Paris' law

Schematic plot of the typical relationship between the crack growth rate and the range of the stress intensity factor. In practice, the Paris law is calibrated to model the linear interval around the center.

Paris' law (also known as the Paris-Erdogan law) relates the stress intensity factor range to sub-critical crack growth under a fatigue stress regime. As such, it is the most popular fatigue crack growth model used in materials science and fracture mechanics. The basic formula reads^[1]

\frac{{\rm d}a}{{\rm d}N} = C \Delta K^m

,

where $a$ is the crack length and ${\rm {d}}a/{\rm {d}}N$ is the crack growth rate, ^[2] which denotes the crack growth for a load cycle. On the right hand side, $C$ and $m$ are constants that depend on the material, environment and stress ratio,^[3] and $\Delta K$ is the range of the stress intensity factor during the fatigue cycle, i.e.,

\Delta K=K_{max}-K_{min}

.^[3]

History and use

In a 1961 paper, P.C. Paris introduced the idea that the rate of crack growth may depend on the stress intensity factor.^[4] Then in their 1963 paper, Paris and Erdogen indirectly suggested the Paris law with the aside remark "The authors are hesitant but cannot resist the temptation to draw the straight line slope 1/4 through the data..." after reviewing data on a log-log plot of crack growth versus stress intensity range.^[5] The Paris equation was then presented with the fixed exponent of 4. Being a power law relationship between the crack growth rate during cyclic loading and the range of the stress intensity factor, the Paris law can be visualized as a linear graph on a log-log plot, where the x-axis is denoted by the range of the stress intensity factor and the y-axis is denoted by the crack growth rate.

Paris' law can be used to quantify the residual life (in terms of load cycles) of a specimen given a particular crack size. Defining the stress intensity factor as

where $\sigma$ is a uniform tensile stress perpendicular to the crack plane and Y is a dimensionless parameter that depends on the geometry, the range of the stress intensity factor follows as

\Delta K=\Delta\sigma Y \sqrt{\pi a}

,

where $\Delta \sigma$ is the range of cyclic stress amplitude. Y takes the value 1 for a center crack of length $2a$ in an infinite sheet. The remaining cycles can be found by substituting this equation in the Paris law.

For relatively short cracks, C can be assumed as independent of a and the differential equation can be solved via separation of variables

\int^{N_f}_0 {\rm d}N = \int^{a_c}_{a_i}\frac{{\rm d}a}{C(\Delta\sigma Y \sqrt{\pi a})^m } =\frac{1}{C(\Delta\sigma Y \sqrt{\pi})^m }\int^{a_c}_{a_i} a^{-\frac{m}{2}}\;{\rm d}a

and subsequent integration

N_f=\frac{2\;(a_c^{\frac{2-m}{2}}-a_i^{\frac{2-m}{2}})}{(2-m)\;C(\Delta\sigma Y \sqrt{\pi})^m}

,

where $N_{f}$ is the remaining number of cycles to fracture, $a_c$ is the critical crack length at which instantaneous fracture will occur, and $a_{i}$ is the initial crack length at which fatigue crack growth starts for the given stress range $\Delta \sigma$ . If Y strongly depends on a, numerical methods might be required to find reasonable solutions.

References

↑ "The Paris law". Fatigue crack growth theory. University of Plymouth. Retrieved 21 June 2010.
↑ N. Pugno, M. Ciavarella, P. Cornetti, A. Carpinteri. "A generalized Paris' law for fatigue crack growth" (PDF). Journal of the Mechanics and Physics of Solids 54 (2006) 1333-1349.
1 2 Roylance, David (1 May 2001). "Fatigue" (PDF). Department of Materials Science and Engineering, Massachusetts Institute of Technology. Retrieved 23 July 2010.
↑ P.C. Paris, M.P. Gomez, and W.E. Anderson. A rational analytic theory of fatigue. The Trend in Engineering, 1961, 13: p. 9-14.
↑ P.C. Paris and F. Erdogen. A critical analysis of crack propagation laws. Journal of Basic Engineering, 1963.

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[plymouth-1] "The Paris law". Fatigue crack growth theory. University of Plymouth. Retrieved 21 June 2010.

[convNaz-2] N. Pugno, M. Ciavarella, P. Cornetti, A. Carpinteri. "A generalized Paris' law for fatigue crack growth" (PDF). Journal of the Mechanics and Physics of Solids 54 (2006) 1333-1349.

[mit-3] 1 2 Roylance, David (1 May 2001). "Fatigue" (PDF). Department of Materials Science and Engineering, Massachusetts Institute of Technology. Retrieved 23 July 2010.

[4] P.C. Paris, M.P. Gomez, and W.E. Anderson. A rational analytic theory of fatigue. The Trend in Engineering, 1961, 13: p. 9-14.

[5] P.C. Paris and F. Erdogen. A critical analysis of crack propagation laws. Journal of Basic Engineering, 1963.