ELASTIC STRESS- STRAIN CONSTITUTIVE EQUATIONS FOR VEGETATIVE MATERIAL Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY HARUHIKO MURASE ~ 1977 "IIIII IIIIIIIIIIIIIII 293 _10751 0848 This is to certify that the thesis entitled Elastic Stress-Strain Constitutive Equations for Vegetative Material presented by Haruhiko Murase has been accepted towards fulfillment of the requirements for Ph. D. degree in Agricultural Engineering @5227 OMajor professor Date4Q7/ /41j77 0-7 639 LIBRA 21’ Michigan State University :17- ABSTRACT ELASTIC STRESS-STRAIN CONSTITUTIVE EQUATIONS FOR VEGETATIVE MATERIAL BY Haruhiko Murase The objective of this work was to develop a continuum model which adequately describes the mechanical behavior of vegetative material. The vegetative material was assumed to be a porous medium comprised of interconnected air-filled pores (intercelluler spaces) and a multiphase medium of solid and liquid (nonporous cell medium) which can be visualized as a group of vegetative cells. It was postulated that the system of the continuum of nonporous cell medium can exchange heat and chemical energy with its surroundings. The water potential concept as a state function was introduced to the constitutive equations. Linear elastic stress-strain constitutive equations were derived through thermodynamics considerations for the Haruhiko Murase vegetative material. It was shown that the set of linear elastic stress-strain equations is analogous to that known as the Duhmel-Neumann relation. Experimental attemps were made to determine the material coefficients of tomato flesh. Approved {éZuaz IaJor ro essbr Approvecgf—J. R epartment a rman ELASTIC STRESS-STRAIN CONSTITUTIVE EQUATIONS FOR VEGETATIVE MATERIAL BY Haruhiko Murase A DISSERTATION Submitted to MICHIGAN STATE UNIVERSITY in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1977 ACKNOWLEDGMENTS The author is indebted to his major professor Dr. George E. Merva (Agricultural Engineering) for draw- ing his attention to this field of study, and would like to acknowledge the council, guidance and time extended by Dr. G. E. Merva. To the other members of the guidance committee. Dr. G. E. Mase (Metallurgy. Mechanics and Material science). Dr. L. J. Segerlind (Agricultural Engineering), and Dr. G. R. Safir (Botany and Plant Pathology), the author expresses his deepest gratitude for their time and pro- fessional interest. The financial assistance given the author by the Agricultural Engineering Department is acknowledged. TABLE OF CONTENTS Page LIST OF FIGURES . . . . . . . v LIST OF TABLES . . . . . . . vi ABBREVIATIONS AND SYMBOLS . . . . . vii I. INTRODUCTION . . . . . . . 1 II. REVIEW OF LITERATURE . . . . . 3 2.1 Mechanical Properties of Plant Tissue . 3 2.2 Vegetative Cell System in a Water Potential Fiels . . . . . 5 2.2.1 Introduction . . . . 5 2.2.2 The Water Potential . . . 5 2.2.3 The Cell Water Potential . . 6 2.3 Mechanics involving the Water Potential 7 III. DEVELOPMENT OF STRESS-STRAIN CONSTITUTIVE EQUATIONS FOR VEGETATIVE MATERIAL . . . 9 3.1 Constitutive Equations for Nonporous C911 MQdium c o c o e e 9 3.2 Linear Stress-strain Equations for Nonporous Cell Medium . . . . 12 3.3 Linear Stress-strain Equations for Porous Vegetative Material . . . . 13 3.3.1 Model Description . . . 13 3.3.2 Derivation of Stress-strain Equations . . . . . 13 iii 3.# The Material Coefficients of the Constitutive Equations . . . 3.5 Discussion of the Linear Stress-strain Equations . . . . . IV. EXPERIMENTAL DETERMINATION OF MATERIAL COEFFICIENTS FOR TOMATO FLESH h.l Procedure and Equipment . 4.2 Experimental Results and Discussion V. CONCLUSIONS . . . . VI. SUGGESTION FOR FURTHER STUDY . APPENDIX I . . . . . APPENDIX II . . . . . BIBLIOGRAPHY . . . . . iv 24 2n 2? 31 32 33 49 67 J ‘Illlf‘ t II, \II III III I'll" all (1‘. l: I I It Figure 3.1 3.2.1 3.2.2 4.1 #.2 #.3 4.1» LIST OF FIGURES Differential Element of Porous Vegetative Material . . . . . . ‘. Nonporous Cell Medium Compressibility . Porous Vegetative Material Compressibility Tomato Tissue Specimen . . . . Schematic Figure of Vacuum Apparatus used for Preparation of Gaseless Specimen . Schematic Figure of Experimental Setup for Measurement of Elastic Coefficients and Expansion Coefficient . . . . Schematic Figure of Device for Measurement of Porous Vegetative Material Coefficients Page l4 l9 l9 25 25 26 28 Table 1. 2. LIST OF TABLES Material Properties of Tomato Flesh . Material Coefficients necessary for the Constitutive Equations for Tomato Flesh vi 29 1&0. 8.1. 8.2 33. a“, atm b. bar ABBREVIATIONS AND SYMBOLS Helmholz free energy of nonporous cell medium Inverse of molal volume of water Arbitrary constants Integral constant Atmospheric pressure Number of moles of solutes in a unit volume of nonporous cell medium Integral constant Barometric pressure Length of a side of cubical specimen Initial length of a side of cubical specimen Material property Young's modulus of nonporous cell medium Apparent elastic modulus of tomato epidermis Elastic modulus of tomato epidermal cell wall Dilatation of nonporous cell medium Strain tensor of nonporous cell medium Strain of porous vegetative material under uniaxial loading Strain tensor of porous vegetative material vii 51 :1: B #6 eezz XX yy Strains of porous vegetative material in the normal directions Material property Body force Surface force Mass of loading weight Traction vector acting on gas part of porous vegetative material Gas porosity Traction vector acting on cell medium part of porous vegetative material Material property Mercury Length of a side of cubical specimen Initial length of a side of cubical specimen The first strain invariant for porous vegetative material The second strain invariant for porous vegetative material The third strain invariant for porous vegetative material ohlz The first strain invariant for nonporous cell medium The second strain invariant for nonporous cell medium The third strain invariant for nonporous cell medium viii Kpa Bulk modulus for nonporous cell medium Kilopascal Nonporous cell medium compressibility Water potential energy source Chemical potential of cell fluid Gram molecullar weight of water Water potential energy flow vector milliliter Mass integral Material property Mole fraction of water Normal unit vector Number of moles of solute Number of moles of water Turgor pressure Average turgor pressure Gas pressure Hydrostatic pressure Water vapor pressure Saturated vapor pressure of water Heat flux Universal gas constant Heat source