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In this paper, a residual correction method based upon an extension of the finite calculus concept is presented. The method is described and applied to the solution of a scalar convection‐diffusion problem and the problem of elasticity at the incompressible or quasi‐incompressible limit. The formulation permits the use of equal interpolation for displacements and pressure on linear triangles and tetrahedra as well as any low order element type. To add additional stability in the solution, pressure gradient corrections are introduced as suggested from developments of sub‐scale methods. Numerical examples are included to demonstrate the performance of the method when applied to typical test problems.

The paper starts with a review of constitutive equations for rubber‐like materials, formulated in the invariants of the right Cauchy—Green deformation tensor. A general framework for the derivation of the stress tensor and the tangent moduli for invariant‐based models, for both the reference and the current configuration, is presented. The free energy of incompressible rubber‐like materials is extended to a compressible formulation by adding the volumetric part of the free energy. In order to overcome numerical problems encountered with displacement‐based finite element formulations for nearly incompressible materials, three‐dimensional finite elements, based on a penalty‐type formulation, are proposed. They are characterized by applying reduced integration to the volumetric parts of the tangent stiffness matrix and the pressure‐related parts of the internal force vector only. Moreover, hybrid finite elements are proposed. They are based on a three‐field variational principle, characterized by treating the displacements, the dilatation and the hydrostatic pressure as independent variables. Subsequently, this formulation is reduced to a generalized displacement formulation. In the numerical study these formulations are evaluated. The results obtained are compared with numerical results available in the literature. In addition, the proposed formulations are applied to 3D finite element analysis of an automobile tyre. The computed results are compared with experimental data.

This work explores the properties of four‐noded quadrilateral elements for which integration is carried out using low‐order integration rules, based on one‐point integration over subelements making up the quadrilaterals. The purpose is to identify those rules which lead to elements having the desirable properties of high coarse‐mesh accuracy, and stability in the incompressible limit. A two‐point rule is investigated in detail, as is its counterpart for problems of incompressible media, in which the volumetric term is integrated using a one‐point rule. Numerical results indicate that the new elements perform well in general when compared with existing enhanced strain or equivalent elements, and appear to be particularly efficient in cases in which meshes are severely distorted.

This study aims to introduce the hybrid finite element (FE) – meshfree method and multiscale variational principle into the traditional mixed FE formulation, leading to a stable mixed formulation for incompressible linear elasticity which circumvents the need to satisfy inf-sup condition.

Using the hybrid FE–meshfree method, the displacement and pressure are interpolated conveniently with the same order so that a continuous pressure field can be obtained with low-order elements. The multiscale variational principle is then introduced into the Galerkin form to obtain stable and convergent results.

The present method is capable of overcoming volume locking and does not exhibit unphysical oscillations near the incompressible limit. Moreover, there are no extra unknowns introduced in the present method because the fine-scale unknowns are eliminated using the static condensation technique, and there is no need to evaluate any user-defined stability parameter as the classical stabilization methods do. The shape functions constructed in the present model possess continuous derivatives at nodes, which gives a continuous and more precise stress field with no need of an additional smooth process. The shape functions in the present model also possess the Kronecker delta property, so that it is convenient to impose essential boundary conditions.

The proposed model can be implemented easily. Its convergence rates and accuracy in displacement, energy and pressure are even comparable to those of second-order mixed elements.

Classical mixed formulations of the boundary‐value problem of linear elasticity are reviewed, and a new three‐field formulation is introduced. The formulation is an extension of the classical Hu‐Washizu approach, and takes the form of a non‐standard mixed problem. Convergence of finite element approximations of both the old and new methods are discussed, with an emphasis on their behaviour in the incompressible limit. Conditions for the stability and uniform convergence of the new method are presented, and it is shown that the Pian‐Sumihara basis, when used in the new formulation, leads to a convergent method.

The purpose of this paper is to describe a general theoretical and finite element implementation framework for the constitutive modelling of biological soft tissues.

The model is based on continuum fibers reinforced composites in finite strains. As an extension of the isotropic hyperelasticity, it is assumed that the strain energy function is decomposed into a fully isotropic component and an anisotropic component. Closed form expressions of the stress tensor and elasticity tensor are first established in the general case of fully incompressible plane stress which orthotropic and transversely isotropic hyperelasticity. The incompressibility is satisfied exactly.

Numerical examples are presented to illustrate the model's performance.

The paper presents a constitutive model for incompressible plane stress transversely isotropic and orthotropic hyperelastic materials.

As known from nearly incompressible elasticity, selective reduced integration (SRI) is a simple and effective method of overcoming the volumetric locking problem in 2D and 3D solid elements. This method of finite elastoviscoplasticity is discussed as are its well‐known limitations. In this context, an isochoric‐volumetric decoupled material behavior is assumed and thus the additive deviatoric‐volumetric decoupling of the consistent algorithmic moduli tensor is essential. By means of several numerical examples, the performance of elements using selective reduced integration is demonstrated and compared to the performance of other elements such as the enhanced assumed strain elements. It is shown that a minor modification, with little numerical effort, leads to rather robust element behaviour. The application of this process to so‐called solid‐shell elements for thin‐walled structures is also discussed.

Engineers have developed robust and efficient incompressible finite element formulations using tools such as the Patch Test and the counting of constraints/variables, the first one aimed at the development of consistent elements and the second one aimed at the development of non‐locking and stable elements. The mentioned tools are rooted in the physics of the continuum mechanics problem. Mathematicians, on the other side, developed complex and powerful tools to examine the convergence of finite element formulations, such as the inf‐sup condition, these methods are based on the properties of the elliptical PDEs that constitute the mathematical model of the continuum mechanics problem. In this paper we intend to understand the inf‐sup condition from an engineering perspective, so as to be able to incorporate it into the package of tools used in the development of finite element formulations.

A new approach to deal with the finite element analysis of incompressible material is presented. The constrained variational problem relating to the analysis of incompressible material is transformed into two unconstrained variational problems in two corresponding displacement subspaces, which are called the incompressible‐deviatoric (Sd) and the compressible‐undeviatoric (Sc) displacement subspaces respectively. The displacement and deviatoric stress, and the pressure fields, are determined by means of variations in the two subspaces respectively. As compared with some current methods, it is found that the present method is capable of solving the problem of incompressible material with v = 0.5, and that there is no problem about the existence of solution. Further, the ill‐conditioning of the global matrix can be entirely eliminated and the computational effort can be considerably reduced as well. The formulation for the finite element analysis of incompressible material with material or geometrical non‐linearity based on the subspace Sd are given in the paper. The numerical results for some examples show the advantages of the approach presented in the paper.

This paper focuses on the development of a new class of eight‐node solid finite elements, suitable for the treatment of volumetric and transverse shear locking problems. Doing so, the proposed elements can be used efficiently for 3D and thin shell applications. The starting point of the work relies on the analysis of the subspace of incompressible deformations associated with the standard (displacement‐based) fully integrated and reduced integrated hexahedral elements. Prediction capabilities for both formulations are defined related to nearly‐incompressible problems and an enhanced strain approach is developed to improve the performance of the earlier formulation in this case. With the insight into volumetric locking gained and benefiting from a recently proposed enhanced transverse shear strain procedure for shell applications, a new element conjugating both the capabilities of efficient solid and shell formulations is obtained. Numerical results attest the robustness and efficiency of the proposed approach, when compared to solid and shell elements well‐established in the literature.