NONLINEAR ELASTIC CONSTITUTIVE RELATIONS FOR CELLULOSIC MATERIALS

Transcription

1 In: Perkins, R.W., ed. Mechanics of cellulosic and polymeric materials: Proceedings, 3d Joint ASCE/ASME Mechanics conference; 1989 July 9-12; San Diego, New York: The American Society of Mechanical Engineers; 1989: NONLINEAR ELASTIC CONSTITUTIVE RELATIONS FOR CELLULOSIC MATERIALS J. C. Suhling Department of Mechanical Engineering Auburn University Auburn, Alabama M. W. Johnson and R. E. Rowlands Department of Engineering Mechanics University of Wisconsin - Madison Madison, Wisconsin D. E. Gunderson U.S.D.A. Forest Products Laboratory Madison, Wisconsin ABSTRACT In this work, the mechanical behavior of paperboard under the action of uniaxial and biaxial states of loading has been investigated analytically and experimentally. Analytical work has concentrated on the formulation of a set of new nonlinear elastic constitutive relations for orthotropic media based upon a hyperelastic material model. An assumed form for the strain energy density function incorporating a single effective strain variable has been proposed. The theory predicts constant Poisson's ratios but allows for nonlinear uniaxial stress-strain response. Extensive uniaxial and pure shear testing has been performed on paperboard to measure the material constants present in the new theory and to determine the functional form of the strain energy density function. The results of additional paperboard biaxial experiments have been utilized to evaluate the predictive capabilities of the constitutive model. INTRODUCTION Unlike laminated fiber-reinforced composite materials, paper (paperboard) is a multiphase composite composed of moisture, fibers, voids, and chemical additives. The fibrous raw materials which compose paper are typically organic with cellulose fibers from wood being the primary source. Since cellulose fibers are self-binding, no matrix material is needed to form a sheet. Wood itself may be thought of as a natural composite consisting of cellulose fibers interconnected by a primarily lignin binder. The three-dimensional physical structure of paperboard is basically an assembly of discrete fiber:, bonded together into a complex network. If the fibers were oriented randomly throughout the sheet, the material behavior would be isotropic. However, a single preferred orientation of the fibers results from the hydrodynamic forces present in the papermaking machine. In the terminology of the paper industry, this direction in the paper sheet with the highest stiffness (x - direction) is referred to as the machine 1 direction (MD). That with the lowest stiffness (x 2 - direction) is called the cross-machine direction (CD). These orthogonal in-plane directions of material symmetry allow for paper to be modeled macroscopically as an orthotropic solid. Paper is by far the oldest and most economically significant wood fiber composite. It is a prime example of a nonlinear orthotropic material with great economic importance and yet a limited design technology database. Its mechanical behavior is highly nonlinear, and strongly influenced by the environmental conditions of temperature and relative humidity. In addition, most papers are capable of demonstrating every known rheological phenomenon. Current uses of paperboard provide several challenging problems in engineering mechanics. For example, paperboard is often utilized in structural applications such as corrugated containers where it is subjected to complicated biaxial stress states, including shear. At present, lack of accurate constitutive relations and reliable strength predictions under biaxial loading and variable environments hampers analysis of such problems. Prior studies have been hindered by the relative unavailability of suitable but convenient theories and lack of other than uniaxial experimental data. Therefore, it has been common practice in the paper industry to use trial and error, and empirical approaches for optimizing the designs of paperboard products. This current lack of technology often limits creative design improvements which could curtail the excess use of materials and energy. As surveyed by Perkins [1983], theoretical modeling of paperboard mechanical behavior has been undertaken utilizing micromechanics and macromechanics (continuum) approaches. From the micromechanics point of view, paper has been modeled as a bonded fibrous network and as a hydrogen-bonded molecular assemblage. The goal of such formulations is to predict important gross properties of a paperboard characteristics and arrangement of sheet from the its constituents. Common moduli, bulk properties of interest include elastic Poisson's ratios, and ultimate strengths. Accurate micromechanics formulas can be used to facilitate improvements in manufacturing processes and allow tailoring of paper materials to meet particular end-use requirements. The uniaxial viscoelastic formulations of 1

