OVERDUE FINES ARE 25¢ PER DAY PER ITEM Return to book drop to remove this checkout from your record. © 1979 INACIO MARIA DAL FABBRO AILRIGTI'S RESERVED STRAIN FAILURE OF APPLE MATERIAL BY Inacio Maria Dal Fabbro A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1979 ABSTRACT STRAIN FAILURE OF APPLE MATERIAL BY Inacio Maria Dal Fabbro The objective of this work was to define a failure criteria for apple material. Cylindrical apple specimens were tested under uniaxial and triaxial state of stress and stress rate controlled uniaxial loading. Cubic apple specimens were subjected to uniaxial, biaxial and triaxial states of stress. Linear elastic and viscoelastic material properties were used to calculate the stress and strain components within the apple flesh. Uniaxial loading of cylindrical specimens showed that normal stress at failure varied for different strain rates. This eliminated the maximum normal stress failure criteria. Triaxial loading of cylindrical specimens indicated that maximum shear stress and normal stress at failure vary for different levels of cylindrical stress. Failure was also observed at zero maximum shear stress, which excludes the maximum shear stress failure criteria. Uniaxial, biaxial and rigid die loading of cubic and cylindrical specimens also excluded the maximum normal stress failure criteria. Inacio Maria Dal Fabbro Stress rate controlled uniaxial loading showed significant variations of normal stress at failure which again discarded the maximum normal stress failure criteria. Experimental results from these tests indicated that the maximum normal strain at failure remained relatively constant for all the loading situations. Total strain energy and its spherical and deviatoric components obtained from stress and strain values calculated from the linear elastic and viscoelastic theories exhibit significant variations. This eliminates the strain energies failure criterium. A non-linear viscoelastic formulation was proposed for apple material based on the convected derivative repre- sentation for the time derivative appearing in the linear viscoelastic equations. The most significant conclusion of this research is that apple material fails when the normal strain reaches a critical value. Approved Ma'or P so Approved £:[(A?.A;&Qflqd~fi45- DepaTtment Chairman ACKNOWLEDGMENTS The author sincerely appreciates the kindly cooperation and guidance of Dr. Larry J. Segerlind (Agricultural Engin- eering) during the development of this research work. Appreciation is extended to Professor Ernest H. Kidder, Dr. George E. Merva, Dr. Haruhiko Murase (Agricultural Engineering), and Dr. Jayaraman Krishnamurthy (Chemical Engineering) for suggestions and particular discussions. The author is particularly indebted to the Department of Agricultural Engineering for assistance and for the general cooperation of its faculty, staff, and graduate students. Special sincere gratitude to Dr. Joalice M. G. Bueno Dal Fabbro (my wife) and her parents whose efforts made this work possible. To my parents, brothers, and sisters, I leave my deepest gratitude. ii TABLE OF CONTENTS LIST OF TABLES. LIST OF FIGURES . LIST OF SYMBOLS . CHAPTERS I INTRODUCTION. II LITERATURE REVIEW . 2.1 General Remarks. . . 2.2 Elastic Behavior of Vegetative Material. 2.3 Linear Viscoelastic Behavior of Vegetative Material. 2.4 Failure Criteria . 2.5 Summary. III FAILURE THEORIES. General Remarks. The Haigh- Westergaard Hyper- space. Stress Conditions. . 3.3.1 Maximum normal stress theory. Maximum shear stress theory . Modified maximum shear stress theory. Internal friction theory. Conditions. . Maximum strain theory . Maximum shearing strain theory. Conditions. . Constant total strain energy theory. . . Energy of distortion theory ' Combined total strain energy and distortion energy theory. Modified energy of distortion theory. . . WWW WNH off. a”. D 3.5 O D. 01 0101 USQQHW WW rfi WM HSNHI—“fii WM W WW WHWWUJW WW iii Page vi .xii CHAPTER IV VI BASIC THEORY. 4.1 4.2 vbobrbthtb ~109th General Remarks. . The Strain Energy Stored in an Apple Specimen for Different Loading Situations . Maximum Shear Stress Conditions. The Linear Elastic Model The General Viscoelastic Model Stress Controlled Uniaxial Loading . The Non-linear Viscoelastic Formulation ’ for Apple Material . 4.7.1 The convected derivative of a covariant strain tensor EXPERIMENTAL PROCEDURE. 01 0101010! 01 thNH 01 O} .8 .9 010101010! General Remarks. . . Apple Selection and Storage. Specimen Preparation Uniaxial Loading of Cylindrical Specimens at Different Strain Rates. Uniaxial Loading of Cylindrical Specimens of Different Height at a Constant Strain Rate: . Triaxial Loading of Cylindrical Specimens at a Constant Strain Rate, and Different Radial Stresses. . Rigid Die Loading of Cylindrical Apple Specimens. . Uniaxial Loading of. Cubic Specimens at a Constant Strain Rate Biaxial Loading of Cubic Specimens at a. Constant Strain Rate .10 Rigid Die Loading of Cubic Specimens . . .11 Stress Rate Controlled Loading of Cylin- drical Specimens of Red Delicious. RESULTS AND DISCUSSION. 6.1 6 2 6.3 6 4 General Remarks. Uniaxial Loading of Cylindrical specimens at Different Strain Rates. . . Triaxial Loading of Cylindrical Apple Specimens. . . Uniaxial, Biaxial and Rigid Die Loading of Cubic Specimens. Uniaxial and Rigid Die Loading of Cylindrical Specimens. . Stress Controlled Loading of Cylindrical Specimens of Red Delicious iv Page 24 24 39 39 41 43 43 45 45 47 47 48 54 60 7O CHAPTER Page 6.6 The Non-linear Viscoelastic Formulation for Apple Material . . . . . . . . . . . 70 6.7 Summary. . . . . . . . . . . . . . . . . . . 73 VII SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . 75 VIII SUGGESTIONS FOR FUTURE RESEARCH . . . . . . . . . 77 REFERENCES. . . . . . . . . . . . . . . . . . . . . . . 79 APPENDICES. . . . . . . . . . . . . . . . . . . . . . . 87 LIST OF TABLES Table Page 5.1 Tests carried on apple material. Imposed strain rates and radial stresses. Shape and dimensions of the specimens . . . . 37 6.1 Uniaxial loading of cylindrical apple specimens for different strain rates. Average values and standard deviations of stress, strain and time at failure . . . . 49 6.2 Uniaxial loading of cylindrical specimens of red delicious at different strain rates. strain energy components obtained from experimental values of 011 and e and calculated values of e . Maximfim shear stress obtained from experimental data and values of €22 calculated from the elastic and viscoelastic theories . . . . 50 6.3 Triaxial loading of cylindrical apple specimens for different radial stresses and constant strain rate. Average values of time, axial stress and strain at failure. . . . . . . . . . . . . . . . . . 55 6.4 Triaxial loading of cylindrical specimens of Red Delicious at different radial stress. Strain energy components obtained from experimental values of 011 and calculated values of 622. Maximum shear stress obtained from experimental data and values of €22 calculated from elastic and viscoelastic theories . . . . . . . . . . 56 6.5 Rigid die loading of cylindrical (l) and cubic (4) specimens, uniaxial (2) and biaxial (3) loadings of cubic specimens of apple values. Average and standard deviations of stress, strain and time at failure. . . . . . . . . . . . . . . . . . 61 vi Table 6.6 A1 A2 A3 A4 A5 A6 A7 Page Uniaxial (1), biaxial (2) and rigid die loadings (3) of cubic specimens. Rigid die loading of cylindrical specimens (4). Axial stress and strain, lateral or radial stress and strain and energy components obtained from elastic and viscoelastic theories. 62 Creep loading of cylindrical specimens of Red Delicious. Calculated values strain energy components and experimental values of axial stress. Strain and maximum shear stress at failure 63 Uniaxial loading of cylindrical apple specimens. Average values and standard deviations of deformation at = -0.11 Mpa . 64 Uniaxial loading of cylindrical apple specimens of different height. Axial strain and stress at o%% = —0.11 MPa a for McIntosh (1), Jona Red Delicious (3) Stress, strain and time uniaxial loading of specimens of apple. Stress, strain and time uniaxial loading of specimens of apple. Stress, strain and time uniaxial loading of specimens of apple. Stress, strain and time uniaxial loading of specimens of apple. Stress, strain and time uniaxial loading of specimens of apple. Stress, strain and time uniaxial loading of specimens of apple. Stress, strain and time uniaxial loading of specimens of apple. n (2), and at failure for cylindrical E11 2 at failure for cylindrical é11 at failure for cylindrical é11 at failure for cylindrical 811 at failure for cylindrical 811 at failure for cylindrical é11 at failure for cylindrical é11 vii -o.002 sec“ = -0.007 sec- = -o.017 sec- = -0.035 sec- = -0.069 sec- = -O.173 sec— = -O.345 sec- 1 l 1 1 1 l 1 65 87 87 88 88 89 89 90 Table A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 Stress, strain and time triaxial loading of specimens of apple. 022 = 0.000 MPa . Stress, strain and time triaxial loading of specimens of apple. 022 = -0.069 MPa. Stress, strain and time triaxial loading of specimens of apple. 022 = -0.138 MPa. Stress, strain and time triaxial loading of specimens of apple. 022 = -0.207 MPa. Stress, strain and time triaxial loading of specimens of apple. 022 = -0.276 MPa. Stress, strain and time triaxial loading of specimens of apple. 022 = -0.395 MPa. Stress, strain and time loading in rigid die of cylindrical é 11 apple specimens. Stress, strain and time uniaxial loading of specimens. éll = Stress, strain and time at failure for cylindrical é11 = at failure for cylindrical 811 at failure for cylindrical 811 at failure for cylindrical é11 at failure for cylindrical 811 at failure for cylindrical 811 at failure for = -0.007 sec-1. at failure for cubic apple -o.007 sec’l. at failure for biaxial loading of cubic appl specimens. éll Stress, strain and time = -0.007 sec” . at failure for "9'90? 565. ‘7‘?"3"? 19?". = '0’907 18$". 7’95”? TS. 1‘93”? $95.. = —o.007 sec" 1 1 1 1 1 1 loading in rigid die of cubiclapple specimens. éll = -0.007 sec' . Uniaxial compression of cylindrical specimens of McIntosh. Axial deforma- tion values at c = -0.11 MPa for five specimen height. éll = -0.007 sec‘l. viii , 7 ’ 3 2 1 Page 90 91 91 92 92 93 93 94 94 95 96 Table Page A19 Uniaxial compression of cylindrical specimens of Jonathan. Axial deforma- tion values at o = -0.11 MPa for 1 five specimen he1ght. €11 = -0.007 sec' 97 A20 Uniaxial compression of cylindrical specimens of Red Delicious. Axial deformation values TA 011 = —0.11 MPa for five specimen height. éll = -0.007 see"1 . . . . . . . . . . . . . . . 98 ix LIST OF FIGURES Figure '3.1 The Haigh-Westergaard hyper-space 3.2 Yield locus for an isotropic material which does not exhibit Bauschinger effect (Hill, 1964) 5.1 Triaxial loading device, showing the longitudinal cross-sectional view (a) and top view (b). 5.2 Top (a) and longitudinal cross-sectional view (b) of the device for loading of cylindrical specimens in rigid die. 5.3 Exploded View of the device used for biaxial and rigid loading of cubic specimens. 5.4 Creep loading apparatus 6.1 Stress at failure versus strain rate for uniaxial loading of cylindrical apple specimens . . . . 6.2 Strain at failure versus strain rate for uniaxial loading of cylindrical apple specimens 6.3 Time versus strain rate for uniaxial loading of cylindrical apple specimens. 6.4 Axial stress at failure versus radial stress for triaxial loading of cylindrical apple specimens 6.5 Strain at failure versus radial stress for triaxial loading of cylindrical apple specimens . 6.6 Time at failure versus radial stress for triaxial loading of cylindrical apple specimens . . . . . . . . . . . Page l4 16 4O 42 44 46 51 52 53 57 58 59 Figure 6.7 Stress at failure for controlled stress rate loading of cylindrical specimens of Red Delicious. 6.8 Strain at failure versus stress rate for controlled stress rate of Red Delicious 6.9 Deformation values of cylindrical apple specimens of different height at constant axial stress value during uniaxial loading. 6.10 Strain values of cylindrical apple specimens of different height at constant stress values during uniaxial loading. xi Page 66 67 68 69 mt‘Ub LIST OF SYMBOLS Cross sectional area Diameter Side Height Stress tensor Strain tensor Deformation vector Strain rate tensor Cartesian coordinate system Gradient velocity along Xi axis Velocity along Xi axis Partial derivative with respect to time Convected derivative with respect to time Convected strain rate tensor Time Deviatoric stress tensor Deviatoric strain tensor Total strain energy Deviatoric component of the strain energy xii MPa mm/mm sec mm/mm sec mm/sec sec sec MPa mm/mm Joules Joules Us max 0111 F(om) Glct) 620:) E(t) X(t) v(t) qot OC Spherical component of the strain energy Maximum shear stress Maximum absolute value of the principal stress tensor Minimum absolute value of the principle stress tensor Stress rate tensor Constant of the linear stress rate function Modulus of elasticity Bulk modulus Shear modulus Poisson's ratio Lame's constant Laplace parameter Function of mean stress Mean stress Time dependent shear modulus Time dependent bulk modulus Uniaxial relaxation function Constrained relaxation function Time dependent Poisson's ratio Lode's parameter Tensile yielding stress Compression yielding stress Yielding stress xiii Joules MPa MPa MPa/sec MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa MPa Invariants of the stress deviator Constants of proportionality Medium absolute value of the principal strain tensor Minimum absolute value of the principal strain tensor Mean strain xiv MPa mm/mm mm/mm mm/mm CHAPTER I INTRODUCTION Bruising is a major problem in the development of new machines for the mechanical harvesting and handling of large quantities of fruits. Bruising is the rupture of the tissue and consequent exposure of the cell sap. The oxidation of the cell sap gives a darkened color to the softened tissue. This undesirable phenomenon is somehow related to the mech- anical loading of the fruit. The knowledge of the fruit tissue response to known loadings may provide the basis of bruise prediction when the fruit is subjected to other loading conditions. Many investigators have studied the mechanical proper- ties of apple tissue through a very broad theoretical formu- lation bringing about non-specific results. It would be reasonable to say that the overall objective of the majority of the research work conducted on apple tissue was to iden- tify its mechanical behavior. Little of the relevant work on the mechanical properties of apples has been directed toward establishing the failure parameters of apple tissue. The failure phenomenon is believed to be