Application of crystal plasticity FEM from single crystal to bulk polycrystal

This article compares results of crystal plasticity FE simulations with experimental results for single, bi- and oligo crystal deformation. It is shown, that while single crystal deformation is very well reproduced by the simulations, the quality of simulation results for bicrystals strongly depends on the orientation of the grain boundary with respect to the external mechanical load. In the second part of the paper an extension of crystal plasticity FEM (CP-FEM) using texture components for the representation of the crystallographic texture of bulk material is shortly introduced and applied to the cup drawing of sheet material.     

Modelling direction-dependent hardening in magnesium sheet forming simulations

In this work, a model capturing anisotropic hardening during plastic deformation under monotonic loading is proposed. For this purpose, the anisotropic plastic potential coefficients are assumed to be functions of a measure of the accumulated plastic strain. This model is applied to describe the plastic behavior of a magnesium alloy (ZM21) sheet at room temperature. The selected plastic potential accounts for the main features of Mg alloy plasticity, i.e., anisotropy and strength-differential (SD) effects. All the accumulated plastic strain dependent coefficients of the phenomenological model are determined from input data generated with a crystal plasticity approach. They are optimized to best capture the accumulated strain dependent potentials computed with crystal plasticity. The R-value (Lankford coefficient) anisotropy is used as an independent measure for the assessment of the approximation quality. This model is implemented into a finite element (FE) code and successfully validated through the numerical simulations of the cup drawing test. The calculated earing profile obtained with the proposed hardening model is compared to results assuming isotropic hardening for various plausible shapes of the plastic potential. Although the ear and valley numbers and positions are similar in all cases, the height differences between peaks and valleys are strongly dependent on the type of constitutive approach used in the simulation.     

On finite deformation plasticity with directional softening

Summary It is well-known that twisting of cylindrical specimens has shown that axial stress or strain are induced in constrained or unconstrained torsion respectively. During monotonic loading in torsion, the axial stress or axial strain do not change monotonically, but tensile/compressive or lengthening/shortening phenomena are observed. In this study, a two-component model to account phenomenologically for coexisting different textures of rate-independent and rate-dependent finite deformation plasticity is proposed to predict axial shortening/lengthening and tensile/compressive phenomena in torsion. Such predictions are compared with available experimental data as well as recent simulations based on crystal plasticity models. In most cases, the results are in reasonable agreement with both experiments and simulations.     

Effects of constraints on lattice re-orientation and strain in polycrystal plasticity simulations

Employing a rate-dependent crystal plasticity model implemented in a novel and fast algorithm, two instantiations of an OFHC copper microstructure have been simulated by FE modelling to 11% tensile engineering strain with two different sets of boundary conditions. Analysis of lattice rotations, strain distributions and global stress–strain response show the effect of changing from free to periodic boundary conditions to be a perturbation of a response dictated by the microstructure. Average lattice rotation for each crystallographic grain has been found to be in fair agreement with Taylor-constraint simulations while fine scale element-resolved analysis shows large deviations from this prediction. Locally resolved analysis shows the existence of large domains dominated by slip on only a few slip systems. The modelling results are discussed in the light of recent experimental advances with respect to 2- and 3-dimensional characterization and analysis methods.     

Finite deformation formulation for embedded frictional crack with the extended finite element method

We reformulate an extended finite element (FE) framework for embedded frictional cracks in elastoplastic solids to accommodate finite deformation, including finite stretching and rotation. For the FE representation, we consider a Galerkin approximation in which both the trial and weighting functions adapt to the current contact configuration. Contact and frictional constraints employ two Kuhn–Tucker conditions, a contact/separation constraint nesting over a stick/slip constraint for the case when the crack faces are in frictional sliding mode. We integrate finite deformation bulk plasticity into the formulation using the multiplicative decomposition technique of nonlinear continuum mechanics. We then present plane strain simulations demonstrating various aspects of the extended FE solutions. The mechanisms considered include combined opening and frictional sliding in initially straight, curved, and S‐shaped cracks, with and without bulk plasticity. To gain further insight into the extended FE solutions, we perform mesh convergence studies focusing on both the global and the local responses of structures with cracks, including the distribution of the normal component of traction on the crack faces. Copyright © 2009 John Wiley & Sons, Ltd.     

