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. Cowin’s analysis was applied to bone. Neither the Cowin nor Hoffman predictions require biaxial data.

Experimental Data Biaxial strength predictions are correlated with experi­ mental results throughout the four quadrants at = 0, 6.9 MPa, 10.3 MPa, and 15.9 MPa. Experimental data were obtained from on- and off-axis uniaxial coupons, cylinders, and cruciform specimens.4 A complete listing of this data has been reported previously.5 The single material tested was machine-made 100-percent Lake State softwood, unbleached Kraft paper, with a basis weight of 205 g/m2 and a mass density of 670 kg/m3. Experiments were conducted at 70°F, 50-percent R.H. Fig. 1– Off-axis uniaxial tensile test. according to TAPPI standard T-402 OS-70. Measured Specimen fails at = input strength and elastic quantities are listed in Table 5.

Fig. 2-Theoreticalstrength envelopesof paperboard with different values of coefficient F12 Strength coefficients F1, F2, F11, F12 and F66 were cal­ many of these coefficient calculations are unsuitable for culated using eqs (7); they are listed in Table 6. Several paperboard. values of F12 are listed in Tables 1-4. A superposition of several zero-shear theoretical- failure envelopes for the material tested is contained in Fig. 2. These curves all utilize the coefficients of eqs (7) but each uses a different value of coefficient F12. Experi­ Analytical-Experimental Correlations mental data are also included in Fig. 2. Each ellipse inter­ sects the stress axes at the same four points. These inter­ Numerical Limitations on F12 cepts are determined by the coefficients of eqs (7),

As mentioned previously, coefficient F12 is the critical independently of the magnitude of F12. Large variations factor in characterizing the plane-stress failure surface of in the predicted strength envelopes are obvious in quad­ eq (3). Twenty-seven unique determinations for this rants I, II and IV. Quadrant I predictions are particularly constant are presented in Tables 1-4. The computed sensitive. On the other hand, all predictions are relatively results of these formulas span several orders of magni­ similar and slightly conservative in quadrant III. This tude on each side of zero. Since the strength prediction is suggests that combined-stress tests from quadrant III are especially sensitive to the value of F12, it is apparent that inappropriate for determining F12. Slight errors in

9 TABLE 1-DETERMINATIONSOF F12 USING SIMPLE COMBINED STRESS STATES SUGGESTED BY TSAl AND WU

'Computed using strengths listed in Table 5

78 • March 1985 TABLE 2-OFF-AXISUNIAXIAL TENSILE STRENGTHS AND GENERATED F12 VALUES

*Computed using eqs (7) and (9) and coefficients listed in Table 6

TABLE 3-OFF-AXISUNIAXIAL COMPRESSIVE STRENGTHS AND GENERATED F12 VALUES

*Computed using eqs (7) and (9) and coefficients listed in Table 6

TABLE 4-OTHERCALCULATIONS OF COEFFICIENT F12

*Computed using strengths listed in Table 5 and coefficients listed in Table 6

Experimental Mechanics • 79 from observing Fig. 2, the use of the biaxial compression

test (Table 1, Case 2) to compute F12 did not produce an acceptable failure surface. The unsuitability of either off- axis tensile or biaxial compressive tests for determining 5 F12 of Hill-type criteria has been noted previously.

Optimum Values of F12

The optimum value of F12 can be considered as that which generates the best prediction of experimental data in the least-square sense. We determined this coefficient numerically using a linear single-regression analysis routine. While fitting F12, the other coefficients of eq (3) Fig. 3 – Zero-shear analytical failure are chosen in the normal manner using eqs (7). envelope of paperboard with F12 evaluated Two regression analyses were completed for paper­ from tension-tension biaxial test; F12 = -2.373 x 10-4 MPa-2 board. The first calculated F12 that best fits the data for = 0. The second computed F12 that best fits the entire set of experimental data from all four levels of shear.

Table 7 lists the computed optimum values of F12.

measuring experimental data from this quadrant could Best Fit for = 0 Data yield a value of F12 which produces large discrepancies in the theory-experiment correlations in the other The optimum value of F12 which best fits the = 0 -4 -2 quadrants. data was found to be F12 = -2.297 × 10 MPa . This is almost identical to the value determined from a tension- Quantitative limits on F12 may be expressed by using the stability condition of eq (4c). To ensure a closed tension biaxial test (Table 1, Case l), F12 = -2.373 × strength surface, eq (4c) requires 10-4 MPa-2. The failure envelopes for these two choices of F12 graph nearly identically. Fellers, Westerbind and (10) de Ruvo recommend using tension-tension biaxial tests 3 for calculating F12 . Their results demonstrate that the Based on Table 6, this becomes Tsai-Wu theory could adequately predict the zero-shear strength envelope in quadrants I, II, and IV. At that time, no experimental data were available in the com­ pression-compression region. (11) Figure 2 exhibits the effects on the accuracy of the pre­

dictions as F12 nears either of these limits. As the upper bound is approached, the failure surface becomes too liberal in tension-compression regions while becoming TABLE 5-INPUT (MEASURED)STRENGTHS AND ELASTIC far too conservative in quadrant I. When near the lower PROPERTIES OF PAPERBOARD'." bound, the predictions well exceed observation in the first quadrant.

