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

On Computational Modelling of Strain-Hardening Material Dynamics

In this paper we show that entropy can be used within a functional for the stress relaxation time of solid materials to parametrise finite viscoplastic strain-hardening deformations. Through doing so the classical empirical recovery of a suitable irreversible scalar measure of work-hardening from the three-dimensional state parameters is avoided. The success of the proposed approach centres on determination of a rate-independent relation between plastic strain and entropy, which is found to be suitably simplistic such to not add any significant complexity to the final model. The result is sufficiently general to be used in combination with existing constitutive models for inelastic deformations parametrised by one-dimensional plastic strain provided the constitutive models are thermodynamically consistent. Here a model for the tangential stress relaxation time based upon established dislocation mechanics theory is calibrated for OFHC copper and subsequently integrated within a two-dimensional moving-mesh scheme. We address some of the numerical challenges that are faced in order to ensure successful implementation of the proposed model within a hydrocode. The approach is demonstrated through simulations of flyer-plate and cylinder impacts

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case

International audienceOne of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in the one-dimensional Lagrangian gas dynamics framework. After briefly recalling how to derive Lagrangian forms of the 1D gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS, and relies on two different techniques: either a particular definition of the local approximation of the acoustic impedances arising from the approximate Riemann solver, or an additional time step constraint relative to the cell volume variation. Then, making use of the work presented in [89, 90, 22], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. This paper is the first part of a series of two. The whole analysis presented here is extended to the two-dimensional case in [85], and proves to fit a wide range of numerical schemes in the literature, such as those presented in [19, 64, 15, 82, 84]

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders

One of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in this Lagrangian gas dynamics framework. To this end, we first focus on the one-dimensional case. After briefly recalling how to derive Lagrangian forms of the gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS. It enables us to derive time step conditions ensuring the desired positivity property, as well as L 1 stability of the specific volume and total energy over the domain. Then, making use of the work presented in [74, 75, 15], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. This whole analysis is finally applied to the two-dimensional case, and shown to fit a wide range of numerical schemes in the literature, such as the GLACE scheme [12], the EUCCLHYD scheme [55], the GLACE scheme on conical meshes [8], and the LCCDG method [72]. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. Finally, let us emphasize that even if this paper is concerned with purely Lagrangian schemes, the theory developed is of fundamental importance for any methods relying on a purely Lagrangian step, as ALE methods or non-direct Euler schemes

Simulation of Wave in Hypo-Elastic-Plastic Solids Modeled by Eulerian Conservation Laws

This paper reports a theoretical and numerical framework to model nonlinear waves in elastic-plastic solids. Formulated in the Eulerian frame, the governing equations employed include the continuity equation, the momentum equation, and an elastic-plastic constitutive relation. The complete governing equations are a set of first-order, fully coupled partial differential equations with source terms. The primary unknowns are velocities and deviatoric stresses. By casting the governing equations into a vector-matrix form, we derive the eigenvalues of the Jacobian matrix to show the wave speeds. The eigenvalues are also used to calculate the Courant number for numerical stability. The model equations are solved using the Space-Time Conservation Element and Solution Element (CESE) method. The approach is validated by comparing our numerical results to an analytical solution for the special case of longitudinal wave motion.Comment: 34 pages, 11 figure

A first order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme

This paper builds on recent work developed by the authors for the numerical analysis of large strain solid dynamics, by introducing an upwind cell centred hexahedral Finite Volume framework implemented within the open source code OpenFOAM [http://www.openfoam.com/http://www.openfoam.com/]. In Lee, Gil and Bonet [1], a first order hyperbolic system of conservation laws was introduced in terms of the linear momentum and the deformation gradient tensor of the system, leading to excellent behaviour in two dimensional bending dominated nearly incompressible scenarios. The main aim of this paper is the extension of this algorithm into three dimensions, its tailor-made implementation into OpenFOAM and the enhancement of the formulation with three key novelties. First, the introduction of two different strategies in order to ensure the satisfaction of the underlying involutions of the system, that is, that the deformation gradient tensor must be curl-free throughout the deformation process. Second, the use of a discrete angular momentum projection algorithm and a monolithic Total Variation Diminishing Runge-Kutta time integrator combined in order to guarantee the conservation of angular momentum. Third, and for comparison purposes, an adapted Total Lagrangian version of the Hyperelastic-GLACE nodal scheme of Kluth and Despr´es [2] is presented. A series of challenging numerical examples are examined in order to assess the robustness and accuracy of the proposed algorithm, benchmarking it against an ample spectrum of alternative numerical strategies developed by the authors in recent publications