Two‐point constitutive equations and integration algorithms for isotropic‐hardening rate‐independent elastoplastic materials in large deformation

This paper presents alternative forms of hyperelastic–plastic constitutive equations and their integration algorithms for isotropic‐hardening materials at large strain, which are established in two‐point tensor field, namely between the first Piola–Kirchhoff stress tensor and deformation gradient. The eigenvalue problems for symmetric and non‐symmetric tensors are applied to kinematics of multiplicative plasticity, which imply the transformation relationships of eigenvectors in current, intermediate and initial configurations. Based on the principle of plastic maximum dissipation, the two‐point hyperelastic stress–strain relationships and the evolution equations are achieved, in which it is considered that the plastic spin vanishes for isotropic plasticity. On the computational side, the exponential algorithm is used to integrate the plastic evolution equation. The return‐mapping procedure in principal axes, with respect to logarithmic elastic strain, possesses the same structure as infinitesimal deformation theory. Then, the theory of derivatives of non‐symmetric tensor functions is applied to derive the two‐point closed‐form consistent tangent modulus, which is useful for Newton's iterative solution of boundary value problem. Finally, the numerical simulation illustrates the application of the proposed formulations. Copyright © 2008 John Wiley & Sons, Ltd.     

Mixed finite element method for coupled thermo‐hydro‐mechanical process in poro‐elasto‐plastic media at large strains

A mixed finite element for coupled thermo‐hydro‐mechanical (THM) analysis in unsaturated porous media is proposed. Displacements, strains, the net stresses for the solid phase; pressures, pressure gradients, Darcy velocities for pore water and pore air phases; temperature, temperature gradients, the total heat flux are interpolated as independent variables. The weak form of the governing equations of coupled THM problems in porous media within the element is given on the basis of the Hu–Washizu three‐filed variational principle. The proposed mixed finite element formulation is derived. The non‐linear version of the element formulation is further derived with particular consideration of the THM constitutive model for unsaturated porous media based on the CAP model. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elasto‐plastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non‐linearity, the co‐rotational formulation approach is utilized. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization and the softening behaviours caused by thermal and chemical effects. Copyright © 2005 John Wiley & Sons, Ltd.     

A semi‐implicit integration scheme for a combined viscoelastic‐damage model of plastic bonded explosives

This paper presents a new implementation of a constitutive model commonly used to represent plastic bonded explosives in finite element simulations of thermomechanical response. The constitutive model, viscoSCRAM, combines linear viscoelasticity with isotropic damage evolution. The original implementation was focused on short duration transient events; thus, an explicit update scheme was used. For longer duration simulations that employ significantly larger time step sizes, the explicit update scheme is inadequate. This work presents a new semi‐implicit update scheme suitable for simulations using relatively large time steps. The algorithm solves a nonlinear system of equations to ensure that the stress, damaged state, and internal stresses are in agreement with implicit update equations at the end of each increment. The crack growth is advanced in time using a sub‐incremental explicit scheme; thus, the entire implementation is semi‐implicit. The theory is briefly discussed along with previous explicit integration schemes. The new integration algorithm and its implementation into the finite element code, Abaqus, are detailed. Finally, the new and old algorithms are compared via simulations of uniaxial compression and beam bending. The semi‐implicit scheme has been demonstrated to provide higher accuracy for a given allocated computational time for the quasistatic cases considered here. Published 2014. This article is a US Government work and is in the public domain in the USA.     

Numerical integration and FEM-implementation of a viscoplastic Chaboche-model with static recovery

This paper is concerned with the implementation of a viscoplastic material model of the Chaboche type in the framework of the finite element method (FEM). The equations of the used constitutive law, that incorporates isotropic hardening, back stress evolution with static recovery terms and drag stress evolution, are introduced. A representation of their numerical integration using the implicit backward Euler method under the assumption of small deformations and an isothermal formulation follows. The use of the backward Euler method leads to a nonlinear algebraic system of three equations, which is solved by a combination of the Pegasus method and a fixed-point iteration. After considering the accuracy of the presented integration algorithm in form of iso-error maps, the derivation of the consistent viscoplastic tangent operator is shown. The integration scheme and the calculation of the consistent viscoplastic tangent operator are implemented in the commercial finite element code ABAQUS, using the possibility of the user-defined material subroutine (UMAT). Finally a numerical example in form of a notched bar under tension is presented.     

