New development in extended finite element modeling of large elasto‐plastic deformations

This paper presents new achievements in the extended finite element modeling of large elasto‐plastic deformation in solid problems. The computational technique is presented based on the extended finite element method (X‐FEM) coupled with the Lagrangian formulation in order to model arbitrary interfaces in large deformations. In X‐FEM, the material interfaces are represented independently of element boundaries, and the process is accomplished by partitioning the domain with some triangular sub‐elements whose Gauss points are used for integration of the domain of elements. The large elasto‐plastic deformation formulation is employed within the X‐FEM framework to simulate the non‐linear behavior of materials. The interface between two bodies is modeled by using the X‐FEM technique and applying the Heaviside‐ and level‐set‐based enrichment functions. Finally, several numerical examples are analyzed, including arbitrary material interfaces, to demonstrate the efficiency of the X‐FEM technique in large plasticity deformations. Copyright © 2008 John Wiley & Sons, Ltd.     

Thermo‐elasto‐plastic finite element analysis of quasi‐state processes in Eulerian reference frames

An Eulerian finite element formulation for quasi‐state one way coupled thermo‐elasto‐plastic systems is presented. The formulation is suitable for modeling material processes such as welding and laser surfacing. In an Eulerian frame, the solution field of a quasi‐state process becomes steady state for the heat transfer problem and static for the stress problem. A mixed small deformation displacement elasto‐plastic formulation is proposed. The formulation accounts for temperature dependent material properties and exhibits a robust convergence. Streamline upwind Petrov–Galerkin (SUPG) is used to remove spurious oscillations. Smoothing functions are introduced to relax the non‐differentiable evolution equations and allow for the use of gradient (stiffness) solution scheme via the Newton–Raphson method. A 3‐dimensional simulation of a laser surfacing process is presented to exemplify the formulation. Results from the Eulerian formulation are in good agreement with results from the conventional Lagrangian formulation. However, the Eulerian formulation is approximately 15 times faster than the Lagrangian. Copyright © 2001 John Wiley & Sons, Ltd.     

On the computation of design derivatives for Huber–Mises plasticity with non‐linear hardening

This paper concerns design sensitivity analysis (DSA) for an elasto–plastic material, with material parameters depending on, or serving as, design variables. The considered constitutive model is Huber–Mises deviatoric plasticity with non‐linear isotropic/kinematic hardening, one which is applicable to metals. The standard radial return algorithm for linear hardening is generalized to account for non‐linear hardening functions. Two generalizations are presented; in both the non‐linearity is treated iteratively, but the iteration loop contains either a scalar equation or a group of tensorial equations. It is proven that the second formulation, which is the one used in some parallel codes, can be equivalently brought to a scalar form, more suitable for design differentiation. The design derivatives of both the algorithms are given explicitly, enabling thus calculation of the ‘explicit’ design derivative of stresses entering the global sensitivity equation. The paper addresses several issues related to the implementation and testing of the DSA module; among them the concept of verification tests, both outside and inside a FE code, as well as the data handling implied by the algorithm. The numerical tests, which are used for verification of the DSA module, are described. They shed light on (a) the accuracy of the design derivatives, by comparison with finite difference computations and (b) the effect of the finite element formulation on the design derivatives for an isochoric plastic flow. Copyright © 2003 John Wiley & Sons, Ltd.     

Enhanced transverse shear strain shell formulation applied to large elasto‐plastic deformation problems

In this work, a previously proposed Enhanced Assumed Strain (EAS) finite element formulation for thin shells is revised and extended to account for isotropic and anisotropic material non‐linearities. Transverse shear and membrane‐locking patterns are successfully removed from the displacement‐based formulation. The resultant EAS shell finite element does not rely on any other mixed formulation, since the enhanced strain field is designed to fulfil the null transverse shear strain subspace coming from the classical degenerated formulation. At the same time, a minimum number of enhanced variables is achieved, when compared with previous works in the field. Non‐linear effects are treated within a local reference frame affected by the rigid‐body part of the total deformation. Additive and multiplicative update procedures for the finite rotation degrees‐of‐freedom are implemented to correctly reproduce mid‐point configurations along the incremental deformation path, improving the overall convergence rate. The stress and strain tensors update in the local frame, together with an additive treatment of the EAS terms, lead to a straightforward implementation of non‐linear geometric and material relations. Accuracy of the implemented algorithms is shown in isotropic and anisotropic elasto‐plastic problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Design sensitivity analysis and optimization of non‐linear transient dynamics. Part I—sizing design

