Large strain elastic-plastic theory and nonlinear finite element analysis based on metric transformation tensors

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-plastic deformations. The formulation relies on the introduction of a mixed-variant metric transformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure whose rate is shown to be additively decomposed into elastic and plastic strain rate tensors. The mixed-variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response in the elastic-plastic solid. Additionally, the plastic material behavior is assumed to be governed by a generalized J ₂ 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 1st and higher order Padé approximations. Estimates of the stress and strain histories are obtained via a highly stable and accurate explicit scalar integration procedure which employs a plastic predictor followed by an elastic corrector step. The development of a consistent elastic-plastic tangent operator as well as its implementation into a nonlinear finite element program will also be discussed. Finally, the numerical solution of finite strain elastic-plastic problems is presented to demonstrate the efficiency of the algorithm. Received: 17 May 1998     

Phenomenological modelling of rate-dependent plasticity for high strain rate problems

The results of two related theoretical investigations for large-scale computations of elasto-plastic deformations at ultrahigh strain rates are summarized in this paper. The first effort concerns the development of a phenomenological constitutive model for finite deformation elasto-plasticity which includes the effects of thermal softening, strain hardening, rate-dependence, as well as the noncoaxiality of the plastic strain rate and the stress deviator, and the incorporation of this model in a large-scale explicit finite-element code. The second effort involves the investigation of localized deformations and shear banding at high strain rates. It is shown that the constitutive model considered, together with standard quadrilateral finite elements with one point integration (piece-wise constant strain and stress fields), can nicely produce the observed intense localized deformations without recourse to any special elements. The results are illustrated in terms of the shear localization observed in uniaxial extension, and of void collapse under uniaxial compression. In addition, the effect of the noncoaxiality of the plastic strain rate and the stress tensor is included in the model, and its influence on strain localization at high strain rates is investigated.     

On the formulation of closest‐point projection algorithms in elastoplasticity—part I: The variational structure

We present in this paper the characterization of the variational structure behind the discrete equations defining the closest‐point projection approximation in elastoplasticity. Rate‐independent and viscoplastic formulations are considered in the infinitesimal and the finite deformation range, the later in the context of isotropic finite‐strain multiplicative plasticity. Primal variational principles in terms of the stresses and stress‐like hardening variables are presented first, followed by the formulation of dual principles incorporating explicitly the plastic multiplier. Augmented Lagrangian extensions are also presented allowing a complete regularization of the problem in the constrained rate‐independent limit. The variational structure identified in this paper leads to the proper framework for the development of new improved numerical algorithms for the integration of the local constitutive equations of plasticity as it is undertaken in Part II of this work. Copyright © 2001 John Wiley & Sons, Ltd.     

Estimation of elastic‐plastic strain response at two‐dimensional notches

It is widely recognized that the accuracy of notch fatigue calculations can be improved significantly when those calculations are based on the elastic‐plastic response strain at the notch root, as opposed to the remotely applied loads or stresses. Two of the most widely used approximations for this response are Neuber's rule and Glinka's equivalent strain energy density method. In the present work, a survey of some of the many published evaluations of these methods was first conducted, and then, additional detailed comparisons with elastic‐plastic finite element analyses for a series of semicircular and V‐shaped notch configurations were performed. Based on the observed limitations of both the Neuber and Glinka approaches, and with the guidance of the elastic‐plastic finite element results, a new (and more robust) approach for the estimation of notch response strains is proposed. This approach calls for the definition of a generalized notch response curve (GNRC), which is dependent on both the material stress–strain curve and the notch geometry. Once defined, the GNRC allows the determination of the response strain for any applied stress.     

BIEM-based variational principles for elastoplasticity with unilateral contact boundary conditions

The structural step problem for elastic-plastic internal-variable materials is addressed in the presence of frictionless unilateral contact conditions. Basing on the BIEM (boundary integral equation method) and making use of deformation-theory plasticity (through the backward-difference method of computational plasticity), two variational principles are shown to characterize the solution to the step problem: one is a stationarity principle having as unknowns all the problem variables, the other is a saddle-point principle having as unknowns the increments of the boundary tractions and displacements, along with the plastic strain increments in the domain. The discretization by boundary and interior elements transforms the above principles into well-posed mathematical programming formulations belonging to the symmetric Galerkin BEM formulations (with features such as a symmetric sign-definite coefficient matrix, double integrations, and hypersingular integrals).     

