, reaching the selected rate only after the peak load had been reached. Output from the various transducers and thermistors were recorded on a data logger strip chart, using four channels. The test set-up is shown in Figure (3-9). Test Procedures: After the frozen ample was mounted in the triaxial cell, the test apparatus was placed in the cold bath. When a sufficient period of time had elapsed for temperature stabilization (6-hours), the following test sequence was followed: 1. The signal conditioners were connected to the recorder, which was allowed to warm up for approximately one-half hour prior to testing. After the warmrup period, the signal conditioners were adjusted to a zero reading. 2. The loading ram of the test frame was brought into contact with the test apparatus, but with no applied load. A small seating stress, approximately 100 psi, was applied to the sample prior to testing. The manual loading feature of the Wykeham-Farrance test frame was used to apply the seating load and the magnitude was monitored with the load transducer. The specimen was not tested until the seating 55 Load Cell Triaxial Cell _.——- Loading Ram Test Frame h \ F LT — L___..-———LVDT ' ~ 1 Coolant In L..-..-.._1 _‘1 I: _,,.—— Insulation - ’r—‘gu Coolant Out Thermistor /— r—‘— _- ET / C01 Bat r—‘h' 1——11 w 1—" "' " ’ "'1 I 1 _ ‘———!' _- : Coolant | _. Recorder 1 .Q__D_J__L lid fl Refrigeration Unit Figure 3-8: Diagram of Test System. .uooamwsom ooUooSoo some A5 .233 announce—non. 33230 new 3:: monoooq on .nbluow anon. sawnmoumaou dingo: 31m 33w: 6. 7. 57 stress had decreased to nearly zero by ample relaxation. This procedure gave a more uniform contact surface between the sample and loading platens and helped minimize data scatter. Temperatures adjacent to the ample (inside the triaxial cell) were measured using a thermistor and monitored by the data logger. The gear box controls for the loading ram were adjusted to give the desired loading rate and the loading rm was engaged. Samples were deformed to at least 7 percent strain (failure or peak stresses normally occurred at greater than 5 percent axial strain). The drive mechanism for the recorder and test frme were then stopped. Circuits from the transducers were disconnected from the recorder and the recorder strips were labeled and filed until the data could be transcribed to data sheets. The ample was unloaded and the triaxial cell removed from the cold bath and disassenbled. The failure mode was sketched and the sample was oven dried at 110 °C to determine the weight of sand. 3.2.3 Compression Test Results Steel fibers used in reinforcing the frozen sand, provided information on the influence of gage size, volume fraction, length, and aspect ration (length to dimeter ratio) on the unconfined compressive strength. A sumary of these tests are given in Table (3-1). For comparison, tests were also run on specimens with no fibers. 5E3 ..0 050 0..0 00.0 .00. 0.0 an 00 ..0 .00 ...5 ...5 -0N «..0 0.0. 0.0 05 00 0.0 0.0 0N.0 5..0 000. «..0 0.0. 0.0 mm ..0 0.0 50.0 50.0 0.0. «..0 0.0. 0N 00 0.0 005 «0.0 .0.0 .00. 0..~ 0N0. 05 55 0.0 000 00.0 0..0 000. 0.0. 0.0. 0N.0 on 05 0.0 ..0 0.0 00.0 00.0 0.0 ..0 N00 .00 500 050 000 000 N00 00.5 05.0 50.0 50.0 00.0 0N.0 0..0 «0.0 50.0 50.0 00.0 00.0 00.0 00.0 000. 000. 000. «00. 050. 000. 000. 5.0. 0.0 0..N 0.0. 0.0. 0.0. n~0. 000. 0N0. 050. 0.0. 050. 0N.0 0N.0 0.0 0~.0 05.0 05.0 puss uazoxa on 5. 0a 0N 0N «N 05 «a .N 05 0. 0. 0. 000 05.0 00.0 0.0. «.0 000. 5. n. ..0 000 0..0 00.0 000. «.0 000. 5. N. 0.0 000 05.0 00.0 000. «.0 000. 5. 0.0 0~.0 00.0 .00. 000. 0~.0 5. 0. 0.0 000 00.0 00.0 0.0. 000. 05.0 5? 500 ~..5 .n.5 000. 0.0. 0.0. 0~.0 0N A00 Iv ouauauoaaoh ..uouv uu:.«wm on 08.8 .50 cannon zaax< vouoouuou .50 sacrum zumx< .uamxunuv noncomm- o>meaounaoo conducouca t 3 > cemuowuu oaazo> uoswa .05.0 cwuau uooan< 5.050600 an.) we uauoadwa Aeosummv on.) 00 some»; Aowwuv on.) .ooum oz ode-um .vawm concuh uaououawou nonmm no auaou nuuaouuo cam-aounloo vacuucooca 0o annulam u.ln 0.9mH 59 The effect of different volume fractions for a given fiber are shown in Figures (3-10), (3-11), and (3-12). In these tests the fiber length was kept constant, either 0.25 inches (6.35 mm), or 0.50 inches (12.7 mm). As shown in these figures, increasing the fiber volume fraction increased the unconfined compressive strength and, in general, increased the modulus of elasticity for the reinforced frozen sand. A steel fiber, gage 28, with volume fraction of 91 (sample 33), increased the compressive strength from 1464 psi (02 fiber) to 2022 psi. Note that the failure strain increased with an increase in volume fraction as shown in Figure (3-11). This behavior can be explained in that frozen soil is a brittle material while steel wire is an excellent ductile material. Reducing the frozen soil volume fraction by adding steel fiber will also improved the energy absorption characteristics of the frozen soil. An increase from.5.32 strain for 0 2 fiber up to 9.6 1 (sample No. 27) with 9 1 fiber was observed. The fiber diameter, as well as the length, have an effect on the unconfined compressive strength of reinforced frozen sand. Figures (3- 13) to (3-15) represent data for a constant fiber volume fraction and fiber length, but different fiber diameters. In these tests the effect of fiber surface area appears to influence the behavior. A failure mechanism for the fiber reinforced frozen sand can be explained as fiber pull out and debonding, which depends on the fiber surface area. Increasing the surface area will increase the total bond between steel fibers and ice, sand or both. This behavior will increase the resistance of the mix to compressive failure loads. The surface area for a given fiber depends on fiber diameter. As shown in Figure (3-16), 60 2000 T I I I NA e i N 1600— - :3 . [3 Sample 10 s — U as 5 3 1200 - a: o 3. .. m G o u geoc- -o Temperature -6 °C 3 ’ Wire Gage # l7 _. 14 ‘g Fiber Length 13 mm 3 Volume Fraction : 400 4 D G) 3% A 6% "‘ El 92 0 l l 1 l 0 2 4 6 8 10 Axial Strain ( Z ) Figure 3-10: Influence of Fiber Volume Fraction on Compressive Strength of Fiber-Reinforced Frozen Sand (steel fiber gage l7). 61 I l l l 2000— Sample 27 — ”’B“‘ -. NA _ ’ // _. s ,43 21600 _ , I _ A z :3 ,’ __ ’ ‘ 20 .5 / " _ “'O\ ‘5 i” (D ‘\'\\ 81200 — ,’ \ _ a . I; O\ 21 .. / ,1 a — o/ / Temperature -6 ° C — 3 I ;] Wire Cage #24 asm— ' _ g l . l’ - Fiber Length 13 mm 2 l 1’ Volume Fraction g I. ’ 3 z (3 ‘- I . E / [j 6 z A: 3 400— I 9 z E] ‘ D / I I I — I — ’l c ’ I 1 1 I 0 2 4 6 8 10 Axial Strain ( Z ) Figure 3-11: Influence of Fiber Volume Fraction on Compressive Strength of Fiber-Reinforced Frozen Sand (steel fiber gage 24). 62 I I I If I II I I I 2000 r- ,G‘—-D--~G~ -g — / /’ Sample 33 I” I— I d [,8 . 31 1600 _- x], ' .1 I, / ‘,,_ -“C) " I], ,r”<>". “\.‘\\ -1 I / / 'z” \\\ 1200 .. I O Unconfined Compressive Strength (lhf/inz) ' Temperature -6°C 800 ’ / _. " I Wire gage #28 6/ Fiber Length 13 mm " I' Volume Fraction ‘ f 0 Z (3 400 _: 6 Z A —I ' 9 z E] 0 l l l l 1 .1 l l l 0 2 4 6 8 10 Axial Strain ( Z ) Figure 3-12: Influence of Fiber Volume Fraction on Compressive Strength of Fiber-Reinforced Frozen Sand (steel fiber gage 28). 63 2000 I I I 1 NA I‘" —-I a ...; \ Qua '3 1600 *- .- ___- . _ “‘\ Sample 19 5 ~— ~\ _ 9: 0___\ \ 3 I // \\G ‘8‘ a: 1200 L- ./ \ \ _q 3 ' /G \ 18 '3 ’ \\ 3 \\ 3. I . C) 3 800 __ ' Temperature -6 c \7 c: 4 _T '0 Fiber length, 6.35 mm g __ /,/; Volume Fraction 3% .4 “g [I Wire Gage U a :3 400 , G 17 .4 AA 24 [J 28 ..0 o 2 4 5 8 10 Axial Strain ( z ) Figure 3-13: Influence of Fiber Diameter on Compressive Strength of ' Fiber-Reinforced Frozen Sand (fiber length - 0.25 inch). 64 oi: _' Sample 4 " ...q E , ,- -a ----- 43-- - a 1600 — .’ — v , I «D g '1' ’0...— a _ [I / \ O\ 25 -| 5 F] 1” :3 O m 1200 _ [I / _ a: / / .H I m , / F3 " ' / 0 3' ,’ Temperature -6°C 24 8 800 ... D ¢ Fiber Length 6.35 m _ I '8 ’ // Volume Fraction 9Z a l E - ,' / Wire Gage _1 o I“ 2 II / A 17 :2 400 _. 7’ Q G) 24 _ .II/ B 28 ..I I/' -' 0 l l 1 1 0 2 4 6 8 10 Axial Strain ( Z ) Figure 3-14: Influence of Fiber Diameter on Compressive Strength of Fiber-Reinforced Frozen Sand (fiber length - 0.25 inch). 65 2000 - A 1600 - N 1200 — Unconfined Compressive Strength (lbf/in I I ' SampiL 33 fl.--D... .8‘ ~- ._1 I, _1 I Temperature -6°C Fiber Length 13 mm 800 Volume Fraction 9Z - Wire Gage 0 17 — A 24 400 [3 28 q o 1 1 1 0 2 4 6 8 10 Figure 3-15: Axial Strain ( Z ) Influence of Fiber Diameter on Compressive Strength of Fiber-Reinforced Frozen Sand (fiber length - 0.50 inch). 66 Sample 33 O 2000 — o o _ O. {3.0’ 1E7 O 3 / .5. 4 E 1600 r- .1 ..D C V/ ‘ at ‘9. ‘ ‘ ‘ a” ’ ' ‘ .8; ‘ l: 25 '5 - ‘0 {égpt 1” \~\\‘§C%‘~5 " no , . , 8 V ’1 I; B/ \ \ \ A ‘1 1 I: 1200 - I,’ / . \ .4 m 1” / \\ g: ”/ o ‘A ‘3 — 40‘; / Temperature -6°C ‘.24 - w I 3 o ,;J Volume Fraction 9% a. v 8 L- ’l _- 8 800 ,7 Fiber Length Fiber Length Wire '§ $2; 13 mm 6.35 mm Cage 83 T" $1 A 6 17 “ 8 '7/ <3 E] 24 a I D 400 '- Jfi] 0 v 28 " l/l ° 0 L l l 0 2 4 6 8 10 Figure 3-16: Axial Strain ( Z ) Influence of Fiber Diameter and Gage Length on the Compressive Strength of Fiber-Reinforced Frozen Sand. 67 for a fiber volume fraction of 31, the steel wire (gage 28) of diameter 0.016 inches (0.4064 m) has a higher modulus of elasticity and failed at a higher compressive strength, as compared to samples reinforced with fiber gages 24 and 17. The effects of a combination of fiber lengths and diameters can be visualized by introducing the aspect ratio effect on the unconfined compressive strength of frozen sand. Increasing the length will increase the aspect ratio of a given fiber, with a known dimeter. Influence of volume fraction and aspect ratio of different fibers on the unconfined compressive strength are shown in Figure (3-17), Al-Houssawi, and Andersland (I988). For both an increase in aspect ratio and volume fraction of steel fiber, all tests showed an increase in the unconfined compressive strength. At a given aspect ratio, the addition of fiber made it more difficult to obtain the same soil density in areas adjacent to the fibers. For this reason little or no strength increase was indicated for 32 fiber volume. 3.3 Thermal Tensile and Stress Relaxation Tests 3.3.] Materials and preparation of Tensile Specimens Silica sand with properties the same as that used for the thermal contraction tests was used in sample preparation. Particle size distribution for the sand is given in Table (3-2). An aluminum mold, with dimensions shown in Figure (3-18), was used to prepare the frozen sand sample. The sample was shaped as shown in Figure (3-19) to minimize end effects. Screw rods at both ends transferred the load fran the end plates (Figure 3-20), to the frozen sample. The three piece mold was first coated with vacuum grease at all 68 2200 I j I I I Volume Fraction of; 32 E] 2 2000 . 62 O ._ 'H . '3 9x A " Temperature -6°C — '5 60 5 1800 L- _- H U m 0) I" _ E o m ’ / 3 1600 _. $3 .— / .— G- I 8 M o/ o 'u " a/’ _. “I / E. O// “a 1400 - ’ -— o l 2 / = _ [If I I I I I- 17 17 24 28 24 28 1200 __ Gage Number _‘ ..I 2 - 1050 in. V“ a o 3 - 0.136in. / 11 222 8 ‘ U) Top View 1 .75 in. / in. I; C / II II I '1 :1 I' II II I' i: :1 :i 15 SE :: I; Rods Length . h 1' Outer . 13.0 mm Screw R°d Dia. 2: Intermediate I 25.4 mm - 3.45 m U Inner - 50.8 mm Side View Figure 3-20: Tensile Test Specimen End Plate with Fabrication Details. 72 Table 3-2: Weight Size Fractions of Sand Used in Preparation of the 2100 gm Samples for Tensile Stress Relaxation Tests. 0.8. Standard Weight Percent Finer Sieve Number (gm) by Weight 30 4.20 96.00 40 1092.00 47.98 50 831.60 20.44 70 140.70 10.36 100 24.78 3.26 140 4.62 0.56 200 1.26 0.07 pan 0.84 0.00 Total 2100.00 joints and assembled. Next, water was poured in the mold to help air bubbles escape during placement of the sand. Tamping the sand during placement helped achieve a relatively homogeneous and dense sample. The mold with the sample inside, waw placed in the freezer for freezing and cooling to -20 'C (-4 °F). Next, the sample was removed from the mold and supported vertically inside the freezer by connecting the end plates to the connector bars attached to the load frame. While in the freezer, the sample temperature was allowed to adjust to -5 °C (23 °F). The test was started at -5 °C. Two metal screens placed around the sample provided support for the dry ice coolant as shown in ZFigure (3-21). Partial control of cooling rates was handled by changing the space between the sample surface and the dry ice. The outer screen diameter“was adjusted so as to provide space for a minimal amount of dry +5» Figure 3-21: 73 Dry Ice /oo’:'\ -\\‘~‘*Er>-.\%Qi A {0 1“ .4. $011 ' '°\ N [:21 . \ O: 99°0/ Insulation-1L 4?” board Metal Screen -/ (a) Top View Freezer 'box Insulatifin’béard __Jlnx .128--- ... .. ... -.. (b) Section A - A Freezer Box Modification for the Thermal Contraction Tests Including Insulation Board Location and Metal Screen for Support of the Dry Ice. 74 ice. An insulation board box placed around the dry ice in the freezer helped reduce external heat effects and vaporization of the dry ice. 3.3.2 Experimental Equipment Temperatures were measured using six thermistors (type 100 OHM Platinum RTD .00385 T.C.) attached to the sample surface, four on the ends and two at the mid cylindrical section as shown in Figure (3-22). A flat load cell (Model FL50(C)-ZSP) with rated capacity of 5000 pounds was used with a strain gage signal conditioner (Model SCM-700) to measure axial loads. Axial deformation was measured.with an LVDT type (GCAr121-500 S/N 3665 with SCM series, S/N 987) strain signal ‘ conditioner as shown in Figure (3-23). Loads, axial defonmation, and temperature were monitored using a Doric Minitrend 205 data logger. Data output from.the various transducers and themmistora were recorded on the logger strip chart, using eight channels. Figure (3-23) shows the test set-up for both tensile and stress relaxation tests. 3.3.3 Test Procedures After mounting the tensile sample as shown in Figure (3-24) and waiting six hours for temperature stabilization, the test sequence described below was followed for both tensile and stress relaxation tests. 1. The signal conditioner was connected to the recorder, and allowed to warm up for approximately one-half hour prior to testing. After warm-up the signal conditioners were adjusted to a zero reading. 2. A.small seating stress, approximately 100 psi, was applied to the sample prior to testing by adjusting the nut on tap of the load 75 Frame Large Nut for Length Adjust- ment and Load Application Insulation Board for Thermal Contraction Tests Frozen Sample Dry Ice for Thermal Contrac- tion tests Thermistor Freezer Wall Screen Freezer Box Supports Load Cell Load Frame Support Figure 3-22: Diagram Showing Frozen Tensile Sample in Position for Stress Relaxation Tests or Loading by Thermal Contraction. 76 .a0 .somuaauowon Hn0n< no unuauuaeeox .00 .oauum veoa one Mom nonouum .aunua uoauewuHom enouum use 000.cop now uaoanwavm "0N1 0 annuah 77 Figure 3-24: Frozen Sample with Thermistors in Freezer Box After Mounting on Load Frame. 78 frame. next the seating stress was allowed to decrease by relaxation to nearly zero. This procedure provided a more consistent load transfer between the end plates and the sample through the connectors, and helped to minimize data scatter. With the sample in tension, dry ice was placed around the sample. This caused thermal contraction in the ample and an increase in tensile stress. With time and sufficient cooling, peak stresses or tensile failure occurred in the samples. In the stress relaxation test all samples were tested at -15 °C (5 ’F) with no dry ice. Temperatures were controlled by the freezer with a small fan providing circulation and a more uniform temperature within the freezer. Initial stresses were introduced to the sample manually by adjusting the load on the top sample end plate. Tests were strain controlled, i.e., stress relaxation occurred only as a result of creep in tension. This creep was allowed to continue to a level where very little change in the stress occurred.with time. After the desired number of strain cycles, the sample was manually loaded in tension to failure. After completion of each test, all transducer circuits were disconnected from.the recorder, and data strips were labelled and filed until they could be transcribed to data sheets. The sample with end plates was removed from.the freezer. The failure mode was sketched or photographed, as shown in Figure (3-25), and the sample was oven dried at 110 °C (230 'F) to determine the weight of sand, and degree of ice saturation. 79 Figure 3-25: Frozen Sand Sample (No. 4) after Tensile Failure. 