Incompressible Isotropic Elasticity Research Articles

Plane Stress Problems for Isotropic Incompressible Hyperelastic Materials

The analysis of plane stress problems has long been a topic of interest in linear elasticity. The corresponding problem for non-linearly elastic materials is considered here within the context of homogeneous incompressible isotropic elasticity. It is shown that when the problem is posed in terms of the Cauchy stress, a semi-inverse approach must be employed to obtain the displacement of a typical particle. If however the general plane stress problem is formulated in terms of the Piola-Kirchhoff stress, the deformation of a particle requires the solution of a non-linear partial differential equation for both simple tension and simple shear, the trivial solution of which yields a homogeneous deformation. It is also shown that the general plane stress problem can be solved for the special case of the neo-Hookean material.

Large acoustoelastic effect for Lamb waves propagating in an incompressible elastic plate.

In this paper, the effect of a large pre-stress on the propagation of small amplitude Lamb waves in an incompressible elastic plate is investigated. Using the theory of incremental elasticity, the dispersion equations, which give the phase velocity of the symmetric and anti-symmetric wave modes as a function of the wavenumber, plate thickness, and pre-stress state, are derived for a general strain energy function. By considering the fourth-order strain energy function of incompressible isotropic elasticity, the correction to the phase velocity due to the pre-stress is obtained implicitly to the second order in the pre-strain/stress, and depends on the second, third, and fourth-order elastic constants. Numerical results are presented to show the dependence of the phase velocity of the Lamb wave modes upon the applied stress. These are compared to the first-order correction, and agree well with the limiting and asymptotic values obtained previously. It is envisaged that the present results may well find important practical applications in various guided wave based ultrasonic techniques utilising gels and rubber-like materials.

Anisotropic separable free energy functions for elastic and non-elastic solids

In incompressible isotropic elasticity, the Valanis and Landel strain energy function has certain attractive features from both the mathematical and physical view points. This separable form of strain energy has been widely and successfully used in predicting isotropic elastic deformations. We prove that the Valanis–Landel hypothesis is part of a general form of the isotropic strain energy function. The Valanis–Landel form is extended to take anisotropy into account and used to construct constitutive equations for anisotropic problems including stress-softening Mullins materials. The anisotropic separable forms are expressed in terms of spectral invariants that have clear physical meanings. The elegance and attractive features of the extended form are demonstrated, and its simplicity in analysing anisotropic and stress-softening materials is expressed. The extended anisotropic separable form is able to predict, and compares well with, numerous experimental data available in the literature for different types of materials, such as soft tissues, magneto-sensitive materials and (stress-softening) Mullins materials. The simplicity in handling some constitutive inequalities is demonstrated. The work here sets an alternative direction in formulating anisotropic solids in the sense that it does not explicitly use the standard classical invariants (or their variants) in the governing equations.

On the third- and fourth-order constants of incompressible isotropic elasticity

Consider the constitutive law for an isotropic elastic solid with the strain-energy function expanded up to the fourth order in the strain and the stress up to the third order in the strain. The stress-strain relation can then be inverted to give the strain in terms of the stress with a view to considering the incompressible limit. For this purpose, use of the logarithmic strain tensor is of particular value. It enables the limiting values of all nine fourth-order elastic constants in the incompressible limit to be evaluated precisely and rigorously. In particular, it is explained why the three constants of fourth-order incompressible elasticity μ, Ā, and D are of the same order of magnitude. Several examples of application of the results follow, including determination of the acoustoelastic coefficients in incompressible solids and the limiting values of the coefficients of nonlinearity for elastic wave propagation.

Heat Conduction and Controlled Deformations in Incompressible Isotropic Elasticity

For incompressible isotropic elastic heat conducting solids we examine the classes of universal deformations that are controllable under homothermal conditions and, in respect of a general law of heat conduction, determine the members of these classes that can support non-homothermal temperature fields. It is shown, in particular, that in some cases the values of the components of the (Cauchy) stress tensor T necessary for the satisfaction of the equilibrium equation ▿·T = 0 in the homothermal problem are retained in the corresponding non-homothermal problem to within an additive constant. These observations have implications for the design of experiments in which a material is subject to a temperature gradient since it is of value to know in advance that a specific deformation can be controlled in the presence of a suitable temperature gradient. The temperature may then be obtained as the solution of the steady state heat equation.

