Generalized Interface Models With Damage in Gradient Plasticity

Gradient plasticity can account for experimentally observed size effects. Here, a previously developed gradient plasticity model is extended to account for interface delamination processes. The crystal plasticity model is based on the gradient of an equivalent plastic strain measure. A modification of the related boundary conditions allows for the formulation of a generalized cohezive zone model which can take into account the effect of interface delamination on the gradient plasticity solution. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micromechanical Simulation of the Hall-Petch Effect with a Crystal Gradient Theory including a Grain Boundary Yield Criterion

A viscoplastic strain gradient crystal plasticity theory based on the gradient of the equivalent plastic strain ∇γ_eq is proposed. A grain boundary yield condition is introduced. The microstructural explanation of the Hall-Petch effect, accounting for notch-like stress concentrations at the grain boundary as a result of discrete slip bands, is reviewed. Periodic tensile test FEM simulation results illustrate the prediction of the numerical model. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Microscale mechanics for metal thin film delamination along ceramic substrates

The metal thin film delamination along metal/ceramic interface in the case of large scale yielding is studied by employing the strain gradient plasticity theory and the material microscale effects are considered. Two different fracture process models are used in this study to describe the nonlinear delamination phenomena for metal thin films. A set of experiments have been done on the mechanism of copper films delaminating from silica substrates, based on which the peak interface separation stress and the micro-length scale of material, as well as the dislocation-free zone size are predicted.     

Strain gradient plasticity solution for an internally pressurized thick-walled cylinder of an elastic linear-hardening material

An elastic-plastic solution is presented for an internally pressurized thick-walled plane strain cylinder of an elastic linear-hardening plastic material. The solution is derived in a closed form using a strain gradient plasticity theory. The inner radius of the cylinder enters the solution not only in non-dimensional forms but also with its own dimensional identity, which differs from that in classical plasticity based solutions and makes it possible to capture the size effect at the micron scale. The classical plasticity solution of the same problem is recovered as a special case of the current solution. To further illustrate the newly derived solution, formulas and numerical results for the plastic limit pressure are provided. These results reveal that the load-carrying capacity of the cylinder increases with decreasing inner radius at the micron scale. It is also seen that the macroscopic behavior of the pressurized cylinder can be well described by using classical plasticity based solutions.     

Strain gradient plasticity solution for an internally pressurized thick-walled cylinder of an elastic linear-hardening material

An elastic-plastic solution is presented for an internally pressurized thick-walled plane strain cylinder of an elastic linear-hardening plastic material. The solution is derived in a closed form using a strain gradient plasticity theory. The inner radius of the cylinder enters the solution not only in non-dimensional forms but also with its own dimensional identity, which differs from that in classical plasticity based solutions and makes it possible to capture the size effect at the micron scale. The classical plasticity solution of the same problem is recovered as a special case of the current solution. To further illustrate the newly derived solution, formulas and numerical results for the plastic limit pressure are provided. These results reveal that the load-carrying capacity of the cylinder increases with decreasing inner radius at the micron scale. It is also seen that the macroscopic behavior of the pressurized cylinder can be well described by using classical plasticity based solutions.     

Spherical indentation into a fcc single crystal – the squaring of the circle?

Spherical microindentation into the (100) surface of a fcc single crystal results in a remaining indent shape which looks rather like a square than like a circle. This contribution aims to elucidate this curious phenomenon observed in a recently performed experiment within SFB 298 at TU Darmstadt. The applied methodology is threefold: (i) On the modelling side, a phenomenological orthotropic finite-strain plasticity model is proposed and applied within finite-element simulations. (ii) On the experimental side, the topography of the indentation crater is reconstructed by means of scanning-electron microscopy along with image-processing. (iii) Experimental observations are explained by plastic slip mechanisms. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of Strain Gradient Plasticity to Micro-torsion Experiments

The applicability of a strain gradient crystal plasticity model to size effects observed on microwires in torsion experiments is discussed. Finite Element simulations of simplified cylindrical grain aggregations are presented and the resulting overall mechanical response is compared to experimental data for gold from the literature. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于应变梯度理论的粘塑性厚壁圆筒和球壳极限内压分析

基于应变梯度塑性理论,分析了内压作用下厚壁圆筒和球壳的塑性极限荷载．结果表明:圆筒内径在微米量级时,存在尺度效应现象,内径减小,其尺度效应增强;变形越大,影响越大;应变速率敏感指数越大,尺度效应越明显．经典塑性理论结果是当前解的特例．     

Incompatibility Induced Nonlocal Crystal Plasticity

Motivated by the fact that locally inhomogeneous elastic or plastic deformations may result in incompatibilities of the fictitious intermediate configuration a strain gradient crystal plasticity model is developed. Thereby incompatibilities can be accounted for and scale dependent material behavior, as also observed experimentally, is predictable. A nonlocal extension of existing local formulations is proposed which does not require additional boundary conditions and thus maintains the classical BVP structure. On the numerical side key developments are an extended FE-formulation for rate-(in)dependent strain gradient plasticity and a local FE-formulation which bases the gradient computation on an operator split combined with a smoothing algorithm. Comparative numerical studies for classical examples proove the superior efficiency of the second approach. (© 2005 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)     