80 3.3.4 Thermal Tensile Test Results To evaluate the effect of temperature and cooling rate on thermal contraction and cracking in frozen sand, tensile specimens (Figure 3- 26), were initially loaded with a seating stress (close to 100 psi) after which the specimen ends were fixed. Some stress relaxation occurred before placement of dry ice around the ample area. Sample temperatures were close to -5 'C during mounting and placement of the seating load. Stress and temperatures were then monitored as a function of time as the specimens were cooled to temperatures as low as -78 'C (-108.4 ‘F). The sublimation temperature for dry ice provided a reasonable lower temperature for these tests and helped provide a larger range of cooling rates. The rate of cooling was partially controlled by the spacing between the dry ice and the specimen. For the most rapid cooling rate {-4.31 °C/min) the specimen dimeter was reduced to 1 inch at its mid-sect ion allowing temperature change to occur more quickly. The change in temperature and tensile stress with time for ample 4 is presented in Figures (3-26) and (3-27). Reasonably constant rates of temperature decrease were achieved over portions of the curve. Smple temperatures are average values for the smaller middle port ion of the test specimens (Figure 3-26). With thermistors placed on the specimen sides, cooling tests on duplicate smples showed that measured temperatures were close to average values when a small adjustment was made in the time scale. For the tensile test results, it appeared that this correction was very minor, and therefore has been omit ted for the data presentation. Towards the end of a test at higher stresses, failure in tension was very sensitive to any vibrations in the load frame caused by adding dry ice or other adjustments in the equipment. 81 .ue59e o0 ounmno «c.3000 .0 0.9aem now eflwa sauna» ounueuonluh “0510 euawwm .ouncwe 0 made 00. 05. 00 00 0 a _ 0 _ a 0 0505mm 0 11 u. 0.1 :56 50... AI 0.0 I u mg :08.0. oza.~u u ue\ee 0 '- ~01 ONI- TI :08 00.05 AI 50... n u II c5850o 005.01 m 00590 \ \ \ 1| , :56 00. AI 00.55 I a II 001 IIII.V\\\. caaxo. oo~.cn u ue\eu 0 . _ . aznaaiadmal atdmes aSezaAv (3°) 82 .0 o.nanm now 0:05 0003 aeouum 000.coa uo unoemo.o>oo “551 n ounwmm . muscua 0 0559 00. 00. 05. 00. 00 00 0c 05 . 0 _ 0 . _ . 1: .c« 00.5 I 0 IL 005 man “ 0:054 wcfifiooo IL 00. 005 000 000 000 (lat/501) 939138 aIIsuai 83 Data for samples 1, 2, and 3 are summarized in Figures (3-28), (3- 29), and (3-30). As expected, the more rapid cooling rates gave higher rates of stress increase since creep and stress relaxation are time dependent. The tensile stress-time curves in Figure (3-28) show an upward curvature as lower temperatures are achieved. The temperature- time curves in Figure (3-29) appear to be more linear. A cross-plot of tensile stress versus temperature in Figure (3-30) for samples 1, 2, and 3, all show an increase in slope with colder temperatures. The greater stiffness of frozen sand at colder temperatures would explain this increase in slope. A decrease in the coefficient of thermal contraction with colder temperatures (Figure 3-5) would lead to a decrease in slope and partially cancel the effect of greater stiffness. This thermal contraction behavior implies that a colder soil sample will experience a larger stress increase per degree drop in temperature when compared to a warmer frozen soil. Field observations reported by Lachenbruch (1961) are in agreement with this observation, in that cracks more readily occur later in the winter season.when the ground is colder. A modified cold storage freezer was used to achieve a cooling rate close to -0.021 °C/min for sample 5. This tensile stress versus time data (Figure 3-31) also shows an upward curvature in agreement with data summarized in Figure (3-28). A stress of 77 psi was reached after 7.75 hr. of slow cooling. A summary of the stress/temperature cooling curves for saturated frozen sand is given in Figure (3-32). Data for sample 5 was limited to temperatures above -15 °C by the freezer box capacity. Data for sample 4 (Figure 3-32) is shown by three groups of data representing different average cooling rates. 84 .auuem una0ooo cough you 0305 no coauom a uo>o anew museum voueunuew a. aaouum oHMesoa mo uooano.u>00 "0510 0u3505 . 0o 0 ounumuomamk 050600 ownuo>< 001 05I 051 0.1 0.1 0| . . . a . . :aaxo. no.0 I 1 ue\ao . mamanm 05650o 00.0 I I 00\Hv 5 0 new a m cas\o. .m.q u u uu\ev m «Haaam 00 00. 00. 005 005 833135 atrsual (aux/501) 85 .o.namm vow 005 5a: monsoon muaoeam vow nouoao00 0.03nm o0 omcano 50 co>owno< uo\av 0o Houuooo .ewuuam .eoanaem anew sonoum voueuauem couch you coaunu 0:0.oo0 ”0510 ounmfim . 3:5... 0 as: m .0 m ... _ _ . . . . .3 a .. a 5.5. 3.0.. 0 5 0505mm 0.0 c.0350. 00.0I a ‘00 :5650o 00.01 mV flu O u0590 acoavnnu Hosanna 0 an as o m 0 .ca . I 0 0 000600 AV n a. .c 5 I 0 n G 41511 G\m Q 0 0.9.80 _ . F _ . 0.I 0.1 051 aznaexadmal atdmss afleianv (Do) 86 .cwa\0o 00.0: .:ME\0. 0.01 .:ME\0o 0.0: II 00000 0:0.oo0 00005 new ouaumuonaoa :0 00000000 0003 0:00 museum 000nu=0nm =0 onnouucu unouum 0.0ncos "0010 ounwmm A ouocwa 0 mafia 353. $6- a 03% l 0 0.0600 ows\0o 00.0 I I 00590 5 caE\0o .0.0I n 00590 0 000800 000500 C In 0 0 _1 O W —I O O N (zur/;qt) ssalas atrsual 87 .aaa\u. _uo.o- no case uawaoou a now «say :00: 0:00 cououm 000003000 a s0 aaouum 0.0-009 0o 0eoano.0>00 . 00:0: 0 0809 0 0 0 0 5 _nun shaman GHE\U¢ . . . 0 . 050.01 I 00590 .:0 5 I 0 0 050500 05 (3 fi' ‘0 \0 888133 BIISUSL ( zuI/JQI ) 88 .0oem sououm 000003000 000 eo>u3o 030.000 003000003095000000 no hueaanm u~m1n «0:000 . 0o 0 00300000509 000300 05000>< 001 001 001 001 0m1 0.1 _ _ 0 . _ 01 M5 90 .0 4\4 b no \ \\ \ \ \ 0 005 5.00 0 00.01 m. \x 0 00000.00 5 50.01 D x.\ N 000 man . .o.mu .0 \O .0 03 5.5 N 3.51 0 Q .0 005 5.5 5 005.01 4 \O .0 005 ..0.— 5 005.01 0 \ .02 000300 .c« 358509 033mm 0:280 .03 $590 00805 _ 5 .1. 5 . 00. 005 000 000 000 ssaaas atrsual ( zuI/th ) 89 Table 3-3: Summary of Tensile Strength Tests on Frozen Saturated Sand. m Sample + dT/dt Maximum. Time to Max. Sample Cooling No. (°C/min) Stress Stress or Dia. Source (psi) Failure (min.) (inch) -.63 258.3 - 2 Dry Ice -3.01 211.0 3.98 1 Dry Ice -4.31 138.5 1.876 1 Dry Ice -2.41 51.83 8.21 2 Dry Ice -.789 247.5 39.78 2 Dry Ice -.200 490.8 152.19 2 Dry Ice -.021 77.05 - 2 Freezer + Average values. A.uniform cooling rate was achieved only over limited temperature ranges due to equipment limitations. 3.3.5 Stress Relaxation Behavior Soil creep involves slow and progressive deformation of the material with time under a constant tensile stress. When deformation (or strain) is held constant, the tensile stress will gradually decrease or relax with time. A combination of these processes occurs during and after thermal contraction in frozen soil. Thermal contraction test results (section 3.3.4) showed that increase in tensile stress was reduced for lower cooling rates due to creep and stress relaxation. To provide more information on the effect of cumulative strain (damage) resulting from cyclic thermal contraction on frozen soil 90 tensile strength, a series of cyclic constant strain tests were run on duplicate samples of the type described in section 3.3.1. After mounting the specimen and allowing the temperature to come to equilibrium.at -15 '6, an axial strain was applied to the specimen after which the strain was fixed. These strains, including 6.6 x 10.4 in/in for the fourth cycle, 12.0 x 10-4 in/in for the second cycle, and 10 x 10.4 in/in for the forth cycle, were applied prior to each cycle so as to simulate a permanent known deformation (plastic strain) representing past sample strain history. Selection of these strains was arbitrary, but did recognize that a total tensile strain of about 100 x 10-4 in/in would likely lead to failure (Bckardt, 1982). The sequence of load applications with the above strains is illustrated in Figures (3-33), (3-35), (3-37), and (3-38). Figure (3-33) illustrates a reference value for stress and strain for a one load cycle. This strain value was used to estimate the. permanent cyclic deformation used for the stress relaxation tests. With strains fixed for a 24-hour period, stresses decreased to the point where only very slow changes were observed. At this point the next cycle was initiated. Fatiguing the sample to a fixed total strain and then breaking it showed effects of stress and strain at failure. This simulated what really happens in the field. In these tests, holding the temperature constant while cyclic strains where imposed in the laboratory simulated the cyclic cooling in the field. The cyclic strain showed the following phenomenon: The series of limited cyclic tensile strains imposed on the frozen sand samples showed that strain controlled fatigue will induce strain hardening and reduce 91 ._ 00050 00o0 .0000 000003000 000o00 no 0009 0000009 000 0om 0309 .0> 000000 000 000000 “0010 003000 00030080 0809 ... 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''''''''' t 1 ON. «I00x0 0N “ - ...... - 00.0.0. .. a _ . ” 0000000: lllllfloflxmo \l 0000000 .I III I 0.0 0 62 000000 0 _ P _ 0 z-OI x “19138 (2) 95 .0000000 000000 0000< 0000 000000 000000000 000000 000 00. 0_|v 00000000500 00000000 0 00 00>000 0000000000 000000 "00:0 000000 000000.00 0.00.0. 0000 000 00 —¢.... . u . ‘0-0041 0 u —¢-4u‘u< . u o I I IIIII 0:0 0/ 1 0: ./I0 00000 0/0/ -Anuvawlllmw n~0 nyx 00,, 0 ,Aww l. Av mu N 000 0 u ///,/z ONN 000 0.00~ . “__- G l E 0.80 JHHLA ,Dva 0 0. 00: 00 0000 000000000 000000 000 00>000 0000000000 000000 3000.25 0.00.0. 0000 000 00 0 0000 000 00 0 0000 000 . 00 . 0 0000 000 00 I. . 0 0.000 .; . wan w NMO 00 «on n.0m . «0 on ..0mw n 000 o o 0 0. o 0 00000 A? 7 o a o o 0 o 0 . 00 000 o 0 o ... . . . . . . . N 00000 . .u 000 0.000 , h 0» o 0 00000 on... 00000 .0 0000.93,: .0 0 . o o 0 . w 000 0.000 .. ..w E m 20. m 62 3.0.00 0 «00 00.000 _ _ _ __ _ _ __ _ ._ __ _ _ “amun 000000 000 000 cam 000 can 000 can (zu;/;q{) 888133 at;sual 97 the tensile strength, i.e., induce stress softening. The data summary (Figure 3-39) showed that failure strains increased from 0.24 2 in 1 cycle to 0.48 1 in 4 cycles, an increase of 100 1. For the same series of 4 cycles the tensile strength decreased by about 3 Z (Andersland and Al-Moussawi, 1988). Table 3-4: Results from Stress Relaxation Tests on Saturated Frozen Quartz Sand at -IS '0. Sample of 2 Reduction 6 f 2 Increase Cycle Number psi in of (X) in/in in 6f (1) Number 1 10" i 722.41 0 24.19 0 1 2 709.9 1.73 38.70 60 2 3 700.80 2.99 48.38 100 4 98 8 100— 8 0 $2 72 {.1 d I 7 I81 c‘” a 'H “-4 u) t; 60- 2 40— o o O) U) m o o o n u 8 8 '5 .u c I: «:3 -a n u a 20— H 0 a u c u o u o u o cn 0 en a u o c o a 9.. NI. :1. «4 0L 4 m L l l 2 3 4 Cycle Number Figure 3-39: Load Cycle Effect on Failure Strain and Rupture Stress for Saturated Frozen Quartz Sand at -15 'C. CHAPTER IV THEOETICAL CONS IDERATIONS 4.] Mathematical Formulation Construction cracks in frozen soil can be considered analogous to a theoretical model of tension cracks in a semi-infinite solid. The effect of the crack in relieving tensile stress at the ground surface relates to the problem of crack spacing. The rate of energy dissipation at the advancing crack tip relates to the problem of crack depth. Even though the stress that causes cracking develops slowly, an elastic model of the stress near a crack can be useful as long as the cracks, once initiated, propagate rapidly. The crack is generally initiated on a plane of great stress (often) at or near the ground surface, and is propagated toward the interior of the medium where the tension diminishes and ultimately passes into compression. It is well known that most soil materials behave elastically only when.defbrmed rapidly. Thus the processes which produce tension (such as thermal contraction) could be modeled elastically. This approach gives a very good approximation to the stress conditions that exist immediately after the crack has formed. This review of theory will serve to present expressions which describe the elastic stress in a semi-infinite solid containing a long crack of finite depth. Before the crack appears, the medium is assumed 99 100 to have a nonuniform stress field with the maxinnm principal stress normal to the plane on which the crack will form. The components of initial stress vary only in the direction of crack depth. The crack will be considered as lying in the plane y - 0 in a strip 0 < x < b for x > 0, Figure (lo-1). The maximum principal stress will be represented by T(x) which is directed parallel to the y axis and varies only with x. The x and z caponents of the initial stress can be assigned later, in conformity with any particular application, as only their component influences the boundary conditions at the crack walls. One can also assume that the stress vector will vanish at the surface x - 0. Under conditions of plane strain the stress in the interior of an elastic body can, in the absence of body forces, be reduced to the Airy stress function U(x, y) such that I. I. 9 3:“ a’xa‘y 3y“ where 2 2 2 a u 3.! a n a U 1’ a -. 2.2 (4—2) " 3yz , 5” axz , ‘Y 3x3y and the Airy stress function U will lead to the prescribed stresses on the boundary. mskhelishvili (1963) defined the analytic function of the complex variable 2, namely, 01(z) and IIIlCz) such that 39 + 1 3‘1 - ¢1(z) + z cp1’(z) + ¢1(z) (4-3) 3: 3y throughout the mass and 101 b/ Inn--.s IIIIICu . /-....I-o /III«II [flluunnh.- r O fiend..- Coordinate System.and Position of the Thermal Crack. 4-1: Figure 102 39 + 1-32 - f(s) (4'4) 3: 3y on the boundary L. The prime denotes differentiation with respect to z - x + iy, the cmplex conjugate is represented by a bar, and f(s) '.f1(s) + if2(8) " i I (Xn + iYn) d8 (4-5) 0 where Xn and Yu are the x and y components of the prescribed surface stresses in the direction of the positive normal on the boundary 1.. The analytic functions 01(z) and ¢l(z) are determined, respectively, within a complex constant and a complex linear function, by the boundary relation. s ¢1(z) + z¢1‘(z) + 1111(2) " i [can + iYn) ds (lo-6) The arbitrariness in 0 and $1, as determined by equation (Ii-6), 1 represents only rigid body motion and hence equation (4-6) completely determines the state of stress in terms of surface force. Applying the caplex transformation z - w(t;) , and setting oplcz) - cplancc» - ¢1 «21(2) - w1 19 C'De for equation (4-5) gives 103 wCC) ¢(C) + ¢’(C) + W(C) w'(t) 8 - 1 I (xn + iYn) ds, 1; - a (4—3) 0 where a is the value of C on the boundary A in the C plane. By setting d¢1(z) ¢'(C) ¢ - - (4-9) dz w'(C) and 2 11m - Ji— I R?) We) + arm] (4-10) 0’ w'(t) one can write Muskehelishvili’s (1963) general results for the 'caaponents of stress referred to the curvilinear coordinates of transformation, as follows oE + on - 4 Re [¢(C)] (4'11) on - OE +211En - H(C) (4-12) This‘approach.is also suggested by Timoshenko and Goodier (1970), and Sih (1973). The components of stress are given explicitly by GE - 1/2 (4Re [0(t)] - Re [H(C)]1 (4-13) 0 - 1/2 {has [0(C)] + Re [H(C)l] (4-14) 11 IE“ . (1/21) In [H(§)1 (4-15) 104 One can use the transformation (Muskhelishvili, 1963) in which 2 - w(l;), is the analytic function. Thus cum - a (c + n/c). R > o, m z o <4-1s) For transformation of the exterior of an ellipse into an exterior of a unit circle a - 1/2 b ' (4-17) So that z - mm - (112) b (c + m/C), o 5 m g 1 ' <4-1a) which maps the exterior of the unit circle in the 1; plane Icl 2 1 into the exterior of the ellipse m g 1/2 b (p - a/p) sin 9 (4-19) M g 1/2 b (p + m/D) cos 9 (4-20) in the 2 plane, as shown in Figure (4-2). The cracking problem centers about the degenerate case m - l, in which the unit circle ICI - 1 corresponds to the double line segment IxI < b, y - i 0. 4.1.1 Case I: The Step Function Consider first the case of a step distribution of self- equilibrating surface stresses on the degenerate ellipse, m - 1, p - 1. This case, shown in Figure (4-3), is described by 105 .Ancm_ .eae>cceaclccxccx aouw counuv2 - 402cos 29 1 l 2 o-PC-[o-ad-a-al-[l--61[1- n 1: 4 1 "0 (p"-292 cos 264-1)2 3 2 zp’(p“-1)ain 29 1 - cos 29)[2p’(coc 29 - cos 290) - (pl-1):] o‘ 1 «(9" - szcos 29 + 1)[p" -, 2p’ccc 2(9—90) + mp" - 2p’coc 2(91-90) + 1) (4-28) 108 and 29’(p‘-1)2cin 29 1 - (2/1090 Tin - M I 2 (0" - 292 cos 26 + 1) (0" - 20 cos 29 + 1) sin 290[2p’(ccc 29 + cos 290) - (ch-1):] - 1} Mo“ - szcos 2(9—90) + 1)(p" - Zozcos 2(6-1-60) + 1) where _1 9 sin 9 + sineo a1 - tan ( 0 cos 9 + cos 00) _1 0 sin 0 - sin 90 a2 - tan ( 0 cos 9’+ cos 90) _1- 0 sin 0 - sin 60 a3 . tan ( 0 cos 6 - cos 60) _1 p sin.9 + sineo a4-tan (pcosG-cosGo) (4-29) (4-30) (4-31) (4-32) (4-33) For the special case a - b, equations 4-27 to 4-29 reduce to those given by Husktelishvili (1963) 4.1.2 Case II: by setting 6 . 0. 0 The Linear-Function Consider the effect of a symmetrical linear distribution of normal pressures over the degenerate ellipse m - 1, p - 1, (see Figure 4-4). For this case Yn " ‘ (Q’b) Xs - + (Q/b) x, x>0 x<0 (4-34) 109 X - 0 n where Q is an intensity parameter. Setting a - e19, obtain from equation.4-5. + if - +1/8 (Qb) (2 + 9‘ + 1/92), -n/2 < 9 < +w/2 f1 2 - -1/s (Qb) (2 + o2 + 1/02), n/z < 9 < 3n/2 (4-35) This leads to the Lachenbruch (1961) derivation of the curvilinear stress components 0 ¢-+QrLu-l+i(t+%znéiil mam 4n c 2 C t - i 2 2 ¢--Qb1_.[_1__(§3+53;+l)+1££_:__1.2.135_111 4n c3- 1 C 2;2 t - 1 and finally a = +Q [2(p“-1) [1 _(pz+1)z sin2 9 + 4p2 sin~ 9] g “(0' - 202 cos 29 + 1) p” - 292 cos 29 + 1) (oz-l)’ p’ + 29 sin 9 + 1 - (sin e + sin 39) 1n[ 1 49 92-29 cm9+1 - .1 [(p‘ + p~ + p2 + 1) cos 6 20 - (o‘ - 39“ - 3p2 + 1) cos 391(91 - 92)) (4-37) 0' = +Q [2(p~_1) [1 40244.): 81112 9 + 402 8111~ 9] n nco“ - 29’ cos 29 + 1) o“ - 2;)2 cos 29 + 1) 110 1 o2 + 2p sin 9 + 1 -«- (pz+l)(D~-l)[3 sin 9 - sin 39] ln[ ] 4p pz-Zpsine-o-l - -1—[(39‘-p"-pz+3)cose 20 - (p6 + p“ + 9’ + 1) cos 391(81 - 82)] (4-38) T = -2Q a (4-42) Interest relative to the thermal crack problem centers about the stress perturbation in the plane x - 0, since this is the plane in which cracks are assumed to originate. The result is obtained by setting 9 - “/2 in equations 4-27, 4-28, and 4-29, thus w 2 (oz-1)’ 2 _1 sin 290 O '- ' -P 1 -'- 9 1 - ---- - - tan ( 5(2) [[ n 011 (p’+1)’] n p2 + cos 290 492(92-1) sin 290 + (4-43) w(p’+1)(p" + 21)2 cos 290 + 1) 1r 2 (oz-1)[(p’+1)+4pzl 2 _1 sin 290 o- -4 1--9 1- --c n(2) [[ n Oll (o2+1)3 ] u an o2 + cos 90 402(92-1) sin 29o ‘ 1 (4-44) 11(pz+l)(p~ + 202 cos 29b + 1) Tfin - 0 (4-45) 112 where y - 1/2 b

