I MSU l LIBRARIES “ RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINEQ wil] be charged if book is returned after the date stamped below. q *1 (1mm ‘3 .-: ABSTRACT THE VISCOELASTIC BEHAVIOR OF THE POTATO, ‘SOLANUM TUBEROSUM, UNDER QUASI-STATIC LOADING by Essex Eugene Finney, Jr. Bruising during mechanical handling is a major problem in the potato industry. An investigation was conducted to obtain a more thorough understanding of the behavior of the potato under the influence of externally applied non-impact forces. The potato was considered as a viscoelastic product, i. e. , a product which combines the effects of an elastic solid and a viscous liquid in response to applied loads. A three-hundred pound capacity testing machine with controlled testing speeds from 1 to 40 inches per minute was developed for'loading and unloading the product. Strain gage transducers and an electronic recorder were used to measure and record the forceflapplied to the product as a function of the deformation of the potato.‘ - The integral of the force-deformation curve was used to 'determine the energy capacity of the product. Apparent elastic constants for potato tissue were determined by uniaxial compression tests and hydrostatic bulk modulus tests. Strength characteristics of the tuber were determined by loading a rigid die acting against the surface of the tuber with the skin intact. Five varieties-~Katahdin, Kennebec, Onaway, Russet Burbank, and Sebago-- were studied from one month before until six months after harvest. Effects of rates of deformation from 1 to 40 inches per minute and tuber temperatures from 40 to 140°F were investigated. - The influence of EssexEugene Finney, Jr. l traction areas from 0. 01 to 0. 50 sq in upon the strength of the tuber was determined. The potato, similar to other viscoelastic materials, exhibits stress relaxation which can be represented qualitatively by a Maxwell model. Stress relaxation within the potato tuber was represented by four parallel connected Maxwell models having time-constants in the form of a geometric series. Only one-half of the initially induced strain within potato tissue is immediately recoverable upon unloading and 70 to 90 per cent of the energy expended during loading is dissipated due to a pronounced hysteresis effect. % . Potatoes do not display a yield (or bio-yield) point. The rupture point on the force-deformation curve coincided with the occurrence of localized tissue failure within the tuber. The capacity of the tuber to resist applied forces was characterized by the force, stress, deform- ation, and energy parameters taken from the force-deformation curve at the point of rupture. Varieties differ significantly in response to applied forces. . The strength of potatoes decreased during the pre- harvest tests and increased with time after harvesting. Increasing the rate of deformation above 20 inches per minute has an effect similar to increasing the temperature of the tuber above lOSoF; both cause a decrease in the strength of the tuber. A decrease in strength from 380 psi to 272 psi as the rate of deformation increased from 20 to 40 inches per minute suggested that potatoes may be more vulnerable to injury under impact than under the corresponding quasi- static loading conditions. Approved w a} % Maj or Profe s sor THE VISCOELASTIC BEHAVIOR OF THE POTATO, SOLANUM TUBEROSUM, UNDER QUASI-STATIC LOADING BY Essex Eugene Finney, Jr. A THESIS Submitted to Michigan. State University in partial fulfillment of the requirements for the degree of. DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1963 ACKNOWLEDGMENTS The author sincerely appreciates the assistance of all who have aided in this study. He is especially appreciative, however, for the counsel and guidance provided by his major professor, ~Dr. Carl W. Hall (Agricultural Engineering), during this graduate program. To the other members of the guidance committee, Dr. -L. E. Malvern and Dr. G. E. Mase (Metallurgy, ‘Mechanics, and Materials Science), and Dr. B.7A. Stout (Agricultural Engineering), the author expresses his deepest gratitude for their time, professional interest, and constructive suggestions. Dr. N. R. Thompson (Farm Crops) and‘Dr- W. J. Hooker (Botany and'Plant Pathology) made helpful comments during various stages of this study. Dr. A. W. Farrall, Chairman of the Agricultural Engineering Department, arranged and approved the as sistantship and the operating funds which made this work possible. Mr. James Cawood and his staff in the Agricultural Engineering Research Laboratory assisted in the development of the experimental apparatus. This dissertation is dedicated to my wife, Ellen, whose unfailing confidence has been a continuing source of encouragement. *************** ii TABLE OF CONTENTS iii Page ABSTRACT O’O O O O O 000000000000 O O O O O O O O O 1 ACKNOWLEDGMENTS .................... . ii LIST OF TABLES. o o o o o o o o o o o o o o ooooo o o o e V LIST OF FIGURES O O O O O ........ O O O O O O O vii LIST OF APPENDIXES TABLES. . . . . . . . . . . . . . xi ABBREVIATIONS AND SYMBOLS O O O O O O O O O O O O Xiv I.‘INTRODUCTION,,,.............. ..... l 1.1 Objective ...... . . ...... . ..... . 2 1.2 Statement of the Thesis Problem. . . . . . . Z 110' REVIEW OF LITERATURE o o o o o o oooooooooo -’3 2. 1 Instrumentation and Techniques . V ...... . 3 2.2 Results of Related Studies . . . . . . . . . . 8 III. THEORETICALCONSIDERATIONS . . . . ........ 15 3O 1 ElaStiCity O O O O O O O O O O O O O O O O O O O 15 3O 2 ViSCOSity O O O O O OOOOOOO O O O O O O O O O 20 3.3 Viscoelasticity . . . . . . . . . . . ....... 23 IV. EXPERIMENTAL TECHNIQUES . . . . . . . ...... . 29 4.1~TheTestingMachine.............. 29 4.2 The Sensing Elements. . . . . . . . . . . . . . 34 4.3 The X‘Y Recorder 0 a to o o o o o o o o o o o o 35 4.4 The Compression Tests . . . . . . . . . 39 4.5 The Uniaxial Compression Tests. . . . . . . . 45 4.6 The BulkModulus Tests . . . . . . . . . . . . . 45 4.7 Growth and Storage of the Test Specimen . . . . 50 4O 8 SPeCific GraVitY O O O O O O O O O O O O O O O O O O 51 TABLE OF CONTENTS - Continued Page V. PRESENTATION. AND DISCUSSION OF RESULTS ..... 53 5. 1 Failure of Cellular Tissue within-Potatoes under MechanicalStress................. 53 5. 2-E1astic Modulus under Uniaxial Compression. . .. . 60 5.3ElasticBulkModulus ................ 69 5.4 Influence of Variety and Maturity Upon the'Rupture Parameters ..... 74 5. 5 Influence of Traction Area Upon the Rupture Parameters.................... 84 5.6 Temperature Effects. . . . . . . . . . . . . . . 93 5.7'Strain Rate Effects. . . . . . . . . . . . . ..... 99 5. 8 Stress Relaxation Within the Potato Tuber .. . . . . . 104 VI.»SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . 112 6.1'Summary.......................112 6.2Conc1usions...................... .116 SUGGESTIONS FOR FURTHER STUDY. . . . . . . . . . . . . 119 REFERENCESO O O O O O O O O O O O O O O O O O O O O O O O O O 120 APPENDIXES. . . . . . . . . ................ . 127 iv TABLE LIST OF TABLES Page Piston speed and hydro-check valve setting for an inletairpressureof44psi. . . . . . . . . . . . . . . Correction multipliers applied to the recorded force- deformation curves to obtain the equivalent rupture parameters for one-half of a potato tuber supported byarigidsurface.................... Degree of elasticity of potato tissue under uniaxial compression below the rupture point. ~Variety: Russet Rural; Test date: 10/17/62 . . . . . . . . . . Hysteresis loss in potato tissue loaded and unloaded below the rupture point. Variety: Russet Rural; Test date: 10/17/62 O O O O O O O O O O O O O O O O O O Modulus of elasticity for cylindrical sections of potato tissue having various cross-sectional areas and a length of one inch. Variety: Russet Rural; Test date: October 17, 1962; Rate of deformation: 1 ipm O O O O O O O O O O O O O O O O O O O O O O O O O O O Hydrostatic pressure versus volumetric strain and other data summarized from the elastic bulk modulus testSO O O O O O O O O O O O O O O O O O O O O O O O O O O Some apparent elastic coefficients for mature potato tiasueO O O O O O O O O O O O O O O O O O O O O O O O O O Experimental product-moment correlation coefficients between certain measured parameters for five potato varieties grown in Michigan . . . . . . . . . . . . . . Rupture parameters for Emmet potato tubers as influenced by the traction area of the loading surface. Each value represents ten replications . . . . . . . . 31 44 64 66 68 72 74 82 85 LIST OF TABLES - Continued TABLE 10.. 11. Page Rupture parameters, as influenced by temperature, for potato tubers under compression with a O. 05 sq in cylindrical rigid die. Each value based upon ten replications........................ 96 Observed versus calculated values of force during relaxation of potato tubers under constant strain between parallel plates . . . . . . . . . . . . . . . . 111 vi FIGURE 1. 11. 12. LIST OF FIGURES Page Sketch of instrument for determining the sensitivity of potato tubers to injury (from-Lampe, 1959). . . . . 5 Force-deformation curves for an apple specimen with skinintact loaded with'i-dnch diameter pin (from Mohsenin and Gohlich, 1962). . . . . . . . . . . . . . 7 Impact testing apparatus (from Gohlich and Mohsenin, 1962)O O O O O O O O O O O O O O O O O O O O O O O O O O O 9 Theoretical stress distribution under a rigid die act- ing against a semi-infinite elastic body . . . . . . . . 19 Velocity distribution in a viscous fluid moving be- tween two parallel flat plates. . . . . . . . . . . . . . 22 -Rate of shear strain as a function of stress for dif- ferenttypesoffluids.................. 22 Some fundamental elements and mechanical models in ViscoelastiCi-ty. O 1O O O O O O O O O O O O O O O. O O O 25 The generalized Maxwell model . . . . . . . . . . . . 27 The testing machine (MSU Photo No.63296-A) . . . . 3O .Displacement-time response of the testing machine as influenced by the force applied by the machine to the test specimen (Inlet air pressure: 44 psi). . . . . 33 A block diagram of the experimental set-up . . . . . . 37 The X-Y recorder and auxiliary devices (MSU Photo ONO-62264'3)ooooooo000.009.000.000. 38 vii - LIST OF FIGURES - Continued ' FIGURE 13O 14. 15. 16. 17O 18O 19. 20. 21. 22. Cork boring machine with cylindrical cutting tools for removing specimens of tissue from the potato tuber (MSU Photo No. 63296-4). (Insert: Cylindrical specimen being removed for uniaxial test. MSU Photo (No.63296'5)ooooooooooooooooooooooo . Force-deformation curves as recorded by the X-Y recorder for the whole tuber and a half-tuber sup- ported upon a flat plate. (Cross-sectional area of theloadingpin: 0.05 sqin). . . . ., . . . . . . . . . . High pressure bulk modulus apparatus, 15 to 500 psi. (MSU PhOtO Noe 63296.11. 0 o o o o o o o o o o o o o o - Low pressure bulk modulus apparatus, 0 to 60 psi. (MSU Photo NO. 63830.3). 0 or. o o o o o ofi>fizmnom on“ mamnmfiugov Ham 2353.59: mo A333 .7 oufimfim —--qp _ w u ___ . _ Hog—H. anon... . - gouge 5&2 iv. 7 _. , vvfivcdm . Romanian m0 230% Ema? mafia gnu undo woman / . .uudfim \ mam . wdSAfloO . H302, cinn— Hoflom _ 035R. uh the skin andthe supporting tissue and penetrated into the tuber. -A direct plot of the load versus pressure pin penetration was obtained by means of a unique arrangement of pulleys and levers synchronized within the testing machine. This technique was tested in a large number of experiments over a two year period and the results were compared with the damage occurring to tubers in-a harvesting machine. A technique similar to that developed by Lampe for potatoes was used by Mohsenin-and G6hlich (1962) to evaluate the resistance of apples to injury. The equipment developed by Mohsenin and Gohlich consisted of a pneumatically operated, hydraulically controlled loading and unloading device with strain gage transducers for measuring force and deformation and auxiliary electronic instrumentation for amplifying and recording the force versus deformation behavior of the apple fruit. The speed of loading the product could be varied from 0.. 01 to l. 5 inches per minute ”(ipm). The mechanical behavior of the apple was character- ized by the force versus deformation behavior of a specimen removed from the fruit. , As a f-inch diameter cylindrical plunger was forced into the apple specimen, an approximate linear relationship between force and deformation of the fruit was observed up to some point where there was a sudden decrease in the force with increasing deformation (Fig. 2). This point was termed the‘ "yield point; " (It has later been referred to as the "bio-yield point. ") As the fruit was deformed beyond the yield point, the force again began to increase until the skin and sup- porting tissue gave away and broke; this was called the "rupture point.- " This apparatus was tested over a period of years and considerable information has been obtained concerning the yield and rupture para- meters for the apple fruit. Gohlich and Mohsenin (1962) developed apparatus for measuring the impact resistance of apples based upon the energy delivered to the fruit by a falling impact lever. A specimen of the fruit with the skin 3 - e Rupture Yield 2: 2 - \ I!) U, 01 s 0. a) [it 1 r- (C) 1 I l g I W1 1 0 20 so 100 DEFORMATION (in. x 10") (a) Loaded through the yield and rupturepoints. (b) Loaded through the yield point and unloaded to zero stress. (c) Loaded up to a point below yield and unloaded to zero stress. Figure 2. Force-deformation curves for an apple specimen with skin intact loaded with f— inch diameter pin. (From. Mohsenin and 661111611, 1962) intact was attached to the free end of the impact arm (Fig. 3) and the impact blow was delivered by the fruit falling against a flat plate, a steel plunger, or another specimen of the fruit. - The impact energy was not measured directly but was calculated based upon the height of fall and the effective mass of the fruit and the impact arm. Deformation of the fruit during impact was measured with a linear variable differential transducer (LVDT) and recorded versus time on an oscillograph chart. Later Mohsenin e_t a_._1. ' (1962) reported that this apparatus had been modified to record impact forceas well as deform- ation by photographing the force-deformation trace from thescreen of an oscilloscope. 2. 2 Results of Related Studies Considerable progress has been made in recent years in studying and defining the mechanical behavior of fruits and vegetables and other agricultural products. ' The techniques and results from other areas of science and engineering have beenliberally applied by the agricultural engineer in his investigations. ' Mohsenin and Giihlich (1962) observed, for example, that the force-deformationcurve for an apple was very similar to the stress-straincurve of steel; that is, .the forceedeformation relationship was approximately linear up to an apparent yield point after which the force suddenly decreased and then continued to increase up to some point of rupture. - Other investigators have examined the strain- rate effect, the creep and relaxation behavior, and the elastic-plastic behavior of various biological products including fruits and vegetables. The yield point in the apple fruit was very significant in that it corresponded to the point where the cells of the fruit were sufficiently damaged to cause. discoloration and deterioration of the fruit. V‘Unless this yield point was exhibited, no bruising of the fruit as indicated by discolor- ation was generally detected. - Mohsenin and G'ohlich (1962) reported a who fl 6/ . E To recorder ‘——-‘%: ‘ A j! A \\\\\\\\\ \m \\\\\\\\\\\\\\\\\\\\\ -- - 1,. Impact arm 4. Plunger or bearing surface 2.‘ LVDT. transducer 5. . Scale ’ 3..Specimen 6. ~ Impact arm release Figure 3. Impact testing apparatus. (From Géhlich and Mohsenin, 1962) 10 similar yield point in theforce-deformation curves for potatoes; however, they did not come to the conclusion that the yield point for this product produced a corresponding discoloration and bruising .of the internal cellular tissue. rBiological products, such as fruits and vegetables, exhibit a very pronounced time-dependent mechanical behavior. Hamson (1953) reported, for example, that the Magness-Taylor pressure tester was not satisfactory for testing the firmness of tomatoes because the "'rate of compressing the spring" as the plunger was forced against the fruit influenced the results. . The influence of rate of straining of the apple fruit was evaluated by Mohsenin, Cooper, and Tukey (1962). They showed that as the rate of deformation of apple specimens in- creased from 1 to 13 ipm, the yield force increased from 4 to 7.6 pounds. The corresponding deformation and energy parameters also increased in a similar manner with increasing rates of deformation. The influence of rate of loading is further illustrated by results obtained under impact loading conditions as Opposed to results observed under quasi-static loading. GEShlich and Mohsenin presented a compari- son of the energy required to initiate bruising of apples under quasi- static versus impact loading using a f-inch diameter pressure pin. It was found that the energy required under impact was 1. 5 to 2. 7 times that required under quasi-static loading at a rate of deformation of 0. 15 ipm. The total deformation under the two types of loading, however, was approximately the same. -Results of studies by Zoerb (1958) indicated that the strain-rate effect for biological material may be influenced by moisture content. He stated, for example, that the energy required for shearing high moisture grain under impact loading was greater than the static shearing energy; the reverse was true for low moisture grain. Low moisture corn kernels exhibited no strain rate effect as the rate of deformation 11 varied from 0. 