Spherical Lame-type problem in second strain gradient theory

Abstract

Second strain gradient theory (SSGT) is employed to examine the spherical single/double-phase Lame-type problem. Due to the capability of SSGT to capture the effects of surface, size, and discrete nature of materials, the pertinent relaxed configuration is sought. SSGT is written in the spherical coordinate system and the corresponding governing equilibrium equations, stress and strain components, constitutive relations, and boundary tractions are derived. With spherical symmetry consideration of the problem, the relaxed configuration is obtained for both the diamond crystalline carbon and carbon coated crystalline silicon shell. Afterwards, the external symmetric loading is applied on the nanostructure relaxed configuration to analyze the mechanical response. The SSGT elastic material parameters for carbon and silicon are calculated via the quantum computations and lattice dynamics combined with the material continuum description. Analysis shows that the nanostructures mechanical response in SSGT is significantly different from that in the classical elasticity. For example, in the single-phase problem with inner and outer radius equal to two and ten lattice parameter, respectively, under an external pressure of about 0.0001 , the classic elasticity predicts an approximately constant radial stress of about -0.0001 in the nanoshell. However, in the framework of SSGT, the radial stress is varying from about -0.001 and -0.0002 in the vicinity of the inner and outer boundaries, respectively, to about 0.0003 in the middle of the hollow nanoshell. With increasing the inner radius, the difference between the two results in the middle points decreases.