0 and the indirect effect of producing new normal stresses, Al'r(x), at the walls of the fracture y - t 0, 0 < x < b. These new stresses A11(x), must be removed from the fracture surface in order to maintain the initial step distribution of normal stress. This requires a method for cmputing Al‘t(x) frc- -an(1r/2). To do this -an(1r/2) is approximated by a continuous even function of y composed of linear segments shown by broken lines in Figure (4-5b). This gives Unfit/2) 9: 2T (4-50) 1 i where T1 - P1(l - |Y|/¢1) for, lyl < c1 and Ti - 0 for, Iyl > c1 Each ten in eqmtion 4-50 corresponds to a triangular distribution of norsml pressures of half-width c centered at y - 0 on the plane x - 0. 1 Starting with equation 4-1, and an assumed function which satisfies the equilibrium and compatibility equations, Carothers (1920) gave the solution for the new stresses at the fracture surface as 2 _1 c1 x oi A1(x)s--ZP (tan ---ln[—+1]} (4-51) 1 i 2 1r 1 1: c1 x Having calculated Al‘r(x), now reaove it from the fracture face by applying normal stresses -A11'(x). ,This is done by approximating -A1‘t(x) by a step function, the effects of which can be described by a linear combination of solutions to case 1 of the preceding section (see Figure 4-5c). Removal of A1T(x) results in a direct contribution Alo€(1T/2) to 117 the tangential stress at x I 0. This is given by a linear combination of expressions of the form of equation 4-43. The horizontal stress then becomes 05(11/2) - On(U/2) + A1°g("/2) for x I 0, and y > 0. In addition, a new normal stress A10n(1r/2), given by equation 4-44, appears on x I 0. However, AIOnUr/Z) decays much more rapidly with y than did the original normal stress Unfit/2). To refine the approximation, - A0 r1(11/2) is represented by an expression of the form of equation 4-50, and the entire process is repeated. All components of the stress in the median bounded by the free surface x I 0 with the prescribed stress -‘t(x) on the crack surfaces y I t 0, 0 < x < b, may be approximted by superposition of the tensor components of the stresses resulting in the infinite medium from the surface stresses -t(x), -on(1r/2), -A1‘t(x), Alana/2), etc. At present our interest centers on the horizontal stress at the free surface x I 0, which will be denoted by 05*(11/2). For N applications of the cycle described above, it is given by N *wz) s z siege/2) - Aianwzn (4-52) 0 E i-o foerO, y>0 where [1095(11/2) I 05(11/2) and AoonUr/Z) . on(1r/2) Values of 05*(11/2) for case 1 are summarized in Table (4-1) and are illustrated in Figure (4-6). For tle linear distribution, case 2, values of 05*(11/2) are given by Table (4-2) and Figure (4-7). These numerical results are of course, approximate and unlike the results of the previous section, which were derived from exact solutions. 118 Table 4-1: Case 1, Step Function. Normalized Tangential Stress o *(n/2)/P, on the Free Surface (x I 0) near a Crack of Dgpth b in a Semi-infinite Medial (Iachenbruch, 1961). a/b y/b We 0.2 03 0.5 0.75 1.0 .05 -0.99 .125 -0.68 -0.98 .25 -0.36 -0.68 -0.98 .5 -0.18 -0.36 -0.68 -0.89 .75 -0.12 -0.24 -0.50 -0.67 -0.87 1.0 -0.087 -0.18 -0.37 -0.51 -0.71 -0.88 -0.94 1.5 -0.051 -0.11 -0.21 -0.31 -0.45 -0.60 -0.65 2.0 -0.033 -0.065 -0.13 -0.20 -0.31 -0.40 -0.45 2.5 -0.024 -0.050 -0.10 -0.15 -0.22 -0.29 -0.32 3.0 -0.017 -0.034 -0.075 -0.11 -0.16 -0.22 -0.24 4.0 -0.010 -0.020 -0.040 -0.059 -0.088 -0.12 -0.15 5.0 -0.006 -0.012 -0.025 -0.036 -0.061 -0.082 -0.097 7.0 -0.003 -0.006 -0.014 -0.021 -0.029 -0.042 -0.049 10.0 -0.001 -0.003 -0.005 -0.008 -0.014 -0.020 -0.023 119 .A_oorl.ao:unoonuoqv Gowns: ouwofiununmaum 4 am Anv NV 0 .+ o s 5v assoc a mo eggs: one so museum communm mo nowusnmuuafia comm < on on: Ac I xv oosuuom sewn on» no anouum usmunowcoa c; S a o o N "ole shaman - ----_— -------- J c.~1 d/I m.OI .0 o.ol.s44 o5- (Z/JL) N.OI u.cl 120 .A_om_lusosuooocosav :33: 3233-3...” < 3 3v xv o ... o .. b .326 4 no edge: can so ouuuum oosuusm mo oofiuscduuuwa nausea s a» can Ac I av oosuusm some on» no enouum uofiusouasa "sue susmwm s: s . . .. _. .... _ _ l H I I... rol b I III L Ms? D x $44 I / l ~61 a. I; \I 1. as ~.on mu 1,; m Ill-IIIIIII‘ - p — — o 121 Table 4-2: Case 2, Linear Function. Normalized Tangential Stress a *(11/2)/Q. on the Free Surface (x I 0) near a Crack of Dgpth b in a Semi-infinite Medim (Lachenbruch, 1961). y/b 0601/2) IQ I0.25 I0.30 .0031 '0.25 I0.17 -0013 I0.10 I0.056 .00040 -00018 e s s e OOOOUIOUICN 11.3 NMO'UNND-‘HOO 4.2 Thermal Contraction and Fracture 4.2.1 The Elastic Problem Vertical cracks form when tensile stresses exceed the frozen soil tensile strength. With colder temperatures and greater contraction at the gromd surface (semi-infinite soil mass), the cracks will propagate dowinlard into the frozen soil. Tensile stresses in the adjacent soil are reduced so as to give a crack spacing with stress over the entire surface reduced below the tensile strength of the frozen ground. With an increase in depth and overburden pressures, hydrostatic compressive stresses increase until they exceed the thermal tensile stresses. For these conditions, to what depth will the crack develOp? How does crack developmnt, with formation of new boundary conditions, alter the stress casponents? The problem can be formally stated in terms of the following boundary conditions : o§(x, 0) I -T*(x), y I 0, 0 < x < b (4-53) cg, 05’ Tin II) 0 x, y -> I (4-54) v(x, 0) I 0 y I 0, x > b (4-55) On(0, y), TE"! (0, y) I 0 x I 0 (4-56) w(x, y) I 0 all x, y (4-57) where v and w are displacements in the y and z directions, respectively. When the crack forms at y I 0 to a depth b, horizontal stress on the crack surface in the y direction must go to zero. This is formally satisfied by superposing a normal stress O€(x, 0) I 'l’*(x) on the two crack surfaces as stated in equation 4-53. Equation 4-55 states that the crack penetrates only to a depth x I b. Equation 4-56 states that stress. goes to zero at the horizontal ground surface. Equation 4-57 states that the medium is constrained in the direction of the crack such that the problem is one of plane strain. Consider the horizontal stress relief at the ground surface in the direction normal to the crack, 05(0, y). Equations 4-53 through 4-57 were solved using Muskehelishvili’s (1963) method as described in section 4.1. Consider first case 1 with a step pressure distribution of magnitude P, to the depth a, on the crack surface, i.e. , 05(x,0)I-P, 0 1 in Figure (4-7). 4.2.2 Crack Stress Intensity Factor For the fracture process, the most important quantity to be derived from a stress analysis is the stress intensity across the plane of fracture in the neighborhood of the crack tip. In particular, the 124 stress component Cy in the crack tip vicinity must be determined. The stress-intensity factor It: for a mode I crack is then defined Li I lim (0 (r, 0) 1’ Zn) (4-59) r + 0 y where r is the radial distance from crack tip (Bellan, 1984). The parameter It depends on the crack length and stress distribution in the I soil medium. To determine KI for problems under consideration, it is necessary to investigate the limit of on(9 I 0) as o + 1 where p I 1 and 9 I 0 are the coordinates of the crack tip. When stresses on the crack surface are represented by a step function for case 1, equation 4-28 yields an(0) I lim 33-— [l - g 90] + constant (4-60) ‘9 + 1 01-1 11 From equation 4-18 and Figure (4-2), it can be shown that as p+l,r+0,and 1 1 CD... .1. (4-61) 0’ - 1 2 f7: Using equation 4-61 and 90 I cos.l a/b gives an I glino pf? [1 - (2/11) cos-1 (a/b)] l//-2-r + constant (4-62) where p is the tensile stress distribution on the crack surface, b is crack depth, and a is depth for stress distribution p. Using equations 4-59 and 4-62 leads to 125 KI - on [1 - (2/11) co." (a/b)] (4-63) When stresses on the crack surface are represented by a linear function ' p2 + l O I lim (2Q/fl) --—- + constant (4-64) n r + 0 92 - l where Q is the tensile stress (linear distribution) and p is the radius of curvature for the crack surface. Use of a linear stress function with equation 4-59 leads to KI - (2./T) (2/11) (4-65) Along with Lachenbruch's (1961) solution for stress intensity factors, Sih (1973) summarized, in his handbook, stress-intensity factors for various cases. A combination of these cases and superposition permit use of these factors with the stress distribution on the crack surface of interest. For example, consider a partially loaded edge crack in a semi-infinite mass (Eartranft and Sih, 1973; Sih, 1973). Introduce K with F(a/b) as given in Table (4-4) for various I values of (a/b). Then ~ KI - 2/11 cos" a/b [1 + F(a/b)1 0 ft? (4-66) Koiter (1965), and Koiter and Benthem (1973) introduced the stress intensity factor KI for stress distribution on a crack surface in a semi-infinite sheet (Figure 4-9) as 111 - (1.1215 0 + 0.439 02) /'1? (4-67) I 126 { 1 T 7 | a 1 b ”L Eli. z; i I ‘L O- f"‘ rI-fi U : : : «>1 . i i E 1 Figure 4-8: Partially Loaded Edge Crack Where F (a/b) is Given in Table (4-3) for Values of (a/b) (Sih, 1973). ,4 A 11 ll '1 '1 I l 1 I 1 \Jr I..— Figure 4-9: Crack in A Semi-Infinite Sheet (Koiter, I973). 127 where b is crack length, a is stress at the crack tip, and (a1 + 02) is 1 stress at the mouth of the crack. For comparison, Sih's (1973), Koiter and Benthemis (1973), and Lachenbruch's (1961) stress intensity factors will be used later in the field example. For the case of a crack extending downward from the surface of a semi-infinite medium, as described in the preceding section, it appears that KI can be approximated as follows: K1 :1 KI(-T) + KI(IAIT) + Lib-A21.) + .... (4'68) where KI(-T) is the value associated with stress distribution of (IT) on the crack surface. Values of K1, for a semi-infinite body, are presented in normalized form in Table (4-4) for both the linear and step functions. Table 4-3: Values of F(a/b) as a Function of (a/b) ' (Hartranft and Sih, 1973). a/b F(a/b) a/b F(a/b) 0.0 0.12147 0.6 0.04624 0.1 0.10984 0.7 0.03408 0.2 0.09733 0.8 0.02244 0.3 0.08443 0.9 0.01383 0.4 0.07150 1.0 0.00000 0.5 0.05874 128 Table 4-4: Normalized Crack-Edge Stress-Intensity Factor Y (Lachenbruch, 1961). 1. Step Function, Case 1 77 Y(a/b) - KI(a/b)/P/—b * Infinite Semi-Infinite a/b Body ° Body ° 0.05 0.0317 0.04 0.1 0.0635 0.08 0.2 0.1285 0.16 0.3 0.1940 0.24 0.5 0.3335 0.41 0.75 0.5399 0.64 1.0 1.0000 1.1 2. Linear Function, Case 2 v(a/b) - ifs/burr? ' Infinite Body Semi-Infinite Body 0.6366 0.68 Notes: + ‘1 - p f1)- [1 - 2/1r cos-la/b] 11 K1 - Q r"? (2/11) * K1 1: KI(-'r) + KI(-Al't) + KI(-A21) + ° Infinite body (Exact solution) Semi-Infinite body (Approximate solution) 129 The stress intensity factor K depends on the assumptions made in I’ derivation of equation 4-59. In general, for uniform tension on the edge crack, the normalized value of RI is in agreement with different methods used to calculate KI from 1958 up to 1973 as shown in Table (4- 5). The calculation for a normalized KI, used by Bartranft and Sih (1973), is more reliable since they used different methods to check the results. Their results will be used later in the field example. Table 4-5: KI Values For Uniform Tension on an Edge Crack (Bartranft and Sih, 1973). Source ILL/o f1)- Irwin, 1958 1.1 Bueckner, 1960 1.13 Eoiter, 1965 1.1215 Wigglesworth, 1957 1.122 Lachenbruch, 1961 1.1 Stallybrass, 1970 1.1215 Senddon, 1971 1.1215 Sih and Bartranft, 1973 1.1215 130 4.2.3 Crack Depth To investigate some implied relationships between crack depth, stress distribution, and material properties, consider the simple case of a semi-infinite solid in which the stress is represented by a uniform tension p to some depth a. The tension can be thought of as resulting from thermal contraction. For a long tension crack in an ideal elastic medium the rate of strain release energy with crack extension G (crack extension force), Young's modulus E, and Poisson's ratio v, are related to the crack stress intensity factor KI (Irwin, 1957; Hellsn, 1984), thus 2 G I (B/E) KI (4-69) where l for plane stress 1 - v2 for plane strain For the step function and plane strain, equations 4-62 and 4-69 give /-' _ I1 _ 2 1/2 p b [1 (Zlfl) cos a/b] ' [G E (l v )1 (4-70) Solve for the crack extension force 2 9 - 29.2.2.1“,(1 - 3- cos"1 9- )12 (4-71) 2 11 b which represents Lachenbruch's (1961) solution. Similarly, Hartranft and Sih (1973) derive the crack driving force as 2 129—12.2 [Z “3‘1 2 [1 + F(a/b)]O]2 (4-72) B U D G- 131 In general, b in equation 4-71 will have two values. The larger value corresponds to the ultimate depth, the smaller value represents the critical crack depth bc necessary for the onset of unstable prOpagation. For a stable crack penetrating a semi-infinite solid with a constant homogenous initial stress, KI will increase as the square root of crack depth. However, if the tensile stress decreases faster with depth x than x1, 2, KI will decrease for a growing crack. Hence, under the present assumptions regarding stress, if It does not assume the critical value KIC by the time the crack reaches depth a, it never will, and propagation will be stable and confined to the surficial layer x < a. If the crack grows unstably to depth bc in the surficial layer x < a, it can generally be expected to prOpagate to some depth b > a if a is not too large. It is likely that a thermal contraction-crack in frozen sand, formed by tension, will propagate unstably in response to surficial tension. If the crack depth is small, it will be very sensitive to Go’ for "crack arresting" which is lower for more brittle and hence colder media. For the case Go I 0, an upper limit to crack depth is given, as it represents an ideal medium in which no energy is consumed by the crackIextension process. Thus the crack continues to pr0pagate until its crackrdriving force falls below Go. In general, Go is expected to be less than the critical crack-driving force Gc needed for crack initiation. 132 4.2.4 Crack Growth Resistance Curve (RICurve) An RIcurve is a continuous record of toughness development in terms of crack-extension resistance Rh plotted against crack extension in frozen soil as a crack is driven under a continuously increasing stress intensity factor KI. Recall that ER is a measure of the resistance of material to crack extension expressed in terms of the stress intensity factor, KI. Assuming that the frozen soil would remain predominantly elastic throughout the duration of a test and restricting our attention to a mode I failure and small-scale yield, the criterion of linear-elastic fracture mechanics can be written as :1 - 51c (443) where RIC is stress fracture toughness. The value of KIC is the value of KR at the instability condition determined from the tangency between the RIcurve and the applied KI curve of the specimen Figure (4-10). The term ‘10 is a function of load, sample geometry, and material proper- ties. Equation 4-73 describes a necessary condition for the onset of crack propagation. The fracture toughness KIC’ the R curve, and the critical length of the crack ac, must be evaluated in order to obtain a complete picture of material behavior during crack propagation. ASTM standards E 616-82, E 399-83, and E 561-86 provide a standard method to evaluate the individual parameter for fracture mechanics of materials. These ASTM standards and information presented by Paris and Sih (1964) provide a procedure which appears to be suitable for determination of the fracture 133 toughness of frozen soil. The RIcurve can then be determined by drawing fracture toughness KIC versus the change in crack length As. When used for metals this procedure for determination of RIC can be summarized as described below. Some modifications in the procedure may be required for frozen soil. 1. 2. 3. 4. 