General Form Of Strain-energy Function Research Articles

Anisotropic residual stresses in arteries.

The paper provides a deepened insight into the role of anisotropy in the analysis of residual stresses in arteries. Residual deformations are modelled following Holzapfel and Ogden (Holzapfel and Ogden 2010, J. R. Soc. Interface 7, 787-799. ( doi:10.1098/rsif.2009.0357 )), which is based on extensive experimental data on human abdominal aortas (Holzapfel et al. 2007, Ann. Biomed. Eng. 35, 530-545. ( doi:10.1007/s10439-006-9252-z )) and accounts for both circumferential and axial residual deformations of the individual layers of arteries-intima, media and adventitia. Each layer exhibits distinctive nonlinear and anisotropic mechanical behaviour originating from its unique microstructure; therefore, we use the most general form of strain-energy function (Holzapfel et al. 2015, J. R. Soc. Interface 12, 20150188. ( doi:10.1098/rsif.2015.0188 )) to derive residual stresses for each layer individually. Finally, the systematic experimental data (Niestrawska et al. 2016, J. R. Soc. Interface 13, 20160620. ( doi:10.1098/rsif.2016.0620 )) on both mechanical and structural properties of the different layers of the human abdominal aorta facilitate our discussion on (i) the importance of anisotropy in modelling residual stresses; (ii) the variability of residual stresses within the same class of tissue, the abdominal aorta; (iii) the limitations of conventional opening angle method to account for complex residual deformations; and (iv) the effect of residual stresses on the loaded configuration of the aorta mimicking in vivo conditions.

Azimuthal Shear of a Transversely Isotropic Elastic Solid

In this paper we study the problem of (plane strain) azimuthal shear of a circular cylindrical tube of incompressible transversely isotropic elastic material subject to finite deformation. The preferred direction associated with the transverse isotropy lies in the planes normal to the tube axis and is at an angle with the radial direction that depends only on the radius. For a general form of strain-energy function the considered deformation yields simple expressions for the azimuthal shear stress and the associated strong ellipticity condition in terms of the azimuthal shear strain. These apply for a sense of shear that is either “with” or “against” the preferred direction (anticlockwise and clockwise, respectively), so that material line elements locally in the preferred direction either extend or (at least initially) contract, respectively. For some specific strain-energy functions we then examine local loss of uniqueness of the shear stress—strain relationship and failure of ellipticity for the case of contraction and the dependence on the geometry of the preferred direction. In particular, for a reinforced neo-Hookean material, we obtain closed-form solutions that determine the domain of strong ellipticity in terms of the relationship between the shear strain and the angle (in general, a function of the radius) between the tangent to the preferred direction and the undeformed radial direction. It is shown, in particular, that as the magnitude of the applied shear stress increases then, after loss of ellipticity, there are two admissible values for the shear strain at certain radial locations. Absolutely stable deformations involve the lower magnitude value outside a certain radius and the higher magnitude value within this radius. The radius that separates the two values increases with increasing magnitude of the shear stress. The results are illustrated graphically for two specific forms of energy function.

Surface waves propagating in a half-space of neo-Hookean elastic material subject to pure homogeneous strain

The propagation of infinitesimal surface waves along a principal axis of strain in a half-space of incompressible isotropic elastic material subject to a homogeneous deformation was discussed in detail by Dowaikh and Ogden (IMA. J. Appl. Math. 44 (1990) 261–284). Here, we consider propagation in an arbitrary direction parallel to the surface of the half-space. The cumbersome algebra involved precludes the possibility of obtaining an explicit form of the secular equation with respect to a general form of strain-energy function, but for the neo-Hookean form of constitutive law and with an arbitrary normal stress on the half-space boundary an explicit secular equation is obtained.

The influence of shear strain and hydrostatic stress on stability and elastic waves in a layer

The theory of incremental motions superimposed on a large static deformation of an elastic solid is used to investigate the propagation of infinitesimal waves along a layer of material of uniform finite thickness. The layer is subject to an underlying simple shear deformation accompanied by an arbitrary uniform hydrostatic stress. In respect of a general form of incompressible, isotropic elastic strain-energy function, the dispersion equation for infinitesimal waves is obtained for two different sets of incremental boundary conditions on the faces of the layer. When the wave speed vanishes the dispersion equation becomes a bifurcation equation, which identifies configurations in which quasi-static incremental deformations can first appear on a path of simple shearing and hydrostatic stressing from the natural (undeforformed, unstressed) configuration of the layer. Explicit bifurcation criteria are obtained for a general form of strain-energy function and their consequences are illustrated by numerical results showing the dependence of bifurcation on certain deformation, stress and layer-thickness parameters. For a particular class of strain-energy functions, dispersion equations are obtained in explicit form for both incremental boundary-value problems, and the dependence of the wave speed on the same parameters is illustrated in detail.

