Relaxed incremental variational formulation for damage at large strains with application to fiber‐reinforced materials and materials with truss‐like microstructures

SUMMARY In this paper, an incremental variational formulation for damage at finite strains is presented. The classical continuum damage mechanics serves as a basis where a stress‐softening term depending on a scalar‐valued damage function is prepended an effective hyperelastic strain energy function, which describes the virtually undamaged material. Because loss of convexity is obtained at some critical deformations, a relaxed incremental stress potential is constructed, which convexifies the original nonconvex problem. The resulting model can be interpreted as the homogenization of a microheterogeneous material bifurcated into a strongly and weakly damaged phase at the microscale. A one‐dimensional relaxed formulation is derived, and a model for fiber‐reinforced materials based thereon is given. Finally, numerical examples illustrate the performance of the model by showing mesh independency of the model in an extended truss, analyzing a numerically homogenized microtruss material and investigating a fiber‐reinforced cantilever beam subject to bending and an overstretched arterial wall. Copyright © 2012 John Wiley & Sons, Ltd.