an indicator of bruise occurrence. It means that a bruise is the result of a tissue failure. This implies that bruises can be predicted in terms of failure parameters. Before this problem can be solved, it is necess- ary to define failure. Failure by yielding or by fracture may occur beyond the elastic limit for common engineering materials (Juvinall, 1967). Vegetative materials exhibit a rupture point close to the elastic limit, which has been referred to as the bio-yielding point (Mohsenin, 1970). The parameters correlated to the bio-yielding point of apple flesh can be studied by imposing different loading conditions on apple specimens. The specific objective of this study was to establish the parameters involved in the failure of apple material. CHAPTER II LITERATURE REVIEW 2.1 General Remarks Research on the mechanical behavior of vegetative mat- erial has as one objective the minimization of bruise damage. Material property determination and a stress-strain analysis seem to be the steps toward complete information on failure parameters. Vegetative material has been generally considered either as an isotropic continuous medium or as a multi-phase medium (Akyurt, 1969; Brusewitz, 1969; Gustafson, 1974; Murase, 1977). Elastic and viscoelastic models had been used to represent the mechanical response to a variety of loading conditions. Mohsenin (1971) cites ample literature on the importance of mechanical properties of agricultural products and the need for study and research in this area. 2.2 Elastic Behavior of Vegetative Material Determination of elastic constants is a frequent subject of research due to the need for basic information on material properties. Modulus of elasticity, bulk modulus, and elastic Poisson's ratio have been determined on cylindrical and whole specimens of potato by uniaxial and hydrostatic compression (Finney, 1963; Finney and Hall, 1968). Modulus of elasticity can also be determined by radial compression of cylindrical specimens (Sherif et a1., 1976). Bulk compression tests directly obtaining the bulk modulus and the calculation of Poisson's ratio yielded reliable results for fruits (White and Mohsenin, 1967). Elastic Poisson's ratio and elastic uniaxial modulus can be simultaneously determined from elastic bulk modulus and Boussinesq solution for cylindrical plunger on a half-space (Morrow, 1965). Elastic Poisson's ratio can also be determined by comparing the axial force- deformation on free and restrained cylindrical specimens of apple (Hughes and Segerlind, 1972). Results from radial s compression loading of cylindrical specimens can be inter- preted using Hertz contact theory to obtain values for the modulus of elasticity (Snobar, 1973). Bulk modulus of a whole-apple specimen can be determined by considering the principle of buoyancy (Chen and Lam, 1975). The stress and strain distribution in an elastic body is also of practical interest for further study on bruise location. Plate and plunger tests have been conducted on whole specimens of peaches and pears correlating deforma- tion and stress distributions with those predicted by elas- tic models (Fridley et a1., 1968). Stress and strain distributions on apples under static axi-symmetric load are similar to those in an elastic sphere subjected to the same conditions (Apaclla, 1973). Potatoes have been consi- dered a nearly incompressible non-linear elastic material to analyze the stress distribution in hemi-spherical specimens (Sherif, 1976). In recent years a more complex approach has started to replace the elastic theory for describing fruits and vegetables. Vegetative material is now considered as a multi-phase medium, having gas, solid, and liquid components (Akyurt, 1969; Gustafson, 1974). A finite element method is then used to obtain strain and stress distributions in spherical bodies under axisymmetric conditions (Gustafson, 1974). Potato tissue was viewed as an interacting combina- tion of solid and liquid phases in determining material properties (Brusewitz, 1969). Cellular and intercellular spaces were interpreted as porous and solid-liquid media (Murase, 1977). Linear elastic stress and strain constitu- tive equations were then derived, analogous to Duhamel's relations (Murase, 1977). 2.3 Linear Viscoelastic Behavior of Vegetative Material Many experimental investigations have indicated a time dependency of the mechanical behavior of plant tissue. Strain rate affects the response to an impact test in biol— ogical materials (Zoerb, 1958). The mechanical damage of potatoes subjected to compressive loads is highly affected by strain rate (Finney, 1963). Non-linear viscoelastic behavior of apples was reported by Morrow and Mohsenin (1966) who approximated it by linear constitutive relations. The stress dependence of material properties of apple material made it impossible to accept a linear approximation (Chappell and Hamann, 1968). Further works have dealt with the non-linear behavior of apple tissue, but the results were interpreted by linear viscoelastic relations (Hamann, 1967, 1970). Tensile tests conducted on apple skin suggested a viscoelastic behavior (Clevenger and Hamann, 1968). The viscoelastic Poisson's ratio can be determined indirectly from the elastic Poisson's ratio constant by the correspon- dence principle (DeBaerdemaeker, 1975). Time dependence of Poisson's ratio was directly noted by measuring lateral and axial displacement of cylindrical specimens (Chappell and Hamann, 1968). Similar results have been reported from tests carried out on sweet potatoes (Hammerle and McClure, 1970). Relaxation functions can be determined by bulk and uniaxial loading (Morrow and Mohsenin, 1966). Similarly, creep functions were determined by applying hydrostatic loads to whole specimens (Morrow, 1965, and Sharma, 1970). Uniaxial loading of cylindrical specimens was reported to yield reliable results for relaxation functions (Finney, 1963; Chappell and Hamann, 1968; Morrow et a1., 1971; Hammerle et a1., 1971). Bulk and shear relaxation functions were experimentally determined for apple tissue and the results were used in a viscoelastic sphere loaded by a flat surface (DeBaerdemaeker, 1975). Rumsey and Fridley (1974) assumed constant bulk modulus and time dependent shear relaxation function. Dynamic methods had also been used to determine viscoelastic properties of biological material (Morrow and Mohsenin, 1968). The parameters of a generalized Maxwell model have been experimentally determined for several different fruits and vegetables (Mohsenin, 1970; Hammerle and Mohsenin, 1970; Chen and Fridley, 1972). Results from bulk loading of apples were compared with a simple Kelvin model to obtain an expression for the creep function. The relationship between the complex moduli and the relaxation functions can be used to calculate the dynamic relaxation and shear relaxation functions from experimental results (Hamann, 1969). Force and deformation dependence on strain rate was reported by Mohsenin et a1. (1963). Creep behavior of papaya was determined under dead load conditions imposed by parallel plates (Wang and Chang, 1969). A viscoelastic stress-strain analysis is the next step once the basic time dependent properties have been deter- mined. A simple Maxwell model can be used to represent the response of two viscoelastic spheres falling onto one another (Hamann, 1970). The viscoelastic sphere subjected to a contact load can be experimentally studied and numer- ically simulated (DeBaerdemaeker, 1975). Vibration analysis in a non-uniform viscoelastic beam has been used to predict the stress-strain distribution in tomato blossoms subjected to similar conditions (De Tar, 1971). 2.4 Failure Criteria One objective of research conducted on the mechanical behavior of vegetative material is to minimize bruise occur- rence. Determination of elastic constants and viscoelastic functions is performed to obtain the constitutive laws for the material. The material properties are needed in order that the stress resulting when external loads are applied to the fruit can be calculated. Impact testing has been used to determine whether a bruise occurs because of the maximum energy absorbed, the maximum stress applied, or the maximum deformation (Mattus et a1., 1960). In this sense it was found that the energy required for bruising was greater under impact conditions than under quasi-state loading conditions (Mohsenin and Gohlich, 1962; Mohsenin et a1., 1965; Nelson et a1., 1968; Fridley et a1., 1964) for apples and peaches. However, for pears and sweet potatoes, it requires more bruising energy under quasi-static loading (Wright and Splinter, 1968; Fridley and Adrian, 1966). Apple-limb impact and its influence in the bruising of apples was investigated by David and Rehkugler (1971). The impact of apples on cushioning mat- erial was studied by Hammerle and Mohsenin, 1966; Simpson and Rehkugler, 1972. Results from impact tests on whole specimens did not reveal any dominance of the force or energy parameter (Fluck and Ahmed, 1972). Analysis of bruise loca- tion indicates a strong possibility of bruise occurrence at maximum shear stress (Fridley and Adrian, 1966). Bruising in peaches due to impact loading can be modeled by applying similar conditions to an elastic sphere (Horsfield et a1., 1972). The problem of potato cracking during handling was experimentally studied using tensile tests (Huff, 1967). However, impact tests had been extensively carried out (Finney, 1963; Park, 1963). Flat plate loading of hemi- spherical specimens of apple and potato have indicated the existence of maximum shear stress near the contact region as well as a tensile stress at the circular boundary of the contact region (DeBaerdemaeker, 1975; Sherif, 1976). There is also indication of a maximum tensile stress or combina- tion of this and shear stress near the center of the white potato (Sherif, 1976). Bruises in peaches may occur at the maximum shear stress on the axis of symmetry (Sherif, 1976). White potatoes and peaches did not fail until large displacements had taken place (Sherif, 1976). Failure strength of apples, referred to as the bio-yield point, have been determined by indentor test as well as by plunger and uniaxial ramp-loading of cylindrical specimens (Van Lancker et a1., 1975). Bruise energy of peaches and apples can be evaluated by measuring the rebounding force in an impact test (Diener et a1., 1977). Attempts have been made in correlating bruise occurrence location to mechanical, thermal, and electrical properties of apples (Holcomb et a1., 1977). Tensile strength of potato and apple tissues increases with increasing water potential levels. The compressive strength of these products, however, decreases with increasing water potential (DeBaerdemaeker et a1., 1978). Maximum shear stress was reported to be the failure parameter of apple flesh (Miles, 1971). Cylindrical speci— mens of apple were subjected to several levels of hydrostatic lO stress superimposed on a uniaxial loading. Failure depended on loading rate and confining pressure; those parameters, however, act independently (Miles, 1971). 2.5 Summary Material properties and stress-strain analysis have been used to characterize the mechanical behavior of vege- tative bodies. The literature discloses a significant amount of research starting from simple assumptions such as a continuous isotropic medium and linear elastic behavior, extending into multi-phase medium, linear viscoelastic, and non-linear elastic behavior. Nevertheless, the failure parameters for a vegetative material have not yet been determined. The triaxial loading of cylindrical specimens (Miles, 1971) can be considered the best attempt toward the deter- mination of failure criteria. In spite of time dependent non-linearities that had been noticed (Hamann, 1967, 1970), no investigation had been reported which assumed non-linear viscoelastic behavior. CHAPTER III FAILURE THEORIES 3.1 General Remarks The limit of the elastic behavior of a body is deter- mined by the existing state of stress, as well as by its material properties. Beyond this limit the material may suffer permanent deformations or fail by fracture. It is commonly agreed that vegetative materials have a rupture point very close to the elastic limit without experiencing any plastic deformations (Mohsenin, 1970). In such condi- tions, failure, yielding, or rupture would have nearly identical meanings. Earlier investigators have attempted to formulate generic yield criteria for metals assuming homogeneous and isotropic condition (Prager, 1942). Some of those theories predict failure under hydrostatic stress conditions (Nadai, 1950). Loading tests conducted on specimens of solid material under high hydrostatic stress did not result in failure (Nadai, 1950). The assumption that hydrostatic loads do not cause failure has a purely experimental basis (Mendelson, 1965). Theories which do not assume failure under pure hydrostatic loads have been modified to fit experimental data from triaxial loading of soil specimens 11 12 (Bishop and Henkel, 1962). Those extended theories assume a contribution of hydrostatic stresses on failing soil specimens (Terzaghi and Peck, 1967). Non-homogeneous materials can exhibit different values for tensile yield stress and compressive yield stress. Under that condition the difficulties in obtaining tensile yield stress values for vegetative materials is the major obstacle in making full use of theories which can account for differ- ences between compressive and tensile yield values. Existing failure criteria by yielding can be formulated in terms of stress, strain, or energy considerations. The theories of failure mentioned and their discussion in this chapter by no means exhaust the available literature. Only those topics pertinent to the present study are included. 