Experimentally Determining Engineering Constants of a Woven Composite Patrick Pollock, Siming Guo Introduction This study describes the linear behavior of a woven glass-epoxy composite under off-axis tension. The material engineering constants E1 , E2 , G12 , and υ12 are unknown and need to be found. The step-by-step theory and experimental procedure are shown so this method can be repeated for future materials. Theory The theory for determining engineering constants and developing a single equation that describes the behavior of a composite regardless of orientation has already been well developed in [1, 2, 3]. Additional explanations of the theoretical work as well as an analysis of nonlinear behavior of woven composites can also be found in [1, 2, 3]. Total strain εij can be split into linear elastic, εl , and nonlinear εn . This relationship is shown ij ij in (1). εij = εl + εn ij ij (1) In the linear elastic region, the strain and applied stress are related by the generalized Hooke’s law shown in (2). εl = aijkl σkl ij (2) Using the quadratic elasticity potential function, the compliance tensor, aijkl , and the relation between applied stress and linear strain can be derived. w= 1 aijkl σij σkl 2 ∂w ∂σij (3) (4) εl = ij In a plane stress state for an orthotropic body, (3) can be reduced to (5) or (6), if engineering constants E1 , E2 , G12 , υ12 = υ21 are used. E1 E2 w= 1 2 2 (a1111 σ11 + a1122 σ12 + a2222 σ22 + a1212 σ12 ) 2 1 2 2 σ11 2υ12 σ2 σ2 − σ11 σ22 + 22 + 12 E1 E1 E2 G12 (5) w= (6) In the case of uni-axial tension, (5) can be reduced to (7), where the applied stress is σθ and the orientation of the specimen with respect to the axis of symmetry is θ. 1 w = 1 2 1 σ cos4 θ + 2 θ E1 2υ12 1 − G12 E1 sin2 θ cos2 θ + 1 sin4 θ E2 (7) Using (4), the linear strain in the direction of σθ is shown in (8). εl = θ where h2 (θ) = cos4 (θ) + c1 sin4 (θ) + c2 sin2 cos2 (θ) l c1 = c2 = E1 E2 (9) σθ 2 ∂w = h (θ) ∂σϑ E1 l (8) E1 − 2υ12 G12 Two new variables are introduced, effective stress, σθ , and effective strain εl . This allows (8) to ˜ ˜θ be independent of angle. εl = ˜θ εl θ hl (θ) (10) σθ = σθ h(θ) ˜ εl = ˜θ 1 σθ ˜ E1 (11) From (11), it is evident that εl /˜θ should equal a constant regardless of the orientation of the ˜θ σ specimen. This can be used to find c1 and c2 values for a material if tensile tests are done at several orientations. Experimental Procedure The composite material Norplex NP130 is being used in future tests so the material engineering constants E1 , E2 , G12 , and υ12 need to be calculated. This material is a 5-layer woven glass fabric combined with an epoxy resin. A 0.04” thick sheet of this material was cut into thin strips measuring 6” x 0.7”. The orientation of the strips are shown in Figure 1 on page 3. Three strips were cut at each orientation. Monotonic tensile tests to failure were done on each strip. The tests were done using an MTS 810 and the TestWare-SX software. An extensometer was used to measure the strain. For each orientation, the results were averaged. The stress-strain curves are shown in Figure 2 on page 3. To find the material engineering constants, only the linear parts of the stress-strain relationship need to be analyzed. For each orientation, the nonlinear strain was subtracted out. The remaining elastic part of each orientation is shown in Table 1 on page 3. Using (11), the unknown parameters E1 , c1 , and c2 can now be discovered. For the θ = 0o case, o h(0 ) = 1 and is not dependent on any unknown parameters. The parameter E1 is simply εl /σθ . For θ the θ = 90o case, h(90o ) = c1 . Because E1 is already known, E2 can be found by calculating E1 /c1 . Now using the experimental data for the remaining orientations c2 can be determined. Matlab was used to solve for c2 using the Nelder-Mead method, which performs an unconstrained nonlinear minimization of the sum of squared residuals. This value was then check visually by plotting the effective stress vs. effective strain. In Figure 3 on page 4, it can be seen that the E1 , c1 , and c2 values that were calculated do cause the curves to line up. It was found that c1 = 1.091 and c2 = 4.22. 2 Figure 1: Composite plate Figure 2: Stress-Strain curve θ Ex (psi ∗ 106 ) 0o 3.646 15o 3.352 30o 2.556 45o 2.247 60o 2.563 90o 3.341 Table 1: Initial longitudinal Young’s Modulus for each orientation 3 Figure 3: Effective stress vs. effective elastic strain A test was also done on a 0o strip using image correlation. The strains were measured in the longitudinal and transverse directions. The results are shown in Figure 4 on page 4. Using (12), (13) and (14), Poisson’s ratio, υ12 , was determined to be 0.15. The parameter c2 can be used now to find G12 . The final calculated properties are shown in Table 2 on page 5. εl = x σx Ex σx Ex (12) (13) (14) εl = −υ y εl = −υεl y x Figure 4: Longitudinal and transverse strains calculated using image correlation 4 E1 (psi ∗ 106 ) 3.646 E2 (psi ∗ 106 ) 3.341 G12 (psi ∗ 106 ) 0.806 υ12 0.15 Table 2: Properties for Norplex NP130 Summary Tensile tests can be performed to obtain engineering constants for an unknown material. The behavior of the material at any orientation can be described by a single curve. The variables of the curve are only two coefficients. The first coefficient describes how the material properties differ in each of the principle material directions. Because of the nature of the woven composite used in this test, the first coefficient had a value close to one. The second coefficient reflects the shear modulus as well as the Poisson’s ratio. To separate these properties, one needs to be determined experimentally. References [1] S Ogihara and KL Reifsnider. Characterization of nonlinear behavior in woven composite laminates. APPLIED COMPOSITE MATERIALS, 9(4):249–263, JUL 2002. [2] V Tamuzs, K Dzelzitis, and K Reifsnider. Fatigue of woven composite laminates in off-axis loading I. The mastercurves. APPLIED COMPOSITE MATERIALS, 11(5):259–279, SEP 2004. [3] Liqun Xing, Kenneth L. Reifsnider, and Xinyu Huang. Progressive damage modeling for large deformation loading of composite structures. COMPOSITES SCIENCE AND TECHNOLOGY, 69(6, Sp. Iss. SI):780–784, MAY 2009. 5
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