User:Sairam Pamulaparthi Venkata/sandbox

Paris' Law
The Paris' law is a power-law relationship which relates the crack growth rate $$(da/dN)$$ to the stress intensity factor range or alternating stress intensity $$(\Delta K)$$ and is given by $$\begin{align} {da \over dN} &= C\left(\Delta K\right)^{m}, \qquad (\Delta K_{\text{th}} < \Delta K < K_{\text{IC}}), \end{align}$$

where $$\Delta K = K_{\text{max}} - K_{\text{min}}$$ is the difference between the maximum and minimum stress intensity factors for each cycle, and $$C$$ and $$m$$ are experimentally determined material constants. The alternating stress intensity at the critical limit is given by $$\begin{align} \Delta K_{\text{cr}} &= (1-R)K_{\text{IC}} \end{align}$$as shown in the figure 1.

It is to be noted that the effect of stress ratio is not included in the Paris' equation $$(R = 0)$$. The Paris' law does not hold for very low values of $$\Delta K$$approaching the threshold value $$(\Delta K_{\text{th}})$$, and for very high values approaching the material's fracture toughness, $$K_{\text{IC}}$$. The slope of the crack growth rate curve on log-log scale denotes the value of the exponent $$m$$ and in general is found to lie between the range $$2$$ to $$4$$. But for the materials of low static fracture toughness like high strength steels, the value of $$m$$ can shoot up to a value of $$10$$.

It is important to note that Paris' law is valid only in linear elastic fracture regime, under uniaxial loading and for long cracks.

Barenblatt and Botvina observed that the values of constants $$C$$ and $$m$$ are not just dependent on the material properties and the nature of the applied loading, but also depends on the characteristic specimen size.

The correlation between the parameters $$C$$ and $$m$$ is observed by Alberto. These relations are validated by Radhakrishnan for steels and aluminium alloys with the experimental data.

Also, we observe that in mid-range of growth rate regime as shown in the figure 1, the size of the plastic zone $$(r_{\text{p}} \approx K_{I}^2/\sigma_{y}^2)$$ is low in comparison to the crack length, $$a$$ (here, $$\sigma_{y}$$ is yield stress). Therefore, we can safely assume the concepts of small scale yielding or linear elastic fracture mechanics. This provides us the liberty to use stress intensity factor as a characterizing parameter for fatigue crack growth rate calculations.

History of crack propagation laws
Fatigue is an important and relevant problem to both the designer and operator of any structure. From the point of view of the designer, primary aspects constitute how the cyclic stresses, material properties, surface quality and other effects that influence the fatigue life. Unfortunately, even after careful consideration of the above factors, fatigue might still occur due to chemical environments, extensive utilization of the structure and so on. All these factors places the users in a position where they have to perform inspections and non-destructive testing techniques. Also, different structures like motor car engines, nuclear pressure vessels, and aircraft structures produce different behaviours under fatigue loading. Therefore, prediction of fatigue crack growth is of extreme importance to understand the failure behaviour.

The fatigue life can be subdivided into nucleation period and crack growth period. The nucleation period consists of crack nucleation and microcrack growth which leads to the next phase or to macrocrack growth.

Several crack propagation laws are proposed in the past. The works of Head, Frost and Dugdale , McEvily and Illg , and Liu on fatigue crack-growth behaviour laid the foundation in this topic. The general form for the above crack propagation laws may be expressed as

$$\begin{align} {da \over dN} &= f\left(\Delta \sigma, a, C_{i}\right), \end{align} $$

where, half of the crack length is denoted by $$a$$, number of cycles of load applied is given by $$N$$, the stress range by $$\Delta \sigma$$, and the material parameters by $$C_{i}$$.

The above propagation laws mostly agree for small samples of data but breakdown for wide ranges of data obtained from different specimens and for varied crack growth rates. In an attempt to fill this gap, Paris, Gomez , and Anderson have proposed an empirical law which fits the broad trend of data.

Crack growth rate in different regimes
The crack growth rate behaviour with respect to the alternating stress intensity can be explained in different regimes (see, figure 1) as follows

Regime A: At low growth rates, variations in microstructure, mean stress (or load ratio), and environment have significant effects on the crack propagation rates. It is observed at low load ratios, crack growth rate is most sensitive to microstructure and in low strength materials it is most sensitive to load ratio.

