Elliptic Hole General Load: Method

Elliptic Hole in an Infinite Plate under General In-plane Load

Problem Statement

A large, thin, flat plate with an elliptic hole (semi-axes a and b, a inclined by β from the x axis) carries a uniform in-plane membrane state far from the hole, given as the stress resultants N_xx, N_yy, N_xy and the thickness t: σ_xx^∞ = N_xx/t, σ_yy^∞ = N_yy/t, τ_xy^∞ = N_xy/t. The plate is linear elastic, isotropic, in plane stress and infinite. The question is the stress at a point P(r, θ), θ from the y axis towards x, and the hoop-stress extremes on the hole edge.

Figure A: Convention (β = 0 shown). N_xx and N_yy positive in tension; N_xy positive when it acts in +x on the top edge; β rotates a counterclockwise from x; θ is measured from the y axis towards x.

Key Assumptions

Infinite plate, linear elastic, plane stress, uniform far field, open traction-free hole. Only membrane resultants are handled; bending moments are a different problem. The material stays elastic at the peak. The solution does not depend on the elastic constants.

Solution by Superposition

The far-field state is resolved into its principal values S₁ ≥ S₂ and the direction φ of S₁. The Inglis solution is linear, so the field is the sum of the uni-axial fields for S₁ at φ and S₂ at φ + 90°. Each is evaluated with the Kolosov–Muskhelishvili potentials of the uni-axial elliptic tool on the map z = R(ζ + m/ζ), R = (a + b)/2, m = (a − b)/(a + b), with the load angle in the hole's frame α = φ − β:

φ(ζ) = (SR/4) [ζ + A/ζ],   A = 2e^2iα − m;    ψ(ζ) = −(SR/4) [1/ζ + Āζ] − (SR/4) (1 + mζ²)(ζ² − A) / [ζ(ζ² − m)]

σ_xx + σ_yy = 4 Re Φ,    σ_yy − σ_xx + 2iτ_xy = 2 [z̄ Φ′ + ψ′] / ω′,    Φ = φ′/ω′.

The two fields are summed in the hole's frame and the tensor is rotated to the plate frame. The result is exact: the only arithmetic is complex algebra. With a = b it reproduces the general Kirsch field of the round-hole tool.

Hole Edge and Stress Concentration Factor

On the free edge only the hoop stress remains, equal to σ_xx + σ_yy = 4 Re Φ there. For an aligned ellipse under a single tension S along y it is (1 + 2a/b)S at the ends of a and −S at the ends of b; under equal biaxial tension 2(a/b)S at the ends of a and 2(b/a)S at the ends of b; for a round hole 3S₁ − S₂ and 3S₂ − S₁. In general (any a/b, β, S₂/S₁) the extremes leave the axis ends, so the tool evaluates the hoop stress around the whole edge (3600 points, refined by a golden-section search) and reports the largest and most compressive values with the polar angle of each.

Reference for K_t. There is no single applied stress, so K_t = σ_edge,max/S₁, referred to the maximum far-field principal stress (uni-axial aligned: 1 + 2a/b; round hole in pure shear: 4). When the far field has no tensile principal stress the ratios are formed with S₂.

Principal Stresses at the Probed Point

From σ_xx, σ_yy, τ_xy at P the tool gives the polar components about the hole center, σ_max, σ_min, τ_max, and the direction of σ_max from the y axis (φ_x) and from the radial direction (φ_p), each within ±90°.

Reading the Plots

• σ_ηη fringe. The tangential stress along the confocal ellipses, the hoop stress on the edge; the color scale spans the two edge extremes in units of the reference stress.
• σ_max fringe. The maximum principal stress; uniform S₁ far from the hole.
• Line plots. The hoop stress around the edge versus the polar angle of the edge point, and the plate-frame stresses along the hole's own a and b axes.

Limits

• Finite width, yielding and very slender holes (a/b above about 20): as for the uni-axial tool.
• Bending resultants are not handled.