Variational bounds and overall shear modulus of nano-composites with interfacial damage in anti-plane couple stress elasticity

It is well known that classical continuum theory has certain deficiencies in capturing the size effects and predicting the nanoscopic behavior of materials in the vicinity of nano-inhomogeneities and nano-defects with reasonable accuracy. Couple stress theory which is associated with an internal length scale for the medium is one of the higher order continuum theories capable of overcoming such difficulties. In this work, the problem of a nano-size fiber embedded in an unbounded isotropic elastic body for three different types of interface conditions: perfect, imperfect (partially damaged), and pure sliding (completely damaged) subjected to remote anti-plane loading is examined in this framework. The physically realistic size-dependent elastic fields for the problem will be derived analytically. The discontinuities of the displacement and rotation fields at the imperfect interfaces are assumed to be proportional to the associated reduced traction and couple traction, respectively. The effect of the interfacial damage on the stress field around the nano-fiber is also examined. Subsequently, the elastic field of a single nano-fiber with a damaged interface condition is employed in conjunction with the Mori–Tanaka theory to estimate the size-dependent overall anti-plane shear modulus of such solids enriched with unidirectional circular cylindrical fibers severely damaged at their interfaces with the matrix. The dependence of the anti-plane elastic shear modulus on several important physical parameters such as size, interface conditions, rigidity of the fiber, and the characteristic length of the constituents is analyzed. Finally, a variational approach for the estimation of the upper and lower bounds of anti-plane shear modulus will be given within couple stress elasticity and, moreover, the dependence of the bounds on the matrix–fiber interface damage and the fiber to the matrix rigidity ratio is examined.