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.     

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.     

Finite element study of the energy dissipation and residual stresses in the closed elastic deformation path

In this paper, energy dissipation and residual stress developments are numerically studied in three‐dimensional closed deformation paths. Different objective stress rates coded in a finite element program are compared. In order to update the stresses, implicit integration algorithm based on mid‐point rule for corotational and non‐corotational objective rates is used. Several corotational objective rates such as Jaumann, Green–Naghdi, Eulerian and Lagrangian triad‐based rates and non‐corotational rates such as Truesdell and Cotter–Rivlin rates are considered. It is shown in this work that in some cases also a non‐integrable model may exhibit no dissipation energy at the end of a closed deformation path. This study underlines some results previously obtained by other researchers, i.e. among all considered stress rates the logarithmic rate manifests the best result in respect of elasticity requirements. Copyright © 2006 John Wiley & Sons, Ltd.     

A particle‐in‐cell solution to the silo discharging problem

The problem of flow of a granular material during the process of discharging a silo is considered in the present paper. The mechanical behaviour of the material is described by the use of the model of the elastic–plastic solid with the Drucker–Prager yield condition and the non‐associative flow rule. The phenomenon of friction between the stored material and the silo walls is taken into account—the Coulomb model of friction is used in the analysis. The problem is analysed by means of the particle‐in‐cell method—a variant of the finite element method which enables to solve the pertinent equations of motion on an arbitrary computational mesh and trace state variables at points of the body chosen independently of the mesh. The method can be regarded as an arbitrary Lagrangian–Eulerian formulation of the finite element method, and overcomes the main drawback of the updated Lagrangian formulation of FEM related to mesh distortion. The entire process of discharging a silo can be analysed by this approach. The dynamic problem is solved by the use of the explicit time‐integration scheme. Several numerical examples are included. The plane strain and axisymmetric problems are solved for silos with flat bottoms and conical hoppers. Some results are compared with experimental ones. Copyright © 1999 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.     

A three‐invariant hardening plasticity for numerical simulation of powder forming processes via the arbitrary Lagrangian–Eulerian FE model

In this paper, a three‐invariant cap plasticity model with an isotropic hardening rule is presented for numerical simulation of powder compaction processes. A general form is developed for single‐cap plasticity which can be compared with some common double‐surface plasticity models proposed for powders in literature. The constitutive elasto‐plastic matrix and its components are derived based on the definition of yield surface, hardening parameter and non‐linear elastic behaviour, as function of relative density of powder. Different aspects of the new single plasticity are illustrated by generating the classical plasticity models as special cases of the proposed model. The procedure for determination of powder parameters is described by fitting the model to reproduce data from triaxial compression and confining pressure experiments. The three‐invariant cap plasticity is performed within the framework of an arbitrary Lagrangian–Eulerian formulation, in order to predict the non‐uniform relative density distribution during large deformation of powder die pressing. In ALE formulation, the reference configuration is used for describing the motion, instead of material configuration in Lagrangian, and spatial configuration in Eulerian formulation. This formulation introduces some convective terms in the finite element equations and consists of two phases. Each time step is analysed according to Lagrangian phase until required convergence is attained. Then, the Eulerian phase is applied to keep mesh configuration regular. Because of relative displacement between mesh and material, all dependent variables such as stress and strain are converted through the Eulerian phase. Finally, the numerical schemes are examined for efficiency and accuracy in the modelling of a rotational flanged component, an automotive component, a conical shaped‐charge liner and a connecting‐rod. Copyright © 2005 John Wiley & Sons, Ltd.     

Floating stress‐point integration meshfree method for large deformation analysis of elastic and elastoplastic materials

A new meshfree formulation of stress‐point integration, called the floating stress‐point integration meshfree method, is proposed for the large deformation analysis of elastic and elastoplastic materials. This method is a Galerkin meshfree method with an updated Lagrangian procedure and a quasi‐implicit time‐advancing scheme without any background cell for domain integration. Its new formulation is based on incremental equilibrium equations derived from the incremental virtual work equation, which is not generally used in meshfree formulations. Hence, this technique allows the temporal continuity of the mechanical equilibrium to be naturally achieved. The details of the new formulation and several examples of the large deformation analysis of elastic and elastoplastic materials are presented to show the validity and accuracy of the proposed method in comparison with those of the finite element method. Copyright © 2012 John Wiley & Sons, Ltd.     

