Evaluation of in-plane orthotropic elastic constants of paper and paperboard T. Yokoyama and K. Nakai Department of Mechanical Engineering, Okayama University of Science - Ridai-cho, Okayama 7-5, Japan ABSTRACT The in-plane orthotropic elastic constants of machine-made paper and paperboard were determined in a compact testing machine equipped with an optical extensometer and using a high-speed digital image sensor. Commercial copy paper, sack paper and paperboard were tested in a standard atmosphere of 23 ±2 C and 5 ±2% relative humidity. A dumbbell-type specimen specified in the JIS Z 22for sheet materials was used in the tension tests. Tensile specimens were cut in three different orientations from each type of paper and paperboard. Their thicknesses were carefully measured with a high-precision digital micrometer under a constant pressure. The in-plane elastic (Young's) moduli, shear modulus and Poisson's ratios as an orthotropic material were determined. It was shown that the in-plane elastic constants of commercial paper and paperboard roughly obey the theory of orthotropic linear elasticity.. INTRODUCTION Paper and paperboard are two of the most commonly used materials in pulp and paper, and packing industries. They generally exhibit anisotropic and non-linear mechanical behavior, strongly affected by moisture and temperature. A sufficient understanding of their mechanical behavior is very important in most converting operations and in end-use applications. However, determination of the mechanical properties is limited to those few properties, which can easily be measured, such as the elastic (Young's) moduli and the ultimate tensile strength in the machine direction (MD) and cross-machine direction (CD). The shear moduli, Poisson's ratios and virtually all of the out-of-plane elastic parameters are not readily measured by normal mechanical methods [], and hence their relationships to the ultimate end-use requirements of paper and paperboard are not known. Measurements of the in-plane orthotropic elastic constants (E MD, E CD, G, ν xy ) of paper and paperboard have been usually conducted using ultrasonic techniques [2-4]. Ultrasonic measurements involve the determination of wave velocities which relate directly to the elastic constants of the paper materials. The primary objective of the present work is to determine the in-plane orthotropic elastic constants of commercial paper and paperboard in a compact testing machine equipped with an optical extensometer and using a high-speed digital image sensor. Tension tests were conducted using dumbbell-type specimens in three different orientations in a standard atmosphere of 23 ±2 C and 5 ±2% relative humidity. The in-plane Young's moduli, shear modulus and Poisson's ratios for each paper and paperboard were determined. The values of the in-plane shear modulus were compared with theoretical values.
2. PAPER MATERIALS AND SPECIMEN PREPARATION Commercial paper and paperboard were chosen for testing: copy paper, sack paper made from unbleached kraft pulp (or UKP-sack paper), and uncoated paperboard. They are made from cellulose fibers. During the manufacturing process, the axes of the fibers tend to be aligned parallel to the paper flow through the paper machine. This phenomenon leads to anisotropy in the mechanical properties of paper. Basically, there are three principal directions which coincide with the directions of the paper machine (see Fig. ). JIS Z 22 dumbbell-type specimens (see Fig. 2) were cut in three different orientations from each type of paper and paperboard on a paper cutter, following a -mm thickness steel template as shown in Fig. 3. The specimen's ends were over-wrapped with chipboard doublers to prevent localized end failure. Their basic properties in both MD and CD in the standard atmosphere are given in Table. It is very difficult to determine the real thickness of paper because of its surface roughness and compressibility. Conforming to the ASTM D645/D645M-97 [5], the thickness of ten sheets of paper was carefully measured with a digital micrometer (Mitutoyo: GMA-25 DM) having a flat ground circular movable face with a 2 mm diameter under a constant pressure of 3 kpa with an accuracy of ±4mm. The surface roughness R a (center-lined average roughness) was non-contactly measured with a confocal laser scanning microscope (Keyence: VK-95). The measured surface roughness R a is almost the same in both MD and CD for any paper and paperboard. The paper tensile strength (S) given in Table was determined on the constant-width strip specimens of 25 mm in width by 25 mm in length specified in the ASTM D828-97 [6]. Note that the paper tensile strength (S) provides only an apparent tensile strength, which is given by the maximum load per unit width (kn/m). ZD (3) MD () 2 SPECIMEN 5 5 6 (DIMENSIONS IN MM) direction of paper flow through paper machine Fig. Definitions of three principal material axes in paper or paperboard where MD is the machine direction, CD the cross-machine direction and ZD the thickness direction. The notations, 2 and 3 are also used Fig. 2 Geometry of JIS Z 22 tensile specimen for commercial paper and paperboard Table Basic properties of commercial paper and paperboard tested at 23 and 5% RH Basis weight (g/m 2 ) Thickness t (µm) Apparent density ρ (kg/m 3 ) Roughness R a (µm) MD CD Tensile strength S (kn/m) MD CD Copy paper (65)* 85 (87) 753 (754) 5.4. 5.3.5 3.9 2.2 UKP-sack paper Paperboard (85) 25 (9) 68 (74) (43) 7 (69) 64 (623) 4.8. 4.7.3 4.7.2 4.5.6 * Note: values in parentheses are provided by paper manufacturers 6.6 23.2 3.4 9.2 Fig. 3 Three orientations in which tensile specimens were cut. The longitudinal direction in paper commonly corresponds to the machine direction
