Overall Solution-Difference Bounds on the Effects of Material Inhomogeneities

In this work variational bounds are developed on the difference between the solutions of two boundary-value problems involving linearly elastic material microstructure. Both problems have the same external geometry, boundary conditions and loading, however, one possesses no microstructure, i.e., it is homogeneous, while the other is “perturbed” with inhomogeneities. The bounds developed are solely in terms of the solution to the easier homogeneous material problem, the microstructure of the inhomogeneous problem and the given loading data. There are no unknown constants in the bounds. Furthermore, no calculation of the harder, typically intractable, inhomogeneous material problem is necessary. The bounds developed are obtained under no assumptions on the character of the microstructure of the inhomogeneous material problem, other than it be pointwise positive-definite, as well as under no assumptions on the external loading and geometry. This revised version was published online in August 2006 with corrections to the Cover Date.