ent material phases is an appropriate and convenient tool for modelling the phase interface in thin-walled shell structures. It is surprising, however, that we are not aware of any complete theoretical model of phase transitions in shells proposed in the literature. A Cosserat-type membrane with one director was proposed in [7, 8] to model single crystal thin films of martensitic materials. However, neither the rotation about the director was included nor dynamic continuity conditions at the curvilinear phase interface were discused in [7, 8]. The aim of this paper is to formulate the general nonlinear theory of shells with an account of occurence of phase transformation in the shell material. One might approach this problem by analysing first some test examples with various simplified shell models. Among such shell models let us mention as examples the linear, the geometrically nonlinear, or the fully nonlinear theories of shells based on the membrane model, the Kirchhoff–Love type model, or the Timoshenko–Reissner type model. However, we want here to forego all possible simplified 2D models of thin-walled shell structures and apply at once the most general approach base

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