Failure Mechanisms

· This initial slope of the stress-strain curve is related to Young's modulus by: (dσ/dx)₀ = [(dσ/dε)(dε/dx)]₀= E (dε/dx)₀

· For small x: ε= Δa/a₀ = x/a₀ , so that:
(dσ/dx)₀ = E/a₀
· Approximating the actual stress-displacement curve by a sinusoidal relationship: σ = σ_th sin(2πx/λ)
(dσ/dx) = (2π/λ) σ_th cos(2πx/λ)

From: Courtney, "Mechanical Behavior of Materials,"
McGraw Hill (1990)

· Taking the limit at x = 0: (dσ/ dx)_{x = 0} = (2πσ_th/λ) , so that E/a₀ = (2πσ_th/λ), and the theoretical fracture stress: σ_F = (λE/2πa₀).
· The Elastic fracture strain corresponds to x =λ/2, and taking λ= a₀ for a strain at fracture of ε_F = 50%, gives: σ_F ~ E/2π ~ E/10.