0777 to 0.478 ipm; the strength of high moisture kernels, however, did show some differences at the five per cent level of sig- nificance. Fruits and vegetables are living organisms which are constantly undergoing physiological and chemical changes (Biale and Young, 1962). Even after maturity, they carry on certain processes as respiration and synthesis of organic compounds. As a result another time-dependent factor is introduced into studies of their mechanical behavior--stage of maturity. Cooper (1962) described the influence of maturity upon some of the physical and mechanical properties of apples. He reported that the capacity of apples to absorb energy without bruising was not a linear function of stage of maturity but had a very rapid rate of change near the harvest period with the direction and rate of change being dependent upon the particular variety under consideration. Mohsenin and G6hlich (1962) further noted that as the fruit ripened, the compressive stresses necessary to cause bruising decreased and the energy capacity followed the same trend as the deformation required to injure the fruit. Cooper also studied the influence of maturity upon the elastic modulus based upon compression of cylindrical specimens removed . from the apple. Similar to the observations of Zoerb with grain, - Cooper found that the elastic modulus was influenced by the number of loading cycles applied to the specimen. Basing his elastic modulus values upon the second loading cycle, Cooper reported values varying from 600 to 1700 pounds per square inch (psi) depending upon variety and stage of maturity. Investigations by Mohsenin, ‘C00per, and Tukey (1962) demonstrated that the elastic-plastic behavior of apples was dependent upon its stage of maturity. They reported that the degree of elasticity (See Section 5. 2) of Golden Delicious apples varied from 80 per cent two weeks before harvest to 55 per cent two months after harvest. Cooper noted, in 12 addition, that the tissue nearest theicenter of the fruit had a higher degree of elasticity than the tissue nearest the surface, and the plastic deformation occurring in the fruit was a function of bothtime and the number of loading cycles. , This time-dependence of the plastic deformation‘is the result of two rheological properties exhibited by most biological materials-- creep and stress relaxation. Zoerb reported, for example, that pea beans held at constant deformation experienced a decrease in force withvincreasing time. He represented this stress? relaxation in pea beans by two Maxwell models in parallel. Creep within the apple fruit was represented by Mohseninet a_._1. . (1962) in terms of the analogous behavior of a Maxwell model in series with a Kelvin-Voigt model. Shtrankfel'd (1957) characterized the elastic-plastic properties of muscles in terms of its creep behavior under various constant loads. The time dependence of the mechanical behavior of biological products is a phenomenon which cannot be ignored. . Even though many previous studies have been conducted on the potato tuber, its mechanical behavior has not been as well defined as that of the apple fruit. The need for a more comprehensive study of the mechanical pr0perties of the potato tuber has been apparent. for many years, since potatoes are very susceptible to injury'during mechanical handling and harvesting operations. Wiant (1945) investigated the nature and causes of internal discoloration (blackspot) of potatoes and showed conclusively that blackspot could be produced by mechanical injury to the tuber at pressure bruises and, conversely, that blac-kspot failed to develop unless the tissues were mechanically injured. ‘ He further observed that pressure bruises occurred only at points of con- tact between tubers, and the size and shape of these bruises varied with the extent of contact and the degree of pressure exerted against » the tuber. Moreover, the smaller pressure bruises tended todisappear and the larger ones tended to become less conspicuous throughthe l3 resumption of their normal shape after the removal of all external loads or forces. Lampe (1959) made an extensive study of the load-bearing capacity of the potato tuber as influenced by climate, weather, variety, position of the tuber in the soil, cultivation practices, and storage. He observed that climatic variations from year to year can have a very strong in- fluence upon the resistance of potatoes to injury. ‘Other factors such as changing soil conditions, depth of planting, and differences in storage conditions were also very significant. 'He noted in addition that there were wide variations within varieties themselves and, hence, it was often necessary to test at least 15 to‘30 tubers in order to come to a statistically significant conclusion concerning the load-bearing capacity and the susceptibility of the potato tubers to injury. Depending upon variety and climate, he reported that the load-bearing capacity of the tubers loaded with a 3-millimeter diameter cylindrical pressure pin varied from 1800 to 2800 grams. -Lampe made another significant observation from his studies. Potatoes which were able to withstand the highest static stress under pressure-pinloading suffered the least number of bruises under dynamic stress during passage through a digging machine. It was found, for example, that tubers which withstood 2400 grams of force under static pres sure-pin loading suffered only one-fifth the number of severe injuries during passage through a harvester as tubers which were able to sustain a static pressure-pin load of only 2100 grams. Hence, rank- ing of potatoes on the basis of a static test served as a good indication of the capacity of the tubers to resist injury due to dynamic stresses during certain handling operations. «Hansen (1952) and Witz (1954) both used similar techniques and came to similar conclusions. -Witz reported that the resistance of varieties to bruising consistently remained in the same order over a l4 period-of, years from 1949 through 1953. There was a highly significant interaction between variety and location indicating that varieties react differently with a. change inlocation. Hansen also reported highly sig- nificant variations between varieties and locations based upon both puncture and abrasion tests. There appeared to be no correlation, however, between the specific gravity, of the potato tuber and the results from the pressure tests. - Most of the results reported for potatoes have been concerned with the factors which influence mechanical behavior, , but only minor progress has been made toward defining those mechanical properties which the engineer needs to know in order to design and improve machinery and equipment for potato handling. The objective of this study was to provide some of that information. - III. THEORETICAL C ONSIDERATIONS The objective of most studies in mechanics is to predict the motion of a material under the influence of various environmental forces‘such as pressure, temperature, time, atmosphere and radiation. All of these environmental factors are considered as forces by Triffet (1962) since they may cause a body to change its motion either by translating, rotating,» or deforming. - In the latter case, the particles within the body experience a change in position relative to each other. rDeformations, in general, may be conservative or dissipative with respect to energy storage within the deformed material. Classical examples of these two types of behavior are elastic deform- ations and viscous deformations, respectively. Intermediate between these two extremes is a type of mechanical deformation which is termed I-".'visceelastic,~ " i. e. '. it is neither ,entirely conservative nor entirely dissipative, but combines these two effects. , Hence, Bland (1960) has defined viscoelasticity as a generalization of elasticity and viscosity. , 3. 1 Elasticity , A body is perfectly elastic if the deformation or strain occurs instantaneously with the application of stress: and this deformation is completely and instantaneously recovered when the stress is removed (Fitzgerald, 1961). It is generally assumed in addition that there is a one-to-one relationship between the state of stress and the state of strain in an ideal linear elastic body; hence, all timeadependent effects are excluded. 15 16 In 1676 Robert Hooke showed that, for small strains, certain bodies under axial stress exhibited ideal elasticity and the stress 0" was directly proportional to strain 6. The proportionality constant has been defined as Young's modulus of elasticity, AE. Hence, Hooke's law has takenthe following form 0': Es (uniaxial stress) (3. 1) A body which obeys equation 3. 1 is sometimes referred to as a "Hookean" body. Equation 3. 1 is a special case of the more generalized form of Hooke's law (sometimes referred to as the constitutive equations of elasticity) ¢L=c e we) 13 ijrs rs where 0‘;j and ers (i,j,rL-, s = 1, 2, 3) denote the components of the tensors of stress and strain with respect to a system of rectangular axes xi (Sokolnikoff, 1956 and Malvern, 1962). The repeated subscript is used to indicate the summation convention, that is, whenever the same letter subscript occurs twice in a term, that subscript is to be given all possible values and the results added together. If ui are the components of the displacement vector, then 6ij = -i--(ui,j + uj,i) (3.3) where the index after the comma denotes differentiation with respect to the corresponding x-coordinate, i. e. , u. . = Bu, /3x.. C.. are 1,3 1 j IJI'S the elements of a fourth order symmetric coefficient matrix which for an isotropic, homogeneous material takes the following simplified form 0. iat +p[a.5, +5.5.J (r4) ijrs ij rs , ir 33 is jr 17 where h and u are arbitrary scalars, and 6mn is the Kronecker delta. For an isotrOpic, homogeneous material, therefore, Hooke's law takes the general form O-ij: )‘Ekk‘sij‘L ZMij (3'5) 01' 0'..= xe a..+2,.c.. (3.6) ij 0 ij IJ w = =3 + + here eo ekk E n €33 €33 / The parameters x and P are two elastic constants which completely determine the elastic properties of the material, and algebraic relations may be derived among the most commonly used elastic constants A, ii, G, v,-E, and the bulk modulus, K K" 3(1- 2v) (3'7) since only two of them may be independently specified for an isotropic material. Irrespective of the mechanical properties of the material, the state of stress in an elastic body must satisfy, in addition to the constitutive equations, the equilibrium conditions. \ O’ik’k+xi=o (3.8) assuming that the components of acceleration may be neglected (quasi- static loading) and where Xirepreaentmthe. body force components. Similarly, the strain components must satisfy the conditions of com- patibility €ik,lm+ 61m,ik= €il,km+ €km,il (3'9) which representssix compatibility equations. The solution of any given 18 problem in elasticity requires the determination of the stress com- ponents, or displacements, which satisfy the differential equations (3.6, 3. 8, 3. 9) along with the appropriate specified boundary con- ditions. .Certain special mathematical techniques have been employed previously torsolve the above equations for certain given boundary conditions and many of these solutions have been presented and dis- cussed by Timoshenko and Goodier“(1951). The solution of the problem of a concentrated force acting on the boundary of a semi-infinite body was solved by Boussinesq :(1885). This solution may be extended by applying the superposition technique to find the displacements and the stresses produced by a distributed load acting on the surface of a semi-infinite body. Timoshenko and Goodier (1951) applied this procedure to determine the relationship between a load distributed uniformly over the area of a circleof radius ‘ "a" and the deflection of a point on the surface of the body at some distance "r" from the center of the loaded area. The pressure-pin loading technique employed by Lampe (1959) for his studies of potato injuries very closely approximated this type of loading since the cross-sectional area of the loading pin (of the order of 0. 01 square inches) was much smaller than the projected cross- sectional area-of the tuber (of the order of 1 square inch). The writer suggests therefore that the elastic solution for the case of a rigid die in the form of a solid circular cylinder pressed against the plane boundary of a semi-infinite body may well be applied with caution to interpret the results from pressure pin investigations of certain fruits and vegetables which have an apparent homogeneous structure, e. 3. potatoes. For the case of a rigid die, the displacement uz is constant over the circular base of the die (Fig- 4). The distribution of pressure is not constant, however, but is given by the expression l9 \\\\\\\\i\\‘ I 0‘, 8.--- db Figure 4. Theoretical stress distribution under a rigid die acting against a semi-infinite elastic body. 20 p = (3.10) where P = load on the die, 1b a = radius of the die, in r = distance from the center of the circular area on which the die acts, in The smallest value of the pressure is at the center, r = 0, where p . ________z_2 (3.11) which is one-half the average pressure acting on the circular area of contact. At the boundary of the loaded area, r = a, the pressure be- comes infinite. Timoshenko and Goodier have pointed out that yielding occurs along this boundary; however, the yielding is localized and does not substantially influence the distribution of pressures (Eqn. 3. 10) at points located at some distance away from the boundary of the circular area. The relationship between the applied force'P (Fig. 4) and the deformation uz of the material under the die is given by the expression u = M (3.12) z 2aE where 'E and v are Young's modulus of elasticity and Poisson's ratio, respectively, for the material. Assumingthat the elastic properties of the body are constants, then for any given size die the force versus deformation relationship is linear. 3. 2 Viscosity A viscous fluid differs from an ideal elastic solid in that the strain resulting from an applied stress is delayed and increases indefinitely with time. Moreover, the strain produced at the end of any period of 21 time is completely irrecoverable when the applied stress is removed. A characteristic property which determines the flow of any fluid is its viscosity which may be defined as its internal friction or resistance to flow. Consider the laminar flow of a fluid between two parallel plates one of which is at rest, the other is moving with a constant velocity "V" parallel to itself as shown in Figure 5. Experiment has shown that the fluid adheres to both walls (Schlichting, 1960), so that the fluid velocity at the lower plate is zero and at the upper plate is V, the velocity of the plate. It can be shown, therefore, that the fluid frictional shear- ing stress, 3' , is of the form 3': TI (dv/dy), (temperature constant) (3.13) where Z’ = the fluid shear stress, lb per ft2 7): the coefficient of viscosity, lb - sec per ftz dv/dy= velocity gradient within the-laminar flow, sec"l This equation is occasionally referred to as Newton's law of friction and fluids which obey it are called "Newtonian fluids. " This expression generally serves as a very useful definition of viscosity. The quantity r) (Eqn. 3. 13) is a measure of the viscosity of the fluid and is temperature dependent. If it is not constant for a given temperature, i. e. , if it varies as a function of the velocity gradient or rate of shear, then the fluid is non-Newtonian and equation 13 takes a more general form ._. 3.1 it 2’ [M330] BY (3.14) where for a constant temperature, 7) is some function of the velocity gradient in the fluid flow stream. Characteristic relationships between shear stress and velocity gradients are shown in Figure 6. 22 l ‘ W I T 7. Y V(y) '3 T l L71/1r/1/7r7 //7 I/IIIIrIII/I/l‘71 Figure 5. Velocity distribution in a viscous fluid moving between two parallel flat plates. Shear Rate , Shear Stress —-—+ A - Ideal fluid (Schlichting, 1960) B - Non—Newtonian, dilatant flow (Jastrzebski, 1959) C -~Newtonian, viscous flow (Alfrey, 1957) D - Non-Newtonian, pseudo plastic flow (Jastrzebski, 1959) E - Bingham body, idealized plastic flow (Alfrey, 1957) F - Quasi plastic flow (Alfrey, 1957) Figure 6. Rate of shear strain as a function of stress for different types of fluids. 23 3. 3 Viscoelasticity Gross (1953) has shown that some of the methods developed for ordinary elastic and viscous bodies may be applied in the study of viscoelastic bodies. The coefficients which appear in elasticity are not considered as constants but are now functions of time or frequency. . Hence, by the use of appropriate mathematical transformations, an elastic solution may be used to develop the stress-strain relations for a viscoelastic material. ’ The development of the theory and the definition of the coefficients used to describe viscoelastic materials have been discussed by numerous authors (Eirich, 1956; Scott-Blair and Reiner, 1957; Reiner, 1960; and Bland, 1960). {(Alfrey (1957) summarized seven distinct methods of specifying the properties of viscoelastic materials and divided these methods into two classes. In Class I he lists the more fundamental theoretical approaches, i. e.,the generalized Voigt model, the generalized Maxwell model, the linear differential operator technique, and the com- plex variable mathematical approach using the mechanical impedance function. Class II methods include the experimental curves used to ‘ "map out" the viscoelastic character of the material, i. e. , the creep curve, consisting of strain as a function of time with constant stress; the relaxation curve, consisting of stress as a function of time at constant strain; and the dynamic modulus curve, consisting of the elastic modulus as a function of the frequency of straining. ~Methods of transforming from one mode of description to another have been developed and are discussed by Gross (1953) and Alfrey (1957). One of the most widely used and most easily interpreted methods of specifying the viscoelastic behavior of materials is in terms of mechanical models. A viscoelastic model is composed of two or more primary elements--the elastic element and the viscous element. .24 These are referred to by some as the Hookean or spring element and as the Newtonian or dashpot element, respectively. If -F is the force applied to the elastic element and u is the corresponding di splac ement, then F = Eu (3.15) where E is a constant known as the elastic modulus for the spring (Fig. 7a). For the viscous element in Figure 7b, the relationship takes the form F = nDu (3.16) where D is the operator denoting differentiation .with respect to time, d/dt, and in is the viscosity of the dashpot fluid. The combination of these two elements in series forms a Maxwell model (Fig. 7c) and it can be shown that the relationship between the force applied to the model and the displacement can be represented by l the differential equation Du = (l/E)DF + (1/n)F (3.17) If, on the other hand, the two primary elements are connected in parallel as shown in Figure 7d, then the resulting force-deformation relationship is of the form F=Eu+nDu . (3.18) Consider a model of a large number of Maxwell models, a Hookean element, and a Newtonian element all connected in parallel (Fig. ,8). If the model is given a sudden deformation, u u= kH(t) (3-19) where H(t) is the Heaviside unit function H(t) = 0 t < 0 H(t) = 1 t 3 0 25 e 1:] n (a) The Hookean elastic (b) The Newtonian viscous element. . element. -E E I "L— r) EFJn (c) The Maxwell model. (d) The Kelvin-Voigt model. Figure 7. Some fundamental elements and mechanical models in viscoelasticity. 