Wrinkling of a Fiber-Reinforced Membrane

2008 ◽  
Vol 76 (1) ◽  
Author(s):  
E. Shmoylova ◽  
A. Dorfmann
Keyword(s):  
Boundary Conditions ◽  
Energy Function ◽  
Mechanical Response ◽  
Transverse Isotropy ◽  
Base Material ◽  
Strain Energy Function ◽  
Transversely Isotropic ◽  
Incompressible Material ◽  
Fiber Reinforced ◽  
Relaxed Energy

In this paper we investigate the response of fiber-reinforced cylindrical membranes subject to axisymmetric deformations. The membrane is considered as an incompressible material, and the phenomenon of wrinkling is taken into account by means of the relaxed energy function. Two cases are considered: transversely isotropic membranes, characterized by one family of fibers oriented in one direction, and orthotropic membranes, characterized by two family of fibers oriented in orthogonal directions. The strain-energy function is considered as the sum of two terms: The first term is associated with the isotropic properties of the base material, and the second term is used to introduce transverse isotropy or orthotropy in the mechanical response. We determine the mechanical response of the membrane as a function of fiber orientations for given boundary conditions. The objective is to find possible fiber orientations that make the membrane as stiff as possible for the given boundary conditions. Specifically, it is shown that for transversely isotropic membranes a unique fiber orientation exists, which does not affect the mechanical response, i.e., the overall behavior is identical to a nonreinforced membrane.

A Simple Transversely Isotropic Hyperelastic Constitutive Model Suitable for Finite Element Analysis of Fiber Reinforced Elastomers

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Leslee W. Brown ◽  
Lorenzo M. Smith
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Transversely Isotropic ◽  
Material Model ◽  
Fiber Reinforced ◽  
Element Analysis ◽  
Material Tests

A transversely isotropic fiber reinforced elastomer’s hyperelasticity is characterized using a series of constitutive tests (uniaxial tension, uniaxial compression, simple shear, and constrained compression test). A suitable transversely isotropic hyperelastic invariant based strain energy function is proposed and methods for determining the material coefficients are shown. This material model is implemented in a finite element analysis by creating a user subroutine for a commercial finite element code and then used to analyze the material tests. A useful set of constitutive material data for multiple modes of deformation is given. The proposed strain energy function fits the experimental data reasonably well over the strain region of interest. Finite element analysis of the material tests reveals further insight into the materials constitutive nature. The proposed strain energy function is suitable for finite element use by the practicing engineer for small to moderate strains. The necessary material coefficients can be determined from a few simple laboratory tests.

scholarly journals A general polyconvex strain-energy function for fiber-reinforced materials

PAMM ◽  
2005 ◽  
Vol 5 (1) ◽  
pp. 245-246 ◽  
Author(s):  
Bernd Markert ◽  
Wolfgang Ehlers ◽  
Nils Karajan
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Fiber Reinforced ◽  
Fiber Reinforced Materials ◽  
Reinforced Materials

Finite Axisymmetric Deformations of Initially Cylindrical Reinforced Membranes

1972 ◽  
Vol 45 (6) ◽  
pp. 1677-1683 ◽  
Author(s):  
A. D. Kydoniefs
Keyword(s):  
Stress Field ◽  
Internal Pressure ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Incompressible Material ◽  
Parameter System ◽  
Axial Section ◽  
Variable Pitch ◽  
Two Parameter

Abstract We consider the axisymmetrie deformations of an initially cylindrical membrane composed of an elastic, homogeneous, isotropic and incompressible material reinforced with a two-parameter system of perfectly flexible and inextensible helicoidal cords of variable pitch. The undeformed configuration is determined so that the deformed membrane has a given axial section under specified internal pressure. The corresponding stress field and cord tensions are obtained. The solution given is exact and valid for the general form of the strain—energy function.