3. TESTING APPARATUS AND PROCEDURE It is virtually impossible to mechanically measure the deformation of a paper specimen in a conventional universal testing machine with a strain-gage extensometer, and hence we used the compact testing machine (JT Tohsi: Little Senstar LSC-/3) equipped with a non-contacting optical extensometer using a CCD camera (see Fig. 4). Tension tests were conducted using dumbbell-type paper specimens at a crosshead velocity of 3mm/min. Before testing, a small pretension load of N was applied to remove slack from the paper specimens. The applied load was measured with a load cell of kn capacity, and the longitudinal elongation over a 5-mm gage length stamped in black on a reduced section of the paper specimen was determined with the optical extensometer. The resulting tensile load-elongation relation was then converted to the nominal tensile stress-strain curve. In the measurement of Poisson's ratios, the transverse strain during tensile loading was simultaneously determined by recording the transverse contraction along the width of the paper specimen using the high-speed digital image sensor at a sampling rate of 5Hz (see Fig. 5). Note that the accuracy of transverse strain measurement was gradually degraded, because the paper specimen tended to curl about a loading axis as the tensile stress increased. CCD CAMERA TESTING MACHINE PERSONAL COMPUTER Fig. 4 Overall view of compact testing machine equipped with optical extensometer and personal computer Fig. 5 Picture of high-speed digital image sensor used for measurement of longitudinal and transverse strains in paper specimen 4. RESULTS AND DISCUSSION 4. In-Plane Tensile Stress-Strain Behavior Typical in-plane tensile stress-strain curves for copy paper in the three different orientations (or MD, 45 and CD) are shown in Fig. 6. It is seen that these stress-strain curves clearly depict the anisotropic elastic, initial yielding and strain hardening behavior. Young's modulus and the tensile strength decrease, and the strain-to-fracture and absorbed energy up to fracture increase with increasing angle from MD. This is due to the fact that machine-made paper and paperboard have more fibers aligned in MD. The tensile data are summarized in Table 2, where the coefficient of variation of both Young's modulus and the strain-to-fracture is lager than that of the tensile strength in any test orientation. The degree of elastic anisotropy is usually defined as the ratio of R = E MD / E CD [2, 4], where E MD and E CD are Young's moduli in MD and CD. The paperboard is found to have the highest anisotropy and the lowest tensile strength in each test orientation. This is because the macrostructure of the paperboard is quite different from that of other two types of paper, i.e., the paperboard has a structure built up of 7 layers of paper. Figure 7 displays fracture appearance of the paper and paperboard specimens, indicating that fractures occurred inside the extensometer gage length. None of the specimens exhibited a necking behavior just before
sudden fracture. Note that fractures in the 45 off-axis specimens took place at planes aligned at about 45 to 8 to the loading axis, which do not correspond to a shearing fracture. Each fracture propagated only along a zigzag path in a direction along the fibers preferentially aligned. 8 6 232 52 % RH (MD) 45 9 (CD) : FRACTURE. -3 ε= /s (V CH= 3mm/min).2. MD LOADING ε ε ε 2 4 2 -ε 2 2 4 6 8 TENSILE STRAIN ε (%) Fig. 6 Typical in-plane tensile stress-strain curves for copy paper in three test orientations 5 TIME t (sec) Fig. 8 Time histories of longitudinal strain and transverse strain in copy paper specimen 5 UKP-SACK PAPER PAPERBOARD MD 45 CD MD 45 CD MD 45 CD Fig. 7 Fracture appearance of dumbbell-type specimen in three test orientations Table 2 Summary of in-plane tensile properties of commercial paper and paperboard tested at a strain rate of -3 /s in three test orientations Copy paper UKP-sack paper Paperboard MD() MD() MD() Young's modulus (GPa) 6.54.64 3.3.54 7.74.75 2.7.3 4.96.52.44.6 Tensile strength σ (MPa) 49.32.4 25.3.6 54.72. 28.2.8 3.3.6 2.7. Strain-to-fracture ε (%) f.7.2 6.3.7.7.2 4.3.3.9.2 4.9.2 Anisotoropy ratio R (=E MD /E CD ) 45 4.45.2 37..3 2.7.4 2.9 45 45 4.28.73 32.. 2.4.2 2.86 2.88.56 7.4.3 2.9. 3.46 MeanS.D. (Standard deviation). No. of specimens tested = 5.
