Finite element analysis of geometrically necessary dislocations in crystal plasticity

We present a finite element method for the analysis of ductile crystals whose energy depends on the density of geometrically necessary dislocations (GNDs). We specifically focus on models in which the energy of the GNDs is assumed to be proportional to the total variation of the slip strains. In particular, the GND energy is homogeneous of degree one in the slip strains. Such models indeed arise from rigorous multiscale analysis as the macroscopic limit of discrete dislocation models or from phenomenological considerations such as a line‐tension approximation for the dislocation self‐energy. The incorporation of internal variable gradients into the free energy of the system renders the constitutive model non‐local. We show that an equivalent free‐energy functional, which does not depend on internal variable gradients, can be obtained by exploiting the variational definition of the total variation. The reformulation of the free energy comes at the expense of auxiliary variational problems, which can be efficiently solved using finite element approximations. The addition of surface terms in the formulation of the free energy results in additional boundary conditions for the internal variables. The proposed framework is verified by way of numerical convergence tests, and simulations of three‐dimensional problems are presented to showcase its applicability. A performance analysis shows that the proposed framework solves strain‐gradient plasticity problems in computing times of the order of local plasticity simulations, making it a promising tool for non‐local crystal plasticity three‐dimensional large‐scale simulations. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite element analysis and algorithms for single‐crystal strain‐gradient plasticity

We provide optimal a priori estimates for finite element approximations of a model of rate‐independent single‐crystal strain‐gradient plasticity. The weak formulation of the problem takes the form of a variational inequality in which the primary unknowns are the displacement and slips on the prescribed slip systems, as well as the back‐stress associated with the vectorial microstress. It is shown that the return mapping algorithm for local plasticity can be applied element‐wise to this non‐local setting. Some numerical examples illustrate characteristic features of the non‐local model. Copyright © 2012 John Wiley & Sons, Ltd.     

A crystal plasticity based methodology for fatigue crack initiation life prediction in polycrystalline copper

Fatigue failure is the dominant mechanism that governs the failure of components and structures in many engineering applications. In conventional engineering applications due to the design specifications, a significant proportion of the fatigue life is spent in the crack initiation phase. In spite of the large number of works addressing fatigue life modelling, the problem of modelling crack initiation life still remains a major challenge in the scientific and engineering community. In the present work, we present a methodology for estimating fatigue crack initiation life using macroscale loading conditions and the microstructural phenomenon causing crack initiation. Microstructure sensitive modelling is used for predicting potential crack initiation life by employing randomly generated representative microstructures. The microstructural parameters contributing to crack initiation life are identified and accounted for by computing lattice level energy dissipation during fatigue crack initiation. This model is coupled with experimental results to improve the predictive capabilities and identification of potentially damaging weak points in the microstructures. The estimated values for crack initiation life were found to be in good agreement with the experimentally observed values of initiation life. The results have shown that this kind of approach could be successfully used to predict crack initiation life in polycrystalline materials. This work successfully provides an approach for estimating crack initiation life based upon numerical computations accounting for the microstructural phenomenon.     

Gradient crystal plasticity as part of the computational modelling of polycrystals

This paper describes how gradient hardening can, in a thermodynamically consistent fashion, be included into a crystal plasticity model. By assuming that the inelastic part of the free energy includes contributions from the gradient of hardening along each slip direction, a hardening stress due to the second derivative of the hardening along each slip direction can be derived. For a finite element model of the grain structure a coupled problem with displacements and gradient hardening variables as degrees of freedom is thereby obtained. This problem is solved using a dual mixed approach. In particular, an algorithm suitable for parallelization is presented, where each grain is treated as a subproblem. The numerical results show that the macroscopic strength increases with decreasing grain size as a result of gradient hardening. Finally, the results of different prolongation assumptions, i.e. how to impose the macroscopic deformation gradient on a representative volume element, are compared. Copyright © 2007 John Wiley & Sons, Ltd.     

