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We have shown in a series of recent papers that the classical theory of thermo-elasticity can be extended to the nano-scale by supplementing the equations of thermo-elasticity for the bulk material with the generalised Young-Laplace equations of surface elasticity. This talk will describe how this has been done in order to capture the often unusual thermo-mechanical and physical properties of nano-structured particulate and porous materials.It will begin with a description of the generalised Young-Laplace equations for surface elasticity. We will then describe how the classical Eshelby formalism can be generalised to nanoinhomogeneities; unlike its classical counterpart the Eshelby tensor now depends on the size of the nano-inhomogeneity and the location of the material point in it. We will demonstrate its application on the calculation of the stress concentration factor of a spherical nano-void. We will next derive the Eshelby tensor for nano-particles consisting of a core surrounded by multiple outer shells. These multi-shell particles are used as novel functional materials as well as stiffeners/toughners in conventional composites and nano-composites. In these nano-heterogeneous particles, the mismatch of thermal expansion coefficients and lattice constants between neighbouring shells induces stress/strain fields in the core and shells, which in turn affect the physical/mechanical properties of the particles themselves and/or of the composites containing them. We will apply this solution to obtain the strain fields in quantum dots (QDs) with multi-shell structures and in alloyed QDs induced by the mismatch in the lattice constants of the atomic species.The next part of the talk will address the generalisation of the micro-mechanical framework for determining the effective elastic properties and effective coefficients of thermal expansion of heterogeneous solids containing nano-inhomogeneities. We will use this generalised framework to calculate the effective elastic constants of nano-porous/cellular materials. It will be shown, in particular that these can be made to exceed those of the parent materials provided the pore surface elastic parameters satisfy certain conditions. These stiff nano-porous materials herald a radical breakthrough in sandwich-type construction. We will also use the generalised framework to study the thermo-elastic properties of heterogeneous materials containing spherical particles or cylindrical fibres. The interface between the matrix and second phase inhomogeneity is imperfect with either the displacement or the stress experiencing a jump across it. We will relate the effective coefficient of thermal expansion (CTE) to the effective elastic moduli and thereby generalise Levin's formula, and reveal two connections among the effective elastic moduli, thereby generalising Hill's connections. In contrast to the classical results, the effective CTE in the presence of an imperfect interface will be shown to be strongly dependent on the size of the inhomogeneity, besides the interf...

SUMMARYThe extended finite element method (XFEM) is improved to directly evaluate mixed mode stress intensity factors (SIFs) without extra post-processing, for homogeneous materials as well as for bimaterials. This is achieved by enriching the finite element (FE) approximation of the nodes surrounding the crack tip with not only the first term but also the higher order terms of the crack tip asymptotic field using a partition of unity method (PUM). The crack faces behind the tip(s) are modelled independently of the mesh by displacement jump functions. The additional coefficients corresponding to the enrichments at the nodes of the elements surrounding the crack tip are forced to be equal by a penalty function method, thus ensuring that the displacement approximations reduce to the actual asymptotic fields adjacent to the crack tip. The numerical results so obtained are in excellent agreement with analytical and numerical results available in the literature.

The Eshelby formalism for inclusion/inhomogeneity problems is extended to the nano-scale at which surface/interface effects become important. The interior and exterior Eshelby tensors for a spherical inhomogeneous inclusion with the interface stress effect subjected to an arbitrary uniform eigenstrain embedded in an infinite alien matrix, and the stress concentration tensors for a spherical inhomogeneity subjected to an arbitrary remote uniform stress field are obtained. Unlike their counterparts at the macro-scale, the Eshelby and stress concentration tensors are, in general, not uniform inside the inhomogeneity but are position-dependent. They have the property of radial transverse isotropy. It is also shown that the size-dependence of the Eshelby tensors and the stress concentration tensors follow very simple scaling laws. Finally, the Eshelby formula to calculate the strain energy in the presence of the interface effect is given.

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