Cost reduction techniques for the design of non‐linear flapping wing structures

This work details a computational framework for gradient‐based optimization of a non‐linear flapping wing structure with a large number of design variables, where analytical sensitivities of the unsteady finite element system are computed using the adjoint method. Two techniques are used to reduce the large computational cost of this structural design process. The first projects the finite element system onto a reduced basis of POD modes. The second uses a monolithic time formulation with spectral elements, and can be used to compute only the desired time‐periodic response. Results are given in terms of the trade‐off between accuracy and computational efficiency of these methods for both system response and adjoint computations, for a variety of mesh/time step refinements, degrees of non‐linearity (i.e. weakly or strongly non‐linear), and harmonic content. The work concludes with the structural design of a flapping wing: the elastic deformation at the wingtip is minimized through the flapping stroke by varying the thickness of each finite element. Significant improvements in computational cost are obtained at little expense to the accuracy of the results obtained via design optimization. Published in 2011 by John Wiley & Sons, Ltd.     

An additive theory for finite elastic–plastic deformations of the micropolar continuous media

In this paper, the method of additive plasticity at finite deformations is generalized to the micropolar continuous media. It is shown that the non-symmetric rate of deformation tensor and gradient of gyration vector could be decomposed into elastic and plastic parts. For the finite elastic deformation, the micropolar hypo-elastic constitutive equations for isotropic micropolar materials are considered. Concerning the additive decomposition and the micropolar hypo-elasticity as the basic tools, an elastic–plastic formulation consisting of an arbitrary number of internal variables and arbitrary form of plastic flow rule is derived. The localization conditions for the micropolar material obeying the developed elastic–plastic constitutive equations are investigated. It is shown that in the proposed formulation, the rate of skew-symmetric part of the stress tensor does not exhibit any jump across the singular surface. As an example, a generalization of the Drucker–Prager yield criterion to the micropolar continuum through a generalized form of the J ₂-flow theory incorporating isotropic and kinematic hardenings is introduced.     

An arbitrary Lagrangian–Eulerian finite element method for finite strain plasticity

This paper presents a new arbitrary Lagrangian–Eulerian (ALE) finite element formulation for finite strain plasticity in non‐linear solid mechanics. We consider the models of finite strain plasticity defined by the multiplicative decomposition of the deformation gradient in an elastic and a plastic part ( F = F ^e F ^p), with the stresses given by a hyperelastic relation. In contrast with more classical ALE approaches based on plastic models of the hypoelastic type, the ALE formulation presented herein considers the direct interpolation of the motion of the material with respect to the reference mesh together with the motion of the spatial mesh with respect to this same reference mesh. This aspect is shown to be crucial for a simple treatment of the advection of the plastic internal variables and dynamic variables. In fact, this advection is carried out exactly through a particle tracking in the reference mesh, a calculation that can be accomplished very efficiently with the use of the connectivity graph of the fixed reference mesh. A staggered scheme defined by three steps (the smoothing, the advection and the Lagrangian steps) leads to an efficient method for the solution of the resulting equations. We present several representative numerical simulations that illustrate the performance of the newly proposed methods. Both quasi‐static and dynamic conditions are considered in these model examples. Copyright © 2003 John Wiley & Sons, Ltd.     

Geometrically non‐linear thermoelastic analysis of functionally graded shells using finite element method

A finite element formulation governing the geometrically non‐linear thermoelastic behaviour of plates and shells made of functionally graded materials is derived in this paper using the updated Lagrangian approach. Derivation of the formulation is based on rewriting the Green–Lagrange strain as well as the 2nd Piola–Kirchhoff stress as two second‐order functions in terms of a through‐the‐thickness parameter. Material properties are assumed to vary through the thickness according to the commonly used power law distribution of the volume fraction of the constituents. Within a non‐linear finite element analysis framework, the main focus of the paper is the proposal of a formulation to account for non‐linear stress distribution in FG plates and shells, particularly, near the inner and outer surfaces for small and large values of the grading index parameter. The non‐linear heat transfer equation is also solved for thermal distribution through the thickness by the Rayleigh–Ritz method. Advantages of the proposed approach are assessed and comparisons with available solutions are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Velocity‐based formulations for standard and quasi‐incompressible hypoelastic‐plastic solids

We present three velocity‐based updated Lagrangian formulations for standard and quasi‐incompressible hypoelastic‐plastic solids. Three low‐order finite elements are derived and tested for non‐linear solid mechanics problems. The so‐called V‐element is based on a standard velocity approach, while a mixed velocity–pressure formulation is used for the VP and the VPS elements. The two‐field problem is solved via a two‐step Gauss–Seidel partitioned iterative scheme. First, the momentum equations are solved in terms of velocity increments, as for the V‐element. Then, the constitutive relation for the pressure is solved using the updated velocities obtained at the previous step. For the VPS‐element, the formulation is stabilized using the finite calculus method in order to solve problems involving quasi‐incompressible materials. All the solid elements are validated by solving two‐dimensional and three‐dimensional benchmark problems in statics as in dynamics. Copyright © 2016 John Wiley & Sons, Ltd.     

