The rule that adds up damage at different stress amplitudes as the sum of the ratios n/N of applied cycles to life, and regards fatigue failure as occurring when the sum reaches 1.
Japanese name: マイナー則
Miner's rule appears in 1 practice question across 1 topic, including Fracture mechanics and fatigue.
Real machines and structures experience repeated stresses of different sizes. Miner's rule takes the life at stress amplitude σi as Ni, regards the damage from ni cycles at that amplitude as ni/Ni, and assumes failure when the summed damage D = Σni/Ni reaches 1. Because Palmgren presented it in 1924 for estimating bearing life and Miner in 1945 for aircraft materials, it is also called the Palmgren–Miner rule. As long as an S-N curve at constant amplitude is available, the life under varying loads can be estimated, so it is widely used in design.
Real load records are irregular, so first the number of cycles of each size is counted with a method such as rainflow counting. Next, the life at each amplitude is read from the S-N curve and the damage is summed. For example, for a part that each day experiences 1000 cycles at stress amplitude σ1 (life 1.0×10⁵ cycles) and 1.0×10⁴ cycles at σ2 (life 1.0×10⁶ cycles), the daily damage is 0.010 + 0.010 = 0.020, and the life is estimated at about 50 days. Although σ2 occurs 10 times as often, its contribution to damage is the same. This shows that large-amplitude cycles, though rare, strongly affect life.
Miner's rule assumes that damage accumulates uniformly in proportion to the number of cycles, regardless of the order of loading. In reality, when a low stress follows a high one, the crack formed earlier can grow even under the low stress, so failure tends to occur before the damage reaches 1. Conversely, in a low-to-high sequence, failure may not occur even beyond 1. A single large tensile overload can also create compressive residual stress at the crack tip that temporarily slows subsequent growth. In experiments, the damage at failure often deviates greatly from 1, so the estimate should be regarded as having a range.
In constant-amplitude tests, stresses below the fatigue limit cause no failure however many times they are applied, so the original Miner's rule counts their damage as zero. Under varying loads, however, a crack formed by large stresses can grow even under stresses below the fatigue limit, and ignoring that damage overestimates life. Methods such as the modified Miner's rule, which extends the S-N curve below the fatigue limit and counts the damage there, are therefore used. Design codes for welded structures and others combine such methods with the idea of taking the damage limit smaller than 1, to give conservative estimates.
Answer: A. About 6.0×10⁵ cycles
The damage at σ₁ is n₁/N₁ = 4.0×10⁴/1.0×10⁵ = 0.40, leaving 1 − 0.40 = 0.60. At σ₂, the damage per cycle is 1/N₂, so the remaining number of cycles is 0.60×1.0×10⁶ = 6.0×10⁵ cycles. About 9.6×10⁵ cycles subtracts the number of cycles itself, rather than the fraction of damage, from N₂, ignoring that the weight of one cycle differs with stress. About 6.0×10⁴ cycles multiplies the remaining fraction by the life N₁ at σ₁. Miner's rule is simple, but it cannot represent the effect of load sequence, and it is known that when the stress steps from high to low, the actual life tends to be shorter than estimated.
No. D = 1 is an average guide, and in experiments the damage at failure can be smaller or larger than 1 depending on load sequence, material and mean stress. The scatter of the S-N curve itself is also large, so in design the rule is used with a margin.
Yes. Because damage is assumed to accumulate linearly, in proportion to the cycle ratio n/N, and can be added up, it is also called the linear cumulative damage rule. Rules that make damage accumulation nonlinear to represent sequence effects have also been proposed, but for its simplicity the linear Miner's rule is the most widely used.
Yes, but the effect of mean stress must be corrected first. The usual procedure is to convert each cycle into an equivalent stress amplitude at zero mean stress with a modified Goodman diagram or similar, read the life from the fully reversed S-N curve and then sum the damage.
Miner's rule is simple, but it cannot represent the effect of load sequence, and it is known that when the stress steps from high to low, the actual life tends to be shorter than estimated.
A steel part fails by fatigue after 1.0×10⁵ cycles at stres…Energy release rate Fatigue striations Griffith theory Modified Goodman diagram Paris law Stress concentration factor Stress intensity factor