Modified gradient elasticity and its finite element method for shear boundary layer analysis

• A modified gradient elasticity theory is proposed by defining internal length vector.
• The MGE can be simplified to the classical elasticity.
• The size effect of bi-material interface can be described by the MGE.
• The pathological mesh-dependence can be avoided by the MGE.

We attempt to present a modified gradient elasticity theory to analyze the shear boundary layer adjacent to the bi-material interface. By defining an internal length scale vector and assuming that the strain energy density depends on both the strain tensor and the strain gradient tensor, a modified gradient elasticity (MGE) theory is derived. When the internal characteristic length vector vanishes, the MGE can be simplified to classical elasticity theory. Based on the principle of virtual work, the corresponding variational principle, boundary condition and finite element formulation of the MGE are derived. The bi-material strip with shear boundary layers is numerically calculated by the MGE. It is found that the thickness of shear boundary layer strongly depends on the internal characteristic length, in addition, the mesh-dependence can be eliminated.