Abstract
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 through-the-thickness procedure applied to a 3D shell-like body.
Highlights
Eshelby tensor known 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]
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]
We show that the 2D Eshelby tensor C can be obtained from its 3D counterpart b using the through-the-thickness integration procedure
Summary
Eshelby tensor known 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]. 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]. Let us consider quasistatic deformations, that is, when the position vectors depend on time-like parameter t r = r(R, t), X = X(t), but we neglect inertial effects In this case, it can be shown, see, e.g. Linear form of (10) was used for the analysis of instabilities of two-phase solids in [35,64]
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