The modified version of strain gradient and couple stress theories in general curvilinear coordinates

• The modified strain gradient theory isderived in general curvilinear coordinates.
• The modified couple stress theory in curvilinear coordinates is readily obtained.
• The results are specialized for cylindrical and spherical coordinates.

It is well-known that conventional continuum theory is not capable of capturing size effects and therefore, many theories have been proposed. Among these theories, strain gradient and modified strain gradient theories have enjoyed great success so far. In the present study, the modified strain gradient theory is in details derived in general curvilinear coordinates. The modified strain gradient theory involves the modified couple stress theory as a special case and therefore, the modified couple stress theory in curvilinear coordinates is readily obtained. The results are then specialized for two practical orthogonal curvilinear coordinates, i.e. cylindrical and spherical coordinates. The expressions including deformation measures, governing equations, boundary conditions and constitutive equations are expressed in terms of physical components which are more common and convenient. The formulations presented here are general and may be appropriately applied to a broad range of problems in which the use of curvilinear coordinates is unavoidable, such as problems of cylindrical and spherical cavity expansion, the analysis of asymptotic crack tip field and the interpretation of micro/nano indentation tests and also bending or twisting tests on small scales.