ABSTRACT SOME MECHANICAL PROPERTIES OF CHERRY BARK AND WOOD by.Robert G. Diener This study was initiated to study the mechanical behavior of fruit tree bark to applied stresses. Since. “the green wood is closely associated with the bark in the tree limb green wood was also included. Also since, during mechanical haryesting different frequencies of Vibration are used, the frequency of the applied stress was also considered. Tests were limited to a single variety and species: Montmorency Cherry, Prunus Crasus. All tests were conducted under controlled temperature and humidity conditions in a testing chamber. Maximum strength of bark specimens was determined from tensile loading in a pneumatic testing machine. Elastic and viscous properties of bark and green wood specimens were measured using elastic and viscoelastic flexure theory. Three loading techniques were used for the flexure tests. TheSe tests were (1) loading at a slow constant strain rate uSing a simple beam arrangement (2) free vibration as a cantileyer beam and (3) forced vibration as a simple beam. As a check on the theory and experimental technique Robert G. Diener several Specimens of aluminum, Plexiglas, and white pine wood were included in these tests. Approximate and exact equations for determining the viscoelastic modulus from dynamic flexure were derived. The accuracy of these equations was described graphically in terms of measured variables. From this it was possible to use the approximate equations with a negligible amount of error. From a transverse microsc0pic section of bark the tissue appeared to be arranged in four major radial layers. These layers consisted of a non functioning phloem, a functioning phloem, and two periderms which made up about 10, 65, 12, and 12 per cent of the total thickness respec- tively. The strength of bark was very dependent on the direction of the applied force. Maximum longitudinal and tangential strengths for the three inner bark tissues were about 6&0 and 70 pounds per square inch respectively. The outer periderm strengths were about 980 and 3250 pounds per square inch in these reSpective directions. Longitudinal cambium shear strength was about 30 pounds per square inch. The storage modulus of bark was very sensitive to frequency of the applied stress and varied from about 10,000 ,to'A0,000 pounds per square inch over a frequency range of Robert G. Diener about 10'".3 to 10-2 cycles per second. The storage modulus of green wood varied a lesser amount from about 500,000 to 700,000 pounds per square inch over a slightly larger frequency range. The loss tangents of bark and wood were about .15 and .0“ respeCtively. . Approved 35% Major Professor Approved 6,0431 Q]. M Department Chairman SOME MECHANICAL PROPERTIES OF CHERRY BARK AND WOOD By Robert G. Diener A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1965 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 pro- vided by his major professor, Dr. F. H. Buelow (Agricultural Engineering), during this graduate program. To the other members of the guidance committee, Dr. B. A. Stout (Agricultural Engineering), Dr. G. E. Mase (Metallurgy, Mechanics and Materials Science), Dr. S. K. Ries (Horticulture) and Dr. E. A. Nordhaus (Mathematics), the author expresses his deepest gratitude for their time, professional interest, and constructive suggestions. Dr. L. E. Malvern (Metallurgy, Mechanics, and Materials) and Dr. L. w. Mericle (Botany and Plant Pathology) made helpful comments during various stages of this study. Mr. James Cawood and his staff in the Agricultural Engi- neering Research Laboratory assisted in the development of the experimental apparatus. 11 TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES LIST OF APPENDIX TABLES Chapter INTRODUCTION. Objective. RELATED LITERATURE. Structural and Mechanical Properties of Plant Tissue. Structural and Rheological PrOperties of Fruit Tree Bark. Concepts in Viscoelasticity and Materials Testing of Hard Viscoelastic Materials . Theory of Flexure Testing for Rigid Visco- elastic Materials THEORETICAL ANALYSIS Development of the Complex Modulus of Elasticity Equations . . . LABORATORY STUDIES Testing Chamber . ' . Maximum Strength Tests Measurement of Elasticity and Damping DISCUSSION OF RESULTS. . . . Effectiveness of the Testing Chamber. Physical Structure of Bark Tissue. iii Page ii vi viii Page Strength of Bark . . 59 Elastic and Viscoelastic Response of Bark and Green Wood to Frequency of the Applied Stress . . . . . 70 Significance of Results . . . . . . . 99 SUMMARY . . . . . . . . . . . . . . . 100 CONCLUSIONS . . . . . . . . . . . . . . 102 REFERENCES . . . . . . . . . . . . . . 104 APPENDIX . . . . . . . . . . . . . . . 109 iv Table LIST OF TABLES Mechanical properties of plant fibers and some other materials . . . . . . . . . . Ultimate stress and elongation for necked bark sections under tensile loading. R = 1.5 inches per minute . . . . . . Ultimate Stress of periderm sections under tensile loading. R = 1.5 inches per minute . . . . . . . . . . . . Maximum longitudinal cambium shear strength. Young's modulus of elasticity and damping ratio for bark and other materials tested for different.frequencies of applied stress . . Comparison of experimental values of loss tangent for Plexiglas with those found by KOppelmann (1958). . . . . . . . . Storage modulus, loss.modulus, and loss tangent . for bark and other materials tested at different frequencies of applied stress . . Page 66 68 70 86 93 96 Figure 1. 2. 10. ll. 12. 13. LIST OF FIGURES A two parameter Kelvin viscoelastic model containing elastic and viscous elements. A Kelvin model with an attached mass subjected to a sinusoidally varying force . . . . Variation between true values of the damping ratio and the values calculated using the approximate equations. . . . . . . . Error resulting from using the approximate equations to calculate the loss and storage moduli. . . . . . . . . . . . . General view of testing apparatus. . . . . Necked bark specimen loaded in tension using the pneumatic testing machine . . . . . Sketch of the bark and wood specimen used to measure the maximum longitudinal cambium shear strength . . . . . . . . . . Bark specimen supported as a simple beam subjected to bending stresses . . . . . Bark Specimen rigidly fixed as a cantilever beam for free vibration tests . . . . . Bark specimen supported at the nodal points for forced vibration tests . . . . . . Transverse bark section showing location and relative thickness of the major tissue regions . . . . . . . . . . . Failure curves for bark sections loaded in tension at a constant rate of extension. Failure of a longitudinal bark specimen subjected to tensile loading . . . . vi Page 30 31 3.8 “3 A5 A6 ‘47 “9 51 56 60 62 Figure I Page 1A. Failure Of a tangential bark specimen sub- jected to tensile loading. . . . . . . 6A 15. Loading and relaxation of bending stresses for bark specimen B5 . . . . . . . . 73 16. Primary relaxation of bending stresses reduced to the same initial values for some ” " "' materials tested. . . . . . . . . . 7” 1?. Damped oscillations at the free end of a canti+ ” 1eVer bark Specimen. . . . . . . . . 76 18. Amplitude reduction of successive cycles at the free end of cantilever specimens . . . 77 19. Farced vibration of a simply supported specimen showing: top, driving wave at half the first mode frequency; bottom; resulting ’osciliation of the beam (fourth order sub— harmonic of the first mode frequency). . . 79 ' 20. Forced vibration resonance at the first mode - for a simply supported specimen showing; top, driving wave; bottom, resultant vibra- tion of beam . . . . . . . . . . . 80 21. Forced vibration resonance for the third mode showing} top, driving wave; bottom, " resultant vibration of beam . . . . . . 31 22a Storage mOdulus for some of the materials tested plotted logarithmically against fre- quency of the applied stress. . . . . . 92 vii LIST OF APPENDIX TABLES Table Page .Al. Values of the damping ratio, n', the frequency ' ratio, Aw/wnd, the logarithmic decrement, A, and the per cent error in the loss and ‘ storage moduli resulting from using the approximate equations which assume small values of damping. (Negative error values indicate the approximate value is larger than the exact value.) . . . . . . . . 112 viii INTRODUCTION Mechanization of harvest operations in recent years has been a practical solution to the problem of the high cost of hand labor facing the American fruit grower. In many cases the cost of harvesting has amounted to as much as half the total cost of production. AlSo, hand labor is nOt always available When needed, and managing and housing laborers requires additional time and expense. ' Several techniques have-been introduced to enable the grower to'mechanize harveSt operations Such as uSe of mobile platforms, picking tubes, picking spindles and tree Vibration. However, the first three methods require selectiVity in their use and do not cover a large area of the tree in a single application. Among the various means and principles Which have been tried for detaching the fruit, vibration appears to be the simplest and most practical methbd. Although vibratory mechanical harvesting is less expensive than hand harVesting, some resultant damage to 'the fruit and tree must be tolerated. Fruit damage occurs in the detachment and collection phases when the fruit come into contact with each other or collide with limbs during the fall through the tree.- Haldersen (1963) categorized the resultant tree damage into three parts; these pro- duced on the limb, trunk, and root systems.' Heweverby thefuse of an accelerometer-attached to the tree during. shaking, he found all motion to cease a few inches below ground level and concluded that root damage was negligible. Injury to the.1imb.occurs by the breaking of fruit bearing spurs, fracture of small limbs from whipping action during shaking, and rupture of the bark tissue in the shaker clamp area. Adrian and Fridley described bark injury as occurring while locating the clamp on the limb, clamping with excessive radial pressures, and improper alignment of the shaker boom with the limb causing exces- sive longitudinal stresses in the bark. Additional stresses in the tangential direction may also be produced if the shaking machine moves while vibrating the limb or if the limb tries to assume a path of oscillation which is not parallel with the applied shaking.force. However, as reported by Adrian and Fridley, what bark damage occurred could be tolerated in view of the savings resulting from mechanical harvesting. Then it was discovered that a Ceratocystis Canker was entering the bruised bark tissue. Devey, S£_Elw (1962), described this canker as a fungus disease which originates in bruised bark tissues and rapidly Spreads to healthy tissues. In a relatively short time the fungus would kill the diseased limb and in many' cases cause the death of the entire tree.» Chaney (1964) reported that orchards so infected may be expected to die out completely within ten years. As a result of loss. in production caused by this infection cooperative studies were initiated by the United States Department of Agri- culture and the University of California Department of Agricultural Engineering to determine allowable shaker clamp stress that could be applied without injuring the bark tissue. From these studies an upper limit of safe radial stress was found as well as general indications of the relative magnitudes of maximum longitudinal and tangential bark strengths. Since mechanical properties of a material such as strength and also elastic modulus and internal damping determine the behavior to applied stresses, it was felt that the study reported in this thesis Should cover these additional properties. Also, since a wide range of fre- quencies is used in mechanical harvesting, it was consid-= ered-advisable that behavior be studied on the basis of the frequencyuof the applied stress. Furthermore, since the bark and green wood are closely associated in the tree, it was felt that the study should be extended to cover green wood as well. Objective The objective of this study was to investigate physi— cal properties and mechanical response to applied stresses in bark and green wood tissues. Specific objectives of this study were: 1. Develop or adapt from existing equations the necessary mathematical formulas for description of mechani- cal behavior of bark and green wood under applied stresses. 2. DevelOp the necessary experimental procedures and equipment-to measure and record the physical properties and mechanical properties such as strength, elasticity, and damping of the materials under study. 3. Develop a testing chamber having controlled temperature and humidity conditions in which to have the testing facilities and to conduct the experiments. RELATED LITERATURE Structural and Mechanical Pronertie; of Plant Tissue The smallest structural uni? »” p;wn1 LISSUS is the cell. The cell is composed of a non-prULOplasmic rigid wall and an inner cytoplasmic fluid. Roelofsen (1959) described cell walls as having largely supportive and protective functions and thus the walls determine the shape of the cell and the texture. Two types of wall structures are present in living plants, a primary wall and a secondary wall. Usually living cells which carry out life processes have only a primary wall whereas non- living cells, whose function is primarily supportive, have an additional secondary wall. Primary walls are composed of a fine mesh network of cellulose fibrils which is filled with pectic and hydrophilic compounds; in woody tissues they are filled with lignin and in cutinited walls with waxes and cutin. The pectic compounds constitute the major portion of the middle lamella between adjacent walls of the cells. In secondary walls most of the cellulose present is crystalline cellulose. In the heavier secondary wall, the cellulose fibrils are grouped into coarser 5 branching strands which are encrusted with pectins, hemi- celluloses and lignins. In walls that are pure cellulose, such as the secondary wall of cotton fibers, the spaces are filled with water. Lignin is one of the most important wall substances in woody tissues. This polymer encrusts the wall providing a stiffening and protective function. Mechanical Properties of Plant Cells.—-Plant cells are classed as parenchyma, collenchyma or sclerenchyma cells depending on their wall structure and function. Parenchyma cells are living cells. They are soft, elastic and are capable of large plastic deformations at certain periods during growth (Heyn, 19A0). Collenchyma cells are similar to parenchyma with the exception of some thickening at the wall edges which contribute some strength to the tissue. In sclerenchyma cells a large secondary wall is present inside the primary wall. This secondary wall may become so thick that the central cavity almost disappears. Both primary and secondary walls in cells exhibit elastic behavior, but primary walls are much more flexible. Meyer et a1. (1963) described the tensile strength of secondary walls as comparable to that of spring steel. Frey—Wyssling (1952) reported that primary walls were capable of up to 50 per cent extension as com- pared to only about two per cent for the secondary walls in fiber cells. Meyer (1950) thought the differences in behavior between primary and secondary walls was due to theVlarge~amounts of amorphous cellulose and pectic compounds in primary walls as contrasted to the crystalline cellulose and lignins in secondary-walls. He also stated that the presence of lignin does not affect the tensile strength, but increases the compression strength by preventing the . cellulb e stran s in the walls from buckling under compres- sive loadings. However, Frey—Wyssling (1952) suggested that the encrusting substances in and between the cells_ and the-arrangement-of the fibrils accounted for the amount of elasticity and strength and not the crystalline and 'amorphous regions in the fibrils themselves. He also stated thatlthe pectins, hemicelluloses and amorphous cellulose.are very sensible to swelling with water and ' that-as a result elasticity and plasticity in the fibers depend markedly on moisture content. «Frey~Wyssling (1952) foundnthat ultimate stress in cotton hairs could be related. to the spiral angle of the fibrils in the cells. He found‘ that the ultimate strength'increased-as the fibrils became orientatedrmore in the direction of pull.' Under high .stressesncellulose fibers and other cells often Show slip planes. 