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)     

On the variationally consistent modeling of material failure

Material failure associated with cracks or shear bands is frequently analyzed by utilizing so-called cohesive models. Such models are based on traction-separation laws. Within such approaches, the stress vector of the considered crack or shear band is related to its conjugate variable being the respective displacement jump (such as the material separation or the crack opening). In the present work, a framework suitable for the analysis of shear bands is discussed. All models belonging to that framework are consistently derived from thermodynamical principles. Hence, the second law of thermodynamics is automatically fulfilled. Furthermore, a variational principle strongly relying on the postulate of maximum dissipation is elaborated leading finally to a variationally consistent implementation. More precisely, all state variables, together with the unknown deformation mapping, follow naturally from minimizing an incrementally defined potential within the presented algorithmic formulation. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Towards variational constitutive updates for finite strain plasticity models based on non-associative evolution equations

In this contribution, first steps towards variational constitutive updates for finite strain plasticity theory based on non-associative evolution equations are presented. These schemes allow to compute the unknown state variables such as the plastic part of the deformation gradient, together with the deformation mapping, by means of a fully variational minimization principle. Therefore, standard optimization algorithms can be applied to the numerical implementation leading to a very robust and efficient numerical implementation. Particularly, for highly non-linear, singular or nearly ill-posed physical models like that corresponding to crystal plasticity showing a large number of possible active slip planes, this is a significant advantage compared to standard constitutive updates such as the by now classical return-mapping algorithm. While variational constitutive updates have been successfully derived for associative plasticity models, their extension to more complex constitutive laws, particularly to those featuring non-associative evolution equations, is highly challenging. In the present contribution, a certain class of non-associative finite strain plasticity models is discussed and recast into a variationally consistent format. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Phase-Field Model of Ductile Fracture at Finite Strains

This work outlines a variational-based framework for the phase field modeling of ductile fracture in elastic-plastic solids at large strains. The phase field approach regularizes sharp crack discontinuities within a pure continuum setting by a specific gradient damage model with geometric features rooted in fracture mechanics. Based on the recent works [1, 2], the phase field model of ductile fracture is linked to a formulation of gradient plasticity at finite strains in order to ensure the crack to evolve inside the plastic zones. The thermodynamic formulation is based on the definition of a constitutive work density function including the stored elastic energy and the dissipated work due to plasticity and fracture. The proposed canonical theory is shown to be governed by a rate-type minimization principle, which determines the coupled multi-field evolution problem. Another aspect is the regularization towards a micromorphic gradient plasticity-damage setting which enhances the robustness of the finite element formulation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterization of energy minimizers in micromagnetics

We present a new characterization of minimizing sequences and possible minimizers (all called the minimizing magnetizations) for a nonlocal micromagnetic-like energy (without the exchange energy). Our method is to replace the nonlocal energy functional and its relaxation with certain local integral functionals on divergence-free fields obtained by a two-step minimization of some auxiliary augmented functionals. Through this procedure, the minimization problem becomes equivalent to the minimization of a new local variational functional, called the dual variational functional, which has a unique minimizer. We then precisely characterize the minimizing magnetizations of original nonlocal functionals in terms of the unique minimizer of the dual variational functional. Finally, we give some remarks and ideas on solving the dual minimization problem.     

Modeling and numerical simulation of multiscale behavior in polycrystals via extended crystal plasticity

The purpose of this work is to exploit the algorithmic formulation of models for multiscale inelastic materials whose behavior is influenced by the evolution of inelastic microstructure and the corresponding material or internal lengthscales. The models for extended crystal plasticity are based on the formulation of rate potentials whose form is determined by (i) energetic processes via the free energy, (ii) kinetic processes via the dissipation potential, and (iii) the form of the evolution relations for the internal-variable-like quantities upon which the free energy and dissipation potential depend. Examples for these latter quantities are the inelastic local deformation or dislocation densities as GNDs. Different algorithmic implementations are discussed, namely the algorithmic variational approach and the dual mixed approach. (© 2011 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)     

