On separation of a layer from the half-plane: Elastic fixation conditions for a plate equivalent to the layer

Fracture Mechanics Solutions for Interfacial Cracks between Compressible Thin Layers and Substrates

“…Analytical expressions for the energy release rate and mode-mixity phase angle associated with the elementary loads in Figure 1b, Equations (9) and (10), have been derived from the results in Reference [28] for materials with β = 0, and depend on…”

“…where, f M = √ 6 and f P = 1/ √ 2 [19,25]. The phase angle ψ P = ω has been derived semi-analytically in Reference [25] for −0.4 ≤ β ≤ 0.4 and −0.8 ≤ α ≤ 0.8 and analytically in [28] for −1 ≤ α ≤ 1 and β = 0; f V and ψ V can be obtained from the analytical results in [28] for β = 0 (see below), and for −0.8 ≤ α ≤ 0.8 and β = 0, they have been defined numerically in [26]. Results for orthotropic materials with α = β = 0 have been presented in References [43,44] for axial forces and bending moments and for shear in [27,45].…”

“…An analytical solution of the problem in Figure 1 for β = 0 has been derived in [28] using the Laplace transform and reducing the problem to a homogeneous matrix Riemann-Hilbert problem following [46,47]; see Appendix A for more details. The solution has been obtained under the assumption that the loads are applied at an infinite distance from the crack tip.…”

“…The analytical solution [28] has been obtained by relating the Laplace transforms of the discontinuous (normal and tangential) displacement derivatives in the strip and in the half plane along the delaminated part with the (normal and shear) stresses acting along the intact part of the interface by means of a 2 × 2 matrix, and reducing therefore the problem under consideration to a matrix Riemann-Hilbert problem. For the particular restriction β = 0, the matrix equation have been solved using the technique in [30] which yields the expressions for the stress intensity factors in Equations (7) and (8).…”

“…The thin layer is subjected to general edge loadings and mode mixity phase angle and energy release rate associated with the collinear propagation of the crackare obtained in terms of crack tip force and moment resultants, Figure 1b. Isotropic bi-material interfaces with a general mismatch of the elastic constants are examined for a wide range of the Dundurs parameters α and β , which measure the mismatch in uniaxial and bulk compliances of the materials (see expands the works in [25][26][27][28] to provide analytical solutions for axial, bending, and shear loading with 0 β = , an efficient and accurate numerical method for the derivation of the fracture mechanics parameters in the numerically challenging film/substrate problem, and numerical coefficients for the application of the semi-analytic solutions in [25,26] to bi-materials with a general mismatch of the elastic constants and For its technological importance, the problem in Figure 1 has been studied and used extensively in the literature since the early work by Obreimoff on the splitting strength of mica [1][2][3][29][30][31][32]. Most studies and applications are limited to conventional bi-material systems with 0.8 0.8 α − ≤ ≤ and neglect the effects of the mismatch of the volumetric stiffness of the layers; this is done by assuming the second Dundurs' parameter 0 β = "because of its secondary role".…”