Effect of surface elasticity in the analysis of nanocracksunder anti-plane load field

Abstract

The influence of surface elasticity on the stress-strain state of a crack type III, which occurs under antiplane shear deformations of a linearly elastic body, is investigated. Me-chanical effects that arise near surfaces, particularly at the crack faces, are taken into ac-count using the Gurtin and Murdoch continuum surface-boundary model. Equilibrium condi-tions on the crack surface are formulated, as well as the relationship between surface and body stresses. Using these relationships, refined boundary conditions are written on the upper and lower faces of the crack, which are further analyzed using the methods of the theory of complex variable functions. As a result of this analysis, a first-order singular integro-differential equation with a Cauchy-type kernel is formulated. For its solution, the representa-tion of unknown functions in terms of Chebyshev polynomials of the first kind and the method of collocation on the nodes of these polynomials are used. The solution of the resulting system of linear algebraic equations allows to obtain the coefficients of the specified expansions. A formula for calculating the stress on the crack extension is found, which is expressed by an integral with a Cauchy type kernel. A comprehensive analysis of the peculiarities of the nu-merical implementation of the developed algorithm is carried out. It includes variations in the number of components in the expansions of unknown functions in Chebyshev polynomials and the number of nodes in Gauss quadrature formulas for calculating the specified integral. The behavior of the stress difference between the upper and lower crack faces as well as the dis-tribution of another stress component on the crack extension is graphically illustrated in the vicinity of the right tip. The dependence of these quantities on the values of the uniform shear stress specified on the crack edges is also illustrated. It is shown that the consideration of sur-face elasticity becomes especially noticeable when the crack length is less than a micrometer. Further decrease of this length leads to significant change of the character of the stress dis-tribution in the vicinity of the crack tip. In particular, the square root singularity of the stresses at the crack tips, which is characteristic for the classical crack model, disappears and the stresses at these tips become finite.