Entropy ix xi' "3 Surface integral Temperature Time Internal energy Displacement of gaseous component Internal energy of water Displacement of nonporous cell medium Volum of porous vegetative material Volume of a vegetative tissue Volume of gas in a vegetative tissue Water potential energy function Volume of nonporous cell medium Original volume of nonporous cell medium Dummy variable for v Volume integral Elastic potential of porous vegetative material Initial elastic potential function Elastic potential function such that V2W1=0 Elastic potential of nonporous cell medium Relative flow vector of gas Rectangular cartesian coordinate Mass of water per unit mass of nonporous cell medium Material coefficient Water potential expantion coefficient Material coefficient Engineering strain tensor Engineering normal shearing strains Kronecker delta Dilatation of porous vegetative material Measure of amount of gas flowing through boundary Porous vegetative material compressibility Lamé’constant Lamé constant Poisson's ratio Mass density of water Mass density of nonporous cell medium Moisture potential Force applied to gas Stress tensor for cell medium part of porous vegetative material Stress tensor for nonporous cell medium Hydrostatic stress Stress tensor of porous vegetative material Water potential Reference level of water potential Dummy variable for’a o-oo xi Cell water potential Gravitational component of water potential Helmholz free energy of a unit mass of cell water Pressure component of water potential Osmotic component of water potential Matrix component of water potential 'd() dt Implies time derivative. i.e.. xii I. INTRODUCTION When one considers the mechanical behavior of vegeta- tive material responding to environmental perturbations in the soil-plant-atmosphere continuum, the use of elastic stress-strain constitutive equations in the traditional continuum mechanics is found unfeasible. The traditional stress-strain equations do not contain any parameters de- finable at all points in the soil-plant atmosphere continuum, and do not incorporate the effects of liquids as well as gases which are found in the cellular matrix of vegetation. The problem of fruit cracking has been under study for many years. generally in a horticultural sense rather than a mechanics sense. Initially. research was concerned with identifying the cause of cracking and. as early as 193#, it was noted that the most severe cracking of tomato occurred following the application of water to the fruit or soil. It has been suggested that the elastic properties of tomato skin are related to skin cracking (Nhuchi gt. gl. 1960). Voisey 23. El- (1964, 1965, 1970) studied such methods as the punc- ture test, bursting test. and skin tensile test to determine the strength of tomato skin in relation to cracking. Also, many studies have shown that fruit cracking is closely related to the absorption of water by fruit (Brown and Chas. 193A: Frazier. 1934. 1935). In spite of these efforts, little progress has been made in relating the phenomenon of fruit cracking to the water status of the fruit. In 1970. when the production value of tomato in Michigan was $5,848,000, the estimated losses due to tomato cracking were in excess of about $909,000. This constituted an overall loss of about 7% of the cash value of the crop. The work reported in this dissertation may be devided into two parts: 1. The development of stress-strain constitutive equa- tions containing parameters necessary to describe the mechanical behavior of vegetative material. 2. The experimental determination of the parameters necessary for use of the constitutive equations. II. REVIEW OF LITERATURE 2.1 Mechanical Preperties of Plant Tissue Studies on the level of a multi-cell structure, or tissue, to include the effect of interaction between cells have been performed by both removing a segment of tissue for analysis and by studying the action of the whole body such as a fruit. Falk et al. (1958) studied the relations between turgor pressure and Young's modulus using the resonant frequency of potato tuber parenchyma. They concluded that the cell wall material follows Hooke's law and that there are changes in elasticity of the whole parenchyma due to turgor pressure which are in turn reversible thanks to the ideal cell wall material. Nilsson et a1. (1958) studied the dependence of Young's modulus of potato tuber parenchyma on turgor pressure using a simple theoretical model. The cells of the parenchyma were approximated by regular geometric cell-forms (spheres or polyhedra). each cell being bounded by an elastic membrane and filled with an incompressible fluid. It was shown that this model yields the correct dependence of cell diameter on turgor 1;, pressure and that certain cell wall constants can be determined using the relation. Clevenger and Hamann (1968) studied the mechanical prOperties of apple skin. They determined material prOper- ties. including'the elastic modulus and Poisson's ratio, for three varieties of apples. All skins were found to be anisotropic with the greatest strength in the longitudinal direction. Relaxation and creep experiments showed that apple skin tends to be viscoelastic in behavior. Four element models were found to describe the action of the material very well. Akyurt (1969) and Akyurt gt §;‘_(l972) attempted to develop methods for studying the stress-strain relations in plant materials. With the cell wall idealized as a shell, the finite element method was proposed for the solution of the corresponding linear equilibrium problem. Akyurt showed that macrodisplacements as well as stresses and couple stresses acting on cellular bodies emerge as solutions of the field equations of the micropolar theory of Eringen (1962). The linear theory of viscoelasticity was also employed. Gustafson (197“) attempted to adopt the mechanics of a fluid-filled porous medium established by Biot (19hla, 1941b, 1942, 1955. 1956. 1962. 1963) to formulate a model containing additional parameters and variables necessary to more 5 adequately represent the mechanical behavior of a vege- tative tissue. Gustafson used the turgor pressure within the plant cells as a stress component. However. in practice, values of turgor pressure are not directly measurable, there- fore, this approach is less than adequate. 2.2 Vegetative Cell System in A Water Potential Field 2.2.1 Introduction Water potential is a true potential function and has the property that it is a single intensive parameter which incorporates all environmental parameters that in- fluence the water potential. Further. water potential is definable at all points in the soil-plant-atmosphere con- tinuum. 