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.     

Optimization of thermo‐elasto‐plastic processes using Eulerian sensitivity analysis

A computational scheme for the analysis and optimization of quasi‐static thermo‐mechanical processes is presented in this paper. In order to obtain desirable mechanical transformations in a workpiece using a thermal treatment process, the optimal control parameters need to be determined. The problem is addressed by posing the process as a decoupled thermo‐mechanical finite element problem and performing an optimization using gradient methods. The forward problem is solved using the Eulerian formulation since it is computationally more efficient compared to an equivalent Lagrangian formulation. The design sensitivities required for the optimization are developed analytically using direct differentiation. This systematic design approach is applied to optimize a laser forming process. The objective is to maximize the angular distortion of a specimen subject to the constraint that the phase transition temperature is not exceeded at any point in the model. Copyright © 2003 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.     

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.     

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.     

Ductile damage analysis of elasto‐plastic shells at large inelastic strains

The objective of this contribution is to model ductile damage phenomena under consideration of large inelastic strains, to couple the corresponding constitutive law with a multi‐layer shell kinematics and to give finally an adequate finite element formulation. An elastic–plastic constitutive law is formulated by using a spatial hyperelasto‐plastic formulation based on the multiplicative decomposition of the deformation gradient. To include isotropic ductile damage the continuum damage model of Rousselier is modified so as to consider large strains and additionally extended by various void nucleation and macro‐crack criteria. In order to achieve numerical efficiency, elastic strains are supposed to be sufficiently small providing a numerical effective integration based on the backward Euler rule. Finite element formulation is enriched by means of the enhanced strain concept. Thus the well‐known deficiencies due to incompressible deformations and the inclusion of transverse strains are avoided. Several examples are given to demonstrate the performance of the algorithms developed concerning large inelastic strains of shells and ductile damage phenomena. Copyright © 2000 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.     

Smooth C1‐interpolations for two‐dimensional frictional contact problems

Finite deformation contact problems are associated with large sliding in the contact area. Thus, in the discrete problem a slave node can slide over several master segments. Standard contact formulations of surfaces discretized by low order finite elements leads to sudden changes in the surface normal field. This can cause loss of convergence properties in the solution procedure and furthermore may initiate jumps in the velocity field in dynamic solutions. Furthermore non‐smooth contact discretizations can lead to incorrect results in special cases where a good approximation of the contacting surfaces is needed. In this paper a smooth contact discretization is developed which circumvents most of the aformentioned problems. A smooth deformed surface with no slope discontinuities between segments is obtained by a C¹‐continuous interpolation of the master surface. Different forms of discretizations are possible. Among these are Bézier, Hermitian or other types of spline interpolations. In this paper we compare two formulations which can be used to obtain smooth normal and tangent fields for frictional contact of deformable bodies. The formulation is developed for two‐dimensional applications and includes finite deformation behaviour. Examples show the performance of the new discretization technique for contact. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

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.     

Finite element modelling of inverse design problems in large deformations anisotropic hyperelasticity

This paper introduces a finite element model for the inverse design of pieces with large displacements in the elastic range. The problem consists in determining the initial shape of the piece, such that it attains the designed shape under the effect of service loads. The model is particularly focused on the design of parts with a markedly anisotropic behavior, like laminated turbine blades. Although the formulation expresses equilibrium on the distorted configuration, it uses a standard constitutive equations library that is expressed as usual for measures attached to the undistorted configuration. In this way, the modifications to a standard finite elements code are limited to the routines for the computation of the finite element internal forces and tangent matrix. Two examples are given, the first one for validation purposes, while the second is an application which has industrial interest for the design of turbine blades. Copyright © 2007 John Wiley & Sons, Ltd.     

Plastic,viscoplastic and creep fracture problems with the boundary element method

The present paper shows the applicability of the dual boundary element method to analyse plastic, viscoplastic and creep behaviours in fracture mechanics problems. Several models with a crack, including a square plate, a holed plate and a notched plate, are analysed. Special attention is taken when the discretization of the domain is performed. In fact, for the plasticity and viscoplasticity cases, only the region susceptible to yielding was discretized, whereas the creep case required the discretization of the whole domain. The proposed formulation is presented as an alternative technique to study these kinds of nonlinear problems. Results from the present formulation are compared to those of the well‐established finite element technique, and they are in good agreement. Important fracture mechanic parameters like K_I, K_II, J‐integrals and C‐integrals are also included. In general, the results, for the plastic, viscoplastic and creep cases, exhibit that the highest stress concentrations are in the vicinity of the crack tip and they decrease as the distance from the crack tip is increased.     

