Incompressible Elastic Research Articles

Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations

The finite element error and adaptive analysis are implemented in finite element procedures to increase the reliability of numerical analyses. In this paper, the mesh-free error-recovery technique based on moving least squares (MLS) interpolation is applied to recover the errors in the stresses and displacements of incompressible elastic finite element solutions and errors are estimated in energy norms. The effects of element types (triangular and quadrilateral elements) and the formation of patches (mesh-free patch, mesh-dependent element-based patch, and mesh-dependent node-based patch) for error recovery in MLS and conventional least-square interpolation-error quantification are also assessed in this study. Numerical examples of incompressible elasticity, including a problem with singularity, are studied to display the effectiveness and applicability of the mesh-free MLS interpolation-error recovery technique. The mixed formulation (displacement and pressure) is adopted for a finite element analysis of the incompressible elastic problem. The rate of convergence, the effectivity of the error estimation, and modified meshes for desired accuracy are used to assess the effectiveness of the error estimators. The error-convergence rates are computed in the original FEM solution, in the post-processed solution using mesh-free MLS-based displacement, stress recovery, mesh-dependent patch-based least-square-based displacement, and stress recovery (ZZ) as (0.9777, 2.2501, 2.0012, 1.6710 and 1.5436), and (0.9736, 2.0869, 1.6931, 1.8806 and 1.4973), respectively, for four-node quadrilateral, and six-node triangular meshes. It is concluded that displacement-based recovery was more effective in the finite element incompressible elastic analysis than stress-based recovery using mesh-free and mesh-dependent patches.

Bifurcation behavior of compressible elastic half-space under plane deformations

A finitely deformed elastic half-space subject to compressive stresses will experience a geometric instability at a critical level and exhibit bifurcation. While the bifurcation of an incompressible elastic half-space is commonly studied, the bifurcation behavior of a compressible elastic half-space remains elusive and poorly understood to date. The main objective of this manuscript is to study the bifurcation of a neo-Hookean compressible elastic half-space against the well-established incompressible case. The formulation of the problem requires a novel description for a non-linear Poisson’s ratio, since the commonly accepted definitions prove insufficient for the current analysis. To investigate the stability of the domain and the possibility of bifurcation, an incremental analysis is carried out. The incremental analysis describes a small departure from an equilibrium configuration at a finite deformation. It is shown that at the incompressibility limit, our results obtained for a compressible elastic half-space recover their incompressible counterparts. Another key feature of this contribution is that we verify the analytical solution of this problem with computational simulations using the finite element method via an eigenvalue analysis. The main outcome of this work is an analytical expression for the critical stretch where bifurcation arises. We demonstrate the utility of our model and its excellent agreement with the numerical results ranging from fully compressible to incompressible elasticity. Moving forward, this approach can be used to comprehend and harness the instabilities in bilayer systems, particularly for compressible ones.

Mixed stress-displacement isogeometric collocation for nearly incompressible elasticity and elastoplasticity

We propose a mixed stress-displacement isogeometric collocation method for nearly incompressible elastic materials and for materials exhibiting von Mises plasticity. The discretization is based on isogeometric analysis (IGA) with non-uniform rational B-Splines (NURBS) as basis functions. As compared to conventional IGA Galerkin formulations, isogeometric collocation methods offer a high potential of computational cost reduction for higher-order discretizations as they eliminate the need for quadrature. In the proposed mixed formulation, both stress and displacement fields are approximated as primary variables with the aim of treating volumetric locking and instability issues, which occur in displacement-based isogeometric collocation for nearly incompressible elasticity and von Mises plasticity. The performance of the proposed approach is demonstrated by several numerical examples.

Large acoustoelastic effect for Lamb waves propagating in an incompressible elastic plate.