A continuum‐based sizing design sensitivity analysis (DSA) method is presented for the transient dynamic response of non‐linear structural systems with elastic–plastic material and large deformation. The methodology is aimed for applications in non‐linear dynamic problems, such as crashworthiness design. The first‐order variations of the energy forms, load form, and kinematic and structural responses with respect to sizing design variables are derived. To obtain design sensitivities, the direct differentiation method and updated Lagrangian formulation are used since they are more appropriate for the path‐dependent problems than the adjoint variable method and the total Lagrangian formulation, respectively. The central difference method and the finite element method are used to discretize the temporal and spatial domains, respectively. The Hughes–Liu truss/beam element, Jaumann rate of Cauchy stress, rate of deformation tensor, and Jaumann rate‐based incrementally objective stress integration scheme are used to handle the finite strain and rotation. An elastic–plastic material model with combined isotropic/kinematic hardening rule is employed. A key development is to use the radial return algorithm along with the secant iteration method to enforce the consistency condition that prevents the discontinuity of stress sensitivities at the yield point. Numerical results of sizing DSA using DYNA3D yield very good agreement with the finite difference results. Design optimization is carried out using the design sensitivity information. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite element formulation for modelling large deformations in elasto‐viscoplastic polycrystals

Anisotropic, elasto‐viscoplastic behaviour in polycrystalline materials is modelled using a new, updated Lagrangian formulation based on a three‐field form of the Hu‐Washizu variational principle to create a stable finite element method in the context of nearly incompressible behaviour. The meso‐scale is characterized by a representative volume element, which contains grains governed by single crystal behaviour. A new, fully implicit, two‐level, backward Euler integration scheme together with an efficient finite element formulation, including consistent linearization, is presented. The proposed finite element model is capable of predicting non‐homogeneous meso‐fields, which, for example, may impact subsequent recrystallization. Finally, simple deformations involving an aluminium alloy are considered in order to demonstrate the algorithm. Copyright © 2004 John Wiley & Sons, Ltd.     

Eulerian elasto‐visco‐plastic formulations for residual stress prediction

A stabilized, Galerkin finite element formulation for modeling the elasto‐visco‐plastic response of quasi‐steady‐state processes, such as welding, laser surfacing, rolling and extrusion, is presented in an Eulerian frame. The mixed formulation consists of four field variables, such as velocity, stress, deformation gradient and internal variable, which is used to describe the evolution of the material's resistance to plastic flow. The streamline upwind Petrov–Galerkin method is used to eliminate spurious oscillations, which may be caused by the convection‐type of stress, deformation gradient and internal variable evolution equations. A progressive solution strategy is introduced to improve the convergence of the Newton–Raphson solution procedure. Two two‐dimensional numerical examples are implemented to verify the accuracy of the Eulerian formulation. 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.     

Least‐squares mixed finite elements for small strain elasto‐viscoplasticity

The main aim of this contribution is to provide a mixed finite element for small strain elasto‐viscoplastic material behavior based on the least‐squares method. The L₂‐norm minimization of the residuals of the given first‐order system of differential equations leads to a two‐field functional with displacements and stresses as process variables. For the continuous approximation of the stresses, lowest‐order Raviart–Thomas elements are used, whereas for the displacements, standard conforming elements are employed. It is shown that the non‐linear least‐squares functional provides an a posteriori error estimator, which establishes ellipticity of the proposed variational approach. Further on, details about the implementation of the least‐squares mixed finite elements are given and some numerical examples are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A rational elasto‐plastic spatially curved thin‐walled beam element

Torsion is one of the primary actions in members curved in space, and so an accurate spatially curved‐beam element needs to be able to predict the elasto‐plastic torsional behaviour of such members correctly. However, there are two major difficulties in most existing finite thin‐walled beam elements, such as in ABAQUS and ANSYS, which may lead to incorrect predictions of the elasto‐plastic behaviour of members curved in space. Firstly, the integration sample point scheme cannot capture the shear strain and stress information resulting from uniform torsion. Secondly, the higher‐order twists are ignored which leads to loss of the significant effects of Wagner moments on the large twist torsional behaviour. In addition, the initial geometric imperfections and residual stresses are significant for the elasto‐plastic behaviour of members curved in space. Many existing finite thin‐walled beam element models do not provide facilities to deal with initial geometric imperfections. Although ABAQUS and ANSYS have facilities for the input of residual stresses as initial stresses, they cannot describe the complicated distribution patterns of residual stresses in thin‐walled members. Furthermore, external loads and elastic restraints may be applied remote from shear centres or centroids. The effects of the load (and restraint) positions are important, but are not considered in many beam elements. This paper presents an elasto‐plastic spatially curved element with arbitrary thin‐walled cross‐sections that can correctly capture the uniform shear strain and stress information for integration, and includes initial geometric imperfections, residual stresses and the effects of the load and restraint positions. The element also includes elastic restraints and supports, which have to be modelled separately as spring elements in some other finite thin‐walled beam elements. Comparisons with existing experimental and analytical results show that the elasto‐plastic spatially curved‐beam element is accurate and efficient. Copyright © 2006 John Wiley & Sons, Ltd.     