To predict the crack growth rate at near threshold region, the following relation is proposed

$${da \over dN} = A_{1}\left(\Delta K - \Delta K_{\text{th}}\right)^{p}.$$

Regime B: At mid-range of growth rates, variations in microstructure, mean stress (or load ratio), thickness, and environment have no significant effects on the crack propagation rates.

To predict the crack growth rate in this intermediate regime, the Paris' law is used

$${da \over dN} = C\left(\Delta K\right)^{m}.$$

Regime C: At high growth rates, crack propagation is highly sensitive to the variations in microstructure, mean stress (or load ratio), and thickness. Environmental effects have relatively very less influence.

Near the fracture toughness region, the static modes of fracture are considered to be leading to the overall crack growth rate and Forman proposed the following relation to predict the crack propagation behaviour

$${da \over dN} = \frac{A_{2}(\Delta K)^n}{(1-R)(K_{\text{IC}} - K_{\text{max}})} \equiv \frac{A_{3}(\Delta K)^r}{(\frac{K_{\text{IC}}}{ K_{\text{max}}} - 1)}.$$

Also, McEvily and Groeger proposed the following power-law relationship which considers the effects of both high and low values of $$\Delta K$$

$${da \over dN}= A_{4}(\Delta K - \Delta K_{\text{th}})^2\Big[1 + \frac{\Delta K}{K_{\text{IC}} - K_{\text{max}}}\Big],$$

here, we notice that the value of exponent is $$2$$. Also, we observe that the load ratio effect is implicitly imbibed in the above relation to consider the near threshold and fracture toughness limits.

Effect of load ratio on the crack growth rate
Practically, it is observed that the load or stress ratio $$ \big(R = P_{\text{min}}/{P_{\text{max}}} \equiv K_{\text{min}}}/{K_{\text{max}}\big)$$does affect the fatigue crack growth rate and is explained using the crack closure concept. When the load is removed, the crack surfaces might come in contact with each other and get locked due to residual compressive stresses. These residual stresses might partially hold the crack surfaces together even when there is some external loading acting on the material resulting in crack closure phenomenon. This reduces both the stress intensity factor and fatigue crack growth rate, which in turn result in longer life for the material.

The crack closure can occur due to the corrosion deposits on the crack surfaces, roughness of the crack surfaces, and other effects.

Modified crack growth rate due to crack closure effect
To account for the crack closure effect, Walker suggested a modified form of the Paris' law which takes the following form

$${da \over dN} = C_{0}\Big(\overline{\Delta K}\Big)^{m} = C_{0}\bigg(\frac{(\Delta K)}{(1-R)^{1-\gamma}}\bigg)^{m} = C_{0}\big(K_{\text{max}}(1-R)^{\gamma}\big)^{m}$$

where, $$\gamma$$ is a material parameter which represents the influence of stress ratio on the fatigue crack growth rate. Typically, $$\gamma$$ takes a value around $$0.5$$, but can vary between $$0.3-1.0$$. In general, it is assumed that compressive portion of the loading cycle $$\big(R < 0\big)$$ have no effect on the crack growth by considering $$\gamma = 0,$$ which gives $$\overline{\Delta K} = K_{\text{max}}.$$ This can be physically explained by considering that the crack closes at zero load and doesn't behave like a crack under compressive loads which results in negligible effect on its growth. But, in very ductile materials like Man-Ten steel compressive loading does contribute to the crack growth according to $$\gamma = 0.22$$.

Comparison of the Walker equation with the Paris' equation will give

$$C = \frac{ C_{0}}{(1-R)^{m(1-\gamma)}}.$$

Both the Walker and Paris' equations are purely empirical.

Fatigue crack propagation in ductile and brittle materials
The general form of the fatigue-crack growth rate in ductile and brittle materials is given by

$${da \over dN} \propto (K_{\text{max}})^{n}(\Delta K)^{p},$$

where, $$n$$ and $$p$$ are material parameters. Based on different crack-advance and crack-tip shielding mechanisms in metals, ceramics, and intermetallics, it is observed that the fatigue crack growth rate in metals is significantly dependent on $$\Delta K$$ term, in ceramics on $$K_{\text{max}}$$, and intermetallics have almost similar dependence on $$\Delta K$$ and $$K_{\text{max}}$$ terms.

