Stress-Induced Phase Transformations of an Isotropic Incompressible Elastic Body Subject to a Triaxial Stress State

Abstract

The present paper concerns the stable multiphase isochoric deformations for an isotropic elastic body subject to a surface traction of uniform Piola stress with two equal principal forces which are opposite to the third. To model the occurrence of such deformations, we consider a strain energy density function which depends on the first principal invariant of deformation through a non-convex function and which has an added linear dependence on the second invariant. We establish existence conditions for equilibrium multiphase deformations which give restrictions on the morphology of the connecting phases as well as on the orientation of the flat interfaces between the phases. Finally, by considering a special, but representative, form for the non-convex strain energy function, we show that there exists a Bcritical'' value of the external load which allows for the emergence of stable coexistent deformation fields.

A lower bound estimate of the critical load in bifurcation analysis for incompressible elastic solids

“…For cases (4.1) and (4.2), we also compare our estimate λ LB to the estimate recently proposed by Del Piero and Rizzoni [8]. For the latter case (4.3), we refer to the special dead-load traction boundary value problem in Foti [12] in order to compare lower bound estimates and the actual critical load.…”

“…In Fosdick et al [9] we provided this kind of analysis for the shear problem studied by Fosdick et al [10], where we had available either the actual critical load or the bifurcation load, considered as an upper bound estimate. Here we consider the effectiveness issue for an equilibrium problem similar to that in Foti [12]; this concerns a state of equibiaxial extension accompanied by an orthogonal uniaxial compression of the same amount.…”

“…Corresponding to a value s cr of the load at which the symmetric branch intersects the asymmetric branch, the asymmetric response function is at its minimum. In Foti [12] it is shown that the symmetric solution is Hadamard stable for s ≤ s cr , and that for s > s cr the asymmetric solution is Hadamard stable. Thus, s = s cr is a bifurcation point, and the related value of the stretch λ cr is the critical load.…”

“…For the cases of uniaxial compression and uniaxial extension of a Mooney–Rivlin cylinder, we discuss the effectiveness of our procedure by comparing our estimates to those obtained by Del Piero and Rizzoni [8]. We then provide some remarks concerning the effectiveness of the considered estimates by evaluating the gap between the lower bound estimate and the critical load for a special boundary value problem inspired by the analyses of Foti [12], D’Ambrosio et al [13] and De Tommasi and Foti [14].…”

Phase Transformations in Isotropic Elastic Materials Induced by Shear Stress States

“…In short, such condition restricts the form of the surface loading allowing for the emergence of multiphase solutions to a shear stress accompanied by an arbitrary transverse traction. A preliminary analysis on multiphase deformations descending from the general results determined in [8] is contained in [11], where a pair of principal forces are equal and opposite to the third force.…”