Problems in nonlinear elastic and elastic-plastic solids

In this thesis we study variational problems in nonlinear elasticity and related problems for elastic-plastic solids. First, we consider a one-dimensional variational problem in nonlinear elasticity. We consider an elastic cylinder of length L subject to axially symmetric deformations which satisfy the boundary condition that the ends of the cylinder are fixed and with no restrictions on the lateral surface of the cylinder. We prove the existence of minimizers under the topological constraint that the ends of the cylinder are twisted through a finite number of turns. A key step is to show that necking does not occur in the deformed cylinder. Secondly, we consider the variational problem of minimising the elastic energy stored in a deforming nonlinear elastic body which occupies the region #OMEGA# in its reference configuration. For the pure displacement problem we let #OMEGA# be the reference configuration of an elastic body. In particular, it is known that minimisers may develop point discontinuities (corresponding to cavities forming in the deformed body). Using the topological condition (INV) and the distributional determinant we prove the existence of minimisers with potential discontinuity at X_0 is an element of #OMEGA#. For the mixed displacement-traction problem we fix X_0 is an element of #OMEGA# and define #OMEGA#_#epsilon# = #OMEGA#/B(x_0, #epsilon#) to be the reference configuration of an elastic body with a pre-existing hole of radius #epsilon# centred at X_0. We prove the corresponding sequence of minimisers of the mixed displacement-traction problem converges weakly to a minimiser of the pure displacement problem as the radius #epsilon# tends to zero. The third problem we consider is the expansion of a spherical cavity in a nonlinearly elastic-perfectly plastic solid under the action of an internal pressure P within the cavity. We calculate (i) the critical pressure required to produce unbounded expansion of the cavity in an infinite medium, and (ii) the critical pressure required to form a cavity from zero initial radius in a finite piece of material and show that both critical pressures are the same. We illustrate these results explicitly for a particular stored energy function. (author)Available from British Library Document Supply Centre-DSC:DXN050331 / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo