Effective moduli of hyperelastic porous media at large deformation

Summary.  Among the parameters affecting the overall material properties of porous media, the most significant involve the micromechanical morphology, the matrix material behavior and the applied load range. Considering a unit cell for the porous medium, several approaches of the material response are developed, which yield the effective properties of the medium. Numerical results are presented and compared with experimental or analytical data available in literature. Proposed formulations impose several material characterizations ranging from linear elastic to incompressible hyperelastic. In the case of nonlinear materials, a special formulation has been developed permitting prediction of the porous material moduli. This formulation considers a special nonlinear form for the strain energy function under specific loading conditions. The proposed method yields simple formulas approximating the effective moduli of porous media, which are useful for design purposes. Received August 3, 2001; revised August 14, 2002 Published online: January 16, 2003 Acknowledgements The first of the authors is grateful to his mentor Dr. Paul J. Blatz for his encouragement all these years for continuous research on the nonlinear theories of hyperelastic materials.     

An updated Lagrangian finite element sensitivity analysis of large deformations using quadrilateral elements

A continuum parameter and shape sensitivity analysis is presented for metal forming processes using the finite element method. The sensitivity problem is posed in a novel updated Lagrangian framework as suitable for very large deformations when remeshing operations are performed during the analysis. In addition to exploring the issue of transfer of variables between meshes for finite deformation analysis, the complex problem of transfer of design sensitivities (derivatives) between meshes for large deformation inelastic analyses is also discussed. A method is proposed that is shown to give accurate estimates of design sensitivities when remeshing operations are performed during the analysis. Sensitivity analysis for the consistent finite element treatment of near incompressibility within the context of the assumed strain methods is also proposed. In particular, the performance of four‐noded quadrilateral elements for the sensitivity analysis of large deformations is studied. The results of the continuum sensitivity analysis are validated by a comparison with those obtained by a finite difference approximation (i.e. using the solution of a perturbed deformation problem). The effectiveness of the method is demonstrated by applications in the design optimization of metal forming processes. Copyright © 2001 John Wiley & Sons, Ltd.     

Axisymmetric three- and four-node finite elements for large strain elastoplasticity

Within the framework of the finite element method an application of the logarithmic strain space formulation of large strain elastoplasticity is illustrated for the examples of axisymmetric three-node triangular and four-node quadrilateral finite elements. The formulation of the large strain elastoplasticity is based on a strain space formulation in conjunction with logarithmic (or Hencky) strain tensors with respect to the reference configuration. It is therefore—from a material point of view—a full Lagrangian formulation. The use of logarithmic strains enables an additive split of finite dilatation and distortion, which are given by the logarithmic strain trace and deviator. As a consequence of the strain space formulation no stress tensors are involved in order to describe the plasticity. The stress which is work-conjugate to the logarithmic strain follows from the stress-strain relations and may be transformed to Cauchy stress. The desired finite element matrices are derived via the principle of virtual work applied to the Cauchy stress distribution of the current configuration. It should be noted that our considerations are not restricted to axisymmetry and that they remain valid for isoparametric, position- (displacement-) based finite elements in general.     

Assumed‐deformation gradient finite elements with nodal integration for nearly incompressible large deformation analysis

An assumed‐strain finite element technique for non‐linear finite deformation is presented. The weighted‐residual method enforces weakly the balance equation with the natural boundary condition and also the kinematic equation that links the elementwise and the assumed‐deformation gradient. Assumed gradient operators are derived via nodal integration from the kinematic‐weighted residual. A variety of finite element shapes fits the derived framework: four‐node tetrahedra, eight‐, 27‐, and 64‐node hexahedra are presented here. Since the assumed‐deformation gradients are expressed entirely in terms of the nodal displacements, the degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials and no volumetric/deviatoric split is required. The consistent tangent operator is inexpensive and symmetric. Furthermore, the material update and the tangent moduli computation are carried out exactly as for classical displacement‐based models; the only deviation is the consistent use of the assumed‐deformation gradient in place of the displacement‐derived deformation gradient. Examples illustrate the performance with respect to the ability of the present technique to resist volumetric locking. A constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Copyright © 2008 John Wiley & Sons, Ltd.     

