Tensorial Strength Analysis of Paperboard by J.C. Suhling, R.E. Rowlands, M.W. Johnson and D.E. Gunderson ABSTRACT–Tensorial-type failure criteria with linear and Introduction quadratic terms are used to calculate the strength of paper­ board under plane stress. Theoretical predictions and experi­ Reliable predictions of paperboard strength under mental data are correlated in all four quadrants of biaxial complicated stress states is mandatory for design pur­ normal stress with various levels of shear. Several methods poses. Controlled testing of paper under biaxial normal are examined for determining the interaction coefficient F12. and shear stresses is presently difficult for most labora­ Comparisons are made with optimum values obtained from tories. Although uniaxial tests are performed more easily, least-squares analyses. The best analytical-experimental agreement at all levels of shear is obtained approximately by they give no physical indication of material strength under more complex loadings. Mathematical failure criteria using coefficient F12 equal to zero. The sensitivity of F12 to errors in experimental input data is also studied. Reliable capable of predicting accurately the strength of paper­ correlation with experiment, as well as operational simplicity, board are of great technological and utilitarian importance. make these criteria attractive for predicting the strength For maximum operational simplicity, these predictions of paperboard. should require only limited experimental input data. Numerous phenomenological anisotropic failure List of Symbols criteria have been suggested. 1 The Tsai-Wu tensorial theory has been widely used with homogeneous ortho­ tropic materials. Fellers, de Ruvo and coworkers have assessed the usefulness of this criterion for prediction of paper strength under zero shear.2,3 Lack of experimental data prevented correlation of theory with experiment in the compression-compression biaxial quadrant. Extensive paperboard experimental data have been obtained recently throughout all four quadrants of biaxial normal stress with varying levels of applied shear.4 The effectiveness of Hill-type criteria to predict these data has been reported.5 Although very satisfactory at low levels of shear stress, Hill-type theories become far too conservative as the shear level approaches the pure-shear strength of the paperboard. Recognizing the shortcomings of Hill-type theories, tensorial-type criteria are correlated here with that same plane-stress failure data. Optimum methods for deter­ mining the critical interaction coefficient F12 are examined. Relative merits of Hill-type criteria and tensorial theories are also assessed. Tensorial-type strength predictions are defined as those which assume a failure surface in six-dimensional stress space in the form of a tensor polynomial: (1) where (2) J.C. Suhling (SEM Member) is Research Assistant, R.E. Rowlands (SEM are the unique components of the stress tensor. F i, Fij Fellow) is Professor and M. W. Johnson is Professor, University of Wisconsin. Mechanics Department, Madison, WI 53706, D.E. Gunderson Fijk, etc., are components in the x, y, z coordinate sys­ is Research Engineer, Forest Products Laboratory, U.S. Department of tem of strength tensors of second, fourth, sixth and Agriculture Forest Service, Madison, WI 53705. higher orders, respectively. In this work, tensorial criteria Final manuscript received: October 9, 1984. with linear and quadratic terms will be considered. Experimental Mechanics • 75 Additional higher order terms may be retained in eq (1).6,7 Substituting eqs (5) into eq (3), five algebraic expressions However, this tends to be impractical from an operational are obtained. standpoint because the number of coefficients to be experimentally determined increases significantly. Such tensorial-strength predictions were proposed by Malmeister8 (1966) and subsequently by Tsai and Wu9 (1971). Although not written in the same form, the tensorial criterion given by Gol’denblat and Kopnov10 (6) (1965) is operationally equivalent to eq (1). The latter authors seem to be the first to recognize that a strength theory should be independent of a coordinate system. Being invariant, eq (1) is valid in all coordinate sys­ tems (x,y,z ) once it is established in a single coordinate system. For orthotropic materials such as paperboard, These equations may be solved for all the strength co­ the coordinate axes are normally aligned with the efficients except F12, i.e., directions of material symmetry (x = MD, y = CD). After components of the strength tensors are obtained in this specially orthotropic coordinate system, com­ ponents may be established in additional rotated co­ ordinate systems by utilizing tensorial transformation laws. Use of other sets of coordinate axes is usually unnecessary with single-ply or uniformly laminated (7) materials such as paper. The coefficients of eqs (7) satisfy eqs (4a) and (4b). The Plane Stress value of F12 utilized must satisfy eq (4c) to ensure a Under plane stress and when coordinate