3.2 The Haigh-Westergaard Hyper-space Failure theories can be generalized by considering the complete state of stress at a point. Since the stress tensor is symmetric, it is possible to describe yielding as a function of the six independent stress components (Mendel- son, 1965). For a material specimen loaded to yield, this function can be written as follows (Prager, 1942): F(o..) = 0 (3.1) Equation (3.1) represents a hypersurface in the six- dimentional stress space formed by yield points. In other words, any point inside of this solid figure represents an elastic state and all the points located on the surface 13 represent the beginning of the plastic deformation or failure (Nadai, 1931). If isotropy is assumed, the rotation of axis will not affect yielding and equation (3.1) would be written in terms of principal stresses, as F (o o (3-2) 11' 22' 033) = 0 Furthermore, since hydrostatic stresses do not affect yielding, the yielding function can be expressed in terms of stress deviators. Since the stress diviators can be written in terms of the invariants, the yielding function can also be expressed in terms of invariants of the stress deviator, as follows F (J J3) = o (3 3) 2, Equation (3.3) is symmetric in the principle axis which indicates that all principle stresses are equally important to the yield condition (Mendelson, 1965). Thus, whatever yield function is proposed it should be symmetric in the principal axis (Hill, 1964). The geometry of the yield sur— face in the Haigh-Westergaard stress-space is a cylinder whose main axis is the hydrostatic axis. Any point Pn (011’ 022, 033) on this surface will have the same deviatoric stress components and different spherical components. Figure 3.1 represents the Haigh-Westergaard yield surface, showing the points P1 and P2 representing state of stress decomposed into spherical parts A1 and A2 and deviatoric parts B1 and B2, respectively. Plane w is the (011 + 022 + 033) = 0 plane where the hydrostatic stress equals zero. 15 The intersection of the yield cylinder with any plane perpen- dicular to it will produce the same curve. This curve is called yield locus (Mendelson, 1965). Yield locus will be sufficient to study the yielding conditions since it is known that hydrostatic stresses do not contribute to failure. The yield locus then can be taken on the plane n. The projec- tions of the principal stress axis on the plane n are lines 60° apart from each other, as shown on Figure 3.2. Since the material is isotropic, the locus is symmetrical about QQ', RR', and SS'. In other words, the yield criteria is a function of the invariants J and J . Similarly, the yield 2 3 locus will be symmetric about the orthogonal lines to the stress axis projections passing through the origin (Hill, 1964). If the Bauschinger effect is neglected, any line- representing unloading, drawn from the locus through the origin, will meet the locus again at the same distance from the origin. This is equivalent to saying that it is only necessary to analyze one of the twelve segments. It is very helpful to think in terms of Lode's parameter y, 2 o — o — o w: 33 11 22 = - 3 tan 9 (3.4) 011' 022 Where a is the angle which defines the stress vector OP. Stress locus can be completely determined by applying stress states such that\y varies from zero to —l or e varies from zero to E/6 radians (Hill, 1964). Existing failure theories do not always agree with the Haigh-Westergaard yield surface. Also, experimental data can l6 Figure 3.2. Yield locus for an isotropic material which does not exhibit Bauschinger effect (Hill, 1964). 17 show yield points whose locus is not symmetric with respect to the axes of principal stresses. 3.3 Stress Conditions 3.3.1 Maximum normal stress theory The literature contains famous names from early times associated with this theory. Galileo Galilei and Leibniz were the first scientists to propose the failure criteria based on the maximum normal stress value (Prager, 1942). Later on, L. Navier, G. Lame, B. P. E. Clapeyron, and Rankine each presented a mathematical formulation for this condition. This theory assumes that yield occurs when the largest of the principal stresses reaches the value of the tensile yield stress 0 or the yield stress value Uoc' For a three- ot dimensional compressive stress configuration, the theory is formulated as: 011 = Ooc 022 = 00c (3.5) 033 00c depending on which one of the principal stresses is the larg- est. For a tensile stress state, equation (3.5) can be written as: (3.6) q N M II Q 0 d q to OJ I Q 0 d 18 3.3.2 Maximum shear stress theory The names of Tresca and Coulomb are related to this theory (Marin, 1953, and Mendelson, 1965). This condition assumes that yielding occurs when the maximum shear stress in the body reaches the shear stress value associated with yielding in simple tension, 0 Mathematically, this theory ot' 'can be expressed as: 011 ‘ 022 = iCot O22 ‘ 033 = toot (3'7) 033 ' 011 = *Oot This condition does not predict failure under hydro- static loading conditions (Hill, 1964). .3.3.3 Modified maximum shear stress theory This theory is a generalization of the maximum shear stress condition, formulated by Mohr. Tresca and Mohr's criteria assume that only the largest and smallest principal stresses influence failure. While the first states that the largest principal circle on the Mohr diagram should have constant radius, the latter assumes that this radius should be a function of the normal stress. The failure will be defined by the envelope of all circles representing yield at different states of stress (Nadai, 1950). This can be analytically expressed as: (all - 022)/2 = F[(o11 + 022)/2] (3 8) If the envelope lines are parallel and horizontal, equation (3.8) will be tranSformed back into equation (3.7), which 19 represents the maximum shear stress condition. 3.3.4 Internal friction theory This condition is related to the names of Mohr, Coulomb, Guest, and Duguet. It can be considered as a special case of Mohr's theory in which the envelopes are two straight lines equally inclined to the normal stress axis (Marin, 1962). In other words, the limiting shear stress can be expressed as a linear function of the normal stress, written as: 011 ' U33 _ Oot ‘ Ooc Cotooc 011 I 033 2 - o + o + o + o 2 (3'9) ot oc ot oc It can be observed that when oc = ot’ this condition is reduced to the maximum shear stress theory. 3.4 Strain Conditions 3.4.1 Maximum strain theory This condition was independently proposed by Saint Venant and Poncelet (Prager, 1942). In a case of combined stress, yielding starts when the maximum value of the principal strains equals the value of the compressive or tensile yielding strain. The analytical expression of this statement can be expressed as: 011 ’ “(022 + 033) = too 022 - v(033 + 011) = :00 (3.10) 033 ‘ V(°11 + 022) = :00 where 00 = doc = Got and v 15 the Po1sson 3 ratio. Th1s y1eld condition does not predict failure under hydrostatic stress 20 state. 3.4.2 Maximum shearing strain theory This condition was proposed by G. Sandel (Prager, 1942). The maximum shearing strain is assumed to be a linear func- tion of the mean strain. The analytical expression for this theory is: El - EII = c - b e (3.11) 3.5 Energy Conditions 3.5.1 Constant total strain energy theory This condition was proposed by Beltrami (Mendelson, 1965). Elastic strain energy is the factor impeding failure. In terms of principal stress it can be expressed as: 2 2 . 2 _ 2 + 022 + 033 ‘ 2G(011022 I O22°33 + “33011) ’ °o (3.12) 011 This condition predicts failure under hydrostatic stress conditions. The representation of the yield surface in stress space is an ellipsoid of revolution whose main axis is coincident with the hydrostatic axis (Prager. 1942). 3.5.2 Energy of distortion theory This condition appears related to the names of Hencky, Von Mises, Hueber, and Maxwell, and it is also known as maximum octahedral stress theory (Juvinall, 1967). This theory assumes that yielding begins when the distortion energy equals the distortion energy at yield in simple tension or compression. Analytically it can be stated in 21 terms of principal stresses as: 2 2 2 2 _ + (022 ‘ 033) + (“33 ’ 011) 1‘ 00 (3°13) 5[(°11 ' 022) where'oo==o0t = doc. This condition does not predict failure under hydrostatic stress states. The failure surface in three-dimensional stress space is a circular cylinder whose main axis is coincident with the hydrostatic axis. 3.5.3 Combined total strain energy and distortion energy theory This condition was proposed by Huber (Prager, 1942). It is assumed that yielding will occur when the energy of dis- tortion reaches the value of the energy of distortion at uniaxial loading when am < 0 or when the total strain energy reaches the value of the total strain energy at uniaxial loading for cm > O. In terms of principal stresses it is stated as: 2 2 _ 2 é[(o11 - 022) + (022 - 033) + (033 - 011)]- do for om < 0 (3.14 o 2 + o 2 + o 2 = 2G(o o + o o + o o ) 11 22 33 ll 22 22 33 33 ll _ 2 . - Go for cm > O (3.15) The failure surface for this condition is represented by a cylinder prolonged by an ellipsoid. 3.5.4 Modified energy of distortion theory This condition assumes that the energy of distortion level which causes failure is also a function of cm (Nadai, 1950). This modification was proposed byIL Von Mises and 22 F. Schleicher (Prager, 1942). The mathematical expression for this condition can be written in terms of principal stresses as: §[(°11 ' ”22)2 + (022 ‘ (’33)2 + (033 ‘ ”11’2]= F(0m) (3.16) Depending on the function F(om), equation (3.16) can represent a circular cone or a paraboloid of revolution (Nadai, 1950). Figure 3.2 shows the projection of some failure surfaces on the 011 - 022 plane. 23 ll 22 Distortion energy Maximum strain energy Maximum strain Maximum normal stress Maximum shear stress UHhWNl—J I Figure 3.3. Comparison of failure surfaces as viewed on the all - 022 plane. CHAPTER IV BASIC THEORY 4.1 General Remarks In the preceeding chapter the failure theories were classified by the parameters, stresses or strains, which are considered to produce a failure. The experimental data, however, must be combined with constitutive equations in order to obtain values for these parameters. Both elastic and viscoelastic equations have been used to represent the mechanical behavior of a vegetative material. The objective of this chapter is to outline the calcu- lation of the stress and strain components and the strain energy stored in the apple flesh for different types of experimental loads. The equations for the triaxial, rigid die, biaxial, and uniaxial tests are presented, first assuming a linear elastic material and then assuming a linear visco— elastic material. 4.2 The Strain Energy Stored in an Apple Specimen for Different Loading Situations It is known from the theory of mechanics of continuous medium (Malvern, 1969) that the deviatoric stress and strain tensors are Sij = Oih - (1/3) dij Okk; (4.1) 24 25 where Oij is the stress tensor, Eij is the strain tensor, Okk and ekk are the spherical components of the total stress and strain tensors, respectively, and 513 is the Kronecker's delta. If a body in equilibrium is deformed by the action of external forces, so that none of the work done goes into kinetic energy, then this work is stored as strain energy of deformation. The total strain energy can be expressed as the summation of the distortional energy and spherical energy components, as U = U + US (4.3) d or in terms of strain and stress tensors U = (1/2) oij eij (4.4) Equations (4.1), (4.2), (4.3), and (4.4) can be com— bined to yield the following expression for the energy of distortion Ud = sij 913/2 (4.5) which can be developed into + (011 ‘ 033) (€11 ‘ €33) + (022 - 033) (822 - 833)] (4-6) Similarly the expression the spherical component of strain energy becomes US = Olleij/6 (4.7) which yields Us = (011+022+°33) (811+522+€33)/6 (4'8) 26 In a stress state in which a22 = a33 and 622 = 833, (4.6) and (4.8) reduce to Ud (1/2) (a11 - 022) (511 - 622) (4.9) Us (1/6) (all + 2022) (all + 2522) (4.10) If the conditions a22 = a33 $ 0 and 522 = 533 = 0 hold, (4.6) and (4.8) yield Ud = (1/2) (a11 - a22) 511 (4.11) US = (1/6) (all + 2a22) 611 (4.12) For the biaxial state of stress in which a22 f 0, a = O, 822 = 0, 833 f 0, all f 0, and 811 f 0 33 Us = (l/6)(a11 + a22)(€11 + 833) Ud = (1/4) [(011-022)€1l + (ell-€33)all] (4.13) When a uniaxial loading is applied, the conditions all f 0,811 f 0, 022 = a33 = 0 and 822 = 833 f 0 define the state of stress. When this occurs, the equations for Ud and Us are C1 ll d (1/2) (311 - £22) 811 (4.14) (1/6) a and U + 28 (4.15) 11(E11 22) 4.3‘ Maximum Shear Stress Conditions The loading tests described in Chapter V develop only normal stresses within the material. For this type of stress state, the maximum shear stress is given by Timoshenko (1970) as rmax = “111 - oI / 2 (4.16) where a and aI are the maximum and minimum values of the III principle stresses, respectively. 27 4.4 The Linear Elastic Model The stress and strain tensors given in (4.1) and (4.2) can be related to each other through a linear material law known as generalized Hooke's law. The stress and strain tensors are related by Sij = 2G eij (4.17) akk = 3K ekk (4.18) Oij = 1 Ekk 6ij + ZG Eij (4.19) The bulk modulus K and shear modulus G are related to the Lame constant A, the modulus of elasticity E and Poisson's ratio v as K = E/3(l-2v) (4.20) K = (31 + zo)/3 (4.21) E = QGK/(3K + G) ' (4.22) In a triaxial loading test in which the state of stress is characterized by holding a22 = a33, 822 = 533 and 1mpos1ng all and 811' Equations (4.1), (4.2), (4.18), and (4.19) yield a11 = E811 + 2va22 (4.23) and 522 = (l/E) [522 - va22 - valll (4.24) In a rigid die loading, the strains 822 and 633 are zero and the expressions for all and a22 are _ E(1-v) O11 ‘ (l+v)(1-2v) 811 (4'25) 022 = (K + % G)e11 (4.26) In a biaxial state of stress, all and all are imposed while a22 f 0, a33 = 0, 622 = 0, £33 # 0. The express1ons 28 for a and a22 in this situation become 11 all = (E/(l-v2)) 811 (4.27) $33 = (-v/(1—v)) 811 (4.28) .229 022 1-v e11 (4.29) The state of stress which describes the uniaxial loading of a specimen (all f 0, all f 0, a22 = a33 = 0, 622 = 833 f 0) combined with equations (4.1), (4.2), (4.17), and (4.18) yield E e (4.30) 011 11 ('V/E) €11 (4.31) 822 4.5 The General Viscoelastic Model The stress and strain tensors formulated by the equations (4.1) and (4.2) can be also related to each other through the relaxation functions G1(t) and G2(t) (Christensen, 1971). The function Gl(t) is the deviatoric relaxation function or the function appropriate to the state of shear while the func- tion G2 is the bulk relaxation function. If a body is in equilibrium and there is no load applied before the time t = 0, the stress and strain relationship can be written as t e . i.) = _ d (1) dr Sij Jr G1(t T) dT (4.32) o t dekk dT 0 Functions Gl(t) and 62(t) can be related to each other by the Laplace transform operation as (Christensen, 1971, and Flugee, 1975): 29 E (35152)/(§1+252) (4.34) and X (261+52)/3 (4.35) where the bar indicates that the function is expressed in terms of the Laplace parameter 5 instead of time t. The function E(t) is called uniaxial relaxation function and X(t) is called constrained relaxation function. Experimental determination of E(t) was carried out in conditions where a22 = 033 = 0, corresponding to a uniaxial loading of cylin- drical specimens. In similar situations X(t) is determined by holding 522 = = 0. The functions E(t) and X(t) 633 are expressed as a summation of exponential terms as given in the generalized Maxwell model relaxation function E(t) = 3' (4.36) IIMD tle (D c+ Experimental