Plane-Strain Fracture with Curvature-Dependent Surface Tension: Mixed-Mode Loading

Abstract

There have been a number of recent papers by various authors addressing static fracture in the setting of the linearized theory of elasticity in the bulk augmented by a model for surface mechanics on fracture surfaces with the goal of developing a fracture theory in which stresses and strains remain bounded at crack-tips without recourse to the introduction of a crack-tip cohesive-zone or process-zone. In this context, surface mechanics refers to viewing interfaces separating distinct material phases as dividing surfaces, in the sense of Gibbs, endowed with excess physical properties such as internal energy, entropy and stress. One model for the mechanics of fracture surfaces that has received much recent attention is based upon the Gurtin-Murdoch surface elasticity model. However, it has been shown recently that while this model removes the strong (square-root) crack-tip stress/strain singularity, it replaces it with a weak (logarithmic) one. A simpler model for surface stress assumes that the surface stress tensor is Eulerian, consisting only of surface tension. If surface tension is assumed to be a material constant and the classical fracture boundary condition is replaced by the jump momentum balance relations on crack surfaces, it has been shown that the classical strong (square-root) crack-tip stress/strain singularity is removed and replaced by a weak, logarithmic singularity. If, in addition, surface tension is assumed to have a (linearized) dependence upon the crack-surface mean-curvature, it has been shown for pure mode I (opening mode), the logarithmic stress/strain singularity is removed leaving bounded crack-tip stresses and strains. However, it has been shown that curvature-dependent surface tension is insufficient for removing the logarithmic singularity for mixed mode (mode I, mode II) cracks. The purpose of this note is to demonstrate that a simple modification of the curvature-dependent surface tension model leads to bounded crack-tip stresses and strains under mixed mode I and mode II loading.

Thermodynamically consistent and scale-dependent phase field approach for crack propagation allowing for surface stresses

“…Using data for the ionic crystals in Chuang (1987), it was shown that the stressdependent part of the surface tension for SiO 2 glass is 3 times larger than the surface energy. Walton (2014) predicted that surface stresses would produce finite crack curvature and, consequently, finite crack-tip stresses and strains for plane strain deformations.…”

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

“…It is well known that the stresses σ 22 and σ 12 , and the derivatives of the displacements u 1 , u 2 in the semiplane S, can be expressed through two complex functions Φ(z), Ψ(z) (complex potentials) analytic in S using the following formulas [10]:…”

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php modified boundary conditions: (2.3) σ 12 (t, 0) = γ 0 u 2,111 (t, 0), t ∈ L 0 , u 2,1 (t, 0) = g 2 (t), t ∈ L 0 , in the contact zone, and (2.4) σ 12 (t, 0) = γ 1 u 2,111 (t, 0), t ∈ L 1 , σ 22 (t, 0) = −γ 2 u 2,11 (t, 0), t ∈ L 1 , outside of the contact zone, where σ 22 , σ 12 are the normal and the shear stresses acting on the boundary of the semiplane, the subindex 1 after the comma means differentiation by x, and g 2 (t) is the derivative of the function which describes the vertical profile of the rigid stamp. The coefficients γ 0 and γ 1 are allowed to have, in general, different values.…”

Frictionless contact of a rigid stamp with a semi-plane in the presence of surface elasticity in the Steigmann–Ogden form

“…Surface elasticity theories of this type have been applied before to the modeling of nanostructures [2, 23–29] and to the modeling of fracture [30–41]. It has been shown that incorporation of the surface energy introduces a length scale into the problems.…”