Remarks on the interpretation of current non‐linear finite element analyses as differential–algebraic equations

For the numerical solution of materially non‐linear problems like in computational plasticity or viscoplasticity the finite element discretization in space is usually coupled with point‐wise defined evolution equations characterizing the material behaviour. The interpretation of such systems as differential–algebraic equations (DAE) allows modern‐day integration algorithms from Numerical Mathematics to be efficiently applied. Especially, the application of diagonally implicit Runge–Kutta methods (DIRK) together with a Multilevel‐Newton method preserves the algorithmic structure of current finite element implementations which are based on the principle of virtual displacements and on backward Euler schemes for the local time integration. Moreover, the notion of the consistent tangent operator becomes more obvious in this context. The quadratical order of convergence of the Multilevel‐Newton algorithm is usually validated by numerical studies. However, an analytical proof of this second order convergence has already been given by authors in the field of non‐linear electrical networks. We show that this proof can be applied in the current context based on the DAE interpretation mentioned above. We finally compare the proposed procedure to several well‐known stress algorithms and show that the inclusion of a step‐size control based on local error estimations merely requires a small extra time‐investment. Copyright © 2001 John Wiley & Sons, Ltd.     

A finite element constitutive update scheme for anisotropic,viscoelastic solids exhibiting non‐linearity of the Schapery type

This study presents a new scheme for performing integration point constitutive updates for anisotropic, small strain, non‐linear viscoelasticity, within the context of implicit, non‐linear finite element structural analysis. While the basic scheme has been presented earlier by the authors for linear viscoelasticity, the present work illustrates the generality of the underlying fundamentals by extending to Schapery's non‐linear model. The method features a judicious choice of state variables, a stable backward Euler integration step, and a consistent tangent operator. Its greatest strength lies in ready incorporation into existing FEM codes. Numerical examples involving homogeneous stress states such as uniaxial extension and simple shear, and non‐uniform stress states such as a beam under tip load, were carried out by incorporating the present scheme into a general purpose FEM package. Excellent agreement with analytical results is observed. Copyright © 1999 John Wiley & Sons, Ltd.     

Efficient thermo‐mechanical model for solidification processes

A new, computationally efficient algorithm has been implemented to solve for thermal stresses, strains, and displacements in realistic solidification processes which involve highly nonlinear constitutive relations. A general form of the transient heat equation including latent‐heat from phase transformations such as solidification and other temperature‐dependent properties is solved numerically for the temperature field history. The resulting thermal stresses are solved by integrating the highly nonlinear thermo‐elastic‐viscoplastic constitutive equations using a two‐level method. First, an estimate of the stress and inelastic strain is obtained at each local integration point by implicit integration followed by a bounded Newton–Raphson (NR) iteration of the constitutive law. Then, the global finite element equations describing the boundary value problem are solved using full NR iteration. The procedure has been implemented into the commercial package Abaqus (Abaqus Standard Users Manuals, v6.4, Abaqus Inc., 2004) using a user‐defined subroutine (UMAT) to integrate the constitutive equations at the local level. Two special treatments for treating the liquid/mushy zone with a fixed grid approach are presented and compared. The model is validated both with a semi‐analytical solution from Weiner and Boley (J. Mech. Phys. Solids 1963; 11 :145–154) as well as with an in‐house finite element code CON2D (Metal. Mater. Trans. B 2004; 35B (6):1151–1172; Continuous Casting Consortium Website. http://ccc.me.uiuc.edu [30 October 2005]; Ph.D. Thesis, University of Illinois, 1993; Proceedings of the 76th Steelmaking Conference, ISS, vol. 76, 1993) specialized in thermo‐mechanical modelling of continuous casting. Both finite element codes are then applied to simulate temperature and stress development of a slice through the solidifying steel shell in a continuous casting mold under realistic operating conditions including a stress state of generalized plane strain and with actual temperature‐dependent properties. Other local integration methods as well as the explicit initial strain method used in CON2D for solving this problem are also briefly reviewed and compared. Copyright © 2006 John Wiley & Sons, Ltd.     

A sub-stepping approach for elasto-plasticity with rotational hardening

Within the framework of the finite element method, this paper presents new algorithms implementing implicit stress integration and consistent tangent matrix calculations for an elasto-plastic model with rotational hardening. The sub-stepping technique is used for both the numerical integration of the constitutive relations and determination of the consistent tangent matrix in order to overcome the convergence difficulty arising from the complexity of the elasto-plastic model with rotational hardening. The integration of the constitutive relations and the computation of the consistent tangent matrix are incorporated into a unique procedure. Numerical tests are carried out and discussed to demonstrate the global accuracy and stability of the presented algorithms.     

