Incompressible Finite Elasticity Research Articles

A kinematically stabilized linear tetrahedral finite element for compressible and nearly incompressible finite elasticity

We propose a stabilized linear tetrahedral finite element method for static, finite elasticity problems involving compressible and nearly incompressible materials. Our approach relies on a mixed formulation, in which the nodal displacement unknown filed is complemented by a nodal Jacobian determinant unknown field. This approach is simple to implement in practical applications (e.g., in commercial software), since it only requires information already available when computing the Newton–Raphson tangent matrix associated with irreducible (i.e., displacement-based) finite element formulations. By nature, the proposed method is easily extensible to nonlinear models involving visco-plastic flow. An extensive suite of numerical tests in two and three dimensions is presented, to demonstrate the performance of the method.

Adaptive stabilized mixed formulation for phase field fracture modeling of nearly incompressible finite elasticity

The recently proposed mixed displacement-pressure-phase field formulation characterized by damage-mitigating incompressibility constraints excels in fracture modeling of nearly incompressible hyperelastic materials. However, such formulations generally call for mixed finite element (FE) allocations that satisfy the inf-sup condition, denying the popular lower equal-order linear FE pairs, like linear polynomial (P1) interpolation for all fields. To overcome this drawback, we develop a stabilized mixed formulation (P1/P1S) by resorting to a pressure projection technique. The lure of this formulation lies in its simplicity and versatility, allowing low-order P1 interpolation for all field variables. With this trait in mind, an efficient adaptive meshing strategy is designed, thereby speeding up computation by more than 15 times. For better coping with the adaptive scenarios involving crack nucleation, we also propose a novel energy-based mesh refinement criterion. The numerical treatment of the stabilized mixed FE formulation is elaborated, as well as the core operations of the adaptive meshing technique. The accuracy, efficiency, and robustness of the proposed formulation are validated by several representative numerical examples. Remarkably, a mixed I/II fracture test thereof is in good agreement with the experiment and thus promises to be a new benchmark.

Variational linearization of pure traction problems in incompressible elasticity

We consider pure traction problems, and we show that incompressible linearized elasticity can be obtained as variational limit of incompressible finite elasticity under suitable conditions on external loads.

A framework for residual-based stabilization of incompressible finite elasticity: Stabilized formulations and [formula omitted] methods for linear triangles and tetrahedra

A new Variational Multiscale framework for finite strain incompressible elasticity is presented. Significant contributions in this work are: (i) a systematic derivation of multiscale formulations that include the classical F¯ method as a particular subclass, (ii) an error estimation procedure for nonlinear elasticity that emanates naturally from within the present multiscale framework, and (iii) robust performance of linear triangular and tetrahedral elements for modeling nearly incompressible materials in the finite strain range. When viewed from the Variational Multiscale perspective, the classical F¯ method is shown to include only a volumetric or diagonal fine-scale deformation gradient while the proposed formulation includes the full spectrum of inter-scale coupling effects. Also, the error estimation procedure readily carries over from developments conducted for small strain linear problems. The formulation is presented first in the context of displacement-based methods and then extended to more general mixed methods accommodating arbitrary combinations of displacement–pressure interpolations. An extensive set of benchmark problems is investigated to show the performance of the method for a variety of hyperelastic materials exhibiting incompressible response.

A macro-element for incompressible finite deformations based on a volume averaged deformation gradient

A three-dimensional 8-node brick continuum finite element formulation for incompressible finite elasticity is presented. The core idea is to introduce a substructure consisting of eight sub-elements inside each finite element, further referred to as macro-element. For each of the sub-elements, the deformation is averaged. The weak form for each sub-element is based on the Hu-Washizu principle. The response of each sub-element is assembled and projected onto the eight external nodes of the macro-element. The introduction of deformable sub-elements in case of incompressible elasticity has two major advantages. Firstly, it is possible to suppress locking by evaluating the volumetric part of the response only in the macro-element instead of in each of the sub-elements. Secondly, no integration is necessary due to the use of averaged deformations on the sub-element level. The idea originates from the Cosserat point element developed in Nadler and Rubin (Int J Solids Struct 40:4585–4614, 2003). A consistent transition between the Cosserat point macro-element and a displacement macro-element formulation using a kinematical description from the enhanced strain element formulation (Flanagan, Belytschko in Int J Numer Methods Eng 17:679–706, 1981) or (Belytschko et al. in Comput Methods Appl Mech Eng 43:251–276, 1984) and the principle of Hu-Washizu is presented. The performance is examined by means of numerical examples.