Stability and vibration of pre-stressed compressible elastic plates

In a recent paper [1], Ogden and Roxburgh Int. J. Engng Sci. have given a detailed account of plane incremental vibrations of a rectangular plate of incompressible isotropic elastic material subject to an underlying homogeneous pure strain. In this paper corresponding results in respect of a compressible elastic material are derived. Specifically, for a general form of strain-energy function, equations governing the frequency of symmetric and antisymmetric modes of vibration are obtained. Depending on the form of strain-energy function, the underlying state of deformation and the aspect ratio of the plate, nine distinct cases arise together with several subcases. This is similar to the situation in the incompressible theory, although, for compressible materials, additional subcases appear. The emergence of quasi-static modes of deformation, corresponding to zero frequency, is accorded special attention since the frequency equations then reduce to bifurcation equations, which provide information about the boundary of stability of the underlying configuration in deformation space. Stability criteria are analysed in detail for a general form of strain-energy function. The dynamic and static results are illustrated by numerical calculations for a number of simple strain-energy functions.

The effect of pre-stress on the vibration and stability of elastic plates

Plane incremental vibrations superimposed on the pure homogeneous deformation of a rectangular block of incompressible isotropic elastic material are studied in detail. Frequency equations, which determine the frequencies of symmetric and antisymmetric modes of vibration in terms of the underlying deformation and stress and of the in-plane aspect ratio of the block, are obtained in respect of a general form of strain-energy function. Different cases, dependent on the properties of the strain-energy function and on the deformation and stress, are enumerated; results analogous to those found in the (compressible) linear theory are recovered for comparison. The case of zero frequency is of special interest since the frequency equations then become bifurcation equations. Each bifurcation equation determines a set of values of the deformation and stress at which, on a path from the undeformed stress-free configuration, stability of the underlying deformed configuration is lost and bifurcation into a (quasi-static) non-homogeneous incremental mode of deformation becomes possible. Results of Sawyers and Rivlin (1974) and Ogden (1984) emerge as special cases of those obtained here. For particular forms of strain-energy function calculations have been carried out to illustrate how the frequency depends on the deformation, stress and aspect ratio.

On the stability of asymmetric deformations of a symmetrically-tensioned elastic sheet

Bifurcation from a symmetrically to an unsymmetrically deformed homogeneous configuration of a plane incompressible isotropic elastic membrane subject to equal in-plane tensions is examined. Critical values of the tension at which bifurcation occurs are obtained for a general form of strain-energy function, and the nature of the deformation in the neighbourhood of the critical point is studied in detail. The stability of the global unsymmetric bifurcation branch is assessed. and secondary branching into either a homogeneous or a non-homogeneous mode of deformation is found to be possible under certain conditions.

Local and global bifurcation phenomena in plane-strain finite elasticity

Bifurcation, global non-uniqueness and stability of solutions to the plane-strain problem of an incompressible isotropic elastic material subject to in-plane dead-load tractions are considered. In particular, for loading in equibiaxial tension, bifurcation from a configuration in which the in-plane principal stretches are equal is shown to occur at a certain critical value of the tension (which depends on the form of strain-energy function). Results concerning the global invertibility of the elastic stress-deformation relations are obtained and then used to derive an equation governing the deformation paths branching from this critical value. The stability of each branch is also examined. The analysis is carried through for a general form of strain-energy function and the results are then illustrated for a particular class of strain-energy functions.

Penny-shaped cracks in a finitely deformed elastic solid

The problem of a penny-shaped crack in an infinite, finitely deformed, compressible elastic material is considered. Solutions for a general form of strain-energy function are given for both circular and elliptic cracks when we have an equi-biaxial prestress in the plane of the crack. We also indicate how solutions to more general problems can be generated from previous work on elliptic cracks in linear anisotropic materials.