'Meyer (1950) explains failure under these condi- tions as a slipping or fracture of the molecular chains. In native fibers where the chains are long, failure occurs by rupture offthe chain. In artificial fibers where the chains are Shorter he suggests that failure occurs by 'slipping between chains and hence these fibers are subject 'to humidity content giving them a low "wet strength." For the purposes of comparison of some of the mechani- cal.pr0perties~of plant fiber cells and other materials Table l is presented below from data presented by Meyer (1950), Frey-Wyssling (1952), and Esau (1965). TABLE l.--Mechanical properties of plant fibers and some other materials. Average property in psi x 10"3 and strain in in/in. Tensile Compression Elastic Ultimate Material strength strength modulus strain Collenchyma (fiber) lu.0 .02 Sclerenchyma (fiber) 28.A .01 Cotton (fiber) 60 7 1,100 Wool (fiber) 25 - 280 Nylon (fiber) 110 17 1,000 Concrete 190 - 2,800 Wood 1A 12 1,750 Aluminum 35 38 9,800 Steel 1A0 <1AO 28,000 In.woody tissues having thickened secondary walls, Frey—Wyssling (1952) thought that the instability of the cell shape determined the mechanical response of the cell to applied stresses. He also thought this to be the reason for anisotrOpy in wood. Wood also has definite isotropic directions with respect to the principal fiber orientations. For example, the average ratio of elastic moduli magnitudes in wood in the tangential, radial and longitudinal directions is about 1:2:20 respectively, (U.S.D.A. Wood Handbook, 1955).‘ Frey~Wyssling (1952) summarized and reported on studies made by other investigators of the anisotrOpy in Douglas fir and spruce. 'They found that the elastic modulus measured between the principal orientations of the fibers varies in a complicated pattern with changes in angular location.' They also found that under compressive loading, the large trachied cells failed by collapsing or structural instability. The walls of the cells deformed in an "S" shape and the edges of the walls remained at right angles to each other. Moisture content in wood is defined as the ratio of water to dry matter (U.S.D.A. Wood Handbook, 1955). The moisture present in wood is divided into two parts: (1) free-water'in'cell cavities and in intercellular spaces and (2) absorbed water in the capillaries of the walls. The fiber saturation point is reached when all the free water is absent at about 30 per cent moisture content. Mechanical properties in wood are affected by moisture content. 'Wood increases in strength as it drys. Longi— tudinal compressive strengths for 12 and five per cent wood, for example, are about two and three times the strength of green wood respectively. The strength of green wood does not begin to increase with drying, however, until the fiber saturation point is reached. The change in strength has lO been associated with moisture content in the following relation (U.S.D.A. Wood Handbook, 1955) as log S = log S 3 l 3/Ml—M2) log 82/81 where 81,82 and S3 are the strengths at moisture contents M1,M2 and M3. Changes in temperature have a slight effect on wood. Generally + (Ml-M wood strength increases with decreasing temperature. The opposite is also true. It is estimated this strength variation is about one—half per cent per degree F. from a base temperature of 77 degrees F. When wood is exposed to temperatures above 150 degrees F. it is permanently weakened. Structural and Rheological Properties of Fruit Tree Bark The nonatechnical term "bark" refers to the tissues in the stem outside the vascular cambium. The bark is composed of the phloem, an inner food conducting tissue, and the periderm, an outer protective tissue. Arrangement of Bark Tissues.-—The periderm consists of three tissue layers. The outer cork*1ayer, a middle single cell layer of cork cambium cells and an inner layer of loosely packed parenchyma cells, which sometimes is not present. The phloem is generated from the vascular cambium in an outward radial direction. The vascular cambium also produces xylem or wood cells to the interior. Generally, less phloem cells are produced than xylem and 11 the crushing of the phloem annually by new growth makes it much thinner than the woody part of the stem. Phloem is composed of several specialized cells such as Sieve cells, fibers, and parenchyma. The sieve cells are long, tubular cells which form the food conducting tissues of the phloem. Parenchyma cells are found both at random in phloem tissue and in radial rays as ray parenchyma° Only a few layers of phloem cells next to the vascular cambium actually function in a given season. The rest are termed non- functioning phloem (Esau, 1965). Schneider (19A5) studied the seasonal changes of peach and cherry phloem. He found sweet cherry produced a new band of phloem each spring to function for the summer. In the fall the new sieve tubes became plugged and the tissue became non-functioning. During each growing season fibers matured in the phloem of the previous year, and fissures occurred along the rays. He also found that less crushing of the old cherry phloem occurred than in some other species with the result that cherry bark was much thicker than peach bark, for example, where the old phloem is completely crushed. Mechanical Properties of Bark.e-Adrian and Fridley (1963) studied bark injury in prune trees as related to shaker clamp design. They reported that bark on the tree could safely withstand a radial pressure of 250 pounds. 12 per square inch without rupture of inner phloem cells. Rupture was indicated by a hairline crack in the functioning phloem thought to be caused by air oxidizing the tissues in the injured cells. 'They found the tangential shear strength of these barks to be about 100 pounds per square inch. Longitudinal strength of bark was found to be about four times the tangential strength. Brown (1965) conducted tests on almond and olive trees to find the effect of moisture content and normal (radial) pressure on the shear strength of bark at the cambium layer. He found that in general bark with high moisture content had low shear strength and bark with low moisture contents had high shear strengths. Shear strength was also found to be affected by the normal pressure depending on the moisture condition of the bark and variety.' At moisture contents less than about 130 per cent an increase in normal pressure caused an increase in the cambium shear strength. These shear strengths were in the order of 1A0 to 200 pounds per square inch for normal pressures of 60 to 300 pounds per square inch. At moisture contents above 160 per cent, increase in“normal pressure did not produce a corresponding increase in the shear strength of the bark, and for some varieties shear strength decreased. He suggested that to reduce'injury during shaking, it would be advisable to harvest when the moisture content of the bark is at a lower level. 13 Concepts in Viscoelasticity and Materials Owing to the tremendous importance of the mechanical prOperties of synthetic high polymers, the science of viscoelasticity has been developed largely in the last 25 years.‘ These-materials have remarkable properties which permit-them to be used aS'rubbers, plastics, and fibers to. replace materials of”plant and animal origin"and for new uses which were impossible before. 'These organi “com— poundsrare'formedwfrom*hydrocarbons and their derivatives ‘with*almost infinite variations in molecular structure and resulting mechanical properties. Unsaturated hydrocarbons undergo polymerization reactions in which molecules couple together to form extended chains. The term high polymer (Sienko and Plane, 1961) is any of these large molecules containing recognizable repeatable units which form extended chains. Polymers are formed in leaf tissue of plants where carbon.dioxideahd.water are combined to form the sugar, and glucose. Thousands of g1uCose molecules unite to form large molecules of cellulose and starch. ‘The cellulose conk- stitutes the framework of the plant while starch serves as a food material. Viscoelastic Behavior of Materials.--The theory of Viscoelasticity, developed to explain the behavior of high p01ymers, allows mathematical representation of such 1A phenomena as stress relaxation during loading, retarded elasticity, creep and non-recoverable deformation. Even metals and glass fibers show a slight degree of viscous behavior under creep, relaxation, and sinusoidal testing. While these materials-are-primarily elastic these tests yield valuable information about the structure and behavior of the material. 'Polymers, however, exhibit a high degree of viscoelastic behavior in tests of the material. In crystalline high polymers, the crystals are rigid for all practical purposes while the amorphous regions are only rigid and brittle at temperatures below the amorphous "brittle point" (Alfrey, 19A8). At higher temperatures these amorphous regions have a viscous and rubbery behavior. Frey-Wyssling (1952) summarized and reported on studies done by other investigators on the mechanical behavior of cellulose fibers. 'These investigators found the tensile strength and modulus of elasticity of native cellulose fibers”to be as high as the finest steels. Their highest experimental values approached the theoretical strength of'a stretched rope of primary valence chains. High values were found to occur with low moisture content, low temperatures, and good orientation of the fiber. They found the large plastic deformation to occur only through consolidation and orientation of the amorphous regions. When extensions reached this point of consolidation, the fiber was rather completely crystallized, and failure occurred. 15 Techniques in Viscoelasticity.--It is common to represent a viscoelastic material with a mechanical model of springs, dashpots, and frictional elements in order to write a-mathematical description of the behavior. These models are useful in one—dimensional studies of strain rate, creep, and stress relaxation.' The oneedimensional model may be extended~by'use'of'the CorreSpondence PrinCiple into a three dimensional stressestrain tensor (Bland, 1960).‘ However, this leads to very complicated expressions for higher parameter models having three or' more elements, and is useful only for homogeneous materials having simple shapes. The behavior of viscoelastic materials may also be studied by applying a sinusoidal stress to the material and observing the resultant strain. *The behavior of a specific viscoelastic model may also be studied in this way by replacing the stress and strain components in the constitutive equation With the corresponding sinusoi- dally varying components of stress and strain. 'The resulting equation may thenwbe solved directly for the elastic modulus by obtaining the stressestrain ratio. The elastic modulus obtained in this way is called the complex modulus of elasticity. The complex modulus is usually arranged in its real and imaginary parts of vectors which represent the elastic and viscous elements respectively. 16 o S i Stu ie in A 1 ur Pr d t .-- Viscoelastic theory has more recently been applied to agricultural products in an attempt to describe their behavior. Mohsenin (1962) e£_al. studied creep in apple fruit using a Kelvin and a Maxwell model in series. They*also recognized the importance of strain rate. Zoerb and Hall (1960), and Finney and Hall (1964) used Maxwell models in parallel arrangements to represent stress relaxation in agricultural products. Zoerb used. a two element model to represent pea beans and Finney used a four element model for potatoes. Both studied the effect of strain rate on the rate of stress relaxation after loading ceased. 'They observed that at slower loading rates less stress relaxation resulted in a given time than that for higher rates. It is obvious here that the material is relaxing in the loading phase. At high loading rates, the material cannot relax as much in the loading phase; therefore, this occurs in the relaxation phase. AS a further step, Zoerb and Finney might have plotted the relaxation time constants against‘loading'rate to determine what relations existedu‘ Interestingly, Zoerb found no relation between moisture*content of pea beans and relaxa- tion time. However, he noted an increase in loading force with an increase in moisture content which demonstrated a viscous effect. l7 Finney (1963) presented a comprehensive introduction‘ to use of models to represent viscoelastic behavior in his Doctoral Thesis. Testing of Hard Viscoelastic Materials Relatively stiff materials such as wood fibers approach the rigidity of the testing apparatus and as such are not suitable for testing in sandwich or block geometry for measurement of pure shear.' Rather, these materials are more suitably tested as long, thin strips in tension, thin rods in torsion or as beams in flexure. Tension and torsion testing are sometimes not prac— tical because of difficulties in gripping the specimen, due to low strengths of the material in the radial direction as compared'to the longitudinal direction. Also it is often difficult to measure the relatively small displace- ments resulting from the applied stresses by direct methods such as the use of strain gages.