Yvan Chastel

Viscoplastic self-consistent and equilibrium-based modeling of olivine lattice preferred orientations : Implications for the upper mantle seismic anisotropy

Anisotropy of upper mantle physical properties results from lattice preferred orientation (LPO) of upper mantle minerals, in particular olivine. We use an anisotropic viscoplastic self-consistent (VPSC) and an equilibrium-based model to simulate the development of olivine LPO and, hence, of seismic anisotropy during deformation. Comparison of model predictions with olivine LPO of naturally and experimentally deformed peridotites shows that the best fit is obtained for VPSC models with relaxed strain compatibility. Slight differences between modeled and measured LPO may be ascribed to activation of dynamic recrystallization during experimental and natural deformation. In simple shear, for instance, experimental results suggest that dynamic re-crystallization results in further reorientation of the LPO leading to parallelism between the main (010)[100] slip system and the macroscopic shear. Thus modeled simple shear LPOs are slightly misoriented relative to LPOs measured in natural and experimentally sheared peridotites. This misorientation is higher for equilibrium-based models. Yet seismic properties calculated using LPO simulated using either anisotropic VPSC or equilibrium-based models are similar to those of naturally deformed peridotites; errors in the prediction of the polarization direction of the fast S wave and of the fast propagation direction for P waves are usually <15°. Moreover, overestimation of LPO intensities in equilibrium-based and VPSC simulations at high strains does not affect seismic anisotropy estimates, because these latter are weakly dependent on the LPO intensity once a distinct LPO pattern has been developed. Thus both methods yield good predictions of development of upper mantle seismic anisotropy in response to plastic flow. Two notes of caution have nevertheless to be observed in using these results: (1) the dilution effect of other upper mantle mineral phases, in particular enstatite, has to be taken into account in quantitative predictions of upper mantle seismic anisotropy, and (2) LPO patterns from a few naturally deformed peridotites cannot be reproduced in simulations. These abnormal LPOs represent a small percent of the measured natural LPOs, but the present sampling may not be representative of their abundance in the Earths upper mantle.

Numerical modelling of crack propagation: automatic remeshing and comparison of different criteria

Modelling of a crack propagating through a finite element mesh under mixed mode conditions is of prime importance in fracture mechanics. Three different crack growth criteria and the respective crack paths prediction for several test cases are compared. The maximal circumferential stress criterion, the strain energy density fracture criterion and the criterion of the maximal strain energy release rate are implemented using advanced finite element techniques. A fully automatic remesher enables to deal with multiple boundaries and multiple materials. The propagation of the crack is calculated with both remeshing and nodal relaxation. Several examples are presented to check for the robustness of the numerical techniques, and to study specific features of each criterion.

Crack propagation modelling using an advanced remeshing technique

The modelling of a crack propagation through a finite element mesh is of prime importance in fracture mechanics. We propose here a solution based on an advanced remeshing technique. A fully automatic remesher enables us to deal with multiple boundaries and multiple materials. The propagation of the crack is achieved with both remeshing and nodal relaxation. A maximal normal stress criterion is used to compute the crack direction. Several tools are developed and presented to obtain accurate results at the crack tip: evolving mesh refinement, crack tip finite elements, ring of elements surrounding the crack. Finally, several applications are presented to show the robustness of this technique.

Forming limits prediction using rate-independent polycrystalline plasticity

The purpose of this paper is the prediction of forming limits computed from an initial defect approach combined with a rate-independent polycrystalline plasticity model. The algorithm used for the integration of the material behaviour inside and outside the localization band is presented. Results are compared with the forming limit curves at necking and at failure for 6116-T4 aluminium.

Estimation of constitutive parameters using an inverse method coupled to a 3D finite element software

Abstract Forming process simulations require a precise knowledge of the input material parameters. These parameters are usually estimated from mechanical tests. The classical analysis of these tests are usually based on a few assumptions: material flow homogeneity, isothermal conditions, etc. But in some cases with strain localisation or self-heating, these assumptions overestimate material strength. Analysis techniques using inverse methods are then good alternatives. This paper deals with the estimation of mechanical parameters using an inverse method. The direct model is a 3D forming process simulation software (FORGE3 ® ). The numerical formulation is based on a mixed finite element method using two unknowns, the velocity and the pressure. The tetrahedron element is linear in velocity and pressure and the thermal problem is solved using a linear element. The inverse problem associated with the estimation of mechanical parameters is expressed as a least square problem. The aim is to obtain output of the direct model which fits experimental data measured during the mechanical test. The optimisation problem is solved using a Gauss–Newton algorithm. At the end of the optimisation, an estimation of confidence intervals is done. A Gauss–Newton algorithm requires the computation of the derivatives of the output with respect to the parameters to be identified. In this work, a semi-analytical differentiation is performed. The proposed method is first validated on artificial experimental data obtained from direct simulations of hot uniaxial compressions for a viscoplastic cylinder. The confidence interval is provided by the algorithm for different configurations with additional random noise. Finally a real steel compression test is analysed to provide parameters for the Norton–Hoff viscoplastic law.