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

In this paper a modified multiplicative decomposition of the right stretch tensor is proposed and used for finite deformation elastoplastic analysis of hardening materials. The total symmetric right stretch tensor is decomposed into a symmetric elastic stretch tensor and a non-symmetric plastic deformation tensor. The plastic deformation tensor is further decomposed into an orthogonal transformation and a symmetric plastic stretch tensor. This plastic stretch tensor and its corresponding Hencky’s plastic strain measure are then used for the evolution of the plastic internal variables. Furthermore, a new evolution equation for the back stress tensor is introduced based on the Hencky plastic strain. The proposed constitutive model is integrated on the Lagrangian axis of the plastic stretch tensor and does not make reference to any objective rate of stress. The classic problem of simple shear is solved using the proposed model. Results obtained for the problem of simple shear are identical to those of the self-consistent Eulerian rate model based on the logarithmic rate of stress. Furthermore, extension of the proposed model to the mixed nonlinear isotropic/kinematic hardening behaviour is presented. The model is used to predict the nonlinear hardening behaviour of SUS 304 stainless steel under fixed end finite torsional loading. Results obtained are in good agreement with the available experimental results reported for this material under fixed end finite torsional loading.     

Analysis of forming rate sensitive materials

Summary To study how the rate of deformation effects forming rate sensitive materials, the Bodner-Partom elastic-viscoplastic constitutive law is incorporated into a finite element program. This law postulates both elastic and plastic components of deformation at any stress level. The stress is a Hookean function of the elastic strain, while the plastic deformation rate is a function of the deviatoric stress and an internal state variable defining the load history. The finite element derivation adopts the small strain assumption with updated coordinates. The equilibrium rate equation is formulated using total velocities with the nonlinearities incorporated into an equivalent plastic load vector depending upon the current stress. The resulting equation explicitly includes time and is a true rate equation. The program calculates the current stress field using the incremental equilibrium equation. An iterative technique is used to ensure that the assumed current load rate used to calculate the current stress field is correct. Convergence of the iteration procedure needs only be monitored at the velocity specified nodes. To demonstrate the applicability of this method, two plane strain problems, a tensile bar and strip rolling, with rate sensitive materials are investigated.With 12 Figures     

Influence of Ramberg–Osgood fitting on the determination of plastic displacement rates in creep crack growth testing

This paper investigates the effect of the Ramberg–Osgood (R‐O) fitting procedures on plastic displacement rate estimates in creep crack growth testing, via detailed two‐dimensional and three‐dimensional finite‐element analyses of the standard compact tension specimen. Four different R‐O fitting procedures are considered: (i) fitting the entire true stress–strain data up to the ultimate tensile strength, (ii) fitting the true stress–strain data from 0.1% strain to 0.8 of the true ultimate strain, (iii) fitting the true stress–strain data only up to 5% strain and (iv) fitting the engineering stress–strain data. It is found that the first two fitting procedures can produce significant errors in plastic displacement rate estimates. The last two procedures, on the other hand, provide reasonably accurate plastic displacement rates and thus should be recommended in creep crack growth testing. Several advantages of fitting the engineering stress–strain data over fitting the true stress–strain data only up to 5% strain are discussed.     

The integrability criterion in finite elastoplasticity and its constitutive implications

Summary In traditional Eulerian rate formulation of finite elastoplasticity, an Eulerian rate equation of hypoelastic type is used as one of the basic constituents to relate the elastic part of the stretching to a stress rate. On account of the fact that the elastic-like behavior should be expected prior to yielding, the foregoing basic elastic rate equation should be exactly integrable to deliver a conventional hyperelastic stress-strain relation. Physically, it requires a consistent combination of hypoelasticity and hyperelasticity into a single Eulerian rate equation. Since this criterion is purely a physical consistency requirement and since the basic elastic rate equation involves no strain concept and allows for any stress rate in its own right, from a physical standpoint it may be of both interest and significance to know what consequences it will imply concerning the stress rate and the finite strain measure. By a simple, straightforward procedure we demonstrate in a general sense that both Hencky strain and the logarithmic rate emerge naturally as direct consequences of the foregoing criterion. This result may be regarded to reveal the physical essence behind Hencky strain and the logarithmic rate in connection with a basic physical consistency requirement in finite elastoplasticity. Constitutive implications are discussed in a few relevant respects concerning representative formulations of finite elastoplasticity.     

Efficient recovery-based error estimation for the smoothed finite element method for smooth and singular linear elasticity

An error control technique aimed to assess the quality of smoothed finite element approximations is presented in this paper. Finite element techniques based on strain smoothing appeared in 2007 were shown to provide significant advantages compared to conventional finite element approximations. In particular, a widely cited strength of such methods is improved accuracy for the same computational cost. Yet, few attempts have been made to directly assess the quality of the results obtained during the simulation by evaluating an estimate of the discretization error. Here we propose a recovery type error estimator based on an enhanced recovery technique. The salient features of the recovery are: enforcement of local equilibrium and, for singular problems a “smooth + singular” decomposition of the recovered stress. We evaluate the proposed estimator on a number of test cases from linear elastic structural mechanics and obtain efficient error estimations whose effectivities, both at local and global levels, are improved compared to recovery procedures not implementing these features.     