Effects of F12 on strength envelopes at other levels of shear have also been examined. 17 Similar tendencies of deteriorating correlation with experiment were noticed

when F12 approached its stability bounds. Upon evaluation of graphs at all four levels of shear, the best analytical- experimental agreement occurs when F12 is confined to the interval

(12)

These bounds include F12 values calculated by the following methods: (1) tension-tension biaxial test (Table 1, Case 1); (2) compression-tension biaxial test (Table 1, Case 5); (3) off-axis uniaxial tensile tests at = 20 deg, 40 deg, 45 deg (Table 2; Table 1, Case 3); (4) off-axis uniaxial compressive tests at = 50 deg, 60 deg (Table 3); (5) Gol’denblat-Kopnov theory (Table 4) and (6) Cowin theory (Table 4). It is observed that off-axis uniaxial tests (compressive and tensile) of paperboard generally provide ineffective TABLE 6-STRENGTH COEFFICIENTS

determinations of coefficient F12. Of the 20 specimens considered in Tables 2 and 3, only five generated adequate predictions for the experimental data. This is unfortunate because of the relatively inexpensive nature of uniaxial testing. Pipes and Cole found similar results for uniaxial tensile tests of boron-epoxy composites. 11 As expected

80 • March 1985 Fig. 4 – Tensorial failure envelopes of paperboard with F12 = 0

Equation (3) is correlated in Fig. 3 with zero-shear 15.9 MPa in Fig. 4. The zero-shear failure prediction in paperboard experimental data throughout the four Fig. 4(a) (F 12 = 0) is slightly less accurate than that of quadrants. F12 was computed from tension-tension biaxial Fig. 3 where F12 was calculated from the tension-tension failure data (Table 1, Case 1). Agreement is good in all biaxial failure point. However, this curve is a very quadrants. The theory is slightly conservative in quadrant acceptable prediction of the data. Analytical-experimental III, where experimental data were previously lacking. At correlation at nonzero shear levels is superior when the nonzero levels of shear, this value of F12 gives strength usingF12 = 0. predictions that are too liberal in quadrant I.17 Relative Merits of Hill-Type and Best Fit for All Data Tensorial-Type Criteria Based on the paperboard data at all four levels of The ability of Hill-type criteria to predict the strength 5 shear, the optimum value of F12 was found to be F12 = of paperboard has been reported. Under plane stress, the +2.362 × 10-5 MPa-f. The failure envelopes in this case tensorial theory differs from the Hill prediction by the are virtually horizontal, with a rotation angle from the inclusion of linear terms + in eq (3). These axis of less than one degree.17 The predicted constant- extra terms allow the analytical prediction to accommodate shear ellipses are exactly horizontal for F12, chosen to be different uniaxial strengths in tension and compression identically zero. When plotted simultaneously, strength with a single ellipsoidal strength surface. In contrast, envelopes with the optimum F12 (Table 7, Case 2) and Hill-type theories (as defined in Ref. 5) form a conglo­ those with F12 = 0 are visually indistinguishable. Optimum merate of four quarter ellipsoids joined together at the fit to the experimental data at all shear levels is therefore quadrant boundaries. obtained approximately by taking F12 = 0. This choice of The added linear terms contribute additional effects.

F 12 greatly simplifies analysis in that no biaxial input data Equation (3) scribes ellipses in planes of constant shear. are needed. Other authors have recommended F12 = 0 for Each of these ellipses shares a common center having fiber-reinforced composite materials.18 coordinates given by' The analytical criterion with F12 = 0 is compared with experiment at = 0, 6.9 MPa, 10.3 MPa, and (13)

TABLE 7 –OPTIMUM VALUES OF COEFFICIENT F12

These coordinates are generally nonzero. Therefore, the analytical-strength envelopes will be offset from the – plane origin. In Hill-type criteria, the linear terms are absent (F 1 = F2 = 0); eq (13) becomes = = 0. The four ellipses that mold together to make the Hill-

Experimental Mechanics • 81 Fig. 5 – Theoretical failure envelopes shrink towards their common center as the level of shear is increased.