- Indirect methods must then be used by measuring the displacement of an element strained~in'serieS'with'the'specimen“being'tested.' Flexure'testing-iS'frequently-more-desirable‘since this method permits large displacements without rupturing- the material. -In addition this method permits moving of the supportsto adjust the amount of force applied to the specimen for a given displacement. Dynamic testing techniques should be very helpful in testing semi—rigid materials. These tests are usually 18 fast, non-destructive, and give good results. By using rapid loading rates these tests can provide information about the material behavior for higher frequencies of applied stresses. In addition, cyclic testing in flexure, for example, bends the specimen in both forward and reverse directions, thus averaging the properties across the section. This often leads to more consistent results. Theory of Flexure Testing for Rigid Viscoelastic Materials Static Testing. For a simply supported beam with a center load P, the elementary elastic equation for average bending stress 0 is 3PL G g. (1) ubhé and for Young's modulus of elasticity, E, is 3 quh3 where E and o are in units of pounds per square inch, P is the applied force in pounds, L is the distance between the supports in inches, Y is the displacement at the center of the beam in inches and b and h are the width and depth of the beam respectively, in inches. However if the material under test is viscoelastic in behavior, then time must also be considered and the resulting stress and modulus become o(t) and E(t) l9 respectivelyi A generalized stress equation of this type haS‘been prOposed by Ferry (1961) for a Maxwellian material under a constant loading rate R as o(e,t) = R/ITHEl—exp(-t/T)] d1 (3) This equation represents a spectrum of Maxwell models each contributing a stress E and a relaxation time between T and d1. T is defined as the ratio of the dashpot constant n to the spring constant E. The term H(§) is a function of the relaxation time I and is generally called the spectrum of relaxation time. This term replaces the E's of the individual models. Thepvariables-H and T are deter- mined experimentally by loading~the-material at different strain rates and long term relaxations;' However, of more. importance, the time derivatiVE'of‘the'ratio of the stress to the loading rate R in equation (3) is equal to the relaxation modulus E(t). The relaxation modulus is defined. .as the ratio of the relaxation stress to the strain at which relaxation is carried-out (Alfrey, l9u8). However, Chang (1964) has shown that the relaxation modulus can also be measured by taking the tangent modulus of the loading curve at time t., This is a useful relation since E(t) has already been related to the storage modulus, E'(m), by Ferry (1961), in the relation E’(w) z E'(l/t) _ (u) 20 This relation is possible for low frequencies or short relaxation times since the relaxation and storage moduli curves are mirror images of each other in the time and frequency axes respectively. Since both E(t) and H'(m) are measures of stored elastic energy, a dynamic measure- ment at frequency w is equivalent to a transient one at t =‘1/w. Nonresonant Dynamic Testing.--The storage modulus E'(w) and the loss modulus E"(w) components of the visco- elastic modulus E* are most simply written E'(w) E“ cosé ’ (5) E"(w) ‘E* sins i _ (6) where 6 is the phase angle between the applied stress and the resulting strain.' A convenientway to measure these moduli would be to apply a sinusoidally varying force to the material and observe the modulus E* and the-phase angle 6. If the material is purely viscous the displacement will lag the applied stress by:a phase angle of 90 degrees” For.pure elastic response the phase angle is zero. Visco- elastic materials fall somewhere between these two extremes. However, considerable-difficulty is usually encountered in measuring the phase angle, 6, accurately. KOppelmann (1858) tested Plexiglas in this way using cantilever beam geometry. ~In this case the modulus, E*, was replaced by IfI/h'lxl, where f and x were the peak 21 values of applied force and resulting displacement at the end of the beam respectively and h' was a form factor depending on the geometry of the beam. In the case of a cantilever beam h' = bh3/uL3 . (7) 'where b, h, and L were the width, depth, and free length of the beam respectively. The beam was driven bya rocker arm. One end of the rocker arm was attached to‘the beam by a wire and the other end of’the arm was driven by a sinuSOidally varying force.' By the use of two small mirrors, ‘one at the rocker arm pivot and one attached to the sinusOidal driver, Koppelmann was able to determine the phase angle from the images traced on a high Speed, light sensitive film strip. Resonance Testing.--Resonance testing techniques. use the frequency and relative amplitudes of the resonance condition and as such are capable of measuring response at much higher frequencies than either static or dynamic techniques.' The use of resonance techniques in measuring ‘the behavior of materials at high frequencies are usually referred to as sonic testing. Sonic testing techniques are very suitable for testing beams in flexure.' This method requires very small deflections and as such there is no danger of exceeding the limits of linear behavior in 22 . the material or the assumptions made in the.derivation of the flexure equations. The basic relation for flexural vibrations in beams is given by Timoshenko (1921) in the equation a 2 33’ JALX - HI «I» —0 (8) 3X; g at2 where E is the elastic modulus, I is the inertia of the cross section, 7 is the density of the material, g is acceleration due to gravity and y and x are distances measured in the depth and length directions of the beam respectively. By assuming a solution of the form y = D (A'sin wt + B'cos ut) the well known expression for the natural frequency, fd’ for flexural vibrations in beams is expressed as . 1 n .KE (9) where E‘is the modulus of elasticity in pounds per square inch, K is a beam constant and fh is the observed natural frequency in cycles per second. The beam constant, K, is given by the relation K = (2n)2 ASLu/kmu lg (10) where L is the length of the beam and km_is a modal cone~ 1 stant based on the beam geometry and the particular mode- of vibration. Modal constants listed by Timoshenko (1955) 23 for the first three modes of vibration are: for a cantie‘ lever beam, 1.875, H.69M and 7.855 respectively and for a free beam simply supported at its nodal points, 4.730, 7.835 and 10.996 respectively. The nodal points are locations along the length of the beam which are stationary .during flexural vibration of the beam at a particular mode. Church-(1964) has given the nodal locations as a non-dimensional ratio of the distance of the node from the end of the beam to the total length of the beam. For beams having different-boundary conditions at the ends of. the beam, such as the cantilever, the particular end of the beam from which measurement is to be made must be specified; For a cantilever beam the nodal positions as. measured from the free end are: first mode, 1.0, second mode 0g226, third mode, 0.499 and 0.132. For a free beam which is simply supported, the nodal locations are symmetrie cal and may be~measured from either end of the beam. The nodal positions are:' for the first mode, 0.22“ and 0.776, for the second mode, 0.132, 0.500 and 0.868, for the third mode,.0.094,.0.356, 0.6M“, and.0.906. Timoshenko (1921, 1922) first recognized the-effect of forces present in beams due to shear and rotation inertia and-developed a new flexure equation to include these terms.' Goens (1931) used the Timoshenko equation to develop an approximate correction expression for shear and rotation inertia. The Goens correction factor, T, was 24 used to calculate a corrected resonance frequency fC from the observed resonance frequency fn by the relation f = f -V T ' (11) However, in practice, the Goens equation was not easy to calculate, since the solution for T involved an iterative solution and a prior knowledge of the elastic modulus, E. Picket (1995) developed a more usable equation for the Goens correction factor T for flexural vibration of beams and cylinders. This relation is 41' 2 ‘02n (r/l) Tn = 1 + Cln (r/2)‘ ’ 2 (12) l + C3n(r/£) where the subscript, n, refers to the particular mode; Cl’ 02, and C3 are constants, r is the radius of gyration and 2 is the length of the beam. In a rectangular beam the ratio r/i is equal to the ratio of the depth squared to twelve times the length and for a cylinder the ratio is equal to the ratio of the diameter to four times the length. The constants calculated by Picket, C C In’ 2n’ and C for the first three-modes are: for n = l, 88.12, 1572, 3n’ and 92.61 respectively; n = 2, 229.81, 9984, and 223.6 respectively; and n = 3, 446, 38,372, and “38.3 respectively. It can be seen from equation (10) that the value of T is always slightly larger than 1.00 and becomes larger when the frequency is increased, but remains approximately equal to one for values of r/2 less than 0.005. 25 In recent years investigators have used various forms ‘ of equations (10),(ll) and (12) to measure the modulus of, elasticity, E, in materialssusing resonant flexure testinga Hearmon (1958), for example, measured Young's modulus in wooden.beams, 18 to 20 inches long with nearly square cross section,using forced resonant vibrationu 'In these tests Hearmon supported the free beams at the nodal points .by'two"thrEads; Hearmon used a coil, driven by a signal generator,tb drive the beam through the use of a small steel shim attached to the end of the beam. He used a shim and cOil at the-other endiof the beam to measure the reSulting amplitude which was displayed on an oscillOSCOpe screen. The resonant frequency was determined as that . frequency which resulted in the maximum amplitude of the‘ driven beam. HearmOn found very good agreement for calcu- lated'valueS‘of E from different modes in hih“tests. At higher mOdes the value ofiT became significantly large. For'efample,~in a typical sample, with a first mode reson— ance at 85 cycles per second the value of T was only 1.007, however,~at'the 16th mode, at 5365 cycles per second, the -va1ue of TTincreased to 1.682. Hearmon also corrected for. moisture content variations'ifiVhislspecimens by using the relation fa = fb(l-0.004AM) (13) where fb and fa were the resonant frequencies in cycles per 26 second before and after a change in moisture content, AM, in per cent. I ‘Bair~(l96h) used-free vibration of hardboard using cantilever beam geometry in an attempt to measure the effect-of moisture content on Youngfs modulus of"elasti- city, E, and the logarithmic decrement, A.* Vibrations were caused by-lightly tapping the-beam. The resulting motion was displayed-on an oscillosc0pe screen using a method similar to that used by (Hearmon, 1958). From the frequency and the rate of decay of the wave form, Bair was able toscalculate Young's modulus and the logarithmic decrement.respectively; The logarithmic decrement, A, was calculated using the relation t> ll Sill—J 1n xO/xn (1“) where the ratio xO/xn was the initial amplitude to the amplitude n cycles later. Bair.found that a linear rela— tion existed between both E and the logarithmic decrement with respect to the moisture contents from zero up to 20 per cent.‘ For spruce and oak, Kellmann (1960) found that the logarithmic decrement generally increased with an increase in moisture-content, but not in a linear relation,up to moisture contents of 40 per cent. He noted'a large-drop ‘in‘the logarithmic decrement at moisture contents from about 6-to 10 per cent. In addition, the logarithmic decrement did not increase for increases in ‘ meisture content above 40 per cent. 27 f.In a study of rubber—like materials Nolle (19u8) extended the flexure solution for the elastic modulus equation (9), to the viscoelastic case. ~In his solution, Nolle assumed the material could be represented by a spring and.viscous element in parallel. By use of solu- tion for a similar a.c. resonant circuit and assuming small values of damping, he developed the two approximate equations for the storage modulus, E'(w) and the loss modulus,'E"(w). These relations were E'(w) K(VO2 +(AV)2/2) (15) E"(a) = KVOAV (16) where K was a constant involving the mass and dimensions of the beam as well as other constants in equation (9), V0 was the observed natural frequency, and AV was the observed resonance frequency band-width at .707 Of the resonance frequency amplitude; Horio and Qnogi (1951) developed a solution similar 5 .. .- 8 to that by Nolle by adding a damping term, n1 ——§¥— 0. 3x St to the original flexure equation given by Timoshenko, equation (8).»‘They also assumed only small values of damping in deriving the approximate Solutions. These solutions were identical with those by Nolle except that the constant, 1/2, in the equation for the storage modulus was replaced-by a smaller value, 1/8. This change is insignificant for low values of damping. 28 Bland and Lee (1955) developed an exact solution using a completely general viscoelastic law rather than the specific model used by Nolle for the determination of the viscoelastic modulus, Their solution used the ampli— tude and the stress-strain phase difference of the beam at resonance.. A specific case of the general solution was developed to cover the case of small damping and . which required only the measurement of the beam resonances for several-different lengths of the beam. Another modification of the original solution allowed for solution- with an end load on the beam. 'As a result of laboratory tests they concluded that reasonable accuracy could be obtained with Nolle's equations for small damping. THEORETICAL ANALYSIS DevelOpment of the Complex Modulus of Elasticity Equations M Before using the equations for forced beam vibration developed by Nolle (1948), it was decided to (a) develop these equations using theory of elementary vibrations and beam mechanics for a two parameter viscoelastic model and extend this solution to cover the case of free beam vibra— tions;,(b) develop a second formulation of the same equa- tions without making the usual assumptions for small damping; and (c) derive error terms for the above two sets of expressions for‘the storage and loss moduli in terms of the .measured quantities;.the frequency bandwidth ratio Aw/wnd -§nd}theqlogarithmicdecrementA. These error terms would be of benefit since the investigator can immediately deter- mine his error for higher damping when using the approximate 'eduations without actually calculating values of the complex modulus. Viscoelastic moduli for small damping.-—For this derivation the two parameter model shown in Figure l is proposed to represent the material. 29 E .flge... n Figure l. A two parameter Kelvin viscoelastic model con- taining elastic and viscous elements. Under an applied tensile stress, 0, a material rep- resented by this model would yield a constitutive equation of the form dc = __ ( o E e + n dt (17) where E is the elastic modulus of the material in psi., n is the damping constant in units of pound seconds per square inch, and e is the resultant strain in units of inches per inch. If we assume the material to be excited by a sinusoidal stress the resultant stress 0 and strain 6 may be represented by iwt _ € ei(wt—6) o = a e and e - o o where w is the frequency in radians per second, t is the elapsed time in a cycle and 6 is the phase angle between the stress 0 and the strain 8. If this stress is now applied to a Kelvin material the resultant stress- strain ratio mayébe shown to be el. 0/8 = O = E -inw (18) 60 From the above expression the real and imaginary parts of the elastic modulus may be written as the storage modulus 31 E'(w) and the loss modulus E"(w) where E'(w) = E and E"(w) = um (19) In the solution obtained by Nolle (19A8) it was assumed that E'(w) = E + nw and E"(m) = 0w (20) so that E'(w) = E +.E"(w) By using a different formulation of the complex modulus as presented in equation 19, together with the theory of vibrations and beam mechanics, one can derive a similar set of expressions similar to equations 15 and 16 developed by Nolle (19MB). ‘In using an electrical analogy Nolle (19A8) replaced mass with inductance, damping force with resistance and the spring force with the reciprocal of the capacitance. The behavior of a spring mass system having a small amount of damping which is driven by a sinusoidal force has been studied in detail and is presented in numberous vibration texts. This system as shown in Figure 2. 211/ ,/“ ./ M ——-—3 F = Fo sin wt Figure 2.--Forced vibration of a damped spring-mass system. 