Paris’ Law describes the rate of fatigue crack growth in materials under cyclic loading. The law is expressed with the following equation:

Where:

• is the crack growth rate per cycle, meaning the increment of crack length per loading cycle .
• is the range of the stress intensity factor, defined as the difference between the maximum () and minimum () stress intensity factors during a loading cycle: .
• and are material constants that are determined experimentally. These constants depend on the material properties and the environmental conditions.

Detailed Explanation

1. Stress Intensity Factor ():
   • The stress intensity factor represents the intensity of the stress field near the tip of a crack and is influenced by the applied load, the geometry of the crack, and the size of the crack.
   • For a given cyclic loading, varies between a minimum () and maximum () value. The range is the key driving force for fatigue crack growth.
2. Crack Growth Rate ():
   • This term represents how fast the crack grows with each loading cycle. It is a direct measure of the fatigue damage occurring in the material.
3. Material Constants ( and ):
   • These constants are specific to each material and must be determined through experimental testing. is a coefficient that scales the relationship, and is the exponent that determines the sensitivity of the crack growth rate to changes in .
   • Typically, is greater than 2, indicating that the crack growth rate increases rapidly with increasing .

Experimental Determination

To determine the constants and , fatigue crack growth tests are conducted on samples made of the material of interest. The tests involve:

• Applying cyclic loading to the sample.
• Measuring the crack length as a function of the number of cycles .
• Calculating the stress intensity range for each cycle.
• Plotting versus .

The resulting plot is typically a straight line, and the slope of this line gives the exponent , while the intercept provides the coefficient .

Importance

Paris’ Law is crucial for predicting the fatigue life of components and structures. By understanding how cracks grow under cyclic loading, engineers can design more durable materials, plan maintenance schedules, and prevent catastrophic failures in critical applications such as aerospace, automotive, and civil engineering structures.