In this paper, the effect of a large pre-stress on the propagation of small amplitude Lamb waves in an incompressible elastic plate is investigated. Using the theory of incremental elasticity, the dispersion equations, which give the phase velocity of the symmetric and anti-symmetric wave modes as a function of the wavenumber, plate thickness, and pre-stress state, are derived for a general strain energy function. By considering the fourth-order strain energy function of incompressible isotropic elasticity, the correction to the phase velocity due to the pre-stress is obtained implicitly to the second order in the pre-strain/stress, and depends on the second, third, and fourth-order elastic constants. Numerical results are presented to show the dependence of the phase velocity of the Lamb wave modes upon the applied stress. These are compared to the first-order correction, and agree well with the limiting and asymptotic values obtained previously. It is envisaged that the present results may well find important practical applications in various guided wave based ultrasonic techniques utilising gels and rubber-like materials.

Behavior of Extreme Auxetic and Incompressible Elastic Materials

We investigate auxetic isotropic elastic materials with the Poisson's ratio of exactly −1 and isotropic materials with the Poisson's ratio of near 0.5 (incompressible). In both cases the energy is not positive definite, which can lead to the material instability. In incompressible materials, the instability manifests itself in the emergence of non‐uniform displacement distribution. Investigation of the behavior of such materials with respect to damage or crack accumulation is based on the notion that during the instantaneous process of fracture formation, the fracture acts as a negative stiffness element; after the fracture is formed, it immediately turns into a conventional fracture with usual positive stiffness. We demonstrate that the cracks formed due to tensile or combined tensile and shear fracturing of the material are capable of momentarily bringing the Poisson's ratio of nearly incompressible material to 0.5, which instantaneously turns the material into unstable. On the other hand, the formation of shear cracks leads to reduction in the Poisson's ratio bringing the material away from the point of instability. Opposite to this, cracks of all kinds do not change the Poisson's ratio in extreme auxetics (the Poisson's ratio equal to −1). Thus the auxetic materials remain stable with respect to fracture formation.

Radial Basis Collocation Method for Nearly Incompressible Elasticity

AbstractStrong form collocation using radial basis functions approximation, which is called the radial basis collocation method, is an efficient method for solving partial differential equations or integral equations because it has implementation simplicity and exponential convergence. In this work, the radial basis collocation method is introduced in conjunction with a least-squares scheme to suppress the volumetric locking in nearly incompressible elastic problems. Radial basis approximation, which possesses infinite continuity, is rich enough to ensure the divergence-free condition. Modal analysis proves that all the nonphysical locking modes can be removed, which eliminates the locking effect. Numerical studies validate the good performance of the proposed method. Exponential convergence is obtained as expected, and no pressure oscillation is observed.

On Formulas for the Velocity of Rayleigh Waves in Prestrained Incompressible Elastic Solids

In the present paper, formulas for the velocity of Rayleigh waves in incompressible isotropic solids subject to a general pure homogeneous prestrain are derived using the theory of cubic equation. They have simple algebraic form and hold for a general strain-energy function. The formulas are concretized for some specific forms of strain-energy function. They then become totally explicit in terms of parameters characterizing the material and the prestrains. These formulas recover the (exact) value of the dimensionless speed of Rayleigh wave in incompressible isotropic elastic materials (without prestrain). Interestingly that, for the case of hydrostatic stress, the formula for the Rayleigh wave velocity does not depend on the type of strain-energy function.

An asymmetric Mixed Finite Element for incompressible Elasticity based on a modified Least‐Squares Formulation

AbstractThe main goal of this work is to derive a mixed finite element for quasi incompressible elastic problems at small strains. Basis for the element formulation is a weak form resulting from a least–squares method, see e.g. Jiang [1]. Therefore, the L2‐norm minimization of the residuals of the given first order system of differential equations leads to a functional depending on displacements, strains and stresses. In this context an enrichment of the strain space is introduced and the respective weak form is derived. The proposed approach differs from a classical least–squares method in the elimination of the additional strains on element level. As a consequence the resulting matrices become asymmetric. This reduction results in a two field formulation related to displacements and stresses. The approximation of these unknows follows the same procedures as for a conventional least–squares method, see e.g. Cai & Starke [2] or Schwarz & Schröder [6]. At the end a numerical examples for quasi incompressible elasticity is presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Effects of Compressibility on Inhomogeneous Deformations for a Class of Almost Incompressible Isotropic Nonlinearly Elastic Materials