32 It is represented by the equation K1€ F081nwt d2(e) Q dE __ _ . ——2_dt +MR+M - M (21) where e is the displacement of the mass center at any time, t, c is the damping constant in pound seconds per inch, K1 is the Spring constant in pounds per inch, M is the mass in pound (seconds)2 per inch and w is the radian frequency. It has also been shown that for small amounts of damping and using the maximum displacement as a criteria of resonance the two following relations will be true: 2 = 2 _ 12 (0nd wn (1 2U ) (22) Aw/w 3 20' (23) where Am is the frequency bandwidth at 0.707 of the maximum amplitude, wnd is the damped natural frequency, wn is the undamped natural frequency and n' is a dimensionless term known as the damping ratio. Equation 22 may now be easily rewritten in the form of a series equation as 2=- 2 '12 Ivlo 18___ _'_. ,,1’l wn “nd (1 + 2n + An + 8n + .t nn ) (2A) If the usual assumptions for small damping are made,terms of n! highethh n the second power are ignored and equations 23 and 2A are combined, the relation that results is 2 2 z A“) 2 mm (1 + Eaggy) wnd (25) For this system wn may also be expressed by the ratio Kl/M. A similar relation also exists resulting from 33 beam theory as discussed by Timoshenko (1921, 1922, 1961), where the natural frequency of a vibrating beam, on, in radians per second, may be represented by the relation “n2 = E/H where E is the Young's modulus of the beam in flexure and H is the beam constant which is given by the relation H = KTr1 (26) where the values of K andlklare given by equations 10 and 12, respectively. Thus the constants for a spring-mass system and a beam undergoing flexural vibration at the same frequency may be related by the equation n2 =Kfi = g (27) (1) Therefore from equation 25, the storage modulus E'(w) may be represented by < l + 3—3—27) (28) E'(w) = Hm n nd d This is the same as equation 15 found by Nolle (19MB). By an argument similar to that used for equation 27, a relation may be written for the damping constant c in the form (29) CHI: 2 M where n is also a damping constant, but in units of pound seconds per square inch. 34 From equations 19 and 29 the expression for the loss modulus, E"(w), becomes H _ E (w) — Cwnd By use of the relations c/Cc = n', given by Thompson (196“), where CC is the critical damping constant in units of pound seconds per square inch, and the relations CC = 2Mcn and M = H, E"(m) = 2Hwnwnd0' (30) If n' is small, equations 23 and 30 may be combined to give the relation Aw ) n = 2 E (w) H wnd (w nd (31) Equations 28 and 31 developed in this manner are identical to equations 15 and 16 developed by Nolle (19A8). For the case of free damped vibrations of a spring- masssnmmxmlthe following two equations have been presented by Thompson (1964), which are analogous to equations 21 and 23. The free vibration relationships are wndz = wn2(l-2n'2) (32) and A = 2fln'//T_W (33) where A is the logarithmic decrement. By a completely analogous method to that used in the case for forced vibrations similar solutions for storage and loss modulus may be derived for the case of free vibrations to give the following equations: 35 E'() H 2on“) (34) w wnd 2?? E'(w) A H wndz (1?) (35) Viscoelastic moduli for large damping.—-Equations will now be derived for the determination of the visco- elastic moduli in beams subjected to forced and free vibration without the assumption thatn'>>1. However, owing to the nature of equation 22 the restriction is imposed that U':0.700. Therefore, a new equation is now derived for the case for high damping in a manner similar to that used by Thompson (196“) for equation 23 where Aw ,, EH‘2n/l+a'2 (36) and by letting A = Aw/wn equation 36 can easily be arranged into the form n'=(/_lfI—T7 -l)/2 (37) If the expression for w' is then substituted in equation 22, ”n becomes w = wnd , (38) / 2- 71 - A2 This equation cannot be solved directly for “n since A is also a function of ”n' However, since this equation is already in the form x = f(x), it can be easily solved for mm by using iteration procedures to a specified interval 36 of convergence such as 10-6, so that for all practical purposes this becomes an exact solution. Thus the storage modulus becomes E'(w) = Hwnz (38) where the value of mm is given by equation 38. The storage modulus E"(w) may now be represented by use of equations 30 and 37 as E"(w) - H wn wnd(/—l:77_?l) (39) where values of mm and A are known from equation 38 and the value of Am. For the case of free vibrations, equation 33 may be rewritten in the form 1 (A0) /hn2 A2 + l 11': From equation 32, wn can be written wnd w - 1.- n Vl-'2/(Lln"‘/A2 + if (“1) The storage modulus may now be written for free vibration as 7 2 . Hm , _ “a nd E (w) ‘ (l -EV(DnZ/AZ + l) (“2) and the loss modulus becomes E'(w) = gHwnwnd (”3) /sz7hz + l Error terms for the approximate solutions. The error resulting from using the approximate solutions may be 37 calculated directly from the equations presented in the preceding two sections. However, it was felt that a graphical presentation of the measured variables A and Aw/mnd versus per cent error would be of decided advantage over this procedure. In order to keep the.same basis of comparison between the forced and free vibration solutions, values of A and Aw/cnd were generated from the same value 0f n'. Values of n' were varied from zero to 0.700 in increments of .01. Values of A were calculated from equation 33 and values of Aw/wnd were calculated from the relation 2n1 1 +‘fi2 /1—2n2 Aw/wnd = The previous solutions for the elastic moduli were redee fined in terms of the measured variables A and Aw/wnd and the damping ratio n'- . The results of these calculations are presented graphically in Figures 3 and A. I ”T I l I I I F . 4.,__ I I I I I I IL 8: I— I .1 H :ch3.01- —1 m $4 _ - m 4.) m E 2.0L— _ x o a 82 (U I:— - E 7r'=.LAw g 2 i363. m. 8 1; L0 :-’ ._:. "g '9:- '2 :3 3? ‘1 O .7__ __ -524 ,_ _, c .GFL— -— "a e - ‘6 .5~—- —— £3 .4'- .— .p m _ _ a g) .3-—- __. '3. 5 - J 'o c... (3 2’F_ ‘__ U) m 5 r1 :3 e _ 1 I 1 I 1 l 1 1 11111 111 .I .2 .3 .§~ .5 .6 .7 True value of the damping ratio n'. Figure 3.--Variation between true values of the damping ratio and the values calculated using the approximate equations. (per cent) Error in loss and storage moduli 39 IOO , ~ ~ ~ ~ _ :_ ' I ' I II'ITI'I'I'I'I 'A ' I 'l'l‘l'l'l'l'l ' I: 5- FREE VIBRATION -,-; .— a (POSITIVE VALUES) -: .— A E L FORGED VIBRATION: P (NEGATIVE VALUES) l 20 -— a — IO 5: :5 — -I - QURVE MODULUS . __ A STORAGE _ a LOSS 2.0 — — l.0 E j 3:: L5 7— 4 - 4 t A .2 — _. J, l l I 1111 lllIlIlll I l I I 1] 11111111111] I l .2 .3 .5 L0 Ii! lOb 2() Experimental values of Aw/wnd and A. (dimensionless) Figure A.--Error resulting from using the approximate equations to calculate the loss and storage moduli. LABORATORY STUDIES To initiate a study of the physical and rheological prOperties of fresh bark and wood tissue, it was first necessary to assemble suitable testing facilities to measure these prOperTies. Also it was recognized from preliminary tests that these tissues were viscoelastic in behavior and as such would be considerably influenced-by temperature and humidity conditions. In an effort to reduce the effects of these variables a testing chamber was constructed to provide environmental control and house the instrumentation. ' Testing Chamber After preliminary calculations it was decided that a chamber six feet wide, eight feet long and seven feet high would be large enough to conduct the tests in and yet would not be too difficult in which to maintain controlled conditions with the equipment available. The frame of the- chamber was constructed of two by two inch studding covered on both sides with one-eighth inch plywood. »F1berglas:n was used as insulation in the walls, ceiling, and floor of the chamber. MO Ml An air conditioner located in the lower front corner of the chamber was used to control the temperature. A deflection duct was fitted to the air conditioner to direct the cool air toward the top of the chamber to minimize temperature gradients and to reduce air movement in the area where the samples were to be tested. Moisture was introduced by releasing low pressure steam through a horizontal 20 inch length of one-half inch pipe. The pipe had small holes drilled at one inch spacings along the top. A small piece of copper tubing was connected On the low pressure end of the manifold to drain away the water condensate. The manifold was located near the floor at the rear of the chamber to allow the steam to rise and diffuse through the air before entering the air conditioner. The controls for the temperature and humidity systems were located at the testing area in the chamber. Thus when the specified conditions were set on the thermostat and the humidistat it was assured that these conditions would be met in the immediate area where the sample was to be tested. A humidistat operated solenoid in the steam line controlled the amount of steam entering the chamber. During Operation the temperature was maintained at 72 :3 degrees F.and.the humidity was held at about 75 per cent. “2 Maximum Strength Tests As a preliminary part of the study of the mechanical properties of bark and green wood, bark specimens were subjected to a series of strength tests; In these studies ‘the maximum tensile strength of bark in the longitudinal and tangential directions and the maximum longitudinal shear strength of the cambium layer were measured. Because of its higher tensile strength, the rupture strength of green wood was not measured in this study since the forces required were beyond the capacity of the testing machine. Maximum strength of wood is more easily determined by obtaining the so called rupture modulus from a bending test. Measurement of the Maximum Tensile Strength.--For these tests bark specimens, one inch wide and ten inches long, were cut from the tree using a rectangular steel cutting frame. Then the samples were cut into necked Specimens, four inches long, using a specially prepared cutter. The specimens were loaded in tension using the Bellows Valvair pneumatic testing machine shown in Figure 5. This machine is capable of producing forces in tension and compression of about 300 pounds at constant strain rates which may be varied from near zero to about 100 inches per minute. Specimens tested using this machine were held by two specially designed vices shown ’4. a- Figure 5.--General view of testing apparatus. an in Figure 6. Each vice had about one square inch of clamping surface. The jaws were designed with horizontal serrated surfaces to reduce slipping. Specimens were loaded at a constant strain rate of about 1.5 inches per minute and the resulting force was measured by a Baldwin- Lima-Hamilton U-lB 50 pound capacity load cell and recorded by a Mosley 135 X-Y recorder. Displacement was measured using a dial gage indicator as shown in Figure 6. An event marker on the X-Y recorder was used to record the ..observed displacement intervals. Measurement of the Longitudinal Cambium Shear Strength.-- In this study the apparatus described in the preceding section was also used to measure the cambium shear strength. Specimens of bark and green wood tissue, with the cambium intact, were cut from whole limb sections which had been removed from the tree. The test specimens were about one- half inch thick, one inch wide and 5 inches long. The thickness of the woody part of the specimen was cut so as to be about equal to the thickness of the bark. This was done to prevent a moment from developing in the shear plane when a tensile force was applied. A sample prepared for testing is shown in Figure 7. Measurement of Elasticity and Damping In the measurement of elastic and damping properties of bark and green wood, freshly collected sections were 145 Figure 6.—-Necked bark specimen loaded in tension using the pnuematic testing machine. A6 \ I mile bark ‘ clamp cambium shear surface Figure 7.——Sketch of the bark and wood specimen used to determine maximum longitudinal cambium shear strength. tested as beams using elastic and viscoelastic flexure theory. These sections were about 0.2 inch deep, one inch wide and ten inches long. The thickness of the bark sections varied slightly depending on the age. To measure the effect of frequency of the applied stress, three separate testing techniques were used: static loading, free vibrations, and forced vibrations. However, before testing any bark or green wood specimens, several check specimens of aluminum, Plexiglas, and white pine wood were tested to check the accuracy Of the experimental apparatus and to gain experience in testing techniques. These check specimens were about the same size as the bark and green wood beams, but were machined to very uniform dimensions. Static Tests.--Static tests were conducted by loading the specimen as a simply supported beam as shown in Figure 8. The force was applied at a constant loading rate of 0.05 117 Figure 8.--Bark specimen supported as a simple beam subjected to bending stresses. #8 inches per minute. Owing to the low forces resulting from this type of loading, additional amplification of the load cell output was provided by using a Brush strain gage bridge amplifier. The beam support frame as shown in Figure 8 was constructed with one movable support for force-displacement ratio. The beam rested on two steel shafts which were force fitted through the supports. In addition, the shaft of the movable support was held in a ball bearing mount to prevent horizontal constraints from occurring during bending of the beams. Tests were conducted in tWO'parts:' (1) a constant strain rate loading, and (2) followed by'a stress relaxa- tion test while the Specimen was held at constant defor- mation. The time base of the recorder was used for both deformation and relaxation measurements. During the loading phase displacements were measured using a dial gage at the load cell. The observed displacements were recorded using an event marker on the X-Y recorder. Free Vibration TestS.--In the free vibration tests the Specimen was clamped at one end while the other end was free to vibrate as Shown in Figure 9. Free vibrations were initiated by lightly tapping the beam; The resulting vibrations were detected by positioning a magnetic coil under a small steel Shim which was fastened to the bottom of the beam. The resulting voltage from the coil was 149 1 " Figure 9.--Bark specimen rigidly fixed as a cantilever beam for free vibration tests. 50 amplified using a Tektronix preamplifier Shown in a rack mounting thigure 5. The resulting wave was displayed on an oscilloscope screen. The wave was photographed with an oscilloscope camera. By knowing the sweep speed the frequency of the vibration could later be calculated. Forced Vibration TestS.-—In forced vibration tests the beam was simply supported at the nodal points by two aluminum knife edges as Shown in Figure 10. The beam was excited by positioning an electromagnetic driver coil under a small metal shim fastened on one end of the beam. The resulting vibrations were detected at the opposite end of the beam by using the same technique as for the free vibration tests. 'Resonance was obtained by adjusting the driver frequency to obtain maximum displacement of the beam. 'At this time the resonance bandwidth frequency was measured by adjusting the driving frequency until an amplitude of about 0.707 of the resonance amplitude was obtained. This occurred at two frequencies, one above resonance, and one below resonance. The supports were then repositioned and the second mode frequency was obtained using the same procedure. 51 . cc“ : - ‘ I I . A...” ‘ “i .‘r’ - l\’-Illl “1;." .. . . "— . 3 8 brush' . Figure 10.