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

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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.

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Notes

  1. One should recall that the appearance of singularities at crack-tips also is predicted within the fully nonlinear theory of finite elasticity. Thus, use of the linearized equations of elasticity in the bulk phase is not necessarily the fundamental issue leading to stress/strain singularities. However, it merits comment that it has been shown recently [15] that stress and strain singularities at crack-tips can be removed within a nonlinear, infinitesimal strain, implicit theory of elasticity while still using the classical LEFM fracture boundary conditions, that is, without introduction of a cohesive-zone or crack-surface mechanics model.

  2. Another appealing strategy for solving fracture problems in two and three dimensional linear elasticity makes use of a weight function approach [13, 14] to formulating boundary singular integral relations through which classical fracture mixed boundary conditions are transformed into singular integral equations on fracture surfaces.

  3. It should be noted that this is not the upper/lower crack-face deformation.

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Acknowledgement

The author gratefully acknowledges support for this research from the Air Force Office of Scientific Research through the task number 12RX13COR and the Air Force Materials and Manufacturing Directorate under contract # FA8650-07-D-5800.

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  1. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Jay R. Walton

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  1. Jay R. Walton
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Walton, J.R. Plane-Strain Fracture with Curvature-Dependent Surface Tension: Mixed-Mode Loading. J Elast 114, 127–142 (2014). https://doi.org/10.1007/s10659-013-9430-9

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