An improved strain gradient plasticity formulation with energetic interfaces: theory and a fully implicit finite element formulation

A fully implicit backward-Euler implementation of a higher order strain gradient plasticity theory is presented. A tangent operator consistent with the numerical update procedure is given. The implemented theory is a dissipative bulk formulation with energetic contribution from internal interface to model the behavior of material interfaces at small length scales. The implementation is tested by solving some examples that specifically highlight the numerics and the effect of using the energetic interfaces as higher order boundary conditions. Specifically, it is demonstrated that the energetic interface formulation is able to mimic a wide range of plastic strain conditions at internal boundaries. It is also shown that delayed micro-hard conditions may arise under certain circumstances such that an interface at first offers little constraints on plastic flow, but with increasing plastic deformation will develop and become a barrier to dislocation motion.     

Solution of the slump test using a finite deformation elasto-plastic Drucker-Prager model

Development of a finite deformation elasto-plasticity model based on the multiplicative decomposition of the deformation gradient is presented and discussed in detail. The formulation presented in this paper includes the derivation of the full set of equations for the Drucker-Prager yield criterion. The equations, which are not available elsewhere, are developed within a framework using a spectral decomposition approach. Further, expressions for the consistent (algorithmic) tangent moduli in the finite strain regime are developed. Since the finite deformation framework employed to obtain the expressions presented here collapses to the classical infinitesimal plasticity framework when the finite strain assumption is no longer necessary, the finite deformation consistent tangent moduli are compared to the consistent tangent moduli valid for use with infinitesimal plasticity. Validation of the implemented finite deformation elasto-plastic Drucker-Prager model is performed through the solution of the concrete slump test. Comparisons between an existing approximate analytical solution and experimental data are presented, and results are discussed in detail.     

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     

A large strain plasticity model for implicit finite element analyses

The theoretical basis and numerical implementation of a plasticity model suitable for finite strains and rotations are described. The constitutive equations governing J ₂ flow theory are formulated using strains-stresses and their rates defined on the unrotated frame of reference. Unlike models based on the classical Jaumann (or corotational) stress rate, the present model predicts physically acceptable responses for homogeneous deformations of exceedingly large magnitude. The associated numerical algorithms accommodate the large strain increments that arise in finite-element formulations employing an implicit solution of the global equilibrium equations. The resulting computational framework divorces the finite rotation effects on strain-stress rates from integration of the rates to update the material response over a load (time) step. Consequently, all of the numerical refinements developed previously for small-strain plasticity (radial return with subincrementation, plane stress modifications, kinematic hardening, consistent tangent operators) are utilized without modification. Details of the numerical algorithms are provided including the necessary transformation matrices and additional techniques required for finite deformations in plane stress. Several numerical examples are presented to illustrate the realistic responses predicted by the model and the robustness of the numerical procedures.     

Assumed‐deformation gradient finite elements with nodal integration for nearly incompressible large deformation analysis

An assumed‐strain finite element technique for non‐linear finite deformation is presented. The weighted‐residual method enforces weakly the balance equation with the natural boundary condition and also the kinematic equation that links the elementwise and the assumed‐deformation gradient. Assumed gradient operators are derived via nodal integration from the kinematic‐weighted residual. A variety of finite element shapes fits the derived framework: four‐node tetrahedra, eight‐, 27‐, and 64‐node hexahedra are presented here. Since the assumed‐deformation gradients are expressed entirely in terms of the nodal displacements, the degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials and no volumetric/deviatoric split is required. The consistent tangent operator is inexpensive and symmetric. Furthermore, the material update and the tangent moduli computation are carried out exactly as for classical displacement‐based models; the only deviation is the consistent use of the assumed‐deformation gradient in place of the displacement‐derived deformation gradient. Examples illustrate the performance with respect to the ability of the present technique to resist volumetric locking. A constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Copyright © 2008 John Wiley & Sons, Ltd.     

Macroscopic theory and nonlinear finite element analysis of micromechanics of single crystals at finite strains

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the macroscopic rate-independent and rate-dependent analysis of micromechanics of metal single crystals undergoing finite elastic-plastic deformations which is based on the assumption that inelastic deformation is solely due to crystallographic slip. The formulation relies on a multiplicative decomposition of the material deformation gradient into incompressible elastic and plastic as well as a scalar valued volumetric part. Furthermore, the crystal deformation is described as arising from two distinct physical mechanisms, elastic deformation due to distortion of the lattice and crystallographic slip due to shearing along certain preferred lattice planes in certain preferred lattice directions. Macro- and microscopic stress measures are related to Green’s macroscopic strains via a hyperelastic constitutive law based on a free energy potential function, whereas plastic potentials expressed in terms of the generalized Schmid stress lead to a normality rule for the macroscopic plastic strain rate. Estimates of the microscopic stress and strain histories are obtained via a highly stable and very accurate semi-implicit scalar integration procedure which employs a plastic predictor followed by an elastic corrector step, and, furthermore, 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 simulation of finite strain elastic-plastic tension tests is presented to demonstrate the efficiency of the algorithm.     