Stochastic finite element method for elasto‐plastic body

This paper proposes a Stochastic Finite Element Method (SFEM) for non‐linear elasto‐plastic bodies, as a generalization of the SFEM for linear elastic bodies developed by Ghanem and Spanos who applied the Karhunen–Loeve expansion and the polynomial chaos expansion for stochastic material properties and field variables, respectively. The key feature of the proposed SFEM is the introduction of two fictitious bodies whose behaviours provide upper and lower bounds for the mean of field variables. The two bounding bodies are rigorously obtained from a given distribution of material properties. The deformation of an ideal elasto‐plastic body of the Huber–von Mises type is computed as an illustrative example. The results are compared with Monte‐Carlo simulation. It is shown that the proposed SFEM can satisfactorily estimate means, variances and other probabilistic characteristics of field variables even when the body has a larger variance of the material properties. Copyright © 1999 John Wiley & Sons, Ltd.     

The least‐squares meshfree method for elasto‐plasticity and its application to metal forming analysis

A new meshfree method for the analysis of elasto‐plastic deformation is presented. The method is based on the proposed first‐order least‐squares formulation for elasto‐plasticity and the moving least‐squares approximation. The least‐squares formulation for classical elasto‐plasticity and its extension to an incrementally objective formulation for finite deformation are proposed. In the formulation, equilibrium equation and flow rule are enforced in least‐squares sense, i.e. their squared residuals are minimized, and hardening law and loading/unloading condition are enforced pointwise at each integration point. The closest point projection method for the integration of rate‐form constitutive equation is inherently involved in the formulation, and thus the radial‐return mapping algorithm is not performed explicitly. The proposed formulation is a mixed‐type method since the residuals are represented in a form of first‐order differential system using displacement and stress components as nodal unknowns. Also the penalty schemes for the enforcement of boundary and frictional contact conditions are devised and the reshaping of nodal supports is introduced to avoid the difficulties due to the severe local deformation near contact interface. The proposed method does not employ structure of extrinsic cells for any purpose. Through some numerical examples of metal forming processes, the validity and effectiveness of the method are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

A stabilised large‐strain elasto‐plastic Q1‐P0 method

The mean dilatation method effectively involves bi‐linear displacements and a constant pressure and is often known as the Q1‐P0 formulation. Its non‐linear implementation was originally derived as a three‐field formulation which included the volume ratio via the Jacobian, J, of the deformation gradient as an additional separate variable. However, the latter term was not directly required in the numerical implementation once J was assumed constant along with the pressure. This formulation will here be termed the non‐linear Q1‐P0 method. It is known to give good solutions for many practical large‐strain elasto‐plastic problems. However, for some problems, it has been shown to be prone to severe ‘hour‐glassing’. With a view to remedying this situation, we here re‐visit the three‐field formulation and derive a modified form, which although variationally valid, is over‐stiff in comparison to the original procedure (here simply called the Q1‐P0 method). However, the concepts lead to a natural method for stabilising the Q1‐P0 technique. The associated tangent stiffness matrix is symmetric. Copyright © 1999 John Wiley & Sons, Ltd.     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

A computational approach for multi‐scale analysis of heterogeneous elasto‐plastic media subjected to short duration loads

The asymptotic expansion homogenization (AEH) approach has found wide acceptance for the study of heterogeneous structures due to its ability to account for multi‐scale features. The emphasis of the present study is to develop consistent AEH numerical formulations to address elasto‐plastic material response of structures subjected to short‐duration transient loading. A second‐order accurate velocity‐based explicit time integration method, in conjunction with the AEH approach, is currently developed that accounts for large deformation non‐linear material response. The approach is verified under degenerate homogeneous conditions using existing experimental data in the literature and its ability to account for heterogeneous conditions is demonstrated for a number of test problems. Copyright © 2004 John Wiley & Sons, Ltd.     

