Effective transient behaviour of inclusions in diffusion problems

This paper is concerned with the effective transport properties of heterogeneous media in which there is a high contrast between the phase diffusivities. In this case the transient response of the slow phase induces a memory effect at the macroscopic scale, which needs to be included in a macroscopic continuum description. This paper focuses on the slow phase, which we take as a dispersion of inclusions of arbitrary shape. We revisit the linear diffusion problem in such inclusions in order to identify the structure of the effective (average) inclusion response to a chemical load applied on the inclusion boundary. We identify a chemical creep function (similar to the creep function of viscoelasticity), from which we construct estimates with a reduced number of relaxation modes. The proposed estimates admit an equivalent representation based on a finite number of internal variables. These estimates allow us to predict the average inclusion response under arbitrary time-varying boundary conditions at very low computational cost. A heuristic generalisation to concentration-dependent diffusion coefficient is also presented. The proposed estimates for the effective transient response of an inclusion can serve as a building block for the formulation of multi-inclusion homogenisation schemes.Comment: 24 pages, 9 figures. Submitted to ZAMM (under review

Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities

This paper presents a homogenisation-based constitutive model to describe the effective tran- sient diffusion behaviour in heterogeneous media in which there is a large contrast between the phase diffusivities. In this case mobile species can diffuse over long distances through the fast phase in the time scale of diffusion in the slow phase. At macroscopic scale, contrasted phase diffusivities lead to a memory effect that cannot be properly described by classical Fick's second law. Here we obtain effective governing equations through a two-scale approach for composite materials consisting of a fast matrix and slow inclusions. The micro-macro transition is similar to first-order computational homogenisation, and involves the solution of a transient diffusion boundary-value problem in a Representative Volume Element of the microstructure. Different from computational homogenisation, we propose a semi-analytical mean-field estimate of the composite response based on the exact solution for a single inclusion developed in our previous work [Brassart, L., Stainier, L., 2018. Effective transient behaviour of inclusions in diffusion problems. Z. Angew Math. Mech. 98, 981-998]. A key outcome of the model is that the macroscopic concentration is not one-to-one related to the macroscopic chemical potential, but obeys a local kinetic equation associated with diffusion in the slow phase. The history-dependent macroscopic response admits a representation based on internal variables, enabling efficient time integration. We show that the local chemical kinetics can result in non-Fickian behaviour in macroscale boundary-value problems.Comment: 36 pages, 14 figure

Consistent incremental approximation of dissipation pseudo-potentials in the variational formulation of thermo-mechanical constitutive updates

International audienceIn this paper, we detail a consistent approximate expression for incremental dissipation pseudo-potentials which appear in the variational formulation of coupled thermo-mechanical boundary-value problems. We explain why the most intuitive expression does not work in the case of an explicit temperature dependence in the dissipation, and propose an alternative expression ensuring consistent results when reducing the time increment towards zero

Model-Free Data-Driven Methods in Mechanics: Material Data Identification and Solvers

This paper presents an integrated model-free data-driven approach to solid mechanics, allowing to perform numerical simulations on structures on the basis of measures of displacement fields on representative samples, without postulating a specific constitutive model. A material data identification procedure, allowing to infer strain-stress pairs from displacement fields and boundary conditions, is used to build a material database from a set of mutiaxial tests on a non-conventional sample. This database is in turn used by a data-driven solver, based on an algorithm minimizing the distance between manifolds of compatible and balanced mechanical states and the given database, to predict the response of structures of the same material, with arbitrary geometry and boundary conditions. Examples illustrate this modelling cycle and demonstrate how the data-driven identification method allows importance sampling of the material state space, yielding faster convergence of simulation results with increasing database size, when compared to synthetic material databases with regular sampling patterns.Comment: Revised versio

A Micromechanical Model of Hardening, Rate Sensitivity and Thermal Softening in BCC Single Crystals

The present paper is concerned with the development of a micromechanical model of the hardening, rate-sensitivity and thermal softening of bcc crystals. In formulating the model we specifically consider the following unit processes: double-kink formation and thermally activated motion of kinks; the close-range interactions between primary and forest dislocations, leading to the formation of jogs; the percolation motion of dislocations through a random array of forest dislocations introducing short-range obstacles of different strengths; dislocation multiplication due to breeding by double cross-slip; and dislocation pair annihilation. The model is found to capture salient features of the behavior of Ta crystals such as: the dependence of the initial yield point on temperature and strain rate; the presence of a marked stage I of easy glide, specially at low temperatures and high strain rates; the sharp onset of stage II hardening and its tendency to shift towards lower strains, and eventually disappear, as the temperature increases or the strain rate decreases; the parabolic stage II hardening at low strain rates or high temperatures; the stage II softening at high strain rates or low temperatures; the trend towards saturation at high strains; the temperature and strain-rate dependence of the saturation stress; and the orientation dependence of the hardening rate.Comment: 27 pages (LaTeX) and 15 Figures (jpg

A variational framework for nonlinear viscoelastic models in finite deformation regime

International audienceThis work presents a general framework for constitutive viscoelastic models in the finite deformation regime. The approach is qualified as variational since the constitutive updates consist of a minimization problem within each load increment. The set of internal variables is strain-based and uses a multiplicative decomposition of strain in elastic and viscous components. Spectral decomposition is explored in order to accommodate, into analytically tractable expressions, a wide set of specific models. Moreover, it is shown that, through appropriate choices of the constitutive potentials, the proposed formulation is able to reproduce results obtained elsewhere in the literature. Finally, numerical examples are included to illustrate the characteristics of the present formulation. MSC: 74C20; 74D10; 74S05; 35J5

Efficient data structures for model-free data-driven computational mechanics

The data-driven computing paradigm initially introduced by Kirchdoerfer & Ortiz (2016) enables finite element computations in solid mechanics to be performed directly from material data sets, without an explicit material model. From a computational effort point of view, the most challenging task is the projection of admissible states at material points onto their closest states in the material data set. In this study, we compare and develop several possible data structures for solving the nearest-neighbor problem. We show that approximate nearest-neighbor (ANN) algorithms can accelerate material data searches by several orders of magnitude relative to exact searching algorithms. The approximations are suggested by—and adapted to—the structure of the data-driven iterative solver and result in no significant loss of solution accuracy. We assess the performance of the ANN algorithm with respect to material data set size with the aid of a 3D elasticity test case. We show that computations on a single processor with up to one billion material data points are feasible within a few seconds execution time with a speed up of more than 10⁶ with respect to exact k-d trees

Computational non-smooth fracture dynamics in nonlinear and heterogeneous materials. Application to fracture of hydrided Zircaloy

This paper presents a new computational method and the associated software dedicated to the study of the dynamic fracture of nonlinear and heterogeneous materials. This method is based on the concept of Frictional Cohesive Zone Model in the framework of Non-Smooth Contact Dynamics. The associated numerical platform, composed of three object-oriented libraries, allows to simulate, in three dimensional finite deformations, the dynamic fracture of both multiphase and functionally graded materials from crack initiation to post-fracture behavior. The ability of this software, developed by the French ’Institut de Radioprotection et de Sûreté Nucléaire’ (IRSN) in the frame of its research program on nuclear fuel safety, is illustrated on the fracture of hydrided Zircaloy-4, constituting nuclear cladding tubes at high burnup, during a reactivity initiated accident (RIA). The macroscopic behavior of this heterogeneous material is deduced as an averaged energy release rate depending on the volume fraction of hydride phase