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By utilizing methods recently developed in the theory of fluid interfaces, we provide a new framework for considering the localization of deformation and illustrate it for the case of hyperelastic materials. The approach overcomes one of the major shortcomings in constitutive equations for solids admitting localization of deformation at finite strains, i.e. their inability to provide physically acceptable solutions to boundary value problems in the post-localization range due to loss of ellipticity of the governing equations. Specifically, strain-induced localized deformation patterns are accounted for by adding a second deformation gradient-dependent term to the expression for the strain energy density. The modified strain energy function leads to equilibrium equations which remain always elliptic. Explicit solutions of these equations can be found for certain classes of deformations. They suggest not only the direction but also the width of the deformation bands providing for the first time a predictive unifying method for the study of pre-and post-localization behavior. The results derived here are a three-dimensional extension of certain one-dimensional findings reported earlier by the second author for the problem of simple shear. * Permanent address:

For ductile solids with periodic microstructures (e.g., honeycombs, fiber-reinforced composites, cellular solids) which are loaded primarily in compression, their ultimate failure is related to the onset of a buckling mode. Consequently, for periodic solids of infinite extent, one can define as the onset of failure the first occurrence of a bifurcation in the fundamental solution, for which all cells deform identically. By following all possible loading paths in strain or stress space, one can construct onset-of-failure surfaces for finitely strained, rate-independent solids with arbitrary microstructures. The calculations required are based on a Bloch wave analysis on the deformed unit cell. The presentation of the general theory is followed by the description of a numerical algorithm which reduces the size of stability matrices by an order of magnitude, thus improving the computational efficiency for the case of continuum unit cells. The theory is subsequently applied to porous and particle-reinforced hyperelastic solids with circular inclusions of variable stiffness. The corresponding failure surfaces in strain-space, the wavelength of the instabilities, and their dependence on micro-geometry and macroscopic loading conditions are presented and discussed.

To investigate the relation between the macroscopic and microscopic instability predictions for certain microstructured media, we study the stability of an axially stretched fiber-reinforced composite under plane strain conditions. The microstructural instability is attributed to a bifurcation buckling of the fibers while the corresponding macroscopic one occurs at the loss of ellipticity in the homogenized incremental equilibrium equations of the material. The effects of geometry and material properties on those phenomena are analyzed. The macroscopic approach consistently and sometimes considerably overestimates the stable region of the composite. The attractive feature in this work is that all the investigations can be done by using analytical solutions instead of the numerical ones employed in similar investigations so far.

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International audienceThe present work is an in-depth study of the connections between microstructural instabilities and their macroscopic manifestations--as captured through the effective properties--in finitely strained porous elastomers. The powerful second-order homogenization (SOH) technique initially developed for random media, is used for the first time here to study the onset of failure in periodic porous elastomers and the results are compared to more accurate finite element method (FEM) calculations. The influence of different microgeometries (random and periodic), initial porosity, matrix constitutive law and macroscopic load orientation on the microscopic buckling (for periodic microgeometries) and macroscopic loss of ellipticity (for all microgeometries) is investigated in detail. In addition to the above-described stability-based onset-of-failure mechanisms, constraints on the principal solution are also addressed, thus giving a complete picture of the different possible failure mechanisms present in finitely strained porous elastomers