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The classical Eshelby's theory, developed based on local linear elasticity, cannot be applied to inclusion problems that involve nonlocal (long range) elastic effects often observed in micromechanical systems. In this study, we introduce the extension of Eshelby's inclusion theory to nonlocal elasticity. Starting from Eringen's integral formulation of nonlocal elasticity, an eigenstress-driven nonlocal Eshelby's inclusion theory is presented. The eigenstress-driven approach is shown to be a valid mathematical extension of the classical eigenstrain-driven approach in the context of nonlocal inclusion problems. Two individual numerical approaches are developed and applied to simulate inclusion problems and numerically extract the corresponding nonlocal Eshelby tensor. The numerical results obtained from both approaches confirm the validity of the derived nonlocal Eshelby tensor and its ability to capture the non-uniform eigenstress distribution within an elliptic inclusion. These results also help reveal the fundamental difference between the mechanical behaviour of the classical local and the nonlocal inclusion problems. The eigenstress-driven nonlocal inclusion theory could provide the necessary theoretical foundation for the development of homogenization methods of nonlocal heterogeneous media. [ABSTRACT FROM AUTHOR]

Flexural-gravity wave blocking occurs in the presence of lateral compressive force. The existing expansion formulae for the velocity potential were limited in the case of distinct real roots of the dispersion relation with the multiplicity one. This limitation is overcome, and we obtain the complete solution for the flexuralgravity wave-maker problem at blocking and saddle points by deriving the expansion formula for velocity potential using the Fourier transform corresponding to the roots of the dispersion relation with multiplicity two and three, respectively. Further, we check the convergence of the newly derived expansion formulae by using the spectral representation of the associated eigenfunction. It has been proved that the eigenfunctions related to the expansion formulae are linearly dependent and satisfy the orthogonal modecoupling relation. Next, we solve a model problem of flexural-gravity wave scattering by a crack using the derived expansion formulae and establish the energy balance relation by applying Green's integral theorem at blocking points for infinite water depth. In addition, we have presented the validation of energy balance relation involving the generalized scattering coefficients. [ABSTRACT FROM AUTHOR]