5. Run a uniaxial tensile test on the material at a strain rate close to 0.02 inch/min and determine the 0.22 offset yield strength Oys and the modulus of Elasticity E. Based on the ratio aye/E, select a crack length from the list given in Table (4-6), taking into consideration that our material is frozen Ottawa sand. Note that Table (4-6) is recommended for metals. Select a crack length and run a fatigue test using stresses less than 1/2 the yield strength 0y so as to give a sharp notch. Run a uniaxial tensile test on a sample with a sharp notch. The load corresponding to a 2 Z apparent increment of crack extension is established by a specified deviation from the linear portion of the record. The RIC value is calculated from this load by an equation that has been established on the basis of an elastic stress analysis of the specimen. Superposition of the elastic solution for RI on the RIcurve (Figure 4-10) will give a value for the critical crack length at which the crack will form.unstably. n no .mQ 134 H can if. u Ao©\s©v 0.. Gov as o o. cannons: 135 Table 4-6: Minimum.Recommended Crack Length (ASTM E 399-82). o [E Minimmn Recommended ys Crack Length (in) (mm) 0.0050 to 0.0057 3 75 0.0057 to 0.0062 2.5 63 0.0062 to 0.0065 2 50 0.0065 to 0.0068 1.75 44 0.0068 to 0.0071 1.5 38 0.0071 to 0.0075 1.25 32 0.0075 to 0.0080 1 25 0.0080 to 0.0085 0.75 20 0.0085 to 0.0100 0.5 12.5 0.0100 or greater 0.25 6.5 The elastic solution for KI is a function of specimen geometry and the applied load. Paris and Sih (1964) introduce an elastic solution for a round bar subjected to tension, similar to the frozen sand tensile sample described in section (3.3). The stress intensity factor KI for the specimen shown in Figure (4-11), is 1/2 KI - onet (U/D) F(d/D) (4-74) where duct is the tensile stress on the sample, D is the outer diameter, d is the notched-section diameter, and F(d/D) is a function of the diameter ratio given in Table (4-7). One can regard equation (4-74) as 136 /l 1 6 -30° to 60° Figure 4-11: A Circumferentially Cracked Round Bar Subjected to Tension, (Paris and Sih, 1964). 137 KI - f (Oi / fla) i I 1, ...... , n (4-75) where a is crack length. If a is variable, say (01, 02, O3, .... ’On)’ then one can generate a series of curves from equation (4-75) in which KI and ae (effective crack length) are variable for a fixed value of ai' As shown in Figure (4-12), the Oi curve, which is tangent to the Rr curve, will give the critical crack length ac. Table 4-7: Stress-Intensity Factor Coefficients for Notched Round Bars (Paris and Sih, 1964). d/D F(d/D) d/D F(d/D) 0 0 0.70 0.240 0.1 0.111 0.75 0.237 0.2 0.155 0.80 0.233 0.3 0.185 0.85 0.225 0.4 0.209 0.90 0.205 0.5 0.227 0.95 0.162 0.6 0.238 0.97 0.130 0.65 0.240 1.00 0 138 c 0 .fi . a seem xoauu fiawumnu so on nooneouuoo a mas o .uHx you nondm> .nofiuomvoum huwuwnwuanu you use: o>uno M nowdoo< new o>u=ULm ecu uo nowueuooaounom owuoaonom u~_1e ounmwm we seem xomuo o>wuoommm Ho>oH mood nouns mo>uoo M vouaan< no1o soeoosom muons“. no momma Nb \ mousaooaou one as ‘ Jonas; Kapsuauul 883133 partddv Ix Xx ‘aonsnsrsaa unmois noeig 139 4.2.5 Experimental Method to Determine the RICurve This method involves testing of notched specimens in tension that have been precracked in fatigue. Load versus displacement across the notch at the specimen edge is recorded autographically. The load, corresponding to a 21 apparent increment of crack extension, is established by a specified deviation from the linear portion of the record. The KI value calculated from this load by equation 4-74 represents the fracture toughness KIC' The validity of this determination of ‘IC depends upon establishment of a sharp-crack condition at the tip of the fatigue crack in a specimen of adequate size. A sharp-crack under severe tensile constraint, will have a small plastic region at the crack tip compared to the crack size. Specimen size required for testing varies as the square of the ratio of toughness to yield strength of the material. More details are given in AS!!! standard E 561-86. 1 As shown in Table (4-6), and based on earlier comments on dependence of fracture toughness on specimen geometry, information is needed to establish adequate size for frozen soil specimens. Note that Table (4-6) gives values for metal specimens. Recall that frozen soil has a lower yield stress and modulus of elasticity than metal. Highly computerized and controllable machines with an adequate load cell and standard fracture mechanics crack opening displacement transducers, such as the Material Test System.(MTS) Units or Instron Units, are recommended for use in.measurement of KIC' This equipment was not available to this project and measurement of k remains for future IC research. At this stage, a comparison of theoretical and experimental fracture mechanics for frozen soil is not available. CHAPTER V FIELD EXAMPLE 5.1 Field Conditions -- Fay, North Dakota Results of the last three chapters and available theories may now be applied to a numerical example. Among three different locations considered, temperature conditions for Fargo, North Dakota, was selected for the field example. No field sites in North Dakota were examined as to where cracking occurred. Thermal cracking has been observed in Michigan for less severe temperature conditions. The data selected may not represent a typical season, but thermal contraction cracking is predicted for the soil conditions assumed, as will be shown later in the mmierical example . 5.1.1 Soil Profile Analyzed The soil profile analyzed is frozen sand. "Frozen sand" is a term applied to those sands in which below freezing temperatures exist and in which at least a part of the water contained in the sand pores is frozen. The sand particle size ranged from 3.36 mm (0.132 inch) to 0.074 mm (0.0029 inch), which consists mainly of sub-angular quartz particles with specific gravity Gs equal 2.65. Unsaturated frozen sand is a complex four phase system consisting of four interrelated components: solid mineral particles and ice, liquid water, and air or gas. The ice matrix serves to cement the soil particles into a much 140 141 more‘coherent mass (Tsytovich, 1975) which can resist large tensile stresses. The physico-mechanical processes present in freezing sand produce properties and a structure which are quite different from those of unfrozen soils. A partial or almost complete change of water into ice is accompanied by the appearance of ice cementation bonds between the sand mineral particles, and by sharp changes in the physical and mechanical properties of the sand. With quartz as the primary component of sand, essentially all moisture will change to ice in frozen sand. During cooling of frozen sand, especially in zones of intensive phase change of water into ice, there occurs continuously a redistribution of moisture and water movement away fran the plane of cooling as the water expands about 9 percent on freezing. Changes in the frozen sand temperatures control the degree of ice caaentation of particles. However, at any temperature below freezing, there always remain small amunts of unfrozen water (Tsytovich, 1975). Since the cementation bond in frozen sand is a function of the ice. content, it is necessary to determine the amunt of unfrozen water at any temperature. Methods for estimating unfrozen water contents were reviewed in section 2.2.1. Below the water table the sand is completely or almost completely saturated. Above the water table the degree of saturation depends primarily on the grain size characteristics of the soil as shown in Figure (5-1). A typical degree of saturation for sand is between 302 to 352. This will 'lead to a water content of 5.72 to 6.72 for sand with a specific gravity C1' of 2.65, an average void ratio e of 0.51, and effective grain size D10 of 0.2 mm to 0.3 mm, as shown in Figure (5-1). 142 Gravel Sand Silt Clay 0.06 0.002 s o - r I g 20 k-- m u . 1‘ a s 1) o u 40 en a L o "a 9 .0 11- 0 C ‘ll ~C ‘ ‘1 "1 I} 11‘ g, 80 ~ ~ 31 «r U ‘75 __190 2 1 .5 .2 .1 .01 .001 .0001 Effective Grainsize D1 in mm, Logarithmic Scale 0 Figure 5-1: Approximate Relationship between Effective Grain Size and Degree of Saturation in the Zone of Soil Moisture in Temperature Zones with Moderate Rainfall (Terzaghi, 1952). 143 5.1.2 Surface and Ground temperatures. "Temperatures at the air-ground interface exhibit random daily and yearly variations due to the partially independent fluctuation of climatological factors influencing these temperatures" (Aldrich and Paynter, 1953). For numerical computations it is convenient to assure a step change in surface temperature, and/or a sinusoidal change in surface temperature. The step change in temperature approximates a sudden change in air (surface) taperature and can be used for calculation of the change in temperature of the soil mass which is assumed to be initially at a constant temperature. To avoid the nonlinearty problem due to latent heat, assume that during the period of interest the depth of frost penetration has a very limited fluctuation. It is convenient to describe the soil relative to two zmes: frozen and unfrozen. For each surface temperature the frost depth represents the lower boundary for the frozen soil. Now consider the sari-infinite soil mass as show in Figure (5-2), maintained at sme initial uniform temperature To. The surface temperature is suddenly lowered and maintained at a temperature Ts‘ Now asstne an expression for temperature distribution as a function of time. The one-dimensional differential equation for tenperature distribution T(x, t) is 2 L! I J— .32 (5-1) 3x2 1: 3t where symbols are defined after equation 5-2. 144 Sudden Change in Temperature Ts, XIO Frozen Soil Figure 5-2: Transient Heat Flow in a Semi-Infinite Frozen Soil. 145 The boundary and initial conditions include T(x, 0) I To, and T(o, t) I Is for t > 0 Equation 5-1 has been solved by Arpaci (1966) using the Laplace transform technique with the result T(x, t) I T x ’ - erf-I——-- (5-2) To - 13 2 / at where T(x, t) is the ground temperature at depth.x and time t, To is the initial uniform ground temperature in degrees Fahrenheit, T8 is the applied step surface temperature in degrees Fahrenheit, (1 is the soil toss coefficient of themal diffusivity in ftzlday, of , x S frost depth n = (5-3) a“ , x > frost depth and t is time after application of the step change in surface temperature in days, x is depth below surface in ft, and erf is the error function. The Cams error function in equation (5-2) is defined .3... No? 2 ecfz‘ - 7%: 1 c“'1 dn (54.) Val: 11 0 where n is a dummy variable, and the integral is a function of its upper limit. It can be shown that _ (5-5) 2 Jo"8 d8: n 2 0 146 andthus ”...; 8,3...2. 6.1 .470 11 2' The error function can be approximated‘as n x2n-1-1 3 5 erfx=-2— z (‘1) --3 sauna unannoum one: we nofiunnfiuunwn "e1n ounwwm \ / .4 \ / .. (..., e .. . - K .I......... ......HE ....p “ ......ixlh-..jn..aw..-_=aa_fi%aF/=_._=— {SNSSW§§N .Ei._m....5=aamnuar..! e . . 1.. . . . .... . II 156 Sea. .ouuom a: on» use he: on» no unset—case 110310133303 Y Season souwwuom name—so: one an unowuwmmooo 4 31m one»: 1 .cupuzcren :23... Q23 3 3 ca 0.. ..o no 34 . Io ..0-2. 86 coo. . So .062. 2/7/ I “v72 _ A _ A _ / /// /r. )/ / ..o / [,7/ . //// / // / // / I .... /V / :// / / _ ////fl7/V///// ... . .rw/ AV/ /// /// o.o // // /yV/K ////1/ ho V... )9) , 27 / , // .. /i )%%V //V )/// ... A .r// M%M )W/ // / OV/V/IWWI/fl [... W final //1 /.//[fl/M/ __ _. sum... __ _ 1W1./..1.%W. (or 157 Table 5-3: n Factor Data, General Surfaces (Lunardini, 1981). Surface - Freeze Location Type n I + Sprue trees, brush moss over peat soil .29 5042 Fairbanks, Alaska Brush and tree cleared, moss in place, peat soil .25 5042 Fairbanks, Alaska Vegetation and 16 in. of soil stripped clean .33 5042 Fairbanks, Alaska Turf . 5 _ A1 aska, and Greenland Snow 1 .0 _ Al sake, and Greenland Sand and gravel .9 ___ Alaska, and Greenland ' Gravel . .6 5042 Fairbanks, Alaska Pavement without snow .9 .__ Fairbanks, Alaska Sandy soil, with snow .49 1908 Lakselv .02 2034 Os, Norway .53 342 Amli, 1974, Norway 1.39 234 Amli, 1975, Norway + Ia I nF I freezing index for the ground surface. 158 n I 0.9 (from Table 5I3), k I 3.657 W/mPk (from Example 1), f ku I 2.182 W/m'k (from Example 1), and kav I 2.915 W/m'k. Compute the freezing index based on local climatological data given in Table (5-4). From Table (5I4) note that the length of the winter season I 30 + 31 + 31 + 28 + 31 I 151 days. Now compute the freezing index PI I I15 x 31 I 12.66 x 28 I 4.44 x 31 I 2.61 x 30 I 10.66 x 31 I I1365.88 'C day. In the same manner compute the thaw index TI I 3021.18 'C days. Using equation 2I38 compute the latent heat for the soil, thus 5 L - w L' - 1693.18 (21.13/100) 333.7 - 1.193 x 10 kJ/m3 f pd From example 1, obtain the volumetric heat capacity C I 1790.8 kJ/m. 'C. vf Using the freezing and thaw indexes compute the mean annual temperature T; I (FI + TI)/365 I 4.535 'C I V0. The equivalent step temperature V, I n FI/t I 0.9 (1365.88)/151 I 8.141 'C. The thermal ratio a I V'O/Vs I 0.5570 and the fusion parameter 0 - v8 x C/L - 8.141 x 1790.9/(1.193 x 105) - 0.122. From Figure (5I6) read A I 0.9 and compute the frost depth x I 0.9 //7200 x 2.915 x 0.9 x 1365.88 x 24/(1.193 x 108) I 2.05 m, which is in close agreement with the 80 inch depth shown in Figure (5-5). It is obvious that the depth of the frost for sand with 301 moisture saturation at Fargo, ND, will be the highest among the three locations, shown in Table (5-5). 159 Table 5I4: Average Monthly Temperature over 54 Years (1900-1954), in Fargo, N.D. + Month Mean Days of Month Temperature (' F) Jan. I15.00 31 Feb. I12.00 28 March I 4.44 31 April 5.66 30 May 12.55 . 31 June 18.05 30 July 20.94 31 Aug. 19.77 31 Sep. 14.5 30 Oct. 7.22 31 Nov. ‘ -2.61 30 Dec. I10.66 ' 31 + Source: Local Climatological Data, U.S. Department of Commerce, National Oceanic and Atmospheric Administration, Environmental Data Service, National Climatic Center. Ashville, NC. 160 Table 5-5: Maximum Frost Depths for the Three Locations during the Year for Sand, Silt, and Clay Soils. T Frost depth, cm. Location Sand Silt Clay yd - 16.6 kN/m3 yd - 14.0 1111/1113 yd - 16.0 1111/1113 w I 21.13 2 w I 26.2 I w I 24.2 2 S I 100 X S I 80.0 2 S I 100 2 Fargo, ND 1973-74 205 155 196 Madison, WI 1978-79 157 110 133 Lansing, MI 1983-84 116 83 100 Notes: ya I soil dry density (kN/ma), w I water content (2, dry weight basis), and S I degree of soil saturation (1), average from Figure (5-1). The ground surface was assumed to have no snow or turf cover. + 1 kN/m3 - 101.99 kg/m3 161 5.2 Numerical Results 5.2.1 Thermal Stresses and Soil Strength Contraction cracking is most likely to occur when surface temperatures are falling rapidly. A period of rapid cooling occurs in late fall, and others occur during winter storms, for example the period of February 2I3 in Figure (5I3). Although cooling rates are not appreciably greater during mid-winter fluctuations, thermal stresses are likely to be on the order of 100 percent greater in mid-winter when the ground is more solidly frozen. For elastic behavior the thermal stress can be computed from 1*(x) I I J— 3 AT (5-12) 1 I v where E is Young's modulus, v is Poisson's ratio, a is the coefficient of thermal contract ion, and AT represents changes in temperature. Values of E, v and tensile strength for frozen Ottawa sand were selected from experimental results reported by Bragg (1980). The coefficient of thermal contraction is based on data reported in Chapter III, Figure (5- 7), (5-8), and (5I9). These values permit calculation of thermal stresses at different soil depths as summarized in Table (5I6), and Figure (5-10). Exapple 3 . The thermal stress and strength at a depth of 20 cm and hour 21 are computed as follows: At hour 21 the frozen soil temperature was I25.97 'C at a depth I 20 cm (Table 5I2). AT I I25.97 I 0 I I25.97 °C 162 From Figure (5-7) and (5I8) the experimental values of E and v are, 9 I I0.08625 (Average value) and 5 5 E I [0.55 I 0.424 T] x 10 I [0.55 I 0.424 (I25.97)] x 10 - 11.561 x 105 psi 60-] From Figure (3—4) obtain a - 25.90 x 10‘ c at T - -25.97 '0 Using Equation (5-12) compute the thermal tensile stress * f(s)---—9—- nAT 1 - v 11.561 x 105 (25.9 x 10'6)(-25.97) - 713.11 psi 1 I (I0.08625) From Figure (5I9) read the tensile strength T(x) I 750 psi From.Figure (3-38) estimate the residual stress to be about 175 psi based on the I15 °C temperature prior to the winter storm. Therefore, the total tensile stress will be about (713.11 + 175) = 888.11 psi which is greater than 750 psi. Based on these calculations tensile failure will occur for frozen saturated sand. Partially saturated sand would have lower tensile strengths and would be much more susceptible to tensile failure. 5.2.2 Crack Depth Prediction Temperature conditions recorded for 2I3 February 1974 at Fargo, North Dakota, have been selected for use in prediction of thermal contraction crack depth at a site with deep cohesionless soil (sand). No snow cover and a bare ground surface will be assumed. Air 163 * Table 5I6: Comparison between Thermal Stresses, T(x)’ and Tensile Strengths, at the Respective Depths, for Temperatures at Fargo, N.D. on Feb. 2I3, 1974. Depth AT E x 105 v a x 10 61 T *(x) T(x) remarks (cm) (deg C) (psi) (deg C) Stress Strength (psi) (psi) I29.44 13.03 I.086 25.11 886 821 I28.62 12.69 I.086 25.27 844 807 I27.83 12.34 I.086 25.43 804 790 T(x) < 1*(x) + TR 16 I26.49 11.78 I.086 25.70 738 780 20 I25.97 11.56 I.086 25.80 713 750 Therefore Tensile 24 I25.55 11.38 I.086 25.89 693 740 failure occurs 30 I25.12 11.20 I.086 26.00 673 735 50 I24.60 10.98 I.086 26.12 649 720 70 I24.56 10.96 I.086 26.12 648 718 90 I24.56 10.96 I.086 26.12 648 718 Notes: E I Young' a modulus extrapolated from data (Bragg, 1980) for warmer temperatures and a strain rate of 1 x 10 sec . T I computed temperatures in the frozen sand at the given depth on Feb. 2I3, 1979, Fargo, N.D. T(x) I tensile strengths for saturated frozen sand extrapolated from data (Bragg, 1980) for warmer temperatures and a strain rate of 1 x 10" sec" . a I linear thermal contraction coefficients, from Figure (3-4). 