Bi-modulus materials consistent with a stored energy function: Theory and numerical implementation

2020 ◽  
Vol 229 ◽  
pp. 106176 ◽  
Author(s):  
Marcos Latorre ◽  
Francisco J. Montáns
Keyword(s):  
Energy Function ◽  
Function Theory ◽  
Numerical Implementation ◽  
Stored Energy

A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and Applications

2011 ◽  
Vol 58 (3) ◽  
pp. 319-346 ◽  
Author(s):  
Cyprian Suchocki
Keyword(s):  
Finite Element ◽  
Energy Function ◽  
Function Theory ◽  
Constitutive Models ◽  
Elasticity Tensor ◽  
Stored Energy ◽  
Energy Potential ◽  
Hyperelastic Materials ◽  
Finite Element Implementation ◽  
Living Tissues

A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and Applications This paper contains the full way of implementing a user-defined hyperelastic constitutive model into the finite element method (FEM) through defining an appropriate elasticity tensor. The Knowles stored-energy potential has been chosen to illustrate the implementation, as this particular potential function proved to be very effective in modeling nonlinear elasticity within moderate deformations. Thus, the Knowles stored-energy potential allows for appropriate modeling of thermoplastics, resins, polymeric composites and living tissues, such as bone for example. The decoupling of volumetric and isochoric behavior within a hyperelastic constitutive equation has been extensively discussed. An analytical elasticity tensor, corresponding to the Knowles stored-energy potential, has been derived. To the best of author's knowledge, this tensor has not been presented in the literature yet. The way of deriving analytical elasticity tensors for hyperelastic materials has been discussed in detail. The analytical elasticity tensor may be further used to develop visco-hyperelastic, nonlinear viscoelastic or viscoplastic constitutive models. A FORTRAN 77 code has been written in order to implement the Knowles hyperelastic model into a FEM system. The performance of the developed code is examined using an exemplary problem.

scholarly journals An efficiency comparison of numerical implementation approaches of hyperelastic constitutive model in ABAQUS/Standard

2018 ◽  
Vol 196 ◽  
pp. 01052
Author(s):  
Aleksander Franus ◽  
Łukasz Kowalewski
Keyword(s):  
Constitutive Model ◽  
Contact Problem ◽  
Computational Efficiency ◽  
Energy Function ◽  
Material Model ◽  
Numerical Implementation ◽  
Stored Energy ◽  
Efficiency Comparison ◽  
Consistent Tangent Operator

The main goal of the article is to compare a computational efficiency of different implementation of a hyperelastic material model in the ABAQUS/Standard v. 6.14 [1]. The software offers basically two ways to proceed, namely UMAT and UHYPER user subroutines [2]. The procedures are employed to implement an isotropic, compressible neo-Hookean material model [3]. Corresponding stored-energy function, constitutive equations and the consistent tangent operator are presented. These are essential to program the subroutines. Some theoretical and numerical aspects of different implementation approaches are discussed. As examples, a tube under axial compression and a contact problem of disc are considered. On the basis of obtained results, selected aspects of computational efficiency and quality of solutions are compared.

The quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function

1995 ◽  
Vol 125 (6) ◽  
pp. 1179-1192 ◽  
Author(s):  
Hervé Le Dret ◽  
Annie Raoult
Keyword(s):  
Explicit Expression ◽  
Energy Function ◽  
Singular Values ◽  
Convex Envelope ◽  
External Loading ◽  
Stored Energy ◽  
Quasiconvex Envelope

We give an explicit expression for the quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function in terms of the singular values. This envelope is also the convex, polyconvex and rank 1 convex envelope of the Saint Venant–Kirchhoff stored energy function. Moreover, it coincides with the Saint Venant–Kirchhoff stored energy function itself on, and only on, the set of matrices whose singular values arranged in increasing order are located outside an ellipsoid. It vanishes on, and only on, the set of matrices whose singular values are less than 1. Consequently, a Saint Venant–Kirchhoff material can be compressed under zero external loading.