Mixed Variational Principles and Robust Finite Element Design of Gradient Plasticity at Finite Strains

The modeling of size effects in elastic-plastic solids, such as the width of shear bands or the grain size dependence in polycrystals, must be based on non-standard theories which incorporate length-scales. This is achieved by models of strain gradient plasticity, incorporating spatial gradients of selected micro-structural fields which describe the evolving dissipative mechanisms. The key aspect of this work is to provide a rigorous incremental variational formulation and mixed finite element design of additive finite gradient plasticity in the logarithmic strain space. We start from a mixed saddle point principle for metric-type plasticity, which is specified for the important model problem of isochoric plasticity with gradient-extended hardening/softening response. To this end, we propose a novel finite element design of the coupled problem incorporating a local-global solution strategy of short- and long-range fields. This includes several new aspects, such as extended Q1P0-type and MINI-type finite elements for gradient plasticity [4]. (© 2013 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 Hardening Description based on a Finite-Deformation Gradient Crystal Plasticity Model: Formulation and Numerical Implementation

In this study, a well-defined finite-deformation gradient crystal plasticity is employed to study size-dependence strengthening behavior of a single crystal under simple shear loading. The constitutive model is implemented in the FEM software ABAQUS via a user-defined element subroutine (UEL). Effect of gradient strengthening, latent hardening and scale-variation in mechanically plastic response of a single crystal subjected to isothermal quasi-static loading is studied and discussed. (© 2015 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)     

A critical review of the state of finite plasticity

The object of this paper is to provide a critical review of the current state of plasticity in the presence of finite deformation. In view of the controversy regarding a number of fundamental issues between several existing schools of plasticity, the areas of agreement are described separately from those of disagreement. Attention is mainly focussed on the purely mechanical, rate-independent, theory of elastic-plastic materials, although closely related topics such as rate-dependent behavior, thermal effects, experimental and computational aspects, microstructural effects and crystal plasticity are also discussed and potentially fruitful directions are identified.A substantial portion of this review is devoted to the area of disagreement that covers a detailed presentation of argument(s), bothpro andcon, for all of the basic constitutive ingredients of the rate-independent theory such as the primitive notion or definition of plastic strain, the structure of the constitutive equation for the stress response, the yield function, the loading criteria and the flow and the hardening rules. The majority of current research in finite plasticity theory, as with its infinitesimal counterpart, still utilizes a (classical) stress-based approach which inherently possesses some shortcomings for the characterization of elastic-plastic materials. These and other anomalous behavior of a stress-based formulation are contrasted with the more recent strain-based formulation of finite plasticity. A number of important features and theoretical advantages of the latter formulation, along with its computational potential and experimental interpretation, are discussed separately.     

Modeling of microstructural pattern formation in crystal plasticity

The mechanical behavior of most materials is dictated by a present or emergent underlying microstructure which is a direct result of different, even competing physical mechanisms occurring at lower length scales. In this work, energetic microstructure interaction via different non-convex contributions to the free energy in metals is modeled. For this purpose rate dependent gradient extended crystal plasticity models at the glide-system level are formulated. The non-convex energy serves as the driving force for the emergent microstructure. The competition between the kinetics and the relaxation of the free energy is an essential feature of the model. Non-convexity naturally arises in finite-deformation single-slip crystal plasticity and the results of the gradient model for this case are compared with an effective laminate model based on energy relaxation. Similarities as well as essential differences are observed and explained. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comparison of strain gradient theories with applications to the functionally graded circular micro-plate

The strain gradient theory of Zhou et al. is re-expressed in a more direct form and the differences with other strain gradient theories are investigated by an application on static and dynamic analyses of FGM circular micro-plate. To facilitate the modeling, the strain gradient theory of Zhou et al. is re-expressed in cylindrical coordinates, and then the governing equation, boundary conditions and initial condition for circular plate are derived with the help of the Hamilton's principle. The present model can degenerate into other models based on the strain gradient theory of Lam et al., the couple stress theory, the modified couple stress theory or even the classical theory, respectively. The static bending and free vibration problems of a simply supported circular plate are solved. The results indicate that the consideration of strain gradients results in an increase in plate stiffness, and leads to a reduction of deflection and an increase in natural frequency. Compared with the reduced models, the present model can predict a stronger size effect since the contribution from all strain gradient components is considered, and the differences of results from all these models are diminishing when the plate thickness is far greater than the material length-sale parameter.     

Application of nonlinear continuum dislocation theory to antiplane shear

We apply the nonlinear dislocation theory to the problem of antiplane constrained shear in a single crystal with one slip system. By taking dissipation into account, the relaxed energy functional has to be minimized. We show that, up to a threshold strain, no dislocations are nucleated and therefore the plastic slip is zero. Since this threshold value depends on the width of the specimen, a size effect takes place. The stress strain curve turns out to be a hysteresis loop exhibiting the work hardening due to the dislocation pile-up. It is shown that the Bauschinger effect holds true. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)