9 I Poisson's ratio (data from Bragg, 1980). E 1 I v a T (x) I I 0 AT I thermal stress assuming elastic soil behavior. TR I residual stress I 165 psi, (based on Figure (3I41) and T I -15°C. Figure 5-7: 164 _. ,7 - /// E I 0.55 + T tan 1 Modulus of Elasticity E, for Saturated Frozen Sand X 105 (psi) b 1V Average E / G B (Data from Bragg) ' 1980 Split-Cylinder test 123106 pcf . I 32.97% 2 Sand By Volume 2; 642 __ Sample D 63 ‘ _. .41“. / Tinfic // dB Where E in psi — /é 1-22.976° .. h— 1 l L l L l -5 -1o -15 -2o -25 Temperature ( °C ) Effect of Temperature on the Modulus of Elasticity of Saturated Frozen Sand (Data from Bragg, 1980). 165 1 71 1 1 I SD A Average ' 0 Data from Bragg, 1980 .0016 — 1 '- | Split Cylinder Test 1 I :x 1 o u G '13 I 9 5‘3 1 G) n V Average I .08625 m | C) ' ' - -0008 _ g . | 1 .- § 0 t 1 o 3 A i o I m I -0.04 "' 0: g -- 3 l l l l l -5 -10 -15 -20 -25 -30 Temperature ( °C ) Figure 5-8: Effect of Temperature on Poisson's Ratio of Saturated Frozen Sand (Data from Bragg, 1980). 166 700 F- _. 600 -— ' ._ 500 -— - -1 N 5'. E 400 f— ' Data from Bragg, 1980... ...-1 é o g / Split Cylinder Tests 300 - 200 - .4 Tensile Strength 100 - _1 o 1 1 1 1 n 0 -5 -10 -15 -20 -25 Temperature ( °C ) Figure 5-9: Effect of Temperature on Tensile Strength of Saturated Frozen Sand (Data from Bragg, 1980). 167 .euso: _N as osm— .huesuaom man no .euoxsn :uuoz .ouueh us canon aaauo> enouum saw-nos fineness uc_|n ouowmm $53.5 .. b . 595qu «:28. 352.5. com cam occ cow 0 _ d fl _ av fiocm .ol. n.hnuL \ . Rs-.. l of laomu lion voasmm<1.l :ouumauxounn< couuocom doom.ulun .L cc mommouum vouoaaoo.llll 13521 puss uazozg ;o undaq (m9) 168 temperature data at 3-hr intervals are listed in Table (5-1) along with data for 2 other locations. Frost penetration in a sand deposit, above the water table, for Fargo, North Dakota, and the 1973-74 winter season would be close to 205 cm based on the modified Berggren equation (Aldrich and Paynter, 1953). Cooling rates for each 3-hr interval are listed in Table (5-1). Computed ground temperatures and thermal properties for the saturated sand (yd I 1693.18 kg/m3 and w I 21.13 I) are given in Table (5-2). The crack stress intensity factor, KI, due to thermal stress, was calculated as a function of crack depth b using equation 4-66 with values tabulated in Table (5-7). The crack driving force, G, computed in terms of K1 using equation 4-69, is included in Table (5-7). Values for ‘1 and G are graphically represented in figure (5-11). The following example illustrates calculations required for estimating the crack depth. Example 4. Calculation of K and G for a crack depth b I 20 cm.involves the I following steps: From Equation 4-66 101 - 2/11 cos-l a/b [1 + F(a/b)] 0.15 and K - 1.12147 05, for a - 0. I For b I 20 cm.I 7.87 inch, and using the superposition method obtain KI - 1.12147 (1140) #587 - 2/11 cos-l 1/2 [1 + r(1/2)] (9s) /7'."s7 From Table (4-3) read 17(1/2) - 0.05874 and x1 - 2454.53 113/1:13” Recall Equation 4-63 where K I p /b [1 - 2/w cos"l (a/b)]. Compute I KI - 90 f_"7.87 [1 - 2/11 cos" (1/10)] + 750 F_7.s7 [1 - 2/11 cos" (1)] + 95 /—7.87 [1 - 2/11 cos" (1/2)] - 2040.62 1.0/1.113” Table 5-7: 169 Stress Intensity Factor X and Crack Driving Force G. I Crack Depth +KI +0 #K1 '0 b 3/2 3/2 (cm) (lb/in ) (lb/in) (lb/in ) (lb/in) 0.0 0.0 0.0 0.0 5 1342.6 2.579 1196.63 2.049 10 1838.7 4.838 1635.63 3.828 15 2176.2 6.778 1953.27 5.460 20 2454.63 8.623 2040.62 5.959 25 2653.3 10.075 2349.719 7.902 30 2822.57 11.402 2516.364 9.062 50 3489.19 17.42 3135.05 14.06 70 4032.30 23.27 3593.62 18.48 90 4552.84 29.66 4027.18 23.21 (1) v I 0.08625 I Poisson's ratio (Bragg, 1980). (2) Eavg I 6.935 x 105 lb/in2 I average Young's modulus. (3) C I [(1 - v2)/E] K12 I crack driving force for plane strain 4 (Fallen, 1984). + Values based on Koiter (1965) and Hartranft and Sih (1973) equations using superposition. # values based on Lachenbruch (1962) equation for a semi-infinite media using a step function distribution where KI I p/F'[1 - (2/fl) cos.l (a/b)] and superposition of stresses. Note: Lachenbruch's solution gives values 10 percent less than the Koiter, Hartranft, and Sih solution due to different assumptions. The latter values are used in the field example. 170 (“I/JQI) 0 ‘ 39105 Sutntza finals .<~m_ .huesuaom MIN no .ouosmn :uuoz .ouusm you unsound uncauwuoou fiwom vow ousuouonaoa any new a suave wo semuoosm m . no .Ha hauuem havenouon weapon can .u «ouch mowoaua xoouo "__Im unamwm A BE v gamma xumuo com ace coo cow _ _ d _ a 4 a mrl o .mouOh wcfi>fiun xomuo q 11 q o 4 o ozT. q o o o o «N11 O H . V x .HOuumm mufimsoucH mmmuum any! _ _ _ _ oco~ OOON coon occc coon (z/cuI/th) Ix ‘ 103395 AntananuI ssazng 171 Using Equation 4-69 G I [(1 - vz)/E] hi For the Koiter (1965) and Hartranft and Sih (1973) solutions, the stress 3/2 intensity factor is RT I 2454.63 Ib/in and c - [(1 - (0.08625)2)/6.935 x 105] (2454.63)2 - 8.623 Lb/in Using the Lachenbruch (1962) solution gives KI . 2040,52 Lb/in3/2 and 0 - [(1 - (0.08625)2)/6.935 x 105] (2040.52)2 - 5.959 Lb/in These values are listed in Table (5-7). In Example 4 an assumed homogeneous layer of frozen sand with a depth of 2050 mn.was analyzed. ‘In the field other sites may involve a combination of different soil strata. Each layer will have different thermal properties with different unfrozen water contents. Hydrostatic pressures, in the form of compressive stresses on crack surfaces, will tend to close the cracks with increase in depth. In general, for more than one layer both KI and the crack driving force will start from.zero for the zero depth, increase to a peak value at some depth and then return to zero at a greater depth. Therefore, the crack depth will be controlled by values of G and C as explained in Chapter III. 0 Table (5-6) shows that thermal tensile stress is higher than the sum of the tensile strength and residual stress down through the 90 cm section which was analyzed. The estimated residual stress of 175 psi was based on a temperature of -15 'C at the 90 cm.depth were the ground temperature is -24.56 ’C. This indicates that the residual stress of 175 psi is low and that failure would occur below the 90 cm.depth. The error function limitation in equation 5-8 prevents computation of ground temperatures below 90 cm. Temperature at the frost line would be at zero degrees Celsius. Lack of experimental work on fracture 172 toughness evaluation limits the analysis of the critical crack depth, but it appears that for a partially saturated soil with lower strength and residual stresses from prior cooling that the crack would continue through the frozen layer. 5.2.3 Horizontal Stress Relief and Crack Spacing Consider now the effect of the crack in relieving tensile stresses near the surface. Apply the multiple step-function approximation of 1*(x) (Figure 5-10) for assuned crack depths b equal to 5, 10, 20, and 30 cm to calculate the stress at a given horizontal distance. The results using Figure (4-6), are given in Table (5-8) and shown in Figure (5-12). As may be expected, all four cracks relieve surface stress in the same way to a horizontal distance of about 5 cm. Beyond 5 cm the constraint at the bottom of the 30 cm cracks becomes apparent, and the stress relief drops sharply to 5 percent at a horizontal distance of about 200 cm. Similarly, the effect of the 20 cm crack does not differ from that for the 30 cm crack for horizontal distances much less than 30 cm. Five percent stress relief is achieved by the 10 cm crack at a horizontal distance of 82 cm and by the 5 cm crack at about 40 cm. Assuming that crack spacing was determined by the distance to points of 5 percent stress relief, the four crack depths would be associated with minimum spacing of 200, 170, 92, and 40 cm. 4 Varying the crack depth will effect the width of the zone of stress relief, but crack depth is a function of soil brittleness (Go) so that in fine-grained soils the cracks will be more closely spaced as compared to coarse-grained soils. Part of the reason is due to the more plastic behavior of fine-grained soils under concentrated stress (higher Go). 173 .Acno_ .hueounom nun .euoan nuuoz .owuom uou mus: enoueuonaoa no runway Joana onu cu guano: nowuoouwa ago a“ ouuwusm vasouu on» us «madam eeouum moundsuawo ”N_1n enough AEBV oucmumfia Hmucouwuo: occn cemw cco~ oomu occ~ com a _ _ _ _ d . c.~1 I. . m.cl I. . o.o1 was one u m .as can u a :3 m2.a.§.oo~un . . an I as I . H c~w m Gag a .J «.01 I Ea Sana .Eomun . I Noel- . p n o- )1.‘ n o o d/(Z/JLkBD ‘1311811 898135 0321121111011 174 , a: Table 5-8: Normalized Stress Relief 0 (1r/2)/P E , o§*(11/2)/P Horizontal b I 5 cm. b I 10 cm b I 20 cm. b I 30 cm Distance (Zn) p I 853 psi p I 840 psi p I 728 psi p I 680 psi 20 -0.150 -0.465 -0.995 -0.998 40 -0.050 -0.158 -0.499 -0.966 60 -0.025 -0.062 -0.267 -0.512 80 ___ -0.051 -0.159 -0.341 100 ___ -0.025 -0.109 -0.227 120 _ _ -0.086 -0. 170 140 _ __ -0.065 -0.136 160 .__ ___ 0.054 -0.101 200 __ _ 0.031 -0.076 240 _ _ _ -0.056 300 -0.028 Although not illustrated in this example, the effect of varying the stress regime is just as important to the configuration of the zones of stress relief as is varying the crack depth. Thus, a crack forming early in the winter will have a narrow zone of stress relief, as most of the thermal tension will be confined to surficial layers. With the advancing season, thermal stress 1*(x), crack depth b, and the surficial stress relief 05(0/2), will change according to the complex interaction of other factors. Calculations for stress relief are illustrated in example 5 . 175 Example 5. The following steps were used to find the normalized stress relief at a given horizontal distance y from the crack. For a crack depth b I 10 cm and y I 40 cm, the average uniform pressure P on the crack surface, Figure (5-10) gives P I 840 psi. From1Figure (5-10), a I 10 cm and b I 10 can Therefore a/b I 1 and y/b I 40/10 I 4. From Figure (4-5) for a/b - 1 and y/b - 4 read ag*(n/2)/r - -0.1556 as shown in Table (5-8). Values of normalized stress relief are given in Table (5- 8) for different value of crack depth b. 5.2.4 Inelastic Effects after Soil Rupture For previous computations horizontal stress relief was computed from Figure (4-6), which applies strictly only to elastic media. Once a crack has been initiated in a solidly frozen soil, it probably propagates rapidly to an equilibrium depth, b, leaving little time for large-scale inelastic effects. The stress relief depicted in Figure (5- 12) shows the stresses whichuwould be obtained immediately after fracturing. Although the crack acts to decrease distortional stresses near the surface, it concentrates them near its tip. Thus, under a constant thermal-stress regime, a crack that propagated rapidly to some initial equilibrium depth would close slowly from the bottom.at a progressively decreasing rate as the large shears near its tip relaxed. This would be accompanied by a corresponding contraction of the zone of stress relief. However, the stress perturbation caused by the crack could still be approximated by superposition using the new reduced value of crack depth. 176 5.3 Discussion with Design Implications In cold regions, special attention must be directed to the disturbing effects of ground freezing, subsequent thermal contraction, and potential cracking. A substatial decrease in effectiveness of soil covers for limiting water and gas migration may result from cracking. Lutton (1982) suggested that cover thickness be 1 ft more than the average annual maximum.depth of freezing to avoid disturbance of the cover. It is likely that thermal contraction cracks will penetrate the frozen layer as shown in Section 5.2.2. 5.3.1 landfill Covers Landfill covers designed for hazardous waste landfills serve multiple functions. The surface layer (Figure 5-13) is intended for vegetation, the clay serves as a barrier for infiltration of water and prevents escape of gases, and the gravel serves as a gas channel. Layering serves as a technique for designing solid waste landfill covers. By combining two or three distinct materials in layers, the designer“mobilizes favorable characteristics of each material together at minimal expense. The cover top soil typically will be a loose, 6-12 inch, loamy soil suitable for supporting vegetation. The underlying clay barrier layer is a critical cover component because it is intended to minimize the transmission of water that would contribute to leachate generation and of gas that might kill vegetation and pose an explosion or other hazard. A.buffer or foundation layer, sand or other soil, protects the barrier from damage. A buffer soil may also be placed above the clay barrier layer to increase depth in areas of deep frost penetration. Soil 177 ooeooooooooo °soswam":101-1:101:11): 333333333333333.33 """"""""" g “““ . 33. 3333 °33°333333: eeooeogegeoeoooeeo GRAVEL ‘GAS CH N~EL1 boogoooooooooooooz eoeeoooaooeooeeozezsz33535333333337 0000000000000‘ 00000000 0000000000000000 000 00 000 0 ' 00000 0000 0000000000 . . 0000000 0000000000 000000000000000 00000000000000 ’ 3: $ANO‘BUF8'IER) Figure 5-13: Two Representative Systems for Layered Solid Waste Covers (Lutton, 1982). 178 densities will correspond to those accomplished during spreading of cover soil with dozers and other compacting equipment. The top soil is placed in a loose condition and not compacted. For surface soils, including landfill covers and exposed soil liners located above the water table, partial soil ice saturation make these earth structures more susceptable to thermal contraction cracking (Andersland and Al-Moussawi, 1985, 1987). These cracks normally'would not be observed because they occur during winter cold periods, may be quickly covered with drifting snow, and may partially close as spring approachs with warmer ground temperatures. Current EPA design (Lutton, et al., 1979; Lutton, 1982) recommends the placement of additional cover soils to prevent freezing and potential cracking of the clay barrier. This can be rather expensive due to the large soil volumes required to adequately cover the landfill. Use of fiber reinforcement to enhance soil behavior in tension appears to be a reasonable technique to reduce the volume of cover soils needed, greatly reduce construction costs, and help insure that hazardous landfills remain sealed as intended. 5.3.2 Highway Subgrade Soils Modern highway design involves the placement of special pavement layers over subgrade soils. These layers are designated as surface, binder, and base courses as indicated in Figure (5-14). The upper layer may consist of a deep layer of asphalt concrete pavement placed directly on an aggregate base course or Portland cement concrete placed on a granular base. These materials generally have higher thermal conductivities than soils and contain little or no moisture, hence little or no latent heat effects or heat conduction are present. The 179 Seal Coat Surface Course Binder Course Base Course Subbase Course ' Compacted Subgrade 1 L_-_-_.".. ----_-__-------J NaturallSubgrade (a) Portland-Cement Concrete r’ 1 1 Base Course may_gr;may_Not be Used! ---------—- --------J (b) ' Figure 5-14: Components of (a) Flexible and (b) Rigid Pavements (Yoder and Witczak, 1975). 180 result is that freezing temperatures will penetrate more quickly and to greater depths into and below the pavement structure. Pavement structures, typically about 600 mm in thickness, are placed directly on suitable subgrade soils. Good drainage is designed into system so that detrimental effects of high water contents on soil subgrade support are avoided. Good drainage will decrease soil water contents to levels shown in Figure (5-14) with values approaching zero percent for coarse sand materials. Compression tests on partially saturated frozen sand (Alkire and Andersland, 1973) show that strength of unconfined samples decreases almost linearly with decrease in ice content. Since the ice matrix is primarily responsible for frozen soil tensile strengths, it is reasonable to assume that tensile strengths for the pavement subgrade soils would normally be very low and conducive to failure in tension. Transverse pavement cracking, caused by thermal contraction at low temperatures, has been reported as the second principal nontraffic associated mode of distress in highway structures (NCHRP synthesis 26, 1974). Host of the highway agencies surveyed in this synthesis reported that contraction cracks significantly affected the serviceability of their roads and that the seriousness of the problem had been recognized only within the past 10 to 15 years. Fromm and Phang (1972) and Nady (1972) called attention to the fact that transverse contraction cracks penetrate into the subgrade soils. The lower tensile strengths of these subgrade soils in the zone of seasonal frost depth penetration can only contribute to the thermal cracking problem. There is clearly a need for extension of the work reported herein on prediction of thermal crack depth to multiple layered frozen soil 181 and/or pavement systems. The highway engineer should be able to predict potential thermal cracking for highway structures in his state based on local freezing indices, pavement structure, and soil conditions. When thermal cracking is found to be a problem, then techniques must be developed on how to avoid the problem or to minimize the cracking effects on pavement structures so as to avoid costly highway maintenance problems. The preliminary compression tests on fiber reinforced frozen sand samples suggest one approach for strengthening frozen subgrade soils in tension. Stress-strain data for saturated frozen sand (Eckardt, 1982) and for silt (Insulin and Carbee, 1987) in tension shows failure at strains ranging from 1/2 to 2 percent. Fiber reinforcement would increase the axial strain at failure. wa add the increase in tensile strength contributed by the fibers and thermal cracking should be limited, possibly prevented. With a reduced post-peak loss in strength, the strain which does occur will help reduce tensile stresses and thus help prevent the formation of thermal cracks. With seasonal (summer) thaw the highway subgrade soils would return to their normal unfrozen stress conditions before a repeat of lowered temperatures and thermal contraction during the following winter season. CHAPTER~VI SUMMARX AND CONCLUSIONS 6.1 Summary The earth's surface temperatures fluctuate on the order of 25 °C about the mean through the combined effects of changing seasons and shorter period random and diurnal changes. For those regions with mean annual temperatures a few degrees above zero degrees Celsius, for example 4.4 ’C for Fargo, ND,'winter temperatures will freeze surface soils to depths of 2 m and more. During periods of decreasing winter temperatures, cooling of surface soils with thermal contraction can increase tensile stresses to levels greater than the tensile strength. 