Investigation of integration algorithms for rate-dependent crystal plasticity using explicit finite element codes

The work presented here concerns the use of rate-dependent crystal plasticity into explicit dynamic finite element codes for structural analysis. Different integration or stress update algorithms for the numerical implementation of crystal plasticity, two explicit algorithms and a fully-implicit one, are described in detail and compared in terms of convergence, accuracy and computation time. The results show that the implicit time integration is very robust and stable, provided low enough convergence tolerance is used for low strain-rate sensitivity coefficients, while being the slowest in terms of CPU time. Explicit methods prove to be fast, stable and accurate. The algorithms are then applied to two structural analyses, one concerning flat rolling of a polycrystalline slab and another on the response of a multicrystalline sample under uniaxial tensile condition. The results show that the explicit algorithms perform well with simulation times much smaller compared to their implicit counterpart. Finally, mesh sensitivity for the second structural analysis is investigated and shows to slightly affect the global response of the structure.     

Incorporating the grain boundary misorientation effects on slip activity into crystal plasticity

The role of grain boundary misorientation angle (GBMA) distribution on slip activity in a high-manganese austenitic steel was investigated through experiments and simulations. Crystal plasticity simulations incorporating the GBMA distribution and the corresponding dislocation–grain boundary interactions were conducted. The computational analysis revealed that the number of active slip systems decreased when GBMA distribution was taken into account owing to the larger volume of grain boundary–dislocation interactions. The current results demonstrate that the dislocation–grain boundary interactions significantly contribute to the overall hardening, and the GBMA distribution constitutes a key parameter dictating the slip activity.     

The meshless finite element method

A meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n, and the condition that the shape functions depend only on the node positions. Furthermore, the method proposed also shares several of the advantages of the finite element method such as: (a) the simplicity of the shape functions in a large part of the domain; (b) C⁰ continuity between elements, which allows the treatment of material discontinuities, and (c) ease of introduction of the boundary conditions. Copyright © 2003 John Wiley & Sons, Ltd.     

A three‐dimensional computational procedure for reproducing meshless methods and the finite element method

A flexible computational procedure for solving 3D linear elastic structural mechanics problems is presented that currently uses three forms of approximation function (natural neighbour, moving least squares—using a new nearest neighbour weight function—and Lagrange polynomial) and three types of integration grids to reproduce the natural element method and the finite element method. The addition of more approximation functions, which is not difficult given the structure of the code, will allow reproduction of other popular meshless methods. Results are presented that demonstrate the ability of the first‐order meshless approximations to capture solutions more accurately than first‐order finite elements. Also, the quality of integration for the three types of integration grids is compared. The concept of a region is introduced, which allows the splitting of a domain into different sections, each with its own type of approximation function and spatial integration scheme. Copyright © 2004 John Wiley & Sons, Ltd.     

An arbitrary Lagrangian–Eulerian finite element method for finite strain plasticity

This paper presents a new arbitrary Lagrangian–Eulerian (ALE) finite element formulation for finite strain plasticity in non‐linear solid mechanics. We consider the models of finite strain plasticity defined by the multiplicative decomposition of the deformation gradient in an elastic and a plastic part ( F = F ^e F ^p), with the stresses given by a hyperelastic relation. In contrast with more classical ALE approaches based on plastic models of the hypoelastic type, the ALE formulation presented herein considers the direct interpolation of the motion of the material with respect to the reference mesh together with the motion of the spatial mesh with respect to this same reference mesh. This aspect is shown to be crucial for a simple treatment of the advection of the plastic internal variables and dynamic variables. In fact, this advection is carried out exactly through a particle tracking in the reference mesh, a calculation that can be accomplished very efficiently with the use of the connectivity graph of the fixed reference mesh. A staggered scheme defined by three steps (the smoothing, the advection and the Lagrangian steps) leads to an efficient method for the solution of the resulting equations. We present several representative numerical simulations that illustrate the performance of the newly proposed methods. Both quasi‐static and dynamic conditions are considered in these model examples. Copyright © 2003 John Wiley & Sons, Ltd.     