Fig. 6 – The effect of variations in experimental combined-stress input data on the value of

F12 for paperboard

type strength envelope therefore all have centers at the Table 1, Case 1) and a Hill-type criterion (Norris theory5). – plane origin. The strength envelopes converge to the single center point

As increases, the theoretical-failure envelopes shrink at = Sc, the shear level which bounds the top of the toward their common center at Figure 5 exhibits closed failure surface. For the tensorial prediction of 17 this phenomenon for the tensorial prediction (F12 from eq (3), this stress is

82 • March 1985 The Hoffman and Malmeister theories use values (14) (Tables 1 and 4) of F12 beyond the limits of eq (12). These criteria overpredict significantly the actual strength in quadrant 1, the correlation deteriorating with increasing 712. For = 0, each of these predictions would occur outside of curve B in Fig. 2. The Cowin and Gol'denblat­ Any state of stress where > Sc will theoretically fail the material. Kopnov theories also tend to overpredict observation, but Equation (14) demonstrates that tensorial criteria can to a lesser extent.17 predict unfailed stress states having > S. Experimental data in this category have been observed for paperboard.5 Predicting Off-Axis Strengths For Hill-type criteria (Fl = F2 = 0), eq (14) simplifies to The unstable characteristics of determining F12 from Sc = S so that these theories do not admit unfailed stress states where > S. Figure 5(a) exhibits plotted off-axis uniaxial tests have been discussed. One might ask analytical-failure envelopes with >S. if eq (3) could then be used to predict accurate off-axis Hill-type criteria do offer some advantages over the uniaxial tensile and compressive strengths. Rearranging prediotions of eq (3). In addition to typically requiring less eq (3) leads to the following mathematical prediction for experimental input data, they predict the zero-shear the off-axis uniaxial failure stresses. failure envelope of paperboard in quadrant III more accurately. (15) Sensitivity of F12 to Experimental Input Data Tsai and Wu recommend studying the effects of varia- where tions in the experimental combined-stress input data on 9 F12 . For graphite-epoxy composites, they found the degree of sensitivity to experimental error varied between (16) the six tests considered (listed in Table 1). In Fig. 6 the six combined stress states of Table 1 are plotted against the values of F12 which they generate for paperboard. The Figure 7 contains plots of the strength predictions of F12 stability bounds and the limits of eq (12) are shown on eqs (15) and the experimental data of Tables 2 and 3. this graph. Curves for P' and U' are nearly horizontal Based on prior arguments, the value of F12 = 0 was which means these tests are unreliable for determining utilized. Good correlation is realized even though the F12. A small inaccuracy in the experimentally measured off-axis test has been shown to be an unreliable deter- value of U', for example, can induce a large variation in mination of F12. This demonstrates the high sensitivity of F12. The curve for P (tension-tension biaxial test) gives eq (9) to small errors in the experimentally measured the most stable determination of F12. It is no coincidence off-axis strengths. 16 that the F12 value from this test gives an acceptable The Hankinson formula was also used to predict the prediction to the experimental data. off-axis compressive strengths of paperboard [Fig. 7(b)].

Fig. 7 – Off-axis uniaxial strength predictions for paperboard, (a) tensile, (b) compressive

Experimental Mechanics • 83 This prediction gives the uniaxial compressive-failure these theories overpredict observed strengths in quad­ stress at angle by the empirical formula rant I. As with the Hill-type theories, sizeable variations in

(17) F12 have relatively little effect on strength predictions in quadrant III.

Compared to eq (15), the Hankinson computation requires Acknowledgment only limited input test data. In particular, it does not involve shear strength, and compressive strengths do not The research is based upon work supported jointly by involve tensile data. This formula has been successfully the National Science Foundation (Grant No. MEA­ used for wood and bone but applications to other 8120393; Dr. C.J. Astill, program director) and the materials are unknown to the authors. The agreement USDA Forest Products Laboratory. Mrs. M.M. Lynch with paperboard data appears to be quite adequate. capably typed the paper.

