- Classical Fracture Mechanics Approaches
Linear elastic fracture mechanics (LEFM) dates back to 1920-1921 when Grifªth proposed his energy approach for the brittle fracture of glass. Any material, including a very smooth homogeneous material such as glass, contains imperfections. These imperfections are the source of stress concentrations, which may lead to failure of the material well below its theoretical strength. Based on a sinusoidal approximation of the atomic bond potential,
σ = σ ⋅ pi
r x rsin ( )max 0
where σmax is the peak stress in the atomic bond stress-spacing diagram and r is the increase of the original lattice spacing r0 of the atoms, it is possible to calculate the theoretical strength of crystalline solids, which leads to (Kelly and MacMillan 1986):
σ = pi
The Young’s modulus E relates stress with strain following σ = E.ε = E.x/r0. For example, for alkali-resistant glass ªber, with a Young’s modulus E = 70 GPa (Gupta 2002), the predicted theoretical strength according to Equation (2.2) would be σmax = 23 GPa, whereas in reality about 70 MPa is measured on a single ªber. The strength of the ªber is very much affected by its diameter. Surface defects result in premature failure at stress levels quite below the maximum attainable value. In the glass rod of Figure 2.1a, the crack seems to have nucleated from the small white line at the bottom of the mirror area. The rod, which was a simple off-the-shelf product, was highly polluted on the outside as can be seen in Figure 2.1b. The crack nucleated from an imperfection, and appeared to have started symmetrically in the beginning. After the rather smooth mirrorzone, surface roughness gradually increased into the mist-and hackle-zones.