Necessary Conditions at the Boundary for Minimizers in Incompressible Finite Elasticity

New necessary conditions for energy-minimizing states of an incompressible, elastic body are found. These must hold at boundary points at which dead-load traction data is prescribed.

On an Incompressible Nonlinearly Elastic Half-Space under a Compressive Point Load

The problem of determining the deformation produced in an elastic half space due to a point force normal to the undeformed boundary has been a subject of considerable interest in the linear theory of elas-ticity and in a variety of engineering and geotechnical applications for well over a century. Recently, this inherently nonlinear problem has been posed within the framework of the nonlinear theory of elasticity for tensile point loads and results stemming from the exact governing equations have been obtained aymptotically In this paper, we consider a compressive point load and address this core problem within the context of incompressible finite elasticity. An asymptotic analysis of the exact governing equations is carried out, and asymptotic tests providing a simple way to determine if an isotropic homogeneous hyperelastic material is capable of sustaining a compressive point load are developed. The results are then applied to several particular constitutive models for incompressible nonlinearly elastic materials, and contrasts with the tensile load problem are made.

An Asymptotic Finite Deformation Analysis for an Isotropic Compressible Hyperelastic Half-Space Subjected to a Tensile Point Load

The nonlinearly elastic Boussinesq problem is to find the deformation produced in a homogeneous, isotropic, elastic half-space by a point force normal to the undeformed boundary, using the exact equations of elasticity. For this core problem of elasticity and engineering, the 1885 linear elasticity solution of Boussinesq is still used in a variety of applications. In [SIAM J. Appl. Math., 62 (2001), pp. 107--128], we addressed the case of a tensile point load under the constraint of incompressible finite elasticity. Here we consider an analogous asymptotic analysis of this problem within the context of compressible finite elasticity. Asymptotic tests are developed to determine whether an isotropic hyperelastic material can support a finite deflection under a tensile point load. The results are then applied to a variety of particular constitutive models for compressible nonlinearly elastic materials. It is found that, for many of the well-known strain energy models for compressible hyperelastic materials proposed in the literature, a tensile point load cannot be supported. For models which may sustain a tensile point load, we determine the remaining equations and conditions for the asymptotic solution, and numerically compute this solution for a particular case.

An Asymptotic Finite Deformation Analysis for an Isotropic Incompressible Hyperelastic Half-Space Subjected to a Tensile Point Load

The nonlinearly elastic Boussinesq problem is to find the deformation produced in a homogeneous, isotropic, elastic half-space by a point force normal to the undeformed boundary, using the exact equations of elasticity. In this paper, we consider a tensile point load and address this core problem within the context of incompressible finite elasticity. The governing equations are first derived from a material formulation of the equilibrium equations for nominal stresses, and then, assuming that the well-known linear Boussinesq solution is valid at large distances from the point load, some implications of conservation laws of elastostatics when applied to the entire half-space are examined. Finally, asymptotic forms for solutions near the point load are hypothesized, and asymptotic tests providing a simple way of determining if an isotropic homogeneous material is capable of sustaining a tensile point load are developed. The results are then applied to several particular constitutive models for incompressible nonlinearly elastic materials.

Some recent improvements in meshfree methods for incompressible finite elasticity boundary value problems with contact

Two major difficulties are encountered in the meshfree solution of incompressible boundary value problems. The first is due to the employment of higher-order quadrature rules that leads to an over-constrained discrete system in incompressible problems. The second is associated with the treatment of essential boundary conditions and contact conditions owing to the loss of Kronecker delta properties in the meshfree shape functions. This paper discusses some recent enhancements in meshfree methods for incompressible boundary value problems, carries out numerical convergence analysis, and compares accuracy and efficiency improvement of these methods. Presented methods are a pressure projection method to remedy the over-constrained discrete system, and a mixed transformation method and a boundary singular kernel method for imposition of essential boundary conditions and contact constraints. Several linear and nonlinear problems were analyzed to demonstrate the effectiveness of the new approaches.