3D elastic-plastic finite element simulation of cold pilgering of zircaloy tubes

In cold pilgering of tubes, a material element undergoes a series of small incremental deformations (≈100 strokes), alternatively under tensile and compressive stresses. This complex history sometimes results in surface damage, seemingly by low-cycle fatigue. Prior to studying the resistance of diverse potential materials to this kind of complex, non-proportional multi-axial, and non-periodic cycling, a thorough mechanical analysis of the stress states is necessary: the finite element method (FEM) software Forge3(®) has been used, with updated Lagrangian formulation due to the transient character of strains and stresses. The process is periodical, except for the ends of a given preform, which are cut off afterwards. One stroke only should thus be sufficient to analyse the whole process, provided the correct initialisations are done in terms of shape, strains and stresses, but these are parts of the unknown of the problem. This point will be particularly addressed in the following, where it is shown that in the non-work hardening case at least, simulating three strokes leads to an invariant geometry and state of stress, starting from a reasonable estimate of the geometry. Strains and stresses thus obtained will be discussed in detail, together with their probable consequences on the damage and fatigue of the material, to be later correlated with defects.

Rate-independent crystalline and polycrystalline plasticity, application to FCC materials

Abstract This paper deals with the simulation of the mechanical response and texture evolution of cubic crystals and polycrystals for a rate-independent elastic–plastic constitutive law. No viscous effects are considered. An algorithm is introduced to treat the difficult case of multi-surface plasticity. This algorithm allows the computation of the mechanical response of a single crystal. The corresponding yield surface is made of the intersection of several hyper-planes in the stress space. The problem of the multiplicity of the slip systems is solved thanks to a pseudo-inversion method. Self and latent hardening are taken into account. In order to compute the response of a polycrystal, a Taylor homogenization scheme is used. The stress–strain response of single crystals and polycrystals is computed for various loading cases. The texture evolution predicted for compression, plane strain compression and simple shear are compared with the results given by a visco-plastic polycrystalline model.

Prediction of intergranular strains in cubic metals using a multisite elastic-plastic model

A novel approach is adopted for determining the elastic and plastic strains of individual grains within a deformed polycrystalline aggregate. In this approach, termed “multisite modeling”, the deformation of a grain does not merely depend on the grain lattice orientation. It is also significantly influenced by the interaction with one or several of the surrounding grains. The elastic-plastic constitutive law is integrated by identifying iteratively which dislocation slip systems are activated within the grains, and the local stress tensor is shown to be the solution of a linear equation set. Several micro–macro averaging schemes are considered for the distribution of the macroscopic load over the polycrystalline aggregate. These averaging schemes are tested by simulating the development of intergranular strains during uniaxial tension of MONEL-400 as well as commercial purity aluminium. Neutron diffraction measurements of the elastic lattice strains are used as a reference in order to discriminate between the various predictions. The results demonstrate the relevance of “multisite” grain interactions in f.c.c. polycrystals.

Sensitivity of α-ZY4 high-temperature deformation textures to the β-quenched precipitate structure and to recrystallization: Application to hot extrusion

Hot extrusion of Zircaloy-4 tubes usually starts from β-quenched microstructures and induces strong textures. Individual crystallographic orientations were investigated by transmission electron microscopy using the electron backscatter pattern (EBSP) technique as well as Kikuchi patterns. Basal poles were found close to the tangential direction of the tubes in regions exhibiting fine and homogeneously distributed precipitates (FHDPs). In contrast, regions with large and isolated precipitates (LIPs) had more variable orientations. Laboratory plane strain compression tests were performed and the induced textures were compared with numerical simulations using a polycrystalline viscoplastic self-consistent model. The β-quenched material was modeled as a mixture of LIP and FHDP regions, each having a different set of slip system hardnesses, with a volume fraction depending on the previous thermal history. The model was subsequently applied to predict the texture evolution during extrusion with metadynamic recrystallization taking place thereafter. The calculation suggests that recrystallization modifies the orientation of those grains where 〈c+a〉 crystallographic slip has been significantly activated during deformation.

Simulation of cup-drawing based on crystal plasticity applied to reduced grain samplings

This paper presents a means of reducing the computational cost of finite element (FE) simulations coupled to polycrystal plasticity theory. One typically assumes that a polycrystal with a large number of grains underlies every integration point of the FE mesh. Instead, it is suggested here using reduced samplings of grains which differ from one integration point to another. On average, every set of 5 to 25 finite elements contains a variety of lattice orientations that is representative of the macroscopic texture. The model is applied to deep-drawing of a cylindrical cup made of steel. In a first set of simulations, grains are assigned orientations representative of a cold rolling texture and the “earing” profile is compared to experiment. In a second set of simulations, lattice orientations are random and an isotropic deep-drawing result is expected. It is demonstrated that using a minimum of 20 grains per integration point allows properly predicting the final shape of the cup and the texture development.