Upper and lower bounds to the overall incremental stiffness of hyperelastic composites

Abstract

Bounds to the overall stiffness of a composite are well-known within the classical theory of elasticity. They are based on the positive-definiteness of the local stiffness. A transfer to a prestressed state is not trivial. We may study the incremental stiffness that connects the nominal stress rate with the velocity gradient. But when there are mainly compressive stresses, then positive-definiteness can only be secured if this stiffness is replaced by a pseudo-stiffness. Its existence is equivalent to a strengthened form of uniform infinitesimal polyconvexity and is independent of the geometry. The same is the case with the crude Voigt and Reuss bounds. More refined kinematic or dynamic approximations do, of course, depend on the geometry. This is demonstrated with the unidirectional reinforcement of a matrix.