Paris' law

Paris' law (also known as the Paris-Erdogan equation) is a crack growth equation that relates the stress intensity factor range to sub-critical crack growth under a fatigue stress regime. The Paris equation isFollowing the procedure adopted by Ritchie, the average crack growth rate is considered asA similar procedure adopted by Alberto is presented here to find the correlation between the parameters C {displaystyle C} and m {displaystyle m} . A tangent to the mid-point of the central linear region of the curve is considered as shown in the figure 1, which intersects with the vertical dash lines at Δ K = Δ K th {displaystyle Delta K=Delta K_{ ext{th}}} along with the corresponding threshold crack growth rate of V th = 10 − 9 mm cycle {displaystyle V_{ ext{th}}=10^{-9}{frac { ext{mm}}{ ext{cycle}}}} and Δ K = Δ K cr = ( 1 − R ) K Ic {displaystyle Delta K=Delta K_{ ext{cr}}=left(1-R ight)K_{ ext{Ic}}} along with the corresponding critical crack growth rate of V cr = 10 − 5 mm cycle {displaystyle V_{ ext{cr}}=10^{-5}{frac { ext{mm}}{ ext{cycle}}}} . The law predicts a better fatigue life for the crack growth rate in the range of 10 − 8 − 10 − 6 mm cycle {displaystyle 10^{-8}-10^{-6}{frac { ext{mm}}{ ext{cycle}}}} . Plugging the critical and threshold limit points into the Paris' law leads to Paris' law (also known as the Paris-Erdogan equation) is a crack growth equation that relates the stress intensity factor range to sub-critical crack growth under a fatigue stress regime. The Paris equation is where a {displaystyle a} is the crack length and d a / d N {displaystyle { m {d}}a/{ m {d}}N} is the fatigue crack growth for a load cycle N {displaystyle N} . C {displaystyle C} and m {displaystyle m} are experimentally determined material constants which also depend on environmental effects, stress ratio, and the characteristic specimen size. The stress intensity factor characterises the stress field around a crack tip and the range has been found to correlate the rate of crack growth from a variety of different conditions. The stress intensity range ( Δ K ) {displaystyle (Delta K)} is the difference between the maximum and minimum stress intensity factors for each load cycle and is defined as 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 (see Figure 1). 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. 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 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. The Paris equation was then presented with the fixed exponent of 4. 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 {displaystyle R} which is defined as or ratio of minimum to maximum stress intensity factors. In the linear elastic fracture regime, R {displaystyle R} is also equivalent to the load ratio 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 equations such as the Forman Equation do explicitly include the effect of stress ratio as does the Elber equation by modelling the effect using crack closure. Paris' law holds over the mid-range of growth rate regime as shown in the figure 1, but does not apply for very low values of Δ K {displaystyle Delta K} approaching the threshold value Δ K th {displaystyle Delta K_{ ext{th}}} , or for very high values approaching the material's fracture toughness, K Ic {displaystyle K_{ ext{Ic}}} . The alternating stress intensity at the critical limit is given by Δ K cr = ( 1 − R ) K Ic {displaystyle {egin{aligned}Delta K_{ ext{cr}}&=(1-R)K_{ ext{Ic}}end{aligned}}} as shown in the figure 1.