Formulation and numerical treatment of incompressibility constraints in large strain elastic–plastic analysis

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate‐independent finite strain analysis of solids undergoing large elastic‐isochoric plastic deformations. The formulation relies on the introduction of a mixed‐variant metric deformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure which can be additively decomposed into the exact isochoric (deviatoric) and volumetric (spheric) strain measures. This fact may be seen as the basic idea in the formulation of appropriate mixed finite elements which guarantee the accurate computation of isochoric strains. The mixed‐variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response whereas the plastic material behavior is assumed to be governed by a generalized J₂ yield criterion and rate‐independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on higher‐order Padé approximations. To be able to take into account the plastic incompressibility constraint a modified mixed variational principle is considered which leads to a quasi‐displacement finite element procedure. Finally, the numerical solution of finite strain elastic‐plastic problems is presented to demonstrate the efficiency and the accuracy of the algorithm. Copyright © 1999 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.     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part II: 3‐D non‐linear behaviour

In a companion paper, the effects of approximations in the flexural‐torsional stability analysis of beams was studied, and it was shown that a second‐order rotation matrix was sufficiently accurate for a flexural‐torsional stability analysis. However, the second‐order rotation matrix is not necessarily accurate in formulating finite element model for a 3‐D non‐linear analysis of thin‐walled beams of open cross‐section. The approximations in the second‐order rotation matrix may introduce ‘self‐straining’ due to superimposed rigid‐body motions, which may lead to physically incorrect predictions of the 3‐D non‐linear behaviour of beams. In a 3‐D non‐linear elastic–plastic analysis, numerical integration over the cross‐section is usually used to check the yield criterion and to calculate the stress increments, the stress resultants, the elastic–plastic stress–strain matrix and the tangent modulus matrix. A scheme of the arrangement of sampling points over the cross‐section that is not consistent with the strain distributions may lead to incorrect predictions of the 3‐D non‐linear elastic–plastic behaviour of beams. This paper investigates the effects of approximations on the 3‐D non‐linear analysis of beams. It is found that a finite element model for 3‐D non‐linear analysis based on the second‐order rotation matrix leads to over‐stiff predictions of the flexural‐torsional buckling and postbuckling response and to an overestimate of the maximum load‐carrying capacities of beams in some cases. To perform a correct 3‐D non‐linear analysis of beams, an accurate model of the rotations must be used. A scheme of the arrangement of sampling points over the cross‐section that is consistent with both the longitudinal normal and shear strain distributions is needed to predict the correct 3‐D non‐linear elastic–plastic behaviour of beams. Copyright © 2001 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.     

A mixed augmented Lagrangian-extended finite element method for modelling elastic–plastic fatigue crack growth with unilateral contact

The complete modelling of fatigue crack growth is still an industrial challenging issue for numerical methods. A new technique for the finite element modelling of elastic–plastic fatigue crack growth with unilateral contact on the crack faces is presented. The extended finite element method (X-FEM) is used to discretize the equations, allowing for the modelling of arbitrary cracks whose geometries are independent of the finite element mesh. This paper presents an augmented Lagrangian formulation in the X-FEM framework that is able to deal with elastic–plastic crack growth with treatment of contact. An original formulation, which takes advantages of two powerful numerical methods, is presented. Next the numerical issues such as contact treatment and numerical integration are addressed, and finally numerical examples are shown to validate the method. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis on the fatigue damage evolution of notched specimens with consideration of cyclic plasticity

This paper presents a damage mechanics method applied successfully to assess fatigue life of notched specimens with plastic deformation at the notch tip. A damage‐coupled elasto‐plastic constitutive model is employed in which nonlinear kinematic hardening is considered. The accumulated damage is described by a stress‐based damage model and a plastic strain‐based damage model, which depend on the cyclic stress and accumulated plastic strain, respectively. A three‐dimensional finite element implementation of these models is developed to predict the crack initiation life of notched specimens. Two cases, a notched plate under tension‐compression loadings and an SAE notched shaft under bending‐torsion loadings including non‐proportional loadings, are studied and the predicted results are compared with experimental data.