The empirical law da/dN = C(ΔK)ᵐ, stating that the fatigue crack growth rate is proportional to a power of the stress intensity factor range. It is used to estimate crack growth life.
Japanese name: パリス則
Paris law appears in 2 practice questions across 1 topic, including Fracture mechanics and fatigue.
Plotting the fatigue crack growth rate da/dN against the stress intensity factor range ΔK = K_max − K_min on log-log axes gives an S-shaped curve. In the region of small ΔK, the rate drops sharply as ΔK approaches the threshold ΔK_th, and below it the crack effectively does not grow. In the intermediate region the curve is a straight line, and expressing it as da/dN = C(ΔK)ᵐ is the Paris law. In the region where ΔK is large and the maximum stress intensity factor approaches the fracture toughness, the rate rises sharply and finally leads to unstable fracture. The region used to estimate life in design is mainly the intermediate one.
The exponent m is roughly 2–4 for metals and can exceed 10 for ceramics. The larger m is, the more the growth rate changes with a small increase in stress or crack length. The value of the constant C changes with the unit system, so the data are used with consistent units, such as da/dN in m/cycle and ΔK in MPa·√m. It is also known that, among steels, differences in the intermediate-region rate due to microstructure and strength are relatively small. On the other hand, the rate is affected by mean stress (the stress ratio R), environment and temperature, so it is important to choose data measured under the conditions of use.
Substituting ΔK = YΔσ√(πa) into the Paris law and integrating from the initial crack length a₀ to the critical crack length a_f gives the number of cycles needed for the crack to grow. The critical length is set by the condition K_max = K_IC. Because growth is slow while the crack is short, most of the life is spent early on, and the result depends strongly on the estimate of a₀. In damage-tolerant design for aircraft and similar structures, a crack of a size that nondestructive inspection could miss is assumed to exist, and the inspection interval is set so that it is found at the next inspection before it reaches the critical size.
Answer: B. About 5.1×10⁴ cycles
For m = 4, da/dN = C(Δσ√π)⁴a², so separating variables and integrating gives N = (1/a₀ − 1/a_f)/[C(Δσ√π)⁴]. C(Δσ√π)⁴ = 1.0×10⁻¹²×200⁴×π² ≈ 1.58×10⁻² m⁻¹ and 1/a₀ − 1/a_f = 1000 − 200 = 800 m⁻¹, so N ≈ 800/(1.58×10⁻²) ≈ 5.1×10⁴ cycles. About 2.5×10⁵ cycles assumes the rate at the initial 1.0 mm, 1.58×10⁻⁸ m/cycle, continues to the end, ignoring that ΔK rises and the crack accelerates as it grows. About 6.3×10⁴ cycles drops the 1/a_f term. Most of the life is spent while the crack is still short, early on.
No, it is an empirical law obtained by organizing a large body of experimental data. There are models that try to explain the exponent from plastic deformation and blunting at the crack tip, but C and m are usually determined by tests on each material. Note also that it holds only in the intermediate region.
Under constant-amplitude loading it is a guide below which the crack can be regarded as effectively not growing. However, when the crack is very short, when occasional large loads are mixed in, or in corrosive environments, cracks can grow even below ΔK_th.
Even at the same ΔK, the larger R = σmin/σmax is (the higher the mean stress), the faster the growth rate tends to be and the smaller ΔK_th becomes. One main reason is crack closure: when the crack faces touch partway through unloading, the ΔK actually acting at the tip becomes smaller, but this effect weakens as R increases.
The intermediate region is a straight line on log-log axes and can be described by the Paris law da/dN = C(ΔK)ᵐ, with an exponent m of roughly 2–4 for metals.
Which statement correctly describes the curve of fatigue cr…Stress intensity factor Fracture toughness Energy release rate Fatigue striations Griffith theory Miner's rule Modified Goodman diagram Stress concentration factor