Crack Propagation — CAE Terminology Glossary

Breaking at once is called brittle fracture or unstable fracture. Crack propagation is a phenomenon where microscopic cracks in materials grow gradually due to repeated loading (fatigue) or environmental factors (corrosion, creep, etc.).

For example, an aircraft fuselage experiences pressurization and depressurization cycles on each flight. Fatigue cracks develop from stress concentration points like rivet holes and grow slowly over thousands of cycles. Once the crack length exceeds a critical value, sudden unstable fracture occurs. That's why we must predict how fast cracks grow to determine inspection intervals. Steel bridge girders, pressure vessels, and turbine blades all use the same principle. This is the foundation of Damage Tolerance Design.

The most famous is Paris law, proposed by Paris and Erdogan in 1963. It is written as:

They are determined through fatigue crack propagation experiments using CT (Compact Tension) or CCT (Center-Cracked Tension) specimens. ASTM E647 is the standard test specification. Plot $\Delta K$ versus $da/dN$ on a log-log scale; the middle region becomes a clean straight line. The slope is $m$, and $C$ is derived from the intercept.

Yes, the actual $da/dN$ versus $\Delta K$ curve is S-shaped (sigmoid). At low ΔK, there is a threshold stress intensity factor range $\Delta K_{\text{th}}$; crack growth is negligible below this. At high ΔK, as $K_{\max}$ approaches fracture toughness $K_{IC}$, it accelerates toward unstable fracture. Paris law describes only the middle stable propagation region (Region II).

The basic form is:

Integrate Paris law. The number of cycles from initial crack length $a_0$ to critical crack length $a_c$ (where $K_{\max} = K_{IC}$) is:

With conventional FEM, the mesh must be remeshed along the crack path as it grows. That is very labor-intensive. XFEM (eXtended FEM) adds enrichment functions to the existing mesh, enabling mesh-independent crack representation without remeshing.

There are two types. First, the Heaviside function $H(\mathbf{x})$ — added to nodes crossing the crack face to represent displacement discontinuity (crack opening). Second, crack tip enrichment functions — singularity functions like $\sqrt{r}\sin(\theta/2)$ that accurately capture the stress singularity at the crack tip ($1/\sqrt{r}$). Mathematically:

Abaqus has XFEM built-in; you can specify crack initiation criteria (e.g., maximum principal stress) and propagation direction (e.g., maximum tangential stress). Ansys Mechanical has SMART crack growth functionality. However, with complex 3D crack paths, convergence issues sometimes arise, so many practitioners use Cohesive Zone Model as an alternative.

CZM employs a traction-separation law (TSL) on the crack surface. Whereas XFEM excels when the crack path is unknown, CZM is particularly effective when the path is roughly known — for example, adhesive layer delamination, composite interface delamination, or weld joint fracture.

The most commonly used is the bilinear (triangular) form. As separation displacement $\delta$ increases, traction $T$ first increases linearly to critical traction $\sigma_c$, then softens until final separation displacement $\delta_f$ where traction becomes zero. The area under the triangle equals fracture energy $G_c$:

That's a major advantage of CZM. Even without an initial crack, once traction exceeds $\sigma_c$, damage begins and progresses through opening. CZM handles crack initiation and propagation in a unified manner. XFEM, by contrast, requires either specifying an initial crack or defining a separate initiation criterion. For CFRP laminate delamination, CZM has become the de facto standard.

Several things to watch out for:

Yes! The Paris Law Crack Propagation Simulator lets you input $C$, $m$, initial crack length, and loading conditions, then interactively compute crack length evolution and residual life. Try plugging in your own material parameters.