A curvature-dependent interfacial energy-based interface stress theory and its applications to nano-structured materials: (I) General theory

Experimental observations have shown the size-dependent residual surface stresses on spherical nanoparticles and their influence on the effective modulus of heterogeneous nanostructures. Based on these experimental findings, this paper proposes a new interface stress theory that considers the curvature effect on the interfacial energy. To investigate this curvature-dependent interfacial energy, we use the Green elasticity theory to describe the nonlinear constitutive relation of the interface at finite deformation, thus explicitly demonstrating the curvature-dependent nature of the interface stress and bending moment. By introducing a fictitious stress-free configuration, we then propose a new energy functional for heterogeneous hyperelastic solids with interfaces. For the first time, both the Lagrangian and Eulerian descriptions of the generalized Young-Laplace equation, which describes the intrinsic flexural resistance of the interface, are derived from the newly developed energy functional. This new interface stress theory is then used to investigate the residual elastic field in a heterogeneous hyperelastic solid containing interfaces. The present theory differs from the existing theories in that it takes fully into account both the curvature-dependence of the interfacial energy and the interfacial energy-induced residual elastic field in the bulk solid. Furthermore, the fundamental equations of the interface are given in Lagrangian description, which are preferable when considering the effects of residual interface stress, residual interface bending moment and interface elasticity. Finally, two examples are presented to shed light on the significance of this new interface stress theory. A more detailed analysis and applications of the new theory will be presented in Part (II) of this paper.