HOME

TheInfoList



OR:

Paris' law (also known as the Paris–Erdogan equation) is a
crack growth equation A crack growth equation is used for calculating the size of a fatigue (material), fatigue crack growing from cyclic loads. The growth of a fatigue crack can result in catastrophic failure, particularly in the case of aircraft. When many growing f ...
that gives the rate of growth of a
fatigue Fatigue is a state of tiredness (which is not sleepiness), exhaustion or loss of energy. It is a signs and symptoms, symptom of any of various diseases; it is not a disease in itself. Fatigue (in the medical sense) is sometimes associated wit ...
crack. The
stress intensity factor In fracture mechanics, the stress intensity factor () is used to predict the Stress (mechanics), stress state ("stress intensity") near the tip of a Fracture, crack or Notch (engineering), notch caused by a remote load or residual stresses. It i ...
K characterises the load around a crack tip and the rate of crack growth is experimentally shown to be a function of the range of stress intensity \Delta K seen in a loading cycle. The Paris equation is :\begin &= C\left(\Delta K\right)^, \end where a is the crack length and a/N is the fatigue crack growth for a load cycle N. The material coefficients C and m are obtained experimentally and also depend on environment, frequency, temperature and stress ratio. The stress intensity factor range has been found to correlate the rate of crack growth from a variety of different conditions and is the difference between the maximum and minimum stress intensity factors in a load cycle and is defined as : \Delta K= K_-K_. Being a
power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a relative change in the other quantity proportional to the ...
relationship between the crack growth rate during cyclic loading and the range of the stress intensity factor, the Paris–Erdogan equation can be visualized as a straight line on a log-log plot, where the
x-axis In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
is denoted by the range of the stress intensity factor and the y-axis is denoted by the crack growth rate. The ability of ΔK to correlate crack growth rate data depends to a large extent on the fact that alternating stresses causing crack growth are small compared to the yield strength. Therefore crack tip plastic zones are small compared to crack length even in very ductile materials like stainless steels. The equation gives the growth for a single cycle. Single cycles can be readily counted for ''constant-amplitude'' loading. Additional cycle identification techniques such as rainflow-counting algorithm need to be used to extract the equivalent constant-amplitude cycles from a ''variable-amplitude'' loading sequence.


History

In a 1961 paper, P. C. Paris introduced the idea that the rate of crack growth may depend on the stress intensity factor. Then in their 1963 paper, Paris and Erdogan indirectly suggested the equation 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. The Paris equation was then presented with the fixed exponent of 4.


Domain of applicability


Stress ratio

Higher mean stress is known to increase the rate of crack growth and is known as the ''mean stress effect''. The mean stress of a cycle is expressed in terms of the ''stress ratio'' R which is defined as :R = , or ratio of minimum to maximum stress intensity factors. In the linear elastic fracture regime, R is also equivalent to the load ratio :R \equiv . The Paris–Erdogan equation does not explicitly include the effect of stress ratio, although equation coefficients can be chosen for a specific stress ratio. Other
crack growth equation A crack growth equation is used for calculating the size of a fatigue (material), fatigue crack growing from cyclic loads. The growth of a fatigue crack can result in catastrophic failure, particularly in the case of aircraft. When many growing f ...
s such as the ''Forman equation'' do explicitly include the effect of stress ratio, as does the ''Elber equation'' by modelling the effect of
crack closure Crack closure is a phenomenon in fatigue loading, where the opposing faces of a crack remain in contact even with an external load acting on the material. As the load is increased, a critical value will be reached at which time the crack becomes ''o ...
.


Intermediate stress intensity range

The Paris–Erdogan equation holds over the mid-range of growth rate regime, but does not apply for very low values of \Delta Kapproaching the threshold value \Delta K_, or for very high values approaching the material's
fracture toughness In materials science, fracture toughness is the critical stress intensity factor of a sharp Fracture, crack where propagation of the crack suddenly becomes rapid and unlimited. It is a material property that quantifies its ability to resist crac ...
, K_. The alternating stress intensity at the critical limit is given by \begin \Delta K_ &= (1-R)K_ \end. The slope of the crack growth rate curve on log-log scale denotes the value of the exponent m and is typically found to lie between 2 and 4, although for materials with low static fracture toughness such as high-strength steels, the value of m can be as high as 10.


Long cracks

Because the size of the plastic zone (r_ \approx K_^2/\sigma_^2) is small in comparison to the crack length, a (here, \sigma_ is yield stress), the approximation of small-scale yielding applies, enabling the use of linear elastic fracture mechanics and the
stress intensity factor In fracture mechanics, the stress intensity factor () is used to predict the Stress (mechanics), stress state ("stress intensity") near the tip of a Fracture, crack or Notch (engineering), notch caused by a remote load or residual stresses. It i ...
. Thus, the Paris–Erdogan equation is only valid in the linear elastic fracture regime, under tensile loading and for long cracks.


References

{{DEFAULTSORT:Paris' Law Materials science Fracture mechanics Mechanical failure Mechanical failure modes Solid mechanics