A robust and efficient substepping scheme for the explicit numerical integration of a rate‐dependent crystal plasticity model

This paper describes the development of efficient and robust numerical integration schemes for rate‐dependent crystal plasticity models. A forward Euler integration algorithm is first formulated. An integration algorithm based on the modified Euler method with an adaptive substepping scheme is then proposed, where the substepping is mainly controlled by the local error of the stress predictions within the time step. Both integration algorithms are implemented in a stand‐alone code with the Taylor aggregate assumption and in an explicit finite element code. The robustness, accuracy and efficiency of the substepping scheme are extensively evaluated for large time steps, extremely low strain‐rate sensitivity, high deformation rates and strain‐path changes using the stand‐alone code. The results show that the substepping scheme is robust and in some cases one order of magnitude faster than the forward Euler algorithm. The use of mass scaling to reduce computation time in crystal plasticity finite element simulations for quasi‐static problems is also discussed. Finally, simulation of Taylor bar impact test is carried out to show the applicability and robustness of the proposed integration algorithm for the modelling of dynamic problems with contact. Copyright © 2014 John Wiley & Sons, Ltd.     

Stress update algorithm for the combined viscoplastic and plastic behaviours of single‐crystal superalloys

A crystallographic constitutive model is developed, which accounts for both rate‐sensitive and rate‐insensitive flow. Single‐crystal plasticity and viscoplasticity are the limiting cases of the model, so that it properly reflects the material response over a wide temperature range. A non‐linear dynamic recovery is included to properly describe ratchetting. We provide a robust integration scheme based on generalization of the return‐mapping algorithm and of the procedure for active set search. The implicit integration and consistent tangent are implemented through the UMAT subroutine in the ABAQUS finite element program. The capability of the model to account for both high and low strain rates is demonstrated in numerical examples. Finally, the stability of integration scheme and quadratic convergence of the global Newton–Raphson equilibrium iterations are demonstrated on the example of a notched bar under tension. Copyright © 2011 John Wiley & Sons, Ltd.     

A robust return‐map algorithm for general multisurface plasticity

Three new contributions to the field of multisurface plasticity are presented for general situations with an arbitrary number of nonlinear yield surfaces with hardening or softening. A method for handling linearly dependent flow directions is described. A residual that can be used in a line search is defined. An algorithm that has been implemented and comprehensively tested is discussed in detail. Examples are presented to illustrate the computational cost of various components of the algorithm. The overall result is that a single Newton‐Raphson iteration of the algorithm costs between 1.5 and 2 times that of an elastic calculation. Examples also illustrate the successful convergence of the algorithm in complicated situations. For example, without using the new contributions presented here, the algorithm fails to converge for approximately 50% of the trial stresses for a common geomechanical model of sedementary rocks, while the current algorithm results in complete success. Because it involves no approximations, the algorithm is used to quantify the accuracy of an efficient, pragmatic, but approximate, algorithm used for sedimentary‐rock plasticity in a commercial software package. The main weakness of the algorithm is identified as the difficulty of correctly choosing the set of initially active constraints in the general setting. Copyright © 2016 John Wiley & Sons, Ltd.     

基于弹塑性损伤本构模型的复合材料层合板破坏荷载预测

在连续损伤力学和塑性力学框架内,建立一个同时考虑塑性效应和损伤累积导致材料属性退化的复合材料弹塑性损伤本构模型。基于最近点投影回映算法,开发本构模型的应变驱动隐式积分算法以更新应力及与解答相关的状态变量,并推导与所开发算法相应的数值一致性切线刚度矩阵,保证有限元分析采用NewtonRaphson迭代法解答非线性问题的计算效率。采用断裂带模型对已开发的本构模型软化段进行规则化,以减轻有限元分析结果的网格相关性问题。对损伤变量进行粘滞规则化,并推导出相应的粘滞规则化数值一致性切线刚度张量,解决了在有限元隐式计算程序中采用含应变软化段本构关系的数值分析由于计算困难而提前终止的问题。开发包含数值积分算法的用户材料子程序UMAT,并嵌于有限元程序Abaqus v6.14中。通过对力学行为展现显著塑性效应的AS4/3501-6V型开口复合材料层合板的渐进失效分析,验证本文提出的材料本构模型的有效性。结果显示,预测结果与已报道的试验结果吻合良好,并且预测精度高于其他已有弹性损伤模型。表明已建立的弹塑性损伤本构模型能够准确预测力学行为,展现显著塑性效应的复合材料层合板的破坏荷载,为其构件和结构设计提供一种有效的分析方法。     

