Paris' Law

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The Paris' law is a power-law relationship which relates the crack growth rate   to the stress intensity factor range or alternating stress intensity   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.

 

where   is the difference between the maximum and minimum stress intensity factors for each cycle, and   and   are experimentally determined material constants. The alternating stress intensity at the critical limit is given by  as shown in the figure 1 [1].

It is to be noted that the effect of stress ratio is not included in the Paris' equation  . The Paris' law does not hold for very low values of  approaching the threshold value  , and for very high values approaching the material's fracture toughness,  . The slope of the crack growth rate curve on log-log scale denotes the value of the exponent   and in general is found to lie between the range   to  . But for the materials of low static fracture toughness like high strength steels, the value of   can shoot up to a value of  .

It is important to note that Paris' law is valid only in linear elastic fracture regime, under uniaxial loading and for long cracks[2].

Barenblatt and Botvina[3] observed that the values of constants   and   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   and   is observed by Alberto[5]. These relations are validated by Radhakrishnan[6] 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   is low in comparison to the crack length,   (here,   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[10].

History of crack propagation laws

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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[11]. 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[12], Frost and Dugdale[13], McEvily and Illg[14], and Liu[15] on fatigue crack-growth behaviour laid the foundation in this topic. The general form for the above crack propagation laws may be expressed as[16]

 

where, half of the crack length is denoted by  , number of cycles of load applied is given by  , the stress range by  , and the material parameters by  .

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[17], Gomez[18], and Anderson[19] have proposed an empirical law which fits the broad trend of data.

Crack growth rate in different regimes

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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[20].

To predict the crack growth rate at near threshold region, the following relation[21] is proposed

 

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

 

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[22] proposed the following relation to predict the crack propagation behaviour

 

Also, McEvily and Groeger[23] proposed the following power-law relationship which considers the effects of both high and low values of  

 

here, we notice that the value of exponent is  . 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

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Practically, it is observed that the load or stress ratio  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[24].

The crack closure can occur due to the corrosion deposits on the crack surfaces[25], roughness of the crack surfaces, and other effects[26].

Modified crack growth rate due to crack closure effect

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To account for the crack closure effect, Walker[27] suggested a modified form of the Paris' law which takes the following form

 

where,   is a material parameter which represents the influence of stress ratio on the fatigue crack growth rate. Typically,   takes a value around  , but can vary between  . In general, it is assumed that compressive portion of the loading cycle   have no effect on the crack growth by considering   which gives   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[28] compressive loading does contribute to the crack growth according to  .

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

 

Both the Walker and Paris' equations are purely empirical.

Fatigue crack propagation in ductile and brittle materials

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The general form of the fatigue-crack growth rate in ductile and brittle materials[29] is given by

 

where,   and   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   term, in ceramics on  , and intermetallics have almost similar dependence on   and   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  
Metals  
Intermetallics  
Ceramics  

Prediction of fatigue life

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Analytical solution

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The stress intensity factor is given by

 

where   is the applied uniform tensile stress acting on the specimen in the direction perpendicular to the crack plane,   and  represent the crack size and width of the specimen respectively, and   is a dimensionless parameter that depends on the geometry of the specimen. The alternating stress intensity becomes

 

where   is the range of the cyclic stress amplitude.

By assuming the initial crack size to be  , the critical crack size   before the specimen fails can be computed using   as

 

The above equation in  is implicit in nature and can be solved numerically if necessary.

Case I

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For   crack closure has negligible effect on the crack growth rate[30] and the Paris' law can be used to compute the fatigue life of a specimen before it reaches the critical crack size   as

 

Crack growth model with constant value of  and R = 0
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Figure 2: Geometrical representation of Center Cracked Tension test specimen

For Griffith-Irwin crack growth model or center crack of length   in an infinite sheet model as shown in the figure 2, we have   and is independent of the crack length. Also,   can be considered to be independent of the crack length. By assuming   the above integral simplifies to

 

by integrating the above expression for   and   cases, the total number of load cycles   are given by

 

Now, for   and critical crack size to be very large in comparison to the initial crack size   will give

 

It is to be noted that the above analytical expressions for total number of load cycles to fracture   are obtained by assuming  . For the cases, where   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  .

