An improved ‘assumed enhanced displacement gradient’ ring‐element for finite deformation axisymmetric and torsional problems

Analyzing axisymmetric solids under torsional loading the 3d(imensional) problem can always be reduced by one dimension, since the displacement field and the rotation field are independent of the cylindrical (angle) co‐ordinate Θ, respectively. For this purpose a four‐node ring‐element for finite deformation axisymmetric and torsional problems was recently developed and is now going to be up‐dated. The original assumption of the enhanced displacement gradient H̃ = α _i⊗ G ⁱ is expanded in two steps according to Simo, Armero and Taylor and to Glaser and Armero, respectively: firstly in defining the additional unknowns (parameters) α _i as objects in the material configuration and pushing forward H̃ by ( 1 + U ⊗ Grad ) ∣ξ=0—this provides ‘objectivity’—and secondly in replacing α _i⊗ G ⁱ by G _i⊗ α ⁱ. Numerical results of three classical benchmarks, the in‐plane torsion test, the copper rod impact and the thermomechanical localization of a rectangular strip are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Three‐dimensional meshfree‐enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites

A three‐dimensional microstructure‐based finite element framework is presented for modeling the mechanical response of rubber composites in the microscopic level. This framework introduces a novel finite element formulation, the meshfree‐enriched FEM, to overcome the volumetric locking and pressure oscillation problems that normally arise in the numerical simulation of rubber composites using conventional displacement‐based FEM. The three‐dimensional meshfree‐enriched FEM is composed of five‐noded tetrahedral elements with a volume‐weighted smoothing of deformation gradient between neighboring elements. The L₂‐orthogonality property of the smoothing operator enables the employed Hu–Washizu–de Veubeke functional to be degenerated to an assumed strain method, which leads to a displacement‐based formulation that is easily incorporated with the periodic boundary conditions imposed on the unit cell. Two numerical examples are analyzed to demonstrate the effectiveness of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite elements for materials with strain gradient effects

A finite element implementation is reported of the Fleck–Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin–Mindlin framework and deals with first‐order strain gradients and the associated work‐conjugate higher‐order stresses. In conventional displacement‐based approaches, the interpolation of displacement requires C¹‐continuity in order to ensure convergence of the finite element procedure for higher‐order theories. Mixed‐type finite elements are developed herein for the Fleck–Hutchinson theory; these elements use standard C⁰‐continuous shape functions and can achieve the same convergence as C¹ elements. These C⁰ elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.     

A finite element displacement formulation for gradient elastoplasticity

We present a second gradient elastoplastic model for strain‐softening materials based entirely on a finite element displacement formulation. The stress increment is related to both the strain increment and its Laplacian. The displacement field is the only field needed to be discretized using a C¹ continuity element. The required higher‐order boundary conditions arise naturally from the displacement field. The model is developed to regularize the ill‐posedness caused by strain‐softening material behaviour. The gradient terms in the constitutive equations introduce an extra material parameter with dimensions of length allowing robust modelling of the post‐peak material behaviour leading to localization of deformation. Mesh insensitivity is demonstrated by modelling localization of deformation in biaxial tests. It is shown that both the thickness and inclination of the shear‐band zone are insensitive to the mesh directionality and refinement and agree with the expected theoretical and experimental values. Copyright © 2001 John Wiley & Sons, Ltd.     

A multiplicative finite strain finite element framework for the modelling of semicrystalline polymers and polycarbonates

This paper presents a phenomenological model for the simulation and analysis of stress‐induced orientational hardening in semicrystalline polymers and polycarbonates at finite strains. The notion of intermediate (local) stress‐free configuration is used to develop a set of constitutive equations, and its relation to the multiple natural (stress‐free) configurations in the class of materials being considered here is discussed. A hyperelastic stored energy function, written with respect to the intermediate stress‐free configuration is presented to model the finite elastic response. It is then combined with the J₂‐flow theory to model the finite inelastic response. The isochoric constraint during inelastic deformation is treated via an exact multiplicative decomposition of the deformation gradient into volume‐preserving and spherical parts. The numerical solution algorithm is based on the use of operator splitting technique that results in a product formula algorithm with elastic‐predictor/inelastic‐corrector components. Numerical results are presented to show the behaviour of the model. Copyright © 2000 John Wiley & Sons, Ltd.     

