A contact analysis approach based on linear complementarity formulation using smoothed finite element methods

Based on the subdomain parametric variational principle (SPVP), a contact analysis approach is formulated in the incremental form for 2D solid mechanics problems discretized using only triangular elements. The present approach is implemented for the newly developed node-based smoothed finite element method (NS-FEM), the edge-based smoothed finite element method (ES-FEM) as well as standard FEM models. In the approach, the contact interface equations are discretized by contact point-pairs using a modified Coulomb frictional contact model. For strictly imposing the contact constraints, the global discretized system equations are transformed into a standard linear complementarity problem (LCP), which can be readily solved using the Lemke method. This approach can simulate different contact behaviors including bonding/debonding, contacting/departing, and sticking/slipping. An intensive numerical study is conducted to investigate the effects of various parameters and validate the proposed method. The numerical results have demonstrated the validity and efficiency of the present contact analysis approach as well as the good performance of the ES-FEM method, which provides solutions of about 10 times better accuracy and higher convergence rate than the FEM and NS-FEM methods. The results also indicate that the NS-FEM provides upper-bound solutions in energy norm, relative to the fact that FEM provides lower-bound solutions.     

A mixed finite element method for bending of plates

A simple mixed finite element method is developed. The finite element is a rectangular triangle and rectangle. In the element the deflections are assumed to be simple four-element polynomials, bending moments, M_x and M_y with a partially linear distribution, and a constant, M_xy, expressed in terms of the node deflections. The element matrix is of the order of 8 × 8. It is derived in a common engineering way. The unknowns are the deflections at the nodes and mid-diagonal, the two moments at the end of the diagonal and the two moments on the cathetus. The results obtained by this method show good convergence and an improvement in the accuracy of the moments as well as in the deflections, compared with results obtained by similar methods, such as those of Herrmann.     

A mixed finite element method for beam and frame problems

 In this work we consider solutions for the Euler-Bernoulli and Timoshenko theories of beams in which material behavior may be elastic or inelastic. The formulation relies on the integration of the local constitutive equation over the beam cross section to develop the relations for beam resultants. For this case we include axial, bending and shear effects. This permits consideration in a direct manner of elastic and inelastic behavior with or without shear deformation. A finite element solution method is presented from a three-field variational form based on an extension of the Hu–Washizu principle to permit inelastic material behavior. The approximation for beams uses equilibrium satisfying axial force and bending moments in each element combined with discontinuous strain approximations. Shear forces are computed as derivative of bending moment and, thus, also satisfy equilibrium. For quasi-static applications no interpolation is needed for the displacement fields, these are merely expressed in terms of nodal values. The development results in a straight forward, variationally consistent formulation which shares all the properties of so-called flexibility methods. Moreover, the approach leads to a shear deformable formulation which is free of locking effects – identical to the behavior of flexibility based elements. The advantages of the approach are illustrated with a few numerical examples. Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.     

Symmetric Galerkin boundary element method in plasticity and gradient plasticity

A boundary integral equation (BIE) formulation and a boundary element (BE) method which confer symmetry to key operators are concisely described with reference to quasi-static plasticity. This formulation is based on the combined use of static and kinematic sources, on Galerkin weighted residual enforcement of integral equations for displacements and tractions along the boundary and for stresses in the potentially yielding domain and on space discretizations in terms of generalized variables in Prager's sense. Typical theoretical results of computational interest not available in conventional nonsymmetric BE methods are surveyed. The subjectivity (mesh-dependence) implied by material instability is illustrated by examples. As a remedy, a symmetric BIE-BE formulation for nonlocal, gradient-dependent plasticity is developed and discussed on the basis of its variationally consistent discretization.Extended version of a key-note lecture at the 4th International Conference on Computational Plasticity, Barcelona, April 3–6, 1995.Dedicated to J. C. Simo     

Variational principles and finite element analysis for polarization gradient theory

The equations of classical polarization gradient theory are studied using variational methods and finite element analysis. Variational principles are derived and specialized to represent the cubic centro-symmetric crystal structure. An isoparametric nine node axisymmetric finite element is developed and used to demostrate the application of the theory. An analysis of the effects of a point charge in a semi-infinite isotropic halfspace including surface tension effects is computed.     

