constituents, and despite the nonlinear shape of the fiber bundles, which means they are not truly transversely isotropic, the failure criteria for these constituents is assumed to be transversely isotropic [17]. MCM allows for a stress or strain based failure criterion [15, 17]. 14.3 Equation of State Coupling As discussed earlier, in large deformation hydrocodes (e.g., CTH [16]), the stress tensor is decomposed into pressure and deviatoric stress, and such codes require an equation of state (EOS) to determine the pressure (change in size or volume) and a constitutive model, sometimes called a strength model, to determine the deviatoric stress (change in shape). In anisotropic materials, pressure and deviatoric strain components are coupled. An applied hydrostatic pressure not only causes a change in size, but also induces strains that may be different in different directions. Likewise, an applied shear stress not only induces a change in shape, but may also lead to a change in size in different directions (volumetric strain). Therefore, the EOS and constitutive equations must be coupled. Segletes [4, 11], O’Donoghue and Anderson et al. [7, 8], and Lukyanov [10, 11] have developed anisotropic constitutive relationships. Segletes decomposed hydrostatic and deviatoric terms [4]. O’Donoghue et al. [7] and Anderson et al. [8] developed relationships for pressure and deviatoric stress as functions of volumetric and deviatoric strains and anisotropic material properties (elastic modulus and Poisson’s ratio). Anderson et al. [8] also made a correction to the Mie-Grüneisen EOS [26, 27], which amounts to the addition of the deviatoric strain contribution to pressure to the pressure-volume relationship of the EOS, the standard form of which is given in Eq. 14.1. In Eq. 14.1, PH is the shocked material pressure state from the Hugoniot, μ is the bulk modulus, Γ is the well-known Gruneisen parameter, a thermodynamic material property, V is volume (or inverse density), and E is the internal energy density. Lukyanov [10] stated that purely hydrostatic pressure should produce only a change in size, and so he decomposed the stress tensor into hydrostatic pressure, pressure dependent on deviatoric strains, and deviatoric stress, as in Eq. 14.2. Deviatoric stresses and strains are determined as described earlier. Then the hydrostatic pressure is determined from a given EOS, and the coupled pressure term is found from deviatoric strains and anisotropic stiffness and isotropic bulk modulus material properties as described by Lukyanov [10]. PEOS ¼PH 1þ Γ 2 μ þ Γ V E ð14:1Þ σij ¼P EOSδ ij þP dev Sij þSij ð14:2Þ As currently implemented in CTH, the equation of state pressure, PEOS, is found from the Mie-Grüneisen EOS [26, 27] given in Eq. 14.1, but theoretically any relevant EOS could be used to provide the hydrostatic pressure. 14.4 Numerical Simulations Gorfain and Key [14] used the MCM implementation in CTH to evaluate prediction of damage extent in rib-stiffened composite structures similar to composite wing ribs and spars. In the present work we model ballistic perforation experiments conducted by Gama and Gillespie [28], and compare the experimental results with the unidirectional and plain weave architecture implementation of MCM in CTH and a micromechanics code, used to calculate the strength and failure properties of the composites as discussed earlier, and we evaluate impact velocity vs. residual velocity (Vi Vr) curves for stress and strain failure criteria. CTH version 11.0 and the micromechanics code version as of October 2015 (updated in 2016, see Conclusions) were used. The target modeled was plain weave (5 5 tows/in.) S-2 Glass / SC15 epoxy composite, 22 layers thick, 17.8 cm by 17.8 cm by 1.32 cm with clamped boundary conditions. Stiffness, strength, and failure material properties for the composite were calculated as described above based on known material properties for thick sections of this composite. The projectile modeled was 4340 steel 0.22 caliber right circular cylinders. In experiments, the hard steel projectiles perforated the softer targets while remaining rigid, undeformed, and uneroded. However, because CTH is an Eulerian code, it is subject to mixed-cell erosion (that is, strength properties can be averaged in mixed cells such that a cell containing both steel and composite material could be effectively softened and thereby eroded, which is 106 C.S. Meyer et al.
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