A simple finite element‐based framework for the analysis of elastic pseudo‐rigid bodies

This article advocates a general procedure for the numerical investigation of pseudo‐rigid bodies. The equations of motion for pseudo‐rigid bodies are shown to be mathematically equivalent to those corresponding to certain constant‐strain finite element approximations for general deformable continua. A straightforward algorithmic implementation is achieved in a classical finite element framework. Also, a penalty formulation is suggested for modelling contact between pseudo‐rigid bodies. Representative planar simulations using a non‐linear elastic model demonstrate the predictive capacity of the pseudo‐rigid theory, as well as the robustness of the proposed computational procedure. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite element analysis and algorithms for single‐crystal strain‐gradient plasticity

We provide optimal a priori estimates for finite element approximations of a model of rate‐independent single‐crystal strain‐gradient plasticity. The weak formulation of the problem takes the form of a variational inequality in which the primary unknowns are the displacement and slips on the prescribed slip systems, as well as the back‐stress associated with the vectorial microstress. It is shown that the return mapping algorithm for local plasticity can be applied element‐wise to this non‐local setting. Some numerical examples illustrate characteristic features of the non‐local model. Copyright © 2012 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.     

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.     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part I: Flexural–torsional stability

In formulating a finite element model for the flexural–torsional stability and 3‐D non‐linear analyses of thin‐walled beams, a rotation matrix is usually used to obtain the non‐linear strain–displacement relationships. Because of the coupling between displacements, twist rotations and their derivatives, the components of the rotation matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the rotation matrix. A simplified small rotation matrix is often used in the formulation of finite element models for the flexural–torsional stability analysis of thin‐walled beams of open cross‐section. However, the approximations in the small rotation matrix may lead to the loss of some significant terms in the stability stiffness matrix. Without these terms, a finite element line model may predict the incorrect flexural–torsional buckling load of a beam. This paper investigates the effects of approximations in the elastic flexural–torsional stability analysis of thin‐walled beams, while a companion paper investigates the effects of approximations in the 3‐D non‐linear analysis. It is found that a finite element line model based on a small rotation matrix may predict incorrect elastic flexural–torsional buckling loads of beams. To perform a correct flexural–torsional stability analysis of thin‐walled beams, modification of the model is needed, or a finite element model based on a second‐order rotation matrix can be used. Copyright © 2001 John Wiley & Sons, Ltd.     

A two‐dimensional BEM/FEM coupling applied to viscoelastic analysis of composite domains

A coupling between the boundary‐element and finite‐element methods is studied for the viscoelastic analysis of reinforced media. The viscous behaviour of the composed body is taken into account by an alternative BEM methodology developed for the Boltzmann model. This methodology is based on differential constitutive relations for viscoelasticity. The reinforcements are modelled by finite elements and are considered elastic. The coupling is based on the sub‐region technique due to its generality and easy implementation. The resulting time‐marching process is able to represent both the instantaneous and the time‐dependent behaviour of a body subjected to general boundary conditions. The method is validated by an experimental result and its accuracy tested by comparing numerical results with analytical solutions. The generality of the method is proved by an infinite domain application. Copyright © 2003 John Wiley & Sons, Ltd.     

Lower‐dimensional interface elements with local enrichment: application to coupled hydro‐mechanical problems in discretely fractured porous media

In this study, we develop lower‐dimensional interface elements to represent preexisting fractures in rock material, focusing on finite element analysis of coupled hydro‐mechanical problems in discrete fractures–porous media systems. The method adopts local enrichment approximations for a discontinuous displacement and a fracture relative displacement function. Multiple and intersected fractures can be treated with the new scheme. Moreover, the method requires less mesh dependencies for accurate finiteelement approximations compared with the conventional interface element method. In particular, for coupled problems, the method allows for the use of a single mesh for both mechanical and other related processes such as flow and transport. For verification purposes, several numerical examples are examined in detail. Application to a coupled hydro‐mechanical problem is demonstrated with fluid injection into a single fracture. The numerical examples prove that the proposed method produces results in strong agreement with reference solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Path‐following analysis of thin‐walled structures and comparison with asymptotic post‐critical solutions

A path‐following non‐linear elastic analysis for structures composed of assemblages of flat slender elastic panels is presented. The proposed path‐following method employs FEM technology and a kinematical model to analyse these structures using a Koiter asymptotic approach. As a result it is possible to verify the accuracy achieved by the asymptotic method. The proposed mixed path‐following formulation is both efficient and robust with regards to the locking extrapolation phenomenon that strongly affects compatible formulations. The use of an HC finite element makes it possible to avoid the problem of the finite rotations in the space, maintaining a high degree of continuity and making the numeric formulation simple and efficient. Copyright © 2002 John Wiley & Sons, Ltd.     

A distributed memory parallel implementation of the multigrid method for solving three‐dimensional implicit solid mechanics problems

We describe the parallel implementation of a multigrid method for unstructured finite element discretizations of solid mechanics problems. We focus on a distributed memory programming model and use the MPI library to perform the required interprocessor communications. We present an algebraic framework for our parallel computations, and describe an object‐based programming methodology using Fortran90. The performance of the implementation is measured by solving both fixed‐ and scaled‐size problems on three different parallel computers (an SGI Origin2000, an IBM SP2 and a Cray T3E). The code performs well in terms of speedup, parallel efficiency and scalability. However, the floating point performance is considerably below the peak values attributed to these machines. Lazy processors are documented on the Origin that produce reduced performance statistics. The solution of two problems on an SGI Origin2000, an IBM PowerPC SMP and a Linux cluster demonstrate that the algorithm performs well when applied to the unstructured meshes required for practical engineering analysis. Copyright © 2004 John Wiley & Sons, Ltd.     