An isogeometric analysis approach to gradient damage models

Continuum damage formulations are commonly used for the simulation of diffuse fracture processes. Implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models. The C⁰‐continuity of traditional finite elements has hindered the study of higher order gradient damage approximations. In this contribution we use isogeometric finite elements, which allow for the construction of higher order continuous basis functions on complex domains. We study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite element formulations for large deformation dynamic analysis

Starting from continuum mechanics principles, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived. The aim in this paper is a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure. The general formulations include large displacements, large strains and material non-linearities. For specific static and dynamic analyses in this paper, elastic, hyperelastic (rubber-like) and hypoelastic elastic-plastic materials are considered. The numerical solution of the continuum mechanics equations is achieved using isoparametric finite element discretization. The specific matrices which need be calculated in the formulations are presented and discussed. To demonstrate the applicability and the important differences in the formulations, the solution of static and dynamic problems involving large displacements and large strains are presented.     

Localized deform behavior of AZ31B magnesium alloy by numerical simulation

The dynamic compression behavior of AZ31B magnesium alloy with hat shaped specimen was investigated at high strain rate in this paper. Based on the Johnson‐cook constitutive model and fracture model, the interaction of temperature, stress and strain fields of AZ31B magnesium alloy with hat shaped specimen were numerically simulated by using ANSYS/LS‐DYNA software under different strain rates, which was validated by experiment. It is found that the plastic strain is highly concentrated on the corner of the hat shaped specimen, which leads to large localized deformation. The voids are nucleated and extended by compression stress. Work harden effect is caused by remained plastic strain, which is located around adiabatic shear band. The stress collapse is discovered in gauge section, which is also discovered in experiment. Thermal soften effect is suppressed with the strain rate increased.     

Dispersion analysis of numerical approximations to plane wave motions in an isotropic elastic solid

 It is well known that isotropic, nondispersive continuous hyperbolic problems become dispersive and anisotropic upon discretization. The purpose of this paper is to conduct a dispersion analysis of the nondissipative numerical approximations to plane wave motions in isotropic elastic solids. The discrete formulations considered are: an explicit, second-order accurate finite difference scheme, a consistent mass matrix formulation with linear quadrilateral elements and the corresponding lumped mass matrix formulation. Dispersion relation is derived for each of these formulations. In the context of the finite difference scheme, expressions for group velocity for both the shear and longitudinal waves are derived and the effect of using meshes of unequal size in x and y directions is studied. Results from numerical experiments confirming the predictions of analysis are also presented. Received 22 October 1999     

Dual finite element formulations for lumped reluctances coupling

A method for coupling magnetostatic and magnetodynamic finite element formulations with lumped reluctances is developed. Two dual h and b-conform formulations are extended to define the necessary magnetic relations between the magnetic fluxes and the magnetomotive forces. Adequate surface scalar potentials are defined and adequate boundary terms in the weak formulations are used. The magnetic relations can then be included in a reluctance network, in which the coupling of finite element regions and lumped regions aims at higher computational efficiency. The methods are developed in three dimensions, using a coupling of nodal and edge finite element approximations for the unknowns, and can easily be particularized in two dimensions.     

Energy flux into process region in elastic-plastic fracture problems

Formulas for the fluxes of total and free energies into a finite fracture process region near a crack tip are derived taking account of the effects of the plastic deformation, thermal strain, body forces and inertia forces. Two different limiting procedures of shrinking the size of the process region are considered in order to obtain the formulas for a crack model with an infinitesimal process region. The formulations are extended to three-dimensional crack problems.     

Plastic intensity factors for cracked plates

An elastic-plastic analysis is performed for two problems relevant to fracture mechanics: a semiinfinite body with an edge crack in a far out-of-plane shearing field and an infinite plate under plane stress conditions containing a finite line crack in a remote tensile field. Amplitudes of the dominant singularity in the plastic region at the crack tip, the plastic stress and strain intensity factors, are calculated for applied stress levels approaching the yield stress. A technique is developed for using the dominant singular solution in conjunction with the finite element method to make accurate calculations for the near-tip fields. Additionally, a comparative study of deformation theory with flow theory is performed for cracks in an anti-plane shear field. Elastic fracture mechanics is extended to high levels of applied stress for which the plastic zone is no longer small compared to the crack length by relating the critical stress for fracture initiation to the plastic intensity factors.