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.     

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.     

Coupled multi‐scale cohesive modeling of failure in heterogeneous adhesives

A multi‐scale cohesive numerical framework is proposed to simulate the failure of heterogeneous adhesively bonded systems. This multi‐scale scheme is based on Hill's variational principle of energy equivalence between the higher and lower level scales. It provides an easy way to obtain accurate homogenized macroscopic properties while capturing the physics of failure processes at the micro‐scale in sufficient detail. We use an isotropic rate‐dependent damage model to mimic the failure response of the constituents of heterogeneous adhesives. The finite element method is used to solve the equilibrium equation at each scale. A nested iterative scheme inspired by the return mapping algorithm used in computational inelasticity is implemented. We propose a computationally attractive technique to couple the macro‐ and micro‐scales for rate‐dependent constitutive laws. We introduce an adhesive patch test to study the numerical performance, including spatial and temporal convergence of the multi‐scale scheme. We compare the solution of the multi‐scale cohesive scheme with a direct numerical simulation. Finally, we solve mode I and mode II fracture problems to demonstrate failure at the macro‐scale. Copyright © 2010 John Wiley & Sons, Ltd.     

Design sensitivity analysis in large deformation elasto‐plastic and elasto‐viscoplastic problems

Implicit time integration algorithm derived by Simo for his large‐deformation elasto‐plastic constitutive model is generalized, for the case of isotropy and associative flow rule, towards viscoplastic material behaviour and consistently differentiated with respect to its input parameters. Combining it with the general formulation of design sensitivity analysis (DSA) for non‐linear finite element transient equilibrium problem, we come at a numerically efficient, closed‐form finite element formulation of DSA for large deformation elasto‐plastic and elasto‐viscoplastic problems, with various types of design variables (material constants, shape parameters). The paper handles several specific issues, like the use of a non‐algorithmic coefficient matrix or sensitivity discontinuities at points of instantaneous structural stiffness change. Computational examples demonstrate abilities of the formulation and quality of results. Copyright © 2005 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.     

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.     

A computational framework for scale‐bridging in multi‐scale simulations

A computational framework for scale‐bridging in multi‐scale simulations is presented. The framework enables seamless combination of at‐scale models into highly dynamic hierarchies to build a multi‐scale model. Its centerpiece is formulated as a standalone module capable of fully asynchronous operation. We assess its feasibility and performance for a two‐scale model applied to two challenging test problems from impact physics. We find that the computational cost associated with using the framework may, as expected, become substantial. However, the framework has the ability of effortlessly combining at‐scale models to render complex multi‐scale models. The main source of the computational inefficiency of the framework is related to poor load balancing of the lower‐scale model evaluation We demonstrate that the load balancing can be efficiently addressed by recourse to conventional load‐balancing strategies. Copyright © 2016 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.     

A node‐to‐node scheme with the aid of variable‐node elements for elasto‐plastic contact analysis

A strategy for a two‐dimensional contact analysis involving finite strain plasticity is developed with the aid of variable‐node elements. The variable‐node elements, in which nodes are added freely where they are needed, make it possible to transform the non‐matching meshes into matching meshes directly. They thereby facilitate an efficient analysis, maintaining node‐to‐node contact during the contact deformation. The contact patch test, wherein the contact patch is constructed out of variable‐node elements, is thus passed, and iterations for equilibrium solutions reach convergence faster in this scheme than in the conventional approach based on the node‐to‐surface contact. The effectiveness and accuracy of the proposed scheme are demonstrated through several numerical examples of elasto‐plastic contact analyses. Copyright © 2015 John Wiley & Sons, Ltd.     

A new multi‐scale scheme for modeling heterogeneous incompressible hyperelastic materials

A computational homogenization scheme is developed to model heterogeneous hyperelastic materials undergoing large deformations. The homogenization scheme is based on a so‐called computational continua formulation in which the macro‐scale model is assumed to consist of disjoint unit cells. This formulation adds no higher‐order boundary conditions and extra degrees of freedom to the problem. A computational procedure is presented to calculate the macroscopic quantities from the solution of the representative volume element boundary value problem. The proposed homogenization scheme is verified against a direct numerical simulation. It is also shown that the computational cost of the proposed model is lower than that of standard homogenization schemes. Copyright © 2015 John Wiley & Sons, Ltd.     

A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non‐linear solids and saturated porous media

A time‐discontinuous Galerkin finite element method (DGFEM) for dynamics and wave propagation in non‐linear solids and saturated porous media is presented. The main distinct characteristic of the proposed DGFEM is that the specific P3–P1 interpolation approximation, which uses piecewise cubic (Hermite's polynomial) and linear interpolations for both displacements and velocities, in the time domain is particularly proposed. Consequently, continuity of the displacement vector at each discrete time instant is exactly ensured, whereas discontinuity of the velocity vector at the discrete time levels still remains. The computational cost is then obviously saved, particularly in the materially non‐linear problems, as compared with that required for the existing DGFEM. Both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non‐linear problems. Numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional Galerkin finite element method using the Newmark algorithm in the time domain. 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.     

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.     

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.     

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.     

Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme

A gradient‐enhanced computational homogenization procedure, that allows for the modelling of microstructural size effects, is proposed within a general non‐linear framework. In this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element (RVE). This enables us to incorporate the microstructural size and to account for non‐uniform macroscopic deformation fields within the microstructural cell. Every microstructural constituent is modelled as a classical continuum and the RVE problem is formulated in terms of standard equilibrium and boundary conditions. From the solution of the microstructural boundary value problem, the macroscopic stress tensor and the higher‐order stress tensor are derived based on an extension of the Hill–Mandel condition. This automatically delivers the microstructurally based constitutive response of the higher‐order macro continuum and deals with the microstructural size in a natural way. Several examples illustrate the approach, particularly the microstructural size effects. Copyright © 2002 John Wiley & Sons, Ltd.     

An energy momentum conserving algorithm using the variational formulation of visco‐plastic updates

In this paper we use the variational formulation of elasto‐plastic updates proposed by Ortiz and Stainier (Comput. Methods Appl. Mech. Eng. 1999; 171 :419– 444) in the context of consistent time integration schemes. We show that such a formulation is well suited to obtain a general expression of energy momentum conserving algorithms. Moreover, we present numerical examples that illustrate the efficiency of our developments. Copyright © 2005 John Wiley & Sons, Ltd.     

A call to arms for task parallelism in multi‐scale materials modeling

Simulations based on multi‐scale material models enabled by adaptive sampling have demonstrated speedup factors exceeding an order of magnitude. The use of these methods in parallel computing is hampered by dynamic load imbalance, with load imbalance measurably reducing the achieved speedup. Here we discuss these issues in the context of task parallelism, showing results achieved to date and discussing possibilities for further improvement. In some cases, the task parallelism methods employed to date are able to restore much of the potential wall‐clock speedup. The specific application highlighted here focuses on the connection between microstructure and material performance using a polycrystal plasticity‐based multi‐scale method. However, the parallel load balancing issues are germane to a broad class of multi‐scale problems. Copyright © 2011 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.