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

Figure 1: Geometrical representation of crack growth with respect to the alternate stress intensity, along with the representation of Paris' curve in the linear region of Regime B.

${\begin{aligned}{da \over dN}&=C\left(\Delta K\right)^{m},\qquad (\Delta K_{\text{th}}<\Delta K<K_{\text{IC}}),\end{aligned}}$

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 ${\begin{aligned}\Delta K_{\text{cr}}&=(1-R)K_{\text{IC}}\end{aligned}}$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.

Dependence of the parameters $C$ and $m$ on the characteristic specimen size

Following the procedure adopted by Ritchie, the average crack growth rate is considered as ${da \over dN}={\bigg (}{\frac {\Delta K}{\sigma _{\text{y}}}}{\bigg )}^{2}\Phi {\bigg (}{\frac {\Delta K}{K_{\text{IC}}}},R,Z,\nu t{\bigg )},$ where, average crack growth rate is dependent on (i) the alternating stress intensity $\Delta K$, (ii) material properties like yield stress $\sigma _{\text{y}}$ and the fracture toughness $K_{\text{IC}}$, (iii) load ratio, $R={\frac {K_{\text{min}}}{K_{\text{max}}}}$, (iv) basic similarity parameter, $Z={\frac {\sigma _{\text{y}}{\sqrt {h}}}{K_{\text{IC}}}}$ with $h$ being the characteristic specimen length scale, and (v) cyclic frequency $\nu$ and time $t$. By neglecting the environmental effects, $\nu t$ can be ignored and by considering the mid-range values of growth rates ${\frac {\Delta K}{K_{\text{IC}}}}<<1$ along with the similarity of the first kind or complete similarity, the power law expression for fatigue crack-growth can be written as ${da \over dN}={\bigg (}{\frac {\Delta K}{\sigma _{\text{y}}}}{\bigg )}^{2}\Phi {\big (}R,Z{\big )},$ which predicts that the value of exponent to be $m=2$ and is suitable for most of the perfectly plastic models. But, if we consider the similarity of the second kind or incomplete similarity along with ${\frac {\Delta K}{K_{\text{IC}}}}<<1$, the power-law expression for fatigue crack-growth can be rewritten as ${da \over dN}={\frac {(\Delta K)^{2+\alpha }}{\sigma _{y}^{2}K_{\text{IC}}^{\alpha }}}\Phi _{1}\left(R,Z\right),\qquad ({\text{where}}\,\Phi {\big (}R,Z{\big )}={\bigg (}{\frac {\Delta K}{K_{\text{IC}}}}{\bigg )}^{\alpha }\Phi _{1}{\big (}R,Z{\big )}),$ which represents the Paris' law with $C={\frac {\Phi _{1}{\big (}R,Z{\big )}}{\sigma _{y}^{2}K_{\text{IC}}^{\alpha }}},\qquad m=2+\alpha \left(R,Z\right),$ where, $\alpha$ is a function of $R$ and $Z$ and we observe that $C$ and $Z$ are dependent on $h$ through the similarity parameter, $Z$. By comparison with the experiments on low to high strength steels, it is noted these relations are justified. It is also observed that $C$ varies proportionally with the characteristic specimen height $h$ and $m$ varies with ${\sqrt {h}}$.

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.