Dual-time scale crystal plasticity FE model for cyclic deformation of Ti alloys

A dual-time scale finite element model is developed in this paper for simulating cyclic deformation in a Titanium alloy Ti-6242. The material is characterized by crystal plasticity constitutive relations. Modeling cyclic deformation using conventional time integration algorithms in a single time scale can be prohibitive for crystal plasticity computations. Typically 3D crystal plasticity based fatigue simulations found in the literature are in the range of 100 cycles. Results are subsequently extrapolated to thousands of cycles, which can lead to considerable error in fatigue predictions. However, the dual-time scale model enables simulations up to a significantly high number of cycles to reach local states of damage initiation leading to fatigue crack growth. This formulation decomposes the governing equations into two sets of problems, corresponding to a coarse time scale (low frequency) cycle-averaged problem and a fine time scale (high frequency) oscillatory problem. A statistically equivalent 3D polycrystalline model of Ti-6242 is simulated by the crystal plasticity finite element model to study the evolution of local stresses and strains in the microstructure with cyclic loading. The comparison with the single time scale reference solution shows excellent accuracy while the efficiency gained through time-scale compression can be enormous.     

Numerical implementation of a neural network based material model in finite element analysis

Neural network (NN) based constitutive models can capture non‐linear material behaviour. These models are versatile and have the capacity to continuously learn as additional material response data becomes available. NN constitutive models are increasingly used within the finite element (FE) method for the solution of boundary value problems. NN constitutive models, unlike commonly used plasticity models, do not require special integration procedures for implementation in FE analysis. NN constitutive model formulation does not use a material stiffness matrix concept in contrast to the elasto‐plastic matrix central to conventional plasticity based models. This paper addresses numerical implementation issues related to the use of NN constitutive models in FE analysis. A consistent material stiffness matrix is derived for the NN constitutive model that leads to efficient convergence of the FE Newton iterations. The proposed stiffness matrix is general and valid regardless of the material behaviour represented by the NN constitutive model. Two examples demonstrate the performance of the proposed NN constitutive model implementation. Copyright © 2004 John Wiley & Sons, Ltd.     

An Experimental/Analytical Compression of Three-Dimentional Deformations at the Tip of a Crack in a Plastically deforming Plate III: Material Characterization and finite element analysis

A comparison between the three-dimensional experimental and numerical displacement fields surrounding a notch/crack in a ductile 4340 steel tested in three-point bending is presented. Excellent agreement between computed and measured deformations exists at load levels below 50 to 75 percent of ultimate loads. Experimentally determined crack tunnel profiles are included in the finite element model through nodal release; the evidence of the crack tunnel appears in the displacements at the surface. It is shown that surface measurements of unloading reveal specimen-internal failure initiation in the form of tunneling. Out-of-plane deformations deviate from analytical values earlier than in-plane values; this observation compromises the accuracy with which predictions of in-plane crack tip variables can be made when they are based on measured out-of-plane deformations (caustics, gradient sensing) once significant plasticity arises. Comparison is made between J-integral values calculated from the external boundary conditions and from a domain integral. The tunneling tests provide a method of estimating a critical value of J. The stress intensity factor governs the deformation in the elastic regime, but, because of the finite notch- tip radius underlying the experimental configuration, the HRR field does not describe the deformation well under plastic conditions. Comparison of numerical simulations with and without tunneling provide insight into criteria that could be used to implement an implicit crack propagation scheme into the numerical model.     

Crystal plasticity with Jacobian-Free Newton–Krylov

The objective of this work is to study potential benefits of solving crystal plasticity finite element method (CPFEM) implicit simulations using the Jacobian-Free Newton–Krylov (JFNK) technique. Implicit implementations of CPFEM are usually solved using Newton’s method. However, the inherent non-linearity in the flow rule model that characterizes the crystal slip system deformation on occasions would require considerable effort to form the exact analytical Jacobian needed by Newton’s method. In this paper we present an alternative using JFNK. As it does not require an exact Jacobian, JFNK can potentially decrease development time. JFNK approximates the effect of the Jacobian through finite differences of the residual vector, allowing modified formulations to be studied with relative ease. We show that the JFNK solution is identical to that obtained using Newton’s method and produces quadratic convergence. We also find that preconditioning the JFNK solution with the elastic tensor provides the best computational efficiency.     

Crystal plasticity modeling of damage accumulation in dissimilar Mg alloy bi-crystals under high-cycle fatigue

Damage accumulation in Mg AZ31–AZ80 alloy bi-crystals under fatigue loading at room temperature is studied using a modified version of the crystal plasticity finite element model of Abdolvand and Daymond. The model accounts for strain accommodation by both slip and tensile twinning, and is first shown to reasonably describe monotonic single crystal Mg experimental data from the literature. The high cycle fatigue behavior was then investigated in misoriented dissimilar alloy bi-crystals through stress-controlled simulations up to 1000 cycles. Nine different orientation combinations were simulated and the fatigue damage evolution, defined as the cumulative shear strain amplitude, were compared and analyzed. The bi-crystal geometry was used to simulate possible microstructure combinations occurring, for instance within an idealized friction stir weld. Findings suggest that when either of the alloy bi-crystal grains is oriented for basal slip, poor fatigue performance can occur by twinning or slip localization depending upon the neighboring orientation.     