2.2.2 The water potential Buckingham (1907) used basic thermodynamics to formu- late an expression for the status of water in soils. Others have since applied the concepts of Gibbs free energy to form- late the intensive parameter of water potential which express- es the availability of water in terms of the parameters that affect its chemical potential. A development of water poten- tial from the view point of the plant physiologist as related to plants and soils is given by Slayter (1967). Merva (1975) 6 took an engineering approach to the subject. He expressed the water potential as with $1. and fig as the matrix component and the gravitational component of the water potential respectively where the matrix component arises from water interaction and water absorption on molecules of constituents of the cellular interior as well as from surface tension effects. is a component of the "'p water potential due to pressure caused by containment of liquid water. Within a cell. Np is the turgor pressure. Us is a component of the water potential resulting from the presence of solutes in water and Operative only when the solution is contained in a semipermeable membrane such as within a cell whose exterior is in contact with a source of water at a different value of water potential. The dimen- sions are energy per unit volume of water or. if we divide by the density of water energy per unit mass of water. 2.2.3 The cell water potential In a cell. the cell water potential, Cc, can be ex- pressed as 1) 0 a w '. we . w o (2e2) 1) Matrix and gravitational contribution are not considered here. The matrix component due to water attracted to cel- 1u1er constituents is considered to be negligible as is the gravitational component. the latter because only small changes in elevation are thought to occun 7 Eq.(2.2) shows the energy status of water in a cell is dependent upon the osmotic potential due to the presence of solutes. and upon the turgor pressure of the fluid with- in the cell. The Helmholz free energy, “'11 , of a unit mass of cell water can be expressed by 1 "H. = U" -TS 4- 513+ M (2.3) where Uw is the internal energy, P is the turgor pressure within the cell. S is the entrapy. and MO is the chemical energy due to the presence of solutes. In Eq.(2.2).u&)and we are 1 RT vs ' c ‘ Mw w where Nw is the mole fraction of water. 2.3 Mechanics involving The Water Potential Mountfort (1972) considered moisture transfer in porous elastic solids and introduced a "Moisture Potential" as a central concept in the theory. Mountfort defined the hydrostatic component (moisture potential).cr. of stressed liquid water in equilibrium with stress free liquid water at the same temperature and elevation by 0 2 m RT 1n p0 = -O' (2.6) where p is the vapor pressure over stressed liquid water, 8 p0 is the saturated vapor pressure of stress free water. R is the universal gas constant. T is the temperature of the system, (Jis the density of liquid water at temperature T. and Mw is the gram molecular weight of water. Mountfort's theory posesses several deficiencies. The potential func- tion is such that moisture must move from areas of low poten- tial toward areas of higher potential. Further, no reference is made to the role of pressure such as might be encountered in plant cells as it affects moisture movement. Nontheless. it recognizes the need for a potential function which would regulate water movement. Murase and Merva (1977a, Appendix I) showed that the apparent elastic modulus. Ea' of the medium (tomato epider- mis) is a function of the water potential of the cell. and assumed that the model is Ea= BC + f(p) = gwcen) (2.7) where Ea is the apparent elastic modulus. Ec the elastic modulus of the epidermal cell wall. f(p) a function of tur- gor pressure and g('”cell) a function of cell water potential. Murase and Merva (1977b, Appendix II) also reported that measurements of hydraulic conductivity of vegetative tissue are possible using water potential as the measurement para- meter and introduced a-new parameter, a,, the water poten- tial expansion coefficient where AV 1 a = ———_. . i . A"’cell (2'8) III. DEVELOPMENT OF STRESS-STRAIN CONSTITUTIVE EQUATIONS FOR VEGETATIVE MATERIAL 3.1 Constitutive Equations for Nonporous Cell Medium The law of conservation of energy for a liquid-filled nonporous cell system in which chemical reactions may occur can be written .9. ° ' d j” . 4f , dtlmguiuidIR'taE’mUdms mFiuidm+Sfiuids tLthm -LQinids+[dem—ISMinids (3.1) where U is the internal energy of a unit mass of the nonporous cell medium, ui the velocity of the nonporous cell medium. F1 the body force per unit mass. fithe surface force, Rh the heat source per unit mass of the medium,Qi the heat flux, L the energy source per unit mass of the nonporous cell medium due to metabolism,and Mi the water potential energy flux through the boundary. The principle of rate of work states that the rate of change of kinetic energy is equal to the rate of work done by every internal and external force. The principle of rate of work can be expressed as fife 1:11 61 dmulFi ‘31 dm-o-f fi 111 ds m S J 5'13 . €13 (W (3.2) v 10 where 531 and 513 are the stress and strain tensor respec- tively. We substitute Eq.(3.2) into Eq.(3.l) to obtain Applying Gauss' theoremto Eq.(3.3) with dma'b'dv where fiis the mass density of the nonporous cell medium, we obtain 6(6 - Rh - L) a 513' £313 " Qi'i " “‘1'1 (3'4) The second law of thermodynamics is 3.3.13de [4:11 dm—fS-Ei—flds (3.5) m m Assuming that the gradient of water potential results in water movement, t“ fiLde-‘lr‘n? dm—Lfl%ds (3.6) where Z is the mass of water per unit mass of the nonporous cell medium. and ois the water potential defined in terms of energy per unit mass of water. Applying Gauss' theorem to Eq.(3.5) and Eq.(3.6), we obtain and __ , __ Mi D¢Z -‘="- 9L "' M191 4' T¢ei (308) We now define the Helmholz free energy for nonporous cell 11 medium as AsU—TS—Vm (3.9) where vm is the chemical energy potential and taken as 02. For the continuous nonporous cell medium we postulate that A depends on only 315' T. and 0. then the total differential for-A is ' .. LA -'- 9A ' eA ' A .. .ij. eij + T + 80° . (3.10) We substitute Eq.(3.10) into Eq.(3.9) and (3.#) to obtain >. a o a e Z)¢ .. ; 8A 0 .— (‘99. Pee” +5PH = ( 2W3 +Me . (3.38) 19 V \n. Figure 3.2.1. Nonporous Cell Medium Compressibility 3 1% / \ \\ R. R. .» Figure 3.2.2. Porous Vegetative Material Compressibility n1 r—q 20 Substituting Eq.(3.35) and (3.36) into (3.38). we obtain In Eq.(3.35) and (3.39). A. p. and 9 are measurable. however. difficulties are involved in the measurement of A andtlfor porous vegetative material. In order to measure the engineering elastic modulus. i.e.. Young's modulus and Poisson's ratio. a technique to extract gases from a vege- tative material is required. The coefficient a. can be deter- mined experimentally or approximated from a knowledge ofHA. p and b (the number of moles of solutes per unit volume of a vegetative material). We derive the latter as follows. Consider a small volume change. v - v0. resulting from a water potential change of a block of the nonporous cell medium whose original volume is v0. Using Eq.(2.2). we can write. (10 :3 dI'p + (“'8 . (3.40) We substitute Eq.(2.#) and (2.5) into (3.#0) to obtain do =- tha ”1}:— d(1n NW) (3.41) where P8 is an average turgor pressure of the block. Introducing a hydrostatic stress. - OP' (the negative sign indicates a comprssive stress for a positive value ofiop). we may rewrite Eq.(3.#l) as a. = _% d op . % d(1n NW) . (3.42) £51 21 Assuming that the magnitude of the internal compressive stress OP is equal to the magnitude of the external pres- sure which relates to the volumetric strain av/v with the bulk modulus defined as - O’p - K-Av/v = K( v - vo)/vo. we may express Eq.(3.92) in terms of the volume change of the block as v - v0 RT V/V' d0 = %.d(——‘;O——) " Fwd(1nns * V/Vw ) (3.93) where n8 is the number of moles of solutes in the block and VV, the molal volume of water. We integrate the both sides of Eq.(3.43) fromaoto a and from V0 to v to obtain a V e _ V . Id¢'=-§-[d( vv'°)+;—T d(ln V/V ), “0 V0 0 VI v0 n8 *VVW (3.“) Using the volumetric strain Sunny/v. Eq.(3.94) becomes RT (Eel)(a+b) K O "' 0° - p 3 9 'M—" ln 3(3*1) * b (3.“5) where a is the inverse of the molal volume of water. and b the number of moles of solutes per unit volume of the non- porous cell medium. We linearize the logarithmic term in Eq.(3.l+5) around ‘e'= 0 to obtain the desired linear rela- tion between a and '6. then Eq.(3.l+5) becomes e - “o = [K/p + RTb/Mw(a + 13)] '5 (3.46) 22 Therefore. the strain tensor éi due only to a water poten- j tial change is OMw ( a + b ) ( ) j ¢-¢ .e ii.3TFM,,(a1-b)o-DRTBJ o 613 (3.4?) Since the strain due to a water potential change is defined as 31:, Res” in Eq.(3-17). OMw ( a + b ) =—L . (3-48) 3xnw(a+b)+pRTt§| pl 3.5 Discussion of the Linear Stress-strain Equations Since Eq.(3.3b) appears to be analogous to the equation known as the Duhmel-Neumann relation. the Navier equation of elasticity for vegetative material can be readily derived. The small strain tensor is expressedin terms of the displace- ment vector as eui eu an: em .3311). (3.49) 91' Using the equilibrium equation-3;?- = 0 and Eq.(3.‘+9) in Eq.(3.34). we can eliminate'Tij and eij to obtain the Navier equation as ' ee 9P pVBui +(Aep)-a—x'i == -Q::i ‘- 5 a}: (3-50) Eq.(3.50) is a linear nonhomogeneous equation. Since it is linear. the principle of superposition can be applied in order to solve the Navier equation (3.50). The solution of 23 the homogeneous equation, .35L == 0 9x1 ' PVIli 4' (ATP) can be obtained by the use of Galerkin vector or Papkovich- Neuber solution. Since-ggi ='ui' Eq.(3.50) may be expressed in terms of the elastic potential as 2 1 V?! = ( -dm+ BPg) . (3.51) A + 2p Assuming an unsteady state without energy source. the solu- tion to Eq.(3.51) becomes 1 t t W =-.. —— da'l dt + fib'ng dt 4» W0 + tWI Air 2p 0 o (3.52) where W0 is an initial elastic potential function and ViewI a: 0. Theoretical verification of Eq.(2.7) can be made by letting in j - l in Eq.(3.3h). Assume a medium (tomato skin) which does not contain pores. For this condition the P term g vanishes. Therefore or T 11 B Ea = EC fi—fl. . e11 °11 With a constant e11. Ea may be considered as a function of the water potential. IV. EXPERIMENTAL DETERMINATION OF MATERIAL COEFFICIENTS FOR TOMATO FLESH #.1 Procedure and Equipment Material coefficients were determined for tomato flesh. Specimens of tomato tissue 7 mm on a side were taken from the hatched region shown in Figure “.1. The specimens were im- mersed in -10 bar mannitol solution for 29 hours to release tension in the cell walls. This value is less than approxi- mately -7 bar which is the average value of osmotic water po- tential for the tomato cell found in earlier work (Murase and Merva. 1977a. Appendix I) and did not appear to cause plasmoly- sis of tomato tissue. To attain gas withdrawal. a force was exerted on the tissue by means of a 2.1 gram mass and the sample was subjected to a 630 mm Hg vacuum. figure 4.2 removed a considerable quantity of gas from the tissue. The treatment was assumed to have yielded a gaseless sample of tis- sue satisfactory for the nonporous cell medium compressibility tests. The setup shown in Figure #.3 was used to determine both a, as well as E and v. To determine& a sample of gaseless medium was immersed in -5 bar mannitol solution and the di- mensional change was computed as the specimen tensed from -10 29 25 Figure 4.1. Tomato Tissue Specimen. MANNITOL SOLUTION \4 .WEIGHT VACUUM seecnmenw . PUMP / / ////// ’/'///////////// Figure 4.2. Schematic Figure of Vacuum Apparatus used for Preparation of Gaseless Specimen. 26 .ecowowhmeoo ceamcmmxm use upcoHoHMhooo oapmsflm no vsosoHSmse: non mavem amazoSasomxm mo shaman cavssemom .m.: ousmwm qumw> szmamwz i. - if i. - . oc\? nozv \Amo\ov I u: 27 bar to -5 bar. Measurements were taken until no detect- able visual changes in the parameters h and c occured as shown.in Figure 4.3. For this test no weight was used. Figure 4.4 shows a device used for the measurement of 9. The mercury not only applies hydrostatic pressure to the specimen but also seals the specimen's pores. 4.2 Experimental Results and Discussion Table 1 capsulizes the measured values of the material coefficients for tomato tissue. Also included is the mea- sured value of b (number of moles of solutes in a unit volume of tomato tissue) which was determined by freezing a represen- tative sample of tomato tissue to destroy the cell walls. and then determining the concentration of solutes in the resulting mixture using a thermocouple psychrometer technique. All ma- terial coefficients of tomato tissue necessary for the linear stress-strain constitutive equations are given in Table 2. The following relations were used to calculate the material coefficients: V A = (1-2Ui(1+v) P " 2 (I+\J) K- E -1 3 ( 1 - §\)) "'E' (1: 5(3A‘I’2P) a: l - O/k 28 ELECTRIC DISPLACEMENT MEASURING SYSTEM MERCURY r SPECIMEN SUPPORTER SPECIMEN -—— Figure 4.4. Schematic Figure of Device for Measurement of Porous Vegetative Material Compressibility. 29 Table 1. Material Preperties of Tomato Flesh. Parameter Dimensions Mean value Standmfl.deviation E Kpa 2275.240 323.740 0 — 0.131 0.0510 a atm“ 0.0069 0.0012 9 atm‘1 0.083 0.0013 b mole-liter” 0. 218 0.0240 Table 2. Material Coefficients necessary for the Constitutive Equations for Tomato Flesh. ugtgpig; coefficient Dimensions Calculated value A Kpa 357.33 p Kpa 1005.67 a. Kpa-bar"1 20.89 B .. 0.158 K Kpa 1027.78 (.34 (0 (“v t" 30 Assuming T-300°k. Eq.(3.48) gives Eta-0.0296 atm . The measured value of Elie four times lower than the cal- culated value. The approximation formula (3.48) was de- veloped based upon the assumption that the difference between the magnitude of the internal pressure and the external hydraulic pressure is very small. Furthermore. the effectscf physiological activities such as osmoregula- tion. metabolism. etc. were neglected. Finally. it is quite likely that all gas was not removed from the tissue. The discrepency may have resulted from the reasons mentioned. l. 2. 3. V. CONCLUSIONS The auther concludes that: Stress-strain constitutive equations capable of describ- ing the elastic response of vegetative material to en- vironmental perturbations in terms of the change in water potential are T1j= 2P81j+ Aekk Gij + a(¢'¢o)61j - BPg 5 '13 All material coefficients involved in the constitutive equations are determinable. The Lame’ constants p and Aof tomato flesh are 1005.67 Kpa and 357.33 Kpa. respectively. The other coefficientsti and Bare 20.89 Kpa/bar and 0.158. respectively. 31 VI. SUGGESTION FOR FURTHER STUDY The present theory should be extended to the visco- elastic theory. Biot (1956) has developed a theory of deformation of a porous viscoelastic solid. In order to employ Biot's theory to develop a viscoelastic theory for vegetative material. the gas strain component should be taken into account. The gaseous diffusion phenomenon in vegetative material must be studied so that the relaxation due to gas displacement in the medium can be comprehended. 32 APPENDIX I STATIC ELASTIC MODULUS OF TOMATO EPIDERMIS AS AFFECTED BY WATER POTENTIAL by l/ H. Murase and G. E. Merva- INTRODUCTION A relationship exists between cracking and the strength of tomato epidermis (Reynard 1960. Voisey and Lyall 1965). Miles £31.2la (1969) concluded that the epidermis is the single most important component of the tomato as related to mechanical strength. Voisey and others (Voisey and MacDonald 1964. Voisey and Lyall 1965. Voisey 25‘.g;; 1970) used punc- ture tests. bursting tests and tensile tests to relate tomato skin strength to fruit cracking and concluded puncture tests might be used as an index to cracking resistance. Batal £5; £1; (1970) suggested that the percent increase in length un- til failure. along with the ultimate force at failure. might be related to cracking resistance. Batal determined a value of elastic modulus from theslope of the stress-strain curve at a selected value of force but concluded that the elastic modulus thus obtained was totally unrelated to cracking. No evidence appears that the viscoelastic properties of the tis- sue were considered. even though others have considered these properties as being vety important (Miles 23‘ 31‘ 1969. Cleland I/G'radu ate Issistant and Professor of Agricultural Engineering respectively. Department of Agricultural Engineering. M.S.U. East Lansing. Michigan. 33 34 1971. Voisey and Lyall 1965). Tomato fruit cracking is closely related to water absorp- tion (Brown and Price 1934. Frazier 1934. 1935). Cell expan- sion results from water absorption (Considine and Krudeman 1972. Chaney and Kozlowski 1971. Boyer 1968. Slatyer 1957) which causes turgor pressure increase within cells. Cell size increase is responsible for fruit expansion and subsequent 'cracking. Water movement in vegetative tissues is known to occur as a result of water potential differences. Changes in cell water potential affect turgor pressure. influencing mechanical properties of tissues (Falk et, al. 1958). Unfor- tunately. even though considerable investigation of rheolo- gical properties with water potential characteristics. pri- marily because the two subjects are generally not studied to- gether (Cleland 1971). In a preliminary investigation the authors observed a definite relation between the water potential of tomato epider- ‘mis and a measured value of elastic modulus. Lack of control in the experimental approach cast some question on the results. however. Therefore. the work was repeated using a single. crack suseptable variety (Tuck-Cross Ohio) of tomato and re- producable experimental procedures in order to assess the ef- fect of water potential on a single mechanical property of tomato epidermis. the static elastic modulus. 35 THEORY Consider a simple uniaxial state of stress. We assume that the stress may be expressed as a two part phenomenon. 030'. +01, (1) where a; is stress due to loading while Up is an initial stress which is due to the turgor pressure within a cell. Figure 1. Using the elastic modulus E we can rewrite Eq.(l) as Ea = Bee 4' Op (2) where Ea is the apparent elastic modulus of the medium. Ec is the elastic modulus of the epidermal cell wall andeais the strain due to a uniaxial load. If we divide both sides of Eq.(z) by we obtain E!3L a EC 0» op/e . 133<3 *- f(p) (3) According to Eq.(3) the apparent elastic modulus is a function of the turgor pressure in the cell(s). Note that f(p) is a monotonically increasing function for Patm‘P> o m.NI o.ml mN I 06 PI . . ‘ 'I ‘ .5 shaman ON Fl ‘ 47 REP ENC Batal. K. M.. J. L. Weigle and D. C. Foley. 1970. Relation of stress-strain preperties of tomato skin to cracking of tomato fruit. Hort. Sci. 5:223. Boyer. J. B. 1968. Relationships of water potential to the growth of leaves. Plant Physiol. 43:1056-1062. Brown. H. B. and V. P. Chas. l93h. Effect of irrigation. degree of maturity and shading upon the yield and degree of cracking of tomatoes. Proc. Amer. Soc. Hort. Sci. 32' 524’528 s Chaney. W. R. and T. T. Kozlowski. 1971. Water transport in relation to expansion and contraction of leaves and fruits of Calamondin orange. J. Hort. Sci. #6:?1. Cleland. R. E. 1971. Cell wall extension. An. Rev. Plant Physiol. 