Circular Inclusion: Method

Circular Inclusion in an Infinite Plate under General In-plane Load

Problem Statement

A large, thin, flat plate (the matrix, modulus E₁, Poisson's ratio ν₁) contains a circular inclusion of diameter D made of another material (E₂, ν₂), perfectly bonded along the interface. Far from the inclusion the plate carries the uniform membrane state N_xx, N_yy, N_xy over the thickness t. Wanted are the stress inside the inclusion, the stress at a point P(r, θ) in either region, and the interface stresses that govern matrix cracking (hoop), debonding (radial) and interface slip (shear). A rigid plug and an empty hole are the two limits.

Figure A: Convention. N_xx, N_yy positive in tension; N_xy positive when it acts in +x on the top edge; θ is measured from the y axis towards x; the shaded disc is the inclusion.

Key Assumptions

Infinite plate, linear elastic isotropic materials, perfect bonding, plane stress or plane strain. No slip and no separation at the interface (the tool reports the radial and shear interface stresses so that the bond can be checked), no interference fit or thermal mismatch (those add a Lamé pressure that superposes), uniform far field. Plane stress uses κ = (3 − ν)/(1 + ν), plane strain κ = 3 − 4ν.

Solution

With the Kolosov–Muskhelishvili potentials the far field is φ → Γz, ψ → Γ′z, where Γ = (σ_xx^∞ + σ_yy^∞)/4 is real and Γ′ = [(σ_yy^∞ − σ_xx^∞) + 2iτ_xy^∞]/2 carries the deviatoric part. The disturbance of a circle is described by the same powers of z as in Kirsch's problem, so

matrix (r ≥ a):   φ₁ = Γz + Aa²/z,   ψ₁ = Γ′z + Ba²/z + Aa⁴/z³

inclusion (r ≤ a):   φ₂ = Dz,   ψ₂ = Fz,    a uniform state: σ_xx + σ_yy = 4D,   σ_yy − σ_xx + 2iτ_xy = 2F.

Continuity of traction, φ + zφ̄′ + ψ̄, and of displacement, (κφ − zφ̄′ − ψ̄)/2μ, on |z| = a gives, power by power in z,

A = Γ̄′ (μ₂ − μ₁)/(κ₁μ₂ + μ₁),    F = Γ′ μ₂(1 + κ₁)/(κ₁μ₂ + μ₁),    D = μ₂(κ₁ + 1)Γ / [2μ₂ + μ₁(κ₂ − 1)],    B = 2D − 2Γ.

The deviatoric and hydrostatic parts decouple: the inclusion sees the far-field mean stress multiplied by f_h = μ₂(κ₁ + 1)/[2μ₂ + μ₁(κ₂ − 1)] and the far-field deviator multiplied by f_d = μ₂(1 + κ₁)/(κ₁μ₂ + μ₁). A matched inclusion gives f_h = f_d = 1 and A = B = 0 (no disturbance); a rigid one gives (κ₁ + 1)/2 and (1 + κ₁)/κ₁; an empty hole gives 0 and the Kirsch potentials φ = Γz − Γ̄′a²/z, ψ = Γ′z − 2Γa²/z − Γ̄′a⁴/z³. The stresses follow from σ_xx + σ_yy = 4 Re φ′ and σ_yy − σ_xx + 2iτ_xy = 2(z̄φ″ + ψ′). The matrix field is exact and closed-form; no mapping or inversion is needed.

Checks built into the development: with E₂ = 0 the field equals the general Kirsch field of the round-hole tool to all digits; with matched materials the field is the far field everywhere; for an elastic inclusion the radial and shear stresses and both displacement components are continuous across the interface to ten digits while the hoop stress jumps; for a rigid inclusion the matrix displacement on the interface is zero; the far field is recovered.

Interface Stresses and Concentration Factors

A stiff inclusion attracts load: the radial stress at the interface on the load axis exceeds the far-field stress (about 1.5S for a rigid plug in plane stress under uni-axial tension, the classical Goodier value), and this radial stress is what tries to separate a bonded plug from the plate; the interface shear peaks at 45°. A soft inclusion sheds load: the matrix hoop stress at the sides rises towards the Kirsch value of 3S as E₂ → 0. The tool scans the interface on the matrix side and reports the hoop, radial and shear extremes with their angles, and refers them to the maximum far-field principal stress S₁: K_hoop for matrix cracking, K_rad for debonding, and K_I = σ_{max,inclusion}/S₁ for the inclusion itself. For the empty hole K_hoop = 3 − S₂/S₁.

Reading the Plots

• σ_θθ and σ_max fringes. Both regions are colored; inside the inclusion the state is uniform, so σ_max is constant there while σ_θθ, a polar component of a uniform state, varies with the angle. The color scale spans the extreme stresses of the problem.
• Interface plot. Radial and shear stresses are continuous across the bond; the hoop stress is drawn for both sides.
• Axis plots. The stresses along the x and y axes from the center outward, uniform inside the inclusion and jumping in the tangential component at r/a = 1.

Limits

• Imperfect bonding (slip, partial debonding) and interference or thermal fit are not included; an interference pressure p adds the Lamé solution (radial −p, hoop +p at the matrix interface) and can be superposed by hand.
• Finite width and yielding: as for the hole tools.
• Three-dimensional inclusions (spheres, fibers loaded along their axis) and elliptic inclusions (Hardiman) are different problems.