On the Maxwell–Eshelby relation in the theory of 6-parameter elastic shells

In this work, we first derive jump conditions in addition to the balance laws and boundary conditions for 6-parameter elastic shells with a discontinuity curve. Furthermore, we obtain Hadamard compatibility conditions that must be satisfied by strain measures in the presence of a discontinuity curve. By using energy-minimizing conditions (rank-one convexity conditions), we derive Maxwell-Eshelby relations that must be satisfied. • We present the balance laws and boundary conditions for 6-parameter elastic shells with a discontinuity curve. • We derive jump conditions for Cosserat shells with a discontinuity curve. • We obtain Hadamard compatibility conditions for shell strain measures in the presence of discontinuity curves. • We use rank-one convexity conditions to derive Maxwell-Eshelby relations for shells.