Minimization- and Saddle-Point-Based Modeling of Diffusion-Deformation-Processes in Hydrogels

Hydrogels have gained importance during the last years due to their wide range of synthetically fabricable elastic properties as well their increasing meaning in biomedical applications. Future exploitation of the vast prospects of hydrogels is however only feasible by establishing reliable material models that precisely capture their behavior in different environments. To this end, we propose a consistent variational framework for deformation-diffusion processes, offering a canonically compact approach to the chemo-mechanical coupling of hydrogels via a saddle-point as well as a new minimization formulation. The work depicts the construction of rate-type potentials for the chemo-mechanical evolution problem and their transformation into time-discrete incremental potentials. In terms of spatial discretization, the finite element method is employed, benefiting from the intrinsic symmetric structure of the variational foundation. While the saddle-point formulation yields the well-known LBB condition as a constraint for finite element interpolations, on the part of its minimizing counterpart H(Div, ℬ︁)-conforming elements have to be chosen. We illustrate appropriate solutions to both challenges, using mixed Taylor-Hood for the saddle-point and Raviart-Thomas elements for the minimization formulation and discuss advantages of the new approach. (© 2016 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.     

Dynamics of discrete screw dislocations via discrete gradient flow

We present variational approaches (developed in [3,4,11]) to the study of statics and dynamics of screw dislocations in crystals. We model the crystal as a cubic lattice and we give the asymptotic Γ-convergence expansion of the elastic energy induced by a finite family of screw dislocations as the lattice spacing goes to zero. We show that the effective energy associated to the presence of a finite system of screw dislocations coincides with the renormalized energy, studied within the Ginzburg-Landau framework and ruling the interactions between the dislocations. As a byproduct of this analysis, we show the existence of many metastable configurations of dislocations pinned by energy barries. Using the minimizing movement approach á la De Giorgi, we introduce a discrete-in-time variational dynamics, referred to as Discrete Gradient Flow, which allows to overcome these energy barriers. More precisely, we show that lettting first the lattice spacing and then the time step of minimizing movements tend to zero, dislocations move accordingly with the gradient flow of the renormalized energy. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite strain inelasticity for isotropy,a simple and efficient finite element formulation

We consider a formulation of associative isotropic J₂‐elastoplasticity at finite inelastic strains and aspects of its numerical implementation. The essential ingredients include the multiplicative decomposition of the deformation gradient in elastic and inelastic parts, the definition of a convex elastic domain in stress space and a material representation of the constitutive equations for general non‐Cartesian coordinate charts. On the numerical side we propose a stress update algorithm for elasto‐plastic response, including isotropic hardening. The finite element formulation is based on assumed strain and enhanced strain variational principles, for a complete outline see [3]. Remarkably the formulation is very similar to the case of infinitesimal plasticity: (i) The scheme of linear return mapping algorithm takes the form of standard return mapping of the infinitesimal theory for the case of isotropic elastic response. (ii) The algorithmic elastoplastic moduli have a similar structure as in the linear case. Together with an exact fulfillment of plastic incompressibility by means of a simple correction one achieves an advantageously efficient finite element formulation. Its performance is documented by a numerical example.     

Thermodynamically consistent modeling of yield surface distortion

From a macroscopic point of view, the deformation of most metals results in an evolution of the symmetry groups characterizing the isotropy of the considered materials. With respect to plastic deformation for instance, the shape of the macroscopic yield surface evolves during deformation. In the present paper, a novel constitutive framework capturing this evolution is proposed. This framework is based on the fundamentals of thermodynamics. Furthermore, it also shows a variational structure such that all state variables follow jointly from minimizing an incrementally defined energy functional. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Surface energy and microstructure in coherent phase transitions

We study a variational problem involving a nonconvex function of Δu, regularized by a higher order term. The motivation comes from the theory of martensitic phase transformation—specifically, a model for the fine scale structure of twinning near an austenite-twinned-martensite interface. It is widely believed that the fine scale structure can be understood variationally, through the minimization of elastic and surface energy. Our problem represents the essence of this minimization. Similar variational problems have been considered by many authors in the materials science literature. They have always assumed, however, that the twinning should be essentially one-dimensional. This is in general false. Energy minimization can require a complex pattern of twin branching near the austenite interface. There are indications that the states of minimum energy may be asymptotically self-similar. © 1994 John Wiley & Sons., Inc.     

