Paris' law
The Paris law describes how fast a fatigue crack grows per load cycle (da/dN). It relates this growth to the stress intensity factor range (ΔK). For welds, it is the most important tool for calculating remaining life and inspection intervals.
Physical-Chemical Mechanism of Action
At the crack tip, a small plastic zone forms with each load cycle. The crack opens and closes, growing a tiny amount each time. The Paris law describes this process with the formula: da/dN = C * (ΔK)^m. ΔK is the difference between the maximum and minimum stress intensity factor. In a double-logarithmic plot, this gives a straight line. The slope m is usually between 2 and 4 for steel. Below a threshold value ΔK_th, the crack practically does not grow.
Metallurgical Relevance & Risks in Welding
Welds almost always contain small defects: porosity, slag inclusions, lack of fusion. These act like pre-cracks. Additionally, welding residual stresses create tensile stresses that keep the crack open. This lowers the threshold value ΔK_th, and crack growth starts earlier. The notching effect at the weld increases local stress and thus ΔK. If growth is ignored, a small defect can become a critical crack and lead to sudden fracture.
Workshop Practice & Action Items
Determine the initial crack size with non-destructive testing: ultrasonic (UT) or magnetic particle (MT). Estimate ΔK from stress analysis or data tables. Calculate the life by integrating da/dN from the initial crack size to the critical crack size. Typical threshold for steel: ΔK_th ≈ 5–6 MPa√m (at R = 0.1). If ΔK is below the threshold, the crack does not grow. Reduce ΔK by smaller stress amplitudes, better weld geometry (grind weld toe, TIG dressing), and compressive residual stresses (e.g., hammer peening). Set inspection intervals by calculation, not arbitrarily. The intervals must be shorter than the time for a detectable crack to grow to critical size. Material constants: for structural steel, m ≈ 3, C is in the range from 10^-11 to 10^-13 depending on units (with da/dN in m/cycle, ΔK in MPa√m). Use only values from verified sources.