Abstract
A criterion to predict crack onset at a sharp V-notch tip in homogeneous brittle materials under a mixed-mode loading was presented and validated by experimental observations in a previous paper by the authors. This criterion slightly underestimates the experimental loads causing failure which is attributed to a small notch tip radius that blunts the sharp corner. This discrepancy is rigorously analyzed mathematically in this paper by means of matched asymptotics involving 2 small parameters: a micro-crack increment length and the notch tip radius. A correction is brought to the initial prediction and a better agreement is obtained with experiments on PMMA notched specimens.
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Abbreviations
- A 1, A 2 :
-
Generalized stress intensity factors (GSIFs), mode I and mode II, associated with the sharp V-notch
- A 1c , A blunt1c :
-
Critical mode I GSIF for sharp/blunt V-notch
- α 1, α 2 :
-
Sharp V-notch singularity exponents
- λ, G :
-
The two Lamé constants
- E, ν :
-
Young modulus and Poisson ratio
- G c , σ c , K Ic :
-
Fracture toughness, stress at brittle failure and plane-strain critical SIF
- ℓ, ℓ 0 :
-
Crack length and critical crack length at the V-notch tip
- r, ρ :
-
The radius measured from the V-notch tip in the outer and inner domains
- θ :
-
Angle measured from the horizontal axis in the inner and outer domains
- θ* :
-
θ − π/2
- δΠ:
-
Change in potential energy due to the presence of a crack at the V-notch tip
- a, μ:
-
V-notch tip radius and normalized crack length μ = ℓ/a
- ω :
-
Solid opening angle of the V-notch
- \({\Omega_\infty^0, \Omega_\infty^\mu}\) :
-
Inner domain without/with a crack
- u, σ :
-
Radial and tangential displacement vector and stress tensor
- u a(0), u a(μ):
-
Displacements vector for the blunt V-notch without/with the presence of a crack
- v 1, v 2 :
-
Mode I and II expansion terms of the displacements vector in the inner domain
- \({\hat{\boldsymbol{v}}^{(1)}, \hat{\boldsymbol{v}}^{(2)}}\) :
-
Mode I and II expansion terms of the displacements vector in the inner domain with finite strain energy
- u(0, 0):
-
V-notch tip displacement vector
- \({\mathcal L}\) :
-
The 2-D plane strain elasticity Navier operator
- \({{\mathcal T}}\) :
-
The traction operator
- g(ρ) :
-
Cutoff function
- Ψ :
-
Path integral defined in (29)
- Γ∞ :
-
The circular boundary of the inner domain at \({\rho \rightarrow \infty}\)
- H ij (μ, θ), H ij (0):
-
Functions used to compute potential energy in cracked/uncracked domain with rounded V-notch tip
- ΔH ij :
-
H ij (μ, θ) − H ij (0)
- m :
-
Mode mixity: \({m=\frac{A_2}{A_1}a^{\alpha_2-\alpha_1}}\)
- σ n :
-
Normal stress
- \({\bar{\bullet}}\) :
-
Quantities associated with the sharp V-notch
- h mixed, H mixed :
-
Correction factors for the computation of A Ic
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Priel, E., Yosibash, Z. & Leguillon, D. Failure initiation at a blunt V-notch tip under mixed mode loading. Int J Fract 149, 143–173 (2008). https://doi.org/10.1007/s10704-008-9234-2
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DOI: https://doi.org/10.1007/s10704-008-9234-2