——Bark Specimen supported at the nodal points for forced vibration tests. DISCUSSION OF RESULTS Effectiveness of the Testing Chamber Mechanical properties of fiberous and woody materials usually vary more with changes in moisture content than temperature. Decreases in either temperature or moisture content tend to increase strength and elasticity, however, changes with moisture content are much larger. For example, wood changes in strength and elasticity about 5 per cent for a one per cent change in moisture content. However, these same prOperties only vary about one-half per cent for a degree F. variation in temperature from a base temperature of 70 degrees F. (Wood Handbook, 1955). Thus to measure the mechanical properties of bark and green wood effectively both the temperatures and moisture contents that the specimen had while still on the tree should be maintained. Thus the function of the chamber was to maintain a constant normal room temperature and keep humidity as high as possible. During the tests the chamber temperature could be regulated successfully between 69 to 73 degrees F. How- ever, the humidity could not be raised over 75 per cent 52 53 without causing frost formation on the air conditioner. Under these conditions 10 to 15 per cent of the moisture content in the bark Specimens was lost in a two hour period. To reduce this effect to a minimum Specimens were tested immediately upon being brought from the field. Testing time for an individual Specimen was about one-half hour. It is assumed that what moisture loss took place during collection and testing increased the strength and elasticity of the bark specimens, but it is not known to what extent. Wood Specimens tested had moisture contents of about 38 per cent which is well above the fiber saturation POiNt Of 30 per cent. It has been reported (Wood Handbook, 1955) that wood strength does not begin to increase until the fiber saturation point is reached. On this basis what moisture loss that occurred in the wood during testing was probably not significant. As a result of the above experience-it is evident that humidity should be kept as high as possible to reduce such moisture loss; Certainly rapid'testing'is necessary. A better method may be to construct a small chamber of Plexiglas around the test fixture. Then high humidities could be maintained without adverse effects to the cooling system and electronic equipment kept from high humidity. It must be emphasized that bark tissue is primarily a living tissue and life processes will stop soon after collection unless additional moisture is supplied. Therefore an 5“ environment of 100 per cent humidity will not eliminate the change in mechanical properties. It may be profitable to study the effect of storage in high humidity and water environments at various tempera- tures on bark tissues for selected time intervals after collection to see how mechanical properties change. The use of the non-destructive vibration tests used in this study would be very suitable for this purpose. This type of information may also be of considerable use in esti- mating the change in properties which takes place over the time interval from collection until the test is com- pleted. Physical Structure of Bark Tissue In an attempt to study the structure and composition of bark tissues a transverse section was made of living tissue using a freezing microtome. Because of the vast difference in texture between the phloem and periderm it was not possible to make a section less than 30 microns without crushing the phloem tissue. The resulting section did not Show as much detail as desired owing to low light transmissivity. HoWever, with the aid of additional obser— vations with a compound microscope and the use of some standard references (Esau, 196“; and Schneider, 19“5) a classification of the tissue on the basis of location and probable structure was made. 55 AS illustrated in Figure 11 the bark was divided into four major radial layers of tissues for the purpose of this study. Starting with the functioning phloem at the interior and proceeding in an outward radial direction tissues appear to gradually decrease in percentage of living cells. The extreme outer tissue, the scale periderm, is almost completely composed of non—living cells. Bark tissues tested were found to have moisture contents of 110 to 1“5 per cent which is a general indication of a large percen— tage of living cells in an average cross section. This is because the living cells contribute a large amount of free moisture to tissue in the form of the cytoplasmic fluid which is about 95 per cent water. The functioning phloem composed about 10 per cent of the thickness of the section. It is shown at the top of the figure as composed of alternating rows of ray parenchyma as dark bands and sieve cells. The ray paren- chyma rays were about six to eight cells wide and can be seen to extend deep into the nonfunctioning phloem. The browning of these cell walls was caused by exposure to air after sectioning. The hollow sieve cells are the fOOd conducting elements in the bark and are shown as white dots between the ray parenchyma. The long axis of the sieve cells is in the longitudinal direction and the long axis of the ray parenchyma is in the radial direction. All cells in functioning phloem are living and character- istically have only primary walls. 56 List-95‘3" - 3 I“ ~‘. .1 ‘,#=‘.';.\ SIEVE CELLS ; .;“*'.u RAN PARENOHYMA CENTRAL AXIS 3. NEW PERIDERM 4. SCALE PERIDERM H 0.020 INCHES Figure ll.—-Transverse bark section showing location and relative thickness of the major tissue regions. 57 The major portion of the bark is composed of the non- functioning phloem region. This tissue is accumulated from layers of functioning phloem of preceding years which ceases to function at the end of the growing season. A considerable amount of consolidation and collapsing of the old tissue takes place as evidenced by the increasing slope of the ray parenchyma with respect to the radial direction. For example in the nonfunctioning phloem, the rays are oriented at an angle of about “5 degrees to the radial axis. This indicates that deformation of at least half the original. thickness has taken place over a period of years. Under this loading many of the sieve cells eventually collapse and parenchyma cells enlarge and fill the void, (Esau, 1960). In a related study of Sweet Cherry (Prunus arium) phloem, Schneider (19“5) found that secondary fibers develop between the rays and intermingle between the conducting tissue of the preceding year. He observed the cells between the rays were a combination of partially collapsed sieve cells, fibers, cavities and parenchyma cells. This appears to be the case also with the Specimen under study. Large fissures are also found along the rays as observed by Schneider. Therefore it is assumed that the living cells in this tissue are com— posed of ray parenchyma between the rays. The Sieve cells and fibers are essentially nonliving. The general appear— ance of this tissue is soft and fibrous with a spongelike 58 response when pressed in the radial direction. This helps the bark cushion and protect the functioning phloem. The outer 25 per cent of the bark is composed of two periderm tissues, the new periderm and an outer scale periderm. In general, the major function of this tissue is to regulate the rate of gaseous exchange and to prevent moisture loss. To form this type of moisture barrier periderm cells are tightly packed and have walls formed from layers of suberins and waxes. Under the light micro- scope the tightly packed nature of these cells was observed. This is shown in Figurelfl.by the dark areas. During collection of the specimens, July 8, 1965, the outer periderm was loosely attached to the bark section. Its removal exposed a living periderm, green in color, under- neath. In mature periderms, the walls become layered with waxes and the cells become isolated from the living tissue and die. It is assumed that this is the case for the scale periderm in this section, with the exception of a few layers of cells on the inner surface which may have been living since they browned some after detachment. After oven drying the scale periderm did not become stiff and brittle as did the other bark tissues but retained much of its original flexibility. This could be explained by the waxy structure of the cell walls. The other tissues having pectic and other compounds in the walls stiffened with drying. However, in the periderm, the cellulose mesh 59 of the primary cell walls is filled with wax and suberin. Suberin and wax are both fatty substances. Because of their chemical nature these substances did not become hardened from the oven drying. Under closer examination, it appeared that the peri- derm cells had their long axis in the tangential direction. In woody tissue maximum strength lies in the direction of the long axes of the cells. Therefore it is probable that the maximum strength of the periderm is in the tangential direction. Strength of Bark AS a further step in the preliminary investigations the maximum tensile strength and maximum cambium shear strength of bark were studied. The purpose of these tests was to obtain some initial information about the response of the material under load and to compare this information to data reported by Adrian and Fridley (1963) and Brown (1965). Mechanical strength of bark.--For the maximum strength tests specimens were cut into necked sections as shown in Figure 12 and subjected to tensile loadings at a constant strain rate. In all tests performed failure occurred in the necked section. Stresses were calculated on the basis of the unloaded area of the necked cross section. A typical failure curve for a longitudinal bark section is shown in Figure 13. As loading is increased 60 600- LONCITIDML SECTION 500 - 400~ llll' :2 O ~I—I 4.) O (D U) 'U (D XA 0 £2 (1) O C1 52 H CI—I O 33 8‘3 SREOINEN UNDER 53 3‘ 300. LOAO I—I S-I Cd 0) C Q. '1 .,—I b0 U) . "—1 'U ‘5 § ONE INCH 0 200‘ C.‘ Q. 0v U i (l) :3 e TANGENTIAL g '°°* SECTION (D 3 II l U) o t V —T V O .04 .08 .12 .14 Displacement of the Testing Crossheads (inches) Figure l2.--Failure of a longitudinal bark specimen subjected to tensile loading. 61 the curve becomes more non—linear until failure of the specimen. The initial non-linear portion of the curve is presumed to be due to some stress relaxation of the bark under load. In the final stages of loading the 31Ope of the curve decreases more rapidly. During this phase actual failure is taking place by incremental failure of fibers and tissues accompanied by sliding and continued failure in the tissue. This is also evident from the jagged failure surface as Shown in Figure 13. The new periderm failed in a smooth surface presumably along the cell boundaries. If an outer scale periderm was present it usually failed about one—half the total elongation. The maximum displace— ments at failure are not particularly meaningful for this type of cross section except for comparison to failure of tissues at other orientations or failures of the periderm The displacements were corrected for deflections of the load cell relief spring and an attempt was made to determine if slip had occurred under load. It is obvious that any Slip while loading could have a large influence on any loading curve. Two visual observations were made, one by observing movement of a line made on the Specimen at the edge of the clamp before loading and secondly by observing the condition of the clamped surface after failure had occurred. Some slight movement with respect to the lime on the specimen took place during loading. However it is difficult to determine whether this was due 62 Figure l3.—-Failure of a longitudinal bark specimen subjected to tensile loading. 63 to some Slight slip or non recoverable strain produced during loading. Examination of the clamp surface revealed distinct impressions of the serrated surface of the clamp indicating Slip had not occurred. In the longi- tudinal direction the strength of the periderm is about 1.5 times that of the bark. However, the periderm fails before the bark. Thus the periderm contributes some initial strength to the bark but soon fails and contri— butes no strength when the bark reaches the rupture point. Tangential sections loaded in the same manner failed at much lower stresses. From the smooth failure surface it is suspected that failure occurred mainly along the cell boundaries as shown in Figure 1“. This is because in the longitudinal sections cells had their long axis in the longitudinal direction. However in tangential sections the force was applied at a right angle to the long axis of the cell. Further there is no known reason to suspect that the cell wall strength is less in the direction of its Short axis than in the direction of its long axis. Therefore the difference in failure strength must be attributed to failure of cell boundaries rather than the cell wall itself. If the above argument is true the strengths listed in Table 2 are comparisons of longitudinal rupture strength to tangential cell boundary strength, the former being about nine times greater. 6“ Figure l“.-—Failure of a tangential bark specimen subjected to tensile loading. 65 The failure curve for the tangential section was initially curved upward. This type of response indicates a material which increases in stiffness as deformation increases. It is proposed that this is not the case and the response can be explained on the basis of the physical structure of the material. In the natural state tangential sections are on an are about the tree limb. Therefore when first loaded, the specimens first go through a period of reorientation and realignment in the direction of the applied force. When a higher level of stress is reached the cells must also start to deform physically. Normally deformation of the cellular tissue Should produce a stress curve of decreasing Slope. It is thought that a combination or reorientation and deformation accounts for the linear appearance of the last part of the failure curve. Failure stresses are in a ratio of about nine to one. Adrian and Fridley (1963) reported a ratio of about four to one for prune trees. However, there is little basis of comparison owing to the difference of Species and possibly moisture content. Since the scale periderm was readily detachable it was tested separately in longitudinal and tangential directions. The directional strength properties of this tissue were opposite to that of the bark tested in the preceding section. In these tests clear specimens without cracks or lenticels were selected. Longitudinal sections 66 TABLE 2.-—U1timate stress and elongation for necked bark sections under tensile loading. R = 1.5 inches per minute. Tensile Sample Stress Elongation Thickness Number (psi.) (inches) (inches) Longitudinal Section L1 820 . —‘ .155 L2 678 .151 .158 L3 “9“ .1“6 .125 L“ 730 .212 .363 L5 563 .181 .358 Average stress 657 psi Tangential Section T1 12“ .120 .165 T2 82.7 .108 .150 T3 58.2 .10“ .125 T“ ““.6 .11“ .230 T5 “2.3 .113 .190 Average stress 70.