An improved reproducing kernel particle method for nearly incompressible finite elasticity

The previously developed reproducing kernel particle method (RKPM) employs a high-order quadrature rule for desired domain integration accuracy. This leads to an over-constrained condition in the limit of incompressibility, and volumetric locking and pressure oscillation were encountered. The employment of a large support size in the reproducing kernel shape function increases the dependency in the discrete constraint equations at quadrature points and thereby relieves locking. However, this approach consumes high CPU and it cannot effectively resolve pressure oscillation difficulty. In this paper, a pressure projection method is introduced by locally projecting the pressure onto a lower-order space to reduce the number of independent discrete constraint equations. This approach relieves the over-constrained condition and thus eliminates volumetric locking and pressure oscillation without the expense of employing large support size in RKPM. The method is developed in a general framework of nearly incompressible finite elasticity and therefore linear problems are also applicable. To further reduce the computational cost, a stabilized reduced integration method based on an assumed strain approach on the gradient matrix associated with the deformation gradient is also introduced. The resulting stiffness matrix and force vector of RKPM are obtained explicitly without numerical integration.

On some mixed finite element methods for incompressible and nearly incompressible finite elasticity

We compare some mixed methods based on different variational formulations, namely a displacement-pressure formulation employed by de Borst and coworkers, the three-field formulation investigated by Simo and Taylor and a two-field formulation which is directly based on an energy functional. It emerges that all these yield the same discrete results if the stored energy function contains a volumetric contribution 1/2k(J−1)2 whereJ is the volume dilatation, i.e., the Jacobian determinant of the deformation, andk is the bulk modulus. The equivalence holds for arbitrary 3D and plane strain elements. In the numerical examples the mixed formulations are discretized by the quadrilateral Q1/P0 and Q2/P1 elements and the triangular Crouzeix-Raviart P2+/P1 element. We also compare with standard displacement elements and the enhanced strain Q1/E4 element.

Parallel Algorithms for the Numerical Solution of Incompressible Finite Elasticity Problems

This paper deals with the numerical solution of three-dimensional equilibrium problems in incompressible finite elasticity on parallel computers. The approximation of the underlying variational problems by appropriate finite-element methods is discussed and an augmented Lagrangian formulation of the resulting discrete systems is given. Starting from this formulation, a combination of a simple gradient method and a block relaxation process is used to obtain an algorithm for the complicated nonlinear problems, which can be implemented on parallel computers in a very efficient way. Several results of numerical experiments on an array processor illustrate the capabilities of the considered methods.

Numerical Solution of Problems in Incompressible Finite Elasticity by Augmented Lagrangian Methods II. Three-Dimensional Problems

In a first paper [SIAM J. Appl. Math., 42 (1982), pp. 400–429] we introduced an augmented Lagrangian methodology for solving equilibrium problems in incompressible finite elasticity, and then applied it to the solution of several two-dimensional and axisymmetric test problems. In this second paper we extend our methods to the solution of genuine three-dimensional problems; actually using these augmented Lagrangian techniques we can reduce the solution of these complicated nonlinear, nonconvex problems, to the solution of a sequence of linear problems and also to the solution of a sequence of minimization problems for convex functionals on convex sets. The capabilities of our methods are illustrated by the results of numerical experiments.

Existence and approximation results for nonlinear mixed problems: Application to incompressible finite elasticity

This paper considers the problems of minimizing Gateaux-differentiable functionals over subsets of real Banach spaces defined by a non-linear equality constraint. The existence of a Lagrange multiplier is proved, together with approximation results on the constrained subset, provided a nonlinear compatibility condition, generalizing the classical inf-sup condition, is satisfied. These ideas are applied to equilibrium problems in incompressible finite elasticity and lead to convergence results for these problems.

Numerical Solution of Problems in Incompressible Finite Elasticity by Augmented Lagrangian Methods. I. Two-Dimensional and Axisymmetric Problems

We discuss in this paper a new approach to equilibrium problems in incompressible finite elasticity. This approach, which is based on the use of a convenient augmented Lagrangian functional, leads to a family of iterative methods which seem very effective for solving this type of elasticity problems once they have been approximated by appropriate finite element methods.The possibilities of the above methods are illustrated by the numerical solution of several problems in two-dimensional and axisymmetric geometries; comparisons with available analytical results are also presented.

Compatibility condition and existence results in discrete finite incompressible elasticity

A theory on the existence of solutions of the discretized equilibrium equations in incompressible finite elasticity is given. A compatibility condition between the space of displacements and the space of pressures is introduced and is studied in a particular case of finite element spaces, using Q 1 and Q 0 isoparametric elements. An original assembling of these elements is presented which leads to discrete spaces satisfying the compatibility condition.