2 Steenberg [1949] appear to be the first application of continuum modeling to paperboard. In that work, it was attempted to interpret paper mechanical behavior in terms of conceptual models embodying ideal spring and dashpotcomponents. Continuum models for paper elastic response have been restricted typically to linear orthotropic analyses. Mann, Baum, and Habeger [1980] have measured experimentally the orthotropic elastic constants of paper by using ultrasonic techniques coupled with the theory of wave propagation in an orthotropic plate. Carlsson, et al [1980a, 1980b] have predicted stiffness and curl in multiply papers by using classical lamination theory. In addition, a three-dimensional linear elastic finite element analysis of the creasing of paper sheets has been performed [Carlsson, et al, 1982]. The orthotropic version of the von-karman large deflection plate theory (incorporating geometric nonlinearities) has been applied to paperboard by Peterson and Fox [1980]. In that work, the Rayleigh- Ritz method was employed to determine the theoretical displacement and stress fields of simplified rectangular panels. Several loadings were considered which are suitable for containers. The obtained analytical predictions were compared qualitatively to experimental data obtained using photoelastic coatings. The large deflection plate theory has also been applied by Hudson, et al [1981] to predict paperboard buckling loads, and by Yang [1981] to analyze postbuckling behavior of paperboard sheets. In all of the previously mentioned investigations, the elastic behavior of paperboard has been modeled using the linear orthotropic theory. Paper behavior is highly nonlinear even at low strains. Therefore, a linear elastic approach is unsuitable if high accuracy is desired. Thorpe [1981] has presented a tangential nonlinear elastic finite element analysis of a paper sheet in uniaxial extension. An incremental approach was used to update the stiffness matrix. The utilized procedure calculated stiffness reductions based upon experimentally measured uniaxial response. Such an approach allows no interaction (coupling) of the stresses in biaxial situations. The method is similar conceptually to earlier efforts for fiber-reinforced composites by Petit and Waddoups [1969], and Sandhu [1976]. Another approach to modeling nonlinear elastic behavior is to use a hyperelastic formulation. In this case, a strain energy density function must be found which accurately characterizes a material's mechanical response. An advantage of such an approach is that it is predictive in biaxial situations while allowing for stress interactions which are not directly dependent on the material's observed uniaxial response. A strain energy approach for fiber-reinforced composites has been presented by Tsai and Hahn [1973] for addressing nonlinear shear behavior. Conceptually similar nonlinear elastic analyses of composites using a complementary energy density function approach have been undertaken by Pindera and Herakovich [1984], and Luo and Chou [1988]. In the present work, a new total strain hyperelastic constitutive model for nonlinear orthotropic media is presented and then applied to paperboard. The proposed nonlinear elastic constitutive equations are based on a special assumed form for the strain energy density function suggested by Johnson and Urbanik [1984]. Extensive experimental data have been utilized to determine material constants and obtain the optimum functional form for the strain energy density function. Results from additional biaxial experiments have been utilized to validate the adequacy of the formulation. A NEW BIAXIAL CONSTITUTIVE THEORY FOR NONLINEAR ORTHOTROPIC MEDIA A hyperelastic or Green-elastic material is defined as one for which a strain energy density function exists. The strain energy density is a function of the strains and is assumed proportional to the free energy per unit mass. Such a material has a natural state to which the body will always return when the loading is released. No energy dissipation is allowed, so that the hyperelastic constitutive model is purely mechanical. Using an energy balance equation obtained from the first law of thermodynamics and the above assumptions, it can be shown that the stress-strain relations for a hyperelastic material in cartesian coordinates are where T ij are the components of the second Piola- Kirchhoff stress tensor and e ij are the components of the material strain tensor. When displacement gradients are small compared to unity, the components of the familiar Cauchy stress tensor a ij can be substituted for T ij in eqs. (1) without loss of significant accuracy For plane stress situations (2) simplify to where W = (1) (2) (3) is now taken to be a function of the in-plane strains, and the conventional notations have been introduced. For isotropic appropriate strain linear elastic materials, the energy density function for plane stress situations is where E is the elastic modulus and v is the Poisson's ratio. For linear orthotropic elasticity, the correct strain energy density function for plane stress situations is where E 1 and E 2 are elastic moduli, v 12 and v 21 are Poisson's ratios, G 12 is the shear modulus, and the inplane strains (4) (5) are now exclusively evaluated in an x 1 -x 2 coordinate system aligned along the 2

