We consider both three-dimensional (3D) and two-dimensional (2D) Eshelby tensors known also as energy-momentum tensors or chemical potential tensors, which are introduced within the nonlinear elasticity and the resultant nonlinear shell theory, respectively. We demonstrate that 2D Eshelby tensor is introduced earlier directly using 2D constitutive equations of nonlinear shells and can be derived also using the throughthe-thickness procedure applied to a 3D shell-like body.Keywords Eshelby tensor · Nonlinear elasticity · Nonlinear shell · Phase transformations · Through-thethickness integration
IntroductionEshelby tensor known also as the Eshelby stress tensor or the energy-momentum tensor or the tensor of chemical potential was introduced originally in pioneering works by Eshelby [27][28][29]. Since it describes the energy release rate for a moving singularity, the Eshelby tensor found various applications in the theory of fracture [39,41,47,48], in the theory of stress-induced phase transitions [2,9,11,38,49], and in the modelling of the stress-assisted chemical reaction fronts propagation [32,34,58]. In particular, the properties of the Eshelby tensor result in several path-independent line integrals, see, e.g. [41,47,48]. For modelling of stress-induced phase transformations, the jump of discontinuity of the Eshelby tensor across the phase interface is used for the formulation of the thermodynamic compatibility condition which is necessary for determination of the interface position, see [2,9,33,38,64]. Motivating by the behaviour of martensitic films [12,13,30,44,50] and biological membranes [3,15,53], Eremeyev and Pietraszkiewicz [24] proposed the 2D model of thin-walled structures undergoing phase transitions within the nonlinear shell theory presented in [16,45,46,55]. In [24], the thermodynamic compatibility condition on the phase interface was derived using the stationarity of the total energy functional expressed through 2D constitutive equations. This description was further extended for more complex cases such as quasistatic deformations [26], viscoelastic properties [25], and the influence of a line tension [56]. The latter case constitutes an example of singular curves in shells discussed in [57] in detail. The analysis performed in [24,26] demonstrated that the developed 2D model can capture essential peculiarities of deformations of thin films made of shape memory alloys observed experimentally, see, e.g. [12,13,30,44,50]. Communicated by Holm Altenbach. V. A. Eremeyev · V. Konopińska-Zmysłowska (B)