Locking in the incompressible limit for the element‐free Galerkin method

Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the element‐free Galerkin method. The modal analysis developed here shows that the number of non‐physical locking modes is independent of the dilation parameter (support of the interpolation functions). Thus increasing the dilation parameter does not suppress locking. Nevertheless, an increase in the dilation parameter does reduce the energy associated with the non‐physical locking modes; thus, in part, it alleviates the locking phenomena. This is shown for linear and quadratic orders of consistency. Moreover, the biquadratic order of consistency, as in finite elements, improves the locking behaviour. Although more locking modes are present in the element‐free Galerkin method with quadratic consistency than with standard biquadratic finite elements. Finally, numerical examples are shown to validate the modal analysis. In particular, the conclusions of the modal analysis are also confirmed in an elastoplastic example. Copyright © 2001 John Wiley & Sons, Ltd.     

An extension of Key's principle to nonlinear elasticity

A variational principle for finite isothermal deformations of anisotropic compressible and nearly incompressible hyperelastic materials is presented. It is equivalent to the nonlinear elastic field (Lagrangian) equations expressed in terms of the displacement field and a scalar function associated with the hydrostatic mean stress. The formulation for incompressible materials is recovered from the compressible one simply as a limit. The principle is particularly useful in the development of finite element analysis of nearly incompressible and of incompressible materials and is general in the sense that it uses a general form of constitutive equation. It can be considered as an extension of Key's principle to nonlinear elasticity. Various numerical implementations are used to illustrate the efficiency of the proposed formulation and to show the convergence behaviour for different types of elements. These numerical tests suggest that the formulation gives results which change smoothly as the material varies from compressible to incompressible.     

On the dynamics of non-linear viscoelastic solids with material moduli that depend upon pressure

We investigate the dynamic response, of a generalization of an incompressible Kelvin-Voigt viscoelastic solid, whose viscosity depends on the pressure. Bodies with pressure-dependent material moduli have relevance to numerous technologically significant problems in geomechanics, the mechanics of granular media and powder compaction. We obtain analytical results for creep and recovery phenomena as well as solutions to the propagation of waves in such bodies. We are able to obtain explicit exact solutions that clearly illustrate the marked difference in the response of bodies with pressure-dependent material moduli as opposed to their counterparts whose moduli do not depend on the pressure. We also show that the governing equations for such materials can change type, and that their solutions exhibit singularities and localization.     

Machine locking of degenerated thin shell elements

The degenerated shell element is one of the most efficient elements for analysing shell structures. However, it is known to result in rather stiff models when used in thin element applications. The phenomena associated with this behaviour are known as locking phenomena. This paper analyses the machine locking mechanism developed in thin to very thin Lagrangian and serendipity elements. The machine related locking phenomenon is distinguished from the shear and membrane locking phenomena. A remedy for the pure machine locking problem is developed for the two elements. The proposed remedy is based on the technique of the modified transverse shear modulus. It is also extended to control shear locking. The proposed technique is shown to completely eliminate machine locking. Also, it is shown to effectively alleviate stiffening effects due to the presence of spurious shear strain.     

Non‐linear analysis of structures using high performance hybrid elements

In this work, we describe the formulation and implementation for stress‐based hybrid elements for conducting non‐linear analysis of elastic structures. The motivation behind developing these elements is that they should be as simple to use as standard displacement‐based isoparametric brick elements, but at the same time, be relatively immune to the shortcoming that these elements suffer from, namely, ‘locking’ problems which occur when they are used to model plate/shell geometries, almost incompressible materials or when the elements are distorted, and so on. The formulation is based on a two‐field mixed variational principle. Numerical examples are presented to demonstrate the excellent performance of the proposed elements on a variety of challenging problems involving very large deformations, buckling, mesh distortions, almost incompressible materials, etc. Copyright © 2006 John Wiley & Sons, Ltd.     

The boundary contour method for two-dimensional Stokes flow and incompressible elastic materials

 A variant of the boundary element method, called the boundary contour method (BCM), offers a further reduction in dimensionality. Consequently, boundary contour analysis of two-dimensional (2-D) problems does not require any numerical integration at all. While the method has enjoyed many successful applications in linear elasticity, the above advantage has not been exploited for Stokes flow problems and incompressible media. In order to extend the BCM to these materials, this paper presents a development of the method based on the equations of Stokes flow and its 2-D kernel tensors. Potential functions are derived for quadratic boundary elements. Numerical solutions for some well-known examples are compared with the analytical ones to validate the development. Received 28 August 2001 / Accepted 15 January 2002     

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Applications to linear nearly incompressible elastostatics and implicit and explicit elastodynamics