26 then the problem of relaxation can be represented in terms of the mathematical equation n F(t)= k E1H(t) + knza (t) + kffi exp(-Ei t/"i) H(t) (3. 20) where 5 (t) is the Dirac delta function, 6 (t) = D H(t). Because of the dashpot element which is connected in parallel with the spring element and the remaining Maxwell models (Fig. 8), it is physically impossible to give the generalized Maxwell model a sudden, . instantaneous deformation as specified by equation 3. 19. Nevertheless, the mathe- matical problem of stress relaxation for such a sudden loading is adequately defined by means of the Dirac delta function in equation 3. 20. The force response to a unit extension u(t) = H(t), but excluding the constant and delta components, is defined by Bland (1960) as the "relaxation function, " denoted by X(t). For the generalized Maxwell model, therefore, it is n X(t) = Z) Ei exp(-Eit/ni) H(t) (3.21) i=3 If the relaxation times tr are defined by 2'r : ("r/Er) then n X(t) = 11:23 Er exp(- t/z’r) H(t) (3. 22) Stress relaxation in materials can be represented by generalized Maxwell models having various numbers of Maxwell models in parallel. If the stress in a material falls to zero for large values of time, then there should be no spring in parallel with the other elements when a model is postulated to simulate the behavior of this material. If on the 27 --—------- Figure 8. - The generalized Maxwell model. 28 other hand, the stress does not approach zero as time approaches infinity, then obviously this type of behavior should be represented by an elastic element in parallel with the remaining elements in the generalized model. After a satisfactory model is postulated, then the relaxation function (Eqn. 3. 22) can be determined. With a suitable relaxation function, the complete viscoelastic behavior of the material under various types of loading can be mathematically defined throughthe appropriate mathematical transformations listed by Gross (1953) and Alfrey (1957). IV. EXPERIMENTAL TECHNIQUES The basic unit of the mature potato tuber is the living cell which has the capacity to synthesize large molecules from simpler substances. Molecules synthesized in the cells of the tuber are found in the starch grains which form a major portion of the bulk of the mature tuber. - The potato tuber, therefore, along with wood, wool, cotton, and silk, is among the oldest and most common of the natural high polymers. _ It might be expected, therefore, , that an investigation of the mechanical behavior of the potato tuber may involve certain complicating factors common to other high polymers, for example, a complex time behavior within the material. High polymers in general possess a series of retarded elastic mechanisms of response to stress, as well as instan- taneous elasticity and flow. Hence, , the mechanical tests most easily interpreted are those which involve simple time sequences; i. e. , those in which either stress or strain is held constant, or in which the rate of change of strain is held constant. With the above complicating factor under consideration, a testing machine was developed to be used in a study of the viscoelastic response of the potato tuber to applied stresses and strains. 4. l The Testing Machine An overall view of the testing machine is shown in Figure 9. The basic power unit of this machine was a 4-inch stroke, double-acting, pneumatically driven air motor with positive, hydraulically controlled piston speed in both directions. High pressure air alternatingly directed from one side of the air cylinder to the opposite side forced the piston of the air motor to move alternatingly in the forward or the 29 30 Load cell Strain gage transducer for displacement measurements Stop-check for stopping and holding the piston stationary Air motor and hydro-check precision control assembly Manually-operated direction control valve‘ Switch for operating stop-check solenoid Valve for adjusting piston speed on the advancing stroke Valve for adjusting piston speed on the retracting stroke Figure 9. The testing machine. 31 reverse directions. -Connected directly in tandem with the air motor was a hydro-check assembly which consisted basically of an oil-filled checking cylinder, a checking piston rod, and adjustable needle valves for regulating the rate of oil flow within the checking cylinder. The function of the hydro-check was to add precision and smoothness to the action of the air motor by opposing the forward and reverse movement of the air cylinder, thereby reducing or compensating for variations in the power thruStof the air cylinder. The speed of the piston was controlled by regulating the rate of oil flow within the hydraulic checking cylinder. This oil flow rate in turn was regulated by means of two needle valves (7, and 8, Fig. 9) which were opened or closed to vary the rate at which oil was permitted to flow through restrictions in the checking cylinder. Table 1 shows the needle valve settings required for the piston speeds used. - The inlet pressure to the air cylinder was controlled at 44 psi. Table l. Piston speed and hydro-check valve setting for an inlet air pressure of 44 psi. Piston speed* Valve opening (inches per minute) (revolutions) l 1. 30 2 1. 80 4 1 . 95‘ 10 Z. 25 20 2. 70 40 3 . 50 * Estimated accuracy: 1'. 10% Since it was necessary during the tests tomaintain a constant rate of change of strain or rate of deformation, it was important that the speed of the piston be kept constant as stress was applied to the test 32 specimen. The loading head should not "run away, " for example, when there is no restraining force and it should not slow down as the force applied to the specimen builds up to a maximum. Figure 10 shows the relationship between the force applied by the testing machine to a linear elastic specimen (rubber) and the displacement-time response of the piston which served as the loading head of the testing machine. The de- rivative of the displacement-time curve with respect to time represents the speed of the piston which was taken to represent the rate of deform- ation of the test specimen. The air motor was designed to deve10p a maximum force of approximately four to five times the inlet air pressure. As shown in Figure 10, the'maximum applied force for the inlet pressure of 44 psi was approximately 200 pounds. More important, however, the slope of the displac ement-time curve, and hence the speed of the loading piston, was independent of the applied force (or constant) within the force range from 0 to 150 pounds. This indicated that the rate of deformation of a test specimen would be constant as long as the force applied to the specimen did not exceed 75 per cent of the capacity of the testing machine. It should be noted that the inlet pressure to the air cylinder could be increased to 80 psi’ to give the testing machine a maximum force capacity of 300 pounds if necessary. For an inlet pressure other than 44 psi, it was necessary to re-calibrate the needle valve settings to provide the desired piston speeds. Another feature of the testing machine--a "stop-check"--made it possible to carry out stress relaxation investigations. The stop-check functioned to stop the piston rod at any point along its stroke, dwell for any desired time interval, and then to continue in either the forward or reverse directions. This action was accomplished by means of a ‘ stop-check valve which interrupted the flow of oil through the needle DISPLACEMENT (in. ) DISPLACEMENT (in. ) .20 .16 .12 .08 .04 .20 .16 .12 33 p r- 240 1- Piston Motion vs Applied Force 4 ipmN ’7 b 180- ‘ fl"'2 ipm . 3120- ‘Z— 1 1pm -18 60 .. a (Displacement proportional to O a lied force. J” <5 PP ) j_ o 1 1 1 1 1 - 1 1 1 1 1 0 '2 4 6 8 10 12 . 14 16 18 20 TIME (sec.) - 240 " 0 4 ipm} - 130L ' A. Q, , éizom 20 1pm“i--10‘ipm. L11 .2 O 60 - (:4 ._ 0 ‘ l 1 l L I am 1 I J I 0 0. 5 1. 0 1. 5 2. 0 TIME (sec.) Figure 10. .Displacement-time response of the testing machine as influenced by the force applied by the machine to the test specimen (Inlet air pressure: 44 -psi). 34 valve restrictions in the hydro-check cylinder which thereby stopped the piston of the air cylinder until the stop-check valve was released. Hence, a test specimen could be loaded up to some point when the pistonrod would be stopped and held in a fixed position by means of the stop-check valve (3, and 6, Fig. 9). This action simulated holding the deformation of the specimen constant; hence, a measurement of the force-time response of the specimen at constant deformation sufficed to define the stress relaxation behavior of the test material. . Since provisions had been made for applying a load to the specimen, for controlling the rate of deformation, and for holding the deformation of the test specimen constant for stress relaxation studies, the next objective‘was to (provide some means for sensing and-recording the force-deformation response of the test specimen as it was subjected to deformation withinthe testing machine. 4. 2 TheSensing Elements The sensing elements used with this testing machine consisted of a load cell and a cantilever beam strain gage transducer for measuring di splac ements . . A load cell was attached to the piston rod of the air motor in the testing machine (1, Fig. 9) and, thus, served as the movable"'cross- head" of the testing machine. For the major portion of these tests, the forces encountered were below 50 lb; hence, a 50 1b'capacity ILH Type U-lB load cell was used. The accuracy of this load cell was withini 0. 25% with a maximum nonlinearity of i 0. 10% valid for both tension and compression. Before each series of tests the calibrationrof the load cell was checked by applying five 10 lb weights accurate toewithin l per cent to the load cell. ~For force measurements above 50 lb, a 1000 lb capacity 35 strain-gage instrumented load ring was used in place of the commercial load cell. The 1000-lb load ring was used within the force range from 50 to 300 lb and was found to be accurate withini 2% when calibrated with known weights. - A cantilever beam with SR-4, Type A-S strain gages mounted at its root and electrically arranged to form a four arm, temperature compensated bridge served as the deformation sensing element for the testing machine. - The base of the cantilever beam was attached to the framework enclosing the load cell (1, Fig. 9) and moved back and forth as the piston rod advanced and retracted on the forward and reverse strokes. A movable thumb screw mechanism on the framework of the testing machine provided a means of holding the free end of the canti- lever beam fixed at any point along the stroke of the piston rod. The cantilever beam strain gage transducer was calibrated with an Ames dial indicator having a range from 0 to 1 inch, graduated in 0.001 inch increments and accurate to within 1' 0. 001 inch. Using this technique, the deformation sensing transducer was calibrated to within an accuracy of;"-. 1%. 4. 3 The X-Y Recorder There are many advantages of making an automatic, permanent record of observed data over the recording of transitory visual observ- ations in a notebook. As Wilson (1952) pointed out, the possibility of human error in recording the data is reduced, there is a much greater chance of detecting and interpreting unexpected finer features which might otherwise be overlooked, the permanent record can always be examined at leisure, and finally, the possibilities of bias are reduced because other observers can examine and check the records. An X-Y recorder was the major instrument used to make an automatic, permanent record of the experimental results of this investigation. 36 The X-Y recorder is a device for recording two variables simul- taneously, one as a function of the other, on rectangular coordinate paper. This instrument also had a calibrated time base on the X-axis used for recording one variable as a function of time, as for example, in a stress relaxation test where the stress decay within the specimen is recorded as a function of time. The only variable which the recorder is capable of sensing is voltage. Hence, it was necessary to change the force and displacement readings into a voltage signal by means of strain gage transducers. Millivolt signals from the load cell and the cantilever beam transducer were channeled directly into the X-Y recorder without amplification as shown in the block diagram in Figure 11. The maximum sensitivity of the Mosley 135 X-Y recorder was 0.5 milli- volt per inch deflection of the recording pen. Using the 50 lb capacity load cell with a 6-volt source across the strain gage bridge, it was calculated that the maximum voltage output to the recorder was 12 milli- volts (mv)-v-an output sufficient to drive the recording pen 24 inches. Similarly, one inch deflection of the free end of the cantilever beam provided an output voltage sufficient to drive the recording pen 32 inches. Hence, no pre-amplification of the output voltages from the strain gage bridges was necessary. The X-Y recorder is shown in Figure 12 along with two other devices constructed by Hendrick (1962) to improve the versatility of the instrument. "B" of Figure 12 is the Strain Gage Balance and Calibration Unit which allows the recorder to be used directly with a strain gage bridge without channeling the signal through an external amplifier. "C" of Figure 12 is a Performance Test Rig which was used to test the response of both axes of the recorder simultaneously. Any irregularities in the resultant trace of the instrument indicated a possible malfunction in the operation of either axis. Wiring diagrams and details of the con- struction of the auxiliary devices used with the recorder are given by 37 .msdom Hmuaofiflnomxo on» no Emsmmflv x003 < 3: unamwh 2MB.me EHHWWm WHZHEHAM EHHmwm OZHQMOOHM Hugh/:4 OZHmZMm , AOMHZOO noofipmnmnfi pace noumamwfl oGEomz wcflmofi 35 823.523 can confirm O O :00 Umod Hopuooom an Figure 12. The X-Y Recorder and auxiliary devices. Figure 13. The whole potato tuber supported by a flat plate and subjected to compression with a cylindrical die. 39 Hendrick (1962). With the experimental apparatus shown in Figure 11 it was possible to obtain direct records of the force versus deformation characteristics of fruits and vegetables and other test specimens. 4.4 The Compression Tests The compression tests carried out during these investigations were of three types: (i) compression tests using cylindrical pressure pins (dies) which were forced into the surface tissue of the tuber; (ii) uniaxial compression tests using cylindrical sections of tissue removed from the tuber; and (iii) three-dimensional compression of the whole potato tuber under hydrostatic pressure. The first type of test repre- sented what Mooney (1937) referred to as a service or "practical" test in which some practical loading condition is imitated. The second and third types of tests represent what Mooney referred to as being "scientific" tests in which the major objective is to measure some property of the material as completely and accurately as possible by iso- lating as many extraneous effects as possible. -Compression of the potato tuber with the cylindrical die repre- sented a stress state which involved a combination of tension, com- pression, shearing, and bending. It was indeed a most complex state of stress resulting from this loading technique. Nevertheless, this cylindrical die loading technique is one which has been universally accepted and is still practiced in the fruit and vegetable industry to evaluate the "firmness, " maturity, and various other properties of agricultural commodities. .A cylindrical die having a cross-sectional area of 0.05 sq in was used in this study to evaluate the influence of maturity, variety, rate of deformation, and temperature upon the force-deformation-energy ' characteristics of the mature potato tuber. -Other cylindrical dies (Fig. 14) having cross-sectional areas from 0.01 to 2. 00 square inches 40 Figure 14. Cylindrical dies used for loading the potato. Figure 15. Cork boring machine with cylindrical cutting tools for removing specimens of tissue from the potato tuber (Insert: Cylindrical specimen being removed for uniaxial test) 41 were used to investigate the influence of the area of the loading surface upon the capacity of the tuber to (resist applied pressure. To conduct a test, a cylindrical die was placed in the threaded end of the load cell (Fig. 13) and a force versus deformation test was carried out with the experimental arrangement shown in Figure 11. -Since it was not desir- able to alter the potato tuber by cutting or altering its anatomical structure in any other way, it was decided to conduct a-"non-destructiive" test using the whole tuber supported upon a flat surface. - With the tuber supported by a flat plate as shown in Figure 13, the cylindrical die was brought into contact with the surface of the tuber at a location which pro- vided uniform contact between the die and the tuber. At this time the free end of the cantilever beam displacement transducer was held fixed and any further downward movement of the piston rod of the testing machine was sensed and recorded as deformation of the tuber. The load cell simultaneously indicated the force applied to the product. The force versus deformation recordings were in fact force versus piston rod displacement indications since the cantilever beam measured the movement of the piston rod. The tuber inevitably suffered a certain amount of deformation due to the reacting force at the supporting plate. Hence, , the total recorded deformation was the sum of the deformation occurring throughout the thickness of the potato tuber which included the deformation directly beneath the rigid die and the relative deformation of the tuber near the supporting surface. To analyze the magnitude of the deformation occurring within the region of the plate upon which the whole tuber was supported, sixty tests were conducted using, on the one hand, the whole tuber supported by the flat plate, and, on the other hand, one-half of a tuber resting with the flat surface on a flat plate as shown in Figure 16. Force-deformation curves were conducted for both cases using a 0.05 sq in cylindrical die and the results are summarized by curves "A" and "B" in Figure 16. FORCE. (lb) 20 15 10 _ 400 P. , 300 - €7.00 3' 1.1 a: [-4 - m 100 ._.