The stress–elastic strain relationship is studied for a composite under a plastic deformation. The constitutive law of each component is described by a deformation theory of strain gradient plasticity which introduces an internal length scale. The conventional deformation plastic theory is obtained when the internal length scale tends to 0. The Hashin–Shtrikman upper bound for a two-phase composite governed by a power law is derived. It is predicted, by differentiating the bounds, that in most cases, the stress and the elastic strain follow a non-linear relation immediately after the elastic range. However, for some particular values of the ratio of the internal length scale and the micro-scale of the composite, this relation is linear. The prediction is illustrated by various numerical examples.

We present a stability result for a class of non-trivial self-similarly vanishing solutions to a 3D Hele-Shaw moving boundary problem with surface tension and single-point suction. These solutions are domains that bifurcate from the trivial spherical solution. The moving domains have a geometric centre located at the suction point and they are axially symmetric. We show stability with respect to perturbations that preserve these properties.

Ventricular assist devices (VADs) provide long- and short-term support to chronically ill heart disease patients; these devices are expected to match the remarkable functionality of the natural heart, which makes their design a very challenging task. Blood pumps, the principal component of the VADs, must operate over a wide range of flow rates and pressure heads and minimise the damage to blood cells in the process. They should also be small to allow easy implantation in both children and adults. Mathematical methods and computational fluid dynamics (CFD) have recently emerged as powerful design tools in this context; a review of the recent advances in the field is presented here. This review focusses on the CFD-based design strategies applied to blood flow in blood pumps and other blood-handling devices. Both simulation methods for blood flow and blood damage models are reviewed. The literature is put into context with a discussion of the chronological development in the field. The review is illustrated with specific examples drawn from our group's Galerkin/least squares (GLS) finite-element simulations of the basic Newtonian flow problem for the continuous-flow centrifugal GYRO blood pump. The GLS formulation is outlined, and modifications to include models that better represent blood rheology are shown. Haemocompatibility analysis of the pump is reviewed in the context of haemolysis estimations based on different blood damage models. Our strain-based blood damage model that accounts for the viscoleasticity associated with the red blood cells is reviewed in detail. The viability of design improvement based on trial and error and complete simulation-based design optimisation schemes are also discussed.