3 directions of material symmetry. The familiar isotropic and orthotropic versions of plane stress Hooke's Law are obtained by substituting eq. (4) and eq. (5) into eqs. (3). Equation (11) can be inverted to give The Nonlinear Constitutive Model The linear orthotropic strain energy density function given in eq. (5) can be rewritten in the form (13) where e is a positive definite effective strain measure given by (6) where symmetric components S ij = matrix are given by of the compliance and constant C is related to the shear modulus (7) (14) For nonlinear orthotropic media under plane stress conditions, a new theory based on an assumed form for the strain energy density function has been suggested by Johnson and Urbanik [1984]. It is proposed that a class of materials exist for which W is a nonlinear function of the single variable e found in the linear orthotropic theory Substitution of eq. (9) into eqs. (3) leads to the stress-strain relations (plane stress) for the proposed nonlinear theory (8) (9) Note that in general, the strains cannot be determined from the stresses using eq. (13) since the nonlinear function W'(e) multiplies each term. Predictions of the Nonlinear Theory for the Case of Uniaxial Loading For uniaxial = 0), eqs. (13) become extension in the x 1 -direction (15) (10) Algebraic manipulation of eqs. (15) yields the following relationships: (16) where constants v 12, v 21, C and the functional form of the strain energy density derivative W' (e) are to be determined from experimental data. These equations are for a coordinate system aligned with the directions of material symmetry. It is often convenient to express eqs. (10) in matrix form (11) (17) (18) Equation (17) demonstrates that the nonlinear theory predicts v 12 is a constant Poisson's ratio just as in the linear orthotropic theory. The effective strain for this loading is simplified by substituting eqs. (17,18) into eq. (7) (19) where symmetric components Q ij of the stiffness matrix are given by Combining eqs. (16,19) using a chain rule yields (20) (12) or 3

4 Theoretical Predictions for the case of Pure Shear Loading Equation (21) can be integrated directly to obtain when is represented by an experimentally determined empirical formula. Function W (e) is obtained from this result by substituting (22) In the case of on-axis pure shear loading = 0), eqs. (10) simplify to (30) (31) Note that differentiating eq. (16) with respect to leads to expressions for the tangent modulus The effective strain for this loading substituting eqs. (30) into eq. (7) is evaluated by (32) (23) Combining eqs. (31,32) using a chain rule leads to (33) and the initial (zero strain) elastic modulus (24) Analogous results can be derived for uniaxial extension in the 2-direction. In this case, the relevant equations are (25) This relation can also be integrated to obtain strain energy density function W(e) if is represented by an experimentally determined empirical formula. However, since shear testing is typically far more difficult than uniaxial testing, utilization of either eq. (21) or eq. (25) is recommended for determining the functional form of W(e). By differentiating eq. (31) with respect to a prediction for the instantaneous (tangent) shear modulus is obtained (26) (27) (28) Equation (25) can also be integrated to determine function W (e) if is represented by an experimentally determined empirical formula. Note that eliminating W'(0) from eqs. (24,28) gives the relation (29) This is an extension of the familiar linear elastic formula since E 1 and E 2 are initial moduli for the nonlinear material being modeled. It is interesting to note that new nonlinear constitutive theory predicts constant Poisson's ratios while allowing for nonlinear uniaxial stress-strain curves. Also, no normal-shear coupling is present with uniaxial testing in the directions of material symmetry. Experimental testing has demonstrated that paperboard Poisson's ratios are nearly constant and that the uniaxial vs. response is highly nonlinear. These facts suggest that the proposed nonlinear constitutive theory has a high potential for accurate modeling of paperboard elastic behavior (and inelastic behavior if only loading situations are considered). An expression for the predicted value of the initial shear modulus is determined by setting = 0 in eq. (34) (35) Therefore, it is seen that material constant C in the nonlinear theory can be evaluated by (36) Procedure for Determination of the Input Parameters to the Nonlinear Theory The input parameters to the nonlinear theory are material constants v 12, v 21, C, and the functional form of the strain energy density function W(e). These quantities should be determined using data obtained from experimental testing. As demonstrated above, uniaxial testing in the 1- and 2-directions, and pure shear testing are sufficient for completing the material characterization. In the first step in the procedure, uniaxial extension tests in the 1- and 2-directions are performed. In these experiments, the axial stress, axial strain, and transverse strain must all be monitored. Poisson's ratios v 12 and v 21 are evaluated using linear curve fits to graphs of the transverse strain vs. axial strain. The functional form of strain 4

