A coupled phase-field model for ductile fracture in crystal plasticity

The objective of this work is to present a simplified, nonetheless representative first stage of a phenomenological model to predict the crack evolution of ductile fracture in single crystals. The proposed numerical approach is carried out by merging a conventional well- stablished elasto-plastic crystal plasticity model and a well-known phase-field model (PFM) modified to predict ductile fracture. A two-dymensional initial boundary-value problem of ductile fracture is introduced considering a single crystal Nickel-base superalloy material. the model is implemented into the finite element context subjected to a one-dimensional tension test (displacement-controlled). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

3D finite elements to model electromechanical coupled solids at failure

This work presents an extension of newly developed 3D finite elements to model failure in purely mechanical based materials [1, 2] to electromechanical coupled materials. Following the approach suggested in [6] for the plane setting, new finite elements are developed, which in addition to resolving strong discontinuities in the mechanical displacement field are capable of modeling strong discontinuities in 3D also in the electric potential. An application to a 3D off-centered three point bending test under combined mechanical and electrical loading outlines the performance of the newly developed finite elements. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Aspects of the theoretical and numerical formulation of single crystal plasticity at finite strains

We give an overview about an incremental variational formulation of single crystal plasticity at finite strains and discuss an approach to the computation of the exponential map and its derivatives based on a spectral representation.     

A gradient model for finite strain elastoplasticity coupled with damage

This paper describes the formulation of an implicit gradient damage model for finite strain elastoplasticity problems including strain softening. The strain softening behavior is modeled through a variant of Lemaitre's damage evolution law. The resulting constitutive equations are intimately coupled with the finite element formulation, in contrast with standard local material models. A 3D finite element including enhanced strains is used with this material model and coupling peculiarities are fully described. The proposed formulation results in an element which possesses spatial position variables, nonlocal damage variables and also enhanced strain variables. Emphasis is put on the exact consistent linearization of the arising discretized equations.

A numerical set of examples comparing the results of local and the gradient formulations relative to the mesh size influence is presented and some examples comparing results from other authors are also presented, illustrating the capabilities of the present proposal.     

On the numerical relaxation of single-slip plasticity in finite strains

The modeling of the elastoplastic behaviour of single crystals with infinite latent hardening leads to a nonconvex energy density, whose minimization produces fine structures. The computation of the quasiconvex envelope of the energy density is faced in this case with huge numerical difficulties caused by the clusters of local minima. By exploiting the structure of the problem, we present a fast and efficient numerical relaxation algorithm as alternative to global optimization techniques usually adopted in literature which are computationally more expensive. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient energy-stable schemes for the hydrodynamics coupled phase-field model

In this article, several efficient and energy-stable semi–implicit schemes are presented for the Cahn–Hilliard phase-field model of two-phase incompressible flows. A scalar auxiliary variable (SAV) approach is implemented to solve the Cahn–Hilliard equation, while a splitting method based on pressure stabilization is used to solve the Navier–Stokes equation. At each time step, the schemes involve solving only a sequence of linear elliptic equations, and computations of the phase-field variable, velocity, and pressure are totally decoupled. A finite-difference method on staggered grids is adopted to spatially discretize the proposed time-marching schemes. We rigorously prove the unconditional energy stability for the semi-implicit schemes and the fully discrete scheme. Numerical results in both two and three dimensions are obtained, which demonstrate the accuracy and effectiveness of the proposed schemes. Using our numerical schemes, we compare the SAV, invariant energy quadratization (IEQ), and stabilization approaches. Bubble rising dynamics and coarsening dynamics are also investigated in detail. The results demonstrate that the SAV approach is more accurate than the IEQ approach and that the stabilization approach is the least accurate among the three approaches. The energy stability of the SAV approach appears to be better than that of the other approaches at large time steps.     

Adaptive finite element procedures for elastoplastic problems at finite strains

A major difficulty in the context of adaptive analysis of geometrically nonlinear problems is to provide a robust remeshing procedure that accounts both for the error caused by the spatial discretization and for the error due to the time discretization. For stability problems, such as strain localization and necking, it is essential to provide a step–size control in order to get a robust algorithm for the solution of the boundary value problem. For this purpose we developed an easy to implement step–size control algorithm. In addition we will consider possible a posteriori error indicators for the spatial error distribution of elastoplastic problems at finite strains. This indicator is adopted for a density–function–based adaptive remeshing procedure. Both error indicators are combined for the adaptive analysis in time and space. The performance of the proposed method is documented by means of representative numerical examples.     

On the modeling of thermo-electro-magneto-mechanical solids at finite strains

A short overview of the mechanics of electromagnetic continua at finite strains is provided in this paper. In this framework objective thermodynamically-consistent constitutive relations are formulated for materials that exhibit thermo-electro-magnetomechanical coupling. In particular a model for an isotropic hyperelastic magnetic solid is considered, which finds application for magnetorheological elastomers (MRE). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A polyconvex strain-energy split for a high-order phase-field approach to fracture

This contribution focuses on a novel phase-field model for a high-order phase-field approach to brittle fracture in the range finite deformation. In particular, two different challenges are tackled in this study: First, we want to establish a polyconvex free energy density to ensure the existence of a minimizer for the coupled problem, second, we have to deal with a fourth-order Cahn-Hilliard type equation for the approximation of the phase-field. Phase-field methods employ a variational framework for brittle fracture and have proven to predict complex fracture patterns in two and three dimensional examples. Basis of the model are the conjugate stresses of the three strain measures deformation gradient (line map), its cofactor (area map) and its determinant (volume map). The introduction of the tensor cross product simplifies the presentation of the first Piola-Kirchhoff stress tensor and its derivatives in elegant manner. The proposed Cahn-Hilliard type equation requires global -continuity. Therefore, we apply an isogeometric framework using NURBS basis functions. Moreover, a general hierarchical refinement scheme based on subdivision projection is used here for one, two and three dimensional simulations. This technique allows to enhance the approximation space using finer splines on each level but preserves the partition of unity as well as the continuity properties of the original discretization. We finally demonstrate the accuracy and the robustness with a series of benchmark problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New three-dimensional finite elements with embedded strong discontinuities to model solids at failure