On the global stability of two-dimensional, incompressible, elastic bars in uniaxial extension

2009 ◽  
Vol 466 (2116) ◽  
pp. 1167-1176 ◽  
Author(s):  
Jeyabal Sivaloganathan ◽  
Scott J. Spector
Keyword(s):  
Elastic Energy ◽  
Energy Function ◽  
Shear Bands ◽  
Uniaxial Extension ◽  
Two Dimensional ◽  
Stored Energy ◽  
Absolute Minimizer ◽  
Local Bifurcations ◽  
Maximum Value ◽  
Incompressible Hyperelastic Material

When a rectangular bar is subjected to uniaxial tension, the bar usually deforms (approximately) homogeneously and isoaxially until a critical load is reached. A bifurcation, such as the formation of shear bands or a neck, may then be observed. One approach is to model such an experiment as the in-plane extension of a two-dimensional, homogeneous, isotropic, incompressible, hyperelastic material in which the length of the bar is prescribed, the ends of the bar are assumed to be free of shear and the sides are left completely free. It is shown that standard constitutive hypotheses on the stored-energy function imply that no such bifurcation is possible in this model due to the fact that the homogeneous isoaxial deformation is the unique absolute minimizer of the elastic energy. Thus, in order for a bifurcation to occur either the material must cease to be elastic or the stored-energy function must violate the standard hypotheses. The fact that no local bifurcations can occur under the assumptions used herein was known previously, since these assumptions prohibit the load on the bar from reaching a maximum value. However, the fact that the homogeneous deformation is the absolute minimizer of the energy appears to be a new result.

Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions

1992 ◽  
Vol 121 (1-2) ◽  
pp. 101-138 ◽  
Author(s):  
Piotr Rybka
Keyword(s):  
Phase Transitions ◽  
Energy Function ◽  
Boundary Data ◽  
Initial Conditions ◽  
Stored Energy ◽  
Dynamical Modelling

SynopsisWe study the equations of viscoelasticity in a multidimensional setting for the ‘no-traction’ boundary data. For the sake of modelling phase transitions we do not assume elliptieity of the stored energy function W. We construct dynamics in W1,2(Ωℝn) globally in time. Next, we study the question of stability for a class of equilibria. Moreover, we show a certain kind of decay in time of solutions for arbitrary initial conditions.

Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber

1951 ◽  
Vol 243 (865) ◽  
pp. 251-288 ◽  
Keyword(s):  
Energy Function ◽  
Pure Shear ◽  
Thin Sheet ◽  
The Other ◽  
Homogeneous Deformation ◽  
Simple Extension ◽  
Stored Energy ◽  
Elastic Deformations ◽  
Isotropic Materials ◽  
Deformation Curves

It is shown in this part how the theory of large elastic deformations of incompressible isotropic materials, developed in previous parts, can be used to interpret the load-deformation curves obtained for certain simple types of deformation of vulcanized rubber test-pieces in terms of a single stored-energy function. The types of experiment described are: (i) the pure homogeneous deformation of a thin sheet of rubber in which the deformation is varied in such a manner that one of the invariants of the strain, I 1 or I 2 , is maintained constant; (ii) pure shear of a thin sheet of rubber (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained at unity, while the other is varied); (iii) simultaneous simple extension and pure shear of a thin sheet (i.e. pure homogeneous deformation in which one of the extension ratios in the plane of the sheet is maintained constant at a value less than unity, while the other is varied); (iv) simple extension of a strip of rubber; (v) simple compression (i.e. simple extension in which the extension ratio is less than unity); (vi) simple torsion of a right-circular cylinder; (vii) superposed axial extension and torsion of a right-circular cylindrical rod. It is shown that the load-deformation curves in all these cases can be interpreted on the basis of the theory in terms of a stored-energy function W which is such that δ W /δ I 1 is independent of I 1 and I 2 and the ratio (δ W /δ I 2 ) (δ W /δ I 1 ) is independent of I 1 and falls, as I 2 increases, from about 0*25 at I 2 = 3.

A Simple w′(λ) Function for The Valanis-Landel Form of Stored Energy Function

1998 ◽  
Vol 71 (2) ◽  
pp. 234-243 ◽  
Author(s):  
Robert F. Landel
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Long Range ◽  
Energy Function ◽  
Good Accuracy ◽  
Linear Range ◽  
Stored Energy ◽  
Element Analysis ◽  
Strong Curvature

Abstract In the Valanis-Landel formulation of the stored energy function W, stresses depend on the function w′(λ)(=dw/dλ). This function exhibits strong curvature, making it difficult to represent analytically with good accuracy. It is found for both SBR and NR that the function λw′(λ) is not only far less curved, it is essentially linear in λ for the range of about 0.4 < λ < 2.0. The long range of simple proportionality to strain invites examination of molecular theories of rubberlike elasticity. Above the linear range the response can be approximated by kλn. These simplifications should make it easier to convert w′(λ) to the W1 and W2 functions employed in finite element analysis.

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