Non‐uniform plastic deformation of micron scale objects

Significant increases in apparent flow strength are observed when non‐uniform plastic deformation of metals occurs at the scale ranging from roughly one to ten microns. Several basic plane strain problems are analysed numerically in this paper based on a new formulation of strain gradient plasticity. The problems are the tangential and normal loading of a finite rectangular block of material bonded to rigid platens and having traction‐free ends, and the normal loading of a half‐space by a flat, rigid punch. The solutions illustrate fundamental features of plasticity at the micron scale that are not captured by conventional plasticity theory. These include the role of material length parameters in establishing the size dependence of strength and the elevation of resistance to plastic flow resulting from constraint on plastic flow at boundaries. Details of the finite element method employed in the numerical analysis of the higher order gradient theory will be discussed and related to prior formulations having some of the same features. Copyright © 2003 John Wiley & Sons, Ltd.     

A modified finite difference sensitivity analysis method allowing remeshing in large strain path‐dependent problems

Finite difference sensitivity analysis is simple and general yet usually inefficient and inaccurate compared to the analytical sensitivity approach. Although its high computational cost is not an issue in iteratively solved problems, its inaccuracies are critical in path‐dependent problems when remeshing is required. In this case, the errors caused by parametric inversion and interpolation in variables transfer to the new mesh can be as large as the gradient components. This paper presents an efficient modified finite difference approach that allows remeshing either in path‐independent or path‐dependent problems, not being affected by the aforementioned errors. The strategy to cope with remeshing is extensive to the semi‐analytical method which, for non‐linear analyses, is shown to be a particular case of the proposed finite difference sensitivity approach. With this implementation, the finite difference, the semi‐analytical and the analytical sensitivity methods all have comparable computational costs. The perturbation of unstructured meshes is performed with an inverse power Laplacian smoothing. The low cost and the accuracy of the sensitivity fields obtained after remeshing are shown in two examples, considering shape and constitutive design variables. Copyright © 2004 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.     

Finite element response sensitivity analysis using force‐based frame models

This paper presents a method to compute consistent response sensitivities of force‐based finite element models of structural frame systems to both material constitutive and discrete loading parameters. It has been shown that force‐based frame elements are superior to classical displacement‐based elements in the sense that they enable, at no significant additional costs, a drastic reduction in the number of elements required for a given level of accuracy in the computed response of the finite element model. This advantage of force‐based elements is of even more interest in structural reliability analysis, which requires accurate and efficient computation of structural response and structural response sensitivities. This paper focuses on material non‐linearities in the context of both static and dynamic response analysis. The formulation presented herein assumes the use of a general‐purpose non‐linear finite element analysis program based on the direct stiffness method. It is based on the general so‐called direct differentiation method (DDM) for computing response sensitivities. The complete analytical formulation is presented at the element level and details are provided about its implementation in a general‐purpose finite element analysis program. The new formulation and its implementation are validated through some application examples, in which analytical response sensitivities are compared with their counterparts obtained using forward finite difference (FFD) analysis. The force‐based finite element methodology augmented with the developed procedure for analytical response sensitivity computation offers a powerful general tool for structural response sensitivity analysis. Copyright © 2004 John Wiley & Sons, Ltd.     

Slope stability analysis based on elasto‐plastic finite element method

The paper deals with two essential and related closely processes involved in the finite element slope stability analysis in two‐dimensional problems, i.e. computation of the factors of safety (FOS) and location of the critical slide surfaces. A so‐called ?–v inequality, sin ??1 – 2v is proved for any elasto‐plastic material satisfying Mohr–Coulomb's yield criterion. In order to obtain an FOS of high precision with less calculation and a proper distribution of plastic zones in the critical equilibrium state, it is stated that the Poisson's ratio v should be adjusted according to the principle that the ?–v inequality always holds as reducing the strength parameters c and ?. While locating the critical slide surface represented by the critical slide line (CSL) under the plane strain condition, an initial value problem of a system of ordinary differential equations defining the CSL is formulated. A robust numerical solution for the initial value problem based on the predictor–corrector method is given in conjunction with the necessary and sufficient condition ensuring the convergence of solution. A simple example, the kinematic solution of which is available, and a challenging example from a hydraulic project in construction are analysed to demonstrate the effectiveness of the proposed procedures. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite element implementation of large deformation micropolar plasticity exhibiting isotropic and kinematic hardening effects

Micropolar theories offer a possibility to model size effects in the constitutive behaviour of materials. Typical feature of such models is that they deal with a microrotation, which is supposed to represent an independent state variable, and its space gradient. As a consequence, the stress tensor is no longer symmetric and couple stresses enter the theory. Accordingly, a micropolar plasticity law exhibiting kinematic hardening effects should account for both, a back‐stress tensor and a back‐couple stress tensor. This has been considered in the micropolar plasticity model developed by Grammenoudis and Tsakmakis. The purpose of the current paper is to specify some constitutive functions in this model, to elucidate the finite element implementation as well as to demonstrate its capabilities in describing size effects. Copyright © 2005 John Wiley & Sons, Ltd.