On the nonlinear theory of two-phase shells

We discuss the nonlinear theory of shells made of material undergoing phase transitions (PT). The interest to such thin-walled structures is motivated by applications of thin films made of martensitic materials and needs of modeling biological membranes. Here we present the resultant, two-dimensional thermodynamics of non-linear theory of shells undergoing PT. The global and local formulations of the balances of momentum, moment of momentum, energy and entropy are given. Two temperature fields on the shell base surface are introduced: the referential mean temperature and its deviation, as well as two corresponding dual fields: the referential entropy and its deviation. Additional surface heat flux and the extra heat flux vector fields appear as a result of through-the-thickness integration procedure. Within the framework of the resultant shell thermodynamics we derive the continuity conditions along the curvilinear phase interface which separates two material phases. These conditions allow us to formulate the kinetic equation describing the quasistatic motion of the interface relative to the shell base surface. The kinetic equation is expressed by the jump of the Eshelby tensor across the phase interface. In the case of thermodynamic equilibrium the variational statement of the static problem of two-phase shell is presented. © Springer-Verlag Berlin Heidelberg 2011.