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EFFECTS OF MOISTURE ON THE BIAXIAL STRENGTH OF -BASEDCOMPOSITES

J. C. Suhling* J.M. Considine ** K. C. Yeh*

* Department of Mechanical Engineering Auburn University Auburn, AL36849-5341 ** U.S.D.A. Forest Products Laboratory One Gifford Pinchot Drive Madison,WI 53705-2398

ABSTRACT applications such as corrugated where it is subjected to complicated biaxial stress states, In the present work, the effects of moisture on including shear. At present, lack of accurate the biaxial strength of wood-based composite materials constitutive relations and reliable strength has been examined. Experiments have been performed on predictions under biaxial loading and variable paperboard to measure the uniaxial and biaxial input environments hampers analysis of such problems. parameters of the tensor polynomial criterion at Therefore, it has been common practice in the several levels of relative humidity. With these data, industry to use trial and error, and empirical the dependence of the zero shear biaxial failure approaches for optimizing the designs of paperboard on humidity level has been predicted and products. The current lack of technology limits trends have been observed. creative design improvements which could curtail the excess use of materials and energy. Accurate predictions of material strength under 1. INTRODUCTION general biaxial normal loading plus shear are needed by the design engineer when considering typical Unlike laminated fiber-reinforced composite structural applications of paperboard. Examples of materials, paper (paperboard) is a multiphase important current applications where biaxial stress composite composed of moisture, fibers, voids, and states exist in paperboard include the quality control possibly chemical additives. The fibers in paperboard burst test (bulged plate), material handling are typically organic with fibers from wood operations during the process, and stacked being the chief material. Since cellulose fibers are corrugated containers. Controlled testing of paper self-binding, no matrix material is needed. Wood over a wide range of combined stress states is itself may be thought of as a natural composite difficult and expensive. Also, few industry consisting of cellulose fibers interconnected by a laboratories have the necessary experimental primarily binder. Paper is by far the oldest equipment. Therefore, simple mathematical failure and most economically significant wood fiber criteria which yield accurate results are of great composite. importance. The three-dimensional physical structure of Numerous biaxial strength criteria have been paperboard is basically an assembly of discrete fibers proposed for orthotropic materials [1], and several of bonded together into a complex network. If the fibers these have been applied to paperboard. Prud'homme and were oriented randomly throughout the sheet, the Robertson [2] applied both the Tsai-Hill and Liu- material behavior would be isotropic. The actual Stowell (maximum stress) criteria to highly oriented nonrandom (anisotropic) arrangement of the fiber paper sheets and found the former to agree with network is caused by hydrodynamic forces which occur experiment. The results of several recent in the papermaking machine. A single preferred investigations have demonstrated that the tensor orientation of the fibers results which is called the polynomial failure criterion (often called the Tsai-Wu machine direction (MD). The in-plane direction Theory) can adequately predict plane stress biaxial perpendicular to the preferred direction is called the failure states of paperboard [3-7]. This theory cross-machine direction (CD). These directions of requires six experimentally measured input parameters material symmetry allow for paper to be modeled macroscopically as an orthotropic solid. for general utilization. Typically, these are the Paper materials tend to exhibit nonlinear tensile and compressive strengths in the machine and mechanical behavior which is affected significantly by cross-machine directions, the pure shear strength, and slight changes in the surrounding environmental a single piece of biaxial failure data. conditions. In addition, most are capable of All previous research on paperboard biaxial demonstrating every known rheological phenomenon. failure has been carried out in environmental rooms Current uses of paperboard provide several challenging which strive to maintain TAPPI standard conditions of problems in engineering mechanics. For example, temperature and relative humidity (73° F, 50% RH). It paperboard is often utilized in structural is well-known that the most critical quantity