This work focuses on the development of new finite elements which can capture strong discontinuities in three-dimensional failure problems. The displacement jumps in the solid are approximated by a linear interpolation obtained by enforcing a new class of enhanced separation modes to exactly be satisfied by the formulation. Efforts are also put towards the development of a proper crack propagation tracking algorithm needed for the complicated crack surfaces appearing in realistic 3D failure simulations, based on a combination of the global tracking algorithm and the marching cubes algorithm. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical approaches to thermally coupled perfect plasticity

The partial differential equations describing viscoelastic solids in Kelvin–Voigt rheology at small strains exhibiting also stress‐driven Prandtl‐Reuss perfect plasticity are considered and are coupled with a heat‐transfer equation through the dissipative heat produced by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Numerical discretization of the resulting thermodynamically consistent model is proposed by implicit time discretization, suitable regularization, and finite elements in space. Numerical stability is shown and computational simulations are reported to illustrate the practical performance of the method. In a quasistatic case, convergence is proved by careful successive limit passage. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Asymptotic behaviour of a phase-field model with three coupled equations without uniqueness

We prove the existence of weak solutions for a phase-field model with three coupled equations with unknown uniqueness, and state several dynamical systems depending on the regularity of the initial data. Then, the existence of families of global attractors (level-set depending) for the corresponding multi-valued semiflows is established, applying an energy method. Finally, using the regularizing effect of the problem, we prove that these attractors are in fact the same.     

Minimizing finite sums with the stochastic average gradient

We analyze the stochastic average gradient (SAG) method for optimizing the sum of a finite number of smooth convex functions. Like stochastic gradient (SG) methods, the SAG method’s iteration cost is independent of the number of terms in the sum. However, by incorporating a memory of previous gradient values the SAG method achieves a faster convergence rate than black-box SG methods. The convergence rate is improved from \(O(1/\sqrt{k})\) to O(1 / k) in general, and when the sum is strongly-convex the convergence rate is improved from the sub-linear O(1 / k) to a linear convergence rate of the form \(O(\rho ^k)\) for \(\rho < 1\). Further, in many cases the convergence rate of the new method is also faster than black-box deterministic gradient methods, in terms of the number of gradient evaluations. This extends our earlier work Le Roux et al. (Adv Neural Inf Process Syst, 2012), which only lead to a faster rate for well-conditioned strongly-convex problems. Numerical experiments indicate that the new algorithm often dramatically outperforms existing SG and deterministic gradient methods, and that the performance may be further improved through the use of non-uniform sampling strategies.     

Cyclic martensitic phase transformation of monocrystals at finite strains

Based on Bain's principle, a C¹-continuous thermo-mechanical micro-macro constitutive model for martensitic phase transformation (PT) of monocrystals at finite strains and hyperelastic free energy function is used. It is represented by a unified non-convex Lagrangian variational functional. The convexification problem is solved here by generalizing the explicit form of the lower Reuss bound for small strains given in [1] to finite strains. Abaqus is used for implementation of 3D finite elements in space, via UMAT-interface which requires Jaumann rate of Kirchhoff stress tensor. Deterministic validation of the model is presented by comparing verified numerical results with experimental data for Cu₈₂Al₁₄Ni₄ [6] for quasiplastic PT. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fully variational strong discontinuity approach at finite strains

A novel, fully variational three-dimensional finite element formulation for the modeling of locally embedded strong discontinuities at finite strains is presented. The proposed numerical model is based on the Enhanced Assumed Strain concept with an additive decomposition of the displacement gradient into a conforming and an enhanced part. The discontinuous component of the displacement field which is associated with the failure in the modeled structure is isolated in the enhanced part of the deformation gradient. In contrast to previous works, a variational constitutive update is used. The internal variables are determined by minimizing a pseudo-elastic potential. The advantages of such a formulation are well known, e.g. the tangent stiffness matrix is symmetric, standard optimization algorithms can be applied and it represents a natural basis for error estimation and mesh adaption. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dual spaces of stresses and strains,with applications to Hencky plasticity

Applied Mathematics & Optimization - We prove the existence of a dual pairing between admissible stress and displacement fields in the context of Hencky plasticity. We apply this to show (i)...     

A variational arbitrary Lagrangian-Eulerian finite element formulation for standard dissipative solids at finite strains

A novel Arbitrary Lagrangian-Eulerian (ALE) finite element formulation for standard dissipative media at finite strains is presented. In contrast to previously published ALE approaches accounting for dissipative phenomena, the proposed scheme is fully variational. Consequently, no error estimates are necessary and thus, linearity of the problem and the corresponding Hilbert-space are not required. Hence, the resulting Variational Arbitrary Lagrangian-Eulerian (VALE) finite element method can be applied to highly nonlinear phenomena as well. In case of standard dissipative solids, so-called variational constitutive updates provide a variational principle. Based on these updates, the deformation mapping follows from minimizing an incrementally defined (pseudo) potential, i.e., energy minimization is the overriding criterion that governs every aspect of the system. Therefore, it is natural to allow the variational principle to drive mesh adaption as well. Thus, in the present paper, the discretizations of the deformed as well as the undeformed configuration are optimized jointly by minimizing the respective incremental energy of the considered mechanical system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)