'For those soils located above the water table with reduced degrees of ice saturation and lower tensile strength, the potential for crack formation is fairly high. This phenomenon creates engineering problems for landfill covers, highway subgrade soils, and hydraulic earth structures (dams, dikes, etc.). Tb provide more information on this problem, preliminary experimental work was conducted which included thermal contraction/ expansion measurements along1with some thermal tensile and stress relaxation tests on a frozen sand. These tests have provided information useful for an analyses of the problem. Compression tests on fiber reinforced frozen sand provided limited information on one 182 183 technique which appears to be suitable for prevention of thermal cracking in surface soils. An example calculation (Chapter V) for prediction of crack depth and stress relief adjacent to the crack is based on theoretical work reported by Huskhelishvili (1963). This work assumes that the crack involves two parallel lines in an infinite media. By pulling these two lines apart, a crack shape will form.an ellipse. The ellipse is mathematically hard to formulate, therefore complex variables are utilized to map an ellipse into a circle. Then a solution for the stress in the region‘was formulated. By mapping the circle back to an ellipse and taking a limit for the manor axis of the ellipse, the problem is returned to the square (one), i.e. (two parallel lines). Using superposition, and considering an attraction free surface for the crack, the solution for the stress, in a semi-infinite media, was formulated from the infinite media solution. Data from experimental work, and theoretical considerations leads to the thermal crack solution presented in Chapter V, the field example. This example illustrates the complexity of the problem and confirms that thermal cracking will occur in the frozen surface soil assumed for the Fargo, ND, example. In the field example local air temperatures were used with the Stefan equation to calculate the depth of frost. The error function, with a sudden step change in temperature was used to calculate temperature of the ground versus time. To calculate the stress intensity factor k1, crack driving force G, and horizontal stress relief distance, stresses were approximated by a step function. For case 1, the step function solution from Muskhelishvili (1963) was utilized. Increasing the crack depth from 50 184 mm to 300 mm, reduced the instability for crack growth, and increased the distance needed for stress relief. For a crack depth of 30 cm, a distance of 180 cm was needed to decrease the stress to 102 of it's initial value, while for a crack depth of 50 cm a distance of 232 cm was needed to decrease the stress to 102 of it's value. Compression tests on reinforced frozen soil, showed that fiber reinforcement improved the compressive strength and increased the strain to failure for frozen sand. Adding fiber to frozen soils will be one of the more promising solutions available to help overcome the thermal cracking problem.in frozen surface soils. 6.2 Conclusions The conclusions from this study involve several areas ranging from soil behavior to fracture mechanics. They are given below under the following headings: thermal cracking, crack depth, thermal contraction coefficients, soil tensile behavior, and soil property enhancement. Thermal Cracking: Frozen surface soils under decreasing winter temperatures are subject to thermal contraction, increase in tensile stresses, and potential crack formation. Recorded air temperatures for three sites -- Fargo, ND, Madison, WI, and Lansing, MI -- all showed temperature changes which have the potential for initiating crack formation. The surface soils for these sites are generally above the water table with only partial saturation. Frozen soil tensile strengths are reduced in proportion to the degree of ice saturation, thus adding to the probability of crack formation when subjected to decreasing temperatures. 185 Crack Depth: An understanding of the fracture process and prediction of crack depth involves use of a "crack stress-intensity factor". This parameter characterizes the elastic stress across the plane of the crack at a small distance in advance of the crack edge. A reasonable crack depth was predicted in the frozen sand using assumed values for the parameter based on infbrmation reported for other materials. Use of crack depth or a high potential for thermal cracking as a limiting criterion in the modification of soils used in landfill covers, liners, and other earth structures appears to be a reasonable future objective. Thermal Contraction Coefficients: Thermal contraction is that change in length, or volume, of a material resulting from a decrease in temperature. Frozen sand is a composite material in.which contraction involves both the quartz particles and the ice matrix, each with its own coefficient of thermal contraction. The effect of temperature on the coefficient for a saturated frozen sand was very small, ranging from 34 x 10.6 "C“1 at -5 l at -25 'C. A change in dry density (1.96 mg/1m3 to 6 '0 to 26 x 10" '0' 2.28 mg/m?) involved a change in the coefficient by less than 2 x 10- 'C-l. For natural sands with densities in the above range, this change in the coefficient can be assumed to be negligible for many projects. A change in the degree of soil saturation will effect the coefficient in proportion to the change in volume fraction of water in the soil. For a sand specific gravity of 2.65 and a change in degree of ice saturation from.60 to 100 percent, the coefficient of thermal contraction would increase by about 8.5 x 106 °C-l. 186 Soil Tensile Behavior: Frozen soil behavior in tension was observed using thermal tensile tests and stress relaxation tests on saturated frozen quartz sand. The cross-plot of tensile stress versus temperature showed the dependence of stress increase on rate of cooling and concurrent uniaxial creep and stress relaxation. The increase in dO/dT at colder temperatures reflects greater'material stiffness with a more rapid stress increase on cooling and agrees with field observations (Lachenbruch, 1961) that thermal crackingtoccurs more readily in colder frozen soils. The stress relaxation tests confirmed that an accumulation of permanent strain over a number of load cycles leads to a decrease in tensile strength. During a series of winter storms, cyclic temperature changes along with an accumulation of strain may lead to rupture when the total tensile strain approaches one percent. The study showed that ' thermal stresses can.be higher than soil strengths, which is the main reason for crack initiation and propagation. But more important than the thermal tensile stress is the thermal history for a given site. Soil Property Enhancement: Frozen soils with partial ice saturation have low tensile strengths, hence they are susceptible to cracking when temperatures decrease by several degrees. Preliminary compression tests on cylindrical frozen sand specimens showed that fiber reinforcement greatly improved their strength properties. The frozen sand with randomly distributed discrete fibers showed an increase in compressive strength, an increase in axial strain at failure, and a reduced post- peak loss of strength. A similar behavior would be anticipated for frozen soils in tension. All three factors would then contribute to a 187 reduction in the probability of cracking‘due to high tensile stresses caused by thermal contraction. 6.3 Recommended Research The application of fracture mechanics to the thermal cracking problems in frozen surface soils has shown the need for information on several topics. The fracture process involves a stress-intensity factor which characterizes the stress across the plane of the crack at a small distance in advance of the crack edge. How should this parameter be evaluated for frozen soils? Other factors involve the energy required to extend a crack, first to overcome cohesive forces and to produce new surfaces, and second, to do the work of plastic deformation at the crack tip. How is this energy related to the strain energy released from the frozen soil when tensile stresses are relieved? Questions remain relative to determination of a crack growth resistance (R) curve for the frozen soil. If the frozen soil mechanical properties are enhanced by fiber reinforcement, howHwill this alter the stress-intensity factor and the crack driving force? A numerical example was used to illustrate factors effecting crack depth and stress relief with distance from the crack in a sand frozen to a finite depth. Parameters are needed which will permit crack depth prediction and stress relief in multiple layered systems, such as highway pavement structures and landfill covers with different soil types. How will the interaction of layers effect stress relief with distance from the crack and crack spacing? Crack formation creates serious engineering problems relative to landfill covers, exposed liners, highway subgrade soils, and hydraulic 188 earth structures exposed to winter climatic conditions. Fiber reinforcement represents a potential method by which the thermal cracking problem can be mitigated in many areas. There is a need for information on the type of fiber reinforcement most suitable and the volume fraction of fibers required to provide the soil properties needed to eliminate or minimize the effects of thermal cracking. Specific information needed relates to fiber volume fraction and (a) increase in tensile strength, (b) increase in axial strain at failure, and (c) the effect on post-peak tensile strength for the fiber reinforced frozen sand. B IBLICXERAPHY BIBLIOGRAPHY Aldrich, Harl P. Jr., and Paynter, Henry M. 1953. Analytical studies of freezing and thawing of soils. Artic Construction and Frost Effects Laboratory, Boston, Mass., Technical Report No. 42. Alkire, B. D., and Andersland, O. B. 1973. The effect of confining pressure on the mechanical properties of sand-ice materials. Journal of Glaciology, Vol. 12, No. 66, pp. 469-481. AL-Moussawi, Hassan M., and Andersland, O. B. 1988. Discussion of "Behavior of fabric versus fiber-reinforced sand", .1. of Geotechnical Engineering, American Society of Civil Engineers, 114 (3). Andersland, 0. B. 1987. Frozen ground engineering. Chapter 8 in GROUND ENGINEER'S REFERENCE BOOK, edited by F. G. Bell. Butterworths, London, pp. 8/1- 8/24. Andersland, O. B., and AL-Moussmvi, Hassan M. 1985. Thermal contraction and crack formation potential in soil landfill covers. Proceedings, International Conference on New Frontiers for Hazardous Waste Management, U.S. Environmental Protect ion Agency. publication EPA/600/9 - 85/025(SEP.). pp. 274-281. Andersland, 0. B., and Al-Moussavi, Hassan M. 1987. Crack formation in soil landfill covers due to thermal contraction. - Waste Management and Research, J. of the International Solid Wastes and Public Cleansing Association, Academic Press Inc. (London) Ltd., 5, pp. 445-452. Andersland, O. B., and Al-Moussmvi, Hassan. 1988. Cyclic thermal strain and crack formation in frozen soils. Fifth international symposium on ground freezing, (26th - 28th, July), Nottingham, England, (Accepted, 1987). Andersland, 0. B., and Anderson, D. M. 1978. GEOTECHNICAL ENGINEERING FOR COLD REGIONS, McGraw-Hill Book Co. , New York. Anderson, D. M., and Tice, A. R. 1972. Predicting unfrozen water contents in frozen soils from surface area measurements. Highway Research Board, Washington, D. C., Record No. 393. pp. 12-18. 189 190 Anderson, D. M., Tice, A. R., and McKim, H. L. 1973. The unfrozen water and the apparent specific heat capacity of frozen soils. PERMAFROST, North Am. Contrib. 2nd. Int. Conf., Yakutsk, U.S.S.R., National Academy of Sciences, Washington, D.C. , pp. 289-295. Arpaci, Vedat S. 1966. CONWCTION HEAT TRANSFER. Addison-Wesley Publishing Company, Reading, Massachusetts. ASTM Desination E 616-82. Standard terminology relating to fracture testing. ASTM Desination E 399-83. Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials. ASTM Desination E 561-86. Standard practice for R-curve determination. Bragg, Richard A. 1980. Material properties for sand-ice structural systems. Unpublished Ph.D. dissertation, Michigan State University, East Lansing, Michigan. Bragg, Richard A.; and Andersland, 0.8. 1981. Strain rate, temperature, and sample size effects on compression and tensile properties of frozen sand. Engineering Geology, Elservier Scientific Publishing Company, Amsterdam, The Netherlands, pp. 35-46, Vol. 18. Butkovich, T. R. 1957. Linear Thermal Expansion of Ice. U.S. Army Snow Ice and Permafrost Research Establishment, Corps of Engineers, Wilmette, Illinois, Research Report 40. Carothers, S. D. 1920. Plane Strain; The direct determination of stress. Proceedings of the Royal Society of London, Series A. Vol. XCVII, pp. 110-123. Department of the Army and the Air Force. 1966. Calculation methods for determination of depths of freeze and thaw in soils. U. S. Governmnt Printing Office, Technical Manual, TM 5-852-6 (Jan.). Department of the Army, Corps of Engineers. 1958. Frost Conditions, Chapter 4, in Airfield Pavement Design. U.S. Government Printing Office, Engineering Manual, EM 1110-345-306 (Mar.). Eckardt, H. 1982. Creep tests with frozen soils under uniaxial tension and uniaxial compression. The Roger J. E. Brown, Memrial Volume, Proceedings, 4th Canadian Permafrost Conference. Calgary, Alberta (March 2-6, 1981), H. M. French (editor), National Research Council of Canada, Ottawa, pp. 394-405. Farouki, O. T. 1981. Thermal Properties of Soils. U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, New Hampshire, CRREI. MONOGRAPH 81-1 (Dec.). 191 Farouki, O. T. 1982. Evaluation of methods for calculating soil thermal conductivity. U.S. Army cold Regions Research & Engineering Laboratory, Hanover, New Hampshire, CRREL Report 82-8. Fromm, H. J.; and Phang, W. A. 1972. A study of transverse cracking of bituminous pavements, Proc. Assoc. Asphalt Paving Tech. Vol. 41, pp. 383-423. Fuchs, H. 0.; and Stephens, R. I. 1980. METAL FATIGUE IN ENGINEERING, John Wiley and Sons, New York. Hartranft, R. J.; and Sih, G. C. 1973. Alternating method applied to edge and surface crack problems. Chapter 4 in MECHANICS OF FRACTURE 1, METHODS OF ANALYSIS AND SOLUTIONS OF CRACK PROBLEMS, edited by G. C. Sih. Nbordhoff International Publishing, Leydon, The Netherlands, pp. 179-238. , Haynes, F. D.; Carbee, D. L.; and Vanpelt, D. J. 1980. Thermal diffusivity of frozen soil, U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, New Hampshire, CRREL, SR 80-38. Haynes, F. 0.: Karalius, J. A.; and Kalafut, J. 1975. Strain rate effect on the strength of frozen silt. U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, New Hampshire. CRREL, RR 350. Haynes, F. D.; and Raralius, Jack A. 1977. Effect of temperature on the strength of frozen silt, U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, New Hampshire, CRREL, Report 77-3. Hellan R. 1984. INTRODUCTION TO FRACTURE MECHANICS, McGRaw-Hill Book Company, New York. Hoekstra, P. 1969. The physics and chemistry of frozen soils. Effects of Temperature and Heat on Engineering Behavior of Soils, Proceedings of an International Conference, Highway Research Board. Washington, D.C., Special Report 103, pp. 78-90. Hooks, C. C. 1965. Laboratory thermal expansion measuring techniques applied to bituminous concrete. Unpublished MSCE thesis, Purdue University, West Lafayette, Indiana. Irwin, G. R. 1957. Analysis of stress and strain near the end of a crack traversing a plate. J. of Applied Mechanics, ASME, V01. 