考虑位错密度和损伤的NiCoCrFe高熵合金晶体塑性有限元分析EI北大核心CSCD

结合实验和晶体塑性有限元方法研究准静态加载NiCoCrFe高熵合金有限变形过程中的宏观和微观力学响应、损伤行为以及微观结构演化。使用电子背散射衍射技术(EBSD)对拉伸实验变形前后NiCoCrFe的微观结构进行表征。通过修改强化模型和流动准则分别在CPFEM模型中引入位错密度内部状态变量和连续介质损伤因子,并结合拉伸实验应力-应变曲线确定NiCoCrFe相关的模型参数。结果表明:考虑位错密度和损伤的CPFEM模型可以有效地描述NiCoCrFe宏观和微观力学响应。CPFEM模型合理预测NiCoCrFe颈缩区域的变形形状和尺寸,其中实验获得的颈缩区域长度比预测结果小7%,CPFEM预测的颈缩区域宽度比实验结果大23%。CPFEM模型预测NiCoCrFe拉伸变形后的织构演化同EBSD表征结果大致相同,均表现为弱的(100)∥RD以及强的(111)∥RD纤维织构。在三维微观结构损伤分析中,CPFEM模型预测的损伤在应力集中以及位错密度集中的晶界处萌生,表现为晶间损伤机制,并且随着变形的增加损伤逐渐向晶粒内部扩展。     

A perturbation method for stochastic meshless analysis in elastostatics

A stochastic meshless method is presented for solving boundary‐value problems in linear elasticity that involves random material properties. The material property was modelled as a homogeneous random field. A meshless formulation was developed to predict stochastic structural response. Unlike the finite element method, the meshless method requires no structured mesh, since only a scattered set of nodal points is required in the domain of interest. There is no need for fixed connectivities between nodes. In conjunction with the meshless equations, classical perturbation expansions were derived to predict second‐moment characteristics of response. Numerical examples based on one‐ and two‐dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method. A good agreement is obtained between the results of the proposed method and Monte Carlo simulation. Since mesh generation of complex structures can be a far more time‐consuming and costly effort than the solution of a discrete set of equations, the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems. Copyright © 2001 John Wiley & Sons, Ltd.     

Simulating small crack growth behaviour using crystal plasticity theory and finite element analysis

Predictions of small crack growth under cyclic loading in aluminium alloy 7075 are performed using finite element analysis (FEA), and results are compared with published experimental data. A double‐slip crystal plasticity model is implemented within the analyses to enable the anisotropic nature of individual grains to be approximated. Small edge‐cracks in a single grain with a starting length of 6 μm are incrementally grown following a node‐release scheme. Crack‐tip opening displacements (CTOD) and crack opening stresses are calculated during the simulated crack growth, and da/dN against ΔK diagrams are computed. Interactions between the crack tip and a grain boundary are also considered. The computations are shown to accurately capture the magnitude and the variability normally observed in small crack fatigue data.     

A method for calculating meshless finite difference weights

An exact method for calculating the finite difference weights for arbitrary distributed points in multiple dimensions is presented. The method avoids the numerical ill conditioning associated with a small‐scale factor (ε) by delaying the limiting of ε → 0. The Gaussian Radial Basis function is approximated by a Taylor's series expansion in the variable ε, and the resultant matrix equation is solved using the Fraction Free LU decomposition method. Copyright © 2007 John Wiley & Sons, Ltd.     

Simulation of magnesium alloy AZ31 sheet during cylindrical cup drawing with rate independent crystal plasticity finite element method

A rate independent crystal plasticity model is developed to analyze the plastic deformation of hexagonal close packed (HCP) materials by incorporating the crystallography of deformation twinning in plasticity model. The “successive integration method” is used to identify active slip/twinning systems and determine plastic strain. This model is implemented in the commercial finite element code ABAQUS/Explicit to simulate the deformation process of the magnesium alloy sheet. For the determination of model parameters, simulated stress–strain curves for uniaxial tension are adjusted to the measured curves. Then the stamping process of a cylindrical cup is simulated with the developed model, and the effect of initial crystal orientation on final cup earing is discussed.     