4.2 In-Plane Poisson's Ratios and Shear Modulus Figure 8 shows typical time histories of longitudinal strain (ε ) and transverse strain (ε 2 ) in the copy paper specimen under MD loading. Both strains increase discontinuously with time due to the non-homogeneous deformation in copy paper. Eliminating time t yields the ε 2 vs ε and ε vs ε 2 relations as given in Fig.9 where the latter relation is obtained from another tension test under the CD loading. In-plane two Poisson's ratios ν 2 and ν 2 are defined as ν 2 = ε 2 / ε : ratio of transverse contraction ε 2 to longitudinal extension ε under -direction loading ν 2 = ε / ε 2 : ratio of transverse contraction ε to longitudinal extension ε 2 under 2-direction loading and determined from the initial slopes of the two relations depicted in Fig. 9. Both Poisson's ratios increase with increasing longitudinal strain. According to the test procedure specified in the ASTM D358-76 [7], the tension test on a 45 off-axis specimen gives the shear stress (τ = σ /2) and shear strain (γ = ε ε 2 ) where ε and ε 2 are, respectively, the longitudinal and transverse strains, which are simultaneously measured with the high-speed digital image sensor. From this relation, we can determine a shear stress-strain curve as depicted in Fig.. The in-plane shear modulus G 2 is determined from the initial slope of the shear stress-strain curve. Typical in-plane orthotropic constants (E,, ν 2, ν 2, and G 2 ) of paper and paperboard are summarized in Table 3. It is observed that Poisson's ratio ν 2 is about.6 to 2.7 times larger than Poisson's ratio ν 2, suggesting the high anisotropy of paper and paperboard. It is also found that the measured G 2 agrees reasonably well with the theoretical values obtained by Eqs. (3) and (4)..2 5 232 52 % RH. -3 ε= /s (MD) LOADING 9(CD) LOADING (V CH = 3mm/min) 4 3 45 SPECIMEN τ = σ 2 γ = ε ε 2. 2 G 2 ν 2 ν 2..2.3.4.5 LONGITUDINAL STRAIN ε or ε (%) Fig. 9 Transverse strain vs longitudinal strain for copy paper 2.6 2 SHEAR STRAIN γ (%) Fig. Shear stress-strain curve for copy paper from tension test on 45 off-axis specimen 3 4 5 Table 3 Summary of typical in-plane elastic constants of paper and paperboard Young's modulus or (GPa) 2 Poisson's ratio ν or ν 2 2 Shear modulus G 2 (GPa) Measured Theoretical (3) Campbell (4) MD () MD () 6.82.56.95.89.94.8.93.8 MD ()..98.98
Following classical laminate theory [8], Young's modulus and Poisson's ratio in the x-y plane can be expressed in terms of an angle θ from MD (see the inset in Fig. ) as = m2 m 2 n 2 ν 2 E x E ( ) + n2 n 2 m 2 ν 2 ( ) + m2 n 2 () G 2 ν xy = m2 m 2 ν 2 n 2 E x E ( ) + n2 n 2 ν 2 m 2 ( ) + m2 n 2 (2) G 2 where m = cosθ ; n = sinθ ; the subscripts and 2 denote MD and CD, respectively. When θ = 45 is substituted into Eq. (), we obtain the following relation: = ν 2 + ν 2 + (3) E x 4E θ=45 o 4 4G 2 Since the left hand side of Eq. (3) is known from Young's modulus measured in the 45 orientation, the shear modulus G 2 can be then determined from the measured four elastic constants E,, ν 2 and ν 2. Campbell [9] proposed a simple expression similar to Eq. (3) for