WIT Press

Nonlinear Hashin–Shtrikman Bounds For Hereditary Problems


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119 - 127




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WIT Press


P. P. Procházka & M. Toman


Classical Hashin–Shtrikman variational bounds are based on the assumption that all phases behave in a purely elastic and locally homogeneous manner. Hashin–Shtrikman principles, from which the bounds are derived, are extended in a paper by Procházka and Sejnoha (Applications of Mathematics, 2004). The extension consists of introducing eigenparameters (either eigenstrain or eigenstress) into the formulation. Moreover, these eigenparameters were used in the estimation of bounds with the result that elastic strain is in a certain relation with plastic strains. This appears to be a very restrictive condition (constraint) and a new approach will be presented in this paper based on additional estimates. In the classical approach of Hashin and Shtrikman the overall energies were compared with local energies (on the micro scale level). From this the procedure for evaluation of bounds on overall material properties involving eigenparameters begins. In our case it is possible to consider the eigenparameters as characterizations of plastic behavior of one or more phases in the composite structure. Following this idea, the eigenparameters can describe the current situation in the composite structure and involved in the estimates. In such a way, the eigenparameters can be considered as plastic strains or relaxation stresses, and then the properties of eigenparameters can be taken into consideration. Basically, the former approach of the paper by Hashin–Shtrikman still remains in the body of the presented derivation of the new principles. Keywords: extended Hashin–Shtrikman principles, bounds on overall properties, hereditary problems.


extended Hashin–Shtrikman principles, bounds on overall properties, hereditary problems.