5 energy density function W(e) can then be found by integrating an empirical representation of the uniaxial 1-direction stress-strain curve using eqs. (21,22) or by integrating an empirical representation of the uniaxial 2-direction stress-strain curve using eqs. (25,27). However, note that the values of the Poisson's ratios must be known before either of these integration procedures can be implemented. In the second step of the procedure, pure shear testing is performed (such as torsion of cylindrical tubes) and the initial shear modulus G 12 is determined from measurements of the initial slope of the shear stress-strain curve. Constant C is then evaluated by substituting the value of G 12 and the zero strain value of the strain energy density derivative into eq. (36). BASIC MATERIAL CHARACTERIZATION OF PAPERBOARD Uniaxial extension and pure shear experiments have been conducted to determine the basic mechanical response of paperboard and to evaluate the input parameters to the proposed nonlinear constitutive relations. The particular paper considered was machine made 100% Lakes State softwood, unbleached Kraft, with basis weight 205 g/m 2 and mass density 670 kg/m 3. Environmental conditions were maintained at 23 C and 50% R.H. according to TAPPI standard T402 OS-70. Poisson's Ratio Data Poisson's ratios v 12 and v 21 were determined from paperboard uniaxial extension experiments where both axial and transverse strains were monitored. Six uniaxial tension tests were performed in both the MD and CD orientations. All tests were run to failure. The experimental procedure involved tensile testing of horizontal specimens which were supported using the lateral vacuum restraint apparatus designed by Gunderson [1983]. Transverse and axial deformations were measured simultaneously with a pair of extensometers. Figures 1 and 2 contain plots of the recorded analogue transverse strain vs. axial strain data for uniaxial extension in the MD and CD, respectively. Figure 2 The shapes of the experimental strain-strain curves obtained in this investigation agree qualitatively with previous Poisson's ratio results for paperboard given by Brecht and Wanka [1963], and Gottsching and Baumgarten [1976]. For MD uniaxial extension, the slopes of these transverse vs. axial strain plots increase slowly and monotonically as the load level rises. In contrast, the strain-strain curves for CD uniaxial extension have slopes which first decrease and then increase. For both cases the variations in slope are slight and the plots are nearly straight lines. Therefore, it is a reasonable approximation to assume that the Poisson's ratios of the paperboard considered are independent of strain. Thirty strain-strain data pairs were digitized from each of the continuous experimental curves in Figures 1 and 2 using a graphics tablet and specially prepared software. Two sets of composite digitized data were then formed by combining all data pairs from the six curves for MD extension (Figure 1), and by combining all data pairs from the six curves for CD extension (Figure 2). For each combined set, a linear regression analysis was performed to find the best straight line fit to the digitized data in the least squares sense. The slopes of these optimum straight line representations are the desired Poisson's ratios. Calculated values are listed below in Figure 3. Figures 4 and 5 contain correlations of the optimum straight line representations to the spread in the strain-strain experimental data. Figure 3 - Experimentally Measured Material Constants for Paperboard Figure 1 As discussed previously, the experimental curves in Figures 1 and 2 deviate slightly from being straight lines. The Poisson's ratios given in Figure 3 are the slopes of the best straight line representations for the experimental data at all levels of strain. These values are suitable for the special nonlinear orthotropic constitutive theory. In contrast, the Poisson's ratios used with the equations of linear orthotropic elasticity should be evaluated for small 5