3D modelling of strong discontinuities in elastoplastic solids: fixed and rotating localization formulations

This paper is concerned with the incorporation of displacement discontinuities into a continuum theory of elastoplasticity for the modelling of localization processes such as cracking in brittle materials. Based on the strong discontinuity approach (SDA) (Computational Mechanics 1993; 12: 277–296) mesh objective 2D and 3D finite element formulations are developed using linear and quadratic 2D elements as well as 8‐noded 3D elements. In the formulation of the finite‐element model proposed in the paper, the analogy with standard formulations is emphasized. The parameter defining the amplitude of the displacement jump within the finite element is condensed out at the material level without employing the standard static condensation technique. This approach results in linearized constitutive equations formally identical to continuum models. Therefore, the standard return mapping algorithm is used to solve the non‐linear equations. In analogy to concepts used in continuum smeared crack models, a rotating formulation of the SDA is proposed in addition to the standard concept of fixed discontinuities. It is shown that the rotating localization approach reduces locking effects observed in analyses based on fixed localization directions. The applicability of the proposed SDA finite‐element model as well as its numerical performance is investigated by means of a three‐dimensional ultimate load analysis of a steel anchor embedded in a concrete block subjected to a shear force. Copyright © 2003 John Wiley & Sons, Ltd.     

Analysis of transfer procedures in elastoplasticity based on the error in the constitutive equations: Theory and numerical illustration

The aim of this work is to illustrate a methodology for the assessment of adaptive strategies for the solution of associative rate‐independent plasticity problems solved by employing the incremental displacement conforming finite element method. This is the first step towards a more rational definition of transfer operators in terms of the ensuing error. The motivating idea is the observation that change of data and/or finite element mesh from one time interval to the other can be both related to a discontinuity jump of the approximate solution across the time instant t_n. Thus, reliable a posteriori estimates will have to depend not only on the time step and finite element mesh size but also on the value of the jump. A new error estimate based on the error in the constitutive equations is developed which allows characterization of the discontinuity jump. Copyright © 2004 John Wiley & Sons, Ltd.     

Finite element analysis of crack mouth opening displacement compliance in crack length evaluation for clamped single edge tension specimens

In the unloading compliance method developed for clamped single edge tension (SE(T)) specimens, six crack mouth opening displacement (CMOD)‐based compliance equations (i.e. a/W = f(BCE′)) were proposed for the crack length evaluation without clearly clarifying the corresponding predictive accuracies. In addition, the effective elastic modulus (E_e) that reflects the actual state of stress should also be introduced in the crack length evaluation for SE(T) specimens, because the actual state of stress in the remaining ligament of the test specimen is neither plane stress (E) nor plane strain (E′). In this study, two‐dimensional (2D) plane strain and three‐dimensional (3D) finite element analyses (FEAs) are carried out to investigate predictive accuracies of the six compliance equations. In both 2D and 3D FEA, specimens with a wide range of crack lengths and geometric configurations are included. For a given specimen, the value of E_e that presents the equivalent stress state in the remaining ligament is calculated on the basis of 3D FEA data. A set of formulae for the clamped SE(T) specimen is proposed that allows to evaluate E_e from the corresponding CMOD compliance. This approach is verified using numerical data. The observations of the numerical verification suggest that the use of E_e instead of E or E′ in CMOD‐based compliance equations markedly improves the accuracy of the predicted crack length for clamped SE(T) specimens.     

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.     

