Material Tailoring in Structures Composed of Functionally Graded Materials

Functionally graded materials are inhomogeneous with material properties varying continuously in one or more spatial directions. The inverse problem of finding the spatial variation of material moduli (or equivalently the microstructure) so as to achieve a desired state of stress within the body is quite challenging and has received less attention than the direct problem of finding deformations and stresses for a known spatial variation of material properties; this is usually called material tailoring. We summarize below results for the material tailoring problem for the following four problems: (i) find spatial variation of the volume fractions of two constituents comprising an isotropic linear elastic plate to optimize the fundamental frequency, (ii) determine through-the-thickness variation of the fiber orientation angle in a fiber reinforced laminated composite plate to optimize one of the first five frequencies, (iii) ascertain the spatial variation of the two moduli in a hollow cylinder made of a Mooney-Rivlin material so that the in-plane shear stress is constant through the cylinder thickness, and (iv) compute the spatial variation of the volume fraction of tungsten and nickel-iron in a rectangular cross-section of a prismatic body to control the adiabatic shear band initiation time. Whereas the first two problems involve infinitesimal deformations for linear elastic materials, the third problem is for finite deformations of nonlinear elastic materials, and the fourth problem has large coupled thermo-mechanical transient deformations for viscoplastic materials.