24, pp e 361-364e Jahansen, 0.: and Frivik, P. E. 1980. Thermal properties of soils and rock materials. Preprints, the 2nd International Symposium.on Ground Freezing. The Norwegian Institute of Technology, Trondheim, Norway, Jun 24-26. Jones, G. M., Darter, M. I., and Littlefield, G. 1968. Thermal expansion of asphalt concrete. Proceedings of the Association of Asphalt Paving Technologists, V01. 37, pp. 56-100. 192 Kachanov, L. M. 1986. INTRODUCTION TO CONTINUUM DAMAGE MECHANICS. Martinus Nijhoff Publishers, Dordrecht, The Netherlands. Kaplar, Chester W. 1971. Some strength properties of frozen soil and effect of loading rate. U.S. Army Cold Regions Research and Engineering Laboratory, Special Report 159, Hanover, New Hampshire. Kay, B. D.; and Goit, J. B. 1975. Temperature-dependent specific heats of dry soil materials. Canadian Geotechnical Journal, Vol. 12, pp. 209-212. Kersten, M. S. 1949. Laboratory research for the determination of the thermal properties of soils, Univ. Minn. Eng. Exp. 5th, Final Rep. Koiter, W. T., and Benthem, J. P. 1973. Asymptotic approximations to crack problems. Chapter 3 in MECHANICS OF FRACTURE 1, METHODS OF ANALYSIS AND SOLUTIONS OF CRACK PROBLEMS, edited by G. C. Sih, Noordhoff International Publishing, Leydon, The Netherlands, pp. 131-178. Koiter, W. T. 1965. Rectangular tensile sheet with symmetric edge cracks. Journal of Applied Mechanics, Vol. 32, p. 237. Lachenbruch, A. H. 1961. Depth and spacing of tension cracks. J. of Geophysical Research, V01. 66, No. 12, pp. 4273-4292. Lachenbruch, A. H. 1962. Mechanics of thermal contraction cracks and ice-wedge polygons in permafrost. Special GSA Papers, No. 70, New York. Lachenbruch, A. H. 1963. Contraction theory of ice-wedge polygons: A qualitative discussion, PERMAFROST, Proceedings International Conference, National Academy of Sciences - National Research Council, Washington, D.C., Publication No. 1287. Ladanyi, B. 1972. An engineering theory of creep of frozen soils, Canadian Geotechnical Journal, Vol. 9, pp. 63-80. Lunardini, J. Virgil. 1981. HEAT TRANSFER IN COLD CLIMATES. Van Nostrand Reinhold Company, New York. Lutton, R. J. 1982. Evaluation Cover Systems for Solid and Hazardous Waste. U.S. Environmental Protection Agency, Cincinnati, Ohio, Report SW-867. Lutton, R. J.; Regan, G. L.; and Jones, L. W. 1979. Design and Construction of Covers for Sold Waste Land Fills. U.S. Environmental Protection Agency, Cincinnati, Ohio, Report EPA- 600/2-79—165. Muskhelishvili, N. I. 1963. SOME BASIC PROBLEMS OF THE MATHEMATICAL THEORY OF ELASTICITY. P. Noordhoff, Groningen, Holland. 193 Nady, R. M. 1972. Discussion to a study of transverse cracking of bituminous pavements. Proc. Assoc. Asphalt Paving Tech. Vol. 41, p. 424. NAVFAC. 1982. Design Manual DM 7.2: Foundations and Earth structures, Department of the Navy, Naval Facilities Engineering Comand, Alexandria, Va. NCHRP Synthesis 26, 1974. ROADWAY DESIGN IN SEASONAL FROST AREAS, Transportation Research Board, National Research Council, Washington, D.C. Paris, Paul C.; and Sih, George C. 1964. Stress analysis of cracks. Fracture Toughness Testing and Its Application, American Society for Testing and Materials, Special Technical Publication, No. 381. Founder, E. R. 1967. THE PHYSICS OF ICE. Pergamon Press, Oxford, England. Roethlisberger, Hans. 1972. Seismic exploration in cold regions. U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, New Hampshire, CRREL, Monograph II-A2a. Sayles, Francis H. 1973. Triaxial and creep tests on frozen Ottawa sand, North Am. Contrib. 2nd Int. Conf. Permafrost, Yakutsk, U.S.S.R., National Academy‘of Sciences, Washington. Sneddon, I. N. 1946. The distribution of stress in the neighbourhood of a crack in an elastic solid. Proceedings of the Royal Society of london, Series A., Vol. 187, pp. 229-260. Sih, C. G. 1973. HANDBOOK OF STRESS-INTENSITY FACTORS. Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, Pennsylvania Terzaghi, R. 1952. Permafrost. J. of the Boston Society of Civil Engineers, Vol. 39, No. 1, Jan. 1952, pp. 1-40. Timoshenko, S. P.; and Goodier, J. N. 1970. THEORY OF ELASTICITY. McGraw-Hill Book Co., New York. Tsytovich, N. A. 1975. THE MECHANICS OF FROZEN GROUND. McGraw-Hill Book Co., New York. Vyalov, S. S. 1986. REHOLOGIAL FUNDAMENTALS OF SOIL MECHANICS. Translated from the Russian by O. R. Sapunov, Elsevier, New York. Yoder, E. J.; and Witczak, M. W. 1975. PRINCIPLES OF PAVEMENT DESIGN, 2nd Ed., John Wiley and Sons, Inc. , New York. Yuanlin, Zhu;, and Carbee, David. 1987. Tensile strength of frozen silt. U.S. Army Cold Regions Research and Engineering Laboratory. Hanover, New Hampshire, CRREL, REPORT 87-15, pp. 1-23. APPENDIX - DATA Thermal Contraction Measurements Fiber Reinforcement Tests. Thermal Tensile and Stress Relaxation Tests. Temperature Data, Fargo, ND. Table A-1: Thermal Contraction Measurements. SAMPLE NO. 1 Type of Sand I Ottawa Sand Water Content 2 I 21.6 Degree of Saturation I 1001 Density I 2.257 g/cm Test Duration I 36 hr. Temperature Coefficient SAMPLE N0. 3 (cont'd.) -15.43 22.18 -17.43 22.83 -20.65 20.85 -23.47 20.25 SAMPLE N0. 4 of Thermal Contraction x 10 <°c> (°c“) -17.17 27.77 -20.18 28.26 -23.47 26.18 5 SAMPLE NO. 2 Type of Sand I Ottawa Sand Water Content 2 I 14. 46 Degree of Saturatiog I 75.132 Density I 2.13 g/cm Test Duration I 36 hr. Temperature Coefficient Type I Ice, form from distilled water Density I 0.916 Test Duration I 36 hr. Temperature Coefficient of Thermal 6 Contraction x 10 (’c) (30") -5.80 28.21 -16.09 23.37 -18.94 22.21 -25.97 21.44 SAMPLE N0. 3 Type of Sand I Ottawa Sand Water Content 2 I 11.34 Degree of Saturatio I 60.121 Density I 1.96 g/cm Test Duration I 36 hr. Temperature Coefficient of Thermal 6 Contraction x 10 ('c) (fc'l) -8.63 19.57 of Thermal 6 Contraction x 10 ('c) ('c") -6.45 58.70 -14.14 57.56 -16.29 56.51 -17.99 57.42 -21.78 55.39 -25.00 54.99 SAMPLE NO. 5 Type I Snow Density I 0.846 g/cm3 Test Duration I 36 hr. Temperature Coefficient of Thermal Contraction x 10 ('0) <°c") -7.55 55.25 -14.35 53.12 -17.26 55.65 -18.77 54.29 -20.44 52.87 -24.03 51.70 SAMPLE NO. 6 Type I Snow Density I 0.856 g/cm3 Test Duration I 36 hr. 196 Table A-1 (cont'd.) SAMPLE NO. 6 (cont'd.) Temperature Coefficient of Thermal Contraction x 106 . ('c) <°c"> -10.36 56.42 -12.66 55.21 -13.83 51.03 -14.52 55.73 -20.53 54.48 Table A-2: Sieve Analysis for Ottawa Sand Used to Measure the Coefficient of Thermal Contraction. U.S. Standard Percent Finer Sieve Number by Weight 30 99.80 40 47.80 ‘ 50 8.20 70 1.50 100 0.32 140 0.10 200 0.04 Pan 0.00 Table B- 1: SAMPLE N0. 1 Data Invalid - SAMPLE NO. 2 Type of the Fiber I Steel Wire Gage 28 Fiber Diameter I 0.016" Fiber Length I 0.250" Temperature I -6.2 'C _ Nom. Strain Rate I 1.11 x 10_ Ave. Strain Rate I 1.65 x 10 Sample Diameter I 1.13" Initial Length I 2.260" Final Length I 2.157" Percent Sand (by V01.) I 61 Percent Fiber (by V01.) I 3 Percent Ice (by Vol.) I 36 Time to Failure I 275 sec 4 4 Strain Stress (1) (psi) 0.0 0 1.27 209 1.45 380 1.63 578 1.91 746 2.09 897 2.36 997 2.73 1067 3.09 1128 3.37 1176 3.73 1216 4.00 1246 4.55 1261 4.73 1259 5.10 1255 5.46 1241 5.83 1222 6.19 1199 6.55 1185 6.74 1173 SAMPLE NO. Data Invalid - Membrane Leak SAMPLE NO. 4 Type of the Fiber I Steel Wire Gage 28 197 Fiber Reinforcement Tests 80¢- 80¢ 1 1 SAMPLE NO. 4 (cont'd.) Fiber Diameter I 0.016" Fiber Length I 0.250" Temperature I -6.1 °C _ Nom. Strain Rate I 1.11 x 10 4 sec_ Ave. Strain Rate I 0.87 x 10- sec Sample Diameter I 1.13" Initial Length I 2.26" Final Length I 1.92" Percent Sand (by V01.) I 55 Percent Fiber (by Vol.) I 9 Percent Ice (by V01.) I 36 Time to Failure I 837 sec. Strain (I) 0.0 0.096 0.096 0.096 0.288 0.480 0.865 1.154 1.346 1.442 1.731 1.923 2.116 2.308 2.404 2.597 2.789 3.078 3.366 3.559 3.751 3.943 4.136 4.426 4.617 4.809 5.098 5.290 5.482 5.675 5.693 6.156 6.348 6.540 6.733 Stress __(2932 0 4 4 8 184 287 559 852 1020 1147 1220 1268 1316 1351 1384 1406 1429 1454 1466 1484 1502 1515 1533 1553 1566 1592 1603 1616 1629 1642 1658 1670 1675 1684 1688 1 1 Table 8-1 (cont'd.) SAMPLE N0. 4 (cont'd.) 6.925 1693 7.118 1696 7.310 1698 7.502 1698 7.791 1693 7.983 1689 8.176 1686 SAMPLE N0. 5 Type of the Fiber I Steel Wire Gage 24 Fiber Diameter I 0.23" Fiber Length I 0.25" Temperature I -6.1 'C N0m. Strain Rate I 1.11 x 10: Ave. Strain Rate I 1.26 x 10 Sample Diameter I 1.13" Initial Length I 2.260" Final Length I 2.099" Percent Sand (by V01.) I 55 Percent Fiber (by V01.) I 9 Percent Ice (by Vol.) I 36 Time to Failure I 562 sec. Strain (1) .0.0 0.192 0.577 0.769 0.961 1.154 1.442 1.635 1.923 2.308 2.693 3.174 3.559 4.136 5.290 5.770 6.156 6.637 7.118 7.406 7.695 8.080 9.040 Stress ps') 0 124 257 413 614 796 947 1060 1137 1187 1229 1260 1288 1359 1359 1377 1387 1392 1389 1389 1380 1367 1313 SAMPLE no.5 (cont'd.) 9.522 1271 10.100 1220 SAMPLE N0. 6 Type of the Fiber I Steel Wire Gage 17 Fiber Diameter I 0.054" Fiber Length I 0.25" Temperature I -6.1 °C _4 _ Mom. Strain Rate I 1.11 x 10_43ec_ Ave. Strain Rate I 1.51 x 10 sec Sample Diameter I 1.13" Initial Length I 2.26" Final Length I 2.14" Percent Sand (by Vol.) I 55 Percent Fiber (by Vol.) I 9 Percent Ice (by Vol.) I 36 Time to Failure I 350 sec. 1 1 Strain Stress (2) (psi) 0.0 0 0.384 17 0.577 111 0.769 202 0.961 337 1.154 532 1.346 820 1.539 1031 1.827 1134 2.212 1167 2.597 1192 2.981 1225 3.462 1252 3.847 1268 4.136 1281 4.617 1282 5.001 1285 5.194 1291 5.290 1286 6.159 1246 6.925 1167 7.887 1048 SAMPLE N0. 7 Type of the Fiber I Steel Wire Gage 17 Fiber Diameter I 0.054" Fiber Length I 0.25" 199 Table 8-1 (cont'd.) SAMPLE NO. 7 (Cont'd.) Temperature I -6.0 °C N0m. Strain Rate I 1.11 x 10:23ec:: Ave. Strain Rate I 1.42 x 10 sec Sample Diameter I 1.13" Initial Length I 2.260" Final Length I 2.146" Percent Sand (by V01.) I 61 Percent Fiber (by V01.) I 3 Percent Ice (by Vol.) I 36 Time to Failure I 350 sec. Strain Stress (2) (ps') 0.0 0 0.096 305 0.288 541 0.480 703 0.769 842 1.058 955 1.250 1047 1.635 1098 2.116 1147 2.500 1189 3.078 1220 3.462 1252 3.943 1283 4.328 1303 4.809 1304 5.001 1314, 5.290 1302 6.252 1228 7.502 1020 8.176 898 SAMPLE N0. 8 Data Invalid - Membrane Leak SAMPLE N0. 9 Type of the Fiber I Steel Wire Gage 17 Fiber Diameter I 0.054" Fiber Length I 0.25" Temperature I -6.1 'C 4 1 M0m. Strain Rate I 1.11 x 10:4sec:l Ave. Strain Rate I 1.38 x 10 sec Sample Diameter I 1.13" Initial Length I 2.26" Final Length I 2.14" Percent Sand (by V01.) I 55 SAMPLE N0. 9 (cont'd.) Percent Fiber (by Vol.) I 9 Percent Ice (by V01.) I 36 Time to Failure I 369 sec. Strain Stress (1) (psi) 0.0 0 0.096 137 0.192 245 0.480 441 0.673 663 0.961 819 1.154 946 1.539 1056 2.020 1119 2.404 1157 2.693 1191 3.078 1220 3.559 1251 3.943 1279 4.328 1290 4.809 1296 5.098 1296 5.482 1291 5.867 1282 6.444 1250 6.733 1230 7.502 1156 8.464 1049 9.138 959 9.619 888 10.58 767 SAMPLE NO. 10 Type of the Fiber I Steel Wire Gage 17 Fiber Diameter I 0.054" Fiber Length I 0.25" Temperature I -6.1 'C _4 -1 Mom. Strain Rate I 1.11 x 10_43ec_l Ave. Strain Rate I 1.44 x 10 sec Sample Diameter I 1.13" Initial Length I 2.26" Final Length I 2.12" Percent Sand (by Vol.) I 58 Percent Fiber (by Vol.) I 6 Percent Ice (by Vol.) I 36 Time to Failure I 413 sec. 200 Table B-1 (cont'd.) SAMPLE N0. 10 (cont'd.) SAMPLE N0. 11 (cont'd.) Strain Stress 0'384 240 (1) (psi) 0.673 406 0.961 559 0.0 0 1.154 707 0.0 198 1.539 831 0.192 387 1.923 980 0.577 578 2.404 1102 0.865 777 2.789 1198 1.250 919 3.270 1271 1.539 1014 3.655 1324 1.923 1086 4.136 1380 2.212 1142 4.520 1420 2.693 1203 4.905 1459 2.981 1242 5.290 1494 3.462 1290 5.579 1510 3.943 1325 5.963 1536 4.232 1354 6.348 1546 4.617 1385 6.637 1545 5.098 1407 6.925 1540 5.290 1420 7.214 1519 5.675 1427 7.599 1489 5.963 1431 8.080 1418 6.252 1430 8.849 1288 7.310 1370 9.522 1150 7.791 1327 9.907 1064 8.272 1249 10.388 958 8.657 1189 11.254 807 10.003 1000 SAMPLE N0. 12 SAMPLE NO' 1' Type of the Fiber I Steel Wire Type of the Fiber I Steel Wire Gage l7 Gage 17 Fiber Diameter I 0.054" Fiber Diameter I 0.054” Fiber Length I 0.5" Fiber Length I 0.5" Temperature I -6.1 °C _4 -1 Temperature I -6.0 'C _4 -1 Mom. Strain Rate I 1.11 x 10_ sec_l N0m. Strain Rate I 1.11 x 10 sec_ Ave. Strain Rate I 1.43 x 10 sec Ave. Strain Rate I 1.42 x 10- sec Sample Diameter I 1.13" Sample Diameter I 1.13" Initial Length I 2.26" Initial Length I 2.26" Final Length I 2.11" Final Length I 2.11" Percent Sand (by V01.) I 58 Percent Sand (by Vol.) I 55 Percent Fiber (by Vol.) I 6 Percent Fiber (by V01.) I 9 Percent Ice (by Vol.) I 36 Percent Ice (by Vol.) I 36 Time to Failure I 463 sec. Time to Failure I 444 sec. Strain Stress Strain Stress (1) (psi) (2) (ps ) 0.0 0 0.0 0 0.096 111 0.96 133 0.384 201 Table B-1 (cont'd.) SAMPLE N0. 12 (cont'd.) 0.673 278 0.961 448 1.154 605 1.346 757 1.635 915 1.827 1041 2.212 1163 2.597 1221 2.885 1264 3.270 1305 3.655 1341 3.993 1374 4.328 1406 4.617 1439 4.905 1463 5.194 1491 5.482 1511 5.771 1527 6.060 1538 6.348 1542 6.637 1545 6.829 1538 7.118 1529 7.406 1516 7.599 1501 8.080 1450 8.272 1427 8.657 1370 9.234 1275 9.619 1207 10.196 1114 SAMPLE NO. 13 Type of the Fiber I Steel Wire Gage 17 Fiber Diameter I 0.054" Fiber Length I 0.5" Temperature I -6.1 ’C _4 N0m. Strain Rate I 1.11 x 10_43ec Ave. Strain Rate I 1.30 x 10 Sample Diameter I 1.13" Initial Length I 2.26" Final Length I 2.12" Percent Sand (by V01.) I 61. Percent Fiber (by Vol.) I 3 Percent Ice (by Vol.) I 36 Time to Failure I 450 sec. 83C. 201 -1 1 SAMPLE N0. 13 (cont'd.) Strain Stress (1) (psi) 0.0 0 0.096 137 0.288 232 0.577 334 0.865 440 1.154 600 1.731 711 1.827 918 1.923 997 2.212 1087 2.404 1182 2.693 1250 2.981 1309 3.270 1363 3.559 1409 3.847 1450 4.040 1485 4.328 1517 4.713 1556 4.905 _1586 5.194 1606 5.386 1615 5.867 1619 5.963 1617 6.540 1607 7.052 1587 7.599 1354 SAMPLE NO. 14 Type of the Fiber I Sand only Temperature I ~6.1 °C 4 Mom. Strain Rate I 1.11 x 10:4sec: Ave. Strain Rate I 1.48 x 10 sec Sample Diameter I 1.13" Initial Length I 2.26" Final Length I 2.13" Percent Sand (by Vol.) I 64 Percent Fiber (by Vol.) I 0 Percent Ice (by Vol.) I 36 Time to Failure I 362 sec. Strain Stress (1) (psi) 0.0“ 0 0.096 99 0.288 189 202 Table 8- 1 (cont 'd.) 3431213 no. 14 (cont'd.) SAMPLE no. 16 (cont'd.) 0.577 317 2.020 650 0.867 547 2.308 779 1.154 801 2.500 924 1.539 1031 2.693 1057 1.923 1217 3.174 1193 2.308 1263 3.559 1279 2.789 1315 3.847 1313 3.174 1344 4.328 1348 3.559 1384 4.713 1376 4.040 1414 5.001 1392 4.424 1433 5.290 . 1408 4.905 1451 5.579 1416 5.386 1464 5.867 1420 5.482 1458 6.060 1421 5.771 1454 6.156 1415 6.060 1441 6.444 1407 6.540 1402 6.829 1389 7.021 1346 7.502 1331 7.695 1265 7.887 1298 7.983 1217 8.176 1266 8.272 1162 9.428 1104 SAMPLE N0. 17 SAMPLE M0. 15 Type of the Fiber I Steel Wire . Gage 28 Data Invalid - Tilted in the Fiber Diameter I 0.016" Loading Cups Fiber Length I 0.25" Temperature I -6.05 '0 _4 -1 SAMPLE N0. 16 80m. Strain Rate I 1.11 x 10_4sec_l Ave. Strain Rate I 1.45 x 10 sec 3:55.32: 31330:“ ...... We - - - s s - 8' Mom. Strain Rate I 1.11 x 10 4sec 1 Initial Length 2.26 . -4 -1 Final Length I 2.11 Ave. Strain Rate I 1.44 x 10 sec . " Percent Sand (by V01.) I 58 Sample Diameter I 1.13 Initial Len th _ 2 26" Percent Fiber (by Vol.) I 6 8 ' Percent Ice (by Vol.) I 36 Final Length I 2.12" . . Percent Sand (by V01.) I 64 Time to Failure - 437 sec. Percent Fiber (by V01.) I 0 Percent Ice (by Vol.) I 36 Sigain Sfres; Time to Failure I 418 sec. -—— —P—8-1— 0.0 0 Strain Stress 0.096 193 (1) (psi) 0.288 318 0.577 458 °'° 0 0.865 553 0.096 77 1.154 822 0.384 107 1.442 968 0.673 115 _ 1.731 1105 1.154 153 1.442 335 2.020 1216 2.308 1280 1.731 495 203 Table 8-1 (cont'd.) SAMPLE 17 (cont'd.) SAMPLE NO. 18 (cont'd.) 2.789 1324 2.885 1402 3.078 1353 3.270 1430 3.462 1373 3.559 1455 3.751 1389 3.751 1481 4.232 1420 4.040 1501 4.520 1436 4.328 1517 4.809 1452 4.617 1533 5.098 1460 5.001 1543 5.386 1472 5.098 1542 5.675 1480 5.579 1530 5.963 1483 6.060 1494 6.252 1483 6.733 1415 6.348 1485 7.118 1357 6.444 1484 7.695 1269 6.637 1481 8.080 1212 7.118 1469 8.176 1203 7.599 1455 _ 7.887 1425 SAMPLE N0. 19 8°364 1385 Type of the Fiber I Steel Wire Gage 28 SAMPLE N°° '8 Fiber Diameter - 0.016" Type of the Fiber I Steel Wire Fiber Length I 0.25" Gage 24 Temperature I -6.05 “C _‘ -1 Fiber Diameter I 0.023" Non. Strain Rate I 1.11 x 10_4sec_l Fiber Length I 0.25" Ave. Strain Rate I 1.36 x 10 sec Temperature I -6.0 'C _4 _1 Sample Diameter I 1.13" N0m. Strain Rate I 1.11 x 10_4sec_l Initial Length I 2.26" Ave. Strain Rate I 1.35 x 10 sec Final Length I 2.13" Sample Diameter I 1.13" Percent Sand (by V01.) I 61 Initial Length I 2.26" Percent Fiber (by Vol.) I 3 Final Length I 2.14" Percent Ice (by Vol.) I 36 Percent Sand (by V01.) I 61 Time to Failure I 394 sec. Percent Fiber (by V01.) I 3 Percent Ice (by Vol.) I 36 Strain Stress Time to Failure I 368 (1) (ps') Strain Stress 3.396 143 (2) (ps') ' ---- 0.288 245 0.0 0 0.673 441 0.096 198 0.865 636 0.384 352 ' 1.154 839 0.577 552 1.346 1029 0.865 760 1.635 1225 1.154 954 1.923 1302 1.346 1097 2.212 1353 1.635 1212 2.597 1381 1.923 1285 2.885 1406 2.212 1323 3.174 1431 2.597 1373 3.559 1463 204 Table 8-1 cont'd. SAMPLE N0. 19 (cont'd.) SAMPLE N0. 