Case II

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For   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   as

 

Numerical simulation

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This scheme is useful when   is dependent on the crack size  . The initial crack size is considered to be  . The stress intensity factor at the current crack size   is computed using the maximum applied stress as

 
Cite error: The opening <ref> tag is malformed or has a bad name (see the help page).Figure 3: Schematic representation of fatigue life prediction process[31]

 
If   is less than the fracture toughness  , the crack has not reached its critical size   and the simulation is continued with the current crack size to calculate the alternating stress intensity as

 

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

 

where   is cycle step size. The new crack size becomes

 

where index   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

 

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

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The stress intensity factor in a SENT specimen (see, figure 4) under fatigue crack growth[32] is given by 

The following parameters are considered for the calculation

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

 

The critical crack length,  , can be computed when   as

 

By solving the above equation, the critical crack length is obtained as  .

Now, invoking the Paris' law gives

 

By numerical integration of the above expression, the total number of load cycles to failure is obtained as  .

See also

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References

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  1. ^ Ritchie, R.O; Knott, J.F (May 1973). "Mechanisms of fatigue crack growth in low alloy steel". Acta Metallurgica. 21 (5): 639–648. doi:10.1016/0001-6160(73)90073-4. ISSN 0001-6160.
  2. ^ http://www.am.chalmers.se/~anek/teaching/fatfract/98-8.pdf. {{cite web}}: Missing or empty |title= (help)
  3. ^ BARENBLATT, G. I.; BOTVINA, L. R. (July 1980). "Incomplete Self-Similarity of Fatigue in the Linear Range of Crack Growth". Fatigue & Fracture of Engineering Materials and Structures. 3 (3): 193–202. doi:10.1111/j.1460-2695.1980.tb01359.x. ISSN 8756-758X.
  4. ^ RITCHIE, R. O. (April 2005). "Incomplete self-similarity and fatigue-crack growth". International Journal of Fracture. 132 (3): 197–203. doi:10.1007/s10704-005-2266-y. ISSN 0376-9429. S2CID 17045951.
  5. ^ Carpinteri, Alberto; Paggi, Marco (October 2009). "A unified interpretation of the power laws in fatigue and the analytical correlations between cyclic properties of engineering materials". International Journal of Fatigue. 31 (10): 1524–1531. doi:10.1016/j.ijfatigue.2009.04.014. ISSN 0142-1123.
  6. ^ Radhakrishnan, V.M. (January 1980). "Quantifying the parameters in fatigue crack propagation". Engineering Fracture Mechanics. 13 (1): 129–141. doi:10.1016/0013-7944(80)90048-x. ISSN 0013-7944.
  7. ^ Carpinteri, Alberto; Paggi, Marco (October 2009). "A unified interpretation of the power laws in fatigue and the analytical correlations between cyclic properties of engineering materials". International Journal of Fatigue. 31 (10): 1524–1531. doi:10.1016/j.ijfatigue.2009.04.014. ISSN 0142-1123.
  8. ^ PUGNO, N; CIAVARELLA, M; CORNETTI, P; CARPINTERI, A (July 2006). "A generalized Paris' law for fatigue crack growth". Journal of the Mechanics and Physics of Solids. 54 (7): 1333–1349. doi:10.1016/j.jmps.2006.01.007. ISSN 0022-5096.
  9. ^ Radhakrishnan, V.M. (January 1980). "Quantifying the parameters in fatigue crack propagation". Engineering Fracture Mechanics. 13 (1): 129–141. doi:10.1016/0013-7944(80)90048-x. ISSN 0013-7944.
  10. ^ Suresh, S. (Subra) (1998). Fatigue of materials. Cambridge University Press. ISBN 0521570468. OCLC 38732204.
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  12. ^ Head, A.K. (September 1953). "XCVIII. The growth of fatigue cracks". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 44 (356): 925–938. doi:10.1080/14786440908521062. ISSN 1941-5982.
  13. ^ Frost, N.E.; Dugdale, D.S. (January 1958). "The propagation of fatigue cracks in sheet specimens". Journal of the Mechanics and Physics of Solids. 6 (2): 92–110. doi:10.1016/0022-5096(58)90018-8. ISSN 0022-5096.
