Thomas Heuzé

A finite element for laminar flow of incompressible fluids with inertia effects and thermomechanical coupling

The Friction Stir Spot Welding (FSSW) process involves large deformations in the neighborhood of the tool. The simulation of this process has to account for a pasty phase in which the material is stirred, and a phase remaining solid. An Arbitrary Lagrangian Eulerian (ALE) approach combined with respectively fluid and solid behaviours in each of those phases may allow to simulate a lot of rotations of the tool into the material while following the boundaries of the sheets. This work focuses on a first stage of this study, the development of a mixed formulation temperature/velocity/pressure of a fluid finite element P1+/P1 in the unsteady case.

Lax–Wendroff and TVD finite volume methods for unidimensional thermomechanical numerical simulations of impacts on elastic–plastic solids

Abstract We present in this work two finite volume methods for the simulation of unidimensional impact problems, both for bars and plane waves, on elastic–plastic solid media within the small strain framework. First, an extension of Lax–Wendroff to elastic–plastic constitutive models with linear and nonlinear hardenings is presented. Second, a high order TVD method based on flux-difference splitting [1] and Superbee flux limiter [2] is coupled with an approximate elastic–plastic Riemann solver for nonlinear hardenings, and follows that of Fogarty [3] for linear ones. Thermomechanical coupling is accounted for through dissipation heating and thermal softening, and adiabatic conditions are assumed. This paper essentially focuses on one-dimensional problems since analytical solutions exist or can easily be developed. Accordingly, these two numerical methods are compared to analytical solutions and to the explicit finite element method on test cases involving discontinuous and continuous solutions. This allows to study in more details their respective performance during the loading, unloading and reloading stages. Particular emphasis is also paid to the accuracy of the computed plastic strains, some differences being found according to the numerical method used. Lax–Wendoff two-dimensional discretization of a one-dimensional problem is also appended at the end to demonstrate the extensibility of such numerical scheme to multidimensional problems.

Very High Strain Rate Range

The classical Split Hopkinson Pressure Bar (SHPB) system is considered to be able to perform tests at strain rates ranging from 102 to 104 s−1 (Zhao and Gary, Mater Sci Eng A 207:46–50, [1]). However, some modifications can be carried out to extend the reachable strain rate within the specimen. The mean strain rate defined within the specimen

A Discontinuous Galerkin Material Point Method for the solution of impact problems in solid dynamics

Abstract An extension of the Material Point Method [1] based on the Discontinuous Galerkin approximation (DG) [2] is presented here. A solid domain is represented by a collection of particles that can move and carry the fields of the problem inside an arbitrary computational grid in order to provide a Lagrangian description of the deformation without mesh tangling issues. The background mesh is then used as a support for the Discontinuous Galerkin approximation that leads to a weak form of conservation laws involving numerical fluxes defined at element faces. Those terms allow the introduction of the characteristic structure of hyperbolic problems within the numerical method by using an approximate Riemann solver [3] . The Discontinuous Galerkin Material Point Method, which can be viewed as a Discontinuous Galerkin Finite Element Method (DGFEM) with modified quadrature rule, aims at meeting advantages of both mesh-free and DG methods. The method is derived within the finite deformation framework for multidimensional problems by using a total Lagrangian formulation. A particular attention is paid to one specific discretization leading to a stability condition that allows to set the CFL number at one. The approach is illustrated and compared to existing or developed analytical solutions on one-dimensional problems and compared to the finite element method on two-dimensional simulations.

Simulation of impacts on elastic–viscoplastic solids with the flux-difference splitting finite volume method applied to non-uniform quadrilateral meshes

The flux-difference splitting finite volume method (Leveque in J Comput Phys 131:327–353, 1997; Leveque in Finite volume methods for hyperbolic problems. Cambridge: Cambridge University Press, 2002) is here employed to perform numerical simulation of impacts on elastic–viscoplastic solids on bidimensional non-uniform quadrilateral meshes. The formulation is second order accurate in space through flux limiters, embeds the corner transport upwind method, and uses a fractional-step method to compute the relaxation operator. Elastic–viscoplastic constitutive models falling within the framework of generalized standard materials (Halphen and Nguyen in J Mech 14:667–688, 1975) in small strains are considered. Many test cases are proposed and two particular viscoplastic constitutive models are studied, on which comparisons with finite element solutions show a very good accuracy of the finite volume solutions, both on stresses and viscoplastic strains.

A DISCONTINUOUS GALERKIN MATERIAL POINT METHOD (DGMPM) FOR THE SIMULATION OF IMPACT PROBLEMS IN SOLID MECHANICS

The material point method is extended in this work to the Discontinuous Galerkin approximation framework for the simulation of impacts on elastic and hyperelastic solids. The formulation is based on the weak form of conservation laws on each cell of an eulerian grid in which volume integrals are discretized on a set of material points lying in that cell, and on the computation of Godunov fluxes at cells faces. The resulting method is first derived within the small strains framework and illustrated on a one-dimensional and a two-dimensional problem of impact on an elastic media. Then a one-dimensional hyperelastic problem of a solid undergoing large strains is presented, and a comparison is performed with an analytical solution.

Benchmark tests based on the Couette viscometer—I: Laminar flow of incompressible fluids with inertia effects and thermomechanical coupling

Abstract The Couette viscometer is a well-known problem of fluid mechanics, well-suited for the verification of numerical methods. The aim of this work is to extend the classical steady state mechanical solution obtained in fluid mechanics, both to strongly-coupled thermomechanical problems in the case of laminar and incompressible fluid flows, and to solid-type nonlinear behaviours. Extended solutions will allow for the verification of new formulations of a mixed P1+/P1 finite element developed both in fluid and solid mechanics, within a temperature/velocity/pressure formulation coupled with an implicit (backward) Euler algorithm in time. In the present Part I, we address the case of the laminar flow of incompressible fluids with inertia effects and thermomechanical coupling. The verification performed on the reference solutions developed clearly evidence the good behaviour of the fluid finite element. The extension to solid-type nonlinear behaviours for strongly-coupled thermomechanical problems will be the subject of Part II.

Benchmark tests based on the Couette viscometer—II: Thermo-elasto-plastic solid behaviour in small and large strains

Abstract The Couette viscometer is a well-known problem of fluid mechanics, well-suited for the verification of numerical methods. The aim of this work is to extend the classical steady state mechanical solution obtained in fluid mechanics and to use the extended solutions to assess new finite elements. Part I was devoted to the case of laminar flow of incompressible fluids with inertia effects and thermomechanical coupling. The present Part II focuses on solid-type nonlinear behaviours; we address the cases of elastic–plastic and thermo-elastic–plastic von Mises materials, both in small and large strains. The extended solutions permit to assess a new formulation of a mixed P1+/P1 finite element in solid mechanics, in a temperature/velocity/pressure formulation coupled with an implicit (backward) Euler algorithm in time. The verification evidences a good behaviour of the solid finite element.

Two experimental set-ups designed for investigation of friction stir spot welding process

Abstract The effects of positioning and clamping conditions of a specimen of friction stir spot welding are investigated in this paper in terms of axial force and torque generated during the process. For this purpose, two special designs of experimental set-ups embedding different positioning and clamping conditions are presented. A four-component mechanical sensor is used for the measurements. First, the effects of the rotational speed of the spindle and the plunge depth of the tool on the axial force and torque are studied. Second, the effects of positioning and clamping conditions are investigated through both set-ups designed, varying the spindle rotation speed. It is shown that the axial force and torque exhibit an important dependence with respect to the rotation speed of the tool and that their maxima depend on positioning and clamping conditions of the specimen.