6 strains. The small strain Poisson's ratios for paperboard were obtained graphically by measuring the slopes of tangent lines drawn to the experimental curves at = 0, and then averaging the results. The calculated values are given in Figure 6. Note that they differ significantly from the values appropriate for all strains. Stress-strain data were recorded in both analogue and digital formats. Initially, analogue curves were recorded using an x-y recorder and input signals from a load cell and an extensometer. Data pairs were then digitized from the continuous experimental curves using a high resolution graphics tablet. Thirty equally spaced (in strain) stress-strain pairs were stored on floppy disk for later use with curve fitting routines. A superposition of the measured MD and CD analogue stress-strain curves is presented in Figure 7. Figure 4 Figure 7 Figure 6 Figure 5 Uniaxial Stress- Strain Testing - Small Strain Poisson's Ratios for Paperboard Uniaxial paperboard stress-strain curves have been measured experimentally for specimens with the MD and CD along the direction of loading. Ten tension and ten compression tests were performed at each of the orientations. All specimens were loaded to failure. Uniaxial tension experiments were performed on an Instron testing machine equipped with a servo-operated x-y recorder. As shown in Figure 8, the die-cut dogbone specimens were clamped vertically in pneumatically activated grips which were free to rotate. Strain was monitored with the extensometer designed by Jewett [1963]. This instrument uses a differential transformer to record deformation automatically and continuously. The gage is applied at the uniformly loaded central part of the specimen and is counterbalanced carefully to eliminate any transmission of gravity force. The loading rate for the tension experiments was = 2.5 x sec -1. This is small enough to preclude any inertial resistance of the gage or pen recorder and fast enough to minimize viscoelastic effects. Uniaxial compression tests were performed with an apparatus (see Figure 9) which prevents buckling using the lateral vacuum restraint technique introduced by Gunderson [1983]. In this method, horizontal specimens are held in place against a set of closely spaced brass rods by a pressure differential. The support elements are free to rotate and do not inhibit deformation in the plane of the sheet. This technique is able to force a true uniaxial compressive failure. Another advantage over previous compression techniques is that the entire top face of the rectangular specimen is available for observation of deformations or attachment of strain measuring devices. In the present program, deformation was monitored with an extensometer. The loading rate was = 2.2 x sec -1 and the pressure differential was 60 kpa. The uniaxial experimental curves in Figure 7 illustrate strongly the the nonlinear behavior of paperboard. Response is elastic into the nonlinear parts of the curves but not all the way to failure. A linear elastic model is clearly inadequate. Comparison of the MD and CD data indicates that the MD has approximately twice the initial stiffness and double 6

7 the strength. Cross machine direction curves are characterized by a much higher breaking strain. The initial elastic moduli are the same for tension and compression in both specimen orientations. However, the compression curves curl to failure at much smaller strains. Compressive strengths are about one half the value of corresponding tensile strengths. Figure 8 - Uniaxial Tensile Testing of Paperboard (38) As noted previously, the paperboard uniaxial data presented in Figure 7 indicate that the tensile and compressive initial elastic moduli are identical. However, the MD and CD compression curves curl over at much smaller strains than the corresponding tension curves. The empirical model presented in eq. (37) is an odd function. Therefore, it cannot model both tensile and compressive data in an optimum manner. After attempting several alternatives, the best compromise fit to a set of tension and compression uniaxial curves was obtained by fitting the hyperbolic tangent model to the tensile data only. Since the compressive curves did not influence the values of the determined coefficients, the model will slightly overestimate the data near compressive failure. Constants C 1, C 2, C 3 were determined for the MD and CD testing orientations by performing nonlinear regression analyses of eq. (37) through digitized points from the tensile experimental curves in Figure 7. The commercially available program NREG77 was utilized on a IBM-PC type computer. This numerical routine calculated the optimum coefficients which gave the best fit of the model to the data in the least squares sense. At each orientation, the digitized data from all ten experimental tensile curves were fit simultaneously in a regression analysis. Another approach would be to calculate sets of coefficients for each of the ten curves in either the MD or CD, and then average the results. This technique was found to work poorly since the coefficients appear nonlinearly in the model. The calculated regression coefficients C i for the MD and CD data, and the values of