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J Integral Crack Growth Region ->>> http://tinyurl.com/y7q4tnv7
Since.the.total.integral.around.the.crack.vanishes.and.the.contributions.along.the.surface.of.the.crack.are.zero,.we.haveFor.Mode.II.loading,.the.relation.between.the.J-integral.and.the.mode.II.fracture.toughness.(.K.I.I.c.{displaystyle.K{rm.{IIc}}}.).isThe.quantity.J.is.not.path-independent.for.monotonic.mode.I.and.mode.II.loading.of.elastic-plastic.materials,.so.only.a.contour.very.close.to.the.crack.tip.gives.the.energy.release.rateLet..us..just..consider..the..expression..for..J..1..{displaystyle..J{1}}..which..is..J..1..:=..∫..Γ..(..W..n..1..−..n..j..σ..j..k....∂..u..k..∂..x..1..)..d..Γ..{displaystyle..J{1}:=int..{Gamma..}left(Wn{1}-n{j}sigma..{jk}{cfrac..{partial..u{k}}{partial..x{1}}}right),dGamma..}..We..can..write..this..as..J..1..=..∫..Γ..(..W..δ..1..j..−..σ..j..k....∂..u..k..∂..x..1..)..n..j..d..Γ..{displaystyle..J{1}=int..{Gamma..}left(Wdelta..{1j}-sigma..{jk}{cfrac..{partial..u{k}}{partial..x{1}}}right)n{j},dGamma..}..From..Green's..theorem..(or..the..two-dimensional..divergence..theorem)..we..have..∫..Γ..f..j....n..j....d..Γ..=..∫..A..∂..f..j..∂..x..j....d..A..{displaystyle..int..{Gamma..}f{j}n{j}dGamma..=int..{A}{cfrac..{partial..f{j}}{partial..x{j}}}dA}..Using..this..result..we..can..express..J..1..{displaystyle..J{1}}..as..J..1..=..∫..A..∂..∂..x..j..(..W..δ..1..j..−..σ..j..k....∂..u..k..∂..x..1..)..d..A..=..∫..A..[..∂..W..∂..x..1..−..∂..σ..j..k..∂..x..j....∂..u..k..∂..x..1..−..σ..j..k....∂..2..u..k..∂..x..1..∂..x..j..]....d..A..{displaystyle..{begin{aligned}J{1}&=int..{A}{cfrac..{partial..}{partial..x{j}}}left(Wdelta..{1j}-sigma..{jk}{cfrac..{partial..u{k}}{partial..x{1}}}right)dA&=int..{A}left[{cfrac..{partial..W}{partial..x{1}}}-{cfrac..{partial..sigma..{jk}}{partial..x{j}}}{cfrac..{partial..u{k}}{partial..x{1}}}-sigma..{jk}{cfrac..{partial..^{2}u{k}}{partial..x{1}partial..x{j}}}right]dAend{aligned}}}..where..A..{displaystyle..A}..is..the..area..enclosed..by..the..contour..Γ..{displaystyle..Gamma..}..Alan.Zehnder.(from.Cornell.University).Nonlinear.Fracture.Mechanics.Notes.by.ProfRice,.."A..Path..Independent..Integral..and..the..Approximate..Analysis..of..Strain..Concentration..by..Notches..and..Cracks",..Journal..of..Applied..Mechanics,..35,..1968,..pp.379386σ..i..j..=..σ..y..(..E..J..r..α..σ..y..2..I..)..1..n..+..1..σ....i..j..(..n..,..θ..)..{displaystyle..sigma..{ij}=sigma..{y}left({frac..{EJ}{r,alpha..sigma..{y}^{2}I}}right)^{{1}..over..{n+1}}{tilde..{sigma..}}{ij}(n,theta..)}..ε..i..j..=..α..ε..y..E..(..E..J..r..α..σ..y..2..I..)..n..n..+..1..ε....i..j..(..n..,..θ..)..{displaystyle..varepsilon..{ij}={frac..{alpha..varepsilon..{y}}{E}}left({frac..{EJ}{r,alpha..sigma..{y}^{2}I}}right)^{{n}..over..{n+1}}{tilde..{varepsilon..}}{ij}(n,theta..)}..Paths..for..J-integral..calculation..around..a..notch..in..a..two-dimensional..elastic-plastic..material
The..quantity..JIc..defines..the..point..at..which..large-scale..plastic..yielding..during..propagation..takes..place..under..mode..one..loading.[1][5]Hence,.σ.j.k.∂.ε.j.k.∂.x.1.=.σ.j.k.∂.2.u.j.∂.x.1.∂.x.k.{displaystyle.sigma.{jk}{cfrac.{partial.varepsilon.{jk}}{partial.x{1}}}=sigma.{jk}{cfrac.{partial.^{2}u{j}}{partial.x{1}partial.x{k}}}}.The.J-integral.may.then.be.written.as.J.1.=.∫.A.[.∂.W.∂.x.1.−.σ.j.k..∂.ε.j.k.∂.x.1.]..d.A.{displaystyle.J{1}=int.{A}left[{cfrac.{partial.W}{partial.x{1}}}-sigma.{jk}{cfrac.{partial.varepsilon.{jk}}{partial.x{1}}}right]dA}.Now,.for.an.elastic.material.the.stress.can.be.derived.from.the.stored.energy.function.W.{displaystyle.W}.using.σ.j.k.=.∂.W.∂.ε.j.k.{displaystyle.sigma.{jk}={cfrac.{partial.W}{partial.varepsilon.{jk}}}}.Then,.if.the.elastic.modulus.tensor.is.homogeneous,.using.the.chain.rule.of.differentiation,.σ.j.k..∂.ε.j.k.∂.x.1.=.∂.W.∂.ε.j.k..∂.ε.j.k.∂.x.1.=.∂.W.∂.x.1.{displaystyle.sigma.{jk}{cfrac.{partial.varepsilon.{jk}}{partial.x{1}}}={cfrac.{partial.W}{partial.varepsilon.{jk}}}{cfrac.{partial.varepsilon.{jk}}{partial.x{1}}}={cfrac.{partial.W}{partial.x{1}}}}.Therefore,.we.have.J.1.=.0.