Large Deformation Isotropic Elasticity—On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids

1973 ◽  
Vol 46 (2) ◽  
pp. 398-416 ◽  
Author(s):  
R. W. Ogden
Keyword(s):  
Mathematical Analysis ◽  
Energy Function ◽  
Strain Energy ◽  
Mechanical Response ◽  
Strain Energy Function ◽  
Usual Practice ◽  
Principal Axes ◽  
Simple Tension ◽  
Isotropic Elasticity ◽  
Strain Invariants

Abstract Many attempts have been made to reproduce theoretically the stress-strain curves obtained from experiments on the isothermal deformation of highly elastic ‘rubberlike’ materials. The existence of a strain-energy function has usually been postulated, and the simplifications appropriate to the assumptions of isotropy and incompressibility have been exploited. However, the usual practice of writing the strain energy as a function of two independent strain invariants has, in general, the effect of complicating the associated mathematical analysis (this is particularly evident in relation to the calculation of instantaneous moduli of elasticity) and, consequently, the basic elegance and simplicity of isotropic elasticity is sacrificed. Furthermore, recently proposed special forms of the strain-energy function are rather complicated functions of two invariants. The purpose of this paper is, while making full use of the inherent simplicity of isotropic elasticity, to construct a strain-energy function which: (i) provides an adequate representation of the mechanical response of rubberlike solids, and (ii) is simple enough to be amenable to mathematical analysis. A strain-energy function which is a linear combination of strain invariants defined by ϕ(α)=(α1α+α2α+α3α)/α is proposed; and the principal stretches α1, α2, and α3 are used as independent variables subject to the incompressibility constraint α1α2α3=1. Principal axes techniques are used where appropriate. An excellent agreement between this theory and the experimental data from simple tension, pure shear and equibiaxial tension tests is demonstrated. It is also shown that the present theory has certain repercussions in respect of the constitutive inequality proposed by Hill.

Stresses in a Transversely Isotropic Slab Having a Spherical Cavity

1973 ◽  
Vol 40 (3) ◽  
pp. 752-758 ◽  
Author(s):  
A. Atsumi ◽  
S. Itou
Keyword(s):  
Stress Distribution ◽  
Boundary Conditions ◽  
Spherical Cavity ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Hankel Transform ◽  
Isotropic Slab ◽  
Infinite Slab

This paper deals with the analysis of the stress distribution arising in a transversely isotropic infinite slab with a symmetrically located spherical cavity under all-around tension. Difficulties in satisfying both boundary conditions on the surfaces of the slab and the surface of the cavity are successfully overcome by using the methods of Hankel transform and Schmidt-orthogonormalization. For some practical materials the influence of transverse isotropy upon stress distribution is presented in the form of curves.

scholarly journals Evolution of anisotropy in soft tissue

2014 ◽  
Vol 470 (2161) ◽  
pp. 20130548 ◽  
Author(s):  
J. G. Murphy
Keyword(s):  
Linear Theory ◽  
Energy Function ◽  
Anisotropic Material ◽  
Strain Energy ◽  
Mechanical Response ◽  
Essential Feature ◽  
Sufficient Conditions ◽  
Strain Energy Function ◽  
Phenomenological Approach ◽  
Reduced Form

The phenomenological approach to the modelling of the mechanical response of arteries usually assumes a reduced form of the strain-energy function in order to reduce the mathematical complexity of the model. A common approach eschews the full basis of seven invariants for the strain-energy function in favour of a reduced set of only three invariants. It is shown that this reduced form is not consistent with the corresponding full linear theory based on infinitesimal strains. It is proposed that compatibility with the linear theory is an essential feature of any nonlinear model of arterial response. Two approaches towards ensuring such compatibility are proposed. The first is that the nonlinear theory reduces to the full six-constant linear theory, without any restrictions being imposed on the constants. An alternative modelling strategy whereby an anisotropic material is compatible with a simpler material in the linear limit is also proposed. In particular, necessary and sufficient conditions are obtained for a nonlinear anisotropic material to be compatible with an isotropic material for infinitesimal deformations. Materials that satisfy these conditions should be useful in the modelling of the crimped collagen fibres in the undeformed configuration.