Discontinuous variational time integrators for complex multibody collisions

The objective of the present work is to formulate a new class of discontinuous variational time integrators that allow the system to adopt two possibly different configurations at each sampling time t_k, representing predictor and corrector configurations of the system. The resulting sequence of configuration pairs then represents a discontinuous—or non‐classical—trajectory. Continuous or classical trajectories are recovered simply by enforcing a continuity constraint at all times. In particular, in systems subject to one‐sided contact constraints simulated via discontinuous variational time integrators, the predictor configuration is not required to satisfy the one‐sided constraints, whereas the corrector configuration is obtained by a closest‐point projection (CPP) onto the admissible set. The resulting trajectories are generally discontinuous, or non‐classical, but are expected to converge to classical or continuous solutions for decreasing time steps. We account for dissipation, including friction, by means of a discrete Lagrange–d'Alembert principle, and make extensive use of the spacetime formalism in order to ensure exact energy conservation in conservative systems, and the right rate of energy decay in dissipative systems. The structure, range and scope of the discontinuous variational time integrators, and their accuracy characteristics are illustrated by means of examples of application concerned with rigid multibody dynamics. Copyright © 2014 John Wiley & Sons, Ltd.     

A total ALE FE formulation for axisymmetric and torsional problems (including viscoplasticity,thermodynamic coupling and dynamics)

This paper—first—extends a recent ‘assumed enhanced deformation gradient’ finite ring(segment) element (Int. J. Numer. Methods Eng. 2001; 50 :899–918.) to Arbitary Lagrange Euler (ALE) computations, by setting up the assumed tensor on the computational configuration, and—second—shows an elegant way of incorporating dynamics into real ALE computations (no splitting into purely—Lagrange steps and then—remeshing steps), by introducing material mesh velocities and accelerations and spatial mesh velocities and accelerations. Copyright © 2006 John Wiley & Sons, Ltd.     

A mixed finite element formulation of triphasic mechano‐electrochemical theory for charged,hydrated biological soft tissues

An equivalent new expression of the triphasic mechano‐electrochemical theory [9] is presented and a mixed finite element formulation is developed using the standard Galerkin weighted residual method. Solid displacement u ^s, modified electrochemical/chemical potentials ϵ^w, ϵ⁺and ϵ⁻ (with dimensions of concentration) for water, cation and anion are chosen as the four primary degrees of freedom (DOFs) and are independently interpolated. The modified Newton–Raphson iterative procedure is employed to handle the non‐linear terms. The resulting first‐order Ordinary Differential Equations (ODEs) with respect to time are solved using the implicit Euler backward scheme which is unconditionally stable. One‐dimensional (1‐D) linear isoparametric element is developed. The final algebraic equations form a non‐symmetric but sparse matrix system. With the current choice of primary DOFs, the formulation has the advantage of small amount of storage, and the jump conditions between elements and across the interface boundary are satisfied automatically. The finite element formulation has been used to investigate a 1‐D triphasic stress relaxation problem in the confined compression configuration and a 1‐D triphasic free swelling problem. The formulation accuracy and convergence for 1‐D cases are examined with independent finite difference methods. The FEM results are in excellent agreement with those obtained from the other methods. Copyright © 1999 John Wiley & Sons, Ltd.     

A three‐dimensional C1 finite element for gradient elasticity

In gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C¹ continuous interpolation in finite element discretizations of the displacement field only. Employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement, due to the scarcity of C¹ finite elements and their perceived computational cost. In this context, the lack of three‐dimensional C¹ elements is of particular concern. In this paper we present a new C¹ hexahedral element which, to the best of our knowledge, is the first three‐dimensional C¹ element ever constructed. It is shown to pass the single element and patch tests, and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity. It is further shown that C¹ elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good‐quality solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

On the C1 continuous discretization of non‐linear gradient elasticity: A comparison of NEM and FEM based on Bernstein–Bézier patches

In gradient elasticity, the appearance of strain gradients in the free energy density leads to the need of C¹ continuous discretization methods. In the present work, the performances of C¹ finite elements and the C¹ Natural Element Method (NEM) are compared. The triangular Argyris and Hsieh–Clough–Tocher finite elements are reparametrized in terms of the Bernstein polynomials. The quadrilateral Bogner–Fox–Schmidt element is used in an isoparametric framework, for which a preprocessing algorithm is presented. Additionally, the C¹‐NEM is applied to non‐linear gradient elasticity. Several numerical examples are analyzed to compare the convergence behavior of the different methods. It will be illustrated that the isoparametric elements and the NEM show a significantly better performance than the triangular elements. Copyright © 2009 John Wiley & Sons, Ltd.     