Simulating small crack growth behaviour using crystal plasticity theory and finite element analysis

Predictions of small crack growth under cyclic loading in aluminium alloy 7075 are performed using finite element analysis (FEA), and results are compared with published experimental data. A double‐slip crystal plasticity model is implemented within the analyses to enable the anisotropic nature of individual grains to be approximated. Small edge‐cracks in a single grain with a starting length of 6 μm are incrementally grown following a node‐release scheme. Crack‐tip opening displacements (CTOD) and crack opening stresses are calculated during the simulated crack growth, and da/dN against ΔK diagrams are computed. Interactions between the crack tip and a grain boundary are also considered. The computations are shown to accurately capture the magnitude and the variability normally observed in small crack fatigue data.     

A mesh-free finite point method for advective-diffusive transport and fluid flow problems

The finite point method (FPM) is a gridless numerical procedure based on the combination of weighted least square interpolations on a cloud of points with point collocation for evaluating the approximation integrals. In the paper, details of a procedure for stabilizing the numerical solution for advective-diffusive transport and fluid flow problems using the FPM are given. The method is based on a consistent introduction of the stabilizing terms in the governing differential equations. One example showing the applicability of the FPM is given.     

A unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations

Piecewise linear finite element approximations to two-dimensional Poisson problems are treated. For simplicity, consideration is restricted to problems having Dirichlet boundary conditions and defined on rectangular domains Ω which are partitioned by a uniform triangular mesh. It is also required that the solutions u ∈ H³ (Ω). A method is proposed for recovering the gradients of the finite element approximations to a root mean square accuracy of O(h²), both at element edge mid-points and element vertices, using simple averaging schemes over adjacent elements. Piecewise linear interpolants (respectively discontinuous and continuous) are then fitted to these recovered gradients, and are shown to be O(h²) estimates for ?u in the L₂-norm, and thus superconvergent. A discussion is given of the extension of the results to problems with more general region and mesh geometries, boundary conditions and with solutions of lower regularity, and also to other second-order elliptic boundary value problems, e.g. the problem of planar linear elasticity.     

A generalized finite element method with global-local enrichment functions for confined plasticity problems

The main feature of partition of unity methods such as the generalized or extended finite element method is their ability of utilizing a priori knowledge about the solution of a problem in the form of enrichment functions. However, analytical derivation of enrichment functions with good approximation properties is mostly limited to two-dimensional linear problems. This paper presents a procedure to numerically generate proper enrichment functions for three-dimensional problems with confined plasticity where plastic evolution is gradual. This procedure involves the solution of boundary value problems around local regions exhibiting nonlinear behavior and the enrichment of the global solution space with the local solutions through the partition of unity method framework. This approach can produce accurate nonlinear solutions with a reduced computational cost compared to standard finite element methods since computationally intensive nonlinear iterations can be performed on coarse global meshes after the creation of enrichment functions properly describing localized nonlinear behavior. Several three-dimensional nonlinear problems based on the rate-independent J ₂ plasticity theory with isotropic hardening are solved using the proposed procedure to demonstrate its robustness, accuracy and computational efficiency.     

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.     

A computational method for frictional contact problem using finite element method

Using the finite element method a numerical procedure is developed for the solution of the two-dimensional frictional contact problems with Coulomb's law of friction. The formulation for this procedure is reduced to a complementarity problem. The contact region is separated into stick and slip regions and the contact stress can be solved systematically by applying the solution technique of the complementarity problem. Several examples are given to demonstrate the validity of the present formulation.     

Least-squares finite element method and preconditioned conjugate gradient solution

A least-squares variational procedure for first-order systems of differential equations and an approximate formulation based on finite elements are developed. Error estimates, a condition number bound and analysis of weighting factors are given. Steepest descent and conjugate gradient solution procedures are examined, and an appropriate preconditioner constructed which is demonstrated to yield rapid convergence and to be insensitive to problem size. Numerical studies of rates of convergence for a test problem are given.     