Computation of limit and shakedown loads using a node‐based smoothed finite element method

This paper presents a novel numerical procedure for computing limit and shakedown loads of structures using a node‐based smoothed FEM in combination with a primal–dual algorithm. An associated primal–dual form based on the von Mises yield criterion is adopted. The primal‐dual algorithm together with a Newton‐like iteration are then used to solve this associated primal–dual form to determine simultaneously both approximate upper and quasi‐lower bounds of the plastic collapse limit and the shakedown limit. The present formulation uses only linear approximations and its implementation into finite element programs is quite simple. Several numerical examples are given to show the reliability, accuracy, and generality of the present formulation compared with other available methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite element formulation for anisotropic coupled piezoelectro‐hygro‐thermo‐viscoelasto‐dynamic problems

A finite element algorithm has been developed for the efficient analysis of smart composite structures with piezoelectric polymer sensors or/and actuators based on piezoelectro‐hygro‐thermo‐viscoelasticity. Variational principles for anisotropic coupled piezoelectro‐hygro‐thermo‐viscoelasto‐dynamic problems have also been proposed in this study. As illustrative studies, dynamic responses in laminated composite beams and plates with PVDF sensors and actuators are obtained as functions of time using the present finite element procedures. The voltage feedback control scheme is utilized. The proposed numerical method can be used for analysing problems in the design of smart structures as well as smart sensors and actuators. Copyright © 1999 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.     

The partition of unity finite element method for elastic wave propagation in Reissner–Mindlin plates

This paper reports a numerical method for modelling the elastic wave propagation in plates. The method is based on the partition of unity approach, in which the approximate spectral properties of the infinite dimensional system are embedded within the space of a conventional finite element method through a consistent technique of waveform enrichment. The technique is general, such that it can be applied to the Lagrangian family of finite elements with specific waveform enrichment schemes, depending on the dominant modes of wave propagation in the physical system. A four‐noded element for the Reissner–Mindlin plate is derived in this paper, which is free of shear locking. Such a locking‐free property is achieved by removing the transverse displacement degrees of freedom from the element nodal variables and by recovering the same through a line integral and a weak constraint in the frequency domain. As a result, the frequency‐dependent stiffness matrix and the mass matrix are obtained, which capture the higher frequency response with even coarse meshes, accurately. The steps involved in the numerical implementation of such element are discussed in details. Numerical studies on the performance of the proposed element are reported by considering a number of cases, which show very good accuracy and low computational cost. Copyright © 2006 John Wiley & Sons, Ltd.     

The spectral element method for elastic wave equations—application to 2‐D and 3‐D seismic problems

A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two‐ and three‐dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, defined with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor‐product of a Gauss–Lobatto–Legendre 1‐D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy‐momentum conserving scheme that can be put into a classical explicit‐implicit predictor/multicorrector format. Long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two‐dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three‐dimensional model. Very accurate modelling of Rayleigh wave propagation and surface diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three‐dimensional surface topographies. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite–infinite element for dynamic analysis of axisymmetrically saturated composite foundations

The formulations of axisymmetrically infinite elements for dynamic analysis of vertical vibration problems in unbounded saturated composite foundations are presented. The theoretical basis as well as the implementation of the elements is discussed, and the element decay functions are derived using the analytical solutions of axially symmetric configurations. Using the proposed finite–infinite element method, the surface vertical displacements of air‐saturated soil (‘dry’ soil) and of water‐saturated soil with extremely low permeability subjected to a surface point excitation (called as the Lamb's problem) are calculated and the results agree very well with the existing theoretical solutions of single‐phase elastic media. As an application, the velocity admittances of a concrete block resting on cement mixing‐pile or gravel‐pile saturated composite foundations are calculated. The influence of soil permeability and pile rigidity on the dynamic response of the composite foundations is investigated. The method proposed by this paper is a simple and reliable numerical one that could be used to study axisymmetrically dynamic problems of layered saturated media and to get the mechanism of dynamic testing on single‐pile saturated composite foundations. Copyright © 2006 John Wiley & Sons, Ltd.     

A systematic construction of B‐bar functions for linear and non‐linear mixed‐enhanced finite elements for plane elasticity problems

In a previous paper a modified Hu–Washizu variational formulation has been used to derive an accurate four node plane strain/stress finite element denoted QE2. For the mixed element QE2 two enhanced strain terms are used and the assumed stresses satisfy the equilibrium equations a priori for the linear elastic case. In this paper an alternative approach is discussed. The new formulation leads to the same accuracy for linear elastic problems as the QE2 element; however it turns out to be more efficient in numerical simulations, especially for large deformation problems. Using orthogonal stress and strain functions we derive B̄ functions which avoid numerical inversion of matrices. The B̄ ‐strain matrix is sparse and has the same structure as the strain matrix B obtained from a compatible displacement field. The implementation of the derived mixed element is basically the same as the one for a compatible displacement element. The only difference is that we have to compute a B̄ ‐strain matrix instead of the standard B ‐matrix. Accordingly, existing subroutines for a compatible displacement element can be easily changed to obtain the mixed‐enhanced finite element which yields a higher accuracy than the Q4 and QM6 elements. Copyright © 1999 John Wiley & Sons, Ltd.