Towards the solution of finite plane-strain problems for compressible elastic solids

A new approach to the solution of finite plane-strain problems for compressible Isotropie elastic solids is considered. The general problem is formulated in terms of a pair of deformation invariants different from those normally used, enabling the components of (nominal) stress to be expressed in terms of four functions, two of which are rotations associated with the deformation. Moreover, the inverse constitutive law can be written in a simple form involving the same two rotations, and this allows the problem to be formulated in a dual fashion.For particular choices of strain-energy function of the elastic material solutions are found in which the governing differential equations partially decouple, and the theory is then illustrated by simple examples. It is also shown how this part of the analysis is related to the work of F. John on harmonic materials.Detailed consideration is given to the problem of a circular cylindrical annulus whose inner surface is fixed and whose outer surface is subjected to a circular shear stress. We note, in particular, that material circles concentric with the annulus and near its surface decrease in radius whatever the form of constitutive law within the given class. Whether the volume of the material constituting the annulus increases or decreases depends on the form of law and the magnitude of the applied shear stress.     

Bifurcation of inflated circular cylinders of elastic material under axial loading—I. Membrane theory for thin-walled tubes

Bifurcations of circular cylindrical elastic tubes subjected to inflation combined with axial loading are analysed. Membrane tubes are considered in detail as a background to the more difficult analysis of thickwalled tubes described in the companion paper (Part II). Our results for membranes reinforce and extend those given by R.T. Shield and his co-workers.Two modes of bifurcation are investigated: firstly, a bulging (axisyrmmetric) mode; secondly, a prismatic mode in which the cross-section of the tube becomes non-circular. Necessary and sufficient conditions for the existence of modes of either type are given in respect of an arbitrary (incompressible isotropic) form of elastic strain-energy function. For a closed tube with a fixed axial loading many features of the results have close parallels with recent findings by D.M. Haughton and R.W. Ogden for spherical membranes. On the other hand, some results for tubes with fixed ends have no such parallel. In particular, bifurcation may, under certain conditions, occur before the inflating pressure reaches a maximum. A combination of the two modes is interpreted in terms of bending for a tube under axial compression, and the relative importance of the bending and bulging modes is discussed in relation to the length to radius ratio of the tube. The analytical results are illustrated for specific forms of strain-energy function. Corresponding analysis is given for thick-walled tubes in Part II.     

A note on duality in finite elasticity

The duality between stress and deformation fields for plane deformations of a compressible isotropic hyperelastic material established by J. M. Hill [1]is generalized to deformations of a homogeneous elastic material without the restrictions of isotropy and hyperelasticity. At the same time a clarification of Hill's results is achieved.     

Local and overall extremal properties of time-independent materials and non-linear elastic comparison materials

For time-independent materials which undergo non-linear deformations from some given reference configuration two (dual) hypotheses are considered. Firstly it is supposed that the work done to a given state of deformation is bounded below and that the bound is attainable on a physically possible path; secondly that the complementary work to a given state of stress is bounded above and that this bound too is attainable on a physically possible path. The consequences of these assumptions are analysed, and the results of Ponter and Martin [1] in the linear theory are generalized to account for non-linear deformations, due attention being paid to questions of stability.A non-linear elastic comparison material is defined whose strain energy is equal to the work done on a minimum path for the time-independent material. Extremum principles for non-linear elastic materials given in [2] are then applied to the comparison elastic material, and bounds are thereby placed on the work and complementary-work functional of the time-independent material. Corresponding overall properties of the time-independent and elastic materials are then compared by defining respective overall constitutive laws and overall stress and deformation variables.Following the definition of strengthening (weakening) of a non-linear elastic solid given by Ogden[2] a time-independent material is said to be strengthened (weakened) when its comparison elastic material is strengthened (weakened). Local and overall aspects of this definition are examined.