A modified cutting-plane time integration scheme with adaptive substepping for elasto-viscoplastic models

Integration of stress-strain-time relationship is a key issue for the application of elasto-viscoplastic models to engineering practice. This article presents a novel adaptive substepping cutting-plane time integration scheme for elasto-viscoplastic models keeping the advantage of original cutting-plane (OCP) with only the first derivatives of loading surface required. The deficiency of OCP time integration algorithm is first discussed taking a simple overstress theory based elasto-viscoplastic modified Cam-Clay model (EVP-MCC) as example. To overcome this, a new algorithm is developed with three features: (1) an evolution function for the hardening variable of dynamic loading surface is innovatively deduced for the Taylor series approximation, (2) the elastic predictor is modified to account for the initial viscoplastic strain rate with more accuracy, and (3) a new adaptive substepping technique for restricting simultaneously both strain and time incremental sizes based on the overstress distance is proposed. For easy understanding, the proposed algorithm is first presented for one-dimensional condition, and then extended to three-dimensional condition. The new integrated EVP-MCC model using the proposed algorithm is examined by simulating laboratory tests at both levels of integration point and finite element with a good performance in terms of accuracy and convergence.     

A generalized midpoint algorithm for the integration of a generalized plasticity model for sands

This paper describes a method of performing the integration of generalized plasticity models, in which, unlike classical elastoplasticity, the yield surface is not explicitly defined. The algorithm is based on a generalized midpoint scheme and is applied to a specific generalized plasticity model for sands, in which a hyperelastic formulation is introduced to describe the reversible component of the soil response instead of the hypoelastic approach originally proposed. In this way, an efficient integration scheme is developed in the elastic strain space. The consistent, algorithmic tangent operator is derived. Isoerror maps are generated to study the local accuracy of the numerical integration algorithm. Results from a series of numerical examples based on the simulation of drained triaxial tests are given to illustrate the accuracy and convergence properties of the algorithm, both at the local and at the global level. Finally an example is given of the simulation of a cyclic triaxial test to illustrate the improvement on accuracy caused by the use of a hyperelastic law into the constitutive equations, as opposed to the hypoelastic formulation initially adopted in the model. Copyright © 2008 John Wiley & Sons, Ltd.     

CONSISTENT LINEARIZATION FOR THE EXACT STRESS UPDATE OF PRANDTL–REUSS NON-HARDENING ELASTOPLASTIC MODELS

This paper presents a consistent algorithm, which combines the advantages of the exact time integration of Prandtl–Reuss elastoplastic models and the quadratic asymptotic convergence of Newton–Raphson iteration strategies. The consistent modulus is evaluated by a full linearization of the exact stress update procedure. Numerical tests for a thin wall tube subjected to combined loads of tension and torsion are performed to illustrate the accuracy and efficiency of the consistently linearized exact stress update algorithm described in the paper. For comparison purpose numerical results of the radial return method are also given.     

A convergent adaptive finite element method for the primal problem of elastoplasticity

The boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional. This paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and, up to higher order terms, the R‐linear convergence of the stresses with respect to the number of loops. Applications include several plasticity models: linear isotropic‐kinematic hardening, linear kinematic hardening, and multisurface plasticity as model for nonlinear hardening laws. For perfect plasticity, the adaptive algorithm yields strong convergence of the stresses. Numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule. Copyright © 2006 John Wiley & Sons, Ltd.     