“ psi 67 failed soon after loading along smooth horizontal failure planes. It is suggested that failure took place along the cell boundaries. From a microscopic examination of the section in FigurelJ.it appeared that the long axis of the periderm cells was in the tangential direction. If this is true, the highest strength of the periderm should be in the tangential direction by a similar argument as presented for the bark. In fact tensile strength in the tangential direction was about three times that of the longitudinal direction. The tangential sections had a non—linear loading curve Similar to that for bark in longitudinal sections. However the periderm tissue did not fail upon reaching the maximum strength but was then capable of large plastic deformations at the same level of applied force. During crass extension of the area the plastic section became considerably less. This means that the true stress at rupture was much larger than that calculated on the basis of the original area. From this behavior under loading some theories of reaction at the cellular level are advanced. The initial part of the curve up to the plastic region is typically nonlinear owing to stress relaxation under small deformation. When the "yield stress" is reached plastic flow takes place over larger deformations. These cells are dead with thin walls encrusted with waxes which possibly act as a lubricant and allow for some slipping between the cells during large deformation of the cells. 68 TABLE 3}--Ultimate stress of periderm sections under tensile loading. R = 1.5 inches per minute. Longitudinal Sections Tangential Sections Sample" Stress Thickness Sample Stress Thickness Number (psi) (inches) Number (psi) (inches) L6 12“0 .023 T6 3520 .018 L7 1058 .0223 T7 3“30 .020 L8 1035 .020 rP8 2780 .020 L9 8“5 .020 L10 615 .021 Average 958.6 psi Average 32“3.3 psi In general the periderm elongations were as much as one inch or about “0 per cent of the original length. Thus from Table 3 it is evident in the longitudinal direction that the periderm is capable of extensions up to 5 times greater than the bark extension before rupturing. If the elastic modulus of the periderm and the bark were the same in the tangential direction, on the basis of the relative thickness, the periderm would support about 1/20 of the total force. They would of course have the same stress. However, since the periderm is capable of supporting at least 5 times as much stress as the bark, the periderm may support one-fourth or more of the total 69 load. In addition, Since the periderm is capable of greater extensions, it serves to help resist rupture of the bark in the tangential direction. Therefore, when the bark and periderm are subjected to large tangential elongations, such as produced in torsion, it would be possible for the bark underneath the periderm to rupture, but give no visible evidence of rupture, Since the outer periderm would still be intact. Longitudinal cambium Shear strength.--In these tests the specimens of bark and green wood were prepared and loaded as Shown in Figure 7. In preparing the samples an attempt was made to minimize the effect of moments acting at right angles to the shear plane by cutting the wood to the same thickness as the bark. The effect of a moment would lower the value of the calculated Shear force. It must also be assumed that some misalignment and prestressing occurred while clamping the Specimen. The average Shear stress for Montmorency cherry bark as determined by this test was the average of the stresses listed in Table “. This test might be improved by supporting the clamps by knife edges to correct alignment during loading. This data could not be compared to that of Brown (1965) since he used a normal (radial) pressure on the section when measuring Shear stress. 70 TABLE “.—-Maximum longitudinal cambium shear strength. Shear Stress (psi.) 33- 27. 30. 31. 25. 35. 27. 27. 26. )-‘\O)—‘\1\1CDU7CI)U7 Average stress 29.6 psi Elastic and Viscoelastic Response of Bark and Green ' Wood to Frequency of the Applied Stress All Specimens tested including the aluminum showed some viscoelastic behavior. All materials with the excep- tion of aluminum exhibited a detectable stress relaxation and all demonstrated a change in the complex modulus of elasticity with changes in frequency of the applied stress. Static Tests.——In these tests all materials tested were supported as Simple beams and center loaded at a constant rate of deformation. The bending stress-displace- ment relation for all materials was visibly non—linear With the exception of the aluminum. The amount of non-linearity of the relation depended on the amount of damping in the ma- terial. Bark for example, had the largest amount of curvature and in later‘tests bark was also found to have the largest 71 value of the dimensionless damping ratio n'. A typical loading curve for bark is Shown in Figure 15. Values of stress were calculated from experimental data using equation 1. An equation to represent this type of loading behavior has been proposed previously by Ferry (1960) and was pre- sented as equation 3 in this Thesis. In this equation the amount of stress at any time t is reduced by an amount exp (-tH(r)/n) where H(T) is related to the elasticity of material and n is the damping constant in pound (second)2 per square inch. Thus for larger values of n the curvature of the stress curve becomes more pronounced as time increases. No attempt was made to determine values of n from the loading curves. To do this would also require determination of H(T) for various values of strain rates, R. Experiments of this type are reserved for future research. However, equation 1 was used indirectly in calcu- lating the relaxation modulus E(t) and extending the loading curve as shown in Figure 15 using equations 3 and “ as proposed by Ferry (1960) and Chang (196“). By use of equa- tion “ a frequency f was calculated for the storage modulus. For example f for the tangent modulus (relaxation modulus, E(t); storage modulus, E'(1/t))in Figure 15 was calculated by the relation t = l/2nt cycles per second. Values of the secant modulus (tangent modulus) were calculated using equation 2. These moduli are listed later in this section in Tables 5 and 7 and are discussed in comparison with 72 .mm soEHoch xsmn mom wcmmomum mcficson mo coapmxmfims use wcflUMQQII.mH omswfim Amcsoocmv oEHp pccefismdxm 6+0. 0— _ QR. 00¢ can 08 co. _ o A q — u — a I— . — — 4 fl 1 —— u d— d o _ ob\\ . e. ...te..e7<.:.e. I 2253mm 9: aim: «:6 . _ \ .,. 225.233 22 2:93.. 0 . \ L 22.... 35.533 335, _\\ 1 Lb .mu=._<> $35 2223233 0. . . . o .335, e .3 . \ 6 «much... 2 3 o mnuwmma \ , I \ .A ../ . , 1 > are . - u u \ 1 o \ «3:3... .. pandas» . 5L5 1 (qout euenbs Jed spunod) ssauis Surpueq SBBJSAV 73 moduli obtained at higher frequencies of applied stress. By use of the equation (Ge) suggested t =A =(C’e'c’r)t = to + A by Chang (196“), the loading curve for bark shown in Figure 15 was extended from the loading and relaxation data. In this relation 0e and or are the loading and relaxation stresses respectively at times A and to + A respectively. Stress relaxation curves for representative samples of the materials tested in this section are shown in Figure 16. Again the amount of non—linearity in these curves depended on the amount of viscosity in the material represented by the damping constant 0. The relation between R and the relaxation stress, Or’ in a material loaded at a constant rate of elongation R to time tO was proposed by Chang (196“). This relation is o = R/1H(-I) exp(-t/T-)[exp(tO/T:—1] (““) r where T is defined as the ratio of the damping constant n to the elastic spectrum H(T). In this relation the term exp (-t/r) again accounts for changes in stress depending on the value of n and t. This means that for the same amount of stress applied in extension the bark has the ability to relax under load at a higher rate than the wood. This is an indication that the bark could undergo larger deformations than wood before rupturing. In the actual bending of a tree limb for example, both the outer wood fibers and the bark undergo 7“ .Uouwop mfimfipcpme oEow pom mozam> Hmflpflcfl osmm one op pcospon mommcspm wcfioson mo coepmxmacs zsmefimeI.mH oszwflm Amocooonv oEHB com com 03 on. no. on o d u — d _ _ _ W w I1 00. b. o n O 8 l D. . mm m m. I] o~;mwm o . s d m. 11 9 J r]. o O S U . l enamel... no“ 0 J n 8 D. O I. I1 Tru o w m. 0 ms ton on x53 0 . . 1 cm. m. be... no coo; zummo o m . . M... new ~ n. 9.6.3.... m P . I s 3... m3 m2... MEI; 4 IO I ./ .9. $me zuzamdm 4332: m>mao < oo~ 23232“! 442.050 75 essentially the same elongation. If both the bark and wood had the same elastic modulus Similar stresses would be produced in both tissues and the above argument would apply to failure. However, since the elastic modulus of bark is only about 1/50 of that of wood, it means that the bark is stressed only 1/50 of the amount of wood for the same elongation. Thus in all probability in Simple bending the wood would fail first. Bark injury would occur as a result of the clamp slipping in the longitudinal direction. This is a result of the stiffness ratio between the bark and wood. Failure in this case results from large elongations being applied to the bark. The wood being much stiffer resists being elongated and shear failure occurs at the cambium layer. Sonic tests.--0f the two testing techniques used, the free vibration technique using cantilever beam geometry was superior to the forced vibration method. The free vibration method was Simpler, required the least amount of instrumentation and produced the most consistent results. The logarithmic decrement A and the damping ratio n' were calculated from data obtained from the wave form in Figure 17. The decay of successive amplitudes of this wave as plotted in Figure 18 gives a curve whose Slope is equal to the logarithmic decrement. II... I'll-I... "WI-l.- IIII‘III' it”. .~ INIIIW “mum! IIIIII'IIII I‘ll-ll Figure l7.——Damped oscillations at the free end of a canti— lever bark specimen. .77 ME. MATERIA W —- A ALUMINUM A2 0 WHITE PINE W2 ' 3 :2 3° “— ° - E BARK as 'N: t . .\:‘ ’3 ' ' : g;ZCI-— . . . . A H 0 : ° 0 O . o 'H g 0 Q U) y— 0 C1 (1) o O N IO— 0 o . e — C . i0 9:— , D o . 8 9': ° ° ° 7 -_ O .3 - E . . 1'3 6— - S-I _ . 8 5+- . :3 .p I— H O a 4 —- s q: " O 3— 2 _. l 1 I 1 l I l 1 I 0 5 l0 IS 20) n = Number of successive cycles Figure l8.—-Aplitude reduction of successive cycles at the free end of cantilever specimens. 78 Values of n' were calculated using the approximate A relation n' = 5?. It was not necessary to use equation “0 Since n' was small. The highest value of 0' was only 0.08“. From Figures 3 and “ and Table l in the Appendix it is evident that the error resulting from using this approximation is negligible. A least squares linear curve fitting technique in fitting a straight line to the data was not used since it was not felt this would add significantly to the accuracy of the experiment. All bark Specimens tested using this technique had nearly the same values of A which is shown indirectly by the calculated values of n' in Table 5. Of course the use of the logar— ithmic decrement assumes an exponential decay of amplitude. This is Shown to be a good approximation for most materials tested. The green wood and to a greater extent the bark Showed some curvature in the logarithmic decrement curve. This error becomes serious for high values of damping and requires the use of approximation procedures as described in the Theoretical Analysis section of this Thesis. Frequencies for these tests ranged from 7.6 to 117 cps depending on-the material. Notice the curves for materials in Figures 16 and 18 fall in the same order. Both these sets of curves are measures of internal damping in the material. Steeper slopes correspond to a relatively higher energy loss per cycle in the material. For a meaningful comparison the values in Figure 16 whould be compared only 79 to initial Slope values in Figure 18, thus using a similar value of experimental time. Notice after one second the relaxation rate of Plexiglas has reduced sufficiently to approach that of green wood. The forced vibration tests were originally intended as a check on the other methods, however, this test was also useful in obtaining information at higher frequencies of applied stress. In these tests a sub—harmonic resonance of order “ of the first mode was found to exist in the Specimens for a driving frequency of half the first mode frequency. This was an aid in correctly determining the first mode frequency since the period of the sub-harmonic wave was easily recognized being twice that of the driving wave as Shown in Figures 19 and 20. The forced vibration method has the added advantage of determining the behavior of the material at still higher frequencies by using higher modes, (Figure 21). However, in practice modes higher than the third were difficult to detect since the resonance amplitude decreased for each higher mode and resonance could not be separated easily from the background noise present in the system. The only satisfactory third mode resonance obtained for the materials tested was for one of the white pine specimens at 1357 cycles per second as shown in Figure 21. This was also the only case in 80 rvvr~rrrr' ' ' . - o o o v v 9 -40Ic Figure l9.-—Forced vibration of a simply supported specimen showing: top, driving wave at half the first mode frequency; bottom, resulting oscillation of the beam (fourth order sub— harmonic of the first mode frequency). 00 H I t I I 6 A 6 § 0 A ‘ ‘. ‘ O v4¢AoM4¢JW“CA-bttt U ' .— Figure 20.——Forced vibration resonance at the first mode for a simply supported specimen showing: top, driving wave; bottom, resultant vibration of beam. 82 o A..4«4H.0‘6£.LA.I Figure 21.--Forced vibration resonance for the third mode showing: top, driving wage; bottom, resultant vibration of beam. 83 which it was necessary to apply the Goens' correction factor Tn‘ In this case the value of E was adjusted by multipli— cation with a value of T of 1.015“ calculated from equation 12. It is also possible to change resonance frequencies for the same Specimen in both free vibration and forced vibration tests by altering the physical dimensions of the beam. Some difficulty was encountered in using the forced vibration method with Specimens of low stiffness such as bark and specimens of high stiffness such as aluminum. Except for two bark specimens, B3 and B“, which were thicker than the others and consequently had a higher resonance frequency the first mode of the bark specimens was below the lower frequency limit of the oscillator. This was not realized at the time of testing and erroneous values of the elastic modulus were calculated which were about twice the correct value. However, since all modal fre- quencies are related by fixed constants which may be cal- culated from elastic theory by assuming a more probable value of E the first two modal frequencies were calculated. Upon comparison with the erroneous data it was apparent that the sub-harmonics of the second mode was mistakenly used for the first mode frequency. This points out a serious danger in this type of testing of incorrectly selecting the wrong mode or a sub-harmonic for the correct 8“ modal frequency. Since the modulus is a function of the square of the frequency this can lead to large errors. If the approximate value of the elastic modulus is known, the approximate value of the modal frequency can be calculated before testing. However if the modulus changes rapidly over a few decades of frequency such as in the bark tissue, this technique is not as helpful. With the aluminum samples a difficulty of a different nature was encountered. Due to the high stiffness of these beams the power required to excite resonance was near the power limitations of the driver. However, this method was very suitable for the other materials tested which were in the intermediate stiffness range such as green wood, white pine, and Plexi- glas specimens. Values of the damping ratio n' were calculated using the frequency bandwidth of the resonance curve and equation 23. This method was not found to be highly accurate Since measurements must be made by visual refer« ence on an oscilloscope screen. Also it was somewhat difficult to adjust the oscillator to the exact frequency to determine the sideband resonance points. Materials having relatively high damping such as the bark had a wide bandwidth extending over 10 to 15 cps. Materials with lower damping such as the white pine had a narrow bandwidth of two to three cps. Since n' is directly prOportional 85 to the square of the bandwidth, the chance for error increases greatly for materials having a narrow band width. Therefore values of n' listed in Table 5 for forced vibra- tions are not to be considered highly accurate for aluminum and white pine. Some of the common engineering constants determined as a result of this investigation for the materials tested are listed in Table 5. Notice that the values of Young's modulus of elasticity are not constant but become larger with increases in frequency of the applied stress. Values of the elastic modulus for some engineering materials such as concrete and wood were obtained previously, but only for a fixed frequency. In these references the value obtained was called the sonic modulus. However, these authors apparently did not recognize that due to the presence of some viscoelastic effect in all these materials that the modulus actually varies with frequency as Shown in Table 5. Values of elastic modulus in this table are in fact only estimates of the true complex modulus E* which is equal to El + iE2. Thus only in cases of wood and aluminum where E2 is small are these values approximately correct. The modulus of elasticity for bark found in these tests was very low compared to green wood and ranged from about 10,000 to “0,000 psi. over the frequency range used. Modulus of elasticity of green wood ranged from about 500,000 to 700,000 psi. 863 mfipsocfi cofiumpos cam smocm Lou pepoopsoO I ms Lo>fisp do muHEfiH Lozoq cocoooxo I * LopmHHHowo no oncodmoe mocczccsu soaop mocmcommp Hmucmempcsu I + Umhjmmmc. 