Three orders of magnitude improved efficiency with high‐performance spectral crystal plasticity on GPU platforms

We study efficient numerical implementations of crystal plasticity in the spectral representation, with emphasis on high‐performance computational aspects of the simulation. For illustrative purposes, we apply this approach to a Taylor homogenization model of fcc poly‐crystalline materials and show that the spectral representation of crystal plasticity is ideal for parallel implementations aimed at next‐generation large‐scale microstructure‐sensitive simulations of material deformation. We find that multi‐thread parallelizations of the algorithm provide two orders of magnitude acceleration of the calculation, whereas graphics processing unit‐based computing solutions allow for three orders of magnitude speedup factors over the conventional model. Copyright © 2014 John Wiley & Sons, Ltd.     

Ductile failure modeling and simulations using coupled FE–EFG approach

In the present work, a ductile fracture model has been employed to predict the failure of tensile specimen using coupled finite element–element free Galerkin (FE–EFG) approach. The fracture strain as a function of stress triaxiality has been evaluated by analyzing the notched tensile specimens. In the coupled approach, a small portion of the domain, where severe plastic deformation is expected, is modeled by EFG method whereas the rest of the domain is modeled by FEM to exploit the advantages of both the methods. A ramp function has been used in the interface region to maintain the continuity between FE and EFG domains. The nonlinear material behavior is modeled by von-Mises yield criterion and Hollomon’s power law. An implicit return mapping algorithm is employed for stress equilibrium in the plasticity model. The effect of geometric nonlinearity as a result of large deformation is captured by updated Lagrangian approach. The coupled approach is used to study the fracture behavior of two different cracked specimens in order to highlight its capabilities.     

Mesoscale Simulation of Recrystallization Textures and Microstructures

The paper presents mesoscale simulations of textures and microstructures formed during recrystallization. It gives an overview of the method and demonstrates how microstructure and texture simulations can be performed by incorporating realistic input data for the boundary character and for the initial deformation microstructure. Particular attention is placed on the simulation of primary static recrystallization in a deformed aluminum polycrystal on the basis of crystal plasticity finite element data. Various nucleation scenarios are discussed also with respect to macroscopic effects such as friction and shear localization.     

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.     

Quantitative re-examination of Taylor model for FCC polycrystals

The quantitative adequacy of the Taylor model for representing the behaviors of FCC polycrystals is discussed through comparison with crystal plasticity analysis using the homogenization-based finite method. The key element of the crystal plasticity theory is the constitutive relation for single crystals. The most classical way to apply it to polycrystals is the Taylor model. This model assumes that all crystal grains in a crystal aggregate are subjected to the same strain under macroscopically uniform deformation. This assumption provides a solution satisfying the continuity of displacement between crystal grains. The effect and evolution of the crystallographic texture can easily be taken into account. However, the assumption of uniform strain, the main idea in the Taylor model, has never been validated quantitatively. On the other hand, the homogenization-based finite element method can represent arbitrary microscopic deformations, i.e., each crystal grain may have nonuniform deformation, and can provide a material response under more realistic boundary conditions. In this paper, we first determine the appropriate size for the representative volume element (RVE) in the homogenization-based finite element method that can represent the macroscopic polycrystalline behavior of FCC. After that, the polycrystalline behaviors obtained using the Taylor model are compared with those obtained using the homogenization-based finite element method. Finally, the quantitative adequacy of the Taylor model is discussed. It is clarified that the Taylor model is qualitatively consistent with the homogenization-based finite element method and can be used as a practical model of polycrystalline FCC metals for a first-order approximation, although it is not quantitatively reasonable even for FCC metals.     

Accelerating cyclic plasticity simulations using an adaptive wavelet transformation based multitime scaling method

Cyclic finite element simulations of complex materials, for example, polycrystalline metals, are widely used to study fatigue failure due to plasticity and damage. Typically, this requires the simulation of a large number of cycles to failure for accurate determination of evolving deformation variables. Modeling cyclic deformation using conventional methods of time integration in semidiscretization techniques can however be computationally challenging. Single time scale integration methods typically follow the high frequency characteristics and discretize each cycle into a number of time steps over which integration is performed. To overcome this computational challenge, the wavelet transformation‐based multitime scale (WATMUS) method proposed in an earlier work by the authors is advanced and validated in this paper to perform accelerated finite element simulations of materials undergoing rate‐dependent plasticity for large number of cycles. Specifically, the WATMUS algorithm is integrated with crystal plasticity finite element method to perform accelerated simulations of polycrystalline alloys. The WATMUS methodology is also endowed with adaptive capabilities to optimally construct the wavelet basis functions and determine coarse‐scale cycle steps. Accuracy and efficiency of the WATMUS methodology is conclusively demonstrated by comparing the results with cyclic single‐time scale crystal plasticity finite element simulations performed on image‐based microstructure of titanium alloys. Copyright © 2012 John Wiley & Sons, Ltd.     

