A mode-III crack with variable surface effects

Abstract

We study the contribution of variable surface effects to the antiplane deformation of a linearly elastic material with a mode-III crack. The surface elasticity is incorporated using a modified version of the continuum based surface/interface model of Gurtin and Murdoch. In our discussion, the surface moduli are not constant but vary along the crack surfaces. Using Green's function method, the problem is reduced to a single first-order Cauchy singular integro-differential equation, which is solved numerically using Chebyshev polynomials and a collocation method. Our results indicate that the gradient of the surface shear modulus exerts a significant influence on the crack opening displacement and on the singular stress field at the crack tips.

A finite crack with arbitrarily varied surface piezoelectricity

“…During the calculation, we set = N 100 for all the symmetric and monotonic variation functions to obtain satisfactory results. proposed by Wang and Schiavone [11]. In the case of linear variation of surface elasticity, the surface effect near the left crack tip is negligible whereas that near the right crack tip is not remarkably high for a value of k very close to 1, say = k 0.99.…”

“…In the case of linear variation of surface elasticity, the surface effect near the left crack tip is negligible whereas that near the right crack tip is not remarkably high for a value of k very close to 1, say = k 0.99. Thus for the linear variation function, the magnitude of Du 3 significantly enlarges for the section very close to the left crack tip when » k 1 [11]. However, for the rapid growth of exponential variation of surface piezoelectricity, a sufficiently high value of > k 0 ( ) in the exponential function = F x e ( ) also results in considerable surface piezoelectricity near the right crack tip in addition to the negligible surface piezoelectricty near the left crack tip.…”

“…Due to the large ratio of surface area to volume, surface effects often play a crucial and even predominant role in the physical properties of such nanostructures. Recently, in the investigation of non-piezoelectric nanostructures, the incorporation of surface mechanics with a goal to explore the feature of the deformation analysis at the nanoscale has received considerable attention [1][2][3][4][5][6][7][8][9][10][11]. The theory of surface elasticity based on continuum mechanics was first established by Gurtin, Murdoch and co-workers [12][13][14] and was later clarified and modified by Ru [15].…”

“…However, in the fracture analysis incorporating surface elasticity or surface piezoelectricity, the case of variable surface properties has been rarely touched. Most recently, the anti-plane deformation of an isotropic elastic solid containing a finite crack with linearly varied surface shear modulus was first investigated in [11].…”