Solution of large deformation contact problems with friction between Blatz-Ko hyperelastic bodies

The present paper is devoted to the analysis of the contact problems with Coulomb friction and large deformation between two hyperelastic bodies. One approach to separate the material nonlinearity and contact nonlinearity is presented. The total Lagrangian formulation is adopted to describe the geometrically nonlinear behavior. Nondifferentiable contact potentials are regularized by means of the augmented Lagrangian method. Numerical examples are carried out in two cases: rigid-deformable contact and deformable-deformable contact with large slips. The numerical results prove that the proposed approach is robust and efficient concerning numerical stability.     

Asymmetric quadrilateral shell elements for finite strains

Very good results in infinitesimal and finite strain analysis of shells are achieved by combining either the enhanced-metric technique or the selective-reduced integration for the in-plane shear energy and an assumed natural strain technique (ANS) in a non-symmetric Petrov–Galerkin arrangement which complies with the patch-test. A recovery of the original Wilson incompatible mode element is shown for the trial functions in the in-plane components. As a beneficial side-effect, Newton–Raphson convergence behavior for non-linear problems is improved with respect to symmetric formulations. Transverse-shear and in-plane patch tests are satisfied while distorted-mesh accuracy is higher than with symmetric formulations. Classical test functions with assumed-metric components are required for compatibility reasons. Verification tests are performed with advantageous comparisons being observed in all of them. Applications to large displacement elasticity and finite strain plasticity are shown with both low sensitivity to mesh distortion and (relatively) high accuracy. A equilibrium-consistent (and consistently linearized) updated-Lagrangian algorithm is proposed and tested. Concerning the time-step dependency, it was found that the consistent updated-Lagrangian algorithm is nearly time-step independent and can replace the multiplicative plasticity approach if only moderate elastic strains are present, as is the case of most metals.     

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     

Eigenvalue analysis of compatible and incompatible rectangular four-node quadrilateral elements

Exact analytical expressions for the eigenvalues of the elastic stiffness matrix are obtained for the four-node, rectangular, quadrilateral element. A procedure is given for identifying alternative hourglass modes and eigenvalues which render the element incompatible but with non-monotonic convergence assured. A convergence study confirms that for the special case of when the hourglass modes coincide with beam bending the element can serve as a beam element. Analytical expressions are given for the resulting element stiffness matrix.     

Solution of large deformation impact problems with friction between Blatz–Ko hyperelastic bodies

Feng et al. [Z.Q. Feng, F. Peyraut, N. Labed, Solution of large deformation contact problems with friction between Blatz–Ko hyperelastic bodies, Int. J. Eng. Sci. 41 (2003) 2213–2225] have proposed a study of contact problems between Blatz–Ko hyperelastic bodies in static cases using the bi-potential method. The extension of this method for dynamic analysis of impact problems is realized in the present work. The total Lagrangian formulation is adopted to describe large strains and large displacements non-linear behavior. A first order algorithm is applied for the numerical integration of the time-discretized equation of motion. Numerical examples are carried out in two cases: rigid–deformable and deformable–deformable–rigid impacts in 2D. Numerical results show that the proposed approach is robust and efficient and the physical energy dissipation phenomena are apparently illustrated.     

A comparative study of hyperelastic and hypoelastic material models with constant elastic moduli for large deformation problems

Many finite element programs including commercial codes for large deformation analysis employ incremental formulations of rate-type constitutive equations which are based on hyperelastic or hypoelastic material models with constant elastic moduli. In this paper, a comparative study is carried out for hyperelastic and hypoelastic material models with constant elastic moduli of a face-centered cubic single crystal of copper. A strain energy function from the inter-atomic potential for single-crystal copper is also considered for the hyperelastic material model to obtain physically based elastic deformations. Numerical results show that constant elastic moduli of hypoelastic material models can cause considerable errors in stress and strain increments when the changes in volume and cross-sectional area of a material are not negligible.     