22:197-222. Considine. J. A. and P. E. Kriedemann. 1971. Fruit splitting in grapes: Determination of the critical turgor pressure. AuSte Jo Agrie R980 23.17-2ue Engmann. K. F. 1934. Studien fiber die Leislungsfahigkeit der Nassergewehe sukkulenter Pflanzen. Beik. Bot. 3bl. A52:38l-817. Falk. 8.. C. H. Hertz and H. 1. Virgin. 1958. On the relation between turgor pressure and tissue rigidity. I. Experi- ments on resonance frequency and tissue rigidity. Physiol. Plants. 11:802-817. Frazier. I. A. 193“. A study of some factors associated with the occurence of cracks in the tomato fruit. Proc. Amer. Soc. Hort. Sci. 328519-523: Mass. G. E. 1970. ontinuum Mec c chaum' Outline §eries. McGraw- oo o. . . . . Miles. J. A.. R. B. Fridley and C. Lorenzen. 1969. Strength characteristics of tomatoes subjected to quasi-static loading. Trans. of ASAE 12:627-630. Nhuchi. K.. F. Honda and S. Ota. 1960. Studies on cracking in tomato fruits. l. Mechanism of fruit cracking. J. Hort. Assoc.. Japan 29:287-293. #8 Reynard. B. G. 1960. Breeding tomatoes for resistance to fruit cracking. Proc. Plant Sci. Seminar. Campbell Soup Co. 93-112. Slayter. R. 0. 1957. The influence of progressive increase in total soil moisture stress on transpiration. growth and internal water relationships of plants. Aust. Biol. 8°1e 10' 3200 Slayter. R. O. 1967. Plant-Water Relationships, Academic Press. N.Y. Thoday. D. 1921. On the behavior during drought of leaves of two cape species of passerina with some notes on their anatomy. Amer. Bot. Lend. 35:585-602. Voisey. P. W. and D. C. MacDonald. 1964. An instrument for measuring the puncture resistance of fruits and vegetables. Proc. Amer. Soc. Hort. Sci. 8#:557-563. Voisey. P. W. and L. H. Lyall. 1965. Methods for determining the strength of tomato skins in relation to fruit cracking. Proc. Amer. Soc. Hort. Sci. 86:597-609. Voisey. P. W.. L. H. Lyall and M. Kloek. 1970. Tomato skin strength. its measurement and relation to cracking. J. Amer. Soc. Hort. Sci. 95(4).u35-uaa. APPENDI X I I HYDRAULIC CONDUCTIVITY 0F VEGETATIVE TISSUE By H. Murase and G. E. Merva.;/ The movement of water through vegetative tissue is an important aspect of the storage. drying. and handling and processing of fruits. grain. or other biological materials (Walton e_t.a_1_._ 1976. Young and Whitaker 1971). Plant physio- logists have performed many studies on water movement in the leaf. the xylem. and in the stem and roots (Aston and Jones 1976. Weatherly 1976. Newman 1976). Although a knowledge of the hydraulic conductivity. the proportionality parameter relating water flux to the gradient of the potential causing flow. is crucial to water movement work. determinations of the hydraulic conductivity for higher plant cells have not been successful. The only values for hydraulic conductivity which are reliable are those obtained from giant algal cells (Kami- yama and Kuroda 1956. Dainty 23.51; 1959. 196“). Data which are available for single. higher plant cells were mostly ob- tained from an analysis of the kinetics of tissue swelling and shrinking. The reliability of these data is questionable. however (Dainty 1976). Dainty (1963) suggested using labeled 1 _/The authors are respectively Graduate Assistant and Professor of ricultural Engineering. Michigan State University. East Lens ng. Michigan. The assistance of Vilma M. Horinkova in performing the water potential measurements is gratefully _acknowledged. “9 50 water exchange between a plant tissue and its bathing medium as a possible way of determining hydraulic conductivity. However. Dainty himself questioned the techique. This work reports on a technique in which the free energy exchange be- tween a tissue and its bathing medium is used to determine hydraulic conductivity. DIFFUSION OF CHEMICAL POTENTIAL ENERGY IN VEGETATIVE MATERIAL The theory of irreversible thermodynamics postulates a relationship between a flux J1 and a generalized force 03 as J1 = ‘13“: ‘1) where the preportionality parameter L1: is termed a pheno- menological coefficient. Plant physiologists commonly use an adoption of the phenomenological relationship called Ohm's law. The form often seen is: Jw - L'AI' (2) where J' is a volume flux of water. Lw is the phenomenological coefficient and MI is the difference in water potential between two regions between which water exchange occurs. Note that in the case of Eq.(2). the gradient of the potential is not used. rather it is assumed that a discontinuity exists between the two regions and the difference in water potential is assumed to be the driving force. In general. it can be stated that nature abhors a discon- tinuity. i.e.. the water potential is a continuous function throughout a tissue. Thus. it would be more appropriate to 51 consider the force driving the flux to be definable as the gradient of potential between two points. In terms of Eq.(l) we may write: J' -.—. L'VV (3) where V is the gradient operator. Eq.(3) is difficult to use since it leads to the diffe- rential equation .39 __ 2 st DWV V where Gis the volumetric water content of vegetative tissue 2/ and D. is the hydraulic diffusivity. Unfortunately. a non- destructive determination of the relationship between 8 andIPis not presently available. even though it is needed to effect a solution to the equation. The movement of water into or out of a vegetative mass is accompanied by a transfer of free energy. thus a measure- ment of change in the free energy of the medium at some point is indicative of a change in water content. We postulate. therefore. that the phenomenological expression. Jc = LcV'I’ (4) can be used as a basis for our study since W. the water poten- tial. is a function of the free energy. Lc is the phenomeno- logical coefficient relating the gradient of free energy to Jc. the free energy flux. Figure 1 depicts the movement of water It is assumed the medium is homogeneous and isotropic in the region over which the equation holds. 52 y INSULATED ///////// //////// WATER FLOW _ CHEMICAL fl ENERGY FLOW “"“" li/7///// //7///// "I\ , 0' Figure 1. Chemical potential energy as well as water is trans- ferred by the driving force due to the water potential gradient in the system. In this case. a solute trans- fer is ignored. It is assumed that the water poten- tial function for the system is continuous. 