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.     

Non‐linear analysis of shells with arbitrary evolving cracks using XFEM

A new formulation and numerical procedures are developed for the analysis of arbitrary crack propagation in shells using the extended finite element method. The method is valid for completely non‐linear problems. Through‐the‐thickness cracks in sandwich shells are considered. An exact shell kinematics is presented, and a new enrichment of the rotation field is proposed which satisfies the director inextensibility condition. To avoid locking, an enhanced strain formulation is proposed for the 4‐node cracked shell element. A finite strain plane stress constitutive model based on the logarithmic corotational rate is employed. A cohesive zone model is introduced which embodies the special characteristics of the shell kinematics. Stress intensity factors are calculated for selected problems and crack propagation problems are solved. Copyright © 2004 John Wiley & Sons, Ltd.     

A new SMA shell element based on the corotational formulation

Aim of this paper is to develop a new shape memory alloy (SMA) facet-shell finite element accounting for material and geometric nonlinearities. A corotational formulation is exploited, able to filter out large rigid-body motions from the element transformation. Accordingly, a geometrically linear core-element is employed, along with a SMA constitutive model formulated in the small strain framework. In particular, in accordance with the formulation of the classical thin shell theory, a plane-stress SMA model accounting for the pseudo-elastic as well as the shape memory effect is adopted. The time integration of the evolutive equation is performed developing a step-by-step backward-Euler numerical procedure. A highly efficient implementation of the corotational machinery is used, endowed with a fully consistent tangent stiffness. Applications are carried out for assessing the performances of the developed computational procedure and to investigate on some interesting engineering examples. The numerical results show the effectiveness of the proposed shell element, whose simplicity makes it attractive for the design of new advanced SMA-based devices undergoing significant configuration changes during their operation.     

Implicit algorithm for finite deformation hypoelastic–viscoplasticity in fcc metals

An implicit objective stress update algorithm is proposed for a hypoelastic–viscoplastic model. A thermal/dynamic yield function, which is derived based on the thermal activation analysis and dislocation interaction mechanisms, is used, along with the Consistency approach and the framework of additive viscoplasticity, in deriving the proposed model for fcc metals. The corotational formulation approach is utilized in developing the proposed model in the finite deformation field. For the case of the Newton–Raphson iteration method, a new expression for the consistent (algorithmic) tangent stiffness matrix of rate‐dependent metals is derived by direct linearization of the stress update algorithm. Finite element simulations are performed by implementing the proposed viscoplasticity constitutive models in the commercial finite element program ABAQUS. Numerical implementation for a simple tensile problem is used for validating the material parameters of the OFHC Copper under low and high strain rates and temperatures. The numerical results of the adiabatic true stress–true strain curves compare very well with the experimental data. The effectiveness of the present approach is tested by studying strain localization in a simple plane strain problem. Results indicate excellent performance of the present framework in describing the strain localization problem and in obtaining mesh‐independent results. Copyright © 2006 John Wiley & Sons, Ltd.     

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     

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.     

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.     

Constitutive modelling and numerical simulation of multivariant phase transformation in superelastic shape‐memory alloys

This work concerns the micromechanical constitutive modelling, algorithmic implementation and numerical simulation of polycrystalline superelastic alloys under multiaxial loading. The model is formulated in finite deformations and incorporates the effect of texture. The numerical implementation is based on the constrained minimization of the Helmholtz free energy with dissipation. Simulations are conducted for thin tubes of Nitinol under tension–torsion, as well as for a simplified model of a biomedical stent. Copyright © 2004 John Wiley & Sons, Ltd.     

Progressive damage modeling in fiber-reinforced materials

This paper presents an anisotropic damage model suitable for predicting failure and post-failure behavior in fiber-reinforced materials. In the model the plane stress formulation is used and the response of the undamaged material is assumed to be linearly elastic. The model is intended to predict behavior of elastic-brittle materials that show no significant plastic deformation before failure. Four different failure modes – fiber tension, fiber compression, matrix tension, and matrix compression – are considered and modeled separately. The onset of damage is predicted using Hashin’s initiation criteria [Hashin Z, Rotem A. A fatigue failure criterion for fiber-reinforced materials. J Compos Mater 1973;7:448; Hashin Z. Failure criteria for unidirectional fiber composites. J Appl Mech 1980;47:329–34] and the progression of damage is controlled by a new damage evolution law, which is easy to implement in a finite element code. The evolution law is based on fracture energy dissipation during the damage process and the increase in damage is controlled by equivalent displacements. The issues related to numerical implementation, such as mesh sensitivity and convergence in the softening regime, are also addressed.     

Axisymmetric three- and four-node finite elements for large strain elastoplasticity

Within the framework of the finite element method an application of the logarithmic strain space formulation of large strain elastoplasticity is illustrated for the examples of axisymmetric three-node triangular and four-node quadrilateral finite elements. The formulation of the large strain elastoplasticity is based on a strain space formulation in conjunction with logarithmic (or Hencky) strain tensors with respect to the reference configuration. It is therefore—from a material point of view—a full Lagrangian formulation. The use of logarithmic strains enables an additive split of finite dilatation and distortion, which are given by the logarithmic strain trace and deviator. As a consequence of the strain space formulation no stress tensors are involved in order to describe the plasticity. The stress which is work-conjugate to the logarithmic strain follows from the stress-strain relations and may be transformed to Cauchy stress. The desired finite element matrices are derived via the principle of virtual work applied to the Cauchy stress distribution of the current configuration. It should be noted that our considerations are not restricted to axisymmetry and that they remain valid for isoparametric, position- (displacement-) based finite elements in general.     