602 influencing the characteristic mechanical response of a specific paper is its current and past history of moisture content. The moisture content or regain of (2) an element of material is defined as the ratio of the difference between current weight and dry weight to For a particular sample of paper, the are the unique components of the stress tensor. the dry weight. Fi, moisture content is determined predominately by the F , F , etc. are components in the (x,y,z) chemical and structural characteristics of its ij ijk component fibers and the relative humidity (RH) of the coordinate system of strength tensors of second, surrounding atmosphere. It is also affected to a fourth, sixth, and higher orders, respectively. In lesser degree by temperature, state of stress, and this work, tensorial criteria with only linear and past histories of stress, temperature, and humidity. quadratic terms will be considered. Additional higher Several experimental studies have been made on order terms may be retained in eq. (1). However, this the effects of relative humidity on paper behavior in tends to be impractical from an operational standpoint uniaxial stress states. Kolseth and de Ruvo [8] have because the number of coefficients to be surveyed several experimental investigations in this experimentally determined increases significantly. area. Moisture contents of test specimens have been Such tensorial strength predictions were first adjusted primarily by controlling the relative proposed by Malmeister [18] and subsequently by Tsai humidity of the surrounding atmosphere while and Wu [19]. Although not written in the same , maintaining a constant temperature. In some studies the tensorial criterion given by Gol'denblat and the actual dependence of a material property on Kopnov [20] is operationally equivalent to eq. (1). moisture content was calculated by performing The latter authors seem to be the first to recognize additional experiments to determine the characteristic that a strength theory should be independent of sorption curve relating relative humidity to coordinate system. paperboard moisture content. Being invariant, eq. (1) is valid in all Extensive measurements of uniaxial tensile coordinate systems (x,y,z) once it is established in a stress-strain curves of paper at several humidity single coordinate system. For orthotropic materials levels have been performed by Andersson and Berkyto such as paperboard, the coordinate axes are normally [9], and Benson [10]. The latter author also related aligned with the directions of material symmetry (× = tensile properties directly to moisture content. The MD, y = CD). After components of the strength tensors effects of the level of relative humidity on the are obtained in this specially orthotropic coordinate initial elastic modulus of paper have been examined by system, components may be established in additional de Ruvo, et a1 [11], and Salmen and Back [12]. rotated coordinate systems by utilizing tensorial Okushima and Robertson [13] have noted the dependence transformation laws. Use of other sets of coordinate of the tensile stress-strain curve on strain rate at axes is usually unnecessary with single-ply or several different levels of humidity. Other uniformly laminated materials such as paper. researchers have reported on the influence of relative Under plane-stress conditions where the humidity on paperboard creep behavior [14], stress coordinate axes are aligned with the axes of material relaxation [15], and impact properties [16]. Humidity symmetry, eq. (1) reduces to and strain rate effects on the uniaxial compressive stress-strain curves of paperboard have been measured recently at the U.S.D.A. Forest Products Laboratory by Gunderson, Considine, and Scott [17]. (3) The literature shows no experimental data or analytical results which relate the moisture content of a paperboard sheet to its mechanical response under biaxial loading. In this work the effects of the where subscripts 1 and 2 on stresses level of moisture content on the zero-shear biaxial exclusively denote the MD and CD, respectively failure envelope of paperboard have been examined. (directions of material symmetry). Terms with The tensor polynomial criterion was used for coefficients F16, F26, F6, are absent in eq. (3) predicting paperboard biaxial strength. At zero because symmetry conditions require these coefficients shear, the input parameters to this theory are the to vanish in the specially orthotropic coordinate uniaxial tensile and compressive strengths in the MD system. The failure surface described by eq. (3) is and CD, respectively, and a single piece of biaxial an ellipsoid provided the following three restrictions failure data. Experiments have been performed to are satisfied measure these parameters at several levels of relative humidity, With these data, the dependence of the total biaxial failure envelope on moisture content has been predicted and trends have been observed. (4)

2. TENSOR POLYNOMIAL FAILURE CRITERIA Tensorial polynomial failure criteria propose that the failure surface in six-dimensional stress Failure in planes of constant shear stress space assumes the form will then be ellipses. If eqs. (4) are not all satisfied, the quadric surface described by eq. (3) (l) will not close and will therefore be unbounded. Such cases are physically unacceptable. where Most of the coefficients in eq. (3) can be found by requiring that the tensile and compressive