Experimental data for simple tension suggest that there is a power–law kinematic relationship between the stretches for large classes of slightly compressible (or almost incompressible) non-linearly elastic materials that are homogeneous and isotropic. Here we confine attention to a particular constitutive model for such materials that is of generalized Varga type. The corresponding incompressible model has been shown to be particularly tractable analytically. We examine the response of the slightly compressible material to some nonhomogeneous deformations and compare the results with those for the corresponding incompressible model. Thus the effects of slight compressibility for some basic nonhomogeneous deformations are explicitly assessed. The results are fundamental to the analytical modeling of almost incompressible hyperelastic materials and are of importance in the context of finite element methods where slight compressibility is usually introduced to avoid element locking due to the incompressibility constraint. It is also shown that even for slightly compressible materials, the volume change can be significant in certain situations.

Fully explicit Agmon's condition for general states of a special incompressible elastic material

Agmon's condition arises as a necessary condition at the boundary for minimizers in compressible and incompressible elasticity. It is commonly formulated as a statement concerning the solution set of a family of ODEs with constant coefficients. As such, it is algebraic “in principle”. In both the compressible and incompressible cases, Agmon's condition may be recast in a more overtly algebraic form, namely the requirement that a certain family of algebraic Riccati equations (parametrized over the tangent plane) should possess positive solutions. In order to reduce Agmon's condition to a fully explicit set of inequalities involving the components of the incremental elasticity tensor, one must be able to solve the algebraic Riccati equation explicitly. Known situations where this can be done tend to involve highly symmetric states of isotropic materials. It is therefore noteworthy that Agmon's condition may be rendered explicit for any boundary-point of an arbitrarily deformed incompressible neo-Hookean body.

Equivalence Between Short-Time Biphasic and Incompressible Elastic Material Responses

Porous-permeable tissues have often been modeled using porous media theories such as the biphasic theory. This study examines the equivalence of the short-time biphasic and incompressible elastic responses for arbitrary deformations and constitutive relations from first principles. This equivalence is illustrated in problems of unconfined compression of a disk, and of articular contact under finite deformation, using two different constitutive relations for the solid matrix of cartilage, one of which accounts for the large disparity observed between the tensile and compressive moduli in this tissue. Demonstrating this equivalence under general conditions provides a rationale for using available finite element codes for incompressible elastic materials as a practical substitute for biphasic analyses, so long as only the short-time biphasic response is sought. In practice, an incompressible elastic analysis is representative of a biphasic analysis over the short-term response deltat<<Delta(2) / //parallelC(4)//K//, where Delta is a characteristic dimension, C(4) is the elasticity tensor, and K is the hydraulic permeability tensor of the solid matrix. Certain notes of caution are provided with regard to implementation issues, particularly when finite element formulations of incompressible elasticity employ an uncoupled strain energy function consisting of additive deviatoric and volumetric components.

On singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems with corners

AbstractIn this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u, p Galerkin–Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem (P+λQ+λ2R)d=0 is obtained, where the saddle‐point‐type matrices P, Q, R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa‐like scheme is used. This technique needs only one direct matrix factorization as well as few matrix–vector products for finding all eigenvalues in the interval 𝓈(λ) ∈ (−0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface‐breaking crack in an incompressible elastic material and the three‐dimensional viscous flow of a Newtonian fluid past a trihedral corner. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Mixed linear/linear simplicial elements for incompressible elasticity and plasticity

This paper exploits the concept of orthogonal sub-grid scales to stabilize the behaviour of mixed linear/linear simplicial elements (triangles and tetrahedra) in incompressible or nearly incompressible situations. Both incompressible elastic and J2-plastic constitutive behaviours have been considered. The different assumptions and approximations used to derive the method are exposed. Implementation and computational aspects are also discussed, showing that a robust application of the proposed formulation is feasible. Numerical examples show that the elements derived are free of volumetric locking and spurious oscillations of the pressure, and that the results obtained compare favourably with those obtained with the Q1P0 quadrilateral.

