An exact two-dimensional model for heterogeneous plates

This work derives an exact two-dimensional plate theory for heterogeneous plates consistent with the principle of stationary three-dimensional potential energy under general loading. We do not take any hypotheses about the shape of the heterogeneity. We start from three-dimensional linear elasticity and by using the Fourier series expansion in the thickness direction of the displacement field with respect to a basis of scaled Legendre polynomials. We deduce an exact two-dimensional model expressed in power-series in the ratio between the thickness of the plate and a characteristic measurement of its mid-plane. Then we can derive an approximative theory by neglecting in the expression of potential energy all terms that contain a power of this ratio that is higher than a given truncation power for getting to an approximative two-dimensional problem. In the last section, we show that the solution of the approximation problem only differs from the exact solution by a difference of the same order of the neglected terms in the potential energy. A similar result when we truncate the displacement field can be also established. This model can be a starting point to formulate a two-dimensional homogenized boundary value problem for highly heterogeneous periodic plates.