values of E(t) and X(t) have been determined for Red Delicious apples and are given by an experimental series representation (DeBaerdemaeker, 1975) as E(t) 0.744 EXP(-4.l52t) + 2.863 EXP(-0.029t) (4.37) X(t) = 2.011 EXP(-4.630t) + 3.325 EXP(-0.028t) (4.38) Discrete values for G1 and 62 were obtained from the relaxa- tion functions X(t) and E(t) DeBaerdemaeker, 1975). Those values were modeled by an exponential representation as follows (in all those equations t is minutes) Gl(t) and Gz(t) 2.554 EXP(-O.318t) (4.39) 10.665EXP(-0.27t) (4.40) The time dependent Poisson's ratio determined by DeBaer- demaeker (1975) can also be represented by a similar equation 30 as v(t) = 0.330 EXP(-O.27t) (4.41) The convolution integrals (4.32) and (4.33) can be expressed in terms of the Laplace transform parameter s, as (Christen- sen, 1971) ij S G1 eij (4.42) and Ekk = s 62 (4.43) Ekk The triaxial loading case expressed by the equations (4.23) and (4.24) can be derived from those equations by the correspondence principle or directly from (4.1), (4.2), (4.42), and (4.43). In either case, the resulting expres- sions for all and 522 in the Laplace domain are (Fodor, 1965): 311 = s E E11 + 2'3'322 (4.44) and 222 = [3/sz(262+61)1022 - (4.45) V811 Equations (4.44) and (4.45) can be expressed in the time domain as follows 011(t) = éll [98.903 - 0.179 EXP(-4.l52t) - 98.724 EXP(-0.029t)] + 0.660 Exp(-0:270t) 022 (4.46 522 = [0.126 + 0.103t ] 022 - 1.222 [1 - EXP(-O.27t) 1 611 (4.47) The state of stress described by (4.25) and (4.26) for loading in a rigid die can be used to obtain its viscoelastic counterpart, resulting in the following expression for all and 022 011 = (l/S) X 511 (4.48) E22 (1/35)(§1+52) é11 (4.49) 31 The inversion of El and 322 results in 1 011(t) = éll[119.180 - 0.434 EXP(-4.630t) - 118.75 EXP(-0.028t)] (4.50) 022 = 611 [11.183 - 11.183 EXP(-0.9t) + 8.031 — 8.031 EXP(-O.318t)] (4.51) The biaxial state of stress associated with (4.27), (4.28), and (4.29) can be represented in the Laplace domain by 311 = [(sE)/<1-2>1'€11 (4.52) E33 = [(-US)/(1-US)] E11 (4.53) a22 = [vS/(l-vS)] (G1) 811 . (4.54) The inversion of (4.52), (4.53), and (4.54) to the time domain gives 011(t) (0.03 + 3.099t) 51 (4.55) l l.225[EXP(-O.4t) - 1] 411 (4.56) 833(t) a22(t) = -l4.800 611[EXP(-0.318t) - EXP(-0.403t] (4.57) Similarly, (4.30) and (4.31) associated with the case of uniaxial loading of a cylindrical specimen yields a11 - S E 811 (4.58) -...v_s. €22 - €11 (4.59) s The inversion of the above equations yields a11(t) = éll[98.903 — 0.179 EXP(-4.15t) -98.724 EXP(-0.029t)] (4-60) 622(t) = -¢ll[1.222 (1 - EXP(-0.270t))] (4.61) 32 4.6 Stress Controlled Uniaxial Loading For a uniaxial loading of cylinder, the elastic represent- ation is given by (4.30) and (4.31). The Laplace transforms of these equations are 611 E S_56311 (4.62) and £22 = §? -§% (4.63) In the time domain they become 611(t) = 611[-0.021 + 0.347t + 0.006t2 + 0.021 EXP(-3.302t)] (4 64) 822(t) = % [0.292t + 0.018t2]° 611 (4 65) 4.7 The Non-linear Viscoelastic Formulation for Apple Material It was seen in Chapter II that the non-linear visco- elastic behavior of vegetative tissues had been approximated by linear viscoelastic constitutive equations for certain cases (Morrow and Mohsenin, 1966; Hamann, 1967, 1970). However, Chappell and Hamann (1968) have reported cases in which such an approximation was not possible. In either case, the real behavior of vegetative tissue in reality is non-linear viscoelastic. Non-linear behavior of viscoelastic bodies is not as well understood as it is for the linear case. The first attempt in giving a mathematical formulation for non-linear viscoelastic phenomena was to represent the time derivatives c>f the linear operator form by convected derivatiVes (Old- royd, 1950). This model was criticized by the resulting 33 differences when contravariant or covariant tensors are used and does not predict non-newtonian viscous flow (Fredrickson, 1964). Further modification of this formula- tion was proposed by the same author by including non-linear terms on the convected operator form as gij’ aij(i$j), and :ikag. The resulting equation would reduce to the linear operator form in cases of small strain rates. The objections raised against this formulation are related to its lack of generality as well as the covariant and contra- variant effects (Fredrickson, 1964). A further step was taken by expressing the stress tensor 13' . . . t a in terms of a non-linear function of e. 1.1 convected derivatives (Rivlin and Ericksen, 1955). The and its N-l condition that ai‘j = 0 whenever gij = D gij/Dt = 0 was assumed in order to derive the non-linear relations. Instead of convected derivatives, one could use Jaumann derivatives (Fredrickson, 1964; Prager, 1961, and Oldroyd, 1950). The covariant and contravariant tensors are equivalent expressions in terms of Jaumann derivatives. Another approach to describe non-linear behavior is to formulate a non-linear superposition principle (Noll, 1958). However, this new theory sometimes yields the same result as the proposed Rivlin-Ericksen model (Coleman and N011, 1959). This theory had been followed by similar approaches (Green and Rivlin, 1960). Non-linear behavior of anisotropic fluids had been treated with a very different approach, by introducing relaxation effects (Ericksen, 1960). 34 Further development on non-linear viscoelastic behavior has been presented by Bychawski (1974), Lockett (1974), and Sobotka (1975). Comparison of experimental data with theoretical results was reported by Yoshiaki (1977). A non-linear viscoelastic formulation for apple material by representing the time derivatives appearing on the here- ditary integral forms by convected time derivatives is proposed in the following discussion. Although it was not used to isolate the failure parameters, it is included to stimulate the possibility of using a non-linear Viscoelastic theory for apple flesh. 4.7.1 The convected derivative of a covariant strain tensor The convected coordinate system can be understood as a reference frame which moves and deforms with the deforming body (Fredrickson, 1964). Some authors use material coordi- nates as a synonym of convected coordinates (Green and Adkins, 1970). If the strain tensor is written as a covariant cartesian tensor Eij’ its convected derivative D eij/Dt can be expressed as (Fredrickson, 1964) k _ k . k. D eij/Dt aeij/at + v eij k + v,1 Ekj + v,j Eik (4.66) where the commas in the subscripts indicate differentiation. The term vk is a velocity along the X1, X2, and X3 axes. The differentiations vgi and vgj are gradient velocities expressed as (Eringen, 1962) 35 213 = (1/2) (vi j + vj i) (4.67) If symmetrical conditions are held in respect to the axis X1 and 612 = 621 813 = 631, equation (4.66) can be wr1tten as De11 = 3811 + V(1) 3811 + 2V(2) 3811 at at at 323" (1) (2) 8v 3v + 2 —§§I— all + 4 3X1 612 (4.68) Recalling the definition of the infinitesimal strain tensor eij = (1/2) (an/axi + an/ij) (4.69) A strain function of time and strain rate 811(811,t) is imposed on the linear viscoelastic model in the X direction. 1 This means that deformation should also be function of time, Ui(Xi’t)‘ The X1 direction is the only important one due to the fact that the strain and deformation parameters are imposed in this direction. If the deformation is considered a linear function with respect to time and X coordinate, equation (4.68) reduces l to (l) (1) 3811 D811 as11 + V 3x1 Dt = at ax 811 (4°70) Once the deformation function has been determined, the non- linear viscoelastic expression for the different loading situations could be found by replacing the linear strain rate tensor Eij by its convected counterpart z.., where 13 * _ D811 € 11 ’ "55' (4.71) CHAPTER V EXPERIMENTAL PROCEDURE 5.1 General Remarks In the first group of experiments the apple specimens were subjected to compressive loads up to failure. Failure was determined by the point on the loading curve which indicates the end of the elastic behavior. As mentioned previously, cylindrical and cubic specimens were loaded uni— axially, biaxially, or triaxially. All specimens were subjected to a uniaxial strain rate (611) unless a radial stress failure occurred prior to the axial loading. Displace- ment and force values at failure were recorded on a strip chart recorder. The axial load was applied using an Instron TM model testing machine which had several different loading speeds, allowing a wide range of strain rates to be imposed on the specimen. The tests were divided into seven groups according to the loading conditions, and the shape and size of the specimens. Table 5.1 shows the loading conditions, shape, and dimensions of the specimens. The mean and standard deviations of the basic dimensions are given. These were calculated from ten measurements taken from each type of specimen. Twenty replications of each individual type of test were conducted. The individual stress (all) 36 TABLE 5. radial stresses. 1. 37 Tests carried an apple material. Imposed strain rates and Shape and dimensions of the specimens. -EE§§"'T'E"" ' """""" (silo 1) (MPa) 1-0.002 -0.007 -0.017 CNIAXIAL -0.035 0.000 -------- W1--------h-------n UNIAXIAL -o.007 0.000 ———————— uL-I—--—--A---—-—-- 0.000 -0 e 067 :RIAXIAL -0000? -0e138 -0.207 RIGID -0.007 DIE 3314x141 -0. 007 3111141 -0.007 -‘OO--OCdb--------------- RIGID P0.007 DIE -n------d-n-a—ndb-------- (*)stsndsrd deviation SPECIMEN SHAPE b ------------ , CYLINDRICAL CYLINDRICAL CYLINDRICAL L CYLINDRICAL- CUBIC --------------- H [0W GM 3- lo H O? 00‘ 0“) Oh! I- so MN 5 O C we Hm HH PH cm 1.) Ha: MU: #- 3b 'h--==- CUBIC L------d 19.30 0. GM 0 0 DUI p------- 153.02 5.25* 38 and strain (ell) values at failure for those replications are given in the appendices. In the second group of experi- ments, cylindrical specimens of apple were subjected to stress rate controlled uniaxial loading. 5.2 Apple Selection and Storage The varieties Red Delicious, Jonathan, and McIntosh were harvested during the 1977 growing season and were stored at 0-2°C in plastic bags. They were removed from storage 24 hours before being tested. 5.3 Specimen Preparation The specimens were prepared by driving a corkborer into the apple parallel to the stem-calyx axis. The specimen was then placed in a cylindrical hole in a plexiglass bar and the ends were cut parallel to the faces of the bar by using a sharp blade. The same procedure was used to obtain cubic specimens. In this case, a square corkborer and a square trimming hole were used. 5.4 Uniaxial Loading of Cylindrical Specimens at Different Strain Rates Cylindrical specimens with a height of 12.20 t0.08 mm, a diameter of 12.58¢O.17 mm, and a cross-sectional area of 124.29:4.84 mm2 were uniaxially loaded to failure in the Instron testing machine at the following strain rates: -0.002, -0.007, -0.0l7, -0.035, -0.069, -0.137, and -0.347 -1 sec . This first group of tests is summarized in Table 5.1. 39 5.5 Uniaxial Loading of Cylindrical Specimens of Different Height at a Constant Strain Rate Cylindrical specimens with a constant cross-sectional area of 292.55:6.06 mm2 and heights of 8.32:0.06, 12.1340.12, 19.17iO.13, 26.55:0.16, 34.9810.12 mm were uniaxially loaded in the Instron testing machine at a strain rate of -0.007 sec- . For these tests the deformation was obtained for each height with a constant force of 36.38 N. This was done to obtain the variation of the deformation with the height (H) at a fixed load level. 5.6 Triaxial Loading of Cylindrical Specimens at a Constant Strain Rate, and Different Radial Stresses This group of specimens is the third row of Table 5.1. In order to impose a constant strain rate of -0.007 sec.1 along the X1 axis and at the same time impose a radial stress, a22, a special apparatus was developed. Figure 5.1 shows the details of this device. The specimen is contained in a very thin wall rubber tube (6). Two aluminum rods (1 and 11) are in contact with the bottom and top of the specimens. Those aluminum rods are axially and radially perforated in order to allow any small quantity of air that might be trapped between the specimen and the rubber tube to escape. Trapped air would transmit the load applied on the outer surface of the rubber tube to the bottom and top surfaces of the specimen. This situation would create a hydrostatic stress state before the specimen was axially loaded. This test creates a radial stress, 6222611. Figure 5.1. Legend: Triaxial loading device, View (b). 1 Aluminum rod 2 Brass tube 3 Bolts 4 Steel frame 5 Rubber cork 6 Rubber tube 7 Specimen 8 Plexiglass tube 9 Brass tube 10 Rubber cork 11 Aluminum rod 12 Opening 13 Plexiglass frame 14 Steel frame 15 Plexiglass frame 16 Plexiglass tube-frame Air pressure valve Air pressure release showing the longi- tudinal cross-sectional View (a) and top 40 bbWNH-J (a) (b) / f l I/ I \ \\ I" \‘ \\ 12 © .7 {ii Figure 5.1. Triaxial loading device, showing the longi- tudinal cross-sectional view (a) and top view (b). 41 In a hydrostatic stress situation, the axial stress all would be always larger than the radial stress 022 (in this case equal to the hydrostatic stress). The aluminum rods are fitted inside of two brass tubes (2 and 9) which were glued to the rubber tube. Two rubber corks (5 and 10) were inserted in the top and bottom of the plexiglass tube (8). The brass tubes (2 and 9) are fitted in the circular holes made in the rubber corks (5 and 10). The cylindrical specimen was placed between the alumi- num rods and lubricated with vaseline to avoid friction. The apparatus was placed on the load cell of the Instron testing machine, keeping the upper aluminum rod in contact with the compressive head. A strain of —0.007 sec.1 was imposed to the specimen through the aluminum rod.- The radial stresses acting as the outer surface of the rubber membrane were created by connecting the opening (12) on the plexiglass tube to an air pressure line before the axial load was applied. The cylindrical specimens had a diameter of 12.58:0.l7 mm, a height of 12.22:0.08 mm, and a cross-sectional area of 2 124.2914.84 mm . The selected radial stresses were equal to 0.000, -0.069, -0.138, -0.207, and -0.345 MPa. 5.7 Rigid Die Loading of Cylindrical Apple Specimens A rigid die as shown in Figure 5.2 was used to obtain axial deformation while constraining the sample in the radial direction. This made it possible to impose an axial strain rate, 611’ wh1le keeping 622 = €33 = O. The spec1men was placed in the cylindrical hole, topped by an aluminum rod. (a) (b) Figure 5.2. 