🩹 Microcracks (Paris Law)

The Paris law is an empirical equation, da/dN = C·(ΔK)^m, describing how fast a fatigue crack grows per load cycle once it is in the stable propagation regime. C and m are material constants fitted from test data: C sets the overall growth rate and m controls how sensitive that rate is to the stress-intensity range ΔK. In this simulator C is entered in units of ×10⁻¹¹ and m typically ranges from about 2.9 for titanium to 4.5 for cast iron.

The simulation uses the standard linear-elastic fracture mechanics formula K = Y·σ·√(πa), where σ is the applied stress, a is the current crack length and Y is a dimensionless geometry factor that accounts for specimen shape (adjustable via the Y geometry factor slider, default 1.12 for an edge crack). ΔK uses the stress range Δσ = σ_max − σ_min, while K_max uses σ_max directly; failure is triggered as soon as K_max reaches K_IC.

Fracture occurs when the peak stress intensity K_max = Y·σ_max·√(πa) reaches the material's fracture toughness K_IC, entered in MPa·√m via the K_IC slider. As the crack a grows, K_max rises even though the applied stresses stay constant, so every material and load combination has a critical crack length a_c (shown as a dashed line on the plot) beyond which the crack becomes unstable and the specimen snaps rather than growing slowly.

It is a simplified engineering model, not a finite-element or full physical simulation. The crack path itself is a randomly jittered line for visual effect, and the specimen geometry is a basic rectangular bar rather than a real fracture-test coupon. The growth physics, however, is the genuine Paris law integrated numerically cycle by cycle with a real stress-intensity formula, so the trends in crack length, growth rate and cycles-to-failure reflect real fatigue behaviour.

Each preset (Steel A36, Aluminium 7075, Titanium Ti-6Al-4V, Cast Iron) has its own C, m and K_IC values drawn from typical published fatigue data, and the estimated Cycles to failure statistic is computed by integrating the Paris law from the current crack length to the critical length a_c. Materials with low K_IC (like cast iron at 20 MPa·√m) reach their critical crack length sooner, while a high Paris exponent m makes growth accelerate much faster once cracking begins, so small differences in material properties translate into large differences in fatigue life.