the initial elastic moduli calculated with eq. (38) are given in Figure 10. Figure 11 shows the correlations of equation (37) incorporating the calculated regression coefficients with the uniaxial experimental curves shown in Figure 7. The shaded regions represent the spread in the experimental data. Correlation is excellent in the tension region and adequate in compression except near compressive failure Figure 9 - Uniaxial Compressive Testing of Paperboard Using Lateral Vacuum Restraint Device The nonlinear experimental data illustrated in Figure 7 have been modeled accurately using the three parameter hyperbolic tangent empirical representation (37) This model was originally suggested by Andersson and Berkyto [1951], and has been used by Urbanik [1982] with C 3 = 0 to model paperboard compressive data. Differentiating eq. (37) leads to an expression for the initial (zero strain) elastic modulus Figure 10 - Hyperbolic Tangent Model Regression Coefficients and Initial Elastic Moduli Determination of the Strain Energy Density Function for Paperboard The MD and CD uniaxial stress-strain curves of paperboard have been modeled by the expressions (39) (40) where calculated values of the regression coefficients 7

8 are listed in Figure 10. An expression for is obtained by substituting eq. (39) into eq. (21) and performing the integration. Changing variables from to e using eq. (22) leads to (41) Another expression for the strain energy density is obtained by performing an analogous procedure with eqs. (42) uniaxial data by balancing the errors between the two predictions. Figure 12 illustrates a superposition of three graphical representations of the strain energy density derivative W'(e). The two extreme versions of function W'(e) were calculated by differentiating eqs. (41,42). Data points on the compromise curve were chosen so that at the level of e considered, the error between the theoretical MD uniaxial prediction and eq. (39) was equal to the error between the theoretical CD uniaxial prediction and eq. (40). A trial and error interactive computer routine was used to find the values of W' (e) where the errors balanced at a set of fifty discrete values of effective strain e. The obtained data points were then formed into a smooth curve using a cubic spline interpolation. In Figure 13, the uniaxial MD and CD predictions of the nonlinear constitutive theory are correlated with the paperboard experimental data from Figure 7. The theoretical curves were generated using eqs. (16,25) and the compromise cubic spline strain energy density function. These theory predictions do not match the data as well as the empirical fits shown in Figure 11. However, the nonlinear constitutive model has the advantage of being able to predict response in biaxial stress states. Figure 11 Figure 12 Either of the functions in eqs. (41,42) could be used with the special nonlinear orthotropic constitutive theory to characterize the mechanical response of paperboard. Since they specify significantly different mechanical behavior, a dilemma occurs when trying to decide which of the two is most appropriate. If the strain energy in eq. (41) is used, the theory degenerates eq. (39) for the case of MD uniaxial extension. That is, the theory matches the MD uniaxial experimental data in an optimum manner. Similarly, the theory will match the CD uniaxial experimental data in an optimum manner if the strain energy in eq. (42) is utilized. A detailed analysis has demonstrated that neither function alone will allow the special constitutive theory to successfully predict both MD and CD uniaxial response. Therefore, a third version of the strain energy density function has been formulated. This "compromise" function allows the theory to correlate satisfactorily with both MD and CD Figure 13 8

9 Shear Testing Experimental shear stress-strain curves for paperboard have been obtained using a modified version of a special torsion apparatus designed and built at the U.S.D.A. Forest Products Laboratory by Setterholm, et al [1963]. Cylindrical specimen assemblies were manufactured and tested according to the procedures outlined in the original reference. The quantities measured continuously during loading were the applied torque and the angle of twist. Shear stress and shear strain were calculated from these quantities using the standard equations for torsion of a cylinder. In the original design of the apparatus, the angle of twist was obtained manually by observing the relative rotation of a pair of long straight rods extending from a pair of plexiglas rings. The rings were attached to the paper tubes by three pointed set screws and were separated by a known gage length. This method was cumbersome and prone to error. In this work, a new technique was used to continuously measure the twist with an extensometer. As shown in Figure 14, the edges of the deformation gage rest on curved aluminum arms attached to the Plexiglas rings. The output