A New Approach to the Modeling and Analysis of Fracture through Extension of Continuum Mechanics to the Nanoscale

Abstract

Approximation of u 2,1 (x, 0) by a finite sum of Chebyshev polynomials (400 terms) for γ = 0.05 and far-field loading σ = 0.01, 0.02, 0.04. 47 3.3 Approximation of u 2,1 (x, 0) by a finite sum of Chebyshev polynomials (400 terms) for far-field loading σ 0 = 0.02 andγ = 0.

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

“…We shall simply refer to notation employed in [16]. Specifically, our starting point for analyzing the fracture problems of interest here will be the Dirichlet-toNeumann and Neumann-to-Dirichlet maps given in equations (14) and (15) of [16]. The forms given here differ slightly from their counterparts in [16] in that here stresses have been non-dimensionalized by Young's modulus, so that the only material parameter appearing is Poisson's ratio, ν.…”

“…Since various methods of solution for this system are very well known, most of the details will be omitted here. We shall simply refer to notation employed in [16]. Specifically, our starting point for analyzing the fracture problems of interest here will be the Dirichlet-toNeumann and Neumann-to-Dirichlet maps given in equations (14) and (15) of [16].…”

“…In the setting of a pure mode I (opening mode), plane strain, finite-length, straight crack, it was shown in [16], that when surface tension is assumed to be a constant, the theory removes the strong (square-root) crack-tip stress/strain singularity, but leaves a weak, logarithmic one, as happens with use of the Gurtin-Murdoch surface elasticity model. Also, the crack-tip opening angle is predicted to be strictly positive but less than π/2 (the value associated with the square-root singularity of classical LEFM).…”

“…Also, the crack-tip opening angle is predicted to be strictly positive but less than π/2 (the value associated with the square-root singularity of classical LEFM). It was subsequently shown in [16], that if surface tension is assumed to have a linear dependence upon a linearized approximation to the crack-surface mean-curvature, the logarithmic singularity is also removed and the crack-opening displacement takes a cusp shape at the crack-tips, as happens with use of a cohesive-zone model. To reiterate, with the curvature-dependent constitutive model for surface tension imployed in [16], the theory predicts the opening displacement to have zero-slope at the crack-tip and the crack-surface mean curvature to be infinitesimal along the entire crack-surface, in contrast to classical LEFM, in which the crack opening displacement is predicted to have an infinite slope at the crack-tip.…”

“…A simpler approach to modeling the surface stress tensor was adopted in [16], wherein it was assumed to be Eulerian, that is, a scalar function (surface tension) multiplied by the projection tensor into the surface tangent plane. In the setting of a pure mode I (opening mode), plane strain, finite-length, straight crack, it was shown in [16], that when surface tension is assumed to be a constant, the theory removes the strong (square-root) crack-tip stress/strain singularity, but leaves a weak, logarithmic one, as happens with use of the Gurtin-Murdoch surface elasticity model.…”