Thermal Stress in a Finitely Deformed Incompressible Elastic Medium Containing a Penny-Shaped Crack

ABSTRACTThe thermal stress for a penny-shaped crack contained in an infinite isotropic elastic solid initially subjected to an axisymmetrical tension of any amount at infinity is investigated using the techniques of Hankel transforms and multiplying factors. The effect that the lateral normal stress has on the thermal stresses is studied on the basis of the theory of small deformations superposed on finite deformation. Symmetrical thermal loadings are applied over the crack surfaces. For the case of constant temperature over the crack surfaces, expressions for the crack shape and thermal stresses in the crack plane are obtained in closed forms. The stress intensity factor is also obtained and shown to be dependent on the lateral stress.

Quantitative elasticity imaging: what can and cannot be inferred from strain images

We examine the inverse problem associated with quantitative elastic modulus imaging: given the equilibrium strain field in a 2D incompressible elastic material, determine the elastic stiffness (shear modulus). We show analytically that a direct formulation of the inverse problem has no unique solution unless stiffness information is known a priori on a sufficient portion of the boundary. This implies that relative stiffness images constructed on the assumption of constant boundary stiffness are in error, unless the stiffness is truly constant on the boundary. We show further that using displacement boundary conditions in the forward incompressible elasticity problem leads to a nonunique inverse problem. Indeed, we give examples in which exactly the same strain field results from different elastic modulus distributions under displacement boundary conditions. We also show that knowing the stress on the boundary can, in certain configurations, lead to a well-posed inverse problem for the elastic stiffness. These results indicate what data must be taken if the elastic modulus is to be reconstructed reliably and quantitatively from a strain image.

Adaptive coupling of boundary elements and mixed finite elements for incompressible elasticity

The coupling of finite elements and boundary elements is analyzed, where in the FEM domain we assume an incompressible elastic material governed by a uniformly monotone operator and use a Stokes-type mixed FEM. In the BEM domain, linear elasticity is considered. We prove existence and uniqueness of the solution and quasi-optimal convergence of a Galerkin method. We derive an a posteriori error estimator of explicit residual type. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 79–92, 2001

Adaptive coupling of boundary elements and mixed finite elements for incompressible elasticity

The coupling of finite elements and boundary elements is analyzed, where in the FEM domain we assume an incompressible elastic material governed by a uniformly monotone operator and use a Stokes-type mixed FEM. In the BEM domain, linear elasticity is considered. We prove existence and uniqueness of the solution and quasi-optimal convergence of a Galerkin method. We derive an a posteriori error estimator of explicit residual type. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 79–92, 2001

A SINGULAR PROBLEM IN INCOMPRESSIBLE NONLINEAR ELASTOSTATICS

This paper represents a contribution to the numerical treatment of problems in incompressible elasticity theory for large deformations. We are especially concerned about the solution of plane problems with corners. A review of the literature on these problems indicates that the behavior of the solution in the vicinity of a corner is given little attention. We investigate the solution of the compressed bonded block problem corresponding to the compression of an incompressible elastic block of rectangular cross-section and infinite transverse length between two opposing bonded rigid surfaces, with the two remaining lateral faces traction-free. We are especially interested in the behavior at a corner where a bonded end is adjacent to a free lateral side. We employ a finite element method based on a reduced and selective integration technique with penalization to construct a numerical solution for this problem. Our computational method converges everywhere except in a small neighborhood of the corner. We appeal to an elementary a priori inequality concerning the angle of shear to show that the numerical calculations in this neighborhood are inaccurate and need a more refined study. Based on the inequality, we offer a conjecture concerning the local shape of the deformed free lateral surface at the corner.

Homogenisation of a class of fourth order equations with application to incompressible elasticity

SynopsisUpon formalising an analogy between two-dimensional Stokes flow and two-dimensional isotropic conductivity, we exhibit a class of fourth order equations which behave “isomorphically” like isotropic conductivity from the standpoint of homogenisation and from that of corresponding bounding methods on possible effective behaviours. In particular, Lipton's result on the G-closure problem for mixtures of two incompressible elastic materials is recovered in the two-dimensional case.