Fracture Mechanics Analysis
LEFM, XFEM, J-Integral & Fatigue Crack Growth

A notch has a finite radius at its tip — the stress concentration factor K_t is large but finite. A crack has a theoretically zero-radius tip. That means the stress is mathematically infinite at the tip, which breaks classical stress analysis. LEFM doesn't try to evaluate the stress at the exact tip — instead it characterises the intensity of the singular field with the stress intensity factor K_I. In Mode I (opening), the stress field ahead of the crack tip behaves as:

The \(1/\sqrt{r}\) singularity means stress blows up as you approach the tip, but K_I tells you how strongly it blows up. Once K_I exceeds the material's fracture toughness K_Ic, unstable crack growth begins.

For a through crack of half-length a in an infinite plate under remote stress σ, the textbook result is:

where F is a geometry correction factor (F = 1 for an infinite plate; tables and FEM give F for real geometries). Notice the \(\sqrt{a}\) dependence — doubling the crack length increases K_I by 41%. This is why fracture is so sudden: a crack that's "just" subcritical can reach K_Ic after a small amount of growth. For common structural steels K_Ic is around 50–200 MPa√m; for high-strength aerospace alloys it can be as low as 30 MPa√m.

Yes — three classical modes. Mode I is the crack opening mode: the two crack faces move apart perpendicularly. This is the most common and most dangerous in engineering structures — tensile stress driving the crack open. Mode II is in-plane sliding: crack faces slide relative to each other in the crack plane direction. You see this in shear-loaded joints. Mode III is out-of-plane tearing (anti-plane shear): crack faces slide perpendicular to the crack front. Common in torsional failures. In real structures you often have mixed-mode loading — K_I, K_II, K_III coexist, and you need criteria like the maximum principal stress criterion or strain energy density criterion to predict crack propagation direction.

The classic trick is quarter-point elements. Take a standard 8-node quadrilateral (serendipity) element and move the mid-side nodes to the quarter-point position (one-quarter of the way from the crack-tip node). This geometrically introduces a \(1/\sqrt{r}\) Jacobian that reproduces the crack-tip singularity exactly within the element — without infinite stress anywhere. You then extract K_I using the displacement correlation method: measure crack-opening displacement a short distance behind the tip and back-calculate K from the known singular field equations. The more robust modern approach is the J-integral, which we'll get to in a moment.

The J-integral is a path-independent contour integral around the crack tip that measures the energy release rate G — how much elastic energy is released per unit crack area advance. The formula is:

Path independence means you can integrate over a large contour far from the crack tip where the mesh is well-behaved, rather than trying to capture the singular field exactly. In linear elastic problems, J = G = K_I²/E' (plane stress) or K_I²(1-ν²)/E (plane strain). Modern FEM codes like Abaqus and ANSYS compute J-integral automatically from the domain integral method over multiple contour rings — convergence with ring index is an excellent built-in accuracy check.

XFEM — eXtended Finite Element Method — enriches the standard FEM displacement field with extra degrees of freedom that carry crack-related functions. For nodes near the crack tip, it adds crack-tip singular functions; for nodes bisected by the crack face, it adds the Heaviside discontinuity function. The FEM mesh doesn't need to conform to the crack at all — the crack is described as a mathematical level-set surface, and it can propagate through element interiors between steps. This is a massive advantage over remeshing: you set up the mesh once and let the crack advance each increment based on the local stress intensity values and a chosen growth criterion (like maximum hoop stress or minimum strain energy density).

That's where the Paris law comes in. It relates the fatigue crack growth rate per cycle to the stress intensity range ΔK = K_max - K_min:

C and m are material constants measured experimentally. For 7075-T6 aluminium (common in aircraft structures), C ≈ 2.3×10⁻¹² (m/cycle, MPa√m units), m ≈ 3.4. To find fatigue life, you integrate this ODE: as the crack grows, ΔK increases, growth accelerates, and you integrate until K_max reaches K_Ic. In practice you discretise: compute K at crack size a_i, advance the crack by Δa, recompute K, repeat — FEM solves the static problem at each crack size, Paris law gives the cycle increment ΔN for each Δa step.