Cardiovascular Solid Mechanics: Cells, Tissues, and Organs

1. INTRODUCTION. 1.1 Historical Prelude, 1.2 Basic Cell Biology, 1.3 The Extracellular Matrix, 1.4 Soft Tissue Behavior, 1.5 Needs and General Approach, 1.6 Exercises, 1.7 References. 2. MATHEMATICAL PRELIMINARIES 2.1 A Direct Tensor Notation, 2.2 Cartesian Components, 2.3 Further Results in Tensor Calculus, 2.4 Orthogonal Curvilinear Components, 2.5 Matrix Methods, 2.6 Exercises, 2.7 References, 3. CONTINUUM MECHANICS 3.1 Kinematics, 3.2 Forces, Tractions and Stresses, 3.3 Balance Relations, 3.4 Constitutive Formulations, 3.5 Boundary and Initial Conditions, 3.6 Exercises, 3.7 References, 4. FINITE ELASTICITY 4.1 Introduction, 4.2 Incompressible Isotropic Elasticity, 4.3 Solutions in 3-D Incompressible Elasticity, 4.4 Compressible Isotropic Elasticity, 4.5 Membrane Hyperelasticity, 4.6 Exercises, 4.7 References 5. EXPERIMENTAL METHODS 5.1 General Philosophy, 5.2 Measurement of Strain, 5.3 Measurement of Applied Loads, 5.4 Testing Conditions, 5.5 Parameter Estimation and Statistics, 5.6 Exercises, 5.7 References 6. Finite Element Methods 6.1 Fundamental Equations, 6.2 Interpolation, Integration, and Solvers, 6.3 An Illustrative Formulation, 6.4 Inflation of a Membrane, 6.5 Inverse Finite Elements, 6.6 Exercises, 6.7 References PART II - VASCULAR MECHANICS 7. THE NORMAL ARTERIAL WALL 7.1 Structure and Function, 7.2 General Characteristics, 7.3 Constitutive Framework, 7.4 Experimental Methods, 7.5 Specific Constitutive Relations, 7.6 Stress Analyses, 7.7 Exercises, 7.8 References 8. VASCULAR DISORDERS 8.1 Hypertension, 8.2 Intracranial Aneurysms, 8.3 Atherosclerosis, 8.4 Aortic Aneurysms, 8.5 Additional Topics, 8.6 Exercises, 8.7 References 9. VASCULAR ADAPTATION 9.1 Mechanical Preliminaries, 9.2 Cellular Responses to Applied Loads, 9.3 Arterial Response to Hypertension, 9.4 Arterial Response to Altered Flow, 9.5 Vessel Response to Injury, 9.6 Veins as Arterial Grafts, 9.7 Aging, 9.8 Exercises, 9.9 References PART III CARDIAC MECHANICS 10. THE NORMAL HEART 10.1 Structure and Function, 10.2 General Characteristics, 10.3 Constitutive Framework, 10.4 Constitutive Relations, 10.5 Stress Analyses, 10.6 Exercises, 10.7 References 11. EPILOGUE APPENDICES I. Nomenclature, Abbreviations, and Conversions II. Results for Curvilinear Coordinates III. Material Frame Indifference 11. CARDIAC DISORDERS 11.1 Ischemia 11.2 Volume Overload 11.3 Hypertrophy 11.4 Cardiac Aneurysms 11.5 Additional Topics

A pseudo–elastic model for the Mullins effect in filled rubber

When a rubber test piece is loaded in simple tension from its virgin state, unloaded and then reloaded, the stress required on reloading is less than that on the initial loading for stretches up to the maximum stretch achieved on the initial loading. This stress softening phenomenon is referred to as the Mullins effect . In this paper a simple phenomenological model is proposed to account for the Mullins effect observed in filled rubber elastomers. The model is based on the theory of incompressible isotropic elasticity amended by the incorporation of a single continuous parameter, interpreted as a damage parameter. This parameter controls the material properties in the sense that it enables the material response to be governed by a strain–energy function on unloading and subsequent submaximal loading different from that on the primary (initial) loading path from the virgin state. For this reason the model is referred to as pseudo-elastic } and a primary loading-unloading cycle involves energy dissipation. The dissipation is measured by a damage function which depends only on the damage parameter and on the point of the primary loading path from which unloading begins. A specific form of this function with two adjustable material constants, coupled with standard forms of the (incompressible, isotropic) strain–energy function, is used to illustrate the qualitative features of the Mullins effect in both simple tension and pure shear. For simple tension the model is then specialized further in order to fit Mullins effect data. It is emphasized that the model developed here is applicable to multiaxial states of stress and strain, not just the specific uniaxial tests highlighted.