42 M] 4:.me _J #03 Circular hole Aluminum rod Brass rigid die Specimen Top (a) and longitudinal cross-sectional view (b) of the device for loading of cylindrical specimens in rigid die. 43 The die was then placed on the load cell of the testing machine and a constant strain rate of -0.007 sec.1 was imposed on the specimen through the aluminum rod. The specimens used in this test had a height of 12.2210.08 mm, diameter of 12.5810.17 mm, and a cross-sectional area of 124.29:4.84 mmz. 5.8 Uniaxial Loading of Cubic Specimens at a Constant Strain Rate Cubic specimens having a dimension of 12.3720.09 mm and cross—sectional area of 153.0235.25 mm2 were uniaxially loaded in a testing machine under the conditions given in Table 5.1. 5.9 Biaxial Loading of Cubic Specimens at a Constant Strain Rate In the biaxial test, a cubic specimen is loaded axially along the X1 axis while constraining the side orthogonal to the X1 axis (622 = O). The side orthogonal to the X3 axis was free to move. The apparatus designed to allow these features is shown in Figure 5.3. The two blocks (1 and 2) kept the bars (3 and 4) at a constant distance from each other (12.3710.09 mm). The specimens had a dimension of 12.37:0.09 mm and a cross-sectional area of 153.0215.25 mmz. This apparatus was then placed on load cell of the testing machine and the strain rate is imposed to the specimen through the square cross-sectional area steel bar (5) (see the sixtieth row of Table 5.1). 44 — Steel block — Steel block - Aluminum plate Aluminum plate - Steel bar - Cubic specimen WU1hWNt-J l 1. ’4 / Figure 5.3. Exploded view of the device used for biaxial and rigid loading of cubic specimens. 45 5.10 Rigid Die Loading of Cubic Specimens The cubic specimens used in this test had the same dimensions as the uniaxial and biaxial specimens. The appar— atus described in 5.9 was used. This time, the block (2) was aligned such that a constant distance between the bars (3 and 4) was obtained. The cubic specimen was the biaxially constrained. The force orthogonal to the axis X1 was loaded by the steel plunger, to which the strain rate of -0.007 see-1 was imposed. The bottom row of the Table 5.1 summarizes the conditions of this test. 5.11 Stress Rate Controlled Loading of Cylindrical Specimens of Red Delicious This experiment was designed to control stress and give freedom to the state of strain. Cylindrical specimens of Red Delicious with a height of 12.22:0.08 mm, a diameter of 12.58:0.17 mm, were uniaxially loaded.to failure by control— ling the stress rate. Six different stress rates were chosen, from 0.0005 MPa/sec to 0.013 MPa/sec (see Table 6.7). Figure 5.4 illustrates the apparatus designed for this test. The specimen is placed between the plate of a scale (1) and a rigid plate (2). Deformation on X1 direction is measured by a LVDT device (3) and recorded on a strip chart recorder. The second plate of the scale supports the loading water container (4). The water reservoir (5) was kept at a constant level by the outlet (6) and inlet (7). By controlling the valve (8) it was possible to control the stress rate being applied to the apple specimen (9). 46 .mspepsann mcfipeoH compo .v.m mhswfim w .AwWI q IfiH :mEHommm I m>Hm> Hope; I pods“ hope: I aoaupo hope; I hfio>nmmop poems I nmcfimpnoo popes wnfipmoq I moa>06 eo>n.u mamas usage I madam may no madam I I-INCOVIDCDL‘WGI CHAPTER VI RESULTS AND DISCUSSION 6.1 General Remarks Values of all and $11 at failure were experimentally obtained for all the loading cases discussed in Chapter V. In the case of triaxial loading, the values of 022 were also known. Remaining parameters such as 522 for uniaxial and triaxial tests, 533 and 022 for biaxial and 022 for rigid die loading were determined using elastic and viscoelastic formulations. This allowed the experimental and theoretical values of 011 to be compared. The availability of 022 a1So made possible the calculation of Tmax and 322 which was needed for the calculation of the strain energy components. Viscoelastic relaxation functions were not available ,for the McIntosh and Jonathan varieties. The functions E(t), X(t), Gl(t), 62(t) and v(t) determined by DeBaerde- maeker, 1975, apply only to Red Delicious. However, the experimental data obtained for McIntosh and Jonathan varie- ties are presented in parallel with those from Red Delicious with the purpose of comparison. 47 48 6.2 Uniaxial Loading of Cylindrical Specimens at Different Strain Rates In the uniaxial compression test of cylindrical speci- mens, strain rates of —0.002 sec-1 to —0.347 sec.1 were imposed. Average values and standard deviations of stress and strain to failure are shown in Table 6.1. This data is also illustrated on Figures 6.1, 6.2 and 6.3, respectively. The axial stress increases exponentially as strain rate increases while strain values at failure do not exhibit significant changes. Figure 6.2 suggests that strain at failure can be represented by a straight line parallel to the horizontal axis. The average axial strain values for the various strain rates are -0.1l:0.008 mm/mm, -0.13t0.017 mm/mm and -O.1210.012 mm/mm for Red Delicious, Jonathan and McIntosh, respectively. Standard deviations for the axial stress at failure varies from 10 to 15 percent. The standard deviation for strain at failure was about 10 percent of its average value. The value of a11 calculated using the viscoelastic formulation has the same general form as the experimental data. A constant value of all was not obtained because the strain rate is a parameter in the viscoelastic formulation. Lateral strain 822 at failure was calculated from the elastic equation (4.31) and its viscoelastic counterpart (4.61). The values obtained from the elastic formulation are higher than the viscoelastic values, however both of them are relatively constant. From Table 6.2 one can see 4 u a Bulls-IE .cofiuafi>oa anaesmumA«V "-I""I"l'l'l'|"""'lll|'l'l"|"l|'ll'llt-l‘i'l|“"ll|l"'ll‘l"| «cc.o «Ho.o «rc.o «co.o «Ho.c «No.0 «co.o «No.0 «No.o mm.o NH.o- se.cu an.o ma.on mm.o- an.o mH.o- -.o- oqm.o- amc.o *Hc.o «oc.o «mo.o «do.o «so.o aeo.o «Ho.c «no.0 cm.o cH.o- mm.c- cm.o OH.on sn.o- sm.o ofi.ou mm.ou naa.o- «mfl.o ch.o «mc.o «mm.o “No.0 aso.o «No. «Ho.o «No.o ss.a NH.o- em.ou mm.a nH.o- «m.o- mm.H ma.ou sm.o- soo.o- «mm.o «Hc.o «mc.o ao¢.o «No.0 zqo.o «ms.o «Ho.o «mo.o ad.m HH.o- mm.c- sq.m NH.ou om.o- es.m NH.o- c~.o- nmo.o: mm.o «Ho.o «mo.o «ma.o «No.0 «no.0 «Ho.o gHo.o «so.o em.o NH.ou on.Oi mm.s HH.o: o~.o: mm.s NH.o: m~.o: ”Ho.o: «Ac.a «Ho.o «No.o aae.a amo.o «No.c #sm.a «no.0 «No.0 so.ms HH.¢- mm.c- o~.c~ «H.o- -.o- am.sa NH.o- s~.o- ~oo.o- gem.“ «No.0 aqc.o «an.m «so.c ado.o «sH.m «Ho.o «No.0 cc.co ma.cu om.ou oc.m~ mH.ou oH.ou oo.o~ ea.on ma.on moo.ou Auomv ¢Eac unaccaum can mosam> owwum>< .mmuwz camuum ucououufin you mcosaomam oaaa< quauccaazo mo madnaoq quxaaca .H.c mam mo wmomza any a can A00 a: .mofiuomce ofiummaooumw> 0cm oaummam ozu Scum %mumaaoamo Hmucmefiumaxm Eouu vocfimuno mmouum ummzm Essfixmz a Anv m: NNw HHD 050.0 000.0 050.0 050.0 050.0 000.0 500.0 “no, : accumuuaa um m=OHUfiHm0 com «o mcmsaomam 000.0 000.0 N00.0 000.0 ~00.0 000.0 ~00.0 Anv e: H00.0 050.0 H00.0 050.0 H50.0 050.0 000.0 'l"l-'I""'l'-'|'l|l'|'|" >HHUH80 0cm @000 u we mosam> vmumaaoamo van ”flu ”H000 mosam> Hmucmsfiumaxm Scum vocaMuno mucoGOQEoo awuocm camuum .mmumm :Hmuum Hmofiueafiaso,uo wcfieaog Hmflxmfica .N.o mgm mazfiuau an mmoaom .q.0 ma:m«m An:onm0-w mean cashew 0m.o: 00.0: 0N.O: o_.o: 00.0: o n . O T. C. .3350... .0 casuacoal . W. W 9532.5 as: o t. m. S 1 . I O a I n s z S C. D T. .I. m m n o a Cu I.\ 0 . . 0 co 0: .mcoswomdm manna ”coast ::wa>o no unawao~ Hafixdficz hen ouch :“auum mamuo> mazmuau ea :fiauom .N.0 onsmwm AH:oom0 «do was: aaauam 00.0: 00.0: 00.0: 0H.0: 00.0: 0 d d d d d m. w : . .I t 4\\\. Au W . s L 0 1+ . J .L t 9 r: u on 2 c 5 z ) nu MW noon—50:. ( 5:30:00 I £5338 to: 4 53 .m:oEuouQm manna Haofihuzumau no usucao~ Hauxaazz yea wash :aahum mzmhm> $509 .m.0 ohzwum own Ham and: zaaaom H: r 00.0: 00.0: 00.0: 00.0: 00.0: 0 0. a u - 1 III] if i], I- I f. I/ L 0 1 T. 0 .7 L . 0 m 1+ \s, .3353... u caaaacowl ( maofiuwgvn send L O I 0 0 54 that the values of Ud are quite constant; this seems reason- able when we consider that Ud is calculated from strain parameters. Remaining parameters, Tmax, Us’ and U vary as strain rate changes. Strain at failure presents a relatively constant value as strain rate varies. From these results it appears that the axial strain is a possible failure parameter. 6.3 Triaxial Loading of Cylindrical Apple Specimens The average values of axial stress, axial strain and time at failure as well as the values of the imposed radial stresses during the triaxial loading studies are displayed on Table 6.3 and illustrated on Figures 6.4, 6.5 and 6.6, respectively. The average normal stress 011 decreases while the normal strain 511 remains relatively constant when the radial stress increases. McIntosh and Jonathan varieties failed for a radial stress loading between -O.345 and -O.4l4 MPa and Red Delicious failed between -O.4l4 and -O.483 MPa when 0 =0 MPa. This means that radial stress at failure 11 is twice or three times larger than the axial stress at failure, which eliminates the maximum normal stress failure criteria. Table 6.4 shows the values of the maximum shear stress calculated from experimental data as well as the strain energy components obtained from elastic and viscoelastic theory. Maximum shear stress decreases to a minimum of -0.028 MPa and increases for consecutive values of radial stress. This indicates that during a continuous variation 55 .sofiumfi>ov vuwcamuma«v "-'-'-"I"|'"||'I"|-Il"""'-'|ll'l'l""“l"lll"""'|l"""‘l' «00.0 00.00 «00.0 00.5H «0~.0 00.50 «00.0 00.5H «00.0 00.0H «00.0 00.na «00.0 00.0 «00.0 NH.0: «00.0 NH.0: «00.0 NH.0: «00.0 0H.0: «00.0 HH.0: «00.0 00.0 «00.0 «0.0: «00.0 00.0: «00.0 00.0: «00.0 00.0: «00.0 N0.0: 00.0 00.0 «00.~ 00.00 «00.0 00.5H «00.0 00.00 «00.0 00.0H «00.H 0N.0N 00.0 00.0 «00.0 HH.0: «00.0 NH.0: «00.0 00.0: «00.0 0H.0: «00.0 0H.0: 00.0 00.0 «00.0 00.0: «00.0 «0.0: «00.0 00.0: «00.0 00.0: «00.0 5~.0: 00.0 00.0 «50.0 00.0H «00.H 00.5H 000.0: 050.0: 500.0: 000.0: 000.0: 000.0 l-"'l'l|""I'I"""'I‘l|l-"l“lI.‘!‘l"‘l|‘l.l|1|'l“‘-‘l‘I‘ll"""|‘lt Auomv uu AEE\EEV ~00 A0020 so Aummv u AEE\EEV ~00 A0020 . HH 0 Aummv u A0020 mm D l"!'ll‘|-'||"l"l"'|"‘|""'l'l'l"|l|"'|“||l'|"l"'|l"“‘t"lt‘ mSOHUHAmn num z<=a<200 00.0 00.0 00.0 00.0 «00.0 «00.0 0H.0: 0H.0: «00.0 «00.0 HH.0: 00.0: «00.0 «00.0 0H.0: «0.0: «H0.0 «00.0 00.0: 00.0: «H0.0 «H0.0 NH.0: 00.0: 2553 3.00 “Hm "no mmOHzH 02 . .musaqmm um :«muum new amouum Hmfix< .mafis no mmsau> ummum>< .0000 :«muum oamumaoo 0:0 mmmmouuo Hmavmx usuumuua0 you msoafiomam mH00< Hmuauchaao no 0000004 Hawxmaua .0.0 mqm ho waomze fi 000 A00 A00 Ass\ssv A0020 A00 A00 Anv Aijesv A0020 Ammzv 00:0 0 00 .0 Na. 00° 0 00 m: N. 000 «as. «No - . EII""""""I"I|'|||Il"""--'|'|l'll'l"|""'I""l|"'||"|"I"|-""E IlllllllllllllllllllllIlllllllllll;:llllllllllll rHHUHHm<0m mo wxomza "'|"'|I"""I'llIll'l'l'|l""'t 0---..-555... .mn.0u> .00: 000.0n0 .mmfiuooze ofiumaamoumfi> 0:0 aflummHm Eouu vmumaaoamo 0flwuo mwsam> 0:0 mumn Hmucmafiumaxu Eoum 00:00000 mmmuum umwzm EDEme: A 00m 00 mmsam> woumasuamo 0:0 0:000 mo:~m> Haucmsaumaxm Eouu wmcfimuno mucucoasou xwuwcm :Hmuum .mmmuum 000000 ucmummmfia um msoHU0Hm0 000 00 mamsfiomam HmUHuccfiflxo «a 0000000 Hafixmfiua .0.0 000 muafiuau ua mmonum ~afix< .v.m ouzmqm Aaaav «mo amouam fiafiua: mm.o- om.ou om.o- o~.o- o 11 d 1 A. H aw QI'O‘ 03'0- 1 98'0“ j/’ (Bdfl) IIo ssazas terxv / 02'0- P—WOH:HU—‘. / :azua:OHl macaofifimn 60: d 98'0' 58 .mcosuomam manna aaoupcaauao no mcfivaoH ~auxa~ua new mmoaum Hafioau mzmuo> onzauau ad :«auam .n.m vgamfim Admsz mm.OI on.o: ON.OI o~.on o . a a 1‘ {IllllllllllliWWMMMMMMWMHHHUHHH/l! 1 $3530 assuacowl 35323 as: QI'O' OI'O' 90'0“ 08°C- (mm/mm) 59 ov.cn .mcoeaomqm manna Haofiun saunao «o m:~oao~ Hauxafiua new mmmuum gauuau mzmuo> mus~fiau ea mama .©.m annuam Augsv «no mmmaam fiafloam on.c- om.ou o~.o- d 1 - .3352; casaaconl macdnvfldg me‘ OZ OI (ass) 1 amt; 08 0? 60 of radial stress, the maximum shear stress would reach values close to zero or even possibly zero at failure. This elimi- nates the possibility of the apple flesh failing when the maximum shear stress exceeds a critical value. Table 6.4 presents a relative constant value for the spherical component of the total strain energy at failure as calculated by the elastic theory. However, the visco— elastic results show a relative variation for the spherical energy component. Remaining strain energy components vary with increasing values of radial stress. Results from tri- axial loading of cylindrical specimens strongly point to the conclusion that apple flesh fails when a critical value of normal strain reaches a critical value. 6.4 Uniaxial, Biaxial and Rigid Die Loading of Cubic Specimens. Uniaxial and Rigid Die Loading of Cylindrical Specimens. Uniaxial, biaxial and rigid die loadings of cubic specimens are formulated by the elastic equations (4.25) to (4.31) and by the viscoelastic equations (4.50) to (4.57), (4.60) and (4.61), respectively. Table 6.4 gives the experimentally obtained parameters and Table 6.6 gives the calculated values for o and the values for the remaining 11 parameters as calculated by the above equations, in addition to maximum shear stress values. The axial normal stress varies for the different loading cases, for both the cubic and cylindrical specimens, while the axial normal strain remains relatively constant. This supports the conclusion 6] .cOHum«>mv wumvcmumAav «oa.~ «No.0 «no.0 «cm.m «eo.o «mo.o (*H~.~ euo.o «no.0 am.aa NH.o- mm.o- o~.o~ «H.o-. mm.ou mm.- NH.o- mn.ou Aqv «ms.~ «Ho.o «qc.o «c~.~ *Ho.o «no.0 «-.H «Ho.o amo.o mm.~H -.o- am.o- «m.mfi m~.o: an.o- mn.~H NH.o- -.o- Amv «mm.~ «No.0 a~o.o «oa.~ «Ho.o xmo.o emm.a «Ho.o amo.o sm.ma HH.o- m~.o- 4m.mH HH.ou -.ou mn.~H NH.o- -.o- ANV «~9.H «Ho.o «mo.o «H~.m awo.o «mo.o «mm.H gHo.o goo.o ea.ma HH.ou se.ca mm.~H NH.ou an.on mn.- NH.ou om.ou Adv Aommv AEEEEé Ammzv Aommv AEEEEE Ammzv Aommv AEE>E¢ Ammzv monnguc can z<=eoa cumccmum can mmmuo>< mozan>.oada< mo mamEfiomdm ofiazu mo mwcfivmoa mmv Hmfixmfim paw ANV Hauxmaca .mcosaumam Ago canzo can AHV Hmoauecfiaao mo cacao; «an camam .m.o mam van oaummam Eoum pocfimuno mucocodsou awumcm can camuum can mmouuw kuvmm no Hmuouwa .cfimuum can mmwuum Hoax< . «v mcmfiuumom mowuvsfiamo mo mcqvmoa can wwwwm .mcmawomdm canso mo Amv m cacao; man a“ «a ecu Amv Hmflxmfim.afiv Hmaxmfiaa .o.o mnm Hmucmsaumaxm .mucmcomEoo swuocm :«muum uo wmzflm> coucHsono.m:ofiofiHoc cod mo mcwsaumdm mo wsfiwmog woaaouucoo mmmuuw m.o Nance 61- «me.H «mm.H «mo.~ «an.~ «ow.H He.mfl- sm.oa- co.m- ew.m- no.o- anoHUfiHmn so“ «Ho.~ «-.~ «mH.N «mm.a «-.~ m~.mfi- um.efi- 0H.NH- mm.m- Hm.m- cmgumcon «oq.~ «he.~ «os.m *MH.N «sm.~ no.o- w~.oa- mm.ma- m~.cH- o~.m- smoucH oz Aaavaa Asavfia Assvfio Assvaa Aasvaa mmfiuofiua> «HH.o «cfi.o «mfi.o «NH.o *oo.o mm.¢m mm.c~ NH.¢H ma.