This can be summarized in a table I as

Analytical solution
The stress intensity factor is given by

$$K = \sigma \sqrt{\pi a} Y\Big(\frac{a}{W}\Big), $$

where $$\sigma $$ is the applied uniform tensile stress acting on the specimen in the direction perpendicular to the crack plane, $$a $$ and $$W $$represent the crack size and width of the specimen respectively, and $$Y\Big(\frac{a}{W}\Big) $$ is a dimensionless parameter that depends on the geometry of the specimen. The alternating stress intensity becomes

$$\begin{align} \Delta K &= \begin{cases} (\sigma_{\text{max}} - \sigma_{\text{min}}) Y \sqrt{\pi a} = \Delta \sigma Y \sqrt{\pi a},  \qquad R \geq 0 \\ \sigma_{\text{max}}Y \sqrt{\pi a}, \qquad R < 0 \end{cases}, \end{align} $$

where $$\Delta \sigma $$ is the range of the cyclic stress amplitude.

By assuming the initial crack size to be $$a_{0} $$, the critical crack size $$a_{c} $$ before the specimen fails can be computed using $$\big(K = K_{\text{max}} = K_{\text{IC}}\big) $$ as

$$\begin{align} K_{\text{IC}} &= \sigma_{\text{max}}Y\Big(\frac{a_{c}}{W}\Big)\sqrt{\pi a_{c}}, \\ \Rightarrow a_{c} &= \frac{1}{\pi}\bigg(\frac{K_{\text{IC}}}{\sigma_{\text{max}}Y\big(\frac{a_{c}}{W}\big)}\bigg)^2. \end{align} $$

The above equation in $$a_{c}$$is implicit in nature and can be solved numerically if necessary.

Case I
For $$R \geq 0.7,$$ crack closure has negligible effect on the crack growth rate and the Paris' law can be used to compute the fatigue life of a specimen before it reaches the critical crack size $$a_{c} $$ as

$$\begin{align} {da \over dN} &= C(\Delta K)^m = C\bigg( \Delta \sigma Y\Big(\frac{a}{W}\Big) \sqrt{\pi a}\bigg)^m, \\ \Rightarrow N_{f} &= \frac{1}{(\sqrt{\pi}\Delta \sigma)^{m}}\int_{a_{0}}^{a_{c}} \frac{da}{(C\sqrt{a}Y\Big(\frac{a}{W}\Big))^{m}}. \end{align} $$

Crack growth model with constant value of $$Y $$and R = 0
For Griffith-Irwin crack growth model or center crack of length $$2a $$ in an infinite sheet model as shown in the figure 2, we have $$Y=1 $$ and is independent of the crack length. Also, $$C $$ can be considered to be independent of the crack length. By assuming $$Y = \text{constant}, $$ the above integral simplifies to

$$N_{f} = \frac{1}{C(\sqrt{\pi}Y\Delta \sigma)^{m}}\int_{a_{0}}^{a_{c}} \frac{da}{(\sqrt{a})^{m}}, $$

by integrating the above expression for $$m \neq 2$$ and $$m =2$$ cases, the total number of load cycles $$N_{f}$$ are given by

$$\begin{align} N_{f} &= \frac{2}{(m-2)C(\sqrt{\pi}\Delta \sigma Y)^{m}}\Bigg[\frac{1}{(a_{0})^{\frac{m-2}{2}}} - \frac{1}{(a_{c})^{\frac{m-2}{2}}}\Bigg], \qquad m \neq 2, \\ N_{f} &= \frac{1}{\pi C (\Delta \sigma Y)^2 } \ln \frac{a_{c}}{a_{0}}, \qquad m = 2. \end{align}$$

Now, for $$m > 2 $$ and critical crack size to be very large in comparison to the initial crack size $$\big(a_{c} >> a_{0}\big) $$ will give

$$N_{f} = \frac{2}{(m-2)C(\sqrt{\pi}\Delta \sigma Y)^{m}}(a_{0})^{\frac{2-m}{2}}. $$

It is to be noted that the above analytical expressions for total number of load cycles to fracture $$\big(N_{f}\big) $$ are obtained by assuming $$Y = \text{constant} $$. For the cases, where $$Y $$ is dependent on the crack size like in Single Edge Notch Tension (SENT), Center Cracked Tension (CCT) and other crack growth models, it is convenient to perform numerical simulations to compute $$N_{f} $$.