  14. ^ McEvily, Arthur J.; Illg, Walter (1960), "A Method for Predicting the Rate of Fatigue-Crack Propagation", Symposium on Fatigue of Aircraft Structures, ASTM International, pp. 112–112–8, doi:10.1520/stp45927s, ISBN 9780803165793, retrieved 2019-05-04
  15. ^ Liu, H. W. (1961). "Crack Propagation in Thin Metal Sheet Under Repeated Loading". Journal of Basic Engineering. 83 (1): 23–31. doi:10.1115/1.3658886. hdl:2142/111864. ISSN 0021-9223.
  16. ^ Anderson, W. E. (1963). "Discussion: "A Critical Analysis of Crack Propagation Laws" (Paris, P., and Erdogan, F., 1963, ASME J. Basic Eng., 85, pp. 528–533)". Journal of Basic Engineering. 85 (4): 533. doi:10.1115/1.3656901. ISSN 0021-9223.
  17. ^ Paris, Paul C. (1962). "Discussion: "Crack Propagation in Aluminum-Foil Laminates" (Mote, Jr., C. D., and Frisch, J., 1962, ASME J. Basic Eng., 84, pp. 257–264)". Journal of Basic Engineering. 84 (2): 264. doi:10.1115/1.3657299. ISSN 0021-9223.
  18. ^ Paris, Paul C., Mario P. Gomez, and William E. Anderson. "A rational analytic theory of fatigue." (1997).
  19. ^ Anderson, W. E., and P. C. Paris. "Evaluation of aircraft material by fracture." Metals Engineering Quarterly 1.2 (1961): 33.
  20. ^ Ritchie, R. O. (1977). "Near-Threshold Fatigue Crack Propagation in Ultra-High Strength Steel: Influence of Load Ratio and Cyclic Strength". Journal of Engineering Materials and Technology. 99 (3): 195–204. doi:10.1115/1.3443519. ISSN 0094-4289.
  21. ^ Allen, R. J.; Booth, G. S.; Jutla, T. (March 1988). "A Review of Fatigue Crack Growth Characterisation by Linear Elastic Fracture Mechanics (Lefm). Part Ii?Advisory Documents and Applications within National Standards". Fatigue & Fracture of Engineering Materials and Structures. 11 (2): 71–108. doi:10.1111/j.1460-2695.1988.tb01162.x. ISSN 8756-758X.
  22. ^ Forman, R. G.; Kearney, V. E.; Engle, R. M. (1967). "Numerical Analysis of Crack Propagation in Cyclic-Loaded Structures". Journal of Basic Engineering. 89 (3): 459. doi:10.1115/1.3609637. ISSN 0021-9223.
  23. ^ McEVILY, A.J.; Groeger, J. (1978), "On the Threshold for Fatigue Crack Growth", Advances in Research on the Strength and Fracture of Materials, Elsevier, pp. 1293–1298, doi:10.1016/b978-0-08-022140-3.50087-2, ISBN 9780080221403, retrieved 2019-05-04
  24. ^ Elber, W (1971), "The Significance of Fatigue Crack Closure", Damage Tolerance in Aircraft Structures, ASTM International, pp. 230–230–13, doi:10.1520/stp26680s, ISBN 9780803100312, retrieved 2019-05-04
  25. ^ Suresh, S.; Zamiski, G. F.; Ritchie, D R. O. (August 1981). "Oxide-Induced Crack Closure: An Explanation for Near-Threshold Corrosion Fatigue Crack Growth Behavior". Metallurgical and Materials Transactions A. 12 (8): 1435–1443. doi:10.1007/bf02643688. ISSN 1073-5623. S2CID 137261193.
  26. ^ Newman, JC; Elber, W, eds. (1988-01-01). Mechanics of Fatigue Crack Closure. doi:10.1520/stp982-eb. ISBN 978-0-8031-0996-4.
  27. ^ Walker, K (1970), "The Effect of Stress Ratio During Crack Propagation and Fatigue for 2024-T3 and 7075-T6 Aluminum", Effects of Environment and Complex Load History on Fatigue Life, ASTM International, pp. 1–1–14, doi:10.1520/stp32032s, ISBN 9780803100329, retrieved 2019-05-04
  28. ^ Dowling, Norman E. (2012). Mechanical behavior of materials : engineering methods for deformation, fracture, and fatigue. Pearson. ISBN 978-0131395060. OCLC 1055566537.
  29. ^ Ritchie, R.O. (1999-11-01). "Mechanisms of fatigue-crack propagation in ductile and brittle solids". International Journal of Fracture. 100 (1): 55–83. doi:10.1023/A:1018655917051. ISSN 1573-2673. S2CID 13991702.
  30. ^ Zehnder, Alan T. (2012). Fracture Mechanics. Lecture Notes in Applied and Computational Mechanics. Vol. 62. Dordrecht: Springer Netherlands. doi:10.1007/978-94-007-2595-9. ISBN 9789400725942.
  31. ^ https://mechanicalc.com/reference/fatigue-crack-growth. {{cite web}}: Missing or empty |title= (help)
  32. ^ Tada, Hiroshi; Paris, Paul C.; Irwin, George R. (2000-01-01). The Stress Analysis of Cracks Handbook, Third Edition. Three Park Avenue New York, NY 10016-5990: ASME. doi:10.1115/1.801535. ISBN 0791801535.{{cite book}}: CS1 maint: location (link)


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