- mm.» AEEvz A550: A550: A550: Asavz ------44 uuuuuuuuuuu ----- uuuuuuuuuuuuuuuuuu - ...... --- .adz HH.c3n o um :ofiumsuommv mo mcofiuma>mw pumvamum cam mosam> owmuo><.mcwafiomam madm< ququvcwa>o mo wcfipwoa Hoaxmucs .m.o mqm ouanau ea cannon .0.0 ouzmfia noom\aazv can: mmmhum m~0.0- -0.0- 000.0- 500.01 000.01 0000.0- 0000.0- I ‘ d. d ‘ I- )- OI'O' II'O‘ (mm/mm) tta urvxzs Intxv fl fl 1 ZI'O‘ SI'O‘ 68 .mcupao~ aaaxawca mnuuzn oaaa> mmmnaw Hawxa assumcoo pa uzmgmn accumuuuc no mcosfiomnm wanna aaofiuusua>o uo no=~a> :ofianquon. .0.0 ousmqm 055 = 232. .3383 0.00 0.Nn 0.0m 0.vN. 0.0m 0.m~ 0.N~ 0.0 111 \ . Lu \ I\\\\I 0 IV . 13X . t. 03 TL .m I LII. 90 On 3 1 T? .0 an 0 . n o OI .m Face—:03. W casudcohl W m:0«o«~00 vom< .mcuuaog «afixauca m:«a=0 no=~a> mayhem usaamzou an unnams acououudc no mumsuooam mamas ~aouu00->o uo m33~a> custom .0~.0 oasuq: A550 a usage: :maaomam 69 O.®m O.Nm O.mN O.VN O.CN O.QH O.N~ 0.x 7 . q + q q d d L m .0 TL fm 0 W 8 t. 3 I “/- o I .nv n. ./ / O B c. I. u 3 . . I w m / . m I 0 I\ £mOH=HO=. .0 . 5:35:00. 6 253:3 .63: 7O drawn from the experimental results presented in 6.2 and 6.3 that axial normal strain is a possible failure parameter. This group of tests discards the possibility of the maximum normal stress being considered as the failure parameter. 6.5 Stress Controlled Loading of Cylindrical Specimens of Red Delicious Normal stress at failure decreases from -O.391 MPa to -0.139 MPa as the stress rate increases from -0.0005 MPa/ sec to -0.013 MPa/sec, while the strain at failure remains relatively constant, averaging -O.12 mm/mm, Table 6.7 and Figures'6.7 and 6.8. Table 6.7 also gives the values of the maximum shear stress and strain energy components at failure. The fact that a creep failure can be induced in apple specimens is additional support to the hypothesis that apple flesh fails when normal strain reaches a critical value. 6.6 The Non-linear Viscoelastic Formulation for Apple Material Cylindrical specimens of apples of different lengths were axially compressed. For each different length, the axial deformation U1(X1,t) at a predetermined stress level 011 = -0.11 MPa, was obtained as explained in Section 5.5. Table 6.9 shows these deformation values for each of the five lengths. These values are also illustrated on Figures 6.9 and 6.10, respectively. Ideal conditions are assumed, i.e., axial stress does not change along the X1 coordinate 71 which is the same as to say that the deformation at 011 = -0.11 MPa for the specimens of different height each, represents the deformation along the X1 axis for the tallest specimen. In this case, X1 assumes values equal to the heights of each individual specimen. Time parameter is referred to the deformation of the tallest specimen (H = 34.98 mm). Data from Table 6.9 can be fitted in the following power function for McIntosh, Jonathan and Red Delicious, respectively, U (x ) = (-1 007 x -0 036X 2+0 007x 3-0 0004K 4) (6 1) 1 1 ° 1 ° 1 ° 1 ' 1 ° U (x ) = (-3 099x +0 454x 2-0 031x 3+0 001x 4) (6 2) 1 1 ' 1 ° 1 ' 1 ' 1 ° U (x ) = (-3 088x +0 579x 2-0 049x 3+0 002x 4) (6 3) 1 1 ° 1 ° 1 ' 1 ' 1 ' with the following respective coefficient of determination: 0.99, 0.99 and 0.99. If the deformation rate of -0.085 mm/sec is imposed at X1 = 34.98 mm (top of the specimen) and time t = 15.84 sec at 011 = -0.11 MPa are computed, equation 6.1 can be rewritten as: 2 3 4 -0.003X +0.0001X +0.036 X1 1 1 U1(X1,t) = (-O.195X )t (6.4) 1 With the deformation function written in terms of spatial and time coordinates the elements of equation (4.70) can be determined as: 72 - 2 3 811(X1,t) - (-0.195+0.072x1-0.009x1 +0.0004x1 )t (6.5) 3811 2 3 ———— = (-o.195+0.072x -o.009x +0.0004x ) (6.6) at 1 1 1 v(1) = (-0.195x1+0.036x12-0.003x13+0.0001x14) (6.7) 3211 = (0 072 - 0 018x + 0 0012x 2)t (6 8) ax1 ° ' 1 ° 1 ° 3v(l) 2 3 3x = (-0.195 + 0.072x1 - 0.009x1 + 0.0004x1 ) (6.9) 1 From equations (4.66), (6.1), (6.2, (6.3), (6.4), (6.5), (6.6), (6.7), (6.8), and (6.9), the convected axial strain rate can be written as: D611 = (-0 195 + o 072x - 0 009x 2 + 0 0004K 3) Dt ° ° 1 ' 1 ° 1 2 3 2 + 2(-0.195 _ 0.072xl - 0.009x1 + 0.0004x1 ) 2 3 4 + (—0.195 + 0.036x1 - 0.003x1 + 0.0001xl ) . (0.072 — 0.018xl + 0.0012X12)t (6.10) Equation (6.4) should be rewritten if the specimen has a total height different from 34.98 mm and it is subjected to a deformation ratio different from -0.085 mm/sec. Equation (6.5) describes the variation of strain along the X1 coordinate and according to time. Equation (6.10) is the convected strain rate to be substituted on linear viscoelastic equations. 73 The introduction of the spatial coordinate in the visco- elastic equations enables one to relate the strain and strain rate parameters to a fixed point in the body being loaded. A valid question could be raised against such experi- mental procedure. By loading different sizes of specimens, contact stress is developed on the surface being loaded. If only the taller specimens were tested and the deformations were obtained at several axial positions, the question raised could be neglected. A quite useful technique to circumvent this problem would be to mark several points along the height of the specimen and to record the positions of the points by taking pictures during several steps of the loading procedure. Further analysis of those pictures would yield the data to describe the deformation function. Another second question is related to the Poisson's ratio effect. If lateral deformation is measured, the chosen stress level could be found to be slightly different for each specimen. Equation 6.5 indicates that strain is larger for higher values of X1. In other words, strain is higher at the top of the specimen. If normal strain is the failure parameter, failure should start on the surface where the load is being applied. Figure 6.10 illustrates the variations of axial strain for different specimen height. 6.7 Summary Experimental results from uniaxial, biaxial, triaxial, rigid die and creep loading were used to study the parameters 74 involved in the failure phenomena of apple material. Elastic and viscoelastic formulations were used to calculate the parameters not experimentally obtained. From the parameters considered -— maximum normal stress, maximum shear stress, maximum normal strain and strain energy components -- the maximum normal strain was found to be the most likely factor producing a failure in apple flesh. A non—linear viscoelastic formulation has been proposed, based on the model described by Oldroyd (1950) and Fredrickson (1964). Such modeling procedure consists in obtaining the deformation vector as a function of time and space from which the convected strain rate tensor was obtained. The substitution of the strain rate tensor from linear visco- elastic equations by the convected strain rate tensor com- pletes the non-linear viscoelastic formulation. CHAPTER VII SUMMARY AND CONCLUSIONS A failure criteria for apple flesh was presented. A new experimental technique has been developed in order to apply biaxial and triaxial loadings on apple specimens. Uniaxial loading of apple specimens showed that normal stress at failure varies with strain rate while the normal strain turned to be relatively constant. This eliminates the possibility of considering normal stress as failure parameter. Triaxial loading of cylindrical specimens also indicates a constant value for normal strain at different levels of cylindrical stress. This experiment also showed significant variations of shear stress and normal stress at failure, including failure of the specimens at zero level of shear stress, which eliminates the maximum shear stress as a failure criteria. Uniaxial, biaxial, and rigid die loadings of cubic and cylindrical specimens indicated that normal strain at failure remained relatively constant for these different loading cases, meanwhile the normal stress at failure varied. This group of experiments also eliminates the maximum normal stress as a failure criteria. Stress controlled uniaxial loading shows decreasing values of normal stress 75 76 at failure and constant values for normal strain at different stress rate values. This test also eliminates the maximum normal stress failure criteria. Calculated values of total strain energy and its spheri- cal and deviatoric components, obtained from viscoelastic and elastic equations showed significant variations. A non-linear viscoelastic constitutive equation, based on the substitution of the strain rate tensor by a convected strain (Oldroyd, 1950) was proposed. For this accomplish- ment a deformation function in terms of time and space had been obtained. This resulted in a time and space dependent strain and strain rate tensors. The following conclusions can be drawn from this study: 1. Apple tissue fails_when a normal strain exceeds a limiting value. The average normal strain values at failure for all the tests conducted was 0.116 0.007 mm/mm for Red Delicious, 0.126 0.014 mm/mm for Jonathan and 0.122 0.013 mm/mm for McIntosh. 2. There exist a noticeable difference in the mechanical behavior of the three apple varieties tested. 3. The developed experimental procedure yields reliable data. 4. The proposed non-linear viscoelastic constitutive equation can be considered as a preliminary step toward more complete formulations. CHAPTER VIII SUGGESTIONS FOR FUTURE RESEARCH In spite of positive conclusions concerning the failure of apple material that has been reached, certain points still remain unclear. The present work shows a visible difference on the mechanical behavior of the varieties tested. It was seen that viscoelastic functions are available only for Red Delicious (DeBaerdemaeker, 1975). The determination of the time dependent functions Gl(t), 02(t), E(t) and v(t) for varieties of economical importance would provide a better understanding of their mechanical behavior. The strain level at failure possibly varies with the physiological state of the apple tissue. This includes ripening, time and conditions of storage, as well as water potential level. DeBaerdemaeker (1975) reported that the failure of cylindrical apple specimens under compressive uniaxial loads varies from -0.49 MPa at the beginning of the storage period up to -0.34 MPa after four months of storage. This information can be useful in determining the best physiological state for mechanical handling of apples. In other words, a certain amount of bruise damage can be expected for different stages of maturation, water 77 78 potential level and time of storage. These factors should be included in future experimental works. Apple material has been considered homogeneous within the same experimental specimen. For the time being this assumption can be considered satisfactory, however the varia- tion of mechanical properties inside of the fruit should be investigated. This topic should encompass the development of a more realistic shape for the apple fruit. The average size and shape for a specific variety should be determined. Now the whole fruit is divided into elements and for each element specific mechanical properties are allotted. This finite element model would yield the strain level distri- bution in the fruit. This concept will guide the handling of whole fruits since bruise could now be predicted and located. The present experimental technique has proved to be successful and viscoelastic theory supports the interpreta- tion of experimental data. The concept of strain failure can be extended to remaining vegetative material. Suggestions to improve the non-linear viscoelastic formulation have already been presented in Section 6.7. REFERENCES REFERENCES Akyurt, M., 1969. Constitutive relations for Plant Materials. Unpublished Ph.D. thesis. Purdue University. Apaclla, R., 1973. Stress analysis in agricultural products using finite element method. Unpublished technical research report. Agricultural Engineering Department, Michigan State University. Bishop, A. W. and D. J. Henkel, 1962. The Measurement of Soil Properties in the Triaxial Test. Second edition. Edward Arnold, Ltd., London. Brusewitz, G. H., 1969. Consideration of plant materials as an interacting continuum. Unpublished Ph.D. thesis. Agricultural Engineering Department, Michigan State University. Bychawski, Z., 1975. Analysis of features and properties of physical non-linearities in viscoelastic behavior: Mechanics of viscoelastic media and bodies. Symposium Gothenburg/Sweden, September 2-6, 1974. Editor Jan Hult. Springer—Verlag, Berlin. 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Cooper, J. R. Hammerle, S. W. Fletcher, and L. D. Tukey, 1965. Readiness for harvest of apple as affected by physical and mechanical properties of the fruit. Pennsylvania Agricultural Experiment Station, Bulletin No. 721. Morrow, C. T., 1965. Viscoelasticity in a selected agri- cultural product. Unpublished M.S. thesis. Pennsyl- vania State University. Morrow, C. T. and N. N. Mohsenin, 1966. Consideration of selected agricultural products as viscoelastic materials. Journal of Food Science, 31(5):686-698. Morrow, C. T. and N. N. Mohsenin, 1968. Dynamic viscoelastic characterization of solid food materials. Journal of Food Science, 33(6):646-651. Morrow, C. T., D. D. Hamann, N. N. Mohsenin, and E. E. Finney, 1971. Mechanical characterization of red delicious apples. ASAE paper no. 71-372. 