53 and free energy one-dimensionally. A second approach as de- picted in Figure 2 was also used as an independent means of evaluation. Here. cylindrical and reflective symmetry are employed and a two-dimensional flow field is utilized. The technique is as follows. DETERMINATION OF HYDRAULIC CONDUCTIVITY Assume that a sample of vegetative tissue is immersed in a large reservoir and that the water potential of the reser- vior. UR. remains constant. The sample is assumed to have an initial constant water potential IFS?! ‘I’R at the time t = O of immersion. A transfer of free energy will occur. leading to some distribution of water potential in the medium which changes with time as shown in Figure 3. The relation between a location in the medium. time. and water potential at the selected loca- tion is described by the differential equation -§¥-==05€W (5) where Dc= L../pc . (6) In the above equation.£>is the mass density of the system while c is the specific chemical energy. i.e.. the chemical energy per unit mass. Dc is the diffusivity for free energy. In terms of polar coordinates. Eq.(5) is 9" 62V _1_ a! 3? -Dc( are + 1, er) (7) with boundary conditions Hr. O)=I (a constant) (8) 5A DISTILLED WATER L L/2 POTATO TUBER , SPECIMEN I! v - A. - : 9% WATER POTENTIAL MEA URN F :2 [(R- '74)! _L \J 'TT‘ ‘ Inn-e "D / ‘\ Figure 2. The thickness of the disk T is very small in com- parison to L so that the effect of the longitudinal direction is negligible. r can be taken as a con- stant to facilitate calculation of De. 55 z EQUIPOTENTIAL LINE )N 4y Figure 3. The water potential at every point in the system shang'es with time until the system equilibrates h R. 56 for the configuration of Figure 2. The solution of Eq.(7) for conditions (8) is given by Churchill (1963). V 00 21 R A Nut): mgl RanflmR) .[rJOUmmdr] Jo( mr)e oi: 21Jo()\mr) -(A:,Dct) m=1 AmRJ1().mR) e -(I::Dct) (9) where Jo and J. are Bessel's functions. For the conditions given. an excellent approximation was obtained from the first five terms. 5 ZIJ (A r) «has t) wk...” 2 2 O m e m c (10) m=1 AmRJ1(AmR) A value of Dc can be obtained from Eq.(lO) by determiningi’at r as shown in Figure 2. and then solving for Dc using Newton's method. Conditions for the one-dimensional case are given in Figure h. The equation is ' £9. _1_92 et =Dc exit (11) with boundary conditions ”(oet) 3 v(Let) = 0 V(x.0)== I (a constant) (12) and solution (Churchill. 1963) co 21 -(D manat/La) v(x.t) = 2 [l - cos(mn)] sin(mnx/L) e c m=1 1M1 57 (A) Figure h. The water potential on the surface (A) and (B) is kept at zero. The surface (C) is insulated from the chemical potential of the surrounding region by vaseline. (a) is the disk used for determining the water potential at x. To avoid contamination a cylinder of diameter d was isolated from the larger cylinder (diameter D) prior to sectioning to obtain the test disk of thickness T. 58 a good approximation is possible using only the first seven terms so that we can write. so 21 -(Dcmanat/Ia) “1.13) = 2 —- l - cos(mnfl sin(mnx/L) e m==l mm (13) Eq.(13) can then be solved for Dc if T. t. and x are known RELATION BETWEEN I.w AND LC PI'O‘ EQ-(B) and (a) W Lw=-3:'°Lc =XLc (11+) assume that some volume of water 3(m3) was transferred across the section y-y of area A(m2) in figure 1 and that the transfer required t(sec) to complete. Further. assume that 2(jou1es) of energy was transferred across section y-y in the same interval. Under thses conditions. J,= a /(At'A) : Jc=2 /(At~A) . (15) If the original tissue is a block of volume V(m3) surrounded by water. 5 is equal to the product of the increment AV(m3) of volume of the block and the parameter f of the block where the f is a constant defined by f _____ Vcells (16) vblock For the same conditions the free energy transfer b,is h :- \mcMI (17) where the obvious analogy to heat energy transfer is apparent. For these conditions. From Eq(15) and (17) x = a / .12. = E New] .(Avm (18> 59 -so that. in terms of a volumetric strain XAvac/f = e which occurs as a result of free energy transfer. It is customary to relate such a strain in terms of an expansion coefficient as is done with metals and heat energy. We there- fore introduce a parameter. a. the water potential expansion coefficient where -21. o... v“, (19) so that x can be defined as x-.-.- gg— . (20) We can substitute Eq‘ZO) into (1“) using also (6) to obtain L.= %%Iem. (21> and Lw can now be determined if fa.is found since Dc can be estimated from the solutions to Eq.(lO) and (13). Note that the product fa is m =(‘Vce1lg/‘vblockH “block/vim ) = ‘vcells/(vblockW) ' If solute transfer is negligible. "cells is equal to the increase in mass of the block divided by the density of water. Given the initial block volume and the change in water potential with its corresponding change inta. which is detect- able as an increase in block volume. fa.is calculable. Although little work is available to justify the statement. the linear 60 region for the saw relation is probably small. In the estimates used for this paper. a block of 7mm on a side was selected and 2 bars was used for AM. The volume. mass. and water potential were measured and the block was immersed in a solution whose water potential was 2 bars greater. and 2h hours were allowed for equilibrium to become established. The increase in volume was determined and used to estimate fa. The parameter Lw was then determined from Eq.(21). Eq.(3) has the well-known form of the Darcy equation if I... = -K'. the hydraulic conductivity. Thus. the hydraulic conductivity of vegetative tissue is determinable from measure- ments made on water potential. It must be noted that the values thus obtained can be related to values determined by plant phy- siologists and others using the discontinuous form of the water _potential function as expressed in Eq.(2). The comparison is determined by dividing Kw' the hydraulic conductivity. by the number of cells per unit length of path. Values thus obtained we term conductances. They relate the volume flux of water to the potential difference across one cell. Values obtained in this fashion are compared to values obtained by other investi- gators in Table 1. RESULTS Table 1 capsulizes the results of both the one-dimensional and two-dimensional experiments. Also included are values of energy conductivity as estimated from Eq.(20) and (1h). 61 Table 1. Experiments conducted with potato tuber numerical values.* One- Two- Parameter Dimensions dimensional dimensional Comments fa volum water E - 0.03788 measured- vqume tissue-Bar st. 0.002%“ 10 values I)c cma/bar SE = 0 .00812 E . 