20 (cont'd.) 3.943 1499 4.328 1431 4.232 1523 4.617 1455 4.520 1543 4.905 1475 4.809 1555 5.194 1495 5.001 1568 5.386 1513 5.290 1571 5.675 1520 5.386 1574 6.060 1530 5.482 1572 6.348 1542 5.771 1563 6.540 1535 5.963 1548 6.829 1530 6.252 1527 7.021 1519 6.540 1506 7.310 1506 7.310 1402 7.502 1491 7.983 1316 8.080 1442 8.176 1274 SAMPLE NO. 21 SAMPLE NO' 20 Type of the Fiber I Steel Wire Type of the Fiber I Steel Wire Gage 24 Gage 24 Fiber Diameter I 0.023" Fiber Diameter I 0.023" Fiber Length I 0.5" Fiber Length I 0.25" Temperature I -6.0 .C _4 -1 Temperature I -6.05 'C _4 -1 Mom. Strain Rate I 1.11 x 10_l.sec_l Mom. Strain Rate I 1.11 x 10_l.sec_l Ave. Strain Rate I 1.29 x 10 sec Ave. Strain Rate I 1.33 x 10 sec Sample Diameter I 1.13" Sample Diameter I 1.13" Initial Length I 2.26" Initial Length I 2.26" Final Length I 2.13" Final Length I 2.11" Percent Sand (by V01.) I 61 Percent Sand (by Vol.) I 58 Percent Fiber (by Vol.) I 3 Percent Fiber (by Vol.) I 6 Percent Ice (by Vol.) I 36- Percent Ice (by Vol.) I 36 Time to Failure I 437 Time to Failure I 475 sec. Strain Stress Strain Stress (2) (psi) __(1)._ _(PL). 0,0 0 0.0 0 0.096 17 0.096 189 0.577 214 0.288 318 0.673 534 0.673 496 0.865 756 0.961 725 1.058 946 1.250 868 1.346 1067 1.539 997 1.635 1115 1.827 1096 1.923 1128 2.116 1177 2.212 ' 1138 2.404 1236 2.500 1147 2.693 1287 2.789 1169 3.078 1328 3.078 1190 3.366 1358 3.366 1212 3.655 1383 3.655 1229 4.040 1410 3.943 1250 205 Table 8-1 (cont'd.) SAMPLE N0. 21 (cont'd.) SAMPLE N0. 22 (cont'd.) 4.136 1276 4.424 1355 4.424 1301 4.713 1367 4.713 1322 5.098 1390 4.905 1340 5.482 1401 5.194 1352 5.771 1413 5.482 1360 5.963 1418 5.675 1366 6.156 1424 5.771 1364 6.444 1427 6.156 1359 6.737 1431 6.829 1345 7.118 1425 7.021 1290 7.790 1411 7.599 1218 7.887 1183 SAMPLE no. 23 8.080 1161 Data Invalid - Membrane Leak SAMPLE W0. 22 Type of the Fiber I Steel Wire Gage 17 Fiber Diameter I 0.023" Fiber Length I 0.5" Temperature I -6.0 'C _4 _ N0m. Strain Rate I 1.11 x 10_4sec_ Ave. Strain Rate I 1.29 x 10 sec Sample Diameter I 1.13" Initial Length I 2.26" Final Length I 2.09" Percent Send (by V01.) I 55" Percent Fiber (by V01.) I 9 Percent Ice (by Vol.) I 36 Time to Failure I 550 sec. SAMPLE N0. 24 Type of the Fiber I Steel Wire Gage 17 Fiber Diameter I 0.054" Fiber Length I 0.25" 1 Temperature I -6.1 'C _4 _ 1 Mom. Strain Rate I 1.11 x 10_ sec_ Ave. Strain Rate I 1.31 x 10 sec Sample Diameter I 1.13" Initial Length I 2.26" Final Length I 2.13" Percent Sand (by V01.) I 55 Percent Fiber (by Vol.) I 9 Percent Ice (by Vol.) I 36 Time to Failure I 431 sec. 1 1 Strain Stress . (I) ( si) Strain Stress -—1L——- (1) (psi) 0.0 0 0.096 215 0'0 0 0.192 181 0.288 373 0.480 296 0.480 557 0.673 423 0.769 804 0.961 555 1.250 1085 1.250 689 1.539 1158 1.442 811 1.827 1201 1.731 940 2.116 1223 2.020 1034 2.404 1236 2.597 1192 2.693 1250 2.885 1243 2.981 1267 3.174 1285 3.270 1280 3.366 1324 3.559 1297 3.655 1358 4.136 1338 ' 4.136 1413 206 Table B-1 (cont'd.) SAMPLE M0. 24 (cont'd.) SAMPLE N0. 25 (cont'd.) 4.424 1437 4.328 1315 4.713 1458 4.617 1344 4.905 1475 4.905 1369 5.194 1487 5.194 1389 5.386 1500 5.480 1409 5.579 1506 5.771 1425 5.675 1508 5.963 1439 5.771 1507 6.252 1450 5.867 1505 6.540 1454 6.060 1498 6.925 1456 6.156 1500 7.021 1450 6.540 1466 7.118 1449 6.925 1424 7.695 1432 7.118 1389 8.178 1405 8.560 1064 SAMPLE NO. 26 SAMPLE ”0° 25 Type of the Fiber I Steel Wire Type of the Fiber I Steel Wire Gage 28 Gage 24 Fiber Diameter I 0.016" Fiber Diameter I 0.023" Fiber Length I 0.25" Fiber Length I 0.25" Temperature I -6.0 '0 Temperature - -6.0 'c 4 1 Mon. Strain Rate - 1.11 x 10'2sec': l0n. Strain Rate I 1.11 x 10:4sec:l Ave. Strain Rate I 1.36 x 10- sec- Ave. Strain Rate I 1.49 x 10 sec Sample Diameter I 1.13" Sample Diameter I 1.13" Initial Length I 2.26” Initial Length I 2.26" Final Length I 2.12" Final Length I 2.10" Percent Sand (by Vol.) I 58 Percent Sand (by Vol.) I 55 Percent Fiber (by Vol.) I 6 Percent Fiber (by V01.) I 9 Percent Ice (by Vol.) I 36 Percent Ice (by Vol.) I 36 Time to Failure I 450 sec. Time to Failure I 462 Strain Stress Strain Stress (2) (psi) (1) (psi) 0.0 0 0.0 0 0.096 116 0.192 116 0.384 190 0.480 188 0.577 304 0.769 286 0.865 440 1.058 396 1.058 592 1.345 531 1.346 748 1.635 665 1.635 894 1.923 799 1.923 1002 2.212 914 2.116 1092 2.500 1021 2.500 1164 2.789 1102 2.789 1219 3.174 1164 3.078 1270 3.462 1206 3.366 1308 3.751 1248 3.655 1345 4.040 1282 3.943 1383 207 Table B-1 cont'd. SAMPLE no. 26 (cont'd.) SAMPLE N0. 27 (cont'd.) 4.232 1416 6.637 1730 4.520 1440 7.021 1767 4.809 1464 7.406 1808 5.098 1481 7.695 1838 5.386 1496 8.080 1862 5.579 1514 8.464 1878 5.771 1523 8.753 1892 6.060 1526 9.138 1895 6.156 1533 9.426 1901 6.252 1531 9.619 1901 6.540 1518 9.715 1899 7.214 1487 10.677 1856 7.599 1449 11.158 1823 7.983 1396 8.176 1365 SAMPLE no. 28 sgupr no. 27 Data Invalid - Mambrane Leak Type of the Fiber I Steel Wire SAMPLE N0. 29 Fiber Diameter . 0 05:52 2‘ Data Invalid - Membrane Leak a - 1! Fiber Length 0-5 SAMPLE no. 30 Temperature I -6.0 'C -4 -1 80m. Strain Rate I 1.11 x 10_l‘sec__l Type of the Fiber I Steel Wire Ave. Strain Rate I 1.29 x 10 sec ‘ Gage 28 Sample Diameter I 1.13" Fiber Diameter I 0.016" Initial Length I 2.26" Fiber Length I 0.5" Final Length I 2.04" Temperature I -6.1 °C _4 -1 Percent Sand (by V01.) I 55 Mom. Strain Rate I 1.11 x 10_ sec_l Percent Fiber (by V01.) I 9 Ave. Strain Rate I 1.09 x 10 sec Percent Ice (by Vol.) I 36 Sample Diameter I 1.13" Time to Failure I 744 sec. Initial Length I 2.26" Final Length I 2.13" Strain Stress Percent Sand (by V01.) I 61 (1) (psi) Percent Fiber (by Vol.) I 3 0.0 0 Percent Ice (by Vol.) I 36 0.192 137 Time to Failure I 519 sec. $°€§2 437 Strain Stress ° (1) ( si) 1.635 644 --—-—- -———£-—- 2.212 918 0.0 0 2.597 1112 0.096 94 3.174 1248 0.384 157 3.655 1337 0.673 244 4.136 1409 0.961 345 4.520 1477 1.250 464 5.001 1531 1.539 589 5.482 1585 1.731 720 5.867 1635 2.020 844 6.252 1685 2.308 951 208 Table B-1 (cont'd.) SAMPLE N0. 30 (cont'd.) SAMPLE N0. 31 (cont'd.) 2.500 1025 3.943 1519 2.789 1089 4.328 1542 3.078 1144 4.713 1565 3.366 1195 5.098 1587 3.559 1226 5.579 1594 3.847 1260 5.963 1617 4.040 1294 6.348 1622 4.328 1319 6.637 1634 4.520 1341 6.925 1641 4.809 1362 7.214 1651 5.001 1380 7.599 1653 5.194 1397 7.983 1658 5.482 1409 8.176 1658 5.675 1419 8.272 1656 5.771 1417 8.752 1652 5.963 1414 9.041 1644 6.060 1413 9.330 1641 6.540 1394 9.522 1638 7.406 1305 9.715 1630 SAMPLE N0. 31 SAMPLE N0. 32 Type of the Fiber I Steel Wire Type of the Fiber I Steel Wire Gage 28 Gage 28 Fiber Diameter I 0.016" Fiber Diameter I 0.016" Fiber Length I 0.5" Fiber Length I 0.5" Temperature I -6.0 '0 Temperature I -6.0 ’C N0m. Strain Rate I 1.11 x 10:2sec:: Mom. Strain Rate I 1.11 x 10:2sec:: Ave. Strain Rate I 1.59 x 10 sec Ave. Strain Rate I 1.26 x 10 sec Sample Diameter I 1.13" Sample Diameter I 1.13" Initial Length I 2.26" Initial Length I 2.26" Final Length I 2.07" Final Length I 2.04" Percent Send (by V01.) I 58 Percent Sand (by V01.) I 55 Percent Fiber (by V01.) I 6 Percent Fiber (by Vol.) I 9 Percent Ice (by Vol.) I 36 Percent Ice (by V01.) I 36 Time to Failure I 513 sec. Time to Failure I 769 sec. Strain Stress Strain Stress (1) (psi) (2) (psi) 0.0 0 0.0 0 0.192 258 0.288 159 0.384 502 0.673 299 0.673 770 1.058 473 1.154 1005 1.442 718 1.442 1185 1.827 884 1.827 1200 2.212 1020 2.308 1360 2.597 1137 2.693 1417 2.981 1237 3.174 1456 3.462 1315 3.559 1488 3.847 1363 Table 3-1 (cont'd.) SAMPLE N0. 32 (cont'd.) 4.232 1395 4.617 1414 5.001 1433 5.386 1443 5.675 1463 5.963 1487 6.348 1505 6.733 1523 7.021 1539 7.310 1558 7.695 1567 7.887 1576 8.176 1587 8.464 1594 8.753 1600 9.041 1607 9.330 1610 9.619 1613 9.715 1615 9.811 1613 9.907 1611 10.196 1610 10.386 1610 10.677 1609 SAMPLE MO. 33 Type of the Fiber I Steel Wire Gage 28 Fiber Diameter I 0.016" Fiber Length I 0.5" Temperature I -6.1 'C _4 Mom. Strain Rate I 1.11 x 10_4 Ave. Strain Rate I 0.80 x 10 Sample Diameter I 1.13" Initial Length I 2.26" Final Length I 2.10" Percent Sand (by V01.) I 55 Percent Fiber (by V01.) I 9 Percent Ice (by Vol.) I 36 Time to Failure I 881 sec. Strain Stress (2) (psi) 0.0 0 0.096 99 0.288 223 0.480 411 0.673 633 0.961 853 sec_ SEC 209 SAMPLE NO. 33 (cont'd.) 1.154 1026 1.442 1151 1.635 1238 1.923 1302 2.212 1365 2.500 1429 2.693 1489 2.981 1551 3.270 1605 3.559 1658 3.751 1709 4.040 1758 4.328 1802 4.520 1839 4.809 1875 5.098 1910 5.386 1937 5.675 1959 6.060 1976 6.252 2000 6.540 2010 6.829 2020 7.118 2022 7.214 2019 7.406 2019 7.983 2003 8.176 1991 SAMPLE N0. 34 Type of the Fiber I Steel Wire Gage 24 Fiber Diameter I 0.023" Fiber Length I 0.5" Temperature I -6.1 'C _4 Mom. Strain Rate I 1.11 x 10_4 Ave. Strain Rate I 0.85 x 10 sec Sample Diameter I 1.13" Initial Length I 2.26" Final Length I 2.14" Percent Sand (by Vol.) I 58 Percent Fiber (by Vol.) I 6 Percent Ice (by Vol.) I 36 Time to Failure I 575 sec. sec_ Strain Stress (2) (ps') 0.0 0 0.096 124 0.288 253 1 1 210 Table B-1 (cont'd.) SAMPLE N0.434 (cont'd) 0.480 416 0.673 603 0.769 817 0.961 1003 1.154 1146 1.442 1236 1.635 1318 1.827 1392 2.020 1457 2.308 1516 2.597 1570 2.885 1624 3.174 1673 3.366 1720 3.559 1758 3.751 1796 4.040 1820 4.232 1841 4.424 1854 4.617 1858 4.905 1861 5.001 1855 5.386 1839 5.771 1795 6.540 1672 Table 3-2: Sieve Analysis for Ottawa Sand Used in Fiber Reinforcement Tests. U.S. Standard Percent Finer Sieve Number by Weight 30 99.85 40 43.41 50 5.30 70 1.32 100 0.24 140 0.04 200 0.01 Pan ‘0.00 Table C-1: SAMPLE N0. Initial Diameter I 2.0" Final Diameter I 2.0" Thermal Tensile Tests 1 Rate of Cooling I -0.63 °C/min Degree of Saturation I 98.71 2 Cooling Agent I Dry Ice Time to Failure I 182.46 min. Temperature at which Failure Occur I -19.5 °C Maxim Thermal Tensile Stress I 258.3 psi Stress (psi) 0.0 14.51 34.63 47.27 51.95 54.76 58.04 58.98 64.13 74.42 118.40 149.30 197.07 240.60 258.30 SAMPLE N0. Initial Diameter I 1" Final Diameter I 1" Temperature (Ave. 'C) -9.00 I10.40 -11.00 -11.10 I11.50 -11.70 I11.80 -12.00 I14.60 -14.90 -16.10 ~16.80 -18.00 -18.20 -19.50 2 Time (min.) 0.00 0.56 1.36 1.70 2.71 3.39 4.60 4.95 5.80 7.51 12.11 25.56 53.46 89.33 182.46 Rate of Cooling I -3.01 °C/min Degree of Saturation I 98.90 2 Cooling Agent I Dry Ice Time to Failure I 3.96 min. Temperature at which Failure Occur I -18.5 °C Maximum Thermal Tensile Stress I 211 psi Stress (psi) 0.00 33.62 46.77 59.77 89.66 121.40 Temperature (Ave. °C) -5.60 -6.50 -8.25 -9.75 -11.60 -13.40 Time (min.) 0.00 0.00 0.70 1.38 1.93 2.55 SAMPLE NO. 2 (cont'd.) 175.50 184.90 211.00 SAMPLE NO. -16.50 -17.60 -18.50 3 Initial Diameter I 1" Final Diameter I 1" Rate of Cooling I -4.31 °C/min Degree of Saturation I 98.50 2 Cooling Agent I Dry Ice Time to Failure I 1.86 min Temperature at which Failure Occur I -13.4 'C Maximum.Thermal Tensile Stress I 138.5 psi Stress Temperature (psi) (Ave. '0) 0.00 -5.3 71.15 -6.5 74.89 -7.0 84.25 -7.6 89.87 -7.8 99.23 -8.7 102.98 -9.7 106.72 -10.2 108.59 -11.0 127.30 -12.3 131.00 -12.7 132.94 -13.0 138.50 -13.4 SAMPLE NO. 4 Initial Diameter I 2" Final Diameter I 2" Time (min.) 0.00 0.216 0.316 0.416 0.516 0.633 0.833 0.933 1.150 1.560 1.660 1.760 1.860 Rate of Cooling I -0.422 °C/min Degree of Saturation I 98.31 1 Cooling Agent I Dry Ice Time to Failure I 152.19 min. Temperature at which Failure Occur I -66.9 °C Maximum.Therma1 Tensile Stress I 490.8 psi Stress (psi) 0.00 14.94 Temperature (Ave. °C) -3.45 -3.60 Time (min.) 0.00 0.85 212 Table C-1 (cont'd.) SAMPLE NO. 4 (cont'd.) SAMPLE NO. 5 (cont'd.) 22.88 -3.65 1.48 Temperature at which 19.61 -3.30 2.11 Failure Occur I -15.05 °C 18.21 -3.00 2.74 Maximum.Thermal Tensile 19.14 -2.40 3.47 Stress I 77.05 psi 15.41 -2.00 4.37 13.54 -3.30 4.98 Stress Temperature Time 14.00 -9.50 5.61 (psi) (Ave. 'C) (min.) 22.88 -14.00 6.24 0.00 _5.05 0.00 30.82 -17.50 6.97 6.00 -6.05 45.00 40.62 -22.50 7.58 7.90 -7.55 120.00 51.83 -28.00 8.21 9.80 -8.55 135.00 62.10 -28.50 8.84 18.20 -9.45 150.00 70.50 -29.50 9.45 21.40 -10.20 165.00 79.85 -30.00 10.18 27.50 ~11.55 195.00 87.79 -30.50 10.81 29.80 -12.05 210.00 95.26- -31.00 11.42 43.80 -13.20 255.00 101.80 -31.50 12.05 54.10 I14.00 300.00 108.80 -31.75 12.68 77 05 ~15 05 465 00 116.28 -32.25 13.29 ° ° ' 124.68 -33.50 14.02 134.02 -33.50 14.46 141.03 -33.50 15.07 148.96 -33.50 15.70 156.44 -34.25 16.31 164.38 I34.50 17.04 169.98 -35.50 17.65 175.12 -36.00 18.28 180.25 I36.50 18.89 186.32 -36.75 19.52 190.53 -37.50 20.25 193.80 I39.00 21.55 193.33 -39.25 22.33 247.50 -47.50 39.78 323.62 -51.00 69.78 374.99 -58.00 84.78 413.75 -61.25 99.78 450.64 -64.50 119.78 482.86 -66.25 144.78 488.93 -66.80 149.78 490.80 -66.90 152.19 SAMPLE N0. 5 Initial Diameter I 2" Final Diameter I 2" Rate of Cooling I -0.021 ’C/min Degree of Saturation I 98.1 2 Cooling Agent I Freezer Time to Failure I 465 min. 213 Table C-2: Stress Relaxation Tests SAMPLE no. 1 SAMPLE NO. 2 (cont'd.) Temperature I -15 °C 269.52 9.60 4.30 Degree of Saturation I 98.1 2 Initial Diameter I 2" Cycle No. -1- Final Diameter I 1.99" Initial Length I 15.25? 250°84 9'60 4'53 . " 236.82 9.60 6.53 Final Length I 15.286 . 218.14 9.60 12.96 80. of Cyclic Strain, . . 212.53 9.60 16.82 Prior to Failure I 1 . . 195.72 9.60 38.11 Time to Failure I 3.75 min. . . 176.07 11.30 98.11 Failure Stress I 722.43 psi Failure Strain I 24 19 x 10- in/in 155'99 11°30 458'}! ' 127.40 12.90 1418.11 Stress Strain x 10 Time 1:8'32 :2'23 :;;:':: ( s') (in/in) (min.) 1 6° ° ' —L— 105.51 14.50 2318.11 116.28 0.000 0.000 104.11 14.50 2419.16 171.38 1.612 0.250 220.88 3.225 0.480 Cycle No. -2- 259.69 4.838 0.730 104 11 14 50 0.00 396.47 8.064 1.100 120.45 14.50 0.133 454.38 8.064 1.333 162.91 16.10 0.245 461.85 12.904 1.583 186.72 16.10 0.361 506.68 12.900 1.816 225.9 17.70 0.611 580.00 14.510 2.186 250.63 17.70 .844 598.68 17.740 2.780 258.06 19.30 1.094 641.64 19.350 3.150 268.79 19.30 1.210 644.44 19.350 3.260 722 43 24 190 3 750 250.43 19.35 1.326 ' ' ' 229.59 19.35 2.892 212.79 19.35 11.942 3mm "0' 2 192.26 19.35 43.942 Temperature I -15 'C 177.79 19.35 93.942 Degree of Saturation I 98.51 1 152.13 19.35 393.942 Initial Diameter I 2" 143.26 19.35 693.942 Final Diameter I 1.95" 128.79 19.35 993.942 Initial Length I 15.250" 122.73 19.35 1533.942 Final Length I 15.309" 122.73 19.35 1804.942 No. of Cyclic Strain, Prior to Failure I 2 Time to Failure I 4233.316 min. Failure Stress I 709.19 psi Failure Strain I 38.7 x 10 in/in Average Cyclic _4 Strain I 10 x 10 in/in Strain x 104 Time Stress (psi) (in/in) (min.) 0.00 0.00 0.0 84.49 3.20 1.88 . 159.60 6.40 2.36 217.90 8.00 3.94 End of Cycle No. 136.73 225.39 337.80 354.08 488.27 534.65 566.40 709.91 Total Time 19.35 19.35 20.96 22.58 25.80 30.64 33.87 38.70 1808.142 1808.375 1808.625 1809.108 1809.319 1811.029 1814.040 1814.156 4233.316 214 Table C-2 (cont 'd.) SAMPLE no. 3 SAMPLE N0. 3 (cont'd.) Temperature I -15 'C 159.91 16.12 1077.680 Degree of Saturation I 98.31 2 153.85 16.12 1257.680 Initial Diameter I 2" 151.52 16.12 1377.680 Final Diameter I 1.92" Initial Length I 15.25" Cycle No. -3- - _ u ginalngngigc S:S;?:3 155.69 17.740 0.000 °§ 30 {c Fa.1 ‘ ‘_'4 157.56 17.740 0.750 . ‘1 ’ °. 1 “re . 252.61 19.350 0.980 Time to Failure I 5596.86 min. . . 363.48 20.960 1.096 Failure Stress I 700.8 psi _ 477 76 22 580 1 229 Failure Strain I 48.38 x 10 in/in ' ° ' Aver e c clic 519.05 22.580 1.345 Sc:3in Z 10 x 10-4 .n/in 596.66 25.806 1.461 ‘ 1 488.68 24.190 1.577 . . 467.72 24.190 1.943 s"’?‘ Str‘l? ‘.10 11““ 466.29 24.190 3.026 —(E—) (“/1“) 519-‘23- 423.93 24.190 5.159 118.61 0.00 0.00 399.24 24.190 9.159 261.0 8.06 4.86 376.41 24.190 15.159 347.53 24.190 28.159 Cycle 80. -1- 325.17 24.190 43.159 288.60 8.06 4.97 302.34 24.190 58.159 284.64 24.190 83.159 288.60 8.06 4.97 262.27 24.190 113.159 245.16 8.06 5.10 233.86 24.190 173.159 201.73 8.06 5.33 204.51 24.190 263.159 187.26 8.06 5.82 186.34 24.190 383.159 166.71 8.06 6.67 163.98 24.190 653.159 154.10 8.06 7.75 143.01 24.190 953.159 146.16 8.06 8.83 133.70 24.190 1103.150 13"95 8'06 ‘2'98 132 77 24 190 1403 150 119.08 8.06 20.98 ' ' ' 107.40 8.06 36.98 105.07 8.06 1466.95 “7°15 N°° '4' 135.09 24.19 0.000 Cycle no. -2- 166.72 27.41 0.980 118.45 11.29 0.000 353"’ 29'03 '°'°° 484.18 30.64 1.216 248.58 12.90 0.366 541.82 32.25 1.332 310.14 12.90 0.616 613.50 32.25 1.465 346.93 14.51 1.099 469.20 32.25 1.948 412.15 16.12 1.332 445.46 32.25 3.148 443.31 17.74 1.448 422.19 32.25 5.514 334.75 16.12 1.698 401.71 32.25 8.614 313.36 14.51 3.031 377.04 32.25 14.614 304.50 14.51 4.347 354.23 32.25 24.614 283.51 14.51 10.680 309.54 32.25 58.614 259.26 14.51 27.680 , 285.80 32.25 88.614 237.81 14.51 57.680 266.72 32.25 118.614 213.10 14.51 117.680 229.01 32.25 238.614 182.29 16.12 207.680 202 95 32 25 448 614 165.51 16.12 387.680 ° ° ° Table C-2 (cont'd.) SAMPLE N0. 3 (cont'd.) 181.53 32.25 658.614 179.64 33.87 1108.614 178.25 33.87 1168.614 172.66 33.87 1228.614 168.94 33.87 1288.614 164.31 33.87 1348.614 End of Cycle No. -4- 168.7 35.48 0.00 365.51 41.93 0.233 700.80 48.38 0.466 Total Time 5596.86 Table C-3: Sieve Analysis for Ottawa Sand Used in Thermal Tensile, and Stress Relaxation Tests. 0.8. Standard Percent Finer Sieve number by Weight 30 96.00 40 47.98 50 20.44 70 10.36 100 3.26 140 0.56 200 0.07 Pan 0.00 216 Table D-1: Air Temperature at Fargo, North Dakota (2-3 February, 1974) Hour Air Temperature + (°F) 00 03 03 03 06 03 09 06 12 2 Feb., 1974 06 15 05 18 -09 21 -21 00 -23 03 -26 06 -23 09 3 Feb., 1974 -25 12 -17 15 -14 18 I20 21 -26 + Data from Local Climatological Data U.S. Department of Commerce