Crysal Plasticity Finite Element Simulations based on Continuum Dislocation Dynamics

Plastic deformation of crystalline materials is the result of the motion and interaction of dislocations. Continuum dislocation dynamics (CDD) defines flux-type evolution equations of dislocation variables which can capture the kinematics of moving curved dislocations. Coupled with Orowan's law, which connects the plastic shear rate to the dislocation flux, CDD defines a dislocation density based material law for crystal plasticity. In the current work we provide simulations of a micro-bending experiment of a single crystal and compare the results qualitatively to those from discrete dislocation simulations from the literature. We show that CDD reproduces salient features from discrete dislocation simulations regarding the stress distribution, the dislocation density and the accumulated plastic shear, which would be hard to obtain from more traditional crystal plasticity constitutive laws. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A crystal plasticity framework based on the Continuum Dislocation Dynamics theory: relaxation of an idealized micropillar

The motion and interaction of dislocation lines are the physical basis of the plastic deformation of metals. Although ‘discrete dislocation dynamic’ (DDD) simulations are able to predict the kinematics of dislocation microstructure (i.e. the motion of dislocations in a given velocity field) and therefore the plastic behavior of crystals in small length scales, the computational cost makes DDD less feasible for systems larger than a few micro meters. To overcome this problem, the Continuum Dislocation Dynamics (CDD) theory was developed. CDD describes the kinematics of dislocation microstructure based on statistical averages of internal properties of dislocation systems. In this paper we present a crystal plasticity framework based on the CDD theory. It consists of two separate parts: a classical 3D elastic boundary value problem and the evolution of dislocation microstructure within slip planes according to the CDD constitutional equations. We demonstrate the evolution of dislocation density in a micropillar with a single slip plane. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Phase Field Modeling of Laminate Deformation Microstructure in Finite Gradient Crystal Plasticity

The macroscopic mechanical behavior of many materials crucially depends on the formation and evolution of their microstructure. In this work, we consider the formation and evolution of laminate deformation microstructure in plasticity. Inspired by work on the variational modeling of phase transformation [5] and building on related work on multislip gradient crystal plasticity [9], we present a new finite strain model for the formation and evolution of laminate deformation microstructure in double slip gradient crystal plasticity. Basic ingredients of our model are a nonconvex hardening potential and two gradient terms accounting for geometrically necessary dislocations (GNDs) by use of the dislocation density tensor and regularizing the sharp interfaces between different kinematically coherent plastic slip states. The plastic evolution is described by means of a nonsmooth dissipation potential for which we propose a new regularization. We formulate a continuous gradient-extended rate-variational framework and discretize it in time to obtain an incremental-variational formulation. Discretization in space yields a finite element formulation which is used to demonstrate the capability of our model to predict the formation and evolution of laminate deformation microstructure in f.c.c. Copper with two active slip systems in the same slip plane. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Simulation of micro-cutting considering finite deformation crystal plasticity

Micro-machining processes on metalic microstructures are influenced by the crystal structure, i. e. the grain orientation. Furthermore, the chip formation underlies large deformations. To perform finite element simulations of micro-cutting processes, a large deformation material model is necessary in order to model the hyperelastic and finite plastic material behaviour. In the case of cp-titanium material with hcp-crystal structure the anisotropic behaviour must be considered by an appropriate set of slip planes and slip directions. In the present work the impact of the grain orientation on the plastic deformation is demonstrated by means of finite element simulations of a finite deformation single slip crystal plasticity model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rate-independent plasticity and statistically stored dislocations

We present here a continuum model for the evolution of the total dislocation density in the framework of rate-independent plasticity. Three basic physical features are taken into account: (i) the role of dislocation densities on hardening; (ii) the relations between the slip velocity and the mobility of gliding dislocations; (iii) the energetics of self and mutual interactions between dislocations. We restrict attention to plastic processes corresponding to single slip. Numerical simulations showing the formation of bands are also presented.     