603 strengths X, X', Y, Y' in the machine and cross- interaction of the two normal stresses. Its value machine directions, respectively, and the pure shear determines the inclination of the failure ellipses strength S be on the failure surface. This implies which are scribed in planes of constant shear, as well that the following states of stress satisfy eq. (3) as the lengths of the major and minor axes of these ellipses. As demonstrated by eqs. (5), the other coefficients in eq. (3) are related to the intercepts of the failure ellipsoid with the stress coordinate axes. Therefore, the value of F12 typically determines the effectiveness of tensorial-type failure (5) criteria. Many different methods for evaluation of this coefficient have been proposed [1,7]. Wu has suggested a method for determining the optimum biaxial stress ratio under which to experimentally evaluate F12 [21]. Five algebraic expressions are obtained upon The effectiveness of the tensor polynomial substitution of the states of stress given in eqs. (5) criterion to predict the strength of paperboard under into eq. (3) biaxial plane stress loading has been evaluated by several investigators [3-7]. Suhling, et a1 [7] considered several methods for determining interaction coefficient F12 using an extensive set of biaxial experimental data as a guide. They found that the optimum analytical fit to zero-shear paperboard data was obtained when F12 was determined from a tension- (6) tension biaxial test When determined in this manner, the coefficient was relatively insensitive to small errors in the biaxial input data (measured value of

3. UNIAXIAL EXPERIMENTAL DATA

These equations may be solved for all the strength Uniaxial tensile and compressive strengths in the coefficients except F , i.e., MD and CD (X, X’, Y, Y’) have been measured for 100% 12 Lakes State unbleached (basis weight 205 g/m2, mass density 670 kg/m3) at humidity levels of RH = 50%, 70%, and 90%. Tests were performed using a specially constructed apparatus developed at the Forest Products Laboratory [22]. As (7) shown in Figures 1 and 2, this device consists of a load frame and vacuum restraint system enclosed in an environmentally controlled chamber. The utilized vacuum restraint procedure was developed by Gunderson [23]. It prevents buckling under compressive loading by using the lateral support The coefficients in eqs. (7) satisfy the first provided by a flat surface formed by the ends of two of eqs. (4). The value of F chosen must satisfy slender vertical rods. A light vacuum forces the 12 specimen into contact with the support rods and draws the third of eqs. (4) to ensure a closed failure air through and around the edges of the specimen. The surface. Since F12 is the coefficient of the product design allows the rods to move freely as the specimen in eq. (3), it must be evaluated using an deforms. Environmental conditions were controlled in the experiment where both of these stresses are non-zero. chamber using a microcomputer-based feedback control An infinite number of such stress combinations are system incorporating input from a humidity transducer available for this purpose. For a single and output to a flow control valve which mixes humid experimentally measured biaxial failure data point and dry air. The computer is also used to control the the value of F is obtained using eq. load application and record the stress-strain data. 12 The load frame is servo-driven so that it achieves (3) computer-specified load values independent of specimen strain magnitudes. A close-up photograph of the load frame is contained in Figure 3. All specimens were preconditioned at low humidity for several days prior to testing. A typical test was begun by placing the specimen on the restraining rods, applying the vacuum, (8) engaging the load frame, and setting the bell to enclose the system. Air at the specified RH was drawn through the specimen for 30 minutes before beginning the test. Figure 4 contains a tabulation of the Coefficient F12 characterizes the strength experimentally recorded uniaxial compressive and

604 tensile strength data in the machine and cross-machine and biaxial failure parameters at three levels of directions at the three humidities considered. Each relative humidity. Obtained results follow expected data point represents the average of six uniaxial trends and additional testing is underway to guide tests. A total of 72 experiments were performed. A future modeling efforts. group of typical MD and CD tensile and compressive curves at RH = 50% is shown in Figure 5. ACKNOWLEDGEMENTS