Yield surface simulation for partially recrystallized aluminum polycrystals on the basis of spatially discrete data

The paper presents simulations of the yield surface evolution of plastically deformed aluminum polycrystals during recrystallization. The yield surfaces are calculated using a viscoplastic Taylor–Bishop–Hill strain rate polycrystal homogenization method. The input data for the yield surface calculations are the crystal orientations, their volume fractions, and their shear stresses. While the crystal orientations determine the kinematic portion of the yield surface the threshold shear stress of each individual orientation determines the kinetic portion of the yield surface. The input data for the homogenization calculations are generated through a spatially discrete simulation, where crystal deformation and primary static partial recrystallization are simulated by coupling a viscoplastic crystal plasticity finite element model with a cellular automaton. The crystal plasticity finite element model accounts for crystallographic slip and for crystal rotation during plastic deformation using space and time as independent variables and the crystal orientation and the accumulated slip as dependent variables. The cellular automaton uses a switching rule which is formulated as a probabilistic analogue of Turnbull's rate equation for the motion of grain boundaries. The actual decision about a switching event is made using a simple-sampling Monte Carlo step. The automaton uses space and time as independent variables and the crystal orientation and a stored energy measure as dependent variables. The kinetics produced by the switching algorithm are scaled through grain boundary mobility and driving force data. The crystallographic texture and the orientation-dependent resistance to shear are for each interpolation point extracted after each time step during recrystallization. The data serve as input for the calculation of discrete yield surfaces.     

An improved strain gradient plasticity formulation with energetic interfaces: theory and a fully implicit finite element formulation

A fully implicit backward-Euler implementation of a higher order strain gradient plasticity theory is presented. A tangent operator consistent with the numerical update procedure is given. The implemented theory is a dissipative bulk formulation with energetic contribution from internal interface to model the behavior of material interfaces at small length scales. The implementation is tested by solving some examples that specifically highlight the numerics and the effect of using the energetic interfaces as higher order boundary conditions. Specifically, it is demonstrated that the energetic interface formulation is able to mimic a wide range of plastic strain conditions at internal boundaries. It is also shown that delayed micro-hard conditions may arise under certain circumstances such that an interface at first offers little constraints on plastic flow, but with increasing plastic deformation will develop and become a barrier to dislocation motion.     

A new class of algorithms for classical plasticity extended to finite strains. Application to geomaterials

A recently proposed methodology for computational plasticity at finite strains is re-examined within the context of geomechanical applications and cast in the general format of multi-surface plasticity. This approach provides an extension to finite strains of any infinitesimal model of classical plasticity that retains both the form of the yield criterion and the hyperelastic character of the stress-strain relations. Remarkably, the actual algorithmic implementation reduces to a reformulation of the standard return maps in principal axis with algorithmic elastoplastic moduli identical to those of the infinitesimal theory. New results in the area of geomechanics included a fully implicit return map for the modified Cam-Clay model, extended here to the finite deformation regime, and a new semi-explicit scheme that restores symmetry of the algorithmic moduli while retaining the unconditional stability property. In addition, a new phenomenological plasticity model for soils is presented which includes a number of widely used models as special cases. The general applicability of the proposed methodology is illustrated in several geomechanical examples that exhibit localization and finite deformations.Partial support provided by the Max Kade Foundation under Grant No. 2-DJA-616 and with Stanford University, and the Naval Civil Engineering Laboratory at Port Huaneme     

Modeling the mechanical behavior of a multicrystalline zinc coating on a hot-dip galvanized steel sheet

Numerical simulations can play a major role in the understanding of deformation mechanisms in zinc coatings of galvanized steel sheets during forming processes. A three-dimensional finite element (FE) simulation of a thin zinc coating on a galvanized steel sheet has been performed taking the multicrystalline structure of the coating into account. Experimental characterization of the gauge length of a real in situ tensile specimen reveals 34 large flat zinc grains; the grain orientations are determined using the electron back-scatter diffraction (EBSD) technique. The geometry and orientation of the grains and the plastic deformation modes specific to hexagonal close-packed (hcp) metals as plastic slip and twinning are incorporated into the modeling using a classical crystal plasticity framework. The constraint effect of the substrate is evidenced by comparing the results to the computation of a zinc layer without substrate under the same loading conditions. Attention is then focused on, respectively, the initiation of plastic activity at the grain boundaries, the multiaxial stress state of the grains, the development of a strain gradient within the thickness.