Rate-independent plasticity and statistically stored dislocations

We present here a continuum model for the evolution of the total dislocation density in the framework of rate-independent plasticity. Three basic physical features are taken into account: (i) the role of dislocation densities on hardening; (ii) the relations between the slip velocity and the mobility of gliding dislocations; (iii) the energetics of self and mutual interactions between dislocations. We restrict attention to plastic processes corresponding to single slip. Numerical simulations showing the formation of bands are also presented.     

On the role of dislocation conservation in higher-order crystal plasticity

The interactions between individual dislocations contribute significantly to size effects as observed at the plastic deformation of miniaturized structures. When employing a crystal plasticity framework, these interactions are usually captured by an additional balance relation. In this paper we study two approaches to capture dislocation interactions in crystal plasticity that differ by the conservation of dislocations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dual-mixed finite element analysis of crystalline sub-micron gold

An extended crystal plasticity model is applied to crystalline sub-micron gold in order to study the mechanical response. Numerical results for different crystal sizes are presented and discussed. The governing equations are discretized and, subsequently, solved via a dual-mixed finite element formulation [1, 2]. The evolution equation of the dislocation density is taken as a global field relation additionally to the balance of linear momentum, whereas the flow rule is solved locally at the Gauß point level [3,4]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application and evaluation of hyper- and hypoelastic-based plasticity models in the FE simulation of metal forming processes

Metal forming processes are usually accompanied by large plastic strains and rotations of the material elements which emphasizes the need for reliable finite strain elastoplasticity models in corresponding FE simulations. In this work, two specific finite strain hyper- and hypoelastic-based plasticity models with combined nonlinear isotropic and kinematic hardening are presented and compared in numerical FE simulations. Although both models led to remarkably different results in a shear-dominated single element deformation test, the structural simulation of a standard deep drawing process delivered nearly congruent results which suggests that both models are equally well-suited for modeling metals in common forming processes. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of coupled deformation-induced twinning and dislocation slip using incremental energy minimization

The complex interplay between dislocations and deformation-induced twinning leads to a relatively poor formability of magnesium at room temperature. For understanding the complicated behavior of this metal, a novel model is presented. It is based on a variational principle. Within this principle based on energy minimization, dislocation slip is modeled by crystal plasticity theory, while the phase decomposition associated with twinning is considered by sequential laminates. The proposed model captures the transformation of the crystal lattice due to twinning in a continuous fashion by simultaneously taking dislocation slip within both, possibly co-existent, phases into account. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Three-dimensional Continuum Theory of Dislocation Plasticity - Modelling and Application to a Composite

The increasing demand for materials with well defined microstructure, which is accompanied by the advancing miniaturization of devices calls for physically motivated, dislocation-based continuum theories of plasticity. Only recently rigorous techniques have been developed for performing meaningful averages over systems of moving, curved dislocations, yielding evolution equations for a higher order dislocation density tensor. Our continuum dislocation theory allows for generalizing the planar system towards a three-dimensional system, where dislocations may have arbitrary orientation and curvature. With the inclusion of curvature, the theory naturally takes into account a deformation-induced increase in the overall dislocation density without having to invoke ad-hoc assumptions about dislocation sources. A numerical implementation and some benchmark tests of this continuum theory for dislocation dynamics has already been discussed in the literature. In this paper, we apply this continuum theory to composite materials, where we analyze a plastically deforming matrix with an elastic inclusion. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Creep of Single Crystals – Modelling and Numerical Aspects

A number of constitutive models, utilizing both microstructural and/or phenomenological considerations, have been developed for the simulation of the creep behaviour of nickel-base single crystal superalloys at elevated temperatures. In this work, emphasis is placed on the rate-dependent single crystal plasticity model [1]. A strategy for the identification of the material parameters of the model to fit the results from experiments has been implemented. The parameter fitting methodology rests upon a two-membered evolution strategy. In addition, a proposal is made for the extension of the Cailletaud model [1] by means of an evolution equation for a damage variable which enables the modelling of the tertiary creep stage. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identification of optimal material models and parameters in finite strain plasticity

