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Abstract:
The paper deals with the notion of stability for thermo-elastoplastic materials
undergoing large strains. The stability analysis is performed by using the
perturbation approach applied to a comprehensive material model derived in a thermodynamic
format. As the main contribution of this paper a stability condition
for a material model incorporating geometrical and material non-linearities under
full thermo-mechanical coupling, without typical simplifying assumptions, is derived,
and a hybrid analytical-numerical verification of the stability condition at a material
point is investigated for the three-dimensional case. Special emphasis is placed on the
quasi-static case, for which a specific stability criterion is derived. The theoretical
analysis is followed by the numerical verification of the obtained condition. The implementation
of the model in the finite element method, using the numerical-symbolic
package AceGen, is also presented in the paper. Two representative three-dimensional
examples are solved, namely a cube under simple shear and a plate with imperfection,
subjected to tension. The obtained results reveal that the type of softening, i.e.,
thermal or material softening, has a significant influence on the stability at a material
point level.

Keywords:
material stability, localization, thermo-elastoplasticity, large strains, finite element method

Abstract:
This paper deals with the numerical investigation of ellipticity of the boundary value problem for isothermal finite strain elasto-plasticity. Ellipticity can be lost when softening occurs. A discontinuity surface then appears in the considered material body and this is associated with the ill-posedness of the boundary value problem. In the paper the condition for ellipticity loss is derived using the deformation gradient and the first Piola-Kirchhoff stress tensor. Next, the obtained condition is implemented and numerically tested within symbolic-numerical tools AceGen and AceFEM using the benchmark of an elongated rectangular plate with imperfection in plane stress and plane strain conditions.