|
|
| Line 1: |
Line 1: |
| [[File:ParisLaw.png|thumb|Schematic plot of the typical relationship between the crack growth rate and the range of the stress intensity factor. In practice, the Paris law is calibrated to model the linear interval around the center.]]
| | Wilber Berryhill is the name his parents gave him and he completely digs that title. Credit authorising is how she makes a living. My spouse and I reside in Mississippi but now I'm considering other choices. To perform lacross is one of the things she enjoys most.<br><br>My weblog - online psychic reading - [http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ http://myfusionprofits.com/], |
| | |
| '''Paris' law''' (also known as the '''Paris-Erdogan law''') relates the [[stress intensity factor]] range to sub-critical crack growth under a [[Fatigue (material)|fatigue]] stress regime. As such, it is the most popular ''fatigue crack growth model'' used in [[materials science]] and [[fracture mechanics]]. The basic formula reads<ref name=plymouth>{{Cite web| title = The Paris law | work = Fatigue crack growth theory | publisher = [[University of Plymouth]] | date = | url =http://www.tech.plym.ac.uk/sme/tutorials/FMTut/Fatigue/FatTheory1.htm | accessdate = 21 June 2010 }}</ref>
| |
| :<math> \frac{{\rm d}a}{{\rm d}N} = C \Delta K^m </math>,
| |
| where ''a'' is the crack length and ''N'' is the number of load cycles. Thus, the term on the left side, known as the ''crack growth rate'',<ref name=convNaz>{{Cite web| last=M. Ciavarella | first=N. Pugno | title =A generalized law for fatigue crack growth | work =XXXIV Convegno Nazionale | publisher =''Associazione Italiana per l'analisi delle sollecitazioni'' | date =14–17 September 2005 | url=http://www.aiasonline.org/AIAS2005/Articoli/art007.pdf | format =PDF | accessdate =21 July 2010 }}</ref> denotes the [[infinitesimal]] crack length growth per increasing number of load cycles. On the right hand side, ''C'' and ''m'' are material constants, and <math>\Delta K</math> is the range of the stress intensity factor, i.e., the difference between the stress intensity factor at maximum and minimum loading
| |
| :<math> \Delta K = K_{max}-K_{min}</math>,
| |
| where <math>K_{max}</math> is the maximum stress intensity factor and <math>K_{min}</math> is the minimum stress intensity factor.<ref name=mit>{{Cite web| last =Roylance | first =David | title =Fatigue | publisher =Department of Materials Science and Engineering, Massachusetts Institute of Technology | date =1 May 2001 | url =http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/fatigue.pdf | format =PDF | accessdate =23 July 2010 }}</ref>
| |
| | |
| ==History and use==
| |
| The formula was introduced by P.C. Paris in 1961.<ref>P.C. Paris, M.P. Gomez, and W.E. Anderson. A rational analytic theory of fatigue. ''The Trend in Engineering'', 1961, '''13''': p. 9-14.</ref> 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.
| |
| | |
| Paris' law can be used to quantify the residual life (in terms of load cycles) of a specimen given a particular crack size. Defining the crack intensity factor as
| |
| :<math> K=\sigma Y \sqrt{\pi a} </math>,
| |
| where ''<math>\sigma</math>'' is a uniform tensile stress perpendicular to the crack plane and ''Y'' is a dimensionless parameter that depends on the geometry, the range of the stress intensity factor follows as
| |
| :<math> \Delta K=\Delta\sigma Y \sqrt{\pi a} </math>,
| |
| where <math>\Delta\sigma</math> is the range of cyclic stress amplitude. ''Y'' takes the value 1 for a center crack in an infinite sheet. The remaining cycles can be found by substituting this equation in the Paris law
| |
| :<math> \frac{{\rm d}a}{{\rm d}N} = C \Delta K^m =C(\Delta\sigma Y \sqrt{\pi a})^m </math>.
| |
| For relatively short cracks, ''Y'' can be assumed as independent of ''a'' and the [[differential equation]] can be solved via [[separation of variables]]
| |
| :<math>\int^{N_f}_0 {\rm d}N = \int^{a_c}_{a_i}\frac{{\rm d}a}{C(\Delta\sigma Y \sqrt{\pi a})^m } =\frac{1}{C(\Delta\sigma Y \sqrt{\pi})^m }\int^{a_c}_{a_i} a^{-\frac{m}{2}}\;{\rm d}a </math>
| |
| and subsequent [[integral|integration]]
| |
| :<math>N_f=\frac{2\;(a_c^{\frac{2-m}{2}}-a_i^{\frac{2-m}{2}})}{(2-m)\;C(\Delta\sigma Y \sqrt{\pi})^m}</math>,
| |
| where <math>N_f</math> is the remaining number of cycles to fracture, <math>a_c</math> is the critical crack length at which instantaneous fracture will occur, and <math>a_i</math> is the initial crack length at which fatigue crack growth starts for the given stress range <math>\Delta\sigma</math>. If ''Y'' strongly depends on ''a'', numerical methods might be required to find reasonable solutions.
| |
| | |
| For the application to adhesive joints in composites, it is more useful to express the Paris Law in terms of fracture energy rather than stress intensity factors.<ref>Wahab, M.M.A., I.A. Ashcroft, A.D. Crocombe, and P.A. Smith, Fatigue crack propagation in adhesively bonded joints. ''Key Engineering Materials'', 2003, '''251-252''': p. 229-234</ref>
| |
| | |
| ==References==
| |
| <references/>
| |
| {{Use dmy dates|date=September 2010}}
| |
| | |
| {{DEFAULTSORT:Paris' Law}}
| |
| [[Category:Materials science]]
| |
| [[Category:Fracture mechanics]]
| |
Wilber Berryhill is the name his parents gave him and he completely digs that title. Credit authorising is how she makes a living. My spouse and I reside in Mississippi but now I'm considering other choices. To perform lacross is one of the things she enjoys most.
My weblog - online psychic reading - http://myfusionprofits.com/,