Expected Residual Minimization Formulation for a Class of Stochastic Vector Variational Inequalities

This paper considers a class of vector variational inequalities. First, we present an equivalent formulation, which is a scalar variational inequality, for the deterministic vector variational inequality. Then we concentrate on the stochastic circumstance. By noting that the stochastic vector variational inequality may not have a solution feasible for all realizations of the random variable in general, for tractability, we employ the expected residual minimization approach, which aims at minimizing the expected residual of the so-called regularized gap function. We investigate the properties of the expected residual minimization problem, and furthermore, we propose a sample average approximation method for solving the expected residual minimization problem. Comprehensive convergence analysis for the approximation approach is established as well.     

The structure of the Euler-Lagrange mapping

The purpose of this paper is to review properties of the Euler-Lagrange mapping in the higher order variational theory on fibred manifolds. We present basic theorems on the kernel of the Euler-Lagrange mapping, describing variationally trivial Lagrangians, and its image, characterizing variational source forms. We discuss invariance properties of Lagrangians and Euler-Lagrange forms, and the Noether’s theory. The text was submitted by the author in English.     

On the effect of different parameterizations of the flow rule on the efficiency of variational constitutive updates

The numerical efficiency of so-called variational constitutive updates for finite strain plasticity theory is analyzed. These updates compute the unknowns such as the plastic strains by minimizing an appropriate functional. Within the present paper, different parameterizations of the flow rule are utilized within the variational constitutive update scheme. It is shown that comparing to the return-mapping algorithm, the variational updates require significantly less iteration steps and thus, is numerically highly efficient. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Configurational-Force-Driven Crack Propagation and Adaptive Mesh Refinement in Finite Inelasticity

We introduce a consistent variational framework for inelasticity at finite strains, yielding dual balances in physical and material space as the Euler equations. The formulation is employed for the simultaneous usage of configurational forces as both driving forces for crack propagation as well as h-adaptive mesh refinement. The theoretical basis builds upon a global balance of internal and external power, where the mechanical response is exclusively governed by two scalar functions, the free energy function and a dissipation potential. The resulting variational structure is exploited in the context of fracture mechanics and yields evolution equations for internal variables. In the discrete setting, we present a geometry model fully separated from the finite element mesh structure that represents structural changes of the material configuration due to crack propagation. Advanced meshing algorithms provide an optimal discretization at the crack tip. Local and global criteria are obtained via error estimators based on configurational forces being interpreted as indicators of an energetic misfit due to an insufficient discretization. The numerical handling is decomposed into a staggered algorithm scheme for the dual set of equilibrium equations in material and physical space and efficient mesh generation tools. Exemplary numerical examples are considered to illustrate the method and to underline the effects of inelastic material behaviour in the presented context. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A variational framework for nonlinear viscoelastic models in finite deformation regime

This 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.     

A New Minimum Principle for Lagrangian Mechanics

We present a novel variational view at Lagrangian mechanics based on the minimization of weighted inertia-energy functionals on trajectories. In particular, we introduce a family of parameter-dependent global-in-time minimization problems whose respective minimizers converge to solutions of the system of Lagrange’s equations. The interest in this approach is that of reformulating Lagrangian dynamics as a (class of) minimization problem(s) plus a limiting procedure. The theory may be extended in order to include dissipative effects thus providing a unified framework for both dissipative and nondissipative situations. In particular, it allows for a rigorous connection between these two regimes by means of Γ-convergence. Moreover, the variational principle may serve as a selection criterion in case of nonuniqueness of solutions. Finally, this variational approach can be localized on a finite time-horizon resulting in some sharper convergence statements and can be combined with time-discretization.