axes are aligned closed failure surface. Methods for evaluating this inter­ with the axes of material symmetry, eq (1) reduces to action coefficient are discussed below. the following. Importance of Coefficient F12 Since F12 is the coefficient of the product of and (3) in eq (3), it must be evaluated from a test where both of these stresses are nonzero. An infinite number of stress Subscripts 1 and 2 on stresses and now exclusively combinations are available for this purpose. F12 charac­ denote the MD and CD, respectively. Terms with co­ terizes the strength interaction of normal stresses and efficients F16, F26, and F6 are absent in eq (3) because Its value determines the inclination of the failure symmetry conditions on requre these coefficients to ellipses which are scribed in planes of constant shear, as vanish in the specially orthotropic coordinate system. well as the lengths of. the major and minor axes of The failure surface of eq (3) is an ellipsoid provided these ellipses. As shown previously, the other coefficients the following three restrictions are satisfied. in eq (3) are related to the intercepts of the failure ellipsoid with the stress-coordinate axes. Therefore, the (4a) value of F12 typically determines the effectiveness of tensorial-type failure criteria. Many different evaluations (4b) of this coefficient have been proposed.8-l3 Wu suggests a method for determining the optimum biaxial stress (4c) 14 ratio, under which to experimentally evaluate F12. Failure envelopes in planes of constant shear stress will then be ellipses. If eqs (4) are not all satisfied, the quad­ Calculations of F12 ric surface described by eq (3) will not close and will Tsai and Wu9 suggest six simple cases of combined therefore be unbounded. Such cases are physically stress for use in F12 calculation. These are listed in Table 1 unacceptable. along with the appropriate mathematical expressions for Requiring that the tensile and compressive strengths F12. Each of these stress states has X, X', Y, Y' in the machine and cross-machine directions, Cases 1, 2, 5 and 6 of Table 1 are zero-shear biaxial respectively, and the pure-shear strength S be on the failure points in stress-space quadrants I, II, III and IV, failure surface, implies that the following states of stress respectively. In stress space, these points occur on lines satisfy eq (3) which are rotated 45 degrees from the and co­ ordinate axes. Testing in these four cases demands sophisticated and expensive experimental equipment. The tensile and compressive hydrostatic loadings (Cases 1 and 2) require a fixed biaxial ratio of one to be maintained. Tension-compression biaxial testing (Cases 5 and 6) (5) necessitates either a fixed biaxial ratio of negative one or off-axis experiments with nonzero shear stress. In contrast, the uniaxial compressive and tensile strengths of a 45-deg off-axis specimen (Cases 3 and 4) are more easily and inexpensively determined. The choice 76 • March 1985 of 45-deg off-axis coupons has been recommended by Constants F1, F2, F11, F12 and F66 are given by eqs (7). many Soviet scientists. Although testing at this angle is Table 2 contains tensile strengths and F12 values com­ most popular, other specimen orientations have also been puted from eq (9) for off-axis paperboard coupons tested 11 used to determine FI2. at a variety of orientations Figure 1 illustrates an off-axis tensile coupon with Compressive off-axis uniaxial tests may also be used to machine direction at general angle The measured off- determine Fl2. While not as easily performed as uniaxial axis uniaxial tensile strength is designated = Co­ tensile tests, Gunderson’s lateral-restraint technique” ordinates of the combined-stress failure point can be enables compressive testing of paper to be accomplished. calculated from the stress-transformation equations Referring to Fig. 1, failure occurs when = (Mohr’s circle): > 0. Corresponding equations for stresses and coefficient F12 are obtained by replacing with (- in eqs (8) and (9). Table 3 lists compressive strengths and appropriate F12 values for paperboard off- (8) axis specimens tested at various orientations. Other calculations of F12 suggested by Gol’denblat- Kopnov,10 Hoffman,12 and Cowin13 are listed in Table 4. Malmeister’s8 recommendation is given in Table 1. The The value of F12. is computed for this condition by sub­ Malmeister and Gol’denblat-Kopnov criteria were stituting eqs (8) into eq (3) to obtain developed to predict the failure of fiber-reinforced com­ posite materials. Although not widely used in the western world, variations of these theories have been applied extensively in the Soviet Union. Hoffman added linear terms to Hill’s orthotropic yield criterion to account for different strengths in tension and compression. His theory is not based on a tensorial formulation, but it has the (9) same form as eq (3). Cowin chose F12 so that a Hankin­ son16-type formula could be derived from the Tsai-Wu theory. His choice of the interaction coefficient guarantees that eq (4c) will be satisfied.