84 Murase, H., 1977. Elastic stress-strain constitutive equa- tions for vegetable material. Unpublished Ph.D. thesis. Agricultural Engineering Department, Michigan State University. Nadai, A., 1931. Plasticity. McGraw-Hill Book Company, Inc., New York. Nadai, A., 1950. Theory of Flow and Fracture of Solids. Volumes I and II. McGraw-Hill Book Company, Inc., New York. Nelson, C. W. and N. N. Mohsenin, 1968. Maximum allowable static and dynamic loads and effect of temperature for mechanical injury of apples. Journal of Agricul- tural Engineering Research, l3(4):305-317. Oldroyd, J. G., 1950. The motion of an elastico-viscuous liquid contained between coaxial cylinders. I. Quart. J. Mech. Appl. Math., 4, Pt. 3 (1950), 271. Park, D., 1963. The resistance of potato to mechanical damage caused by impact loading. Journal of Agricul- tural Engineering Research, 8(3):l73-177. Peterson, C. L. and C. W. Hall, 1973. Consideration of the Russett Burbank potato as a thermorheological simple material. ASAE paper no. 73-303. Prager, W., 1942. Theory of Plasticity. Advanced Instruc- tion in Mechanics, Brown University. Providence, RI. Prager, W., 1959. An Introduction to Plasticity. Addison- Wesley Publishing Company, Inc., London. Prager, W., 1961. Introduction to Mechanics of Continua. Chicago: Ginn, 1961. Rivlin, R. W. and J. L. Ericksen, 1955. Stress-deformation relations for isotropic materials. J. Rat'l. Mech. Anal., 4(1955), 323. Rivlin, R. W., 1975. On the foundations of the theory of non-linear Viscoelasticity. Mechanics of viscoelastic media and bodies. Symposium Gothenburg/Sweden, September 2-6, 1974. Editor Jan Hult. Springer-Verlag, Berlin. Rumsey, T. R. and R. B. Fridley, 1974. Analysis of visco- elastic contact stresses in agricultural products using finite element method. ASAE paper no. 74-3513. Seeley, F. B. and J. S. Smith, 1967. Advanced Mechanics of Materials. John Wiley & Sons, Inc., New York. 85 Sherif, S. M., 1976. The quasi-static contact problem for nearly-incompressible agricultural products. Unpub- lished Ph.D. thesis. Agricultural Engineering Depart- ment, Michigan State University. Sherif, S. M., L. J. Segerlind, and T. S. Frame, 1976. An equation for the modulus of elasticity of radially compressed cylinder. Transactions of the ASAE (to be published). Simpson, J. B. and G. E. Rehkugler, 1972. Forces and apple damage during impact. ASAE paper no. 72-307 MI-H. Snobar, B. A., 19 . Engineering parameters related to the hardness of carrots. Unpublished Ph.D. thesis. Agri- cultural Engineering Department, Michigan State Univer- sity. Sharma, M. G. and N. N. Mohsenin, 1970. Mechanics of deform- ation of fruits subjected to hydrostatic pressure. Journal of Agricultural Engineering Research, l5(1):65-74. Sobotka, Z., 1975. Non-linear constitutive equations of viscoelastic bodies. Mechanics of viscoelastic media and bodies. Symposium Gothenburg/Sweden, September 2-6, 1974. Editor Jan Hult. Springer-Verlag, Berlin. Stuart, H. A., 1956. Die Physik Der Hochpolymeren. Theorie und Moleunlare Deutung Technologischer Eigenschaften Von Hochpolymeren Werkstoffen. Springer-Verlag, Berlin. Gotlingen-Heilderberg. Terzaghi, K., and R. B. Peck, 1967. Soil Mechanics in Engineering Practice. Second edition. John Wiley & Sons, Inc., New York. Thomas, T. Y., 1961. Plastic Flow and Fracture in Solids. Academic Press, New York. Timbers, G. E., L. M. Staley, and E. L. Watson, 1966. Some mechanical and rheological properties of the Netted Gem potato. Canadian Journal of Agricultural Engineering, February. Timoshenko, S. P. and J. N. Goodier, 1970. Theory of Elasticity. McGraw-Hill Book Company, New York. Van Lancker, J. V., L. Kermis, J. DeBruyn, F. DeSmet, G. Otter- mans, and A. Calus, 1977. Mechanical behavior and compression tests on apples (Golden Delicious). Land- bouwtigschrift, 10(1977), Nr. 1. Wang, J. K. and H. S. Chang, 1970. Mechanical properties of papaya and their dependence on maturity. ASAE paper no. 69-389. 86 Weber, T. D., 1975. Hypothese des variables internes et viscoelasticite non-lineaire: Etude du cas particulier d'une seule variable interne tensorielle. Mechanics of viscoelastic media and bodies. Symposium Gothenburg/ Sweden, September 2-6, 1974. Editor Jan Hult. Springer-Verlag, Berlin. White, R. K. and N. N. Mohsenin, 1967. Apparatus for determination of bulk modulus of compressibility of materials. Transactions of the ASAE, 10(5):670-67l. Wright, F. S. and W. E. Splinter, 1968. Mechanical behavior of sweet potatoes under slow loading and impact loading. Transactions of the ASAE, 11(6):?65-770. Yoshiaki, I. and J. L. White, 1977. Investigation on failure during elongation flow of polymer melts. Journal of Non-newtonian fluid Mechanics, 2(1977)281-298. Zoerb, G. C., 1958. Mechanical and rheological properties of grain. Unpublished Ph.D. thesis. Agricultural Engineering Department, Michigan State University. APPENDICES TABLE A1.STRESS- 87 STRAIN AND TIME AT FAILURE :09 UNIAXIAL LCADING =-o.002 sec-1. SE CYLINDRICAL SPECIMENS 0F APPLE. 611 MC INTDSH JONATHAN RED DELICICUS 311 611 t °11 811 t “11 811 t (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) -o.21 -0.127 63.79 -O.16 -O.137 68.69 -0.2 -0.091 45.90 -O.19 -O.132 66.24 -0.23 -O.159 79.73 -0.31 -O.127 63.33 -0.18 -0.130 65.01 -O.17 -O.142 7 .15 -O.30 -0.132 66.28 -O.2 -0.147 73.60 -O.23 -O.161 80.96 ~0.30 -0.117 38 92 -0.20 -0.147 73.60 -O.16 -0.132 76 03 -0.2 -O.111 33.97 ~O.17 ~0.137 68.69 -O.16 -0.161 80.96 .0.2 -0.130 65.30 -0.20 -0.142 71.13 -O.13 -0.147 73.60 -0.29 -O.116 38.18 -O.15 -O 137 68.69 -O.14 -o.166 83.41 -0.31 -0.120 60.39 90.17 ~0.147 73.60 ~O.14 -O.176 88.32 -0.2 -O.114 .7.44 '0.13 -O.137 68.69 -O.21 ~0.181 90.77 -O.23 -0.114 57.44 -0.17 -o.137 68.69 -O.26 -o.161 80.96 -0.2 -O.115 57.93 -0.17 -0.132 76.03 -O.22 -0.137 78.31 -0.2 -0.131 65.79 -O.18 -O.142 71 13 ~O.22 -0.132 66.2 -0 2 ~0.122 61.37 -0.13 -0.142 71.13 -O.24 -O.132 76.03 -0.31 ~0.126 63.33 -0.17 -0.127 63.79 -O.18 -O.147 73.60 -0.2 -0.120 60.14 ~0.16 -0.142 71.13 -0.21 -0.137 78.31 -0.27 -0.123 62.60 -0.18 -0.137 78.31 -O.13 -0.127 63.79 -0.33 .0. 33 67.31 90.1 -0.137 78.31 -O.21 -0.142 71.13 -0.37 -0.137 68.74 -0.17 -0.137 68.69 -O.14 -0.137 68.69 -0 41 -O.132 76.10 -0.16 -O.134 67.47 -O.17‘ -0.157 73.51 -0.27 -0.117 5 .92 TABLE A2.STRESS- STRAIN AND TIME AT FAILURE FDR UN AXIAL LOADING 0F CYLINDRICAL SPECIMENS 0F APPLE. ella-O.oo7 SEC‘ nc INTOSH JONATHAN RED DELICIOUS “11 E11 t °11 611 c 011 611 c (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) -0.23 -O.123 17.96 -O.27 -O.113 16.77 -0.29 -0.111 16.17 -0.23 -O.123 17 96 -o.3o -0.148 2 36 -o.29 -o.107 15.57 -O.24 -O.148 21.36 -O.26 -O.144 2 .96 -O.33 -0.12 18.36 -0.24 -0.123 17.96 -0.27 -O.144 20.96 -0.33 -0.107 13.57 ~0.24 -0.111 16.17 -O.23 -0.136 19.76 -0.30 -0.113 16.77 -O.25 -0.136 19.76 -O.27 -o.148 21.56 -0.31 -0.115 16.77 -0.24 -O.144 20.96 ~0.24 -O.136 19.76 -O.31 -0.107 15.57 -0.25 -o.119 17.37 -O.26 -O.144 20 96 -0 33 -o.123 17 96 -0.23 -O.136 19.76 -O.26 -O.14O 20.36 -0.34 -0.107 15 7 -O.25 -O.136 19.76 -O.28 -0.132 19.16 -O.34 -O.111 16.17 -O.2O ~O.136 19.76 ‘O.24 -O.111 '16.17 -0.29 -0.107 13.37 -0.18 -O.103 14.97 -O.26 -0.140 2 .36 -O.29 -O.111 16.17 -0.19 -O.103 14.97 -O.23 ~0.123 17.96 -O.33 -0.113 16.77 -o.24 -0.123 17.96 -O.28 -0.132 19.16 -o.33 -C.107 15.57 -O.23 -0.111 16.17 -0.2 -0.119 17 37 -O.30 -0.113 16.77 -O.22 -O.123 17.96 -O.28 -0.157 22.76 -O.31 -O.107 15.57 -o.22 -o.123 17.96 -O.26 -o.132 19.16 -o.31 -0.123 :7 =6 -O.24 -O.119 17.37 -O.31 -0.144 20.96 -0.3 -O.119 17.37 -0.23 -O.111 16.17 -O.27 -0.132 19.16 -O.34 -O.115 16.77 -0.23 ~0.107 13.37 -O.23 -0.123 17 96 -0.34 -0.099 14.37 88 TABLE A3.STRESS- STRAIN AND TIME AT FAILURE FOR UNiAXIAL LOADING 0F CYLINDRICAL SPECIMENS OF APPLE. Ella-0.017 sac . no INTOSH JONATHAN RED DELICIOUS 0 e 0 e 0 ' e 11 11 t 11 11 t 11 11 t (MPa) (mun/Inn) (sec) (MPa) (ma/mm) (sec) (MPa) (mu/min) (sec) -o.22 -0.121 7 03 -O.3O -o.121 7.09 -0.29 -0.106 6.13 -o.19 -o.131 7 62 -o.29, -o.101 3.86 -o.31 -o.106 6.13 -o 22 -0.116 6 74 -o.30 .-o.091 5.27 -o.31 -O.116 6.74 -o.24 -O.126 7 33 -0.26 -o.101 3.86 -o.2 -0.086 4.99 -0.19 -o.101 5 86 -o.29 -o.111 6.43 -o.25 -o.101 3.86 -o.22 -o.111 6 45 -0.29 -o.142 9.21 -0.25 -o.101 3.86 -O.26 -O.106 6 15 -O.26 -0.106 6.15 -o.5o -o.151 7.62 -o.2o -o.111 6 45 -o.29 -o.131 7 62 -o.29 -0.142 9.21 -0.20 -o.131 7 62 -O.26 -o.101 5 86 -o.29 -o.137 7.91 -0.26 -O.106 6 15 -o.32 -0.121 7 05 --o.27 -o.101 3.86 -o.25 -0.111 6 45 -o.29 -o.1o1 5 86 -o.3o -o.121 7.09 -o.19 -o.111 6 45 -o.29 -o.101 5 86 -0.30 -0.126 7153 -o.19 -o.126 7.33 -0.26 -o.142 9 21 -o.25 -0.126 7.33 -o.14 -o.142 9 21 -O.26 -0.106 6.13 -0.25 -0.116 6.74 -o.19 -o.111 6 45 -o.29 -o.121 7 09 -o.25 -o.121 7.03 -o.29 -O.116 6 74 -o.ao -o.111 6 45 -o.29 -0.106 6.13 -o.29 -o.121 7 03 -o.31 -o.101 3.86 -0.29 -o.1o1 3.86 -o.25 -o.152 9 79 -o.24 -0.076 4 59 -0.36 -o.121 7.03 -o.27’ -o.125 7 15 -O.26 -o-101 5 86 -0.36 -o 126 7.3a -O.26 -o.126 7 33 -O.36 -o.111 6 45 -o.37 -0.106 ' 6.13 TABLE A4.STRESS- STRAIN AND TIME AT FAILURE FOR UNIAXIAL LOADING OF CYLINDRICAL SPECIMENS OF APPLE. élli-0.033 SEC MC INTUSH JONATHAN ' ' RED DELICIOUS “11. “11 t “11 “11 t “11 811 t (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) -0.26 -O.138 3.99 -0.28 -0.138 3.99 -0.32 -0.099 2.83 -O.29 ~0.138 3.99 -O.30 -0.099 2.83 -O.33 -O.109 3.14 -O.28 -O.109 3.14 -O.31 -0.089 2.37 -0.37 -0.109 3.14 -0.31 -O.128 3.71 -O.36 -0.109 3.14 -0.33 -O.118 3.42 -O.29 -O.138 3 99 -O.33 -O.109 3.14 -O.36 -0.099 2.83 -O.29 -O.118 3 42 *0.39 -O.118 3.42 -O.26 -O.118 3.42 -0.23 -O.128 3 71 -O.29 -O.109 3.14 -0.34 -O.109 3.14 -O.29 -0.138 3 99 -O.23 -O.109 3.14 -0.33 -0.109 3.14 -0.27 -O.128 3.71 -O.23 -0.118 3.42 -0.36 -0.089 2.37 ~0.26 -O.128 3.71 -O.29 -O.128 3.71 -0.33 -0.099 2.83 -O.27 -O.128 3.71 -O.23 -0.118 3.42 -O.29 -0.099 2.83 -O.28 -O.138 3 99 -O.26 -0.109 3.14 -0.33 ~0.128 3.71 -O.22 -0.109 3 14 -O.2B -O.138 3.99 -0.34 -O.109 3.14 -0.23 -O.109 3 14 -O.24 ~O.138 4.37 -0.33 -0.099 2.83 -O.23 -0.109 3 14 -0.29 -O.118 3.42 -0.34 -0.109 3.14 -O.18 -O.148 4 28 -O.31 -O.118 3.42 -0.34 -0.099 2.83 -0.22 -O.109 3 14 -O.32 00.118 3.42 -O.37 -0.099 2.83 -O.22 -0.109 3 14 -O.30 -0.128 3.71 -0.37 -0.111 3.19 ‘0.23 -0.099 2 83 -O.26 -O.138 ‘3.99 -O.37 -0.138 3.99 -O.23 '0.109 3 14 -O.27 -0.118 3.42 -O.37 -0.099 2.83 89. TABLE A3.STRESS. STRAIN AND TIHE AT FAILURE FOR UN AXIAL LOADING OF CYLINDRICAL SPECIHENS OF APPLE.é11"-0.069 SEC- NC INTOSH JONATHAN RED DELICIOUS 11 11 t 11 11 t 11 11 t (MP4) (mu/mu!) (sec) (MPa) (min/nun) (see) (MPa) (mun/mu!) (sec) 90.29 -0.124 1 79 o0.29 -0.165 2.39 -0.34 -0.092 1.19 80.25 -0.165 2.39 -0.39 -0 165 2.39 -0.34 -0.124 1.79 40 29 -0.144 2 09 -0.34 -0 144 2.09 -0.35 -0.124 1.79 -0.27 —0.144 2.09 -0.29 -0 144 2.09 -0.29 -0.144 2.09 .0.29 -0.144 2 09 -0.27 -0 124 1.7 -o.33 —0.124 1.79 -0 29 -0.124 1 79 -0.39 -0 196 2.69 -0.41 -0.124 1.79 ‘-0.24 -0.124 1 79 -0 33 -0 092 1.19 -O.34 -0.103 1.49- -0.27 .-0.124 1 79 -0.31 -0 103 1.49 -o.31 -0.124 1.79 -0.24 —0.124 1 79 -0 40 -0 144 2.09 -0.32 -0.103 1.49 -0.25 -0.144 2 09 -0.33 -0 124 1.79 -0.32 -0.124 1.79 -0 27 -0.124 1 79 -0.32 -0 124 1.79 -0.30 -0.124 1.79 -0 26 -0.103 1 49 -0.34 -0 165 2.39 -0.34 -0.115 1.67 -0.25 -0.124 1.79 -0.31 -0 144 2.09 -0.26 -0.124 1.79 -0.25 -0 124 1.79 —0.31 -0 124 1.79 -0.3 —0.103 1.49 -0.27 -o.124 1.79 -0.27 -0 103 1.49 -0.34 -0.124 1.79 -0.24 -0.124 1.79 -0.31 -0 124 1.79 -0.39 -o.124 1.79 -0 24 -0.124 1.79 -0.29 -0 144 2.09 -o.37 -0.124 1.79 -0.25 =ro.124 1.79 -0.26 —0 124 1.79 -0.37 -0.124 1.79 -0.24 40.103. 1 49 -0.30 -0.124 1.79 -Q.37 10.124 1.79 -0:24 -0.124 1 79 -0.30 ~0.124 1.79 ' 1.79 -0.38' -0.124 TABLE A6.STRESS. STRAIN AND TINE AT.FAILURE FOR UN AXIAL LOADING OF’CYLINDRICAL SPECIHENS OF APPLE. Elli-0.173 SEC. . nc INTOSH JONATHAN RED DELICIOUS °11 E11 1 C11 611 c 011 811 t (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) (W3) (mm/mm) (sec) .0324 .0.090 0.32 -0.34 -0.099 0.37 -0.37 -0.090 0.32 ‘0f23 '0.099 0.37 .0.33 -0.090 0 32 -0.41 -0.090 0.32 '0323 -0.090 0.32 '0.36 -0.126 0 7 -0.43 -0.099 0.37 -0.28 -0.081 0.47 -0.32 .0.090 0 32 -O.43 -0.090 0.32 '0.23 -0.090 0.32 -O.28 ‘0.090 0 32 -0.41 -0.108 0.62 -0.28 -0.081 0.47 -0.28 -0.081 0 47 -0.43 -0.108 0.62 .-0:30 -0.090 0.32 -0.28 .0.090 0 32 -O.28 -0.081 0.47 '-0:26 -0.090 0.32 -0.37 '0.090 0 32 -0.26 -0.072 0.41 .0.26 '0.099 0.37 -0.41 -0.126 0 73 -0.28 -0.072 0.41 -0.34 -0.099 0.37 -0.37 -0.081 0 47 -0.30 -0.103 0.60 -0.32 -0.099 0.37 -0.37 -0.117 0 68 -O.28 -0.090 0.32 '0.32 -0.090 0.32 -O.31 -0.099 0 37 -0.28 ~0.099 0.37 -0.29 -0.090 0.32 -0.32 -0.099 0 37 -0.28 .0.099 0.37 -0.30 -0.099 0.37 -0.30 -0:099 0 37 -0.34 -0.081 0. 7 “0.30 '0.108 0.62 ‘0.31 .0.099 0 37 ~0.33 -0.099 0.37 90.28 -0.099 0.37 -0.34 -0.108 0 62 -O.34 -0.099 0.37 -0.24 -0.108 0.62 -0.32 -0.099 0 37 -0.32 90.108 0.62 .0.27 90.090 0.32 -0.36 -0.099 0 37 ~0.37 -0.099 0.37 -O.23 -O.108 0.62 -0.37 -0.090 0 32 -0.36 -0.099 0.37 -O.28 .0.099 0.37 '0.34 -0.108 0 62 -0.36 -0.094 0.34 90 TABLE A7.STRESS. STRAIN AND TIME AT FAILURE FOR UN AXIAL LOADING OF CYLINDRICAL SPECIMENS OF APPLE.E --0.343 SEC 11 MC INTOSH JONATHAN RED DELICIOUS 0‘ E O E C C 11 11 t 11 11 t 11 11 t (MPa) (min/mm) (sec) (MPa) (max/mm) (sec) (MPa) (ma/min) (sec) -0.27 -0.067 0.19 -0.30 -0.134 0.39 -0.37 -0.134 0.39 -0.29 -0.134 0.39 -0.32 -0.134 0.39 -0.37 -0.112 0.32 -0.24 -0.112 0.32 -0.34 -0.134 0.39 -0.37 -0.134 0.39 -0.26 -0.134 0.39 -0.34 -0.134 0.39 -0.37 -0.112 0.32- -0.24 -0 134 0.39' -o.29 -0.125 0.36 —0.41 -0.134 0.39 -0.23 -0.112 0.32 -0.29 -0.112 0.32 -O.37 -0.112 0.32 -0.26 -0.134 0.39 -0.30 -0.157 0.45 -0.36 -0.112 0.32 -0.26 -0.112 0.32 -0.30 -0.112 0.32 -0.34 -0.094 0.27 -0.29 -0.134 0.39 —0.33 -0.134 0.39 -0.34 ~0.112 0.32 -0.29 -0.157 0.45 -0.34 -0.157 0.45 -0.34 -0 134 0.39 -0.26 -0 134 0.39 -0.32 -0.134 0.39 -0.3 -0.112 0.3 -0.26 -0 157 0.45 -0.34 -0.125 0.36 -0.3 -0.;12 0.32 -0.30 -0.134 0.39 -0.34 -0.112 0.32 -0.32 -0.134 0.39 60.29 -0.157 0.45 —0.30 -0.134 0.39 -O.32 -O. 12 0.32 30.26 -0.134 0.39 c0.33 —0.112 0.32 ~O.32 -0.112 0 32 -0.30 -0.134 0.39 -0.33 -0.112 0.32 -0.32 -O.112 3.32 -0.29 -0.134 0.39 -0.30 -0.134 0.39 -0.30 -o.112 0.32 -0.29 -0.134 0.39 -0.37 -0.134- .0.39 -O.32 -o.134 0.39 -0.29 -0.112 0332 -0.34 -0.134 0.39 -0.30 -0.099 0.26 -0 26 -0.134 0.39 -0.37 -0.134 0.39 -0.32 -0.134 0.39 TABLE A8.STRESSo STRAIN AND TIME AT FAILURE FOR IAXIAL LOADING OF CYLINDRICAL SPECIMENS OF APPLE.ellI-0.007 SEC' .022-©.000 MP3. MC INTOSH JONATHAN RED DELICIOUS “11 11 t °11 811. t “11 611 c (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) (MP3) (mm/mm) (sec) -0.23 '0.123 17.96 -0.27 -0.113 16.77 -0.29 -0.111 16.17 -0.23 '0.123 17.96 -0.30 -0.148 21.36 -0.29 -0.107 13.37 70.24 ‘0.148 21.36 '0.26 -O.I44 20.96 .0.33 30.12 13.36 ‘0.23 '0.123 17.96 -0.27 -0.144 20.96 '0.33 .0.107 13.37 -0.24 '0.111 16.17 -O.24 .0.136 19.76 .0.30 “0.113 16.77 .0.23 -0.136 19.76 ‘O.26 .0.148 21.36 -0.31 -0.113 16.77 -O.24 '0.144 20.96 -0.24 -0.136 19.76 -0331 .0.107 13.37 -O.23 '0.119 17.37 -O.26 -0.144 20.96 -0.33 -0.123 17.96 -O.23 -0.136 19.76 ‘0.26 ‘0.140 2 .36 -0.34 -0.107 13.37 -0.23 -0.136 19.76 '0.28 ‘0.132 19.16 -0.34 -0.111 16.17 '0.24 '0.123 17.96 '0.24 -0.111 16.17 -0.29 .0.107 13.37 .0.19 '0.103 14.97 -0.26 .0.140 20.36 -O.29 -0.111 16.17 -0.24 '0.103 14 97 -0.23 -0.123 17.96 -0.33 -0.113 16.77 -0.23 '0.123 17.96 -O.2S .0.132 19.16 -0.3 '0.107 3.37 “0.22 ‘0.111 16.17 '0.23 -O.119 7.37 -0.30 -0.113 16. 7 -0.23 '0.123 17 96 -0.28 -0.137 22.76 ‘0.3 -0.107 13.37 -0.24 -0.123 17.96 -0.26 '0.132 19.16 -0.31 -O.123 17.96 -0.23 ‘0.119 17.37 '0.30 -0.144 20.96 -0.3 -0 119 17. 7 .0.23 '0.111 16.17 -0.27 -0.132 19.16 .0 34 -0.113 16.77 '0.23 '0.107 13.37 -0.23 ‘0.123 17.96 -O.34 -0.099 14. 7 91 TABLE A9.STRESS. STRAIN AND TIME AT FAILURE FOR IAXIAL LOADING OF CYLINDRICAL SPECIMENS OF APPLE.911=-0.007 SEC' 0228-0.O69 MP9. no INTOSH JONATHAN RED DELICIOUS 11 11 t 11 11 t 11 11 t (MP4) (min/m) (sec) (MP4) (nun/mu!) (sec) (MP4) (nu/w) (sec) -0.21 -0.113 16.77 -0.26 -O.113 16.77 -0.30 -0.137 22.76 .0.26 '0.136 19.76 -O.2S -O.113 16.77 -0.34 -0.148 21.36 '0.24 -0.148 21.36 -0.26 -0.113 16.77 -0.33 -0.144 20.96 -0.26 '0.140 20.36 -0.27 -0.113 16.77 -0.23 -0.113 16.77 -0 26 '0.132 19.16 ‘0.23 -0.113 16.77 -0.26 “0.113 16.77 ‘0 24 .0.144 20.96 *0.21 ‘0.103 14.97 -0.3 70.132 19.16 -0 22 -O.132 19.16 -0.28 '0.136 19.76 -0.27 -0.144 20.96 '0 24 '0.132 19.16 '0.30 '0.144 2 .96 ~0.22 -0.103‘ 14.97 '0 23 '0.123 17.96 -0.26 ~0.132 19.16 -0.39 -0.169 24.33 .0 22 ‘0.123 17.96 '0.27 -0.103 14.97 -0.34 -0.140 2 .36 ‘0 23 '0.123 17.96 -O.23 .0.132 19.16 -O.30 -0.123 17.96 -0 17 -0.103 14.97 -0.23 -0.144 20.96 -0.27 -0. 