Bounds on elastic constants for random polycrystals of laminates

A well-known result due to Hill provides an exact expression for the bulk modulus of any multicomponent elastic composite whenever the constituents are isotropic and the shear modulus is uniform throughout. Although no precise analog of Hill’s result is available for the opposite case of uniform bulk modulus and varying shear modulus, it is shown here that some similar statements can be made for shear behavior of random polycrystals composed of laminates of isotropic materials. In particular, the Hashin-Shtrikman-type bounds of Peselnick, Meister, and Watt for random polycrystals composed of hexagonal (transversely isotropic) grains are applied to the problem of polycrystals of laminates. An exact product formula relating the Reuss estimate of bulk modulus and an effective shear modulus (of laminated grains composing the system) to products of the eigenvalues for quasicompressional and quasiuniaxial shear eigenvectors also plays an important role in the analysis of the overall shear behavior of the random polycrystal. When the bulk modulus is uniform in such a system, the equations are shown to reduce to a simple form that depends prominently on the uniaxial shear eigenvalue—as expected from physical arguments concerning the importance of uniaxial shear in these systems. Applications of the analytical results presented here include benchmarking of numerical procedures used for studying elastic behavior of complex composites, and estimating coefficients needed in upscaled equations for elasticity and∕or poroelasticity of heterogeneous systems.