Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity

The aim of this paper is to compare a hyperelastic with a hypoelastic model describing the Eulerian dynamics of solids in the context of non-linear elastoplastic deformations. Specifically, we consider the well-known hypoelastic Wilkins model, which is compared against a hyperelastic model based on the work of Godunov and Romenski. First, we discuss some general conceptual differences between the two approaches. Second, a detailed study of both models is proposed, where differences are made evident at the aid of deriving a hypoelastic-type model corresponding to the hyperelastic model and a particular equation of state used in this paper. Third, using the same high order ADER Finite Volume and Discontinuous Galerkin methods on fixed and moving unstructured meshes for both models, a wide range of numerical benchmark test problems has been solved. The numerical solutions obtained for the two different models are directly compared with each other. For small elastic deformations, the two models produce very similar solutions that are close to each other. However, if large elastic or elastoplastic deformations occur, the solutions present larger differences.Comment: 14 figure

The u‐p approximation versus the exact dynamic equations for anisotropic fluid‐saturated solids. I. Hyperbolicity

The numerical solution of dynamic problems for porous fluid-saturated solids is often performed with the use of simplified equations known as the u-p approximation. The simplification of the equations consists in neglecting some acceleration terms, which is justified for a certain class of problems related, in particular, to geomechanics and earthquake engineering. There exist two u-p approximations depending on how many acceleration terms are neglected. All comparative studies of the exact and u-p formulations are focused on the question of how well the u-p solutions approximate those obtained with the exact equations. In this paper, the equations are compared from a different point of view, addressing the question of well-posedness of boundary value problems. The exact equations must be hyperbolic and satisfy the corresponding hyperbolicity conditions for the boundary value problems to be well posed. The u-p equations are not of the form to which the conventional definition of hyperbolicity applies. A slight extension of the approach makes it possible to derive hyperbolicity conditions as necessary conditions for well-posedness for the u-p approximations. The hyperbolicity conditions derived in this paper for the u-p approximations are formulated in terms of the acoustic tensor of the skeleton. They differ essentially from the hyperbolicity conditions for the exact equations

A generalisation of J2-flow theory for polar continua

A pressure-dependent J2-flow theory is proposed for use within the framework of the Cosserat continuum. To this end the definition of the second invariant of the deviatoric stresses is generalised to include couple-stresses, and the strain-hardening hypothesis of plasticity is extended to take account of micro-curvatures. The temporal integration of the resulting set of differential equations is achieved using an implicit Euler backward scheme. This return-mapping algorithm results in an exact satisfaction of the yield condition at the end of the loading step. Moreover, the integration scheme is amenable to exact linearisation, so that a quadratic rate of convergence is obtained when Newton's method is used. An important characteristic of the model is the incorporation of an internal length scale. In finite element simulations of localisation, this property warrants convergence of the load-deflection curve to a physically realistic solution upon mesh refinement and to a finite width of the localisation zone. This is demonstrated for an infinitely long shear layer and for a biaxial specimen composed of a strain-softening Drucker-Prager material

Numerical implementation of the eXtended Finite Element Method for dynamic crack analysis

A numerical implementation of the eXtended Finite Element Method (X-FEM) to analyze crack propagation in a structure under dynamic loading is presented in this paper. The arbitrary crack is treated by the X-FEM method without re-meshing but using an enrichment of the classical displacement-based finite element approximation in the framework of the partition of unity method. Several algorithms have been implemented, within an Oriented Object framework in C++, in the home made explicit FEM code. The new module, called DynaCrack, included in the dynamic FEM code DynELA, evaluates the crack geometry, the propagation of the crack and allow the post-processing of the numerical results. The module solves the system of discrete equations using an explicit integration scheme. Some numerical examples illustrating the main features and the computational efficiency of the DynaCrack module for dynamic crack propagation are presented in the last section of the paper

Modeling of Slow Plasticity Waves

Quasi-static uniaxial loading of a bar with a length L is considered. Mechanical properties of a material in a point are defined by the segment of negative slope on stress-strain diagram which follows the section of elastic deformation. The deformation in specimen is uniform until the stress exceeds the peak yielding stress. The analytical solution shows that stress-strain diagram of the specimen has a yielding plateau. It is shown that the time for a slow wave to advance by a distance equal to the localized band width S is the same as it is required for a plastic wave to run along the whole bar length