The effect of a curvature-dependent surface tension on the singularities at the tips of a straight interface crack

Abstract

SummaryA problem of an interface crack between two semi-planes made out of different materials under an action of an in-plane loading of general tensile-shear type is treated in a semi-analytical manner with the help of Dirichlet-to-Neumann mappings. The boundaries of the crack and the interface between semi-planes are subjected to a curvature-dependent surface tension. The resulting system of six singular integrodifferential equations is reduced to the system of three Fredholm equations. It is shown that the introduction of the curvature-dependent surface tension eliminates both classical integrable power singularity of the order 1/2 and an oscillating singularity present in a classical linear elasticity solutions. The numerical results are obtained by solving the original system of singular integro-differential equations by approximating unknown functions with Taylor polynomials.

A Rigid Stamp Indentation into a Semiplane with a Curvature-Dependent Surface Tension on the Boundary

“…Therefore, it has been shown that similarly to the conclusions of the papers [16], [17], [25], [26], introduction of the surface tension on the boundary of the semiplane leads to elimination of the power singularities of the stresses and the derivatives of the displacements. Only weaker logarithmic singularities may still be present.…”

“…The considered surface tension model has been applied previously to the examples of the interface and the noninterface brittle fracture in the studies [16], [17], [25], [26]. It has been shown here that, similarly to the fracture problems, introduction of the surface tension on the boundary of the semiplane leads to elimination of the integrable power singularities of the order 1/2 at the ends of the contact zone.…”

“…From the previous studies [16], [17], [25], [26], it is known that taking into account the curvature-dependent surface tension in the form (2.2) on the boundary of an interface or a noninterface crack leads to the elimination of the classical power singularities of the order 1/2 at the crack-tips and oscillating singularities of the pure imaginary order in the case of an interface crack. Weaker logarithmic singularities may still be present at the tips of the crack.…”

“…This method has been previously employed in [25], [26] and proven effective in these studies in comparison with the spline collocation method and with the results obtained by approximating the unknown functions by Fourier series.…”

Curvilinear mode-I/mode-II interface fracture with a curvature-dependent surface tension on the boundary

“…However, it has been shown later on the example of a mode-I/mode-II non-interface curvilinear plane fracture [52] and on the example of a straight mode-I/mode-II interface fracture [51] that this conclusion does not hold for non-symmetric configurations. The main conclusion of the papers [33], [51], [52] is that incorporation of the curvature-dependent surface tension allows to eliminate the classical singularity of the order 1/2 and also oscillating singularity in the case of a straight interface crack. However, some components of the stresses and strains may still retain a weaker logarithmic singularity.…”