—._..1, 0 Figure 16. 42 Loading "A"- Loading "'B" . Half tuber “Whole tuber ,._ _ A — B 1 1 1 | 1 l l I 0 40 ' 80 120 ' 160 -D EFORMATION (milli-inch) Force-deformation curves as recorded by the X-Y recorder for the whole tuber and a half-tuber sup- ported upOn a flat plate. (Cross-sectional area of loading pin: 0.05 Sq in) 43 The average force required to rupture the skinof the tuber was the same in both bases (13. 87 i 0. 01 1b). Hence, . as expected, the force readings were not affected by the relative deformation of the tuber near the reaction support. 1 On theeother hand, however, the average piston displac ement necessary to rupture the whole tuber was 0. 121 inches as compared with 0. 104 inches for the half-tuber loading con- dition. It Was concluded, therefore, that for the whole tuber supported upon a flat surface and-loaded with a rigid 0. 05 sq in cylindrical die, an average of 15 per cent of the deformation occurred within the bottom half of the tuber near the supporting plate and 85 per cent of the re- corded deformation occurred within the upper half of the tuber directly beneath the-area of contact with the rigid die. The deformation in the bottom half of the tuber due to the force at the surface supporting the whole tuber contributed .to a systematic type of error which could be and was evaluated and taken intoaccount in the final analysis of data. - An estimate of the relative amount of ~ deformation occurring within the tuber near the region of the support- ing surface, was ~made both mathematically and statistically and the results are summarized in Table 2. Based upon a statistical analysis, the correction factors in Table 2 may vary within an absolute value of i 0. 05; i. e. , the correction value for the recorded deformation using the 0. 05 sq in die could be between 0. 80 and 0. 90 with the average value being 0. 85. ~Note that for parallel plate loading, the correction factor is 0. 50 since it was assumed that the tuber was symmetrically shaped andhomogeneOus in structure and mechanical response. Therefore, one-half of the deformation occurred throughout the top half of the tuber and the other one-half of the total deformation occurred within the bottom half of the tuber. Thus, the equivalentdeformation cf one-half of the potato tuber under parallel plate compression would be 0. 50 times the total deformation recorded by the X-Y recorder. 44 Table 2. « COrrection multipliers applied to the recorded force- deformation curves to obtain theequivalent rupture para- meters for one-half of a pctato tuber supported by a rigid surface. 3 Area of the Correction multipliq‘rs for: * ’ loading surface Force Deformation Energy 0.01 sq in 1.00 0.95 0.92 0.02- sq in 1.00 0.91 0.88 0.05 sq in 1.00 0.85 0.82 0.10 sq in 1.00 0.81 0.78 0.20 sq in 1.00 0.77 0.74 0.50 sq in 1.00 p 0.71 0.68 Parallel plates 1 . 00 , '0. 50 0. 50‘ 1... * q . . Estimated error in the correction multipliers: i” 0. 05 To obtain anestimate of the energycapacity of the product, the area under the force-deformation curve up to the point of rupture was, measured with a planimeter accurate to within: 2 per cent. 'As wasthe case for deformation, a certain amount of the recorded energy, (or work) was absorbed by part of the tuber near the reaction support. This per- centage varied from an average of 8 per cent for the 0. 01 sq in die to .50 per (cent for parallel plateloading (Table 2). _ The deformation and energy. correction multipliers differed . slightly because the force-deformation curves were not truly. linearly related but exhibited a slight curvature. If all of the force-deformation curves had been linearly related, the correction factors for both para- meters would have been the same. 1 The deviations from linearity were quite small, however, and the Confidence limits for the two parameters did overlap indicating that some tubers might indeed show-a linear. relationship between force and deformation. 45 4.5 The Uniaxial Compression Tests Uniaxial compression tests were conducted with the same experi- mental arrangement shown in Figure 11. For these tests, it was necessary to remove a specimen of tissue from the potato tuber. This was done by means of a cork boring machine (Fig. 15) with cylindrically shaped cutting tools having cross-sectional areas from 0. 20 to 2. 00 sq. in. The cutting edge of the cutting tools was chamfered to-a 20- degree included angle. The insert in the upper left corner of Figure 15 shows the type of specimen obtained from the tuber. This specimen was cut to a length of l-inch for an investigation of the force-deformation or stress-strain properties of potato tissue. 1 Of primary interest during these studies was the modulus of elasticity of the tuber tissue, which was determined for small strain conditions from the-slope of the stress- strain curve. The same experimental techniques used for the cylindrical pin investigations were also used for the uniaxial compression tests except that the cylindrical specimen was loaded between parallel plates in- stead of with the smaller pins. To reduce the effects of shear stresses due to contact between, the plates and the flat ends of the specimen, the loading plates were coated with athin film of SAE 20 lubricating oil before each test. 4.6 The Bulk Modulus Tests From the uniaxial tests of the preceding section it was possible to determine the elastic modulus of potato tuber tissue. Since only two of the elastic constants of a homogeneous, isotr0pic elastic body are independent, another technique was needed to evaluate a second elastic property of the potato tuber. It is not suggested that the potato tuber 46 is an elastic body, but that some solutions in elasticity may be useful in interpreting and analyzing the behavior of the tuber under applied stress. Thus, it is important to have at least an estimate or an approximation of at least two of the apparent elastic "constants" of the potato. The most economical (even though not perhaps the easiest) method of determining a second elastic property of the potato tuber was by means of hydrostatic pressure tests which would provide another property of the tuber known as the elastic bulk modulus, K, Eqn .3. 7. The objective was to measure the change in volume of the tuber (or volumetric strain) as a function of the applied hydrostatic stress (or volumetric stress). Hence, analogous to the uniaxial stress-strain curve, we would obtain a three-dimensional or volumetric stress-strain curve. The original volmne of the tuber was measured to within 1 per cent by weighing the water displaced by the submerged tuber. Figure 17 shows the apparatus used for preliminary studies of the behavior of the potato tuber under hydrostatic stresses within the range from 15 to 500 psi. The tuber was submerged in a steel cylinder containing SAE 20 oil. The cylinder was then closed and sealed and a dead weight tester, designed to calibrate pressure gages, was then connected to the cylinder containing the tuber. Air was forced from the system through an air drain plug (E, Fig. 17). As weights, calibrated to give 50 psi pressure increments, were added to the weight tester platform (P, Fig. 17), the volume of the tuber changed by an amount equal to the volume of the fluid displaced by the piston of the weight platform minus the expansion of the cylinder containing the stressed fluid. Hence, the dial indicator (F, Fig. 17) served to measure the volume change of the specimen. The effect of cylinder expansion was first evaluated by means of tests conducted without the potato in the cylinder. 47 Results of preliminary tests indicated pressures within the 0-100 psi range were of the greatest significance from the standpoint of volume-changes under hydrostatic stress. As stresses approached 500 psi, the tuber became incompressible; i. e. , the bulk modulus was verylarge andiPoisson's ratio-approached 0. 50. -On this basis, therefore, the bulk modulus apparatus was redesigned to operate with increased sensitivity within the pressure range from 0 to 60 psi. The low pressure bulk modulus apparatus is shown in Figure 18. The potato tuber in this case was immersed ina cylinder of water. Extreme care was exercised to drain air from pockets in the assembly through‘an air drain plug in the top of the cylinder and at the top of the glass gage assembly. Water was poured into the glass gage tube through a valve at the top of the assembly until theliquid reached the desired level in the glass tube. All valves were then closed and pressure was applied to the liquid from a 60 psi airline through a pressure regulating valve and gage indicator. Pressures above 10 psi could be regulated within: 1 psi of the pressure gage-indication. The glass tube was used to measure changes in volume of the potato tuber under pressure. By measuring the rise and fall of the liquid level in the glass, tube, it was possible to detect volume changes of _-_I-_ 0. 0475 cubic centimeters. A series of tests were carried out without the potato tuber in the cylinder toedetermine the expansion of the cylinder and the compression of air entrained in the water as pressure was applied to the fluid. The results of these tests are shown in Figure 19. This curve represents the volume which had to be subtracted from the indicated total volume change to get the change in volume of the tuber when tests were con- ducted with the tuber submerged in water in the cylinder. Tests indicated that this correction .curve could be consistently reproduced within :1; 10 per cent. 48 r I; '; . ' 32:. Figure 17. High pressure bulk modulus apparatus, 15 to 500 psi. Figure 18. Low pressure bulk modulus apparatus, 0 to 60 psi. 49 Correction Curve for Bulk Modulus Apparatus 2. 4 _ Volumetric Correction - vs Applied Pressure 2. 0 - 1. 6 L VOLUMETRIC CORRECTION (cc.) o I I I I I I 0 10 20 30 40 50 60 PRESSURE (psi. ) Figure 19. , Correction curve to account for the expansion of the cylinder and air entrained in'the water for the low pressure bulk modulus apparatus. 50 4. 7 Growth and Storage of the Test Specimen Potatoes used in these investigations were grown at the Agri- cultural Experiment Station at Lake City, Michigan. Ten varieties were planted on May 22, . 1962 in a randomized block design as shown in Table A1, Appendix A with each variety having three replications. The soil type was a sandy loam. Rainfall during the growing season along with the corresponding dates is tabulated in Table A2,: Appendix A. In addition, water was supplied five different times by irrigation beginning on July 2, 1962. On September 25, 1962 the potatoes were harvested and placed in a 55-600F curing room until October 19, 1962 after which they were removed and stored in the40°F potato storage room located in the Farms Crops Building on Mount HOpe Road in East Lansing. Storage temperature, through its influence upon respiration and water loss, is responsible for changes in weight and specific gravity of potatoes. At 40°F,, potatoes experience a minimum percentage loss in weight and a minimum increase in specific gravity. Over a four and one-half month storage period at a storage temperature of 40°F, Jacob (1959) reported that potatoes experienced only a four per cent loss in weight and an increase in specific gravity of 0. 006. . Experimental investigations were conducted with these potatoes from a pre-harvest date of August 23, 1962 through a post-harvest date of March 20, 1963. The experimental equipment illustrated in Figure 11 was taken to Lake City, Michigan for the pre-harvest tests conducted on August 23,-September 11, and September 25, 1962. This was done to reduce the influence of physiological changes which might otherwise have occurred during the elapsed time interval for transporting the test specimen 150 miles from Lake City to East Lansing, ‘ Michigan. 51 For tests conducted after October 19, 1962,.the potatoes for a test series were removed from the 40°F storage room at least 24 hours before testing and were placed in a neutral atmosphere at a room temperature within the range from 68 to 78°F. For the tests designed to study the influence of temperature upon the mechanical behavior of the tuber, the potatoes were placed in temperature—controlled boxes and a mercury-in-glass thermometer was inserted into one of the tubers with the sensing element of the thermometer embedded within the central tissue of the tuber. Open pans of water were kept in the temperature boxes to maintain a high relative humidity and to minimize moisture loss from the tuber during the heating period. In the instances where specimens were to be removed from the potato tuber, the cutting tools (Figure 15) were also placed in the temperature box and brought to the same temperature as the tuber from which the specimen was to be removed. During the experimental investigations, the following factors were observed and recorded: variety, stage of maturity as indicated by days after planting, weight of the tuber, volume of the tuber, smallest tuber dimension, and the specific gravity of the tuber. .4. 8 Specific Gravity Specific gravity is defined as the weight of a product in relation to the weight of an equal volume of water. Specific gravity has found universal acceptance as an indication of the dry matter content and of starch in potato tubers (Burton, 1948). A chart for converting specific gravity readings .to percentage total solids and starch cOntent is given in Table A3, Appendix A. This chart was adapted from Behrend, Maercker, and Morgen (1880) and is the official conversion chart for the United States Department of Agriculture. 52 The two most common methods of determining specific gravity of potatoes is by weighing the tubers in air and in water, and by flotation of the tubers in brine solutions. The first method is more time- cOnsuming and it was the method used to determine the specific gravities reported in this thesis because of its better accuracy. Specific gravity was computed after weighing the tuber in air and weigh- ing the tuber submerged in water according to the following relationship weight in air weight in air - weight in water specific gravity = All weight measurements were made with a balance read to within it 0.1 gram. V. PRESENTATION AND DISCUSSION OF RESULTS 5. 1 Failure of Cellular Tissue Within Potatoes Under Mechanical Stress Results of over 700 tests have indicated that potato tubers do not exhibit a well-defined yield or bio-yield point such as that reported for apple fruit (Fig. 2) by Mohsenin and Gohlich (1962). 1The force- deformation curves for potatoes under certain conditions deviated from linearity depending mainly upon the type of loading surface and the stage of maturity of the product. In general, however, the force-deformation curves (Fig. 20) and the stress-strain curve (Fig. 21) gave approximately a linear relationship. Figure 20 shows typical force-deformation curves for potato tubers loaded with a rigid die in the form of a solid circular (0. 05 sq in) cylinder for two different dates. For the earlier test date of August 23, 1962, the force-deformation relationship had a noticeable curvature which became less conspicuous after harvesting and storage (B, Fig. 20). Neither of the two curves showed any distinguishing characteristics which might have been associated with a yield or bio-yield point in the material. As a general rule, the force-deformation curves exhibited either a smooth non-linear relationship (A, Fig. 20) or an approxi- mately linear relationship as shown in B of Figure 20 without any apparent discontinuities until point "R" where the load bearing capacity of the tuber was a maximum for the particular loading surface. Mohsenin and Tukey (1962) suggested that potatoes seemed to show the initial break-down of tissues under the skin, associated with the bio- yield point, by a change in the slope of the force-deformation curve well in advance of the rupture point. Results of these studies did not support 53 54 R RN 15 "' A\ 10 v- 7E III P 3 Curve Test Date 0 In 5 )- \ A 8/23/62 B B 1/19/63 Lu- 0 l I I I I J I I l J 0 0.04 0.08 . 112 .1,6 DEF-ORMATION (in) Figure 20. Force-deformation curves for Kennebec tubers loaded with a rigid die in the form of a solid circular (.05 sq in) cylinder. 150 100 a .9: i a: 50 [-1 u: o ‘ 1 I 1 I 1 1 u o 0.10 0.20 0.30 0.40 STRAIN (in/in) Figure 21. Conventional stress-strain curve for a l-in x 1 sq in section of tissue removed from a RussetRural tuber - 10/17/62 55 such an observation. The only distinguishing characteristic consistently observed on the force-deformation curves was the point "R" which appeared to correspond more closely to the ultimate strength of the tuber than the yield strength of the product. Triffet (1962) defined the ultimate strength of a unit cube of material as the point on the force-deformation curve where the force becomes a maximum. Apparently the ultimate strength of apple fruit corresponded to the point in Figure 2 which Mohsenin referred toas the rupture point; i. e. , the point at which the plunger punctured or ruptured the skin of the fruit. For the sake of consistency and to reduce the possibility of confusion and misunderstanding in this area of study, the definitions espoused by Mohsenin (1962) will be used in this thesis. 1 The point where the force applied to the specimen reached a maximum for any given loading surface will hereafter be referred to as the "point of rupture" or _"rupture point. " Corresponding to this point of rupture are three other parameters of primary interest in this investi- gation; namely, the rupture force, the rupture deformation, and the rupture work or energy. (The rupture force (stress) is defined as the force (stress) resisted by the product at its point of rupture and the corresponding deformation (strain) at that point is the rupture deformation (strain). The energy of rupture is the integral or the area under the force-deformation curve up to the point of rupture; i. e. , u=ur gr: deu (5.1) u=0 where ¢r = rupture energy P , applied traction force 11 = deformation of the product under the force P ur= deformation at the point of rupture 56 The rupture point for the potato tuber corresponded tothe point where there occurred some localized failure or injury of the tuber tissue. The severity of damage and the location of this point of failure of the tuber tissue was primarily dependent upon the geometry and size of the loading surface. Failure of tissue within the tuber varied from minor injury of the surface tissue to severe breakage of thetuber along a fault or plane extending into the central tissue. Loading the tubers with cylindrical pins simulated the boundary conditions illustrated in Figure 4. ~ Subsequent observations of tubers loaded by this technique (Fig. 22) indicated localized failure of the surface tissue immediately under the skin around the periphery of the loaded area in contact with the rigid die (cylindrical pin). As predicted from theoretical considerations (Eqn 3. 10), the stress concentrations generated in the region of the periphery of the rigid die induced localized injury of the tuber skin and the tissue directly beneath the ruptured skin. This type of failure was localized. The effect of the stress concentration as indicated by the location of bruised or dis- colored tissue was confined within a small circumference around the area of the surface of contact (Fig. 22). It was possible, however, that this type of loading weakened the cellular structure of the tuber without indicating discoloration and thereby pre-disposed the loaded area to internal blackspot injury such as that discussed by Wiant (1945). Loading the tuber with a rigid sphericallye shaped body resulted in tissue injury as shown in the photograph of Figure 23. The amount of damaged tissue, as indicated by the discolored tissue, was con- siderably greater than when loading with the cylindrical die or pin. The spherical loading surface has been discussed by Timoshenko and Goodier (1951). The maximum stress is the compressive stress at the center of the surface of' contact. 'It has been pointed out, however, that the maximum shearing stress, upon which the failure of some 57 _ 0 0 1 3: memos 0.40 10 DEFORMATION (in) Tissue failure for tuber loaded with rigid solid circular (0. 50 sq in) cylindrical die. Figure 22. (631139-4) — 0 5 .l. 100 - 3: moron 0.20 0.30 0.40 0.50 DEFORMATION (in) 0.10 Tissue failure for tuber loaded with 1-in diameter Figure 23. (631 139-3) rigid sphere. 58 materials such as steel depends, is comparatively small at the center of the surface of contact. The shearing stress reaches a maximum along the axis of loading at a depth of approximately one-half the radius of the surface of contact. Assuming that the radius of the tuber was large in comparison with the radius of the sphere (0. 5 in) used to load the tuber, then the maximum shearing stress in the tuber should have occurred within oneufourth inch of the surface of the tuber. Observations of injured tissue (Fig. 23) confirmed that the depth of the region of most severe injury was located just below the surface of the contact area. Further evidence of the influence of shear stress upon the failure of potato tissue is shown in Figure 24. Tubers were loaded between parallel plates until the force-deformation curve exhibited a dis- continuity or a point where an increment of deformation resulted in a non-positive increment of force, i. e. , a rupture point. After observ- ing the rupture point as shown in Figure 24, the load was removed from the tuber and the tubers were stored at room temperature. Six days after loading, a cross-section of the tubers indicated the development of discolored tissue in the central region of the tuber as shown in the photograph of Figure 24. Applying the theoretical pre- dictions from the Hertz problem discussed by Timoshenko and Goodier (1951), it was apparent that the discolored tissue, which was evidence of cell injury, developed within a region of high shear stress located within the central tissue of the tuber. This observation was quite significant in that it points out the danger which might result from ex- cessive depths in bulk storing of potatoes. Excessive stresses - distributed over the external surface of potatoes may induce internal shear stresses sufficient to cause failure of the internal tuber tissue without giving any visible external evidence of damage to the product. FORCE (1b) 59 300 - 200 ' 150- 100— 0 I l l l 0 0.10 0.20 0.30 0.40 DEFORMATION (in) Figure 24. Failure of internal tissue of tuber compressed between parallel plates. (631139-1) 60 Compression of the tuber between parallel plates, in certain instances, resulted in tuber failure due to induced tensile stresses developed along the outer surface of the tuber (Fig. 25). It is suggested that the mode of failure of the tuber under'parallel plate loading may , depend upon the structure of the tuber in addition to the tuber geometry. Minor deviations from the dominant structure within the tuber may weaken a localized area of the tuber and pre-dispose this area of the tuber to cellular injury or mechanical failure. In summary, therefore, no evidence was found to indicate that potatoes possessed a yield or bio- yield point such as that reported for apple fruit. The point of rupture on the force-deformation curve for potatoes under stress coincided with the structural break-down of cellular tissue within a localized area of the potato tuber. The force, deformation, and energy capacity of the potato tuber up to the point of rupture was influenced by the shape and size of the loading surface. ~When the loading surface was a rigid die in the form of a solid circular cylinder with a cross-sectional area equal to or less than one-half square inch, the rupture parameters coincided with the occurrence of localized tissue failure confined within a small region near the surface of the tuber. As the area and radius of curvature of the load- ing surface increased, there was an increased probability of serious damage to the internal central tissue of the potato tuber. 5. 2 Elastic Modulus Under Uniaxial Compression In Section 3. 1 modulus of elasticity was defined as the proportion- ality constant relating stress to strain within an elastic material. Strain is defined as the change inlength of a uniaxial specimen divided by its current length, or in differential notation 61 FORCE (1b) I l 0 0.10 0.20 0.30 0.40 DEFORMATION (in) Figure 25. Tension failure of tuber under compression between parallel plates. (631139-2) 62 dL d6 - L (5. 2) where dc is the differential element of strain due to the change in length dL of a specimen having a current length L. Integrating this expression, 61 3 In (L) + C (5.3) where C is an arbitrary constant depending upon the boundary con- ditions. When the specimen is in the unstrained state, for example, the strain is zero and the length is the original length of the specimen Lo. Thus, C is evaluated to be - 1n (L0) and the strain as expressed in equation 5. 3 becomes 61 = ln(L/Lo) (5.4) The length‘L' at any time is equal to the original length L0 plus the change in the original length AL. Hence, equation 5.4 may be given as + e. = 1n [in—ALL] :11. [1+5 “‘1 (5.5) LO LO Expanding equation 5. 5 in terms of the appropriate series, it becomes AL, , AL.z 1 AL. 3 6 : —— -T —-—-—- +7 [— "' ° ° ' (5‘6) 1 I Lo] [LO] LO] Assuming that the change in the original length of the specimen is small compared to the original length, the first term on the right hand side of equation 5. 6 L (5. 7) 63 becomes a first order'approximation of the strain defined by equation 5.4. The strain defined by equation 5.4 is usually referred to as logarithmic strain and that in equation 5. 7 is called engineering or conventional strain. Throughout this thesis the term strain refers to conventional or engineering strain. For the case of uniaxial compression, it can be shown that logarithmic strain is related to conventional strain by the expression 61 = ln[ 1 .-‘€ ] (5.8) where 6, refers to logarithmic strain and 5 refers to conventional strain and both are positive in compression. Note that as the length of the specimen approaches zero under compression, logarithmic strain approaches infinity and conventional strain approaches unity. In addition to using the conventional strain definition, modulus of elasticity values were computed based upon nominal stress observ- ations. That is, the stress was based upon the original cross- sectional area A0 of the compression specimen. The increase in cross-sectional area due to lateral expansion under axial compression was negle'cted in comparison with the original area. The use of both conventional strain and nominal stress approximations is satisfactory for small strains such as those encountered for the modulus of elasticity investigations. It was stated previously that the stress-strain curve for cylindri- cal sections of tissue removed frompotato tubers gave an approxi- mately linear relationship such as that shown in Figure 21. This satis- fies one of the requirements of a linear elastic material as discussed in Section 3. l. The potato, however,.i falls far short of meeting the other criteria of an elastic material. For example, as shownin Figure 26, during the unloading cycle of the stress-strain curve, the 64 relationship between stress and strain was noticeably non-linear and deviated considerably from the loading stress-strain relationship. In addition, the unrecovered deformation and loading energy were by no means negligible in comparison with the total deformation-and the total energy expended during the loading process. There are two terms which are very useful in explaining this anelastic behavior of materials, namely, We of elasticity and elastic hysteresis. Frey-Wyssling (1952) defined degree of elasticity as being the ratio of elastic (recovered) deformation to total deformation when a material is loaded to a certain stress and then unloaded to zero stress. A perfectly elastic material has a degree of elasticity of unity and a viscous or perfectly plastic material has a degree of elasticity of zero. Table 3 shows some values for the degree of elasticity of mature potato tissue under uniaxial compression. On the average, only 46 per cent of the total deformation of the product was recovered during the unloading cycle. Hence, potato tissue is considerably anelastic during the unloading of stressed tissue. Table 3. Degree of elasticity of potato tissue under uniaxial compression below the rupture point. Variety: Russet Rural; Test date: 10/17/62 Total strain, 6 Recovered strain, 5e Degree of elasticity (in/in) (in/in) A (cg/E) 0.10 0.06 0.60 0.12 0.06 0.50 0.21 0.10 0.48 0.23 0.09 0.39 0.25 0.12 0.48 0.25 0.10 0.40 0.28 0.09 0.32 0.28 0.13 0.46 Mean O O O OOOOO O O O OOOOOO O O O O O O O O O 0O 46 Coefficient of variation . . . . . . . . . .16. 5% 65 The elastic hysteresis of a materialis defined as the amount of energy dissipated as heat during a cycle of loading and unloading the material (Jastrzebski, 1959). This was measured by determining the stress-strain curve through a cycle of loading and unloading as shown in Figure 26. The larger the area enclosed by the stress-strain load- ing and unloading loop, the greater the energy dissipated internally within the loaded material. For an ideally elastic body, the paths followed during loading and unloading coincide and there is no hysteresis loop. For steel, the loop is small; for high polymers, the hysteresis effect is much more pronounced. The hysteresis loss for potato tissue is also very pronounced (Fig. 26). As shown in Table 4, the energy dis- sipated within loaded potato tissue varied from 72 to 90 per cent of the total energy expended during the loading process. ‘ It is evident, therefore, that as far as unloading of the potato is concerned, it does not behave elastically. The major portion of the loading energy was dissipated due to a hysteresis effect and the unrecovered deformation was of the same order of magnitude as the elastic (recovered) deform- ation. In summary, therefore, potato tissue exhibited a very satisfactory stress-strain relationship during loading, but it had a pronounced anelastic behavior during the unloading phase. It should be noted that the stress-strain relationship in Figure 26 also approximates the behavior of a linear strain-hardening incompressi- ble plastic material having a, negligible initial elastic range. Under uniaxial loading the slope of the stress-strain curve, instead of repre- senting the elastic modulus E, is related to the strain-hardening coefficient H' as defined by Hill (1960) in the expression a? E1:— = H' (5. 9) where 0'- is the generalized equivalent stress and d_p is the 66 Table 4. Hysteresis loss in potato tissue loaded and unloaded below the rupture point. Variety: Russet Rural; Test date: 10/17/62 m Expended loading energy Dissipated energy Hysteresis (in-lb) (in-lb) (per cent) 5.80 4.20 72 11.45 9.42 82 14.35 10.90 76 18.42 15.68 85 8.35 7.05 84 5.12 3.80 74 7.12 6. 35 89 7. 35 6.65 90 Mean..... ........... 81.5 Coefficient of variation . . . . . . . ......... 8. 4% generalized plastic strain-increment. Again, however, it would be necessary to limit the use of the strain-hardening coefficient H' to the initial loading condition with no unloading; and, if no unloading is considered, the distinction between linear strain-hardening plasticity with no elastic range and linear elasticity is not very meaningful. The stress-strain behavior of potato tissue was further influenced by its previous loading history (Fig. 27). Tissue which was loaded, unloaded, and then reloaded, displayed a noticeable change in the slope of the subsequent loading stress-strain curve. This indicated that modulus of elasticity (ratio of stress to strain) was influenced by previous loading of the tissue. For example, during the initial loading along O-A (Fig. 27), the elastic modulus was approximately 500 psi. Subsequent loading from zero stress to 100 psi resulted in modulus of elasticity values which varied from approximately 100 psi to 1000 psi. Upon reach- ing point "A" during the subsequent loading, the stress-strain curve reassurned the slope exhibited during the maiden loading cycle with an 67 - Rate of deformation: 1 ipm 'Variety: Russel Rural 150 __ Test date: 10/17/62 A and B: Below the rupture point C: 'Beyond the rupture point P 100 1—- fig 3: 1— i M 50... 5.. U) 0 ' Jr A 0 0.10 0.20 0.30 0.40 STRAIN (in/in) Figure 26. Conventional stress—strain curves for potato tissue loaded both below and beyond the point of rupture. — Rate of deformation: 1 ipm Variety: Russet Rural I— Test date: 10/17/62 200 _ "" " "' Subsequent loading of previously loaded tissue .1? R U] 3* .. m B IS A , g 100 — I , 1n ,’ "' / l/ l/ // / o l [I 0 m ('I'” ’1’ l I 1 I 0 0.10 0.20 0.30 0.40 STRAIN (in/in) Figure 27. Influence of previous loading upon the stress-strain response of potato tissue. 68 apparently unchanged elastic modulus for stresses above that attained during the initial loading. On the basis of the preceding discussions, therefore,“ the values reported in this thesis for modulus of elasticity assume that the tissue taken from the central portion of the potato tuber had no previous loading history. In addition, the reported modulus of elasticity values do not account for the changes in the ratio of stress to strain during the unloading process. Table 5. Modulus of elasticity for cylindrical sections of potato tissue having various cross-sectional areas and a length of one inch. Variety: Russet Rural, Test date: October 17, 1962 Rate of deformation: 1 ipm Replication Cross-sectional area (sq in) number 0.20 ‘.0.50 1.00 2.00 ------------------- psi-------------------- 1 467 570 550 500 2 450 510 550 480 3 560 620 540 560 4 490 650 590 510 5 560 620 530 550 Mean ......... 505 594 552 520 GRANDMEANOOOOOOOOOOOOOOOOOOOOOO543p81 COEFFICIENT 0F VARIATION 7.9% Values for the modulus of elasticity of potato tissue are presented in Table 5. Individual indications varied from 450 to 650 psi with an average value of 543 psi and a coefficient of variation of 7. 9 per cent. On the basis of a statistical analysis of variance (Ostle, 1958) and a Duncan's Multiple Range Test (LeClerg, 1957), it was found that cross- sectional areas of 0. 20, 1. 00, and 2. 00 sq in did not significantly influence the values obtained for modulus of elasticity. For some 69 unexplained reason, the 0. 50 sq in cross-sectional area specimens indicated an elastic modulus which was significantly higher than that obtained for the 0. 20 sq in specimens. It is believed, nevertheless, that the grand mean of 543 psi with a coefficient of 7.9 per cent for all of the tests is a good estimate of the uniaxial compressive elastic modulus of potato tissue. These tests were conducted at room temperature and at a rate of deformation of 1 ipm. The influence of temperature and certain other factors upon the elastic modulus of potato tissue is discussed later in this thesis. Rates of deformation from 1 to 40 ipm did not significantly affect the elastic modulus (See Section 5. 7). 5. 3 Elastic Bulk Modulus If a hydrostatic pressure is applied to the exterior surface of a body, its volume V will be reduced and its density increased. Let Vo be the volume of a body under a corresponding hydrostatic pressure p = 0. Now if AV is the change in volume of the body produced by a compressive isotropic (hydrostatic) pressure p, then the elastic bulk modulus K is usually defined (Reiner, 1960) by the relationship AV V O _ p = K I (5.10) Note the analogy between the above equation and the uniaxial or one- dimensional form of Hooke's law. The above equation also relates stress to strain, but in the above case, a specialized three-dimensional state of stress is considered. The volumetric strain (sometimes referred to as cubical dilatation) term in equation 5. 10 is an approxi- mation similar to that made for conventional uniaxial strain (Eqn. 5. 