Axisymmetric elastodynamic infinite elements for multi-layered half-space

This paper presents three axisymmetric infinite elements for the elastodynamic problems in a multi-layered half-space. They are the horizontal, the vertical and the corner infinite elements that are developed by using the wave functions in the function spaces derived from approximate expressions of the analytical solutions. An efficient integration procedure is proposed for calculating the element matrices involving multiple wave components. Numerical example analyses are presented for rigid disks on homogeneous and layered half-spaces, and for an embedded caisson. The numerical results obtained show the effectiveness of the proposed infinite elements.     

Adaptive finite element analysis with quadrilateral elements using a newh-refinement strategy

The theory and mathematical bases ofa-posteriori error estimates are explained. It is shown that theMedial Axis of a body can be used to decompose it into a set of mutually non-overlapping quadrilateral and triangular primitives. A mesh generation scheme used to generate quadrilaterals inside these primitives is also presented together with its relevant implementation aspects. A newh-refinement strategy based on weighted average energy norm and enhanced by strain energy density ratios is proposed and two typical problems are solved to demonstrate its efficiency over the conventional refinement strategy in the relative improvement of global asymptotic convergence.     

Bending of bilinear quadrilateral shell elements

Formulae are derived for exact inextensional bending solutions for arbitrary quadrilateral shell finite elements of bilinear parametric representation. It is found that the polynomial degree of parametric representation of the rectangular components of displacement requires to be at least cubic in order to describe any inextensional bending modes.     

Finite element formulations for large deformation dynamic analysis

Starting from continuum mechanics principles, finite element incremental formulations for non-linear static and dynamic analysis are reviewed and derived. The aim in this paper is a consistent summary, comparison, and evaluation of the formulations which have been implemented in the search for the most effective procedure. The general formulations include large displacements, large strains and material non-linearities. For specific static and dynamic analyses in this paper, elastic, hyperelastic (rubber-like) and hypoelastic elastic-plastic materials are considered. The numerical solution of the continuum mechanics equations is achieved using isoparametric finite element discretization. The specific matrices which need be calculated in the formulations are presented and discussed. To demonstrate the applicability and the important differences in the formulations, the solution of static and dynamic problems involving large displacements and large strains are presented.     

A simplified co‐rotational method for quadrilateral shell elements in geometrically nonlinear analysis

This paper presents a simplified co‐rotational formulation for quadrilateral shell elements inheriting the merit of element‐independence from the traditional co‐rotational approach in literature. With the objective of application to nonlinear analysis of civil engineering structures, the authors further simplify the formulation of the geometrical stiffness using the small strain assumption, which is valid in the co‐rotational approach, with the warping effects considered as eccentricities. Compared with the traditional element‐independent co‐rotational method, the projector is neglected both in the tangent stiffness matrix and in the internal force vector for simplicity in formulation. Meanwhile, a quadrilateral flat shell element allowing for drilling rotations is adopted and incorporated into this simplified co‐rotational algorithm for geometrically nonlinear analysis involved with large displacements and large rotations. Several benchmark problems are presented to confirm the efficiency and accuracy of the proposed method for practical applications.     

Derivation of Lagrangian and Hermitian shape functions for quadrilateral elements

This paper introduces a general theory for the derivation of the shape functions for the quadrilateral family of finite elements. The first section deals with the Lagrangian shape functions for the cases of uniform and boundary-described elements. Two basic procedures are introduced; the first by linear combinations of side-interpolations and the second by superposition. The remainder of the paper introduces a theory for the general uniform Hermitian element of any order. Details for quadrilateral elements, with first order derivatives are explained. All of the shape functions presented here were derived in the interval [0,1]. The shape functions, developed by such an engineering approach, have been used successfully in the ABSEA Finite Element System of Cranfield Institute of Technology.     

B spline-based method for 2-D large deformation analysis

In this paper, quadratic cardinal B spline functions are used for solution of 2-D large deformation problems. Because the B spline functions are directly used in function approximation, no meshes are needed and the mesh distortion issues in nonlinear analyses are avoided in this method. Using the B spline functions, the solution formulations based on total Lagrangian (TL) approach for two dimensional large deformation problems are established. The numerical examples of 2-D large deformation problems indicate that the B spline method is effective and stable for solving complicated problems.