Arch Ration Mech Anal Research Articles

Cauchy’s stress theorem for stresses represented by measures

A version of Cauchy’s stress theorem is given in which the stress describing the system of forces in a continuous body is represented by a tensor valued measure with weak divergence a vector valued measure. The system of forces is formalized in the notion of an unbounded Cauchy flux generalizing the bounded Cauchy flux by Gurtin and Martins (Arch Ration Mech Anal 60:305–324, 1976). The main result of the paper says that unbounded Cauchy fluxes are in one-to-one correspondence with tensor valued measures with weak divergence a vector valued measure. Unavoidably, the force transmitted by a surface generally cannot be defined for all surfaces but only for almost every translation of the surface. Also conditions are given guaranteeing that the transmitted force is represented by a measure. These results are proved by using a new homotopy formula for tensor valued measure with weak divergence a vector valued measure.

Self-gravitating relativistic fluids: The formation of a free phase boundary in the phase transition from soft to hard

In the 1990s Christodoulou introduced an idealized fluid model intended to capture some of the features of the gravitational collapse of a massive star to form a neutron star or a black hole. This was the two-phase model introduced in ‘Self-gravitating relativistic fluids: a two phase model’ (Demeterios, Arch Ration Mech Anal 130:343–400, 1995). The present work deals with the formation of a free phase boundary in the phase transition from hard to soft in this model. In this case the phase boundary has corners at the null points; the points which separate the timelike and spacelike components of the interface between the two phases. We prove the existence and uniqueness of a free phase boundary. Also the local form of the shock near the null point is established.