7). Again the conventional volumetric strain notation is a good first order 70 approximation of the logarithmic volumetric strain since the change in volume is small compared to the original volume. During these studies, the volumetric strains were always less than 0.01. Results of the hydrostatic elastic bulk modulus tests are pre- sented in Table 6 along with certain other physical properties of the tubers tested. The typical shape of the volumetric stress-strain curve is illustrated in Figure 28. During the tests, a small amount of creep and retarded elasticity was exhibited by the tubers. Hence, a certain percentage of the deviations in the recorded values of volumetric strain was due to these time-effects and the judgment of the observer. The volumetric stress-strain curve (Fig. 28) was curved toward the stress axis. Thus, for equal increments of increasing stress, the corresponding increments of strain decreased. This indicated that the potato tuber becomes relatively incompressible under large hydro- static pressure and confirmed preliminary observations obtained using the high pressure bulk modulus apparatus (Fig. 17). This type of behavior might be expected since a very large portion of the potato tuber (85% by weight) is composed of water which has an elastic bulk modulus of 300, 000 psi. The elastic bulk modulus for the mature potato tuber (Kennebec) varied from 9, 650 to 15, 000 psi (Table 6). These values were calcu- lated based upon the change in volumetric strain of the tuber within the finite difference range of applied pressure from 10 to 50 psi. A statisti- cal analysis indicated that the average bulk modulus for the series of nine tests was 11, 300 psi with a coefficient of variation of 15 per cent. Values have been presented for the uniaxial elastic modulus E and the elastic bulk modulus K of potatoes. Most of the theoretical relations in elasticity, however, have been presented in terms of the uniaxial elastic modulus and Poisson's ratio v. In Section 3. 1, equation 3. 7, a relationship was given between the uniaxial elastic modulus, the elastic bulk modulus, and Poisson's ratio, i. e. , 71 Test No. 2 Test date: April 24, 1963 1- Variety: Kennebec Original volume: 25.4 cubic in 50 F- f“: 40 '— U) ~93 m . m 1— [:1 m E-4 U) 30 t— I?-4 < 7 [-1 U) 2 Q 20 _ >4 II: 10 —— I... 0 I g , l I 0 1.5 3.0 4.5 6.0x10-3 VOLUMETRIC STRAIN (in3/in3) Figure 28. -A representative volumetric stress-strain curve for a potato tuber under hydrostatic pressure. 4mm cm 3 3 50.5 ouammona Hogan—coupe»: 5 omsmgo m. 3 mcducommounoo 5.93m 023530.» E swam“? 05 com: comma con—253.8 one? dogma, 35605 0:45 gunman... 72 8m .2 coo .3 com .2 8o .2 8:: 2:. .o on: com .2 88.3 .23.: disses. $3 dsndfi NNN SN Sm «mm 3... e2. , .. cum 3... 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Needs Lends came And ”Hummuum amumumouv>m 73 . _ E K‘ 3 (1 - 2..) (5.11) Solving equation 5. 11 explicitly for Poisson's ratio, .one obtains 3K-E E V— T-0.50--6—k- (5.12) On the basis of previously established values of E = 543 i 43* psi and K == 11, 300 i 1, 680* psi, the average value of Poisson's ratio was calculated to be 0.492. -Considering the deviations specified for the uniaxial and bulk elastic modulus values, what deviations might be expected for the calculated Poisson's ratioparameter? The absolute value of the standard deviation in Poisson's ratio Av corresponding to the above specified standard deviations in the measured parameters was calculated from the following expression. - 3v— ' .12. lAvl - 13E” AEI +t 3K IIAKI (5.13) The probable standard deviation in the value given for Poisson's ratio is given by the expression M =f[ 3’; . AEJZ + [—51%— - AK]; (5.14) Substituting the appropriate values in equations 5. l3 and 5. 14, it was found that the maximum absolute value for the standard deviation in Poisson's ratio was 0. 0018 and the probable standard deviation was 0.0013. ‘Hence, it was estimated that the value of Poisson's ratio for potatoes was 0.492 with a probable standard deviation of 0.0013. For many of the solutions in elasticity, it may very well suffice to assume a: Plus or minus values in this case refer to standard deviations. 74 the incompressibility condition and use a Poisson's ratio of 0. 50 for mathematical calculations. The apparent elastic properties of potato tissue are summarized in Table 7. Table 7. Some apparent elastic coefficients for mature potato tissue. Parameter Mean value Standard deviation Uniaxial elastic modulus. 543psi........ .43psi Bulk elastic modulus . . . . . . .11,300psi . . . . . . . . . 1,680 psi Poisson's ratio 0 O O O O O O O O 0.492 C O 0 O O O O O 0 0.0013 5.4 Influence of Variety and Maturity Upon the Rupture Parameters The potato tuber is a biological product which experiences con— tinuing physiological and biochemical changes. It is not only sensitive to enviromnental conditions such as time of planting, rainfall, soil type, and temperature, but many of its properties are influenced to a large extent by its inherited ancestral characteristics. These inherited characteristics are particularly dominant within potato varieties. The objective of this phase of the investigation was to study the influence of variety and stage of maturity upon the capacity of potatoes to resist applied forces, deformation, and energy up to the point of rupture. Five varieties were studied: Katahdin, Kennebec, Onaway, Russet Burbank, and Sebago. These varieties were selected for two primary reasons: .(i) they were representative of varieties grown, not only in 75 Michigan but, throughout a considerable portion of the United States; and (ii) past experience of their behavior under field handling condi- tions provided a possible basis for evaluating the effectiveness of the principles and techniques used in this thesis. Based upon days after planting, potato varieties are generally sub-divided into three maturity classifications: (1) early, 90 days; (ii) mid-season, 105-110 days; and (iii) late, 120-130 days. Table A4, Appendix A, shows the probable dates of full maturity for the varieties studied. Hereafter, the effect of stage of maturity upon the various varieties will be specified in terms of "days after planting. " To evaluate the influence of variety and maturity upon the rupture parameters of potatoes, tubers were loaded with a rigid (steel pin) solid circular cylinder having a cross-sectional area of 0.05 sq in. Rate of deformation was held constant at 1 inch per minute (ipm) for all tests. Force-deformation tests were conducted for the following stages of maturity as indicated by days after the planting date of May 22, 1962: 93, 112., 125, 150, 186, 242, and 302 days. These test dates spanned the interval from August 23, 1962 through March 20, 1963. Twenty replications were made for each test date and each variety, which gave a total of 700 tests. Results of these tests were coded, evaluated with the aid of the Michigan State University digital computer (MISTIC), and are summarized and tabulated in Tables A5- A9, Appendix A. These results have been presented in terms of the means and standard deviations for each of the measured parameters. (Original data is filed in the Agricultural Engineering Department under Project 912, 05-160, 1961-1963. Mechanical and rheological properties of agricultural products.) A product-moment correlation (Ostle, 1958), using the MISTIC Library Program K5-M, was computed to determine whether or not a significant linear relationship existed 76 between various parameters measured for the five potato varieties. These product-moment correlation coefficients are presented in Table 8. Results, in terms of the mean values, are presented graphically in Figures 29 to 31 for the variety-maturity tests. The rupture para- meters for potatoes decreased with time during the pre-harvest period. The applied force, deformation, and energy at the point of rupture for all varieties, with one exception, were less at the time of harvest than for the test date one month earlier. The exception was Katahdin which showed an increase in its stress capacity (Fig. 29), but a decrease in both the allowable deformation (Fig. 30) and the absorbed energy (Fig. 31) at the rupture point. The overall trend, however, was a de- crease in the capacity of the tubers to resist applied stress during the pre-harvest period. Hence, at the time of harvest, the potatoes were more vulnerable to injury resulting from applied mechanical stresses than at any other pre-harvest test date. This may explain why such a large percentage (Larsen, 1962) of injuries to potatoes occurs during harvesting and subsequent handling operations on the grower's premises. It does not necessarily follow, however, that potatoes in general are less resistant to injury at their date of maturity than at any other time. Onaway, for example, is an early variety and reaches full maturity 90 days after planting (Table A4, Appendix A). As indicated in Figures 29-31, its capacity to resist applied stress, deformation, and energy continued to decrease beyond the probable date of full maturity. Thus, leaving the product in the soil beyond the date of full maturity did not increase, but decreased, its resistance to bruising. It was possible that soil moisture interacted with time to cause a decrease in the capacity of the tubers to resist applied stresses because the stress capacity of the tubers continued to decrease as long as the potatoes remained in the soil. The influence of soil moisture upon the mechanical behavior of potato tubers is a factor worthy of future study. 77 .doud GM Um mo .o a wagon .3651“. seasons pSOm d mo 5.3m on» GM 030 gm?” .m no“? mcwvmg ofimumufimmfiv Hopes muons» 330m MOM Ammouumv some «in: non 00.3w one coma anon-mom .wm oudmwh 02573.an mmsm< mid cum owN 3% com of 0.: cm 4 _ T n A a V0 , #0 Amcoflmoflmou om mo c605 3 “50m pofioa gummy W . . 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Assam: sunicm‘a (u; bS/QI'UF) RUBENS: EHflLdle 80 Since the time of harvest was the most critical period from the standpoint of the resistance of the tubers to applied forces, the signifi- cance of differences between varieties on this particular date was evaluated. The objective was to determine which varieties would prob- ably be most resistant and which least resistant to injuries under similar conditions. An analysis of variance was performed assuming a completely randomized statistical design. On the one hand, varieties were separated based upon their resistance to applied forces and, on the other hand, rupture energy was used as the basis for separation. In each case, an analysis of variance indicated that varieties differed significantly at the 99 per cent confidence level. The significance between mean values for the five varieties was evaluated by means of a Duncan's Multiple Range Test (LeClerg, 1957). Results of these calculations are given in Appendix B. On the basis of the capacity of the tuber to resist surface pressure or stress, it was found that Kennebec had a significantly lower resist- ance than the four remaining varieties. This agreed with knowledge gained from previous handling experiences (Thompson, 1963). Differ- ences between means indicated that the remaining four varieties did not differ significantly in their response to applied stress at the rupture point. When rupture energy was used as a basis of comparison, it was 99 per cent probable that, under similar conditions, both Kennebec and Onaway would be more susceptible to injury during mechanical loading than Russet Burbank, Katahdin, and Sebago. In conclusion, therefore, some varieties differed significantly in response to applied mechanical stresses and, thus, in their resistance to injury. This con- clusion agrees with those of Lampe (1959), Witz (1954), and Hansen (1952). It is of interest to note that even though varieties may exhibit relatively small differences in their resistance to quasi-static forces, the differences between the same varieties in their resistance to injury 81 under dynamic handling conditions may be much more dramatic. Lampe (1959), for example, ranked five potato varieties on the basis of their capacity to resist applied quasi-static forces. Representative samples of these same varieties were then subjected to dynamic load- ing by passing the tubers through a harvesting machine. It was re- ported that the variety which resisted the highest quasi-static force (2400 gm) had the least percentage of serious bruises (10 bruises per 100 tubers) under dynamic loading. The variety with the lowest quasi- static strength (2100 gm) had the highest percentage of serious bruises (50 bruises per 100 tubers) under dynamic handling. Thus, a difference between varieties of only 14 per cent, based upon quasi-static force capacity, was considerably magnified to a 400 per cent difference when serious bruising under dynamic handling conditions was used as a basis of comparison. Results in Figures 29 to 31 also point out the importance of proper curingof potatoes before subsequent handling after harvesting. The mean energy capacity (Fig. 31) of potatoes increased by 30 to 80 per cent during a one month curing period at a temperature of 55 to 600F. Subsequent storage at a temperature of 40°F resulted in further in- creases in the energy capacity of the tubers; however, the rate of in— crease was less than that during the curing period. Product-moment correlation coefficients between parameters measured during the force-deformation tests are presented in Table 8. These correlation coefficients measured the degree “15.51233 association existing between compared parameters. Any non—linear relationship-- quadratic, cubic, sinusoidal, etc. --would not necessarily be detected if such existed. Results in Table 8 indicated it was not probable that a linear relationship existed, with one exception, between specific gravity and the rupture parameters of potato tubers. The exception was Sebago 82 3M .0 u :6: #0039805 0.6033 H0>0H E00 .609 0660 063 60 00:63:16me 660603ch on: .o u .76: 1603060013 0.60:3 H0>0H “Goo 00m 03m 013 um 0ocmofificmflm 606.006ch ~11- aw ""1 666.6 (owmnom 1:66 1:330 :16 666.516 666 . 6 613.115 368on 666 . 6 8698me 66 6 . 6 .. 5666636 11666.6 666.6 066666 116.666 16666 1:66.30 :6 656.126 6: 6.. 6666- .663ch 1601.. 666a 1666 6 666 .6 869886 11666 6.. 666 6.. £66386 666.6 11666.6 11666.6 066666 16: .6 6666 116666 1166.80 :6 66.816 16666- 6666.. 116666 663615 1.13636 666.6 666.6 116666 0662536. 666 6.. 6.666 11666 .6 £66336 616.6 666.6 11666.6 11666.6 066666 636 3.66 11666.6 11666.6 1613.30 .6 183688666 116666- 6666.. 116666 116666 6631.15 6.13636 6666 666.6 11666 116666 omnchom 116666- 6666.. 116666 116666 536636 6: .6 $6 .6 11666 .6 11666 .6 11666 .6 06366 113.66 666.6 11666.6 11666.6 11666.6 1:31:66 .6 386 166 6- 666 6.. 1.66 .6 11666 .6 11666 .6 61325 613636 666 6- 666.6 6: .6 11666 .6 11666 .6 862836 16: .6 666 6- 666 6- 116666 6.3 .6 56.6336 6mm03 330 .6th $3:de 16w .6 0:0 1603093306 Mr HHHM<> wmm Ewe/533$ 1 smash. 0630065 .6063 .6th 0.63msm 0.666699% . QmMDm9 woodedfig mm wcfipmoH uflmumufimmaw .3de 3055an when?— Oumuom >9 UoJHOmnn >wnosm .mm onfimfim 6: 63 4mm... 20:60.,me om.o o~.o 3.0 mod no.0 god 0 _ _ 6 . _ _ 3 msofimgov pnmvcmum 03» H n X I m H m03d> can: .. O D t X I. I m A d / 3 /« I H n r . m \ L \\ . L . '1 DA V X H H mm. mm mozmfimzou m J ) m. - / 9 b - m II on / X 89 rupture energy of the tuber but, on the basis of the mean minus two standard deviations. From Figure 33, for example, one should not use the mean energy value of 10 in-lb/sq in for design purposes; rather, one should use the 5 in-lb/sq in value which is the mean minus two standard deviations. This procedure should, theoretically, reduce injuries to within the two and one-half per cent range. Such a procedure is equivalent to the introduction of a safety factor of 2 in conjunction with the use of the mean energy value for the 0. 05 sq in traction area. For larger traction areas, the necessary safety factor to be used in conjunction with the mean energy values might be less according to the information given in Figure 33. If injuries were to be reduced below the two and one-half per cent level, then the required safety factor would have to be increased. It seemed appropriate at this point to evaluate the usefulness of equation 3. 12, Section 3. 1, to predict any two remaining rupture para- meters when one rupture parameter (for example, deformation) is known or specified. Assuming the rupture deformation 111. to be as specified in Table 9, how accurately does equation 3. 12 predict the rupture force Pr and how precisely can the rupture energy be approxi- mated for various cross-sectional area traction surfaces ? The rupture force P1. was calculated from the following expression P1. = (2a E ur)/(l - v2) (5.16) which is analogous to equation 3.12 where "a" denotes the radius of the suI':Eace of contact. To estimate the corresponding rupture energy 01., a linear relationship was assumed between the force and deformation and equation 3. 12 was expanded to take the form gr: (l/Z)(Pr ur) = a Eurz/(l -vz) (5.17) where ¢r is the rupture energy corresponding to the rupture deformation 90 ur and again '"a" is the radius of the surface of contact. The uniaxial elastic modulus E was taken from Table 7 as being 543 psi and the tuber was assmned to be incompressible, i. e. , Poisson's ratio was taken to be 0. 50. .Since the elastic modulus E was determined from a single loading without considering the effects of unloading, the Boussinesq solution may be applied even if the potato were assumed to be a linear strain- hardening plastic material behaving according to the Levy-Mises equations dei?’ 0C 013.! (Hill, 1960). Since the principal axes of stress do not rotate at a point (according to Boussinesq) and all stresses in- crease in proportion to the load, the Levy-Mises equations, under the assumption of linear strain-hardening, may be integrated to give the Hencky equations 613) ccafj' which are equivalent to Hooke's Law. Since the plasticity equations imply incompressibility, the "equivalent Hooke's Law" would have Poisson's ratio equal to O. 50. Calculated results are compared with the experimentally measured mean rupture force and the mean rupture energy in‘Figures 34 and 35, respectively. The calculated and measured values were in good agree- ment for traction areas from O. 01 to 0. 50 sq in. An error analysis indicated plus or minus two standard deviations from the mean valueof the uniaxial elastic modulus of 543 psi would in most cases account for the deviations between the calculated and the measured values in Figures 34 and-35. The compared values tended to diverge for loading areas approaching 0. 50 sq in and greater. This might be expected since equation ‘3. 12 was developed for a die acting on a semi-infinite body. Hence, as the surface of contact approaches the same order of magnitude as the projected area of the tuber itself, then it is to be expected that the validity of equation 3. 