4. BIAXIAL DATA FROM CYLINDRICAL TESTS This research is based upon work supported by the U.S.D.A. Forest Products Laboratory, Madison, WI, and Biaxial tension failure data were measured for the Auburn University and Paper Research and the same kraft paperboard by using pressurized Education Center. cylindrical tube burst tests. As shown in Figure 6, paper tubes were formed from flat sheets using a thin glue seam and the ends were reinforced using REFERENCES additional material. The tubes were then firmly clamped to aluminum end caps and a thin rubber bladder was used to transmit the internal pressure loading. 1. Rowlands, R. E., "Strength (Failure) Theories and axial expansion of the tube was allowed using roller their Experimental Correlation," in Handbook of bearing supports. Figure 7 shows the complete Composite Materials. Vol. III, edited by G. C. configuration utilized for biaxial cylindrical tube Sih and A. M. Skudra, Elsevier Scientific testing. Publishers, 1985. In this study, the tubes were constructed with 2. Prud'homme, R. E. and Robertson, A. A., the CD along the cylinder's axial direction and the MD "Composite Theories Applied to Oriented Paper along the hoop direction. Using the approximate Sheets," TAPPI, Vol. 59(1), pp. 145-148, 1976. strength of materials relations for thin cylindrical 3. de Ruvo, A., Carlsson, L. and Fellers, C., "The pressure vessels leads to an expression for the Biaxial Strength of Paper," TAPPI, Vol. 59(1), measured biaxial failure point pp. 133-136, 1980. 4. Fellers, C., Westerlind, B. and de Ruvo, A., "An Investigation of the Biaxial Failure Envelope of Paper - Experimental Study and Theoretical (9) Analysis," in The Role of Fundamental Research in Papermaking, pp. 527-560, 1983. where pf is the measure failure pressure, and R, t are 5. Suhling, J. C., "Constitutive Relations and the radius and thickness of the cylinder, Failure Predictions for Nonlinear Orthotropic respectively. Media," Ph.D. Thesis, University of Wisconsin, Five cylindrical tube burst tests were performed 1985. at humidities RH = 50%, 70%, and 90%. Average values 6. Suhling, J. C., Rowlands, R. E., Johnson, M. W. of the biaxial failure point are tabulated in Figure and Gunderson, D. E., "Evaluation of Strength 8. A typical failed specimen is shown in Figure 9. Predictions for Paperboard," in the Proceedings of the 1985 SEM Spring Conference on Experimental Mechanics, pp. 134-141, Las Vegas, NV, June 9-14, FAILURE ENVELOPE PREDICTIONS 1985. 7. Suhling, J. C., Rowlands, R. E., Johnson, M. W. and Gunderson, D. E., "Tensorial Strength The zero shear failure envelopes of paperboard at Analysis of Paperboard," Experimental Mechanics, RH = 50%, 70%, and 90% were predicted using the tensor Vol. 25(1), pp. 75-84, 1985. polynomial criterion incorporating the failure data 8. Kolseth, P. and de Ruvo, A., "The Measurement of tabulated in Figures 4 and 8. Figures 10-12 show the Viscoelastic Behavior for Characterization of the calculated envelopes obtained using eq. (3) with Time-, and Temperature-, and Humidity-Dependent Coefficients found using eqs. (7-9). The open circles Properties," in Handbook of Physical and on these plots indicate the data used to position the Mechanical Testing of Paper and Paperboard, elliptical envelopes. A composite of all results is Edited by R. E. Mark, Marcel Dekker, pp. 255-322, found in Figure 13. The obtained curves show the 1983. expected trend that the failure envelopes will reduce 9. Andersson, O. and Berkyto, E., "Some Factors in size without overlapping as the humidity is Affecting the Stress-Strain Characteristics of increased. The biaxial tension quadrant is most Paper," Svensk Papperstidning, Vol. 54(13), pp. affected. Current efforts are directed towards 437-444, 1951. expanding the experimental data base to include 10. Benson, R. E., "Effects of Relative Humidity and results at RH = 20%, 30%, 40%, 50%, 60%, and 80% so Temperature on Tensile Stress-Strain Properties that well-based models can be formulated which of Kraft Linerboard," TAPPI, Vol. 54(5), pp. 699­ describe the humidity dependence of paperboard biaxial 703, 1971. strength. 11. de Ruvo, A., Lundberg, R., Martin-Lof, S. and Soremark, C., "Influence of Temperature and Humidity on the Elastic Properties of Paper and SUMMARY AND CONCLUSIONS the Constituent Fiber," in The Fundamental Properties of Paper Related to its Uses, Edited In this work the effects of moisture on the zero- by F. Bolam, Technical Section of the B.P. & shear biaxial failure envelope of paperboard have been B.M.A., pp. 785-806, 1976. examined. The tensor polynomial criterion was used 12. Salmen, N. L. and Back, E. L., "Moisture- for predicting paperboard biaxial strength. Several Dependent Thermal Softening of Paper, Evaluated experiments have been performed to measure uniaxial by its Elastic Modulus," TAPPI, Vol. 63(6), pp.