On the Constitutive Equations of Biological Materials

1975 ◽  
Vol 42 (1) ◽  
pp. 242-243 ◽  
Author(s):  
H. Demiray
Keyword(s):  
Energy Function ◽  
Constitutive Equations ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Transversely Isotropic ◽  
Biological Materials ◽  
Biological Tissues

This paper deals with a simple possible form of the strain-energy function for biological tissues which are assumed to be transversely isotropic. Also the solution of a problem is studied and the result is compared with experiments.

scholarly journals Residually Stressed Fiber Reinforced Solids: A Spectral Approach

Materials ◽  
2020 ◽  
Vol 13 (18) ◽  
pp. 4076
Author(s):  
Mohd Halim Bin Mohd Shariff ◽  
Jose Merodio
Keyword(s):  
Constitutive Equation ◽  
Energy Function ◽  
Strain Energy ◽  
Finite Strain ◽  
Strain Energy Function ◽  
Spectral Approach ◽  
Fiber Reinforced ◽  
Elastic Solids ◽  
Preferred Direction ◽  
The Right

We use a spectral approach to model residually stressed elastic solids that can be applied to carbon fiber reinforced solids with a preferred direction; since the spectral formulation is more general than the classical-invariant formulation, it facilitates the search for an adequate constitutive equation for these solids. The constitutive equation is governed by spectral invariants, where each of them has a direct meaning, and are functions of the preferred direction, the residual stress tensor and the right stretch tensor. Invariants that have a transparent interpretation are useful in assisting the construction of a stringent experiment to seek a specific form of strain energy function. A separable nonlinear (finite strain) strain energy function containing single-variable functions is postulated and the associated infinitesimal strain energy function is straightforwardly obtained from its finite strain counterpart. We prove that only 11 invariants are independent. Some illustrative boundary value calculations are given. The proposed strain energy function can be simply transformed to admit the mechanical influence of compressed fibers to be partially or fully excluded.

scholarly journals A Bimodular Polyconvex Anisotropic Strain Energy Function for Articular Cartilage

2006 ◽  
Vol 129 (2) ◽  
pp. 250-258 ◽  
Author(s):  
Stephen M. Klisch
Keyword(s):  
Articular Cartilage ◽  
Strong Interaction ◽  
Energy Function ◽  
Strain Energy ◽  
Mechanical Response ◽  
Deformation Gradient ◽  
Strain Energy Function ◽  
Finite Deformations ◽  
Orthotropic Materials ◽  
Interaction Terms

A strain energy function for finite deformations is developed that has the capability to describe the nonlinear, anisotropic, and asymmetric mechanical response that is typical of articular cartilage. In particular, the bimodular feature is employed by including strain energy terms that are only mechanically active when the corresponding fiber directions are in tension. Furthermore, the strain energy function is a polyconvex function of the deformation gradient tensor so that it meets material stability criteria. A novel feature of the model is the use of bimodular and polyconvex “strong interaction terms” for the strain invariants of orthotropic materials. Several regression analyses are performed using a hypothetical experimental dataset that captures the anisotropic and asymmetric behavior of articular cartilage. The results suggest that the main advantage of a model employing the strong interaction terms is to provide the capability for modeling anisotropic and asymmetric Poisson’s ratios, as well as axial stress–axial strain responses, in tension and compression for finite deformations.

Application of Finite Elastic Theory to the Behavior of Rubberlike Materials

1963 ◽  
Vol 36 (5) ◽  
pp. 1459-1496 ◽  
Author(s):  
Paul J. Blatz
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Mechanical Response ◽  
Finite Elasticity ◽  
Strain Energy Function ◽  
Elastic Theory ◽  
Mechanical Parameters ◽  
Molecular Interpretation ◽  
Rubberlike Materials

Abstract A brief review of the theory of finite elasticity is presented. The theory is applied to the characterization of the mechanical response parameters of a polyurethan foam. The incorporation of compressibility and anisotropy effects into the strain energy function are discussed. An example of the behavior of a composite or filled foam is presented. Finally some of the problems associated with the molecular interpretation of mechanical parameters are discussed.

Sign in / Sign up

Export Citation Format

Share Document