a variety of paper as G 2 = + ν 2 E + + ν 2 under the assumption that the following expression given in [] is satisfied: E x = m2 E + n2 Figure indicates the variation of Young's modulus with fiber (or test) orientation. Young's modulus decreases monotonically from its maximum value E at θ = to its minimum value at θ = 9. Measured Young's moduli are in excellent agreement with the theoretical values calculated from Eq. (). Similarly, Fig. 2 presents the variation of Poisson's ratio with fiber (or test) orientation. Poisson's ratio varies monotonically from its maximum value ν 2 at θ = to its minimum value ν 2 at θ = 9. Measured Poisson's ratios also agree well with the theoretical values derived from Eq. (2). (4) (5) 8 6 E UKP-SACK PAPER PAPERBOARD EXPERIMENT THEORY.3.2 ν 2 UKP-SACK PAPER PAPERBOARD EXPERIMENT THEORY 4 y 2 MD() θ x. y MD() θ x ν 2 (MD) 2 3 4 5 6 7 ANGLE θ FROM MD (degree) Fig. Young's modulus as a function of fiber (or test) orientation 8 9 (CD) (MD) 2 3 4 5 6 7 ANGLE θ FROM MD (degree) Fig. 2 Poisson's ratio as a function of fiber (or test) orientation 8 9 (CD) 5. CONCLUSIONS The in-plane orthotropic elastic constants of machine-made copy paper, UKP-sack paper and uncoated paperboard were determined in the compact testing machine equipped with the optical extensometer and using the high-speed digital image sensor. From the present experimental results, the following conclusions can be drawn:
() The in-plane elastic constants of machine-made paper and paperboard roughly obey the theory of orthotropic linear elasticity. (2) Machine-made paper and paperboard may be regarded as a kind of composite material, although pulp (or wood) fibers do not have matrix structure. Acknowledgement The authors wish to thank Mr. Takafumi Odamura, formerly Graduate Student, Okayama University of Science, for his assistance in the experimental work. References [] Jones, A.R.: An experimental investigation of the in-plane elastic moduli of paper, Tappi J., Vol. 5, No. 5, 23-29 (968). [2] Baum, G.A. and Bornhoeft, L.R.: Estimating Poisson ratios in paper using ultrasonic techniques, Tappi J., Vol. 62, No. 5, 87-9 (979). [3] Mann, R.W., Baum, G.A. and Habeger, C.C.: Determination of all nine orthotropic elastic constants for machine-made paper, Tappi J., Vol. 63, No. 2, 63-66 (98). [4] Baum, G.A., Brennan, D.C. and Habeger, C.C.: Orthotropic elastic constants of paper, Tappi J., Vol. 64, No. 8, 97- (98). [5] ASTM D645/D645M-97: Annual Book of ASTM Standards 25, Vol. 5.9, 374, ASTM International (25). [6] ASTM D828-97: Annual Book of ASTM Standards 25, Vol. 5.9, 2, ASTM International (25). [7] ASTM D358-76: ASTM Standards and Literature References for Composite Materials, 5, ASTM (99). [8] Daniel, I.M. and Ishai, O.: Engineering Mechanics of Composite Materials, 64, Oxford University Press, Oxford (994). [9] Campbell, J.G.: The in-plane elastic constants of paper, J. Australian Appl. Sci., Vol. 2, No. 3, 356-357 (96). [] Horio, M. and Onogi, S.: Dynamic measurements of physical properties of pulp and paper by audiofrequency sound, J. Appl. Phys., Vol. 22, 97-977 (95).