12, and of the subsequent relations resulting from it, may very well be nullified. 91 .mmonm 65300.3 maownm.) no.9” newsman mm? GoflmgnoMov 0.25639 on» Sea? 63.35030 «as» 63?? 00.3w mnnumsu 60.25008 0:”. mo GOmMummEoo 4 . .wm 0.20th 6: 63 Na: .405. 206,603: om.o om.o 36 mod Nod Hod _ n a A QEwdoflmfiou “60.3.8005 O 0 6263on H mm? as :05? Mg .m .cwm 89G 600030160 XI I I I IX fl , L n H 3 J O l om H D .3 1 1d I 3. I - m I cc \\ I 663 \ an . .. NHoM odvm .6. Am fly "m .DMMN 92 RUPTUREENERGY, 91,. (in-lb) I I O m ea sur ed experimentally x———- --K calculated from the equation. 2 rupture energy, (11, .=: 1/2 [2_ai_1‘3_\5;_ ] __ / / 0'06 1 l 1 1 n 0.01 0.02 0.05 0.10 0.20 0.5 TRACTION AREA, Traz (sq in) Figure 35. A comparison of the measured rupture energy with that calculated when the rupture deformation was specified for various traction areas. 93 It has been shown that variations in the size of the surface of contact (traction area) exercised a significant influence upon the rupture parameters of potato tubers. Even though other rupture parameters increased with increasing traction area, the allowable stress capacity of the tuber at the point of rupture decreased according to an inverse logarithmic relationship. Tubers exhibited considerable variations in their energy capacity which made it necessary to introduce safety factors when designing on the basis of known mean or average values of rupture parameters. Finally, a theoretical equation was used to pre- dict, with reasonable accuracy, the remaining rupture parameters when only one of the rupture parameters was known or specified. 5 . 6 Temperature “Effects The mechanical behavior of high polymers which display a visco- elastic response to applied stress is influenced to a large extent by temperature variations. Eley (1961) cited references which showed that the uniaxial stress in rubber, a three-dimensional, cross-linked polymer, was influenced by temperature according to the relationship 0': K0 T (M - l/Mz) (5.18) where 0': the tensile (or unidirectional compressive) stress, referred to the unstrained cross-sectional area M = the extension or compression ratio T = the absolute temperature R0 = a constant determined by the structural details of the network. FOI‘ any given compression ratio M, the stress in the material was directly related to the absolute temperature of the material. For materials in which viscous effects predominate, the effect of heating 94 is to increase the vibrational energy of the molecules in the material and thereby to assist in freeing them from the attractions of their neighbors. Under such conditions, viscous flow may be very sus- ceptible to temperature changes. The viscosity-temperature relation- ship may be expressed as an Arrhenius function (Schmidt and Marlies, 1948) n = B exp(U/RT) (5.19) where n = the viscosity coefficient, U = the molar activation energy ‘ constant, R = the gas constant, T = the absolute temperature, and B is a constant. Since stress or force is directly related to viscosity by equations 3. 13 and 3. 14, the effect of increasing temperature would be to reduce the strength of a material in which viscous effects are predominant. The major component of the potato tuber is starch (65 to 80% of , the dry weight of the tuber) which exhibits a viscous effect when in solution. Veselovsky (1940) reported the viscosity of a solution made of 0. 3 percent starch from Katahdin potato tubers to be 12. 9 times the viscosity of water. Since both water and starch are present in large quantities in the potato tuber, it might be expected that viscous effects and thus temperature might have-a pronounced influence upon the resistance of potatoes to applied stress. The potato tuber freezes at 29°F, is subject to low temperature injury below 40°F, and becomes gelatinized above 147 to 160°F. Hence, the range of temperatures from 40 to 1400F were investigated during this study. The influence of temperature upon both the rupture para- meters of the whole tuber and upon the elastic modulus (stiffness) of tuber tissue was investigated. To investigate the rupture parameters, a O. 05 sq in cross-sectional area cylindrical die was used for loading the tuber. For the elastic modulus investigations, lsq in by l in long 95 sections of tissue were loaded under compression between rigid parallel plates. Rate of deformation was held constant at 1 ipm for all tests. Results are given in Table 10 and in Appendix B, Tables B7 through B9. The influence of temperature upon the rupture parameters of potato tubers under rigid die loading is illustrated in Figure 36. Temperatures within the range from 40 to lOSoF, in general, displayed only a very small effect upon the rupture parameters. The average force and deformation at the rupture point increased only by about 10 per cent as the temperature increased from 40 to lOSOF. Accordingly, the energy required for rupture of the potato tuber increased by approxi- mately 20 percent. Above the 105°F temperature, there was an abrupt change in both the rupture force-temperature and the rupture deformation- temperature curves (Fig. 36). The strength of the tuber, as indicated by its rupture stress, suddenly decreased and the deformation required for rupture increased logarithmically with rising temperature. As a result, the energy required for rupture, ire, , the integral of the force- deformation curves, was apparently not significantly influenced by the increase in temperature above 105°F.‘ Even though theory does not account for the rise in the strength of the tuber previous to the sudden drop inlstrength, Houwink (1958) also noted, without presenting an explanation, a small increase in the strength of steel, colophoniurn mixtures, and various other materials with rising temperature before attaining a maximum strength and then decreasing in strength with rising temperature. The following explanation is offered for the potato tuber: As temperature increased from 40 to 105°F, the vibrational energy of the atoms in the molecules within the cells of the tuber in- creased. Yet the atoms were constrained by their structural bonds, in a similar manner as the molecules of a gas in an enclosed heated container. This resulted in an increase in the stress and energy capacity of the tuber within the 40 to lOSoF temperature range. , Above lOSoF 96 Table 10. Rupture parameters, as influenced by temperature, for potato tubers under compression with a 0. 05 sq in cylindrical rigid die. (Each value based upon ten replications. TEMPOERATURE RUPTURE PARAMETERS ( F) Force (lb) Deformation (in) Energy (in-lb) 40 :1: 2 16.8 0.118 0.94 (6.9%)* (16. 1%) (17.3%) 60 i 2 17.4 0.130 1.10 (8.7%) (12.0%) (16.6%) 80:1:2 18.0 0.130 1.16 (10.3%) (15.6%) (25.4%) 105 43 18.5 0.126 1.17 (10.3%) (17.7%) (18.7%) 140 :l: 5 13.3 0.206 1.26 (12.3%) (13.6%) (22.2%) * Values in parentheses are coefficients of variation corresponding to the means directly above them. (at 1220F, Talburt and Smith, 1959), water passed from the non-starchy parts of the cells into the molecular structure of the starch granules which then started to swell until separation and rupture of the cells, due to heating and swelling, occurred. This thermal rupture of the cells resulted in a sudden decrease in the structural strength of the tuber and also in the increased deformation required to produce a mechanically- induced point of rupture. The effect of temperature upon the elastic modulus (stiffness) of potato tissue is shown in Figure 37. Similar to other high polymers (Alfrey, 1957), temperature caused a decrease in the elastic modulus of potato tissue. . Within the temperature range from 60 to lOSoF, the difference in the mean elastic modulus values were not statistically significant (Tables 137-139, Appendix B). Below 60°F, however, and 97‘ em; .mndumdn mo «Eon 06a Hm muong 3.30m omdonxw mo 68.350 663 .cofidgu0w06 .00u0m 6029mm 0:“ com: mudguvmgeu mo 00:05am"; .om chumwh E $66on manam 05 m“ uqflom 63va u I u s 8 n n n n d d d . d L 03 I m L I om L I m o L n n n n H H H H 3 3 3 3 S I 3 I G I 3 L O 3 N n m M m Ass 666163 3 I663 8 l6; 9 ) ) w A d m V fl). m. I ( I L l u ( I — m m 6661.: I62 .m) l6; 4. l I m J oovIom Ioom Join. n L m 98 .cofimmmumgoo 33335 .395 353a 3.30m omcmfiw mo 9:355 3330 9.5 coma mudumnomfiop mo moavsfifi .wm oudwwh Q 3333 mmsam mg mg ”Egon gummy l . 3 L o2 m D 083QO :ofimmmnmgou W b I can m n n1 If n S m I om“V U. SEQE "8% $3. MDmmHH. OHm 0:..— mm “Eon gome I N .o I, ¢.o nonhuman I o.o b commmoumfioo 3039310 I, m..o IIVQHII II I: I. ""'-"-'| '- OOH I N; >mhocm OI llllll I. :omumgugofl m m“ “509 gummy vouch >wnmnm~ 4‘\ COSMEHOHoD o 0% 33.6530 5 3 mo .o .395» $08 co; om; ) WdP'I .I.V uzLawvuch sunLdnu HELHWVHVd gunman ( 103 as the rate of deformation increased from 20 to 40 ipm, the rupture force (stress) decreased from 19 1b‘(380 psi) to 13.6 lb (272 psi) and the rupture deformation increased from 0. 100 to 0.150 inches. Thus, an increase in the rate of deformation had a similar effect as increas- ing the temperature of the tuber. (See Section 5.6). A statistical analysis indicated that this decrease in the strength of the tuber with increasing strain rate was highly significant (Tables BIO-B12, ‘Appendix B). Rates of deformation from 1 to 20 ipm did not significantly influence the strength of the tuber, i. e. , its rupture stress; however, the ~ tuber strength at the 40 ipm loading rate was significantly lower than the corresponding strength at the lower rates of deformation. Theoretical considerations of viscoelastic materials suggest that the material should display greater strengthat increasingly higher rates of strain. Experimental results from many of the investigations mentioned earlier confirm such a behavior in many viscoelastic high polymeric materials. Results of tests by Zoerb (1958) indicated, however, that an increase in strain rate may not always result in an increase in the energy capacity of a material. He reported, for example, that even though the energy capacity of high moisture grain was greater under impact loading than under quasi-static loading, the reverse was true for'low moisture grain. This suggested that in natural high poly- mers, potatoes, and grain for example, some unevaluatedlmechanism may be active which, under certain conditions, causes a decrease rather than an increase in the strength of the product at increasingly higher rates of strain. *On the basis of the evidence presented in this thesis, it may be quite possible that under higher rates of deformation (impact, for example), the potato tuber may exhibit a lower strength than that reported in this thesis. Limitations of the testing machine and record- ing equipment prevented the study of rates of deformation greater than 40 ipm. However, the decrease in the strengthof the tuber above the 104 20 'ipm loading rate was significant in that it warned of the type of trend which might be expected for higher rates of deformation‘and under impact loading. This type of tuber behavior may also explain the dif- ferences,» as reported by Lampe (1959)--see Section 5. 4--in the resistance of the tuber to injury under quasi-static loading as opposed to the results obtained under dynamic loading during field handling conditions. The terminal velocity of the potato tuber is of the order of 100 ft per sec (Gilfillan and Crowther, 1959). Twenty ft per sec is approxi- mately equal to the velocity of the tuber falling freely from a height of 6 ft. Hence, it would be of interest in future work to investigate the influence of rates of deformation up to the range of approximately 20 to lOvat per sec. Summarizing, under uniaxial compression, strain rates from 1 to 40 in/in/min did not significantly influence the strength or energy capacity of potato tissue. Under rigid die compression which involved a complex state of stress, rate of deformation did significantly influence the strength of the tuber. Even though rates of deformation from 1 to 20 ipm did not significantly influence the behavior of the tuber, a rate of deformation of 40 ipm caused a 20 per cent drop in the strength of the tuber. This drop in tuber strength under rigid die compression at the highest rate of deformation indicated that the strength of the potato tuber under impact loading may be significantlyolower than its corresponding strength under quasi- static loading. 5. 8 Stress Relaxation Within the Potato Tuber Stress relaxation is a characteristic property exhibited by most viscoelastic materials under constant strain. In the stress relaxation test, an external force was applied to the potato tuber until the load 105 reached a pre-determined value (35 :l: 1 1b). After this time, the deformation of the product was maintained constant and the force re- quired to maintainthis deformation was measured and recorded directly as a function of time. Since the deformation of the product was held constant, the loaded area of contact was constant during the relaxation tests and the recorded force-time curves were representative of the stress relaxation process within (the potato tuber. The whole tuber was used instead of specimens of tissue in order to minimize the influ- ence of moisture loss from the product during the relaxation test. All tests were conducted at room temperature and, unless otherwise specified, the 'rate of deformation during the loading process was 1 ipm. For the ideal relaxation test, it is desirable to load the specimen by means of some step-change loading technique, i. e. , a loading which increases from zero to the desired value within an infinitely small time interval. Such a loading technique is difficult to simulate but would minimize the effects of relaxation of stresses during the loading cycle. During this investigation, it was only possible to control the rate of loading within certain confined limits of the testing machine. The influence of rate of deformation during the loading process upon subsequent stress relaxation within the potato tuber is shown in Figure 40. It was apparent that during the loading process, time was allowed for the stresses within the material to relax (decay) even before the loading process ended and the observation of the relaxation process was begun. Rate of deformation had the greatest influence upon the observed relaxation process during the first few seconds of observation after stopping the loading cycle (Fig. 40). After five seconds, the re- corded force (stress)-time curves appear to display a parallel relation- ship to-each other. This indicated that after five seconds the relaxation time (or time-constant) was relatively unaffected by the rate of loading. 106 453 05 mo nowumoflmmmmfihsp GOSMEHOHoU mo 3mm >0. voosodficw mm :oflmmonmfioo 3.3m Hedonmm nova: when?» 330m omsouaw c333 £03383." much—m .o¢ oufimfim 33V was 2. 3 om 3 on om . S o q _ q A q — _I . O 683 n: m .o a mm 3 332 3335 :53 scan 1% .. om .. mN 8.: on 5.: S I and v I 83 N En: a I on Fig zofiiéommo mouse; .3. voosodndw mm ZOHB<§MM mmfldhm mm. (qt) 30110.1 107 The influence of rate of deformation upon the relaxation of stresses ‘ within the material during the first seconds of observation indicated the importance of using a high frequency response load cell for detecting applied forces during impact loading. The mathematical formulation of the problem of stress relaxation within a viscoelastic material was discussed in Section 3. 3. If the stress withina body under constant strain decreases with time accord- ing to an exponential relationship, then the time required for the stress to decrease to the value (l/e) times its initial value is referred toas the relaxation time, relaxation constant, or time-constant, 2'. For many materials it is not sufficient to define their relaxation behavior in terms of only one exponential term, but it is necessary to represent the relaxation process by a series of exponential terms, each term having a corresponding relaxation time and an exponential co- efficient as shown in equation 3. 20 and Figure 8 of Section 3. 3. Relaxation curves for potato tubers under parallel plate compression over various intervals of observation are shownin Figure 41. Relaxation curves over time intervals from 10 sec to 10 min did not provide sufficient evidence for selecting the type of model required to represent the relaxa- tion behavior of the product. From the short-time curves, it was not apparent whether or not the stress within the tuber approached zero as the time of observation approached infinity. This question was resolved by means of a long-time test over a period of hours (Fig. 41). It was found that the stress did not level off but continued to decrease with time. This indicated that no elastic element was needed in parallel with the other elements of the viscoelastic model. Sucha behavior also indicated that after a very long period of observation the potato tuber would not exhibit any elastic after-effect upon removal of the loading plates. The dashpot element was excluded from the model (Fig. 8) since there was no indication that the tuber responded as a rigid body for increasingly {, 108 453 H Hoflmauofiwp no 3m.“ 3333 soflm>uomno mo 3.95.35 053 msowumxp Ho>o GowmmouQEou 3.29 #03922” H093 manna 3.30m GEN—w? son—mad?“ mmonum .dw ousmmh c. o .m w m N s o _ J fl _ q _ _ we .oouown n: H u." mm 59G wouumum 95.90 50mm (<11) souos 1 cm mhfio A a was mm message I3 A V m2? on Ava000MV “EH. lg . // _ I: mm . ZOHB<>MMmmO .mO HEB .3. co oauagm. no I ZOHBMaercker, M. , and Morgen, A. 1880 Uber den Zusammenhang des spezifischen Gewichts mit dem Starkemehl - und Trockensubstanzgehalt der Kartoffeln, sowie fiber die Methode der Starkebestimmung in den Kartoffeln. . Landwirtschaftlichen Versuchsstationen. 27:107-165. Biale, J. B. and Young, R.-~ E. 1962 The biochemistry of fruit maturation. Endeavor. 21(83-84):164-l74. Bland, D. R. 1960 The Theory of Linear Viscoelasticity. New York: Pergamon Press. 125 pp. Boussinesq, J. 1885 Applications des Potentiels a l'Etude de l'Equilibre et du 'Mouvement des Solides Elastiques. Paris. ~ British Rheologists' Club. 1 947 ‘Essays in Rheology. - London: Sir Isaac Pitman and Sons, Ltd. 103 pp. Burton, W. - G. 1948 The Potato. -London: Chapman and Hall, -Ltd. 319 pp. Coleman, - B. D. 1957. A rstochastic process model for mechanical breakdown. Transactions of the Society for Rheology. 1:153-168. 