A variational constitutive model for porous metal plasticity

This paper presents a variational formulation of viscoplastic constitutive updates for porous elastoplastic materials. The material model combines von Mises plasticity with volumetric plastic expansion as induced, e.g., by the growth of voids and defects in metals. The finite deformation theory is based on the multiplicative decomposition of the deformation gradient and an internal variable formulation of continuum thermodynamics. By the use of logarithmic and exponential mappings the stress update algorithms are extended from small strains to finite deformations. Thus the time-discretized version of the porous-viscoplastic constitutive updates is described in a fully variational manner. The range of behavior predicted by the model and the performance of the variational update are demonstrated by its application to the forced expansion and fragmentation of U-6%Nb rings.     

Application of generalized measures to an orthotropic finite elasto-plasticity model

A numerical study of finite orthotropic elasto-plasticity based on generalized stress–strain measures is presented. The anisotropic constitutive equations are represented by isotropic tensor functions. A simple additive decomposition of strains can be performed due to the formulation in generalized measures. Furthermore, the plasticity model does not depend on special properties of any particular measure. The required projection tensor is constructed exploiting the coaxiality of the generalized deformation tensor with the right Cauchy–Green tensor. An efficient algorithmic implementation is proposed. Finally, we discuss representative numerical examples for orthotropic elasto-plasticity, where finite deformations occur.     

A finite‐strain gradient‐inelastic beam theory and a corresponding force‐based frame element formulation

This paper extends the gradient‐inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations; (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials; and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates that introduce accumulated computational error during their numerical time integration, and (b) directly integrates the strain‐displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length and, thus, address the coupling between equilibrium and strain‐displacement equations. This approach achieves high accuracy and mesh convergence rate and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a corotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations/imperfections of the initial member geometry.     

Eulerian constitutive model for finite deformation plasticity with anisotropic hardening

A constitutive model is presented for finite strain plasticity. The model incorporates both isotropic and kinematic hardening of the Ziegler type. The corotational rate used here is in line with the theory suggested by Paulun and Pecherski (1985) but not necessarily confined to the von Mises type yield criterion and the Prager hardening rule. The aspect of integration of the corotational rates is also discussed here. The use of the integration of the material rate of tensors with time as a substitute for the proper integration with time of corotational rates leads to mathematical inconsistencies of the theory of Lie derivatives. The problem of simple shear is investigated and compared with other works.     

Energy and momentum conserving elastodynamics of a non‐linear brick‐type mixed finite shell element

The paper presents aspects of the finite element formulation of momentum and energy conserving algorithms for the non‐linear dynamic analysis of shell‐like structures. The key contribution is a detailed analysis of the implementation of a Simó–Tarnow‐type conservation scheme in a recently developed new mixed finite shell element. This continuum‐based shell element provides a well‐defined interface to strain‐driven constitutive stress updates algorithms. It is based on the classic brick‐type trilinear displacement element and is equipped with specific gradient‐type enhanced strain modes and shell‐typical assumed strain modifications. The excellent performance of the proposed dynamic shell formulation with respect to conservation properties and numerical stability behaviour is demonstrated by means of three representative numerical examples of elastodynamics which exhibit complex free motions of flexible structures undergoing large strains and large rigid‐body motions. Copyright © 2001 John Wiley & Sons, Ltd.     

A 3D anisotropic elastoplastic‐damage model using discontinuous displacement fields

This paper is concerned with the development of constitutive equations for finite element formulations based on discontinuous displacement fields. For this purpose, an elastoplastic continuum model (stress–strain relation) as well as an anisotropic damage model (stress–strain relation) are projected onto a surface leading to traction separation laws. The coupling of both continuum models and, subsequently, the derivation of the corresponding constitutive interface law are described in detail. For a simple calibration of the proposed model, the fracture energy resulting from the coupled elastoplastic‐damage traction separation law is computed. By this, the softening evolution is linearly dependent on the fracture energy. The second part of the present paper deals with the numerical implementation. Based on a local and incompatible additive split of the displacement field into a continuous and a discontinuous part, the parameters specifying the jump of the displacement field are condensed out at the material level without employing the standard static condensation technique. To reduce locking effects, a rotating localization zone formulation is applied. The applicability and the performance of the proposed numerical implementation is investigated by means of a re‐analysis of a two‐dimensional L‐shaped slab as well as by means of a three‐dimensional ultimate load analysis of a steel anchor embedded in a concrete block. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Study of Mode I Fracture of Graphene Sheets Using Atomistic Based Finite Element Modeling and Virtual Crack Closure Technique

In this letter a finite element based atomistic model is proposed. This model was employed to study the mode I fracture characteristics of graphene monolayers. The proposed model was found to be less numerically intensive with fewer degrees of freedom in comparison to other numerical techniques, such as Monte Carlo and molecular dynamics modeling. An approach based on the virtual crack closure technique was employed to develop the atomistic formulation for estimating strain energy release rates for graphene sheets under opening mode.