90C l .E.C candofiadam poc I .m.: * ooo.awz.m man once. ooo.mme.d m.mHH ooo.oee.d mm o.w©H me .SoHHa. .e.c .e.: .e.: .e.c .s.: .e.c ooe.mam.d mm o.moe Ha Enszaae .e.: eeo.HHm.H mom .e.: .e.c .e.c oeo.:eH.H no e.:m m3 . xeooo.mem.e shed heed .e.c eoo.wmm.H Hon mac. ooo.eeH.H mad coo.m:H.H me e.mm m3 mesz .e.c ooo.mme.e om: moo. ooo.mofi.e m.MHH oeo.eeo.H so o.em H3 awe. oew.:ms mom Hao. ooe.sdm m.h: ooe.mas me o.me ma oeaoexmed Hmo. gee.mae new one. ooe.msm m.m: ooa.wm: He o.me ea c . see. o:w.o:s cam ego. eeb.mem m.me oew.mmm abs m.mm No S .sweeeo due. eem.ehe esm has. eme.mae e.eh eee.ee: ems a.am He eeeeoeheoz. , . coo; 2mMmo + + + oxo. omw.dm m:.e ems.fim w.wb mm + oe>.:m mm mho. asa.em am Hes. 0mm.am mm.m emm.a mmm s.oe an .seebeo + eho.m: mm SechOESeox. mac. omm.m: am 003. aea.sm ..e eem.ma new m.:e mm mammHe amen + . + + see. osm.mm x.m eme.HH mam m.me mm + + + wee. omo..m b.s omo.me arm d.ms He macacofimcmefia .qu .wao mmoflcoflmcoefio .Hma .mao .Hmd .omw mpp\pa .% .e . m e .e a e massed: mafia seaweed m oHpmm mafispoz zocosconm ofiumm mSHspoz mocmsvmpm ucmoom mcfipmoq I wcfiaeca ofipmmflm wcfidewm ofipwwflm w U .EQH mo. u cums :ofipmsnfi> pcosom cofiuwsofi> comm beenSmIHmase Iona uzbzmeefip Lee Scent» m.masvme .mmcpum Umwflqow do wmfioccSU Locoo meow ocm gems soc oflpms mcHQEmc pcm zpfiOMSmmHm co nsezeoe.nmcsosuu.m mamep 87 over a similar range of frequencies. However, as a result of the large increase of the elastic modulus for bark as compared to the green wood in this frequency range the stiffness ratio between bark and wood decreases from about 50 to one at quasi-static conditions to about 15 to one at frequencies between 100 to 200 cycles per second. The behavior of these materials is discussed in detail in the next section using the more applicable theories of viscoelasticity. However, it is apparent that owing to the much higher stiffness of wood as compared to the bark that the properties of the wood largely determine the response of the limb to applied stresses. When a shaker clamp is attached to the limb the bark is compressed between two media which are much stiffer than the bark. Reduction in bruising would of course result from using a shaker pad which has less stiffness than the bark. The internal damping of bark was about five times higher than that of green wood. This ratio remained at about the same value for the two comparisons made at different frequencies for the free and the forced vibration tests. Thus in free vibration the bark has a damping effect on free vibrations of the limb. Since the bark is at the maximum distance from the neutral axis a thin layer of bark is in a position where it can produce the maximum possible damping effect. 88 The elastic parameters discussed above for bark and green wood can not be compared to existing values Since they are not available in literature. However, comparisons may be made for the secant modulus (Youngs' modulus) of the other materials to existing published values. These values are averages and dependent on the exact structure and composition of the individual material. Values used for this comparison were Plexiglas- “00,000 to 600,000 pounds per square inch (Hodgman, 1952), white pine - 1,170,000 pounds per square inch (USDA Wood Handbook, 1955), and aluminum - 9.3 - 10.6 x 10.6 pounds per square inch, (Baumeister, 1958). Values of Plexiglas are given as a range of values Since the Slope of the stress—strain curve changes during loading;va1ues of aluminum are variable depending on the per cent of alloys present. These values compare very well with those obtained in Table 5 with the exception of the aluminum for which a value of about 9,200,000 was obtained for Youngs' modulus. Since the composition of these aluminum Specimens was unknown, an additional beam of 6061—T6 aluminum was tested. Using the test equipment a value for the elastic modulus of 10,0“0,000 was obtained which is the correct value for this material. As a result of this it was suspected that the low value for the other aluminum beams was due to differences in composition and residual stresses resulting from machining. 89 The density of bark was quite high and Showed a large variation from 68 to 78 pounds per cubic foot. Bark was about twice as heavy as green wood. However, after drying several weeks in an oven bark densities stabilized at 31 to 32 pounds per cubic foot based on the original dimensions. The close agreement in dry densities suggests that the variation in wet densities was due to differences in moisture content and not differences in structural density of the bark. It is of interest to find the same structural density in these bark specimens since they were collected at random from several different cherry trees. The thickness of bark ranged from .098 to .161 inches. Since in cherry bark the old phloem from the preceding year is not severely crushed but becomes part of the non functioning phloem and the outer periderm scales off yearly, increases in thickness are directly related to the age of the bark. The resulting indication is that mechanical properties of cherry bark would not be expected to have much dependence on age. In fact no effect of thickness was apparent on values of Youngs'modulus for bark. Dry weight density of the green cherry wood stabilized at about 26 pounds per cubic foot. The differences of dry weight densities in bark and wood is explained on the basis of the large amount of hollow conducting cells in wood as compared to a high percentage of smaller living cells in the bark having cellular conclusions. 90 Viscoelastic considerationS.--At best theory of elasticity can only be used to calculate general trends in material constants but cannot be used effectively to explain behavior of materials which are highly viscoelastic. For this reason the data in Table 5 is recalculated in terms of viscoelastic components and presented in Table 7 and Figure 22. In this manner it was possible to separate and examine separately elastic and viscous effects by the use of the components of the complex modulus; E'(w), the storage modulus and E"(w), the loss modulus. From the use of the relation E'(w) = E(l/t) described earlier, values of the storage modulus were calculated for very low frequen- cies using the "static" loading data. These values are listed in the third column of Table 7 and shown in Figure 22. These values are Significantly lower than values obtained at higher frequencies particularly for bark. Plexiglas was the only one of the three check materials used in this test whose viscoelastic properties over a wide range of frequencies was known. For this reason a detailed comparison of results of this study are made before discussing the behavior of the other materials. Mechanically, Plexiglas is classed as an uncross-linked amorphous polymer. Physically this means that Plexiglas does not display instantaneous elasticity on loading and that E(t) tends to zero for large values of time. Ferry (1960) describes this material as responding to external 91 ALUMINUM SPECIMEN A 2 o- —————— ----oe WHITE PINE SPECIMEN W3 0. __________ ____,o.e—-O- { GREEN WOOD SPEGINENS GI AND 62 5.8 .9 O-——_———-¢-‘“’f 5.6 O" PLEXIGLAS SPECIMENS PI AND P! ‘(pounds per square inch) ' 3 J 5.3 3 5.6 EL.) . S BARK SPECIMENS B3 AND B4 3%? E "I 46 _t_ O 6 0 4.4 I. ,/ 0 L ”/ 4,2 :. // o- STATIO VALUES E- o ,z’ 0- FREE VIBRATION 4'0 E: ’o e- FORCED VIBRATION, IST NODE t. O— FORCED VIBRATION, 2 ND NODE L 1 I, l I 1 I 1 I 1 I 1 I 1 I, L I 1 I I I I I0‘4 Io‘z IO° IO2 10“ Frequency of the applied stress (cycles per second) — __I Figure 22.--Storage modulus for some of the materials tested plotted logarithmically against frequency of the applied stress. 92 stress by local adjustments at the molecular level through entanglement connections between the molecules. As shown in Figure 22 the storage modulus for Plexiglas increases gradually an amount about 50 per cent of its original value over about five decades of frequency. Since E(t) is nearly the mirror image of the storage modulus one would therefore expect the relaxation modulus to decrease Slowly over several decades of time. This is in fact shown to be the case in relaxation curves presented by Ferry for Plexiglas. Fortunately a rather complete set of dynamic data by Koppelmann (1958) was available to compare to data obtained in this study. Koppelmann used a non-resonant technique of testing in obtaining this data as described in the Related Literature section of this Thesis. Values of E'(w) for Plexiglas as shown in Figure 22 are in very good agreement with those by Koppelmann. Note that this material has a distinct inflection point in the storage modulus curve near 10 cycles per second. KOppelmann found that the loss tangent of this material increases gradually from low values of about .0“5 at both ends of the frequency spectrum to a maximum value of about .10 at “5 cycles per second. These values are compared with those found in this experiment in Table 6. 93 TABLE 6.--Comparison of experimental values of loss tangent for Plexiglas with those by Koppelmann (1958). frequency (cps) 10‘3 “5 200 103 Koppelmann .038 .10 .065 .0“8 Experimental - .077—.082 .060-.068 Values Ferry (1960) commented on these values of loss tangent as being very good with the exception of the maximum value. He felt that on the basis of more recent studies the actual peak value may be about .08 to .09. Experimental values of storage modulus and loss tangent of Plexiglas for several different frequencies were identical to published values within the limits of experimental error. This also served as an additional check on the theory used since the published values were found using a non-resonant technique. In general, Young's modulus and the storage modulus were very similar in numerical value. The storage modulus was slightly larger, however, depending on the amount of damping present in the system. The storage modulus was mOre sensitive to changes in the frequency of the applied stress than other viscoelastic parameters measured for all materials tested. The storage modulus increased with increases in frequency. In addition, the storage modulus 9“ increased much faster over a given frequency range for materials having higher loss tangents such as bark. For example, the storage modulus of bark having a loss tangent of about .15, increased from about 10,000 pounds per square inch at 10-3 cycles per second to over “0,000 pounds per square inch at about 30 cycles per second. The storage modulus for green wood with a loss tangent of about 0.0“ only increased from about 500,000 pounds per square inch to 700,000 pounds per square inch over a slightly larger frequency range. The loss tangent is a ratio of the loss modulus to the storage modulus. Thus the loss tangent is directly proportional to the energy loss owing to heat and internal friction at a given frequency. This is a direct non-dimensional ratio of the amount of damping in the material. At medium frequencies, in the free vibration range, this ratio for the bark is about four times larger than that for the green wood. The amount of energy lost in a cycle may also be compared on the basis of the loss tangent 6 by the relations AW w where W is the work done on the specimen. Therefore for Energy Loss Factor = = 2n Tan<5= 2A = “nn' bark the energy recovered and lost in a half cycle was about 55 and “5 per cent respectively and about 88 and 12 per cent respectively for green wood. Thus for a given stress level bark absorbs about four times as much energy as wood. However, in actual bending of a tree limb the 95 stress levels of the bark and wood are not the same, but are directly dependent on the respective moduli of elasti— city and the distance from the neutral bending axis. At the outer fibers of the wood the elongations for the wood and bark would be about the same. In this case stresses produced in the bark and wood would be directly proportional to the vector sum of the loss and storage moduli. In actual values the storage modulus of the bark is quite small being only 1/50 to 1/15 of that of wood. However, owing to the large difference in the loss tangents ; the loss modulus of bark is almost 1/3 that of wood depending on the frequency as shown in Table 7. The loss tangent and the storage modulus were also found to be somewhat sensitive to frequency. However, the direction of the change did not depend on the direction of change frequency, but was apparently dependent on the type of material being tested. For example, for an increase in frequency the storage modulus in green wood and white pine increased while in bark it remained constant and in Plexi- gglas it actually decreased. However, Koppelmann (1958) :found that for frequencies below “5 cycleS‘per second the :Storage modulus and the loss tangent increase with :increases in frequency. 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"re-,1. 12 «ML; e. L 1 ... . 1 .. . ., 1 T1,... ...,...._-J.M.