A computational model for elasto-viscoplastic solids at finite strain with reference to thin shell applications

This work extends a previously developed methodology for computational plasticity at finite strains that is based on the exponential map and logarithmic stretches to the context of isotropic elasto-viscoplastic solids. A particular form of the strain-energy function, given in terms of its principal values is employed. It is noticeable that within the proposed framework, the small strain integration algorithms, and the corresponding consistent tangent operators, automatically extend to the finite strain regime. Central to the effort of this formulation is the derivation of the closed form of a tangent modulus obtained by linearization of incremental non-linear problem. This ensures asymptotically quadratic rates of convergence of the Newton–Raphson procedure in the implicit finite element solution. To illustrate the performance of the presented formulation, several numerical examples, involving failure by strain localization and finite deformations, are given. © 1998 John Wiley & Sons, Ltd.     

An improved EAS brick element for finite deformation

A new enhanced assumed strain brick element for finite deformations in finite elasticity and plasticity is presented. The element is based on an expansion of shape function derivatives using Taylor series and an extended set of orthogonality conditions that have to be satisfied for an hourglassing free EAS formulation. Such approach has not been applied so far in the context of large deformation three-dimensional problems. It leads to a surprisingly well-behaved locking and hourglassing free element formulation. Major advantage of the new element is its shear locking free performance in the limit of very thin elements, thus it is applicable to shell type problems. Crucial for the derivation of the residual and consistent tangent matrix of the element is the automation of the implementation by automatic code generation.     

Three-dimensional finite element analysis of finite deformation micromorphic linear isotropic elasticity

A finite deformation micromorphic materially linear isotropic elastic model is formulated and implemented for three dimensional finite element analysis. The model is based on the kinematics, balance equations and thermodynamic equations proposed by Eringen and Suhubi (1964). The constitutive equations are calculated in the reference configuration, and the resulting stresses are mapped to the current configuration. The balance of linear momentum and the balance of first moment of momentum are linearized to construct the consistent tangent for three dimensional finite element implementation for solution by the Newton–Raphson method. Three dimensional numerical examples are analyzed to demonstrate preliminarily the implementation.     

Elastoplastic orthotropy at finite strains: multiplicative formulation and numerical implementation

A constitutive model for orthotropic elastoplasticity at finite plastic strains is discussed and basic concepts of its numerical implementation are presented. The essential features are the multiplicative decomposition of the deformation gradient in elastic and inelastic parts, the definition of a convex elastic domain in stress space and a representation of the constitutive equations related to the intermediate configuration. The elastic free energy function and the yield function are formulated in an invariant setting by means of the introduction of structural tensors reflecting the privileged directions of the material. The model accounts for kinematic and isotropic hardening. The associated flow rule is integrated using the so-called exponential map which preserves exactly the plastic incompressibility condition. The constitutive equations are implemented in a brick-type shell element. Representative numerical simulations demonstrate the suitability of the proposed formulations.     

Micropolar hyper‐elastoplasticity: constitutive model,consistent linearization,and simulation of 3D scale effects

A computational model for micropolar hyperelastic‐based finite elastoplasticity that incorporates isotropic hardening is developed. The basic concepts of the non‐linear micropolar kinematic framework are reviewed, and a thermodynamically consistent constitutive model that features Neo‐Hooke‐type elasticity and generalized von Mises plasticity is described. The integration of the constitutive initial value problem is carried out by means of an elastic‐predictor/plastic‐corrector algorithm, which retains plastic incompressibility. The solution procedure is developed carefully and described in detail. The consistent material tangent is derived. The micropolar constitutive model is implemented in an implicit finite element framework. The numerical example of a notched cylindrical bar subjected to large axial displacements and large twist angles is presented. The results of the finite element simulations demonstrate (i) that the methodology is capable of capturing the size effect in three‐dimensional elastoplastic solids in the finite strain regime, (ii) that the formulation possesses a regularizing effect in the presence of strain localization, and (iii) that asymptotically quadratic convergence rates of the Newton–Raphson procedure are achieved. Throughout this paper, effort is made to present the developments as a direct extension of standard finite deformation computational plasticity. Copyright © 2012 John Wiley & Sons, Ltd.     

Computational aspects of tangent moduli tensors in rate-independent crystal elastoplasticity

The computational efficiencies of the continuum and consistent (algorithmic) tangent moduli tensors in rate-independent crystal elastoplasticity are examined in conjunction with the available implicit state update algorithms. It is, in this context, shown that the consistent tangent moduli associated with the state update algorithm with the exponential mapping coincide with the continuum tangent moduli. After verifying the reported performance of the exponential mapping algorithm in preserving the incompressibility of plastic deformation in a single crystal grain, we carry out numerical experiments to understand the convergence trends of the global Newton–Raphson iterative procedure with different kinds of tangent moduli tensors. Having done this, we are concerned with the performance of those tangent moduli tensors for the micro-scale analysis of a polycrystalline aggregate, which is regarded as a representative volume element, subjected to macro-scale uniform deformation in the context of the two-scale homogenization method.     