of the extensometer has been related to the relative angle of twist between the two rings by using trigonometry and the alternating series convergence theorem [Suhling, 1985]. Using the new technique, continuous torque-twist plots can be obtained on an x-y recorder by using the electrical signals from the extensometer and a load cell attached to a torque arm. The shear stress and shear strain are related linearly to the torque and twist, respectively. Therefore, the shear stress-strain curves can be obtained directly. pure shear failure (fracture) could not be obtained with the device because the tubes buckled at approximately 60% of the ultimate load. The buckles occur perpendicular to the maximum compressive stress (at an angle of 45 degrees from the axis of the cylinder). Figure 15 shows a typical shear stressstrain curve recorded for the test paperboard. Typical Shear Stress-Strain Curve for Paperboard Figure 15 The initial shear modulus for each experiment was obtained graphically by drawing a tangent line to the stress-strain curve. The value obtained by averaging the results from all curves was G 12 = 1686 MPa. Referring to Figure 10, this value is approximately one half the amount of initial elastic modulus E 2 and one quarter the amount of initial elastic modulus E 1. The value of nondimensional constant C in the nonlinear theory was found using eq. (36) and the experimentally determined value of G 12. It is listed in Figure 3 along with the measured Poisson's ratios. Note that eq. (36) also requires as input the initial value of the strain energy density derivative W'(0). This was found to be W'(0) = MPa using the cubic spline interpolation shown in Figure 12. The elastic constants in isotropic Hooke's law are related by the expression Figure 14 - Paperboard Torsion Apparatus with an Extensometer Used to Measure Relative Angle of Twist A group of ten paperboard specimen assemblies was tested with the modified version of the F.P.L. torsion apparatus. The set included five specimens with the MD along the generators of the cylinder and five specimens with the CD along the generators of the cylinder. Since G 12 = G 21, all of the shear stress-strain curves were used to evaluate the initial shear modulus. A (43) It is well-known that no analogous relation exists for the material constants in the linear elastic orthotropic theory. Nevertheless, empirical relations have been proposed by Campbell [1961], Szilard [1974], and Darwin and Pecknold [1977] which express the orthotropic shear modulus G 12 in terms of elastic moduli E 1, E 2 and Poisson's ratio v 12, v 21. All of these empirical formulas degenerate to eq. (43) when E 1 9

10 = E 2 = E and v 12 = v 21 = v. Figure 16 contains the empirical formulas and the values of G 12 which they generate for paperboard. The Poisson's ratios and initial elastic moduli tabulated in Figures 6 and 10 were used as input data. Also noted is the error between the empirical result and the average experimental value of G 12 = 1686 MPa. The smallest error was obtained when using the expression proposed by Campbell [1961]. As noted by Jones [1968], the Campbell formula implies that the value of the shear modulus is independent of the in-plane coordinate system considered. Significant time and effort can be saved if the shear modulus can be predicted empirically. performed at each of the ten orientations. All specimens were loaded to failure. The same Kraft paper and uniaxial testing techniques were utilized as in the previously discussed material characterization experiments Figure 17 - Off- Axis Uniaxial Testing Shear-normal coupling is typically present in the off-axis stress-strain relations for orthotropic media. A nonzero shear strain will then be formed for the case of off-axis uniaxial extension in the x-direction. To accommodate this predicted strain, the experimental apparatus should have grips which are free to rotate in the plane of the specimen. Nonrotating grips could restrict the shear strain of the off-axis uniaxial coupon and thereby impose an unwanted shear stress. For the data discussed here, the grips were free to rotate only in the tension experiments. Therefore, the compression data must be evaluated with some caution. Predictions of the Nonlinear Constitutive Theory * Calculated Using Values in Figures 6 and 10 Figure 16 - Empirical Formulas for the Initial Shear Modulus and Calculated Values for Paperboard APPLICATION OF THE NONLINEAR CONSTITUTIVE THEORY TO OFF-AXIS UNIAXIAL TESTING Uniaxial testing of an off-axis specimen in the x direction is shown in Figure 17. The x and y axes are rotated by a clockwise angle from the in-plane directions of material symmetry denoted by x 1 and x 2. Although more complicated to model analytically than the analogous on-axis case, off-axis uniaxial testing is one of the easiest ways to experimentally generate biaxial loading of a material (biaxial in the coordinate system aligned with the directions