In this paper, we present novel techniques of using quadratic Bézier triangular and tetrahedral elements for elastostatic and implicit/explicit elastodynamic simulations involving nearly incompressible linear elastic materials. A simple linear mapping is proposed for developing finite element meshes with quadratic Bézier triangular/tetrahedral elements from the corresponding quadratic Lagrange elements that can be easily generated using the existing mesh generators. Numerical issues arising in the case of nearly incompressible materials are addressed using the consistent B -bar formulation, thus reducing the finite element formulation to one consisting only of displacements. The higher-order spatial discretization and the nonnegative nature of Bernstein polynomials are shown to yield significant computational benefits. The optimal spatial convergence of the B -bar formulation for the quadratic triangular and tetrahedral elements is demonstrated by computing error norms in displacement and stresses. The applicability and computational efficiency of the proposed elements for elastodynamic simulations are demonstrated by studying several numerical examples involving real-world geometries with complex features. Numerical results obtained with the standard linear triangular and tetrahedral elements are also presented for comparison.     

One-dimensional wave propagation in materials with different moduli in tension and compression

One-dimensional wave propagation in compressible and incompressible elastic materials behaving differently in tension and compression is investigated. The constitutive equations of these materials are nonlinear even though small deformations are considered. The characteristic wavespeeds are derived, the hyperbolicity condition is investigated, and analytical simple wave solutions are obtained in a compressible and incompressible semi-infinite half-space. The presented solutions exhibit interesting phenomena of wave propagation, like that of a coupled shear-normal plane wave propagating steadily with a constant velocity in a compressible medium with different moduli in tension and compression.     

Dynamic wave propagation in infinite saturated porous media half spaces

From a macroscopic perspective, saturated porous materials like soils represent volumetrically interacting solid–fluid aggregates. They can be properly modelled using continuum porous media theories accounting for both solid-matrix deformation and pore-fluid flow. The dynamic excitation of such multi-phase materials gives rise to different types of travelling waves, where it is of common interest to adequately describe their propagation through unbounded domains. This poses challenges for the numerical treatment and demands special solution strategies that avoid artificial and numerically-induced perturbations or interferences. The present paper is concerned with the accurate and stable numerical solution of dynamic wave propagation problems in infinite half spaces. Proceeding from an isothermal, biphasic, linear poroelasticity model with incompressible constituents, finite elements are used to discretise the near field and infinite elements to approximate the far field. The transient propagation of the poroacoustic body waves to the infinity is thereby modelled by a viscous damping boundary, which, for stability reasons, necessitates an appropriate treatment of the included velocity-dependent damping forces.     

Electrodynamic equations for heterogeneous media and structures on the length scales of their constituents

A new theoretical approach is proposed for addressing the electrodynamics of heterogeneous media and structures, which takes into account the interrelation between physical phenomena on different length scales. Using volume averaging theory (VAT), parabolic partial differential equations containing integral or integro-differential terms are obtained for electromagnetic wave propagation in heterogeneous media. This theory allows one to accurately describe physical phenomena owing to its ability to take into account spatial scales, interfacial effects, the structural features of heterogeneous media such as high-T_c superconducting ceramics and absorbing composite materials, and heat and mass exchange in porous media. The approach is exemplified by analysis of functional structures with electrodynamic properties, in particular absorbing structures, and its conclusions are supported by experimental data. Issues pertaining to the anisotropy in the properties of composite materials and related functional structures are discussed.     

Finite-element analysis of hyperelastic thin shells with large strains

The objective of this contribution is the development of theoretical and numerical models applicable to large strain analysis of hyperelastic shells confining particular attention to incompressible materials. The theoretical model is developed on the basis of a quadratic displacement approximation in thickness coordinate by neglecting transverse shear strains. In the case of incompressible materials this leads to a three-parametric theory governed solely by mid-surface displacements. The material incompressiblity is expressed by two equivalent equation sets considered at the element level as subsidiary conditions. For the simulation of nonlinear material behaviour the Mooney-Rivlin model is adopted including neo-Hookean materials as a special case. After transformation of nonlinear relations into incremental formulation doubly curved triangular and quadrilateral elements are developed via the displacement method. Finally, examples are given to demonstrate the ability of these models in dealing with large strain as well as finite rotation shell problems.The present study is supported by a research grant of the German National Science Foundation (DFG) under Ba 969/3-1.dedicated to Prof. Dr. Dr. Erwin Stein for his 65th birthday anniversary     

New enhanced strain elements for incompressible problems

A new enhanced strain element, based on the definition of extra compatibles modes of deformation added to the standard four‐node finite element, is initially presented. The element is built with the objective of addressing incompressible problems and avoiding locking effects. By analysing at the element level the deformation modes which form a basis for the incompressible subspace the extra modes of deformation are proposed in order to provide the maximum possible dimension to that subspace. Subsequently another new element with more degrees of freedom is formulated using a mixed method. This is done by including an extra field of variables related to the derivatives of the displacement field of the extra compatible modes defined previously. The performance of the elements proposed is assessed in linear and non‐linear situations. Copyright © 1999 John Wiley & Sons, Ltd.     