A mixed-penalty finite element formulation of the linear biphasic theory for soft tissues

Hydrated soft tissues of the human musculoskeletal system can be represented by a continuum theory of mixtures involving intrinsically incompressible solid and incompressible inviscid fluid phases. This paper describes the development of a mixed-penalty formulation for this biphasic system and the application of the formulation to the development of an axisymmetric, six-node, triangular finite element. In this formulation, the continuity equation of the mixture is replaced by a penalty form of this equation which is introduced along with the momentum equation and mechanical boundary condition for each phase into a weighted residual form. The resulting weak form is expressed in terms of the solid phase displacements (and velocities), fluid phase velocities and pressure. After interpolation, the pressure unknowns can be eliminated at the element level, and a first order coupled system of equations is obtained for the motion of the solid and fluid phases. The formulation is applied to a six-node isoparametric element with a linear pressure field. The element performance is compared with that of the direct penalty form of the six-node biphasic element in which the pressure is eliminated in the governing equations prior to construction of the weak form, and selective reduced integration is used on the penalty term. The mixed-penalty formulation is found to be superior in terms of tendency to lock and sensitivity to mesh distortion. A number of example problems for which analytic solutions exist are used to validate the performance of the element.     

Dynamic analysis of cracked linear viscoelastic solids by finite element method using singular element

The finite element method for the dynamic problem of cracked linear viscoelastic solids is developed using the singular element where the displacement function is taken from the analytical solution near the crack-tip. The time variation of the dynamic stress intensity factors is determined for a center crack and an oblique crack in standard linear viscoelastic rectangular plates subjected to dynamic loading.

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Résumé La méthode par éléments finis permettant d'aborder le problème dynamique des solides linéaires viscoélastiques fissurés est développée en recourant à un élément singulier pour lequel la fonction de déplacement est prise dans une solution analytique au voisinage del'extrémité de la fissure. La variation dans le temps des facteurs d'intensité de contrainte dynamique est déterminée pour une fissure centrale et pour une fissure oblique dans des plaques rectangulaires standard en matériau linéaire viscoélastique soumises à une sollicitation dynamique.

     

A gradient flow theory of plasticity for granular materials

Summary A flow theory of plasticity for pressure-sensitive, dilatant materials incorporating second order gradients into the flow-rule and yield condition is suggested. The appropriate extra boundary conditions are obtained with the aid of the principle of virtual work. The implications of the theory into shear-band analysis are examined. The determination of the shear-band thickness and the persistence of ellipticity in the governing equations are discussed.     

A finite element method for contact using a third medium

The numerical simulation of contact problems is nowadays a standard procedure in many engineering applications. The contact constraints are usually formulated using either the Lagrange multiplier, the penalty approach or variants of both methodologies. The aim of this paper is to introduce a new scheme that is based on a space filling mesh in which the contacting bodies can move and interact. To be able to account for the contact constraints, the property of the medium, that imbeds the bodies coming into contact, has to change with respect to the movements of the bodies. Within this approach the medium will be formulated as an isotropic/anisotropic material with changing characteristics and directions. In this paper we will derive a new finite element formulation that is based on the above mentioned ideas. The formulation is presented for large deformation analysis and frictionless contact.     

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.     

A mixed finite element formulation for plate problems

A mixed triangular finite element model has been developed for plate bending problems in which effects of shear deformation are included. Linear distribution for all variables is assumed and the matrix equation is obtained through Reissner's variational principle. In this model, interelement compatibility is completely satisfied whereas the governing equations within the element are satisfied ‘in the mean’. A detailed error analysis is made and convergence of the scheme is proved. Numerical examples of thin and moderately thick plates are presented.     

A mixed variational principle for finite element analysis

A mixed variational principle is developed and utilized in a finite element formulation. The procedure is mixed in the sense that it is based upon a combination of modified potential and complementary energy principles. Compatibility and equilibrium are satisfied throughout the domain a priori, leaving only the boundary conditions to be satisfied by the variational principle. This leads to a finite element model capable of relaxing troublesome interelement continuity requirements. The nodal concept is also abandoned and, instead, generalized parameters serve as the degrees-of-freedom. This allows for easier construction of higher order elements with the displacements and stresses treated in the same manner. To illustrate these concepts, plane stress and plate bending analyses are presented.     

The mixed finite element method in plane elasticity

Studies of the convergence and performance of the mixed finite element method in plane elasticity are reported. A completeness criterion is proposed, and convergence rates for stresses and strain energy, as predicted elsewhere, are quoted. An eigenvalue analysis of the mixed element matrix is carried out for various combinations of interpolations for displacements and stresses in a triangular and a rectangular element, and the results are discussed in relation to the completeness criterion. Finally, a triangular element with linear interpolations for stresses and displacements is formulated and its use is demonstrated on several numerical examples.