Miner's Rule In 1945, M A Miner popularised popularised a rule that had first been proposed by A. by A. Palmgren in 1924. The rule, ariously !alled Miner's rule or rule or the Palmgren-Miner linear damage hypothesis, hypothesis , states that "here there are k different different stress magnitudes in a spe!trum, Si #1 #1 $ i $ $ k %, %, ea!h !ontributing ni #Si % !y!les, then if N i i# Si % is the number of !y!les to failure of a !onstant stress reersal Si , failure o!!urs "hen&
' is e(perimentally found to be bet"een ).* and 2.2. +sually for design purposes, ' is assumed to be 1. This !an be thought of as assessing "hat proportion of life is !onsumed by a linear !ombination of stress reersals at arying magnitudes.
Though Miners rule is a useful appro(imation in many !ir!umstan!es, it has seeral ma-or limitations&
1.
It fails to to re!ognise the probabilisti! nature of fatigue and there is no simple "ay "ay to relate relate life predi!ted predi!ted by the the rule "ith the !hara!teristi!s of a probability distribution. Industry analysts often use design !ures, ad-usted to a!!ount for s!atter, to !al!ulate N i i# Si %. %.
2.
There is sometimes sometimes an effe!t in in the order in "hi!h the the reersals o!!ur. In some !ir!umstan! !ir!umstan!es, es, !y!les of lo" stress stress follo"ed by high stress !ause more damage than "ould be predi!ted by the rule. It does not !onsider the effe!t of an oerload or high stress "hi!h may result in a !ompressie residual stress that may retard !ra! gro"th. /igh stress follo"ed by lo" stress may hae less damage due to the presen!e of !ompressie residual stress.
Paris' Law
Typi!al fatigue !ra! gro"th rate graph
In 0ra!ture me!hani!s, me!hani!s, Anderson, ome and Paris deried relationships for the stage II !ra! gro"th "ith !y!les 3, in terms of the !y!li!al !omponent of the 6tress Intensity 0a!tor 0a!tor 718
"here a is the !ra! length and m is typi!ally in the range 8 to 5 #for metals%.
This relationship "as later modified #by 0orman, 19:* 714% to mae better allo"an!e for the mean stress, by introdu!ing a fa!tor depending on #1;<% "here < = min stress>ma( stress, in the denominator.
Goodman Relation In the presen!e of a steady stress superimposed on the !y!li! loading, the oodman relation !an be used to estimate a failure !ondition. It plots stress amplitude against mean stress "ith the fatigue limit and the ultimate tensile strength of strength of the material as the t"o e(tremes. Alternatie Alternatie failure !riteria in!lude 6oderberg and erber . erber .715
Low-cycle fatigue ?here the stress is high enough for plasti! deformation to o!!ur, the a!!ounting of the loading in terms of stress is less useful and the strain in the material offers a simpler and more a!!urate des!ription. This type of fatigue is normally e(perien!ed by !omponents "hi!h undergo a relatiely small number of straining !y!les. @o";!y!le fatigue is usually !hara!terised by the Coffin-Manson relation #published independently by @. 0. 'offin in 1954 and 6. 6. Manson 1958%&
"here,
•
p >2 is the plasti! strain amplitudeB
•
f is an empiri!al !onstant no"n as the fatigue ductility coefficient , the failure strain for a single reersalB
•
2N is the number of reersals to failure # N !y!les%B
•
c is an empiri!al !onstant no"n as the fatigue ductility exponent , !ommonly ranging from ;).5 to ;).* for metals in time independent fatigue. 6lopes !an be !onsiderably steeper in the presen!e of !reep or enironmental intera!tions.
A similar relationship for materials su!h as Cir!onium, is used in the nu!lear industry.
Fatigue and fracture mechanics The a!!ount aboe is purely empiri!al and, though it allo"s life predi!tion and design assuran!e, life improement or design optimisation !an be enhan!ed u sing 0ra!ture me!hani!s. It !an be deeloped in four stages.
1.
'ra! nu!leationB
2.
6tage I !ra!;gro"thB
8.
6tage II !ra!;gro"thB and
4.
+ltimate du!tile failure.
Design against fatigue Dependable design against fatigue;failure reEuires thorough edu!ation and superised e(perien!e in stru!tural engineering, me!hani!al engineering, or materials s!ien!e. There are four prin!ipal approa!hes to life assuran!e for me!hani!al parts that display in!reasing degrees of sophisti!ation&71*
1.
Design to eep stress belo" threshold of fatigue limit #infinite lifetime !on!ept%B
2.
fail;safe, gra!eful degradation, and fault;tolerant design& Instru!t the user to repla!e parts "hen they fail. Design in su!h a "ay that there is no single point of failure, and so that "hen any one part !ompletely fails, it does not lead to !atastrophi! failure of the entire system.
8.
6afe;life design& Design #!onseratiely% for a fi(ed life after "hi!h the user is instru!ted to repla!e the part "ith a ne" one #a so;!alled lifed part, finite lifetime !on!ept, or Fsafe;lifeF design pra!ti!e%B planned
obsoles!en!e and disposable produ!t are ariants that design for a fi(ed life after "hi!h the user is instru!ted to repla!e the entire dei!eB
4.
damage tolerant design& Instru!t the user to inspe!t the part periodi!ally for !ra!s and to repla!e the part on!e a !ra! e(!eeds a !riti!al length. This approa!h usually uses the te!hnologies of nondestru!tie testing and reEuires an a!!urate predi!tion of the rate of !ra!;gro"th bet"een inspe!tions. The designer sets some air!raft maintenan!e !he!s s!hedule freEuent enough that parts are repla!ed "hile the !ra! is still in the Fslo" gro"thF phase. This is often referred to as damage tolerant design or Fretirement;for;!auseF.