Correlation between the parameters $C$ and $m$

A similar procedure adopted by Alberto is presented here to find the correlation between the parameters $C$and $m$. 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 $\Delta K=\Delta K_{\text{th}}$ along with the corresponding threshold crack growth rate of $V_{\text{th}}=10^{-9}{\frac {\text{mm}}{\text{cycle}}}$and $\Delta K=\Delta K_{\text{cr}}=\left(1-R\right)K_{\text{IC}}$ along with the corresponding critical crack growth rate of $V_{\text{cr}}=10^{-5}{\frac {\text{mm}}{\text{cycle}}}$. The law predicts a better fatigue life for the crack growth rate in the range of $10^{-8}-10^{-6}{\frac {\text{mm}}{\text{cycle}}}$. Plugging the critical and threshold limit points into the Paris' law leads to ${\begin{aligned}C&={\frac {V_{\text{cr}}}{{\Big [}\Delta K_{\text{cr}}{\Big ]}^{m}}}={\frac {V_{\text{cr}}}{{\Big [}{\big (}1-R{\big )}K_{\text{IC}}{\Big ]}^{m}}},\qquad \log C=\log {V_{\text{cr}}}+m\log {\Bigg [}{\frac {1}{{\big [}{\big (}1-R{\big )}K_{\text{IC}}{\big ]}}}{\Bigg ]},\\C&={\frac {V_{\text{th}}}{{\Big [}\Delta K_{\text{th}}{\Big ]}^{m}}},\qquad \qquad \qquad \qquad \qquad \log C=\log {V_{\text{th}}}+m\log {\Bigg [}{\frac {1}{{\big [}\Delta K_{\text{th}}{\big ]}}}{\Bigg ]},\\\Rightarrow m&={\frac {\log V_{\text{th}}-\log V_{\text{cr}}}{{\bigg [}\log \Delta K_{\text{th}}-\log {\Big [}{\big (}1-R{\big )}K_{\text{IC}}{\Big ]}{\bigg ]}}},\qquad \log {\bigg (}{\frac {\Delta K_{\text{th}}}{K_{\text{IC}}}}{\bigg )}=\log(1-R)+{\frac {1}{m}}\log {\bigg (}{\frac {V_{\text{th}}}{V_{\text{cr}}}}{\bigg )}.\end{aligned}}$ The above correlations are validated by Radhakrishnan for steels and aluminium alloys and proposed the following least square fit relationships which make a good agreement with the experimental data. ${\begin{aligned}\log C&=\log(7.6\times 10^{-7})+m\log(1.81\times 10^{-2}),\qquad {\text{for steels}},\\\log C&=\log(2.5\times 10^{-6})+m\log(4.26\times 10^{-2}),\qquad {\text{for aluminium alloys}}.\end{aligned}}$

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

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

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

Table I: Comparison of fatigue crack growth properties of ductile, intermetallics, and brittle materials

Material	Crack-growth rate exponents $(p,n)$
Metals	$p>>n$
Intermetallics	$p\approx n$
Ceramics	$p<<n$

Prediction of fatigue life

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

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

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

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

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

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

Crack growth model with constant value of $Y$and R = 0

Figure 2: Geometrical representation of Center Cracked Tension test specimen

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

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

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

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

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

Figure 3: Schematic representation of fatigue life prediction process

${\begin{aligned}K_{\text{max}}&=Y\sigma _{\text{max}}{\sqrt {\pi a}}.\end{aligned}}$
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${\begin{aligned}K_{I}&=\sigma {\sqrt {\pi a}}\,Y{\Big (}{\frac {a}{W}}{\Big )}=\sigma {\sqrt {\pi a}}{\Bigg [}0.265{\bigg [}1-{\frac {a}{W}}{\bigg ]}^{4}+{\frac {0.857+0.265{\frac {a}{W}}}{{\big [}1-{\frac {a}{W}}{\big ]}^{\frac {3}{2}}}}{\Bigg ]},\\\Delta K_{I}&=K_{\text{max}}-K_{\text{min}}=\Delta \sigma {\sqrt {\pi a}}\,Y{\Big (}{\frac {a}{W}}{\Big )}.\end{aligned}}$

The following parameters are considered for the calculation

Figure 4: Geometrical representation of Single Edge Notch Tension test specimen

${\begin{aligned}a_{0}&=5{\text{mm}},\qquad W=100{\text{mm}},\qquad h=200{\text{mm}},\qquad K_{\text{IC}}=30{\text{MPa}}{\sqrt {\text{m}}},\qquad R={\frac {K_{\text{min}}}{K_{\text{max}}}}=0.7,\\\Delta \sigma &=20{\text{MPa}}\qquad C=4.6774\times 10^{-11}{\frac {\text{m}}{\text{cycle}}}{\frac {1}{({\text{MPa}}{\sqrt {\text{m}}})^{m}}},\qquad m=3.874.\end{aligned}}$

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 $N_{f}=1.2085\times 10^{6}{\text{cycles}}$.

See also

• Stress intensity factor
• Fracture toughness
• Fatigue (material)
• Nucleation
• Microstructure
• Corrosion fatigue
• Ductility
• Brittleness
• Intermetallic

References

Category:Fracture mechanics Category:Mechanical failure Category:Mechanical failure modes Category:Solid mechanics