Design of microstructures of periodic composite materials.

Abstract

The design of composite materials with prescribed thermoelastic properties opens many possibilities in the fields of mechanical design and optimization. We are designing a composite material with a prescribed linear anisotropic elastic constitutive tensor E and a thermal expansion tensor $\alpha.$ We restrict ourselves to periodic composite materials, so we can use the homogenization method to calculate the composite's average properties. We homogenize the properties using the finite element method. The homogenized properties are written in sums of element mutual energies for the independent thermoelastic coefficients, which are nine for anisotropic 2D cells (27 for 3D cells). We solve the model of the unit cell applying the basic unit pre-strains $\varepsilon\sbsp{ij}{0}$s to get the correspondent strains $\varepsilon\sbsp{ij}{\*}$s. These solutions are obtained using non-conforming elements and a sparse pre-conditioned conjugate gradient equation solver. The homogenization problem is inverted; we find a microstructure that matches a given set $E\sbsp{ijkl}{H}.$ We formulate the following topology optimization problem:(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\vbox{\halign{#\hfil&&\quad#\hfil\cr minimize &{\it J\/}\quad over all $x\sp{e}$\cr subject to &C\sbsp{I}{H}-D\sb{I}=0,$\quad I=1,ND\cr and to &$x\sb{min}\le x\sp{e}\le x\sb{max}\cr}}$$(TABLE/EQUATION ENDS)where J is the objective function (amount of resource), and $x\sb{min}$ and $x\sb{max}$ are the bounds of the design variable (density of resources). The desired properties $D\sb{I}$ are imposed as constraints. We solve the problem with a Sequential Linear Programming approach, where the moving bounds are filtered for smooth solutions, without checkerboards or hinges. We show some results to some specific problems, as designing negative Poisson's ratio materials. Some of the resulting materials were actually built with a rapid prototyping machine.