0.02% measured- s - 0.000233 3 - 0.000108 6 values x. mi/soc-bar 8.55xlo'12 2.59210'12 calculated Conductance m/sec-bar 5.93x10'8 l.81xlO"'8 calculated Moan cell microns 5:- = 110.1 measured- diameter a a: 16.17 20 values *3? .-.-.mean value. s = standard deviation Table 1 shows the hydraulic conductivity as determined from one-dimensional experiments to be the same order of magni- tude as estimated by the two-dimentional technique although the latter value is about 69$ lower. The discrepency is the same for the conductance. The conductance values as determined by other investigators are given in Table 2. Table 2. Conductance of vegetative tissue. ?:7§:§3§:§I Investigator Tissue 1x10'7 Kelly gt. 81.1.: (1963) algal cells 1x10‘7 Klepper 25. gig. (1973)* cotton 3::10'8 Wolley (1955).. maize lxlo'll Glinka a Reinhold (1972)* carrot 1.81::10"8 Murase a Merva potato *Values used were calculated by Dainty (1976). 62 It is obvious that discrepencies between investigators exceed the discrepency as determined by the two approaches used here as based upon water potential changes. DISCUSSION . The concept of using water potential to determine hyd- raulic conductivity is left to be a positive approach to an important vegetative tissue parameter. The differences ob- served using the one- and two-dimensional approaches are not felt to be significant inasmuch as the two dimensional approach requires measurements of water potential made over the face of a disk of tissue across which a water potential gradient is known to exist. It is felt that the agreement in order of magnitude is most encouraging and. further. that the overall agreement in order of magnitude between various investigators related to conductance values is of even greater value. We expect that such agreement should be expected based on the nature of the path assumed. It is clear that the transport of water through vegetative tissue must be either through the cell wall into the cytoplasm and thence into a neighboring cell. or along the microfibrilic structure of the cell wall proper. At any rate. it does not take place through the inter- cellular spaces are so small that surface tension prevents water from entering the spaces and displacing the air which presently occupies the spaces. The application of a vacuum to a tissue submerged under water reveals that a significant 63 amount of air can be withdrawn from the tissue. Upon release of the vacuum. water is drawn into the spaces and immediately becomes available to the cells. A dramatic change in tissue turgor and volume can usually be noted as a rapid change in water potential of the cells comprising the tissue. This con- trases markedly with the changes as notes in simple diffusion type experiments. Of the two other methods of water transfer. it is not possible at this time to determine which is correct. It must be noted that all work thus far is predicated on minimum solute transfer. Undoubtedly. the model is incorrect in that some solute transfer must occur. Stuart (1973) indi- cated that using mannitol as an osmotic agent for vegetative tissue was satisfactory as long as the experiment did not persist over several days. If a long time span was used. some mannitol was detectable within the cell boundaries. In our case. little solute diffusion was assumed to occur. although no detailed check of this assumption was conducted. CONCLUSIONS We conclude that: 1. Measurements of hydraulic conductivity of vegetaive tissue are Ioseible using the chemical potential of water as the measurement parameter. 2. Values obtained in this flnfldon.agreed for potato regardless of the configuration assumed for the experiment. 64 3. The hydraulic conductivity of potato tissue is about u. 3 x lO-Bmé/sec-bar. The conductance. obtained by dividing the conductivity by the number of cells per unit length of path is about 2 x lo'am/sec-bar. avalue in agreement with previous estimates made by other researchers. 65 w Aston. M. J. and M. M. Jones. 1976. A study of the trans- piration surfaces of Aygna ftgzilig L. var. Algerian leaves using monosilic c ac d as a racer for water movoment. Planta 130:121-129. Churchill. R. V. 1963. Eguzigz Series and Boundary Value .Zzghlggg. 2nd ed. McGraw-Hill Book Company. N.Y. Dainty. J. and A. B. Hape. 1959. The water permeablity of cells of Chara_angtzg11n_fig Br. Aust. J. Biol. Sci. 12:136-145. Dainty. J. 1963. Water relations of plant cells. Advan. Bate Reae Is 279-326e Dainty. J. and B. Z. Ginsbur . l96fl. The measurement of hydraulic conductivity Tosmotic permeability to water) of internodal characean cells by means of transcellular osmosis. Biochem. Biophys. Acta 79:102-111. Dainty. J. 1976. Water relations of plant cells. 32 Trans- port in Plant 0311s 11. Part A: Cells. Luttge. U. and M. G. Pitman eds. Spring Verlag. 12-35. Glinka. Z. and L. Reinhold. 1972. Induced changes in per- meability of plant cell membranes to water. Plant Phy'iOIe “98602-606e Kamiyama. N. and K. Kuroda. 1956. Artificial modification of the osmotic pressure of the plant cell. Protoplasma “6.“23.u36e Kelly. R. B.. P. G. Kohn and J. Dainty. 1963. Water relations 0 f m§tg1%g_1zgnglggggg. Trans. Bot. Soc. Edinburgh 3983 3-3 lb . Klepper. B.. F. J. Molz and C. M. Peterson. 1973. Tempera- ture effects on radial propagation of water potential in cotton stem bark. Plant Physiol. 52:565-568. Newman. I. I. 1976. 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Elastic modulus of tomato epidermis as affected by water potential. ASAE Paper 77-3030 presented at Summer Meeting of ASAE. Raleigh. North Carolina. Murase. H. and G. E. Merva. 1977b. Hydraulic conductivity of vegetative tissue. In review for publication by Trans. of ASAE. Nhuohi. K.. F. Honda and S. Ota. 1960. Studies on cracking in tomato fruits. l. Mechanism of fruit cracking. J. Hort. Assoc.. Japan. 29:287-293. Nilsson. Se Be. Ce He Hertz and Se Falke 1958s On the rela- tion between turgor pressure and tissue rigidity. II. Theoretical calculations on model systems. Physiol. Plant. 11:818-837. 69 Slatyer. R. 0. 1967. Plant-Water Relationships. Academic Press. London and New York. Voisey. P. W. and MacDonald. 1964. An instrument for measure- ing the puncture resistance of fruits and vegetables. Proc. Am. Soc. Hort. Sci. 84:557-563. Voisey. P. W. and L. H. Lyall. 1965. Methods for determining the strength of tomato skins in relation to fruit crack- ing. Proc. Am. Soc. Hort. Sci. 86:597-609. Voisey. P. W.. L. H. Lyall and M. Kloek. 1970. Tomato skin strength -- its measurement and relation to cracking. J. Am. Soc. Hort. Sci. 95(4):485-488. "IIIIIILIIIIIIIIIIIIIIII