References Summary, Discussion and Conclusions 1. Rowlands. R.E., “‘Strength (Failure) Theories and Their Experi­ The effectiveness of tensorial failure theories to predict mental Correlation,’’ Handbook of Composite Materials, III, ed. G.C. the strength of paperboard under plane-stress loading is Sih and A.M. Skudra, Elsevier Scientific Publishers (1985). 2. de Ruvo, A., Carlsson, L. and Fellers, C., “The Biaxial Strength evaluated. Several methods for determining interaction of Paper, TAPPI, 63 (5), 133-136 (1980). coefficient F12 are considered. Predictions are compared 3. Fellers, C., Westerbind, B. and de Ruvo. A., “An Investigation throughout all four quadrants to experimental data ob­ of the Biaxial Failure Envelope of Paper,” to appear in Proc, 1981 tained from on- and off-axis uniaxial coupons, cylinders, Cambridge Symp. (in press). and cruciform specimens. Data at four levels of applied 4. Gunderson, D.E. and Rowlands, RE., “Determining Paperboard Strength-Biaxial Tension, Compression, and Shear,’’ Proc. Int. Paper shear had been collected. Physics Conf., TAPPI Press, 253-263 (1983). The optimum analytical fit to the zero-shear paper­ 5. Rowlands, R.E., Gunderson, D.E.. Suhling, J.C. and Johnson, M. W., “Biaxial Strength of Paperboard Predicted by Hill-Type Theories,” board data was obtained approximately when F12 was to appear in J. Strain Anal. determined from a tension-tension biaxial test. F1 deter­ 2 6. Tennyson, R.C., MacDonald, D. and Nanyaro, A.P., “Evaluation mined in this manner was relatively insensitive to small of the Tensor Polynomial Failure Criterion for Composite Materials,” J. errors in the biaxial input data. Unfortunately, this Comp. Mat.. 12 (I). 63-75 (1978). experiment is not yet easily performed on paper. 7. Tennyson. R.C., Nanyaro, A.P. and Wharram, G.E., “Applica­ tion of the Cubic Polynomial Strength Criterion to the Failure Analysis of The optimum analytical fit to experimental data from Composite Materials,” J. Comp. Mat. Supplement, 14, 28-41 (1980). all levels of shear was approximately obtained by taking 8. Malmeister, A.K.. “Geometry of Theories of Strength,” Mekhanika F12 = 0. In this case, analytical-experimental correlation Polimerov, 2 (4). 519-534 (1966). [English translation: Polymer Mechanics, was very good although the theory tended to be slightly 2 (4), 324-331, Faraday Press.] conservative in quadrant I. A great advantage of this 9. Tsai, S. W. and Wu, E.M., “A General Theory of Strength for Anisotropic Materials.” J. Comp. Mat., 5 (1). 58-80 (1971). choice of F12 is that no combined stress testing of any 10. Gol’denblat, I.I. and Kopnov, V.A., “Strength of Glass-Reinforced kind must be done to utilize the theory. Plastics in the Complex Stress State,” Mekhanika Polimerov, 1 (2), 70-78 (1965). [English translation: Polymer Mechanics, 1 (2), 54-59, Faraday Several determinations of F12 proved unsatisfactory. Off-axis uniaxial testing was an especially unstable method. Press.] 11. Pipes, R.B. and Cole, B. W., “On the Off-Axis Strength Test for The coefficient values computed from these experiments Anisotropic Materials.” J. amp. Mal.. 7 (2), 246-256 (1973). could not consistently generate adequate theory pre­ 12. Hoffman, O., “The Brittle Strength of Orthotropic Materials,” Ibid.. 1 (2), 200-206 (1967). dictions. However, the choice F12 = 0 provided an effective analytical prediction for the experimental off- 13. Cowin, S.C., “On the Strength Anisotropy of Bone and Wood,” J. Appl. Mech.. 46 (4), 832-838 (1979). axis uniaxial failure stresses. 14. Wu, E.M., “Optimal Experimental Measurements of Anisotropic Several advantages of the tensorial-type criteria over Failure Tensors,” J. Comp. Mat., 6 (4), 472-489 (1972). Hill-type criteria are noted here. The tensorial theory 15. Gunderson, D.E., “Edgewise Compression Testing of Paperboard- A New Concept of Lateral Support,” APPITA, 37 (2), 137-141 (1983). uses only a single ellipsoid to construct the failure sur­ 16. Hankinson, R.L., “Investigation of Crushing Strength of Spruce at face. Also, it better predicts paperboard strength as the Varying Angles of Grain,” U.S. Air Service Information Circular, level of shear approaches the pure shear strength S. 3 (259), (1921). Unlike Hill-type criteria, the tensorial prediction allows 17. Suhling, J.C., “Failure Predictions for Paper under Plane Stress,” unfailed stress states for > S. Internal Rep., Mech. Dept., Univ. of Wisconsin, Madison (1982). 18. Narayanaswami, R. and Adelman, H.M., “Evaluation of the The criteria of Hoffman and Cowin evaluate F12 Tensor Polynomial and Hoffman Strength Theories for Composite explicitly without the need of biaxial input data. Both of Materials,” J. Comp. Mat., 11 (4), 366-377 (1977).

84 • March 1985