An efficient 3D implicit approach for the thermomechanical simulation of elastic–viscoplastic materials submitted to high strain rate and damage

This paper aims at presenting a general consistent numerical formulation able to take into account, in a coupled way, strain rate, thermal and damage effects on the behavior of materials submitted to quasistatic or dynamic loading conditions in a large deformation context. The main features of this algorithmic treatment are as follows:

• A unified treatment for the analysis and implicit time integration of thermo‐elasto‐viscoplastic constitutive equations including damage that depends on the strain rate for dynamic loading conditions. This formalism enables us to use dynamic thermomechanically coupled damage laws in an implicit framework.
• An implicit framework developed for time integration of the equations of motion. An efficient staggered solution procedure has been elaborated and implemented so that the inertia and heat conduction effects can be properly treated.
• An operator split‐based implementation, accompanied by a unified method to analytically evaluate the consistent tangent operator for the (implicit) coupled damage–thermo‐elasto‐viscoplastic problem.
• The possibility to couple any hardening law, including rate‐dependent models, with any damage model that fits into the present framework.

All the developments have been considered in the framework of an implicit finite element code adapted to large strain problems. The numerical model will be illustrated by several applications issued from the impact and metal‐forming domains. All these physical phenomena have been included into an oriented object finite element code (implemented at LTAS‐MN ²L, University of Liège, Belgium) named Metafor.Copyright © 2013 John Wiley & Sons, Ltd.     

A computational model for the finite element analysis of thermoplasticity coupled with ductile damage at finite strains

An updated Lagrangian implicit FEM model for the analysis of large thermo‐mechanically coupled hyperelastic‐viscoplastic deformations of isotropic porous materials is considered. An appropriate framework for constitutive modelling is introduced that includes a stress‐free thermally expanded configuration and a plastically deformed unstressed damaged configuration. A two‐level iterative scheme is employed at each time increment to solve the field equations governing the conservation of momentum (mechanical step) and the conservation of energy (thermal step) for the coupled thermo‐mechanical problem. Exact linearizations for the calculation of the tangent stiffness are performed in each of these solution steps. A fully implicit, thermo‐mechanically coupled and incrementally objective Euler‐backward radial return based map is developed for the time integration of the constitutive equations. The present model is used to analyse a number of benchmark examples including metal forming processes wherein temperature and the accumulated damage play an important role in influencing the deformation mechanism and the nature of the deformed workpiece. Copyright © 1999 John Wiley & Sons, Ltd.     

循环稳定材料的棘轮行为:II.隐式应力积分算法和有限元实现

针对第一部分发展的、能够合理描述循环稳定材料棘轮行为的粘塑性本构模型,详细讨论该模型的数值计算方法和有限元实现。在径向回退(RadialReturn)和向后欧拉积分方法的基础上,结合连续迭代(SuccessiveSubstitution)方法,推导并建立了针对循环粘塑性本构模型的、新的隐式应力积分算法。为了本构模型在大型有限元分析程序(如ABAQUS等)中的实现,针对有限元的整体节点迭代计算,推导和确立了一个新的、考虑率相关塑性的一致切线刚度矩阵(ConsistentTangentModulus)表达式。通过对一些算例的有限元分析,讨论了建立的隐式应力积分算法的优越性,同时对特定构件的棘轮行为进行了数值模拟,进而检验了有限元实现的合理性和必要性。     

A simple integration algorithm for a non‐associated anisotropic plasticity model for sheet metal forming

In this paper, an anisotropic material model based on a non‐associated flow rule and nonlinear mixed isotropic‐kinematic hardening is developed. The quadratic Hill48 yield criterion is considered in the non‐associated model for both yield function and plastic potential to account for anisotropic behavior. The developed model is integrated based on fully implicit backward Euler's method. The resulting problem is reduced to only two simple scalar equations. The consistent local tangent modulus is obtained by exact linearization of the algorithm. All numerical development was implemented into user‐defined material subroutine for the commercial finite element code ABAQUS/Standard. The performance of the present algorithm is demonstrated by numerical examples. Copyright © 2015 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 fast convergent and accurate temperature model for phase‐change heat conduction

This work proposes a temperature‐based finite element model for transient heat conduction involving phase‐change. Like preceding temperature‐based models, it is characterized by the discontinuous spatial integration over the elements affected by the phase‐change. Using linear triangles or tetrahedrals, integration can be performed in a closed analytical way, assuring an exact evaluation of the discrete balance equation. Because of its unconditional stability, an Euler‐backward time‐stepping scheme is implemented. A crucial fact is the computation of the exact tangent matrices for the Newton–Raphson solution of the non‐linear system of discretized equations. Efficiency of the model is tested by means of the results obtained for the Neumann problem and the solidification of a steel ingot. Copyright © 1999 John Wiley & Sons, Ltd.     

