According to Miner's Rule, when damage is equal to “1”, failure occurs. This article explains how Miner's Rule is used to measure damage and predict failure in a part.
Contents: 1. Introduction 2. SN-Curve 3. Damage Accumulation Simple Example 4. Formula 5. History 6. "Real World" Considerations 6.1 Cycle Sequence 6.2 Cycle Counting Complex Signals
The definition of failure for a physical part varies. It could mean that a crack has initiated on the surface of the part. It could also mean that a crack has gone completely thru the part, separating it. In this article, a conservative approach to failure will be used: a crack starts to appear on the surface of the part.
Applying a constant amplitude, cyclical stress to a metal coupon causes it to fail (a crack appears) after a specific number of cycles. When the crack appears, the accumulated damage is considered to be equal to 1. A brand new part that has never had any stress loads applied is considered to have no accumulated damage (damage equal to 0).
A material sample has four stress (force over area) cycles applied
A stress is a force (F) applied over a surface area (A). Note that a complete cycle starts at a specific value (zero in this illustration), cycles above and below the initial value (or below and above), and then returns to the initial value.
By repeating this test at different stress levels, one could develop a SN-Curve. A SN-Curve relates cyclic stress levels to a number of cycles until failure. The "S" in SN-Curve is "Stress", and the "N" in "Number of Cycles".
Higher levels of cycle stress require fewer cycles until failure. As a result, the SN-Curve slopes downward as shown in the figure below.
SN-Curve: Cyclic Stress Level versus Number of Cycles to Failure
SN-Curves are developed with testing machines that apply constant amplitude cyclical loads.
Axial fatigue test machine for material coupons
They can be axial loads, torsional loads, bending loads, etc. Different stress levels are tested and the number of cycles to failure are recorded.
When a physical part undergoes stress cycles, Miner’s Rule works like this:
On the left graph, there is a loading time history. The SN-Curve is the middle graph, and a damage tally is kept on the right side.
In this case, two cycles at a specific amplitude are applied to the part. At this amplitude the part could take 6 cycles before it would fail. Dividing two cycles by six cycles, the accumulated damage is 0.33. A third of the life of the part has been used.
Two more cycles of a higher amplitude are applied. At this higher amplitude, four cycles would be required for failure to occur. Dividing two cycles by four cycles, an additional 0.5 of damage has occurred. The total accumulated damage is now 0.83. According to Miner’s Rule, no failure has occurred.
One more cycle of the higher amplitude is now applied. The accumulated damage is now 1.08. Failure has occurred!
The equation for Miner's Rule is:
D is total damage. When damage is equal to one, failure occurs.
n is the number of cycles of a given amplitude that the part or object is subjected to in the field or during its lifetime.
N is the total number of cycles of a give amplitude that a material can survive, as determined by laboratory testing.
k is the number of different amplitude levels of the cycles from the field or lifetime data.
In 1945, M. A. Miner popularized a rule that had first been proposed by A. Palmgren in 1924. The rule is variously called Miner's rule or the Palmgren-Miner linear damage hypothesis.
6. "Real World" Considerations
Unlike the simple example presented in this article, real SN curves for metals are log-log curves and easily range into millions of cycles. Further "real world" considerations include:
6.1 Cycle Sequence
Miner's Rule does not take into account the sequencing or order in which in the cyclic loads are applied. For example, if loads are applied in the plastic region, the endurance limit is no longer in effect for any cycles that occur afterwards in time.
6.2 Cycle Counting Complex Signals
The loads applied to product in the real world are usually not constant amplitude cycles that look like a sine wave. Loads can be applied in a haphazard manner by end users of a product as shown below:
To break down a "real world" complex load history into cycles, a cycle counting method called rainflow analysis is employed.
More information on breaking down complex loading signals into specific cycles in the knowledge article: Rainflow Counting.