23 17.96 -0 23 .0.123 17.96 “0.22 -0.123 17.96 -0.24 -0.093 13.77 .0 23 -O.132 19.16 .0.27 -0.132 19.16 -0.31 *0.113 1 .77 ‘0.19 '0.103 14.97 -0.26 -0.148 21.36 “0.31 -0.119 17. 7 -O.24 -0.113 16.77 90.24 -0.113 16.77 -0.29 -0. 28 18.36 -0.22 -0.113 16.77 -0.28 -0.123 17.96 ~0.31 -0.123 17.96 -O.23 '0.137 22.76 '0.23 ‘0.132 19.16 -0.26 -0.119 7.37 -0 20 -0.107 13.37 -0.23 ~0.123 17.96 -0.24 -0.107 13.37 -0 '0.113 16.77 ‘0.21 -0.132 19.16 -O.2 '0.113 16.77 TABLE A10.STRESS.STRAIN AND TIME AI FAILURE FOR TRIAXIAL LOADING OF CYLINDRICAL SPECIMENS OF APPLE.5 t-0.007 SECJ-fl --0.138 MPa. 11 2;: MC INTOSH JONATHAN RED DELICIOUS 0 E 0 e 0 e 11 11 t 11 11 t 11 11 t (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) (MP9) (mm/mm) (sec) -O.26 -0.140 20.36 -O.24 -0.123 17 96 -0.18 -O.103 14.97 -O.29 -O.132 22.16 -O.22 -O.107 13.57 -0.20 -0.140 20.36 -O.17 ~0.136 19.76 -0.27 -0.132 19.16 -O.33 -O.165 23.93 -O.13 -O.103 14.97 -O.22 -O.144 20.96 -0.26 -0.113 16.77 -O.18 -0.140 20.36 -O.23 -O.123 17.96 -O.30 -0.I63 23.95 -O.13 -O.111 16.17 -O.29 -0.123 17.96 -O.22 -O.107 15. 7 -O.22 -0.119 17.37 -O.29 -0.107 13.37 -O.33 -O.130 21.86 -0.20 -O.123 17.96 -O.28 -0.132 19.16 -0.26 ~O.128 18.36 -O.20 -0.123 17.96 -O.26 -O.123 17.96 -O.23 -O.107 13.37 -O.22 ~O.132 19.16 -O.28 -0.136 19.76 -0.27 -0.123 17.96 -0.23 -O.123 17.96 -O.2B -0.093 12.77 -O.23 -O.128 1 .56 -O.23 -O.132 19.16 -O.23 -O.132 19.16. -0.28 -0.123 17.°6 -0.23 -O.132 19.16 -O.26 -0.140 20.36 -0.25 -0.123 17.96 -O.26 -O.132 19.16 -0.23 -0.148 21.36 -0.28 -0.165 23.°5 -O.24 -O.163 23.93 -O.22 -O.173 23.15 -0.22 -0.123 17.96 -O.22 -0.136 19.76 -0.23 -0.132 19.16 -O.34 -O.183 26.95 90.22 -O.123 17.96 -O.26 -O.163 23.93 -0.2 -O.163 23.95 -0 20 -0.132 22.16 -0.23 -O.157 22.76 —O.28 90.132 19.16 -O.13 ~03243 33.341 —O.12 —o.247 33.92 -O.12 -0.074 .1 .7 -O.13 -O.163 23.93 ~O.22 -0.163 23.23 -O.2 ~0.140 2 .26 92 TABLE A11.STRESS. STRAIN AND TIME AT FAILURE FOR T IAXIAL LOADING OF CYLINDRICAL SPECIMENS OF APPLE. 6118-0. 007 SEC', 0228-0. 207 MPa. MC INTOSH JONATHAN RED DELICIOUS °11 611 t “11 611 t “11 611 c (3P4) (mm/min) (sec) (MP4) (nun/mm) (sec) (MPa) (mm/min) (sec) -0.21 -0.123 17 96 -0.24 -0 132 19.16 -0.17 -0.090 13.17 -0 13 -0.079 11.39 -0.22 -0.103 14.97 -o.27 -0.134 19.46 -0.19 —0.090 13.17 -o.27 -0.123 17.96 —o.27 -0.119 17.37 -0.19 -0.090 13.17 -0.20 -0.144 20.96 -0.29 -0.144 20.96 -0 22 -0.107 15.57 -0.17 -0.107 15.57 -0.25 -o.123 17.96 -0 23 -o.132 19.16 -o.22 -o 103 14 97 -0.27 -0.107 15.57 -0 20 -0.103 14.97 -0.24 -o.144 20.96 -o.30 —0.136 19.76 -0 20 —0.103 14.97 -o.29 -o.132 19.16 -o.15 -0.070 10.19 -0 20 -0.103 14.97 -0.17 -0.129 19.56 —0.29 -0.132 19.16 -0 19 -o.107 15 57 ~0.20 -0.123 17.96 -0.29 —0.144 20.96 -0 22 -0.111 16.17 -0.24 -o.157 22.76 -0.27 -0.119 17.37 -0 20 -0.095 13.77 -0.19 -0.095 13.77 -o.2o -0.107 15.57 .0 20 -0.099 14.37 —0.19 -0.107 15.57 -0.16 -0.074 10.79 -0 20 -0.095 13.77 -0.19 -0.132 19.16 -o.29 -0.149 21.56 -0 21 -0.107 15.57 -0.17 -0.092 11.97 -0.30 -0.134 19.46 .0 30 -0.144 20.96 -o.17 -0.092 11 97 —o.29 -o.140 20.36 -0 30 -0.144 20.96 -0.25 -0.099 14.37 -0.29 —0.136 19.76 -0.23 -0.111 16.17 -0.20 -0.095 13.77 -o.26 -0.103 14.97 -0 25 -0.132 19.16 -0.22 -0.099 14.37 90.26 -0 107 15.57 -0 22 -0.101 14.67 -0. -0.23 «0. 24 -0. 107 13. 37. 136 19.76 TABLE A12. STRESS. STRAIN AND TIME AT FAILURE FOR IAXI'AL LOADING OF CYLINDRICAL SPECIMENS OF APPLE. 9113-0. 007 SEC" 0223-0. 276 MPa. MC INTOSH JONATHAN RED DELICIOUS °11 .511 c °11 611 t °11 611 t (MPa) (mm/mm) (390) (MPa) (mm/111m) (sec) (MPa) (mm/mm) (sec) -0.19 -0.144 20.96 —0.12 -0.123 17.96‘ -0.17 —0.070 10.19 —0.19 -0.090 11.69 —0.16 -0 115 16.77 -0.19 -0.096 12.57 -0.19 -0.119 17.37 -0.12 -0.092 11.97 -o.13 -0.057 9.39 -0.19 —0.092 11.97 -o.19 -0.123 17.96 -0.29 -0.123 17.96 -0.23 -0.095 13.77 -0.19 -0.095 13.77 -0.17 -0.074 10.79 -0.14 -0.095 13.77 -0.16 -0.115 16.77 -0.15 -0.066 9.59 -0.14 -0.096 12.57 -0.23 -0.074 10.79 o0.22 -0.095 13.77 90.19 -0.107‘ 15.57 -0.20 .-0.090 13.17 ~0.30 -0.123 17 96 —0.17 —0.092 13.47 -0.22 -0.115 16.77 -0.27 -0.115 16.77 -0.13 -0.074 10.79 -0.17 -0.115 16.77 -0.31 -0.129 19.56 -0.19 -0.095 13.77 -0.17 -0.115 16 77 -0.29 -0.115 16.77 -0.17 -0.099' 14.37 -0.20 -0.099 14.37 -o.26 -0.111 16.17 -0.21 -0.090 11.69 -0.23 -0.107 15.57 —o.29 -0.129 19 56 -0.23 -0.111 16.17 -0 23 -0.107 15.57 -0.17 -0.074 10.79 -0.21 -0.074 10.79 -0.19 -0 115 16.77 -o.23 -0.103 14.97 -0.19 —0.103 14.97 -0.17 -0.103 14.97 -0.26 -0.123 17.96 -0.22 -0.123 17.96 -0.20 -0.103 14.97 ~0.2o -0.099 14.37 -0.17 -0.097 14.07 -0.19 -0.173 25.15 —0.21 -0.099 14.37 -0.14 -0.095 13.77 —0.19 -0.123 17.96 -o.13 -0.061 9.99 -0.19 -0.103 14.97 -0.15 -0.107 15.57 ~O.16 -0.074' 10.7 93 TABLE A13.STRESS.STRAIN AND TIME AI FAILURE FOR TRIAXIAL LOADING 0F CYLINDRICAL SPECIMENS OF APPLE.€Lli-0.OO7 SEC"922--0.395 MPa. MC INTOSH JONATHAN RED DELICIOUS C11 811 t 011 611 t O11 E11 t (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) 0.00 0.000 0.00 0.00 0.000 0 00 -0.15 -0.059 9.69 0.00 0.000 0.00 0.00 0 000 0 00 -0.16 -0.066 9.59 0.00 0.000 0.00 0 00 0.000 0 00 -0.09 -0.066 9.59 0.00 0 000 0.00 0 00 0.000 0 00 -0.15 -0.079 11.39 0.00 0.000 0:00 o 00 0.000 0 00 -0.15 -0.070 10.1 0.00 0.000 0.00 0 00 0.000 0 00 -0.22 -o.092 11.97 0.00 0.000 0.00 0 00 0.000 0 00 -0.25 -0.092 11.97 0.00 0.000 0.00 0 00 0.000 0 00 -0.30 -0.103 14.97 0.00 0.000 0.00 0.00 0.000 0 00 -0.22 -0.092 11.97 0.00 0.000 0.00 0 00 0 000 0.00 -0.22 -0.095 13.77 0.00 0.000 0.00 0 00 0.000 0.00 -o.25 -0.115 16.77 0.00 0.000 0.00 0 00 0.000 0 00 -o.22 -0.092 11.97 0.00 0.000 0.00 0 00 0 000 0 00 -o.26 ~0.103 14.97 0.00 0.000 0.00 0 00 0 000 o 00. —0.11 -0.096 12.57 0.00 0.000 0.00 0 00 0.000 0 00 -0.22 -o.107 15.57 0.00 0.000 0.00 o 00 0.000 0 00 -o.26 -0.107 15.57 0.00 0.000 _0.00 0 00 0 000 0 00 -0.19 .0 092 11.97 0.00 0.000 0.00 0 00 0.000 0 00 -0.22 -0.092 11 97 0.00 0.000 0.00 0 00 0.000 0 00 -o.19 -0.066 9.59 0.00 0.000 0.00' 0 00 0.000 0 00 -0.11 -0.041 5.99 TABLE A14. STRESS: STRAIN AND TIME AT FAILURE FOR LOADING IN RIGID DIE OF CYINDRICAL APPLE SPECIMENS. éllI-O. 007 SEC'l. MC INTOSH JONATHAN R99 DELICIOUS 311 S11 t 511 811 t °11 E:11 t (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) (MPa) (mm/mm) (sec) -0.24 -0.103 14.97 -0.32 -0.134 19.46 —0.41 -0.111 16.17 -0.36 —0.111 16.17 -0 42 -0.107 15.57 -0.33 -0.096 12.57 -0.29 -0.099 14.37 -0 35 -0.105 15.27 ~o.4o -0.o92 13.47 -0.25- -0.136 19 76 -o 49 -o.119 17.37 -o.42 -0.111 16.17 —0.32 ~0.129 19.56 -0 29 -0.144 20.96 -0.57 -0.103 14.97 —o.3o -0.103 14.97 -0 41 -0.157 22.76 -o.44 -0.099 14.37 -0.36 -0.119 17 37 -o 66- -0.165 23.95 -o.36 -0.096 12.57 -0.34 -0.099 14.37 -0 40 -0.140 20.36 -0.36 -0.115 16.77 -o.37 -0.119 17.37 -0 27 -0.103 14.97 -o.34 «0.101 14.67 -0.36 —0.115 16.77 -0 49 -0.144 20.96 -o.23 -0.113 16.47 -0.43 -0.107 15.57 -0 26 -o.103 14.97 -o.42w -0.107 15 57 -0.50 -0.144 20.96 -0 30 -0.095 13.77 -0.44 -o.119 17.37 -0.32 -0.111 16.17 -0 50 -0.149 21.56 -0.53 -0.129 19.56 -0.37 -o.115 16.77 -0 31 -0.o99 14.37 -0.45 -o.117 17.07 -0.37 -0.111 16 17 -o 29' -0.099 14.37 -o.57 -0.103 14.97 -0.47 -0.140 20.36 -0 34 -0.107 15.57 -o.47 -0.111 16.17 -0.40 -o.119 17.37 -0 37 -0.095 13.77 -0.55 -0.109 15.97 -0.36 ~0.123 17.96 -o.34 -o.099 14.37 -0.49 -0.129 19 56 -0.40 -o.111 16.17 -0.32 -0.115 16.77 —0.56 -0.111 16.17 -0.34 -0.111 16.17 -0.29 -0.123 17.96 -0.44 -o.103 14.97 94 TAaLE A15.STREES.STRAIN AND TIME AT FAILURE FDR UNIAXIAL LOADING OF CUBIC APPLE SPECIMENS.ell--0.007 SEC-1. V '7 MC INTOSH JONATHAN RED DELICIOUS ‘11 ‘11 t ‘11 ‘11 t ‘11 ‘11 t (MP4) (mm/mu) (sec) (MP4) (mm/mm) (sec) (MP4) (mm/mm) (sec) 90.27 90.113 16.77 90.29 90.113 16.47 90.27 90.123 17 96 90.26 90.123 17.96 90.24 90.099 14.37 90.27' 90.123 17.96 90.27 90.123 17.96 90.24 90.113 16.77 90.23 90.111 16.17 90.23 90.121 17.67 90.24 90.092 13.47 90.22 90.097 14.07 90.26 90.113 16.77 90.23 90.082 11.97 90.26 90.070 10 18 90.23 90.123 17.96 90.26 90.123 17.96 90.23 90.099 14 37 90.23 90.090 13.17 90.30 90.093 13.77 90.23 90.128 18 36 90.23 90.121 17.67 90.26 90.107 13.37 90.23 90.107 13.37 90.23 90.111 16.17 90.28 90.090 13.17 90.29 90.111 16.17 90.23 90.113 16.77 90.33 90.111 16.17 90.26 90.109 13.87 90.26 90.111 16.17 90.26 90.119 17.37 90.24 90. 28 18.36 90.30 90.136 19.76 90.26 90.119 17 37 90.27 90.111 16.17 90.29 90.123 17.96 90.26 90.123 17.96 90.27 90.119 17.37 90.30 90.136 19.76 90.23 90.082 11.97 90.23 90.119 17.37 90.26 90.144 20.96 90.23 90.097 14.07 90.26 90.123 17.96 90.26 90.134 19.46 90.23 90 132 19 16 90.23 90.103 14 97 90.29 90.144 20.96 90.23 90 090 13 17 90.21 90.132 19 16 90.29 90.103 14.97 90.28 -90.086 12.37' 90.19 90.074 10 78 90.23 90.128 18.36 90.29 90.093 13.77 90.23 90.093 13 77 90.26 90.132 19.16 90.28 90.113 16.77 90.26 90.103 14 97 TABLE A16. STRESSoSTRAIN AND TIME AT FAIEURE FOR BIAXIAL LOADING OF CUBIC APPLE SPECIMENS.€3J:90.007 SEC" . MC INTOSH JONATHAN RED DELICIOUS ‘11 ‘11 t ‘11 ‘11 t ‘11 ‘11 t (MPa) (run/um) (sec) (MPa) (nun/mm) (sec) (MPa) (mu/M) (sec) .40 90.24 90 128 18.36 90.33 90.144 20.96 90 90.136 19.76 90.29 90 123 17.96 90.37 90.126 18.26 90.36 90.132 19.16 90.23 90 117 17.07 90.37 90.132 19.16 90.36 90.119 17.37 90.22 90 107 13.37 90.31 90.123 17.96 90.43 90.119 17.37 90.30 90 119 17.37 90.34 90.140 20.36 90.36 90.107 13.37 90.23 90 099 14.37 90.30 90.123 17.96 90.39 90.111 16.17 90.28 90 103 14.97 90.34 90.119 17.37 90.33 90.123 17.96 90.23 90 113 16.77 90.37 90.144 20.96 90.43 90.119 17.37 '90.28 90 103 14.97 90.34 90.119 17 37 90.43 90.119 17.37 90.29 90 140 20 36 90.44 90.132 22.16 90.40 90.128 18.36 90.29 90 136 19 76 90.31 90.140 20.36 90.40 90.119 17.37 90.27 90 111 16.17 90.42 90.144 20.96 90.33 90.113 16.77 90.26 90 119 17 37 90.42 90.161 23.36 90.33 90.093 13.77 90.34 90 113 16.77 90.33 90.103 14.97 90.41 90.123 17.96 90.26 90 090 13.17 90.29 90.111 16.17 90.34 90.136 19.76 90.24 90 113 16.77 90.26 90.111 16.17 90.38 90.111 16.17 90.23 90 113 16.77 90.29 90.107 13.37 90.49 90.128 18.36 90.23 90 113 16.77 90.30 90.123 17.96 90.36 90.111 16.17 90.26 90 123 17.96 90.34 90.132 .19.16 90-34 90.132 19.16 90.23 90 103 14 97 90.31 901113 16.47 90.41 90.132 22.16 95 TABLE A17.STRESS.STRAIN AND TIME AT FAILURE FOR LOADING IN RIGID --0. 007 SEC-1 . DIE OF CUBIC APPLE SPECIMENS.E11 MC INTOSH JONATHAN RED DELICIOUS ‘11 11 t ‘11 ‘11 t ‘11 ‘11 t (MP3) (mm/mm) (sec) (MPa) (mm/mm) (390) (MPa) (mm/nun) (sec) 90.26 90.097 14.07 90.37 90.123 17.96 90.46 90.093 13.77 90.29 90.111 16.17 90.47 90.183 26.93 90.30 90.103 14.97 90.29 90.103 14.97 90.29 90.113 16.77 90.30 90.119 17.37 90.26 90.111 16617 90.38 90.163 23.93 90.31 90.113 16.77 '90.27 90.099 14.37 90.34 90.163 23.93 90.32 90.148 21.36 90.36 90.113 16.77 90.38 90.113 16.77 90.33 90.128 18.36 ’90.34 90.140 20.36 90.34 90.144 20.96 90.63 90.123 17.96 "90.29 90.107 13.37 90.34 90.281 40.73 90.32 90.103 14.97 '90.34 90.113 16.77 90.37 90.169 24.33 90.34 90.107 13.37 90.23 90.119 17.37 90.34 90.113 16.77 90.63 90.128 18.36 90.34 90.119 17.37 90.34 90.113 16.77 90.63 90.128 18.36 90.30 90.107 13.37 90.43 90.144 20.96 90.32 90.103 14.97 90.33 90.123 17.96 90.46 90.163 23.93 90.38 90.128 18.36 90.38 90.119 17.37 90.34 90.123 17.96 90.43 90.132 22.16 90.39 90.140 20.36 90.36 90.132 19.16 90.30 90.132 19.16 90.33 90.161 23.36 90.29 90.111 16.17 90.33 90.128 18.36 90.31 90.113 16.77 90.38 90.113 16.77 90.33 90.128 18.36 90.49 90.144 20.96 90.34 90.123 17.96 '90.33 90.111 16.17 90.49 90.163 23.93 90.34 90.111 16.17 90.30 90.136 19.76 90.33 90.132 19.16 90.31 90.082 11.97 90.43 90.136 19.76 96 TABLE A19. UNIAXIAL COMPRESSION OF CYLINDRICAL SPECIMENS OF nc INTOSH . AXIAL DEFORNATION VALUES AT‘ 011--0.11 MP2 FOR FIVE SPECIMEN HEIGHT. Ellf90.007 SE04- H (mm) 8.32 12.13 19.17 26.33 34.98 0.069 ' 0.129 0.13. 0.16. 0.119 U1 U1 U1 U1 U1 (ma) (nu) (mm) (mm) (mm) 9 6.60 -10.16 913.71 913.71 913.74 9 7.62 9 8.12 914.22 916.76 917.27 9 8.12 9 8.12 911.68 .914.22 918.79 911.68 - 7.62 912.70 912.19 919.81 9 8.63 - 8.63 913.74 918.28 924.89 9 9.63 -11.17 911.17 919.30 920.32 9 8.12 910.92 912.70 918.28 918.28 9 9.63 - 9.63 918.79 922.86 919.30 9 7.62 -13.20 913.71 916.76 920.82 9 9.63 912.70 911.17 913.24 926.92 9 6.60 914.22 913.20 917.27 921.84 9 7.11 910.66 913.20 913.71 919.81 . 910.16 - 9.14 912.44 914.22 920.82 9 8.12 911.68 912.70 914.22 921.32 9 6.60 914.22 913.20 913.71 921.33 . 9 8.63 9 7.62 911.17 917.78 921.84 9 3.38 912.19 913.74 916.23 920.32 9 9.63 910.16 920.32 913.24 922.33 9 7.11 -13.20 913.24 916.76 922.33 9 7.62 911.68 913.71 916.76 920.32 (9) standard deviation 97 TABLE A19. UNIAXIAL COHPRESSICN 0F CYLINDRICAL SPECIMENS 0F JONATHAN. AXIAL OEEORnATION VALUES AI 011 890.11 MPa FDR FIVE SPECInEN HEIGHT.en --o. 007 SEC' #- H (mm) 8.32 12.13 19.17 26.33 34.98 0.06. 0.129 0.139 0.160 0.11* U1 U1 U1 U1 U1 (nu) (an) (an) (an). (an) 9 9.90 9 7.11 9 8.12 913.24 921.84 9 7.11 9 8.89 913.49 913.20 920.32 912.19 912.19 917.27 913.71 916.76 9 7.62 9 7.11 912.19 917.27 913.24 913.20 9 9.63 914.22 917.27 919 30 913.20 9 8.63 9 9.63 914.22 920.32 9 8.63 913.71 912.70 913.20 914.22 9 7.62 910.16 912.70 914.73 919.30 9 7.11 9 9.14 911.17 913.24 923.87 9 3.38 9 9.14 910.92 912.19 920.82 9 3.38 9 8.63 913.20 911.68 917.27 9 9.63 9 7.11 913.20 913.24- 921.84 9 7.11 9 7.11 914.73 _ 912.70 917.32 9 6.60 9 8.12 911.17 912.19 918.79 9 8.12 9 8.12 9 9.63 916.23 917.27 9 7.11 9 8.63 910.66 913.71 919 30 9 8.12 910.16 912.70 912.70 913.74 9 6.60 9 6.60 910.16 917.27 913.24 9 8.63 910.16 ' 912.19 913.71 913.24 9 6.60 912.19 911.17 913.74 921.84 (9) standard deviation 98 TABLE A20. UNIAXIAL COHPRESSIDN OF CYLINDRICAL SPECIMENS 0F RED DELICIOUS. AXIAL pEFURHATION VALUES AT4H1990.11 MPa FOR FIVE SPECIMEN HEIGHT.EllI-O.OO7 SEC- . H (mm) 3.3: 12.13 19.17 26.33 34.93 0.069 0.124 0.134 0.164 0.114 U1 U1 U1 01 01 (mm) (no) (on) (mm) (mm) - 7.11 - 3.34 - 7.11 -12.70 -13.24 - 3.33 - 3.33 910.16 - 9.14 -13.71 - 3.03 - 3.03 - 7.11 910.66 -12.19 - 6.09 - 3.33 - 7.11 -12.19 -14.73 910.66 - 7.62 - 7.11 910.66 —14.73 - 3.33 - 7.62 - 8.63 -11.17 911.68 - 3.33 - 4.06 - 3.12 -11.17 912.19 - 6.09 - 4.37 - 3.12 9 910.16 -13.71 - 3.03 - 3.12 - 9.14 - - 8.63 ~12.7o - 3.03 - 4.06 - 6.09 - 9.63 -12.19 - 3.33 - 3.03 — 8.63 -12.19 -14.22 - 4.06 - 4.37 ~13.20 -12.19 -13 20 - 3.12 - 3.12 - 9.63 910.16 -13.71 - 3.33 - 7.62 - 9.14 -12.70 -17.73 - 3.03 - 6.60 - 9.63 910.16 913.20 - 3.12 - 4.37 - 7.62 - 3.63 -11.17 - 4.37 - 3.03 - 3.33 — 9.14 -13.71 - 4.06 - 6.60 -12.19 - 9.14 -12.70 - 3.03 910.66 ~12.19 - 9.14 -12.19 - 3.03 - 6.33 - 3.63 - 9.14 -13.20 (9) standard deviation ATE U mum "WNW WIFNWIIHYIYNWWII!“ 3 1293 30561961