一种适用于率相关晶体塑性模型的准隐式积分算法

参考经典的切线系数法,采用向前梯度法对修正的超弹性晶体塑性模型进行本构方程积分,提出了一种新的准隐式积分算法。该算法采用基于中间构型的超弹性晶体塑性模型,无需更新晶粒取向相关的状态变量;采用修正的超弹性框架,无需进行由Green-Naghdi材料共旋引起的旋转变换。这种准隐式积分算法兼有超弹性模型和切线系数法的优点。通...     

A globally convergent numerical algorithm for damaging elasto‐plasticity based on the Multiplier method

In the paper it is proposed a new method for the solution of equilibrium problems based on a F.E. displacement formulation, that, at least in principle, is globally convergent as opposed to the classical Lagrangian method that presents only local convergence. The method, which appears to be particularly useful for plasticity models characterized by yield surfaces with regions of sharp curvature or corner points, is based on the Multiplier method. The structure of the procedure is presented and the consequent constrained optimization scheme is implemented for the case of associated plasticity coupled with damage. The main aspect of originality of the proposal is that it is not applied to the ‘return algorithm’, but to entire equilibrium iteration. At first, the local convergence properties of the constitutive equations are examined at the Gauss point level. It is proved that, also for involved constitutive models (a generalized Ottosen yield surface including isotropic hardening and damage is used in the applications), the convergence of a classical Newton's scheme is always reached with few iterations, ensuring a quadratic rate of convergence in the solution, provided a conversion of the inequality plastic constraint into an equality one is introduced, using an augmented Lagrangian functional for exactly evaluating the slack constraints. However, it is observed that the converged stresses are often attracted far from the initial trial point, towards regions with sharper curvature, and the main reason for the lack of convergence of the procedure is found in a divergence of the solution of the non‐linear equilibrium equations. It is shown that the Multiplier method allows to enlarge the radius of convergence with respect to classical iterations based on the Lagrangian method. The price for the enlargement of the convergence radius is a higher number of iterations, since the Multiplier method presents only a linear rate of convergence. Indeed, the exact fulfilment of the compatibility and admissibility equations is not attained simultaneously, once an equilibrated solution is reached, but it is iterative. In the closure of the paper a convergence analysis of an elastic–plastic problem characterized by a yield criterion resulting from the convex hull of crises surfaces, and as a consequence, presenting regions of non‐differentiability, is presented. It is shown how the ability of the Multiplier method in finding the solution of the structural problem for large loading step with respect to the classical Lagrangian technique compensate its slower convergence rate. Copyright © 2005 John Wiley & Sons, Ltd.     

Damage model and implementation in nonlinear dynamic problems

Microcracking, damage and subsequent softening in materials introduce higher levels of nonlinearity than those for materials characterized by nonlinear elastic or classical plasticity models. Hènce, implementation of such advanced models that allow for the foregoing effects require special considerations in terms of the analysis of the characteristics of the model, convergence during plastic deformations, and time integration schemes that consider the nonlinearity.This paper describes a damage model, a special scheme involving drift correction and the generalized time finite element (GTFEM) scheme for time integration for dynamic analysis. The main objective is to examine the model and develop schemes that can lead to consistent and reliable predictions from computational procedures. Toward this aim, (1) the damage model is analyzed with respect to its convergence behavior with mesh refinement, (2) a special drift correct scheme is implemented for the plasticity based model, (3) the generalized time finite element method (GTFEM) is implemented in the nonlinear dynamic finite element procedure for time integration and compared with the Newmark method, and (4) the damage model, the drift correction scheme and the GTFEM are verified by solution of representative static and dynamic problems involving a material (concrete) that experiences damage and softening, including verification with respect to behavior of concrete in the laboratory.     