{displaystyle.J{1}=0}.for.a.closed.contour.enclosing.a.simply.connected.region.without.any.elastic.inhomogeneity,.such.as.voids.and.cracks{displaystyle..{boldsymbol..{nabla..}}cdot..{boldsymbol..{sigma..}}=mathbf..{0}..qquad..implies..qquad..{cfrac..{partial..sigma..{jk}}{partial..x{j}}}=0.}..Also,..[..ε..]..=..1..2..[..∇..u..+..(..∇..u..)..T..]..⟹..ε..j..k..=..1..2..(..∂..u..k..∂..x..j..+..∂..u..j..∂..x..k..)....where...K...is...a...stress...amplitude,...(r,)...is...a...polar...coordinate...system...with...origin...at...the...crack...tip,...s...is...a...constant...determined...from...an...asymptotic...expansion...of...the...stress...field...around...the...crack,...and...I...is...a...dimensionless...integral^..a..b..Yoda,..M.,..1980,..The..J-integral..fracture..toughness..for..Mode..II,..Int{displaystyle...[{boldsymbol...{varepsilon...}}]={tfrac...{1}{2}}left[{boldsymbol...{nabla...}}mathbf...{u}...+({boldsymbol...{nabla...}}mathbf...{u}...)^{T}right]qquad...implies...qquad...varepsilon...{jk}={tfrac...{1}{2}}left({cfrac...{partial...u{k}}{partial...x{j}}}+{cfrac...{partial...u{j}}{partial...x{k}}}right).}...Therefore,...σ...j...k...∂...ε...j...k...∂...x...1...=...1...2...(...σ...j...k...∂...2...u...k...∂...x...1...∂...x...j...+...σ...j...k...∂...2...u...j...∂...x...1...∂...x...k...)...{displaystyle...sigma...{jk}{cfrac...{partial...varepsilon...{jk}}{partial...x{1}}}={tfrac...{1}{2}}left(sigma...{jk}{cfrac...{partial...^{2}u{k}}{partial...x{1}partial...x{j}}}+sigma...{jk}{cfrac...{partial...^{2}u{j}}{partial...x{1}partial...x{k}}}right)}...From...the...balance...of...angular...momentum...we...have...σ...j...k...=...σ...k...j...{displaystyle...sigma...{jk}=sigma...{kj}}...K..=..(..β..π..α..I..)..1..n..+..1..(..σ..far..)..2..n..+..1..{displaystyle..K=left({frac..{beta..,pi..}{alpha..,I}}right)^{frac..{1}{n+1}},(sigma..{text{far}})^{frac..{2}{n+1}}}..The..model..is..parametrized..by..,..a..dimensionless..constant..characteristic..of..the..material,..and..n,..the..coefficient..of..work..hardeningRice.also.showed.that.the.value.of.the.J-integral.represents.the.energy.release.rate.for.planar.crack.growthProof.that.the.J-integral.is.zero.over.a.closed.path.To.show.the.path.independence.of.the.J-integral,.we.first.have.to.show.that.the.value.of.J.{displaystyle.J}.is.zero.over.a.closed.contour.in.a.simply.connected.domain
F.,..&..Donovan,..JHutchinson,..Rice..and..Rosengren..[7][8]..subsequently..showed..that..J..characterizes..the..singular..stress..and..strain..fields..at..the..tip..of..a..crack..in..nonlinear..(power..law..hardening)..elastic-plastic..materials..where..the..size..of..the..plastic..zone..is..small..compared..with..the..crack..lengthJ-integral.and.fracture.toughness[edit]J...I...I...I...c...=...G...I...I...I...c...=...K...I...I...I...c...2...(...1...+...ν...E...)...{displaystyle...J{rm...{IIIc}}=G{rm...{IIIc}}=K{rm...{IIIc}}^{2}left({frac...{1+nu...}{E}}right)}...Hetnarski,...Springer,...ISBN...978-9400727380The.J-integral.was.developed.because.of.the.difficulties.involved.in.computing.the.stress.close.to.a.crack.in.a.nonlinear.elastic.or.elastic-plastic.material503512{displaystyle..J{Gamma..{1}}=-J{Gamma..{2}},.}..Advances..in..simulation..of..arbitary..3D..crack..growth..using..FRANC3Dv5..
The...J-integral...represents...a...way...to...calculate...the...strain...energy...release...rate,...or...work...(energy)...per...unit...fracture...surface...area,...in...a...material.[1]...The...theoretical...concept...of...J-integral...was...developed...in...1967...by...Cherepanov[2]...and...in...1968...by...Jim...Rice[3]...independently,...who...showed...that...an...energetic...contour...path...integral...(called...J)...was...independent...of...the...path...around...a...crackof...Fracture,...16(4),...pp379386See...also[edit]Cε...ε...y...=...σ...σ...y...+...α...(...σ...σ...y...)...n...{displaystyle...{frac...{varepsilon...}{varepsilon...{y}}}={frac...{sigma...}{sigma...{y}}}+alpha...left({frac...{sigma...}{sigma...{y}}}right)^{n}}...References[edit]Fracture.Mechanics.Notes.by.Prof 583ae2174f
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