Center Crack: Method

Center Crack in an Infinite Plate under General In-plane Load

Problem Statement

A through crack of length 2a, centered at the origin and inclined by β from the x axis, lies in an infinite plate carrying the uniform membrane state N_xx, N_yy, N_xy over the thickness t. Wanted are the stress intensity factors, the T-stress and energy release rate, the full elastic stress field at a point and the near-tip field.

Key Assumptions

Linear elastic, infinite plate, straight through crack with traction-free faces, uniform far field. The faces must open: a compressive σ_y′y′ gives a negative K_I and the elastic solution overlaps the faces, which is not physical (the tool flags it). Plasticity is confined to a small zone at the tips (small-scale yielding) for K to govern.

Solution

In the crack frame (x′ along the crack, y′ normal) the far field is σ_x′x′, σ_y′y′, τ_x′y′. Westergaard's function Z(z) = z/√(z² − a²), with the branch cut on the crack, gives

Mode I (S = σ_y′y′): σ_xx = S(Re Z − y Im Z′) + (σ_x′x′ − S),   σ_yy = S(Re Z + y Im Z′),   τ_xy = −Sy Re Z′

Mode II (T = τ_x′y′): σ_xx = T(2 Im Z + y Re Z′),   σ_yy = −Ty Re Z′,   τ_xy = T(Re Z − y Im Z′)

with Z′ = −a²/(z² − a²)^3/2. The uniform term (σ_x′x′ − S) is Sih's correction for a far field that is not equal-biaxial: the stress parallel to the crack passes through undisturbed and shows up only as the T-stress T = σ_x′x′ − σ_y′y′. The field is exact and closed-form. Near a tip it reduces to the Irwin K field, σ_ij = K_I f_ij(φ)/√(2πr) + K_II g_ij(φ)/√(2πr), with

K_I = σ_y′y′√(πa),    K_II = τ_x′y′√(πa),    G = (K_I² + K_II²)/E′,   E′ = E (plane stress) or E/(1 − ν²) (plane strain).

For a uni-axial stress σ and a crack inclined by β to it, K_I = σcos²β√(πa) and K_II = σsinβcosβ√(πa).

Checks built into the development: the crack faces carry no normal or shear stress; the far field is recovered; ahead of the tip σ_yy and τ_xy approach K/√(2πr) (ratio 1.0001 at r = 10⁻⁴a); on the faces of a uni-axially loaded crack σ_xx = −σ (the classical result); the inclined decomposition above is reproduced.

Plastic Zone

With a yield strength the tool gives Irwin's first estimate of the plastic-zone size ahead of the tip, r_p = (K_I/F_ty)²/(2π) in plane stress and /(6π) in plane strain; the elastic field, and K as a governing parameter, hold only while this is small against a and the ligament.

Reading the Plots

• The fringes are capped at three times the largest far-field component because the field is singular at the tips; the lobes ahead of the tips and the unloaded wake beside the faces are the features to look at.
• Figure 3 compares the full σ_y′y′ ahead of the tip with the one-term K field: they agree close to the tip and part beyond about r = 0.1a, where the far-field stress and the higher-order terms matter.

Limits

• Finite width: the secant and Feddersen corrections (Tada) multiply K; not applied.
• Crack closure, friction on the faces, kinking under mixed mode (the maximum tangential stress criterion gives the kink angle from K_I, K_II): not included.
• Fracture: compare K_I with K_Ic (plane strain) or K_c; fatigue growth uses ΔK.