Enrichment and coupling of the finite element and meshless methods

A mixed hierarchical approximation based on finite elements and meshless methods is presented. Two cases are considered. The first one couples regions where finite elements or meshless methods are used to interpolate: continuity and consistency is preserved. The second one enriches a finite element mesh with particles. Thus, there is no need to remesh in adaptive refinement processes. In both cases the same formulation is used, convergence is studied and examples are shown. Copyright © 2000 John Wiley & Sons, Ltd.     

磷酸二氢钾(KDP)晶体纳米压痕过程的有限元分析

为了求得KDP晶体的应力-应变曲线以及材料的屈服应力,基于圣维南定理和实验得到的材料性能参数建立了KDP晶体的压痕过程仿真模型,利用ABAQUS有限元分析软件对KDP晶体压痕过程进行了有限元仿真,得到了KDP晶体的载荷-位移曲线和加/卸载过程中的等效应力变化规律.仿真结果表明:加载过程中最大应力集中在压头尖角处,卸载后最大应力分布在压头棱边所留下的压痕处,KDP晶体材料的屈服应力为120MPa.     

A micromechanical composite approach for finite element crashworthiness simulation

The present article deals with micromechanical composite modeling. Both analytical and computational micromechanics approaches are described as well as micromechanical modeling of damage. Based on micromechanics of failure theory, a user subroutine including a progressive damage algorithm is programmed for finite element analysis. Three theory-experiment correlations of tubes under a three-point bending test have been carried out using the bi-phase material model developed along with this project. These studies include three-ply schedules.     

Bending of single crystal thin films modeled with micropolar crystal plasticity

The scale-dependent mechanical response of single crystal thin films subjected to pure bending is investigated using a dislocation-based model of micropolar single crystal plasticity via finite element simulations. Due to the presence of couple stresses, the driving force for plastic slip in a micropolar crystal contains an intrinsic back stress component that is related to gradients in lattice torsion-curvature. Strain gradient-dependent back stresses are a common feature of various types of generalized crystal plasticity theories; however, it is often introduced either in a phenomenological manner without additional kinematics or by designating the plastic slips as generalized degrees-of-freedom. The treatment of lattice rotations as fundamental degrees-of-freedom instead of plastic slips greatly reduces the complexity (computational expense) of the single crystal model, and leads to the incorporation of additional elastoplastic kinematics since the lattice torsion-curvature is taken as a work-conjugate continuum deformation measure. A recently proposed single criterion micropolar framework is employed in which the evolution of both the plastic strains and torsion-curvatures are coupled through the use of a unified flow rule. The deformation behavior is characterized by the moment-rotation response and the dislocation substructure evolution for various slip configurations and specimen thicknesses. The results are compared to analogous simulations carried out using a model of discrete dislocation dynamics as well as a statistical-mechanics inspired, flux-based model of nonlocal crystal plasticity. The micropolar model demonstrates good qualitative and quantitative agreement with the previous results up to certain inherent limitations of the current formulation.     

Variational projection methods for gradient crystal plasticity using Lie algebras

A computational method is developed for evaluating the plastic strain gradient hardening term within a crystal plasticity formulation. While such gradient terms reproduce the size effects exhibited in experiments, incorporating derivatives of the plastic strain yields a nonlocal constitutive model. Rather than applying mixed methods, we propose an alternative method whereby the plastic deformation gradient is variationally projected from the elemental integration points onto a smoothed nodal field. Crucially, the projection utilizes the mapping between Lie groups and algebras in order to preserve essential physical properties, such as orthogonality of the plastic rotation tensor. Following the projection, the plastic strain field is directly differentiated to yield the Nye tensor. Additionally, an augmentation scheme is introduced within the global Newton iteration loop such that the computed Nye tensor field is fed back into the stress update procedure. Effectively, this method results in a fully implicit evolution of the constitutive model within a traditional displacement‐based formulation. An elemental projection method with explicit time integration of the plastic rotation tensor is compared as a reference. A series of numerical tests are performed for several element types in order to assess the robustness of the method, with emphasis placed upon polycrystalline domains and multi‐axis loading. Copyright © 2016 John Wiley & Sons, Ltd.