(.;1I. H.713 101x1~3m {E .nll .11. .............L 97 In terms of models Plexiglas below 10 cycles per second behaves like a Kelvin material (spring and dashpot in parallel) where the loss tangent equals Tm and a Maxwell model above 10 cycles per second where the loss tangent equals l/Tw. The response of these two models could be replaced by a Standard Solid model consisting of a spring and Maxwell element in parallel. By adjusting the constants of this model to have a peak value for loss tangent of about .10 at 45 cycles per second, a better approximation for this material may be made. The loss tangent could be even better by making I = 1(m) in the Standard Solid. In the data available for the green wood the loss modulus increases with frequency. This gives the impression that the viscous mechanism is forced to undergo displacement which produces a correspondingly large dissipation energy. This behavior is similar to that of the Kelvin model. Behavior of this type is also somewhat similar to that of cross linked polymers. In these cases the loss tangent reaches peak values at frequencies of about 105 cycles per second. The behavior of the bark was in-direct contrast to that of the green wood. The loss modulus of the bark was relatively flat over the frequency range and was relatively large compared to the storage modulus. The low elasticity and high loss modulus of bark is in keeping with one of its main functions which is acting as a shock absorbing protective 98 tissue for the tree. However, as shown in Figure 22 the storage modulus of bark increases spectacularly over the values at low frequencies. Therefore the bark becomes much stiffer and more subject to damage from forces applied at higher frequencies. This characteristic large increase in storage modulus with lesser increases in loss modulus, in bark tissue is similar to the behavior of polyvinyl acetate. Ferry (1961) describes this polymer as an amorphous polymer of high molecular weight. Over a similar frequency range the storage modulus for this material follows an "S" shaped curve such as suggested in this case for bark in Figure 22. Surprisingly the storage modulus was most affected by changes in moisture content of the bark Specimens. Ideally in_aviscoelastic system represented by springs and dashpots one would expect little change in the storage modulus and an increase in the storage modulus for a decrease in moisture content. Actually the behavior is equivalent to the stiffening of the spring in the model with decreasing moisture content. In this respect the storage modulus appears to be a better indicator of moisture change than the loss tangent or the loss modulus. This points out the validity of studies which have been conducted in the past attempting to relate moisture content in biological materials such as wood and plant fibers to Young's modulus of elasticity. 99 Significance of Results The results in this Thesis have defined some of the more important physical constants necessary to mathematically describe the behavior of bark and greed wood to applied stresses. To determine these constants, viscoelastic models were proposed to represent the material. The time dependent model constants were determined from experimental results. Variability in the constants between samples resulted from experimental error and variations between samples. Tests were conducted over a range of three fixed frequencies for each specimen tested. As a next step in this investigation a complete con- stitutive equation which considers frequency (strain rate) and moisture content should be stated. Actually with the constants known for a particular frequency this equation has already been stated in this Thesis by the expressions for the complex modulus. It remains, however, to adjust this expression to fit a range of frequencies and moisture contents. A further step in this investigation, of course, would be finally to apply these equations to a two dimen- sional study of a vibrating limb. This problem must be formulated to include the taper of the limb and satisfy the boundry conditions at the wood-bark interface. This would provide a direct experimental check on the original con- stants, the constitutive equations and the stress analysis solution for the limb. SUMMARY This study was initiated to study the mechanical behavior of fruit tree bark to applied stresses. Since the green wood is closely associated with the bark in the tree limb green wood was also included. Also since, during mechanical harvesting different frequencies of Vibration are used, the frequency of the applied stress was also considered. Tests were limited to a single variety and species; Montmorency Cherry: FFUHUS CfdsuS- All tests were conducted under controlled tempera- ture and humidity conditions in a testing chamber. Maximum strength of bark specimens was determined from tensile loading in a pneumatic testing machine. Elastic and viscous properties of bark and green wood Specimens were measured using elastic and viscoelastic flexure theory. Three loading techniques were used for the flexure tests. These tests were (1) loading at a slow, constant strain rate using a simple beam arrangement; (2) free vibration as a cantilever beam and (3) forced vibration as a simple beam. As a check on the theory and experimental technique several Specimens of aluminum, Plexiglas, and white pine wood were included in these tests. 100 101 Approximate and exact equations for determining the viscoelastic modulus from dynamic flexure were derived. The accuracy of these equations was described graphically in terms of measured variables. From this it was possible to use the approximate equations with a negligible amount of error. ._ .-— LOG“.H'T From a transverse microscopic section of bark the tissue appeared to be arranged in four major radial layers. These layers consisted of a functioning, and a spongy,non-functioning phloem, and two periderms which made up about 10, 65, 12, and 12 per cent of the total thickness respectively. The strength of bark was very dependent on the direction of the applied force. Maximum longitudinal and tangential strengths for the three inner bark tissues were about 640 and 70 pounds per square inch respectively. The outer periderm strengths were about 980 and 3250 pounds per square inch in these respective directions. Longitudinal cambium shear strength was about 30 pounds per square inch. The storage modulus of bark was very sensitive to frequency of the applied stress and varies from about 10,000 to “0,000 pounds per square inch over a frequency range of about lO-3 to 10‘2 cycles per second. The storage modulus of green wood varied a lesser amount from about 500,000 to 700,000 pounds per square inch over a slightly larger frequency range. The loss tangents of bark and wood were about .15 and .0“ respectively. CONCLUSIONS Both bark and green wood exhibit common viscoelastic effects such as non—linear loading curves, stress relaxation under load and high internal damping. Ema For the purpose of mechanical study bark may be divided into four major radial layers; a cambium, a nonfunctioning phloem and two periderms. a Strength properties of bark tissues are directional. Maximum strength for the inner bark tissues are about 6A0 and 70 pounds per second in the longitudinal and tangential directions respectively. The outer periderm strength is about 980 and 3250 pounds per square inch respectively, in the same direction. Average longitudinal shear strength was about 30 pounds per square inch as measured on July 8, 1965. However, it is suspected that this property is subject to large changes owing to moisture content and seasonal activity of the tree. Viscoelastic flexure theory and testing techniques can be successfully applied to measure the behavior of bark and green wood. The storage modulus of bark was quite low compared to green wood and increased quite rapidly with increases 102 103 in frequency of the applied stress from about 10,000 to “0,000 pounds per square inch. Over a similar frequency range the storage modulus varied from about 500,000 to 700,000 pounds per square inch. It is desirable to have controlled temperature condi- tions in a testing chamber. The most effective method Tm* of preventing moisture loss is rapid testing after collection. REFERENCES REFERENCES Adrian, P. A. and R. B. Fridley. 1963 Shaker-Clamp Design as Related to Allowable Stresses of Tree Bark. American Society of Agricultural Engineers Paper Number 63— 121. Saint Joseph, Michigan. Alfrey, Turner Jr. 1948 Mechanical Behavior of High Polymers. Interscience Publishers, Inc., New York, N. Y. Bair, G. 196A Static and Dynamic Modulus of Elasticity of Hard- board. Unpublished M. S. Thesis (Forest Pro- ducts). The Michigan State University, East Lansing, Michigan. Baumeister, T. (editor) 1958 Mark's Mechanical Engineers' Handbook. Sixth Edition. McGraw-Hill Book Co. Inc., New York, N. Y. Bland, D. R. 1960 Theory of Linear Viscoelasticity. Pergamon Press, New York, N. Y. Bland, D. R. and E. H. Lee. 1955 Calculation of the Complex Modulus of Linear Viscoelastic Materials from Vibrating Reed Measurements. Journal of Applied Physics. 26(12): 1947. Brown, 0. C. 1965 Some Factors Affecting the Longitudinal Shear of Bark From Fruit Tree Limbs, Unpublished M. S. Thesis (Agricultural Engineering). The University of California, Davis, California. Chaney, D. H. 1964 Canker Disease- Setback for Mechanical Harvesting. Western Fruit Grower. 8(6): 23 Chang, F. s. c. 1964 Stress Relaxation and Hysteresis at Various Strain Rates. Journal of Applied Polymer Science. 8:37. 105 -106 Church, Austin H. 196“ Mechanical Vibrations. John Wiley and Sons, Inc., New York, N. Y. Devay, J. E., Lukezic, F. L., English, H., Uriu K., and C. J. 1962 Hansen. Ceratocystis Canker. California Agri— culture. 16:2. Esau, Katherine. 1965 Plant Anatomy. John Wiley and Sons, Inc., New York, N. Y. Ferry, John D. 1961 Viscoelastic PrOperties of Polymers. John Wiley and Sons Inc., New York, N. Y. Finney, E. E. 1963 The Viscoelastic Behavior of the Potato, Solanum Tuberosum, under Quasi-Static Loading. Unpublished Ph. D. Thesis (Agricultural Engineering). The Michigan State University, East Lansing, Michigan. Finney, E. E., Hall, C. W. and G. E. Mase. 196“ Theory of Linear Viscoelasticity Applied to the Potato.Journal of Agricultural Engineering Research. 9(a):3o7. Frey-Wyssling, Albert. 1952 Deformation and Flow in Biological Systems. Inter— science Publishers, Inc., New York, N. Y. Goens, E. 1931 Uber die Bestimmung des Elastizitasmodulus von Staben mit Hilfe von Biegungsschwingungen. Annalen der Physik. 11(6):u2. Halderson, J. L. \ ‘ \ American Society of Agricultural Engineers Paper * Number 63-120. Saint Joseph, Michigan. Hearmon, R. F. S. 1958 The Influence of Shear and Rotary Inertia on the 3 Free Flexural Vibration of Wooden Beams. British Journal of Applied Physics. 9(10):381. Hail; H. G. and w. H. Bennett. 1939 Fundamental Principles of Physics. Prentice-Hall, 1 Inc., New York, N. Y. z 1963 Fundamental Factors in Mechanical Cherry Harvesting. P‘ 107 Heyn, A. N. J. 1940 The Physiology of Cell Elongation. Botany Review. 6:515. Hodgman, C. D. (editor) 1952 Handbook of Chemistry and Physics. 34th Edition. Chemical Rubber Publishing Co., Cleveland, Ohio. Horio, M. and S. Onogi. 1951 Forced Vibration of Reed as a Method of Determining r“: Viscoelasticity. Journal of Applied Physics. 1 22(7):977. Kollmann, F. and H. Krech. 1960 Dynamiche Messung der elastischen Holzeigenschaften und der Dampfung. Holz Als RoH-Und Werkstoff. Koppelmann, J. 1958 Uber die Bestimmung des dynamischen Elastizitats- modulus und dynamischen Schubmodulus im Frequenzbereich von 10‘5 bis 10"1 Hz. Rheology Acta. 1:20. ‘Jxl .. :1 Meyer, B. S., Anderson, D. B. and R. H. Bohning. 1963 Introduction to Plant Physiology. D. Van Nostrand Co., Inc., New York, N. Y. ‘ Meyer, K. H. 1950 Natural and Synthetic High Polymers. Interscience Publishers, Inc., New York, N. Y. Mohsenin, N. N. and H. Gohlich. 1962 An Engineering Approach to Evaluation of Textural Factors in Fruits and Vegetables. American Society of Agricultural Engineers, Paper Number 62- 321. Saint Joseph, Michigan. Nolle, A. W. 1948 Methods for Measuring Dynamic Mechanical Properties of Rubber-Like Materials. Journal of Applied Physics. l9(8):753. ' Picket, G. , of 1945 Equations for Computation‘Elastic Constants from Flexural and Torsional Resonant Frequencies of Vibration of Prisms and Cylinders. Proceedings of the American Society for Testing Materials. 45:846. fioelofsen, P. 1959 The Plant Cell Wall. Handbuch der Pflanzenatomie. 3(4):58. 108 Schneider, H. 1945 The Anatomy of Peach and Cherry Phloem. Bulletin of The Torrey Botanical Club. 72(2):l37. Sienko, M. J. and R. A. Plane 1961 Chemistry. McGraw-Hill Book Co., Inc., New York, N. Y. Thompson, W. T. 1965 Vibration Theory and Applications. Prentice-Hall, 19*- Inc., Englewood Cliffs, New Jersey. Timoshenko, S. P. 1921 On the Correction for Shear of the Differential Equation for Transverse Vibrations of PriSmatic Bars. PhilOSOphical Magazine. 41:744. Timoshenko, S. K. :4, 1922 On the Transverse Vibrations of Bars of Uniform ' Cross-Section. Philosophical Magazine. 43:125. Timoshenko, S. K. 1955 Vibration Problems in Engineering. D. Van Nostrand Co., Inc., New York, N. Y. Wood Handbook Number 72. 1955 Forest Products Laboratory, U. S. Department of Agriculture. Government Printing Office, Washington, D. C. Zoerb, G. C. and C. W. Hall. 1960 Some Mechanical and Rheological Properties of Grains. Journal of Agricultural Engineering Research. 5(1):83. APPENDIX 110 Equations for Error Using the Approximate Solutions The per cent error for storage and loss moduli can be calculated by the equation per cent Error = 100 (l-EA/EE) (46) where EA is the approximate value of the loss or storage modulus and EE is the exact value. Since a loss and storage modulus may be calculated for both forced and free vibration, four error terms will be possible. These equations may be easily derived from equations 28, 31, 34, 35, 38, 39, 42, and 43. As a first step wn in the exact equations was replaced by wnd using equation 38. As a typical example of this equation 39 would become E"(w) = Hwnd2(ji:17;i) /2- V 1—12 By using equation 46 the per cent error may be represented by; for forced vibration, Per Cent , Error [E (w)] 100 [l-(l + 82/2)(2- /l-A2)l (47) Per Cent — - Error [E"(w)] - 100 [l'ég/2 JVl—Az ] (“811 /T-:’IT?-1 A) and for free vibration: Per Cent — 2 Error [E'(w)] — 100 [l - (l + a /2) J (”9) (1 — 2/(1 + 4/a2) 111 Eiiogent [E"(w)] = 100 [l—a' /1-2/(1+4/az)' M/a2+11 (50) where B = w/wnd , A = w/wn and a = A/n. However, these error terms are somewhat involved and the error must be recalculated each time when the value of B or a increases. In addition, equations 47 and 48 require values of wn which are unknown. Therefore it was decided that graphical presentation of error for a range of B and A would be more useful. In order to compare the accuracy of the approximate equations in forced and free vibrations values of B and A were generated using the same values of the damping ratio, n'. Values were generated for n' in increments of .01 as n' varied from zero to 0.700. The error equations for these calculations were: for forced vibrations, Eggogent [E'(w)] = 100 (1- (1 + Rg/g) 52) (51) Eggogent [E"m0m.0 0H:H.0 no.0: 50.0: :00m.0 m::a.0 00:H.0 00.0 00.0 mm0m.0 HHOH.0 30.0: 00.0: S00m.0 000H.0 000H.0 00.0 30.0 Sazm.0 00ma.0 H~.0I :0.0I mm:m.0 0mma.0 00ma.0 H0.0 No.0 0HNN.0 00HH.0 00.0: No.0: 0:mm.0 0NHH.0 00HH.0 00.0 No.0 0H0m.0 0000.0 03.0: No.0: 000m.0 0HOH.0 000H.0 0:.0 H0.0 ~00H.0 0000.0 00.0: H0.0I mm0H.0 HH00.0 0000.0 m0.0 00.0 000H.0 m000.0 H0.0I 00.0: 0HOH.0 5000.0 0000.0 :m.0 00.0 00:H.0 H0w0.0 nm.0| 00.0: 0H:H.0 0000.0 0000.0 0H.0 00.0 m0ma.0 H000.0 SH.0I 00.0: 00NH.0 0000.0 0000.0 ma.0 00.0 HO0H.0 0000.0 NH.0I 00.0: 000H.0 H000.0 0000.0 00.0 00.0 0000.0 0030.0 00.0: 00.0! H000.0 H0:0.0 0000.0 30.0 00.0 0000.0 0000.0 30.01 00.0: 0000.0 0000.0 0000.0 No.0 00.0 0030.0 00m0.0 No.0: 00.01 0000.0 0000.0 00m0.0 00.0 00.0 00m0.0 00H0.0 00.0: 00.0: 00m0.0 00H0.0 00H0.0 00.0 00.0 0000.0 0000.0 00.0 00.0 0000.0 0000.0 0000.0 33d HHJ )0? )H0 33d Had )HH )80 801 nae -Je meo 02m nae -Js Wea 02m eeJ )JJ )JJ .eoB 9:4 .)JJ )HJ 939 an. an mo mo eae IId mo mo er IId Ids (J0 (J0 SSJ DOT. (J0 {\J0 SON OCT: CI. 9 as n wTe n u a e n a n u u Ju Tia 191 I 3 Ju Tia a u I B B 01. 01. SUU. 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