Micro-mechanical analysis and modelling of granular materials loaded at constant volume

The present paper is devoted to the analysis and the modelling of the local phenomena, which are observed in a two-dimensional granular media loaded at constant volume. A micro-mechanical approach is followed here combined with a computer simulation method. Local phenomena observed during test are used to provide some necessary elements to build realistic models. The importance of the induced anisotropy during the test is especially shown, as well as its necessary link to the dilatancy. To illustrate the study a model based on a micro-mechanical approach is validated.     

A stabilized nodally integrated tetrahedral

A stabilized, nodally integrated linear tetrahedral is formulated and analysed. It is well known that linear tetrahedral elements perform poorly in problems with plasticity, nearly incompressible materials, and acute bending. For a variety of reasons, low‐order tetrahedral elements are preferable to quadratic tetrahedral elements; particularly for nonlinear problems. But the severe locking problems of tetrahedrals have forced analysts to employ hexahedral formulations for most nonlinear problems. On the other hand, automatic mesh generation is often not feasible for building many 3D hexahedral meshes. A stabilized, nodally integrated linear tetrahedral is developed and shown to perform very well in problems with plasticity, nearly incompressible materials and acute bending. The formulation is analytically and numerically shown to be stable and optimally convergent for the compressible case provided sufficient smoothness of the exact solution u ∈ C² ∩ (H¹)³. Future work may extend the formulation to the incompressible regime and relax the regularity requirements; nonetheless, the results demonstrate that the method is not susceptible to locking and performs quite well in several standard linear and nonlinear benchmarks. Published in 2006 by John Wiley & Sons, Ltd.     

Analysis of the vacuum infusion moulding process: I. Analytical formulation

The present work is primarily concerned with the analytical formulation of governing equations for flow of incompressible fluids through compacting porous media and their application to vacuum infusion (VI) of composite materials. The literature on VI and the effects of compacting media on permeability and flow is reviewed. A complete development of the proposed governing equation is shown along with a suggested numerical solution. The proposed model is subsequently used to quantify the effect of process parameters such as inlet and outlet pressures, fibre architecture and lay-up. Implications for industrial production are discussed.     

A coupled hp‐finite element scheme for the solution of two‐dimensional electrostrictive materials

As part of the ongoing research within the field of computational analysis for the coupled electro‐magneto‐mechanical response of smart materials, the problem of linearised electrostriction is revisited and analysed for the first time using the framework of hp‐finite elements. The governing equations modelling the physics of the dielectric are suitably modified by introducing a new total Cauchy stress tensor (A. Dorfmann and R.W. Ogden. Nonlinear electroelasticity. Acta Mechanica, 174:167–183, 2005), which includes the electrostrictive effect and a staggered partitioned scheme for the numerical solution of the coupling phenomena. With the purpose of benchmarking numerical results, the problem of an infinite electrostrictive plate with a circular/elliptical dielectric insert is revisited. The presented analytical solution is based on the theoretical framework for two‐dimensional electrostriction proposed by Knops (R.J. Knops. Two‐dimensional electrostriction. Quarterly Journal of Mechanics and Applied Mathematics, 16:377–388, 1963) and uses classical techniques of complex variable analysis. Our presentation, to the best of our knowledge, provides the first correct closed form expression for the solution to the infinite electrostrictive plate with a circular/elliptical dielectric insert, correcting the errors made in previous presentations of this problem. We use this analytical solution to assess the accuracy, efficiency and robustness of the hp‐formulation in the case of nearly incompressible electrostrictive materials. Copyright © 2012 John Wiley & Sons, Ltd.     

Stresses in nearly incompressible materials by finite elements with application to the calculation of excess pore pressures

A number of problems are analysed by the displacement method to assess the stress accuracy at very low compressibilities. ‘Parabolic’ isoparametric elements are used. It is found that the mean stress becomes grossly in error at the centre and edges of each element as the compressibility is reduced whereas the deviatoric stress components do not. All stress components retain good accuracy at the ‘reduced’ integration sampling points (2 × 2 Gauss). ‘Exact’ integration yields a similar stress distribution to ‘reduced’ but the mean stress is grossly in error at the integration points (3 × 3 Gauss). Exceptions, however, occur. These findings are interpreted, and a rule for predetermining whether or not accurate stresses can be obtained at the integrating points is suggested. Thus it is shown that the displacement method is suitable for analysing materials which for practical purposes are incompressible. A procedure is then presented for analysing porous media-both linear and non-linear-by separating the stiffness into ‘effective’ and ‘pore fluid’ components. This allows excess pore pressure to be calculated explicitly. Applications to saturated soils are given which make use of the findings of the first part of the paper.