The numerical simulation of the behaviour of a workpiece during manufacturing depends to a large extent on the quality of the applied material model. In this work, a method for the identification of constitutive models and material parameters in engineering applications is proposed. The presented method is used in the setting of optimal experimental design and is based on successive optimization of a set of finite strain plasticity models with kinematic and/or isotropic hardening. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pyramidal indentation into fcc single crystals and quasi-isotropic polycrystals: similar phenomena,different driving mechanisms

Pyramidal microindentation into the (001) surface of an fcc single crystal made of the Ni–base superalloy CMSX-4 has shown indent shapes which strongly depend on the azimuthal orientation of the pyramid. This observation is experimentally elucidated by digital surface models obtained from high resolution electron back–scatter diffraction (EBSD) technique along with digital image processing. Predictions of crystal plasticity finite element simulations agree with the experimental observations; the observed surface deformation patterns are due to pile–up formation, which is invariantly maximum in <110> directions thus being independent of the azimuthal orientation of the pyramid. The material pile–up is locally accommodated to the indenter faces leading to a convex, a concave contact rim at the faces of the indenter depending on the orientation. The result of maximum pile–up in <110> directions suggests that the driving mechanism for pile–up is purely crystallographic in that the influence of stress concentrations due to different indenter orientations and indenter shapes is negligible. The present findings for fcc single crystals are contrasted to well known observations for quasi–isotropic polycrystals. The different driving mechanisms resulting in phenomenologically similar material response are identified for both materials. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence in rate-independent infinitesimal strain gradient plasticity

We propose a model of infinitesimal strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. The model is a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the non-symmetric plastic distortion p. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied. Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion p are introduced: ṗ.τ = 0 on the microscopically hard boundary Γ_D ⊂ ∂Ω and [Curl p ].τ = 0 on the microscopically free boundary ∂Ω\Γ_D, where τ are the tangential vectors at the boundary ∂Ω. Moreover, a weak reformulation of the infinitesimal model allows for a global in-time solution of the corresponding rate-independent initial boundary value problem. The method of choice are a formulation as a quasi-variational inequality with symmetric and coercive bilinear form. Use is made of new Hilbert-space suitable for dislocation density dependent plasticity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Distribution of dislocation density and residual stresses in plastically deformed specimens: numerical studies

A two-scale approach to the simulation of mechanical properties of metallic materials is considered. On the macroscopic level, the material behavior is described by a phenomenological model of finite strain viscoplasticity with nonlinear kinematic hardening. In particular, the process-induced plastic anisotropy is captured by backstresses. On the microstructural level, the so called “load path sensitive two-population dislocation cell model” is implemented. It describes an evolving dislocation cell structure with dislocation populations for dislocation cell walls and the cell interior. Owing to the coupling with the phenomenological plasticity model, it can describe the evolution of the dislocation densities depending on the load path. The applicability of the multiscale approach to the FEM simulation of severe plastic deformation processes such as Equal Channel Angular Pressing is demonstrated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

From Dislocation Motion to an Additive Velocity Gradient Decomposition, and Some Simple Models of Dislocation Dynamics

A mathematical theory of time-dependent dislocation mechanics of unrestricted geometric and material nonlinearity is reviewed. Within a ``small deformation" setting, a suite of simplified and interesting models consisting of a nonlocal Ginzburg Landau equation, a nonlocal level set equation, and a nonlocal generalized Burgers equation is derived. In the finite deformation setting, it is shown that an additive decomposition of the total velocity gradient into elastic and plastic parts emerges naturally from a micromechanical starting point that involves no notion of plastic deformation but only the elastic distortion, material velocity, dislocation density and the dislocation velocity. Moreover, a plastic spin tensor emerges naturally as well.