A continuum sensitivity method for finite thermo‐inelastic deformations with applications to the design of hot forming processes

A computational framework is presented to evaluate the shape as well as non‐shape (parameter) sensitivity of finite thermo‐inelastic deformations using the continuum sensitivity method (CSM). Weak sensitivity equations are developed for the large thermo‐mechanical deformation of hyperelastic thermo‐viscoplastic materials that are consistent with the kinematic, constitutive, contact and thermal analyses used in the solution of the direct deformation problem. The sensitivities are defined in a rigorous sense and the sensitivity analysis is performed in an infinite‐dimensional continuum framework. The effects of perturbation in the preform, die surface, or other process parameters are carefully considered in the CSM development for the computation of the die temperature sensitivity fields. The direct deformation and sensitivity deformation problems are solved using the finite element method. The results of the continuum sensitivity analysis are validated extensively by a comparison with those obtained by finite difference approximations (i.e. using the solution of a deformation problem with perturbed design variables). The effectiveness of the method is demonstrated with a number of applications in the design optimization of metal forming processes. Copyright © 2002 John Wiley & Sons, Ltd.     

Computational structural and material stability analysis in finite electro‐elasto‐statics of electro‐active materials

Dielectric materials like electro‐active polymers (EAPs) exhibit coupled electro‐mechanical behavior at large strains. They respond by a deformation to an applied electrical field and are used in advanced industrial environments as sensors and actuators, for example, in robotics, biomimetics and smart structures. In field‐activated or electronic EAPs, the electric activation is driven by Coulomb‐type electrostatic forces, resulting in Maxwell stresses. These materials are able to provide finite actuation strains, which can even be improved by optimizing their composite microstructure. However, EAPs suffer from different types of instabilities. This concerns global structural instabilities, such as buckling and wrinkling of EAP devices, as well as local material instabilities, such as limit‐points and bifurcation‐points in the constitutive response, which induce snap‐through and fine scale localization of local states. In this work, we outline variational‐based definitions for structural and material stability, and design algorithms for accompanying stability checks in typical finite element computations. The formulation starts from stability criteria for a canonical energy minimization principle of electro‐elasto‐statics, and then shifts them over to representations related to an enthalpy‐based saddle point principle that is considered as the most convenient setting for numerical implementation. Here, global structural stability is analyzed based on a perturbation of the total electro‐mechanical energy, and related to statements of positive definiteness of incremental finite element tangent arrays. We base the local material stability on an incremental quasi‐convexity condition of the electro‐mechanical energy, inducing the positive definiteness of both the incremental electro‐mechanical moduli as well as a generalized acoustic tensor. It is shown that the incremental arrays to be analyzed in the stability criteria appear within the enthalpy‐based setting in a distinct diagonal form, with pure mechanical and pure electrical partitions. Applications of accompanying stability analyses in finite element computations are demonstrated by means of representative model problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Simulation of strength difference in elasto‐plasticity for adhesive materials

Experimental evidence of certain adhesive materials reveals elastic strains, plastic strains and hardening. Furthermore, a pronounced strength difference effect between tension, torsion or combined loading is observed. For simulation of these phenomena, a yield function dependent on the first and second basic invariants of the related stress tensor in the framework of elasto‐plasticity is used in this work. A plastic potential with the same mathematical structure is introduced to formulate the evolution equation for the inelastic strains. Furthermore, thermodynamic consistency of the model equations is considered, thus rendering some restrictions on the material parameters. For evolution of the strain like internal variable, two cases are considered, and the consequences on the thermodynamic consistency and the numerical implementation are extensively discussed. The resulting evolution equations are integrated with an implicit Euler scheme. In particular, the reduction of the resulting local problem is performed, and for the finite‐element equilibrium iteration, the algorithmic tangent operator is derived. Two examples are presented. The first example demonstrates the capability of the model equations to simulate the yield strength difference between tension and torsion for the adhesive material Betamate 1496. A second example investigates the deformation evolution of a compact tension specimen with an adhesive zone. Copyright © 2005 John Wiley & Sons, Ltd.     

Quadrilateral elements for the solution of elasto‐plastic finite strain problems

In this paper two plane strain quadrilateral elements with two and four variables, are proposed. These elements are applied to the analysis of finite strain elasto‐plastic problems. The elements are based on the enhanced strain and B‐bar methodologies and possess a stabilizing term. The pressure and dilatation fields are assumed to be constant in each element's domain and the deformation gradient is enriched with additional variables, as in the enhanced strain methodology. The formulation is deduced from a four‐field functional, based on the imposition of two constraints: annulment of the enhanced part of the deformation gradient and the relation between the assumed dilatation and the deformation gradient determinant. The discretized form of equilibrium is presented, and the analytical linearization is deduced, to ensure the asymptotically quadratic rate of convergence in the Newton–Raphson method. The proposed formulation for the enhanced terms is carried out in the isoparametric domain and does not need the usually adopted procedure of evaluating the Jacobian matrix in the centre of the element. The elements are very effective for the particular class of problems analysed and do not present any locking or instability tendencies, as illustrated by various representative examples. Copyright © 2001 John Wiley & Sons, Ltd.