605 117-120, 1980. 13. Okushima, S. and Robertson, A. A., "Humidity and the Tensile Behavior of Paper, in Proceedings of the 1979 International Paper Physics Conference, pp. 83-88, Harrison Hot Springs, British Columbia, September 17-19, 1979. 14. Brezinski, J. P., "The Creep Properties of Paper," TAPPI, Vol. 39(2), pp. 116-128, 1956. 15. Johanson, F. and Kubat, J., "Measurements of Stress Relaxation in Paper," Svensk Papperstidning, Vol. 67(20), pp. 822-832, 1964. 16. Kimura, M., Usuda, M. and Kadoya, T., "Moisture Dependence of Impact Rupture Properties of Paper," in Fibre- Interactions- in Papermaking, Technical Section of the B.P. & B.M.A., pp. 671-682. 1978. 17. Gunderson, D. E., Considine, J. M. and Scott, C. T., "The Compressive Load-Strain Curve of Paperboard: Rate of Load and Humidity Effects," Journal of Pulp and Paper Science, Vol. 14(2), pp. 37-41, 1980. Figure 2 18. Malmeister, A. K., "Geometry of Theories of Strength," Polymer Mechanics, Vol. 2(4), pp. 324­ 331, 1966. 19. Tsai, S. W. and Wu, E. M., "A General Theory of Strength for Anisotropic Materials," Journal of Composite Materials, Vol. 5(1), pp. 58-80, 1971. 20. Gol'denblat, I. I. and Kopnov, V. A., "Strength of -Reinforced Plastics in the Complex Stress State," Polymer Mechanics, Vol. 1(2), pp. 54-59, 1965. 21. Wu, E. M., "Optimal Experimental Measurements of Anisotropic Failure Tensors," Journal of Composite Materials, Vol. 6(4), pp. 472-489, --1972. 22. Considine, J. M. and Gunderson, D. E., "A Method for Mechanical Property Testing of Paperboard During Compressive Creep in a Cyclic Humidity Environment," Experimental Techniques, Vol. 11(9), pp. 18-21, 1987. 23. Gunderson, D. E., "Edgewise Compression Testing of Paperboard -- A New Concept of Lateral Support," APPITA, Vol. 37(2), pp. 137-141, 1983.

Figure 3

TABLE: EXPERIMENTALLY MEASURED UNIAXIAL FAILURE PARAMETERS FOR PAPERBOARD

RH = 50% RH = 70% RH = 90%

X (MPa) 51.77 47.41 36.40

X' (MPa) 21.42 20.20 14.74

Figure 1 Y (MPa) 27.88 21.82 15.36

Y' (MPa) 14.70 13.64 9.70

Figure 4

606 Figure

Figure 7 Figure 6

607 TABLE: BIAXIAL FAILURE POINTS FOR PAPERBOARD MEASURED EXPERIMENTALLY USING PRESSURIZED CYLINDRICAL TUBES

Figure 9

Figure 8

PAPERBOARD FAILURE ENVELOPE USING PAPERBOARD FAILURE ENVELOPE USING THE TENSOR POLYNOMIAL CRITERION THE TENSOR POLYNOMIAL CRITERION ( RH = 50 ) ( RH = 70 )

Figure 10 Figure 11 PAPERBOARD FAILURE ENVELOPE USING THE TENSOR POLYNOMIAL CRITERION ( RH = 90 )

Figure 12

PAPERBOARD FAILURE ENVELOPE USING THE TENSOR POLYNOMIAL CRITERION (RH = 50,70,90)

Figure 13

609 Citation 1 of 1

USDA Forest Products Laboratory Library Citation format = AMA

Suhling JC, Considine JM, Yeh KC. Effects of Moisture On the Biaxial Strength of Wood-based Composites. 1989 SEM spring conference on experimental mechanics, 1989 May 29-June 1, Cambridge, MA Bethel, CT: Society for Experimental Mechanics, 1989

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