Implementation of cyclic plasticity models based on a general form of kinematic hardening

This paper deals with implementation of cyclic plastic constitutive models in which a general form of strain hardening and dynamic recovery is employed to represent the multilinear, as well as non‐linear, evolution of back stress. First, in order to incorporate such a general form of kinematic hardening in finite element methods, successive substitution and its convergence are discussed for implicitly integrating stress; moreover, a new expression of consistent tangent modulus is derived by introducing a set of fourth‐rank constitutive parameters into discretized kinematic hardening. Then, the constitutive parameters introduced are specified in three cases of the general form of kinematic hardening; the three cases have distinct capabilities of simulating ratcheting and cyclic stress relaxation. Numerical examples are given to verify the convergence in successive substitution and the new expression of consistent tangent stiffness. Error maps for implicitly integrating stress under non‐proportional as well as proportional loading are also given to show that the multilinear case of the general form provides high accuracy even if strain increment is very large. Copyright © 2002 John Wiley & Sons, Ltd.     

An implicit unitless error and step‐size control method in integrating unified viscoplastic/creep ODE‐type constitutive equations

A series of numerical analyses are carried out to investigate the difficulties in numerical integration of unified viscoplastic/creep constitutive equations, which are normally represented as a system of ordinary differential equations (ODEs). The problems of numerically integrating the constitutive equations are identified and analysed. To overcome the stiffness problems, implicit methods are used for the numerical integration and a generic technique is introduced to calculate the Jacobian matrix. A normalization technique is introduced in the paper to convert the integration errors for each time increment to unitless errors. Thus, a single tolerance can be used to control the accuracy and step size in integrating a set of unified viscoplastic/creep constitutive equations. In addition, an implicit step‐size control method is proposed and used in the integrations. This method reduces the possibility of rejection of an integration increment due to poor accuracy. Copyright © 2007 John Wiley & Sons, Ltd.     

A novel accurate and computationally efficient integration approach to viscoplastic constitutive model

A novel, accurate, and computationally efficient integration approach is developed to integrate small strain viscoplastic constitutive equations involving nonlinear coupled first-order ordinary differential equations. The developed integration scheme is achieved by a combination of the implicit backward Euler difference approximation and the implicit asymptotic integration. For the uniaxial loading case, the developed integration scheme produces accurate results irrespective of time steps. For the multiaxial loading case, the accuracy and computational efficiency of the developed integration scheme are better than those of either the implicit backward Euler difference approximation or the implicit asymptotic integration. The simplicity of the developed integration scheme is equivalent to that of the implicit backward Euler difference approximation since it also reduces the solution of integrated constitutive equations to the solution of a single nonlinear equation. The algorithm tangent constitutive matrix derived for the developed integration scheme is consistent with the integration algorithm and preserves the quadratic convergence of the Newton–Raphson method for global iterations.     

A time‐integration method for the viscoelastic–viscoplastic analyses of polymers and finite element implementation

The present study introduces a time‐integration algorithm for solving a non‐linear viscoelastic–viscoplastic (VE–VP) constitutive equation of isotropic polymers. The material parameters in the constitutive models are stress dependent. The algorithm is derived based on an implicit time‐integration method (Computational Inelasticity. Springer: New York, 1998) within a general displacement‐based finite element (FE) analysis and suitable for small deformation gradient problems. Schapery's integral model is used for the VE responses, while the VP component follows the Perzyna model having an overstress function. A recursive‐iterative method (Int. J. Numer. Meth. Engng 2004; 59 :25–45) is employed and modified to solve the VE–VP constitutive equation. An iterative procedure with predictor–corrector steps is added to the recursive integration method. A residual vector is defined for the incremental total strain and the magnitude of the incremental VP strain. A consistent tangent stiffness matrix, as previously discussed in Ju (J. Eng. Mech. 1990; 116 :1764–1779) and Simo and Hughes (Computational Inelasticity. Springer: New York, 1998), is also formulated to improve convergence and avoid divergence. Available experimental data on time‐dependent and inelastic responses of high‐density polyethylene are used to verify the current numerical algorithm. The time‐integration scheme is examined in terms of its computational efficiency and accuracy. Numerical FE analyses of microstructural responses of polyethylene reinforced with elastic particle are also presented. Copyright © 2009 John Wiley & Sons, Ltd.