Case II
For $$R < 0.7,$$ crack closure phenomenon has an effect on the crack growth rate and we can invoke Walker equation to compute the fatigue life of a specimen before it reaches the critical crack size $$a_{c} $$ as

$$\begin{align} {da \over dN} &= C\bigg(\frac{\Delta K}{(1-R)^{1-\gamma}}\bigg)^m = \frac{C}{(1-R)^{m(1-\gamma)}}\bigg( \Delta \sigma Y\Big(\frac{a}{W}\Big) \sqrt{\pi a}\bigg)^m, \\ \Rightarrow N_{f} &= \frac{(1-R)^{m(1-\gamma)}}{(\sqrt{\pi}\Delta \sigma)^{m}}\int_{a_{0}}^{a_{c}} \frac{da}{(C\sqrt{a}Y\Big(\frac{a}{W}\Big))^{m}}. \end{align} $$

Numerical simulation
This scheme is useful when $$Y$$ is dependent on the crack size $$a$$. The initial crack size is considered to be $$a_{0}$$. The stress intensity factor at the current crack size $$a$$ is computed using the maximum applied stress as $$\begin{align} K_{\text{max}} &= Y \sigma_{\text{max}} \sqrt{\pi a}. \end{align}$$ If $$K_{\text{max}}$$ is less than the fracture toughness $$K_{\text{IC}}$$, the crack has not reached its critical size $$a_{c}$$ and the simulation is continued with the current crack size to calculate the alternating stress intensity as

$$\Delta K = Y\Big(\frac{a}{W}\Big)\Delta \sigma \sqrt{\pi a}.$$

Now, by plugging the stress intensity factor in Paris' law, the increment in the crack size $$\Delta a $$ is computed as

$$\Delta a = C(\Delta K)^m \Delta N,$$

where $$\Delta N $$ is cycle step size. The new crack size becomes

$$a_{i+1} = a_{i} + \Delta a, $$

where index $$i $$ refers to the current iteration step. The new crack size is used to calculate the stress intensity at maximum applied stress for the next iteration. This iterative process is continued until

$$K_{\text{max}} \geq K_{\text{IC}}. $$

Once this failure criterion is met, the simulation is stopped.

The schematic representation of the fatigue life prediction process is shown in figure 3.

Example
The stress intensity factor in a SENT specimen (see, figure 4) under fatigue crack growth is given by$$\begin{align} K_{I} &= \sigma \sqrt{\pi a} \, Y\Big(\frac{a}{W}\Big)= \sigma \sqrt{\pi a}\Bigg[0.265\bigg[1 - \frac{a}{W}\bigg]^{4} + \frac{0.857 + 0.265 \frac{a}{W}}{\big[1 - \frac{a}{W}\big]^{\frac{3}{2}}}\Bigg], \\ \Delta K_{I} &= K_{\text{max}} - K_{\text{min}} = \Delta \sigma \sqrt{\pi a} \, Y\Big(\frac{a}{W}\Big). \end{align}$$

The following parameters are considered for the calculation $$\begin{align} a_0 &= 5 \text{mm}, \qquad W = 100 \text{mm}, \qquad h = 200 \text{mm}, \qquad K_{\text{IC}} = 30 \text{MPa}\sqrt{\text{m}}, \qquad R = \frac{K_{\text{min}}}{K_{\text{max}}} = 0.7, \\ \Delta \sigma &= 20 \text{MPa} \qquad C = 4.6774 \times 10^{-11} \frac{\text{m}}{\text{cycle}}\frac{1}{(\text{MPa}\sqrt{\text{m}})^m}, \qquad m = 3.874. \end{align}$$

The critical crack length, $$a = a_{c}$$, can be computed when $$K_{\text{max}} = K_{\text{IC}}$$ as

$$a_{c} = \frac{1}{\pi}\Bigg(\frac{0.45}{Y\Big(\frac{a_{c}}{W}\Big)}\Bigg)^2.$$

By solving the above equation, the critical crack length is obtained as $$a_{c} = 26.7 \, \text{mm}$$.

Now, invoking the Paris' law gives

$$N_{f} = \frac{1}{C (\Delta \sigma)^{m}(\sqrt{\pi})^{m}}\int_{a_{0}}^{a_{c}} \frac{da}{a^{\frac{m}{2}}\Bigg[0.265\bigg[1 - \frac{a}{W}\bigg]^{4} + \frac{0.857 + 0.265 \frac{a}{W}}{\big[1 - \frac{a}{W}\big]^{\frac{3}{2}}}\Bigg]^m} $$

By numerical integration of the above expression, the total number of load cycles to failure is obtained as $$N_{f} = 1.2085 \times 10^{6} \text{cycles}$$.