Paris’ law (also known as the Paris–Erdogan equation)

Paris’ law, also known as the Paris-Erdogan equation, is a mathematical relationship that describes the growth of fatigue cracks in materials subjected to cyclic loading, such as in mechanical or structural components. The equation is named after the French scientist Paul C. Paris and the American engineer Turan P. Erdogan, who developed it in the mid-1960s.
The Paris-Erdogan equation can be expressed as follows:

da/dN=C(ΔK)m

Where:

• da/dN represents the rate of crack growth per cycle.
• C is a material constant that depends on the specific material and loading conditions.
• ΔK is the range of stress intensity factor, which characterizes the stress field at the crack tip during cyclic loading.
• m is a material-dependent exponent that typically falls in the range of 2 to 4.

Case Study: Assessing Fatigue Life in an Aircraft Component

Background: Aircraft components, especially critical ones like wing structures, are subjected to repetitive cyclic loading during flight. Understanding the fatigue behavior and predicting crack growth is crucial for ensuring the safety and longevity of these components.

Objective: Determine the remaining fatigue life of a wing component by applying the Paris-Erdogan equation.

Steps in the Case Study:

1. Data Collection: Gather necessary data for the analysis, including material properties, stress history, and known crack size. For this case, we assume the following:
   • Material: Aluminum alloy 7075-T6
   • Stress amplitude (Δσ): 30 MPa
   • Initial crack size (a0): 0.5 mm
2. Material Properties: Obtain the material properties specific to Aluminum alloy 7075-T6, including the material constant (C) and the exponent (m) for the Paris-Erdogan equation. These values may be obtained from material testing or literature.
3. Calculation of Stress Intensity Factor (ΔK): Calculate the range of stress intensity factor (ΔK) for each loading cycle based on the stress amplitude (Δσ) and crack size (a) using appropriate equations or methods.
4. Crack Growth Rate (da/dN): Utilize the Paris-Erdogan equation to compute the crack growth rate (da/dN for each cycle.
5. Cumulative Crack Growth: Integrate the crack growth rate over time to determine the cumulative crack growth.
6. Remaining Fatigue Life: Compare the cumulative crack growth to a critical crack size or the component’s allowable crack size. If the crack size reaches a critical limit, the component may need maintenance or replacement. If not, calculate the remaining fatigue life based on the current crack size and known stress conditions.
7. Safety Assessment: Assess whether the component can safely continue in service until the end of its estimated fatigue life or if immediate maintenance or replacement is required.

This case study demonstrates how the Paris-Erdogan equation can be employed to estimate the remaining fatigue life of a critical aircraft component. The results of such analyses are essential for ensuring the safety and reliability of aerospace structures and making informed decisions about maintenance and replacement schedules.

Applications of the Paris-Erdogan Equation

• Predicting the rate of crack growth in materials under cyclic loading conditions.
• Estimating the remaining life of structural components and machinery.
• Ensuring the safety and reliability of structures and equipment.