Finite Inhomogeneous Shearing Deformations of a Transversely Isotropic Incompressible Material

The present paper deals with finite inhomogeneous shearing deformations of a slab of a special anisotropic solid. Two cases according to the directions of the anisotropic director of the medium are examined. In one case the solution reduces to a quadrature and gives an exact deformation field for particular values of the material constants. In the other case an exact solution is obtained. All such solutions reduce to the same existing solution for the corresponding isotropic elastic material. This revised version was published online in August 2006 with corrections to the Cover Date.     

Invariance of the Equilibrium Equations of Finite Elasticity for Compressible Materials

For homogeneous, isotropic, compressible nonlinearly elastic materials, a wide class of strain-energy density functions are obtained that leave the equations of equilibrium invariant under simple scaling transformations of the material and spatial coordinates. These strain-energy densities are homogeneous functions of the principal stretches. Several illustrative examples of particular strain-energies are provided. For axisymmetric problems, the invariance discussed here ensures that the equations of equilibrium can be solved by quadratures and thus often leads to analytic solutions in parametric or closed-form. Mathematics Subject Classifications (2000) 74B20, 74G55.     

End Effects in Anti-plane Shear for an Inhomogeneous Isotropic Linearly Elastic Semi-infinite Strip

The purpose of this research is to further investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This is carried out within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. In previous work [1], the elastic coefficients were assumed to be smooth functions of the transverse coordinate so that the material was inhomogeneous in the lateral direction only. Here we develop a new technique, based on a change of variable, to study generally inhomogeneous isotropic materials. The governing partial differential equation is transformed to a Helmholtz equation with a variable coefficient, which facilitates analysis of the influence of material inhomogeneity on the diffusion of end effects. For certain classes of inhomogeneous materials, an explicit optimal decay estimate is established. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Saint-Venant Decay Rates for an Isotropic Inhomogeneous Linearly Elastic Solid in Anti-Plane Shear

The purpose of this research is to investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This question is addressed within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. The elastic coefficients are assumed to be smooth functions of the transverse coordinate. The estimated rate of exponential decay with distance from the loaded end (a lower bound for the exact rate of decay) is characterized in terms of the smallest positive eigenvalue of a Sturm–Liouville problem with variable coefficients. Analytic lower bounds for this eigenvalue are used to obtain the desired estimated decay rates. Numerical techniques are also employed to assess the accuracy of the analytic results. A related eigenvalue optimization question is discussed and its implications for the issue of material tailoring is addressed. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite Elastic Deformations of a Tyre Modelled as an Ideal Fibre-Reinforced Shell

Vehicle tyres are anisotropic inhomogeneous fibre-reinforced shells which undergo finite elastic deformations. Calculation of their stress and deformation fields is a difficult task and is normally performed using the finite element technique. In this paper an attempt is made to provide an approximate analysis of the deformation field modelling the tyre as an ideal fibre-reinforced material. Radial-ply tyres are reinforced by a belt of fibres running around the wheel in the circumferential direction under the tread of the tyre. A second set of fibres lies in each radial cross-section, of the tyre and runs from the bead wire which seats against one wheel rim to the bead wire at the other wheel rim. We shall assume each radial cross-section of the tyre is in a state of plane strain and is formed from an arch of fibre-reinforced composite material which is reinforced in the hoop direction. This composite is assumed to be an ideal material which is inextensible in the fibre-direction and is incompressible. The plane-strain deformations of this section are examined and then used to analyse the deformation of the tyre as a whole.