A method for smoothing multiple yield functions

Many models of plasticity are built using multiple, simple yield surfaces. Examples include geomechanical models and crystal plasticity. This leads to numerical difficulties, most particularly during the stress update procedure, because the combined yield surface is nondifferentiable, and when employing implicit time stepping to solve numerical models, because the Jacobian is often poorly conditioned. A method is presented that produces a single C² differentiable and convex yield function from a plastic model that contains multiple yield surfaces that are individually C² differentiable and convex. C² differentiability ensures quadratic convergence of implicit stress-update procedures; convexity ensures a unique solution to the stress update problem, whereas smoothness means the Jacobian is much better conditioned. The method contains just one free parameter, and the error incurred through the smoothing procedure is quantified in terms of this parameter. The method is illustrated through three different constitutive models. The method's performance is quantified in terms of the number of iterations required during stress update as a function of the smoothing parameter. Two simple finite-element models are also solved to compare this method with existing approaches. The method has been added to the open-source “MOOSE” framework, for perfect, nonperfect, associated, and nonassociated plasticity.     

On a new integration scheme for von‐Mises plasticity with linear hardening

Limiting the discussion to an associative von‐Mises plasticity model with linear kinematic and isotropic hardening, we compare the performance of the classical radial return map algorithm with a new integration scheme based on the computation of an integration factor. The numerical examples clearly show the improved accuracy of the new method. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical integration of lattice rotation in polycrystal plasticity

A numerical algorithm to calculate the lattice rotation in polycrystal plasticity models, that is based on the analytical integration of the lattice rotation update equation, is presented. Implicit and explicit implementations perform equally well and are superior to the currently used numerical integration schemes. Copyright © 2001 John Wiley & Sons, Ltd.     

Trust‐region based return mapping algorithm for implicit integration of elastic‐plastic constitutive models

A new method for the solution of the non‐linear equations forming the core of constitutive model integration is proposed. Specifically, the trust‐region method that has been developed in the numerical optimization community is successfully modified for use in implicit integration of elastic‐plastic models. Although attention here is restricted to these rate‐independent formulations, the proposed approach holds substantial promise for adoption with models incorporating complex physics, multiple inelastic mechanisms, and/or multiphysics. As a first step, the non‐quadratic Hosford yield surface is used as a representative case to investigate computationally challenging constitutive models. The theory and implementation are presented, discussed, and compared with other common integration schemes. Multiple boundary value problems are studied and used to verify the proposed algorithm and demonstrate the capabilities of this approach over more common methodologies. Robustness and speed are then investigated and compared with existing algorithms. Through these efforts, it is shown that the utilization of a trust‐region approach leads to superior performance versus a traditional closest‐point projection Newton–Raphson method and comparable speed and robustness to a line search augmented scheme. Copyright © 2017 John Wiley & Sons, Ltd.     

An interior‐point algorithm for elastoplasticity

The problem of small‐deformation, rate‐independent elastoplasticity is treated using convex programming theory and algorithms. A finite‐step variational formulation is first derived after which the relevant potential is discretized in space and subsequently viewed as the Lagrangian associated with a convex mathematical program. Next, an algorithm, based on the classical primal–dual interior point method, is developed. Several key modifications to the conventional implementation of this algorithm are made to fully exploit the nature of the common elastoplastic boundary value problem. The resulting method is compared to state‐of‐the‐art elastoplastic procedures for which both similarities and differences are found. Finally, a number of examples are solved, demonstrating the capabilities of the algorithm when applied to standard perfect plasticity, hardening multisurface plasticity, and problems involving softening. Copyright © 2006 John Wiley & Sons, Ltd.     

On the numerical implementation of 3D rate‐dependent single crystal plasticity formulations

In this paper, several important numerical issues are addressed for three‐dimensional (3D) rate‐dependent single crystal plasticity. After a thorough comparison of different constitutive algorithms, we classify the integration methods into three approaches, namely, the implicit elastic/plastic deformation gradient approach, the implicit slip‐rate approach, and the explicit slip‐rate approach. As part of this algorithmic study, we focus on five different schemes to enforce the plastic incompressibility, four ways to update the texture, and one convergence criteria. The numerical performance of these different methods is illustrated. The contribution of this study is three‐fold: a stable scheme for the incompressibility enforcement, an improved implicit algorithm, and a fully explicit algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

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.