Analytical Solutions of Model Problems for Large-Deformation Micromorphic Approach to Gradient Plasticity

The objective of this work is to present analytical solutions for several 2D model problems to demonstrate the unique plastic fields generated by the implementation of micromorphic approach for gradient plasticity. The approach is presented for finite deformations and several macroscopic and nonstandard microscopic boundary conditions are applied to a gliding plate to illustrate the capability to predict the size effects and inhomogeneous plastic fields promoted by the gradient terms. The constitutive behavior of the material undergoing plastic deformation is analyzed for softening, hardening and perfect plastic response and corresponding solutions are provided. The analytical solutions are also shown to match with the numerical results obtained by implementing a user element subroutine (UEL) to the commercial finite element software Abaqus/Standard.

The Nonlinear evolutionary problem for self-stressed multilayered hyperelastic spherical bodies

The present paper studies the evolutionary problem for self-stressed multilayered spherical shells. Their stress-strain state is characterized by incompatible local finite deformations that arise due to the geometric incompatibility of the stress-free shapes of the individual layers with each other. In the considered problem, these shapes are thin-walled hollow balls that cannot be assembled into a single solid without gaps or overlaps. Such an assembly is possible only with the preliminary deformations of individual layers, which cause self-balanced stresses in them. For multilayered structures with a large number of layers, a smoothing procedure is proposed, as a result of which the piecewise continuous functions defining the preliminary deformation of the layers are replaced by continuous distributions. The reference stress-free shape of a body constructed in this way is defined within the framework of geometric continuum mechanics as a manifold with a non-Euclidean (material) connection. For the problem in question, this connection is determined by the metric tensor and its deviation from the Euclidean connection is characterized by the scalar curvature. Generalized representations for Cauchy and Piola stresses are also obtained by the methods of geometric continuum mechanics. Computations, provided for the discrete structure and body with a non-Euclidean reference shape defined by the approximation of deformation parameters, numerically illustrate the convergency of the solution for the discrete model to corresponded solution for the continuous problem if the number of layers is increasing while their total thickness is constant. In modelling it is assumed that the material of the layers is compressible, homogeneous, hyperelastic, and determined by the first-order Mooney - Rivlin elastic potential. Individual layerwise finite deformations are supposed to be centrally symmetric.