Mixed variational principles and robust finite element implementations of gradient plasticity at small strains

This work outlines a theoretical and computational framework of gradient plasticity based on a rigorous exploitation of mixed variational principles. In contrast to classical local approaches to plasticity based on locally evolving internal variables, order parameter fields are taken into account governed by additional balance‐type PDEs including micro‐structural boundary conditions. This incorporates non‐local plastic effects based on length scales, which reflect properties of the material micro‐structure. We develop a unified variational framework based on mixed saddle point principles for the evolution problem of gradient plasticity, which is outlined for the simple model problem of von Mises plasticity with gradient‐extended hardening/softening response. The mixed variational structure includes the hardening/softening variable itself as well as its dual driving force. The numerical implementation exploits the underlying variational structure, yielding a canonical symmetric structure of the monolithic problem. It results in a novel finite element (FE) design of the coupled problem incorporating a long‐range hardening/softening parameter and its dual driving force. This allows a straightforward local definition of plastic loading‐unloading driven by the long‐range fields, providing very robust FE implementations of gradient plasticity. This includes a rational method for the definition of elastic‐plastic‐boundaries in gradient plasticity along with a post‐processor that defines the plastic variables in the elastic range. We discuss alternative mixed FE designs of the coupled problem, including a local‐global solution strategy of short‐range and long‐range fields. This includes several new aspects, such as extended Q1P0‐type and Mini‐type finite elements for gradient plasticity. All methods are derived in a rigorous format from variational principles. Numerical benchmarks address advantages and disadvantages of alternative FE designs, and provide a guide for the evaluation of simple and robust schemes for variational gradient plasticity. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite element evaluation of mixed mode stress intensity factors in functionally graded materials

This paper is directed towards finite element computation of fracture parameters in functionally graded material (FGM) assemblages of arbitrary geometry with stationary cracks. Graded finite elements are developed where the elastic moduli are smooth functions of spatial co‐ordinates which are integrated into the element stiffness matrix. In particular, stress intensity factors for mode I and mixed‐mode two‐dimensional problems are evaluated and compared through three different approaches tailored for FGMs: path‐independent J^*_k‐integral, modified crack‐closure integral method, and displacement correlation technique. The accuracy of these methods is discussed based on comparison with available theoretical, experimental or numerical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

An accurate four‐node shear flexible composite plate element

An efficient and accurate four node shear flexible composite laminated plate element with six degrees of freedom per node, viz. three displacement (u, v, w) along the x‐, y‐and z‐axis, two rotations (θ_x and θ_y) about y‐ and x‐ axis and twist (θ_xy) is proposed in this paper. A coupled displacement field is derived using moment–shear equilibrium and in‐plane equilibrium of composite strips along the x‐ and y‐axis. The displacement field so derived not only depends on the element co‐ordinates but is a function of extensional, bending–extensional coupling, bending and transverse shear stiffnesses as well. The element assumes bi‐cubic polynomial distribution with sixteen generalized undetermined coefficients for the transverse displacement. The element stiffness matrix and load vector are computed numerically by employing 3×3 Gauss–Legendre product rules. The element is found to be devoid of shear locking and does not exhibit any spurious modes. A series of numerical examples are solved to demonstrate the efficacy of the proposed element. Further, the element is found to yield consistently accurate results even with coarse mesh sizes over a wide range of thick plate regimes. Copyright © 2000 John Wiley & Sons, Ltd.     

Overload analysis and Je-based fatigue life prediction of spot-welded auto seat belt anchors

We demonstrate the validity of J_e, which comprehensively describes the effects of specimen geometry and loading type, in predicting the fatigue life of auto seat belt anchor panel. We first simplify the heat‐affected zone model to reduce the number of finite elements. We then establish finite‐element models reflecting the actual overload behaviour of three types of seat belt anchor specimens. Using elaborate finite‐element models, we obtain the effective crack‐driving parameter J_e composed of its ductility‐dependent modal components. It is confirmed that the J_e concept successfully predicts the fatigue life of multi‐spot‐welded panel structures represented by auto seat belt anchors here.