Estimation of very narrow bounds to the behavior of nonlinear incompressible elastic composites

Abstract

Variational bounds for the effective behavior of nonlinear composites are improved by incorporating more-detailed morphological information. Such bounds, which are obtained from the generalized HashinShtrikman variational principles, make use of a reference material with the same microstructure as the nonlinear composite. The geometrical information is contained in the effective properties of the reference material, which are explicitly present in the analytical formulae of the nonlinear bounds. In this paper, the variational approach is combined with estimates for the effective properties of the reference composite via the asymptotic homogenization method (AHM), and applied to a hexagonally periodic fiber-reinforced incompressible nonlinear elastic composite, significantly improving some recent results.

Variational bounds in composites with nonuniform interfacial thermal resistance

“…Integration by parts and the subsequent application of the Lax-Milgram lemma shows that the minimization problem in (20) is equivalent to the problem r Á s ¼ 0 in Y 1 S Y 2 , ss Á nt ¼ Àm on C. Replacing the latter conditions in (18) …”

“…Relevant parameters are the Biot reference number, Bi ¼ bd=r 1 , where d represents the diameter of the fibers crosssection, and k 2 f2; 50g. Results in Tables 1 and 2 were obtained by taking the estimates for m 0 given by formulas (2.3), p. 1553, of [19] and (C.4), p. 230, of [20], respectively.…”

Negligible microstructure in a class of porous nonlinear materials via variational bounds

“…In the incompressible realization of the J 2 deformation theory of plasticity, is the equivalent stress, p 0 ! 1, and 1 3pl is the initial shear modulus and p n is a reference nonlinear compliance in the Ramberg-Osgood model with hardening exponent equal to 3 [6,9,26].…”

“…In the context of nonlinear inclusions in a linear matrix, we have found that, when the microstructure is accounted for in detail, the Ponte Castañeda lower bound is capable of providing excellent estimates for the effective energy density for different geometrical configurations and physical frameworks [5][6][7][8][9] that improve remarkably over previous uses [10,11]. The accuracy of such an estimate is guaranteed by the availability of a nontrivial upper bound by Talbot [10,11] which, together with the lower bound, provides very narrow ranges of effective behavior [5][6][7][8][9].…”