120 121 Cooper,-H. E. 1962 Influence of Maturation on the Physical and Mechanical Properties of the Apple Fruit. Unpublished M. S. - Thesis (Agricultural Engineering). The Pennsylvania State University. Eirich, F. R. (editor) 1956 Rheology--Theory and Applications, Vol. I-III. New York: Academic Press, Inc. 761, 591, 680 pp. Eley, D. D. 1961 Adhesion. New York: Oxford University Press. 290 pp. Fitzgerald, E. R. 1961 Yield strength of cyrstalline solids from dynamic mechanical measurements. ‘Developments in‘Mechanics. 1:10-38. Edited by J. E.’ Lay and L. E. Malvern. ‘New York: ' Plenum Press. 611 pp. Frey-Wyssling, Albert. 1952 Deformation and Flow in Biological Systems. New York: Interscience Publishers, Inc. 552 pp. Gilfillan, G. and Crowther, A. J. 1959 i The behaviour of potatoes, stones, and clods in a vertical airstream. ~ Journal of Agricultural Engineering Research. - Glaves, Archie H. 1963 Personal correspondence dated May 8. Agricultural Research Service, U. S. a Department of Agriculture, East Grand Forks, Minnesota. 5 pp. Gohlich, H. and Mohsenin, N. N. 1962 Untersuchungen fiber mechanische Eigenschaften von Obst‘xunter Berficksichtigung einer maschinellen Ernte. Landtechnische Forschung. 12(4):103-107. -Green, H. C. 1956 Potato damage. Journal of Agricultural Engineering Research. 1(1):56-62. Gross, Bernard. 1953 'Mathematical Structure of the Theories of Viscoelasticity. ~Paris: Hermann and Cie. 74 pp. 122 Hamson, A. R. 1953 Measuring the firmness of tomatoes in a breeding program. Proceedings of the American-Society for Horticultural Science. 58:423-433. Hansen, R. W. 1952 The Development and Testing of Equipment to Measure the Resistance of Potatoes to Bruising and Injury. Unpublished M. S. Thesis (Agricultural Engineering). North Dakota Agricultural College. Hendrick, J. G. III. 1962 The Application of Tillage Energy by Vibration. Unpublished Ph. D. Thesis (Agricultural Engineering). Michigan State University. 104 pp. Hill, R. 1960 The Mathematical Theory of Plasticity. New York: Oxford University Press. .355 pp. Houwink, Roelof. 1958 Elasticity, Plasticity, and Structure of Matter. New York: Dover Publications, Inc. 368 pp. Jacob, Walter C. 1959 Studies on internal blackspot of potatoes. - New York Agricultural Experiment Station’(lthaca) Memoir 368. 86 pp. Jastrzebski, Z. D. 1959 Nature and Properties of Engineering Materials. New York: John Wiley and Sons, Inc. , 571 pp. Jones, - P. G. and Moore, H. F. 1940 An investigation of the effect ofrates of strain on the results of tension tests of metals. -Proceedings of the American Society for Testing Materials. 40:610. -Kunke1, R. and Edmunson, W'. C. 1957 A modified Witz test of the toughness of potato skins. Proceedings of the American Society for Horticultural Science. 70:397-402. -Lampe, Klaus. . 1959 Entwicklung und Erprobung einer Methode zur Bestimmung Wider- standsfahigkeit von-Kartoffelknollen gegen Beschadigungen. Dissertation. Bonn, Germany. .125 pp. 123 Lampe, Klaus. 1959 Mbhlichkeiten zur Messung der Beschadigungsempfindlichkeit von Kartoffelknollen und anderen Frfichten. ‘Landtechnische Forschung. 9(2):50-54. , Lampe, - Klaus. 1960 Die Widerstandsfahigkeit von Kartoffelknollen gegen Besch'adi- gungen. - European Potato Journal. 3(1):13-29. -Larsen, F. E. 1962 External and internal (blackspot) mechanical injury of Washington Russet Burbank potatoes from field to terminal market. American Potato Journal. . 39(7):249-260. Leaderman, Herbert. 1943 Elastic and Creep Properties of Filamentous‘Materials and Other High-Polymers. Washington, D. C.: Textile Foundation. 278 pp. 'LeClerg, -E. L. 1957 Mean separation by functional analysis of variance and multiple comparisons. Agricultural Research Service Publication ARS 20-3. Washington, D.C.: U. S. Department of Agriculture. 33 pp. Lissner, Herbert R. 1963 Biomechanics--What is it? Mechanical Engineering. 85(1):25-29. Magness, J. R. and Taylor, G. F. 1925 An improved type pressure tester for the determination of fruit maturity. U. S. Department of Agriculture Circular 350. 8 pp. Malvern, L. E. 1962 Constitutive equations of elasticity, the elastic potential or strain energy function, and generalized Hooke's Law. Mimeo- graphed class notes for Applied Mechanics 813--Elasticity. rMichigan State University. 9 pp. Maxwell, B. and Harrington, J.‘P. 1952 Effect of velocity on tensile impact properties of polymethyl methacylate. Transactions of the American Society of Mechanical Engineers. 74:579-587. 124 Mohsenin, Nuri N. 1963 A testing machine for determination of mechanical and rheo- logical properties of agricultural products. ' Pennsylvania Agricultural Experiment‘Station Bulletin 701. 26 pp. Mohsenin,- N. N. and Gfihlich, H. 1962 Techniques for determination of mechanical properties of fruits and vegetables as related to design-and development of harvest- ing and processing machinery. Journal of Agricultural Engineer- ing Research. 7(4):300-315. Mohsenin, N. N. , - Gc'ihlich, H. , and Tukey, ~L. D. 1962 Mechanical behavior of apple fruit as related to bruising. Proceedings of the American Society for Horticultural Science. 81:67-77. Mohsenin,~ N.‘ N., .Cooper, H. E. and Tukey, L. D. 1962 An engineering approach to evaluation of textural factors in fruits and vegetables. -AmericanSociety of Agricultural ‘ Engineers Paper Number 62-321. .Saint Joseph, Michigan. .17 pp. -Mohsenin, N.’ N. and Tukey, 1L. D. 1962 Annual report of Cooperative Regional Project 1397 (NE-44). . Pennsylvania Agricultural Experiment Station. 23 pp. Mooney, M. 1937 Consistency measurements in the paint industry. Symposium on Consistency. American Society for Testing Materials. . 73 pp. - Ostle, Bernard. 1958 Statistics in-Research. Ames: The Iowa State College 'Press. 487 pp. Reiner, Markus. 1960 Deformation, Strain, and Flow. . London: .Lewis and Company. . 347 pp. .Schlichting, Hermann. 1960 Boundary Layer Theory. (Translated by J.' Kestin). New York: McGraw-Hill Book Company, Inc. 647 pp. 125 Schmidt, A. X. and Marlies,,C. A. 1948 Principles of High-Polymer Theory and Practice. New York: McGraw-Hill Book Company, Inc. , 743 pp. Scott-Blair, G. W. and Reiner,- Markus. 1957 Agricultural Rheology. London: Routledge and Kegan Paul, Ltd. 222 pp. Shtrankfel'd, I. G. 1957 The viscous-elastic properties of different types of muscles. Biophysics. 2:167-176. Sokolnikoff, I. S. 1956 Mathematical Theory of Elasticity. ~New York: McGraw-Hill Book Company, Inc. 476 pp. Talburt, W. F. and Smith, 0. 1959 Potato Processing. - Westport, Connecticut: The Avi Publish- ing Company, Inc. 475 pp. . Thompson, ’N. R. 1963 Personal communication during an interview. Farm Crops Department, Michigan State University. Thompson, N. R. and Chase, R. W. 1962 (Undated mimeographed sheets on Arenac and Emmet potato varieties based upon the 1962 potato trials under Michigan growing conditions). Farm Crops Department, Michigan State University. Timoshenko, S. and Goodier, ' J. N. 1951 Theory of Elasticity. New York: McGraw-Hill Book Company, Inc. 506 pp. Triffet, T. 1962 Introduction to the mechanics of discontinuous media. (Undated mimeographed class notes used in the winter of 1962 for Applied Mechanics 835 lectures). Michigan State University. 144 pp. Veselovsky, I. A. 1940 Biochemical and anatomical properties of starch of different varieties of potatoes and their industrial purposes. American Potato Journal. 17:330-339. 126 Whitehead, J. B. 1935 Impregnated Paper Insulation. New'York: John Wiley and‘Sons, Inc. 221 pp. Whittenberger, R. T. and Marshall, R. E. 1950 Measuring the firmness of Red Tart cherries. [Food Technology. 4:311-312. Wiant, J.-S. 1945 Internal blackspot of Long Island potato tubers. 'American-Potato Journal. 22:6-11. Wilson,1Eeu B0 1952 An Introduction to Scientific Research. New York: McGraw-Hill Book Company, Inc. 375 pp. Witz, R. L. 1954 Measuring the resistance of potatoes to bruising. Agricultural Engineering. . 35(4):241-244. -Zoerb, G. C. 1958 Mechanical and Rheological Properties of Grain. Unpublished Ph. D. Thesis (Agricultural Engineering). Michigan State University. 139 pp. 'Zoerb, G. C. and Hall, C. W. 1960 Some mechanical and rheological properties of grains. Journal of Agricultural Engineering Research. 5(1):83-93. APPENDIX A 127 128 A m .02 oumofimom IIIV AIIIIN .oZ oumoflmom /) H .02 oumuSQoMIllv AIIIIllum omllfl, owmnom once .3. cwpnmumfi NJ H A H 59.33% BdfiooBoM Uvfloscov“ >m3mn0 canon/q. owenom 358nm Msmnusm Hommsm xamnusm «nomad fidhdm.¢ommdm xcmnnsm nommdm , Baaoospovm Guam uemmfim owmnom Eugen—mm $58M >m3mco oonoccovm umnou< >m3ms0 N.,“ A : BosooBoM “088m Guam pom mam , oonoéovm NJ A A H T 1.. ' EBmOZ ZOHH .NeS .- .32. "33 mcwunmanm £33033“ Edam .uoxoom .HQ paw mmouU gush .GOmmEocB .uQ new? :oflmHomooo CH .umoQ £de .um4 Sonar.” .nw xommm >9 .Gofimu—m >fi0 0034 um moopouom some new 3on mo uzotamq 44‘ QEMB 129 Table A2. Precipitation in inches corresponding to the dates of observation during the growing season at Lake City, Michigan for 1962. May 24-.07 July.8-.76 Aug. 4- 1.59 May 29 - .07 July 10 - .07 Aug. '7 - .06 May 30 - .25 July 12 - .02 Aug 12 - 1.09 July 20 - .75 Aug 13 - . 14 June l-..03 July 21 - .22 Aug. 25 - .90 June 5 - .14 July 22 - .05 Aug 26 - .01 June 9 - .03 July 23 - .48 Aug. 27 - .14 June 10;- .80 July 25 - .56 Sep 9 - .64 June 11 - .70 July 26 - .02 Sep. 10 - 1.08 June 17- .32 July 28 - .06 Sep. 11 - .07 June 18 -1.30 July 29 - .10 Sep. 13 - 1.09 June 22- .01 July 30 - .10 Sep. 18 - . 16 June 24 - .02 Sep. 20 - .03 Sep. 22- .41 130 Table A3. Chart for converting specific gravity readings to total solid and starch readings. Adapted from M. Maercker - Landwirths Band 25 - 1880 - pg. 107 ‘ v—r Specific Total Starch - Specific Total Starch Gravity Solids Gravity Solids 1.040 11.2 1.077 19.0 13.3 1.041 11.4 1.078 19.2 13.5 1.042 11.6 1.079 19.4 13.7 1.043 11.8 1.080 19.7 13.9 1.044 12.0 1.081 19.9 14.1 1.045 12.2 1.082 20.1 14.3 1.046 12.5 1.083 20.3 14.5 1.047 12.7 1.084 20.5 14.7 1.048 12.9 1.085 20.7 14.9 1.049 13.1 1.086 20.9 15.1 1.050 13.3 1.087 ’21.2 15.4 1.051 13.5 1.088 21.4 15.6 1.052 13.7 1.089 21.6 15.8. 1.053 .13.9 1.090 21.8 16.0 1.054 14.1 1.091 22.0 16.2 1.055 14.3 1.092 22.2 16.4 1.056 14.5 1.093 22.4’ 16.6‘ 1.057 14.8 1.094 22.7 16.9 1.058 '15.0 1.095 22.9 17.1 1.059 15.2 1.096 23.1 17.3 1.060 15.4’ 1.097 23.3 17.5 1.061 15.6 1.098 23.5 17.7 1.062 15.8 10.2 1.099 23.7 17.9 1.063 16.0 10.4 1.100 24.0 18.2 1.064 16.2 10.6 1.101 24.2 18.4 1.065 16.5 10.8 1.102 24.4 18.6 1.066 16.7 11.0 1.103 24.6 18.8 1.067 16.9 11.2 1.104 24.8 19.0 1.068 17.1 11.4 1.105 25.0 19.2 1.069 17.3 11.6 1.106 25.2 19.4 1.070 17.5 11.8 1.107 25.5 19.7 1.071 17.7 12.1 1.108 25.7 19.9 1.072 18.0 12.3 1.109 25.9 20.1 1.073 ~18.2 12.5 1.110 26.1 20.3 1.074 18.4 12.7 1.111 26.3 20.5 1.075 18.6 12.9 1.112 26.5 20.7 1.076 18.8 13.1 1.113 ’26.? 20.9 Continued 131 Table A3 -- Continued _ _- — _4 1.128 30.0 24.2 Specific Total Starch Specific Total Starch ‘ Gravity Solids GravitL Solids 1.114 26.9 21.1 1.129 30.2 24.4 1.115 27.2 21.4 1.130 30.4 24.6 1.116 27.4 21.6 1.131 30.6 24.8 1.117 27.6 21.8 1.132 30.8 25.0 1.118 27.8 22.0 1.133 31.0 25.2 1.119 28.0 22.2 1.134 31.3 25.5 1.120 28.3 22.5 1.135 31.5 25.7 1.121 28.5 22.7 1.136 31.7 25.9 1.122 28.7 22.9 1.137 31.9 26.1 1.123 28.9 23.1 1.138 32.1 26.3 1.124 29.1 23.3 1.139. 32.3 26.5 1.125 29.3 23.5 1.140 32.5 26.7 1.126 29.5 23.7 1.141 32.8 27.0 1.127 29.8 24.0 1.142 33.0 27.2 .2232 .8232 2:8 223.2 n22222228222 68.2-5222 r.62 68.2-3222 "maoHumoHuHommHU 313.20 uHmHoonHm .onH .936 omHuoNH “condomupHE .oHHumoH 23.2mm .m>mHu oo "msoHuouHmHoomHU 333m: 502.223 23582.8me. 82-32 N22 22 60.2.3922 >Huomonm Goa? 00253392 HuooU mho .H HooHumHHHo .Hosnom omHnoNH omonom mchHfiuo. 0» ”Eggnog >333. ooo .H pocouumd .Hm>O omHuoNH Hound Hammad , dedeuo. on quHmHmou >233 ooo.H HmoHnHeGHH>o .wsoaH omHuoNH, uHcmnusm Hommdm .02. m3 .2 2058.2 8 232222222222 co 22838220 oNHuoH H BMGoOBoM _ >u2umcH 9330 o» oHnHunHoomno «mow/H ooo .H OH HmoHfiHHHHo .omumd omHuoNH oonoGnovH , canon .2288 3.3.82 228.2 8 2322222222.. 633. 32.2222 52.22832 28 .2 28222222222 82 .2 2 euEEH Hueo2 .0230 Head acmmaoshv msUHoosHonnHo 3. 30326 Hoo .H H0250.H 0» HmoHumHHHfl oNHuoHH umaou< mchHfium ooo .o H. 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Rate of Rupture Parameters Deformation Force Deformation Energy (ipm) (1b) (in) (in-1b) 19.5 0.128 1.305 16.5 0.116 0.995 1 19.0 0.100 0.995 18.2 0.124 1.226 15.6 0.096 0.760 hAean . 17.76 0.113 1.056 17 7 0.114 1.097 22.0 0.128 1.466 2 19.6 0.120 1 230 21.7 0 134 1.613 18.0 0.092 0 954 hAean . .. . 19.8 0.118 1 272 16 0 0.084 0.737 16.2 0.090 0.775 4 21.7 0.114 1.147 19.2 0.148 1.393 20.5 0.160 1.692 hdean . .. . 18.7 0.119 1.149 22.3 0.128 1.470 18.5 0.084 0.854 10 23.0 0.112 1.310 17.2 0.074 0 688 17.6 0.086 0 867 hdean... .. . . 19.7 0.097 1.038 17.9 0.088 0.800 19.3 0.098 0.921 20 19.2 0.100 0.872 19.0 0.126 1.192 19.5 0.090 0.863 Ldean . . .. . 19.0 0.100 0.930 13.5 0.144 0.974 14.2 0.144 1.017 40 14.8 0.156 1.207 13 5 0.160 1.217 12.0 0.144 0.835 bdean... . .. . . 13.6 0.150 1.050 APPENDIX B 137 138 Table B1. Rupture forces (lb) for five potato varieties tested 125 days after planting with a rigid die in the form of a solid circular cylinder having a cross-sectional area of 0.05 sq. in. Rate __ of deformation: 1 inch per minute (ipm) izplication Variety number Katahdin Kennebec Onaway Russet Burbank Sebago 1 15.0 14.7 15.5 15.0 15.0 2 14.7 11.7 13.7 15.5 14.2 3 16.5 13.0 13.0 16.6 16.5 4 13.2 13.2 19.0 16.5 15.0 5 13.7 14.0 13.0 14.0 13.2 6 15.2 10.7 13.7 14.0 13.5 7 15.5 14.2 12.7 15.8 14.2 8 16.7 15.0 12.3 14.6 17.5 9 16.0 12.5 13.6 14.2 15.3 10 13.0 12.5 14.5 15.0 15.4 11 14.0 13.0 15.7 17.0 14.2 12 15.5 12.0 15.3 17.5 15.0 13 14.5 11.7 14.0 15.0 13.2 14 15.0 10.0 13.3 15.5 15.5 15 13.1 14.2 12.2 13.0 14.2 16 16.2 14.0 16.0 17.0 15.0 17.‘ 14.0 13.7 16.7 13.6 14.4 18 15.0 11.0 15.2 15.3 14.8 19 16.7 13.7 14.2 14.6 13.2 ___ 20 16.0 _ 13:0 14.2 _ ___-_14.7_ _ 15.6 _ Means 14 98 12.89 14.39 15.22 14.74 Table B2. An analysis of variance table based upon the rupture forces r e s ented in T ab11. Source of Degrees of . . F -ratio variation freedom sluares sguare Total (N) 99 234. 27 2. 366 Varieties (n) 4 67. 85 16.96 9.69** Error 95 166.42 1.75 ** Indicates significance at the 99 per cent confidence level. Table B3. Duncan's Multiple Range Table for mean separation based upon means in Table B1 and the analysis in Table B2. Error degrees of freedom: 95 w Range between ranked means: 2 3 4 5 Differences at the 95% level: 0. 90 Differences at the 99% level: 1. 18 1. 20 139 Table B4. Rupture energy (milli-inch-lb) for five potato varieties tested 125 days after planting with a rigid die in the form of a solid circular cylinder having a cross-sectional area of 0.05 sq in. Rate of deformation: 1 inch per minute (ipm) Replication Varietj number Katahdin Kennebec Onaway Russet Burbank Sebiig 1 772 555 459 596 529 2 593 422 481 668 483 3 714 479 382 692 666 4 448 549 864 601 473 5 487 427 371 781 382 6 540 373 486 487 478 7 599 467 501 585 551 8 724 537 303 754 760 9 765 448 367 619 547 10 460 353 552 638 664 11 549 387 467 823 548 12 513 425 501 769 560 13 622 344 444 513 444 14 473 282 365 604 560 15 414 585 289 459 555 16 507 492 574 760 587 17 437 587 620 471 604 18 642 364 346 463 567 19 685 577 416 706 598 ___ 20 -_§}3 __ _461 _______ 586 _______ 571 ________ 596____ hAeans 573 456 469 628 558 Table B5. An analysis of variance based upon the rupture energy values presented in Table B4. Source of gDegrees of Sum of Mean . . . F - ratio variation freedom scLuares square Total (N) 99 1, 531, 591 Varieties (n) 4 425, 285 106, 321 9.13** Error 95 1, 106, 306 11, 645 *fslndicates significance at the 99 per cent confidence level. Table B6. -Duncan's Multiple Range values for separation of ranked means based upon the values in Tables B4 and B5. Error degrees of freedom: 95 Range between ranked means: 2 3 4 5 Differences at the 95% level: 68 71 74 75 Differences at the 99% level: 90 93 96 98 140 Table B7. Elastic modulus (psi) values for 1 sq in by 1 in cylindrical sections of Arenac potato tissue as influenced by temperature. T est date: 12/20/62 Rate of deformation: 1 ipm Replication Temperature, degrees F number 40 3:2 60 3:2 80 3:2 105 $3 135 is 1 625 588 500 500 322 2 625 500 435 500 357 3 526 488 454 465 333 4 , 556 526 476 526 417 5 556 526 513 526 400 6 556 526 425 417 333 7 476 513 488 476 345 8 526 476 556 526 333 9 513 500 556 500 322 __ __1_O_ -- _ 488 __ _ 465 454 _ 500 385 - Means 545 511 486 494 355 Table B8. An analysis of variance to determine the significance of the influence of temperature upon the elastic modulus values presented in Table B7. Source of Degrees of Sum of-i Mean , . . F - ratio variation freedom squares square Total 49 283, 949 Temperature 4 210, 303 52, 576 32** Error 45 73, 646 1, 636 ** Indicates significance at the 99 per cent confidence level. Table B9. Duncan's Multiple Range values for separation of ranked means based upon the modulus values presented in Tables B7 and B8. Error degrees of freedom: 45 Range between ranked means: 2 3 4 5 Differences required for significance, (1) at the 95% confidence level: 37 39 40 41 (2) at the 99% confidence level: 49 51 53 54 141 Table B10. ‘Rupture forces (1b) for potato tubers under compression with a 0. 05 sq in cross-sectional area cylindrical rigid die as influenced by rate of loading. Test date: 8/23/62 Variety: 1111-2 (Experimental) Replication ‘Rate of Deformation, inches per minute (ipm) number 1 2 4 10 20 40 1 16.5 17.7 16.0 22.3 17.9 13.5 2 19.0 22.0 16.2 18.5 19.3 14.2 3 18.2 19.6 21.7 23.0 19.2 14.8 4 15 6 21.7 19.2 17.2 19.0 13.5 5 19 5 18.0 20.5 17.6 19 5 12.0 Means 17.76 19.80 18.72 19.72 18.98 13.60 Table B11. An analysis of variance to test the significance of rate of deformation upon the rupture force or strength of the potato tuber based upon the values in Table B10. m m ===== Source of Degrees of Sum of Mean . . . F - ratio variation freedom squares square Total 29 224.09 ' Rates of deformation 5 135.19 27,038 7.13 ** Error 24 88. 90 3. 704 >10: Indicates significance at the 99 per cent confidence level. Table‘B12. Duncan's Multiple Range values for separation of ranked means presented in Table B10. 'Error degrees of freedom: 24 Range between ranked means: 2 3 4 5 6 Differences required for signifi- cance at the, 1(1) 95% confidence level: (2) 99% confidence level: \ a «an. :1 us: 0 ("1f\ V’U R i M .L. .. Tic! law... A»... n a .41. .1 RTE UN . “1111 IV LIBRARIES 111111111111111111111 178443 l1 E01