of material symmetry). In this section, predictions of the nonlinear constitutive relations are correlated with paperboard off-axis uniaxial experimental data. Since the theory is based entirely on on-axis uniaxial and shear experiments, the accuracy of this correlation represents a true test of its biaxial predictive ability. The general off-axis stress-strain behavior (plane stress) predicted by the new nonlinear constitutive theory is derived by considering an off-axis coordinate system (x,y,z) obtained from the orthotropic material symmetry coordinates (x 1, x 2, x 3 ) through a clockwise rotation of angle about the x 3 = z axis. The stresses and strains in the directions of material symmetry are related to those in the off-axis coordinate system by familiar tensorial transformation laws. When these relations are substituted into eqs. (13), the off-axis relations are obtained (44) where the components of the transformed compliance matrix are calculated by (45) Experimental Data Uniaxial paperboard stress-strain curves have been measured experimentally for specimens with their machine direction at angles of = 0, 10, 20, 30, 40, 50, 60, 70, 80, and 90 from the direction of loading. Ten tension and ten compression tests were 10

11 and effective strain e is evaluated in terms of the off-axis strains by (52) Analytical -Experimental Correlation (46) For the case of uniaxial extension of an off-axis specimen in the x-direction eqs. (44) become Manipulating eqs. (47) leads to (47) (48) where v xy is the off-axis Poisson's ratio and is the mutual influence coefficient of the second kind given by (49) Note that the theory predicts that off-axis uniaxial extension produces a shear strain. A relation between the effective strain and the axial strain can be obtained by substituting eqs. (48) into eq. (46) where (50) (51) The off-axis uniaxial curves of paperboard have been predicted by the nonlinear theory using eq. (52) while incorporating the material constants in Figure 3 and strain energy density function derivative shown in Figure 12. Figure 18 shows the correlations of these predictions with the measured off-axis uniaxial data. Again, the shaded regions represent an outline of all of the experimental stress-strain curves. Agreement is very good considering that no off-axis data is imbedded in the model (it is totally predictive), and that paperboard data shows significant scatter. discrepancies occurred in the on-axis cases The largest = 0, MD and = 90, CD). Data from these two orientations were used to establish the compromise cubic spline strain energy density function. off-axis cases and is the best at Agreement improves for = 50. SUMMARY AND CONCLUSIONS In this investigation, a set of new nonlinear elastic constitutive relations for orthotropic media have been presented and then applied to paperboard. The proposed theory is based upon an assumed form for the strain energy density function incorporating a single effective strain variable. It predicts constant Poisson's ratios but allows for nonlinear uniaxial stress-strain response. Extensive uniaxial and pure shear testing has been performed on paperboard to measure the material constants present in the new theory and to determine the functional form of the strain energy density function. The results of offaxis uniaxial experiments were then utilized to evaluate the predictive capabilities of the constitutive model. It should be noted that there are several built-in restrictions that presently limit the range of problems for which the proposed theory is applicable to paperboard. al 1 experimental data presented in this work were collected at fixed environmental conditions of 23 C and 50% R.H. Since these data are reflected in the material constants and the strain energy density function, the theory cannot be applied to problems in other environments. Also, paperboard shows inelastic behavior at higher strain levels. The special theory assumes elastic response and will not model unloading correctly at loads near failure. However, it should successfully model purely loading situations (even in inelastic regions) if the loading rate is high enough so that rheological effects are negligible. All results obtained thus far suggest that the new theory represents a promising technique for analyzing nonlinear elastic paperboard material behavior. ACKNOWLEDGEMENTS Finally, the uniaxial off-axis stress-strain curve predicted by the nonlinear theory is obtained by substituting eq. (50) into the first of eqs. (47) This research is based upon work supported by the National Science Foundation (Grant No. MEA ), the U.S.D.A. Forest Products Laboratory, Madison, WI, and the Auburn University Pulp and Paper Research and Education Center. 11

12 Figure 18 - Correlation of Off-Axis Uniaxial Stress- Strain Curves for Paperboard Predicted by the Nonlinear Constitutive Theory and Paperboard Experimental Data 12

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