Stress intensity factor KI · Paris law fatigue · Crack-tip stress field
Interactive linear elastic fracture mechanics simulator. Compute the stress intensity factor K_I, visualise stress fields near a crack tip, and simulate Paris-law fatigue crack growth cycle by cycle.
The stress intensity factor K_I = Yσ√(πa) quantifies singularity strength at a crack tip. When K_I reaches K_IC (fracture toughness), fast fracture occurs. In fatigue, each load cycle advances the crack by da/dN = C(ΔK)^m — the Paris law.
Set initial crack length, applied stress, and material properties (K_IC, Paris constants). Run the fatigue simulation to watch the crack grow. The stress field around the tip updates live.
The Paris law exponent m is typically between 2 and 4 for metals. A crack in an aircraft fuselage can grow from invisible (0.5 mm) to catastrophic in as few as 10,000 cycles under typical pressurisation loads.
This simulation applies Linear Elastic Fracture Mechanics to a centre-cracked plate under uniform remote tension. It computes the Mode I stress intensity factor KI = σ√(πa) with a finite-width correction √(sec(πa/2W)), then compares it against each material's fracture toughness KIC. The canvas draws the specimen, the crack, the applied stress, and shaded half-circular contours that approximate the elevated stress field clustered around the crack tip.
You choose a material (steel, aluminium or ceramic, each with its own KIC and Paris constants) and adjust three sliders: applied stress σ (10–400 MPa), crack half-length a (1–80 mm) and specimen width W (40–400 mm). A status badge flags SAFE, WARNING or CRITICAL as KI/KIC rises. Pressing Run advances Paris-law fatigue, growing the crack cycle by cycle — the same physics that governs aircraft, pressure vessels and bridges.
What is the stress intensity factor K_I?
K_I is a single parameter that measures how strongly stress is concentrated at a crack tip in Mode I (tensile opening). Here it is computed as K_I = sigma times the square root of pi times a, with a finite-width correction. Its units are MPa root-metre, and it depends only on the applied stress, crack size and geometry — not on the material.
How does the simulation decide if the crack is safe or critical?
It compares K_I against the material's fracture toughness K_IC. The ratio K_I/K_IC drives the colour bar and badge: below 0.6 is SAFE (green), between 0.6 and 1.0 is WARNING (amber), and at or above 1.0 is CRITICAL (red), meaning fast fracture is predicted.
What is the Paris law and how is it used here?
The Paris law describes fatigue crack growth as da/dN = C(deltaK)^m, where deltaK is the cyclic stress intensity range and C and m are material constants. When you press Run, the simulation steps forward 500 cycles at a time, growing the crack by this rate until K_I reaches K_IC or the crack spans most of the specimen.
Applied stress sigma sets the remote tension from 10 to 400 MPa. Crack half-length a sets the size of the central crack from 1 to 80 mm. Specimen width W sets the plate width from 40 to 400 mm, which feeds the finite-width correction term — narrower plates raise K_I for the same crack.
The basic result K_I = sigma root(pi a) assumes an infinite plate. Real specimens are finite, so the crack interacts with the free edges. This model multiplies by root of sec(pi a / 2W), which grows sharply as the crack approaches the plate width, capturing the rising stress intensity in a bounded geometry.
Each material carries its own toughness and Paris constants: steel (K_IC 50 MPa root-m), aluminium (30) and ceramic (3). The brittle ceramic fractures at very low K_I and grows cracks fast, while tough steel tolerates much higher stress intensity before failing. Switching material instantly re-evaluates the safety margin.
They are a visual approximation of the near-tip stress field, which in LEFM scales with K_I divided by the square root of distance from the tip. The contours are drawn as half-circles because the field is symmetric about the crack plane, and they intensify as K_I rises, illustrating the mathematical singularity at the tip.
It uses textbook LEFM equations and realistic material constants, so it is qualitatively faithful and good for learning. However, the contour rendering is a simplified illustration rather than a full finite-element field, and the Paris stepping uses a fixed cycle increment. Treat the numbers as educational estimates, not certified engineering design values.
Because K_I depends on the square root of crack length, even small cracks raise the local stress dramatically, and under fatigue they grow ever faster as K_I increases. A sub-millimetre flaw can advance to catastrophic size in a finite number of cycles, which is why inspection and damage tolerance are central to engineering safety.
It underpins damage-tolerant design of aircraft structures, pressure vessels, pipelines, ship hulls, turbine discs and bridges. Engineers use K_I, K_IC and the Paris law to set inspection intervals, predict remaining life and decide when a cracked component must be repaired or retired before fast fracture occurs.