A penalty finite element analysis for nonlinear mechanics of biphasic hydrated soft tissue under large deformation

The non-linear response of soft hydrated tissues under physiologically relevant levels of mechanical loading can be represented by a two-phase continuum model based on the theory of mixtures. The governing equations for a biphasic soft tissue, consisting of an incompressible solid and an incompressible, inviscid fluid, under finite deformation are presented and a finite element formulation of this highly non-linear problem is developed. The solid phase is assumed to be hyperelastic, and the stress-strain relations for the solid phase are defined in terms of the free energy function. A finite element model is formulated via the Galerkin weighted residual method coupled with a penalty treatment of the continuity equation for the mixture. Using a total Lagrangian formulation, the non-linear weighted residual statement, expressed with respect to the reference configuration, leads to a coupled non-linear system of first order differential equations. The non-linear constitutive equation for the solid phase elasticity is incrementally linearized in terms of the second Piola-Kirchhoff stress and the corresponding Lagrangian strain. A tangent stiffness matrix is defined in terms of the free energy function; this matrix definition can be applied to any free energy function, and will yield a symmetric matrix when the free energy function is convex. An unconditionally stable implicit predictor-corrector algorithm is used to obtain the temporal response histories. The confined compression mechanical test of soft tissue in stress relaxation is used as an example problem. Results are presented for moderate and rapid rates of loading, as well as small and large applied strains. Comparison of the finite element solution with an independent finite difference solution demonstrates the accuracy of the formulation.     

Investigation of integration algorithms for rate-dependent crystal plasticity using explicit finite element codes

The work presented here concerns the use of rate-dependent crystal plasticity into explicit dynamic finite element codes for structural analysis. Different integration or stress update algorithms for the numerical implementation of crystal plasticity, two explicit algorithms and a fully-implicit one, are described in detail and compared in terms of convergence, accuracy and computation time. The results show that the implicit time integration is very robust and stable, provided low enough convergence tolerance is used for low strain-rate sensitivity coefficients, while being the slowest in terms of CPU time. Explicit methods prove to be fast, stable and accurate. The algorithms are then applied to two structural analyses, one concerning flat rolling of a polycrystalline slab and another on the response of a multicrystalline sample under uniaxial tensile condition. The results show that the explicit algorithms perform well with simulation times much smaller compared to their implicit counterpart. Finally, mesh sensitivity for the second structural analysis is investigated and shows to slightly affect the global response of the structure.     

An explicit finite element approach with patch projection technique for strain gradient plasticity formulations

Implicit and explicit finite element approaches are frequently applied in real problems. Explicit finite element approaches exhibit several advantages over implicit method for problems which include dynamic effects and instability. Such problems also arise for materials and structures at small length scales and here length scales at the micro and sub-micron scales are considered. At these length scales size effects can be present which are often treated with strain gradient plasticity formulations. Numerical treatments for strain gradient plasticity applying the explicit finite element approach appear however to be absent in the scientific literature. Here such a numerical approach is suggested which is based on patch recovery techniques which have their origin in error indication procedures and adaptive finite element approaches. Along with the proposed explicit finite element procedure for a strain gradient plasticity formulation some numerical examples are discussed to assess the suggested approach.     

Finite deformation plasticity and viscoplasticity laws exhibiting nonlinear hardening rules

 This paper deals with plasticity and viscoplasticity laws exhibiting nonlinear kinematic hardening as well as nonlinear isotropic hardening rules. In Tsakmakis (1996a, b) a constitutive theory has been formulated within the framework of finite deformations, which is based on the concept of so-called dual variables and associated time derivatives. Within two families of dual variables, two different formulations have been proposed for kinematic hardening, referred to as Models 1 and 2. In particular, rigid plastic deformations without isotropic hardening have been considered. In the present paper, the constitutive theory of Tsakmakis (1996a, b) is appropriately extended to take into account isotropic hardening as well as elastic deformations. Care is taken that the evolution equations governing the hardening response fulfill the intrinsic dissipation inequality in every admissible process. For the case of small elastic strains combined with a simplification concerning kinematic hardening, to be explained in the paper, an efficient, implicit time-integration algorithm is presented. The algorithm is developed with a view to implementation in the ABAQUS Finite Element code. Also, explicit formulas for the consistent tangent modulus are derived. Received 22 September 1999     

Numerical implementation of constitutive integration for rate-independent elastoplasticity

In this paper, constitutive integration for rate-independent, small deformation elastoplasticity is studied. Smooth yield surfaces and work/strain hardening are assumed. Both associative or non-associative flow rules are considered. An Euler backward algorithm is applied for constitutive integration. Tangent moduli that are consistent with the Euler backward algorithm, i.e. a so-called consistent tangent operator, are derived. Emphasis is placed on numerical implementation of the Eular backward algorithm into finite element codes using such a consistent tangent operator. In particular, a commercial code ANSYS is considered. Numerical examples, including materials sensitive and insensitive to hydrostatic stress, are used for the verification of the implementation. A comparison of the algorithmic performance to an explicit Euler forward algorithm is given and the superiority of the Euler backward algorithm is demonstrated.The work described in the present paper has been sponsored by The Research Council of Norway, The North Calotte Education and Research Council, Statoil and Norsk Hydro.