A new general approach for deriving the weight functions for 2D interfacial cracks in anisotropic bimaterials has been developed.For perfect interface conditions, the new method avoid the use of Wiener-Hopf technique and the challenging factorization problem connected. Both symmetric and skew-symmetric weight functions can be derived by means of the new approach. Weight functions can be used for deriving singular integral formulation of interfacial cracks in anisotropic media.  The proposed method can be applied for studying interfacial cracks problems in many materials:monoclinic, orthotropic, cubic, piezoelectrics, poroelastics, quasicrystals

A. B. Movchan

MORINI, Lorenzo

N. V. Movchan

RADI, Enrico

Archivio istituzionale della ricerca - Università di Modena e Reggio Emilia

1 / 1923rd International Congress of Theoretical and Applied Mechanics,August, 19-24, 2012, Beijing, ChinaInterfacial cracks in bi-material solids:Stroh formalism and skew-symmetricweight functionsLorenzo Morini, Enrico Radi,Dipartimento di Scienze e Metodi dell’ Ingegneria,Università di Modena e Reggio Emilia,Via Amendola 2, 42100, Reggio Emilia, ItalyAlexander B. Movchan, Natalia V. Movchan,Department of Mathematical Sciences, University of Liverpool,Liverpool L69 3BX, U.K.August 7, 2012Outline• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References2 / 19• Interface cracks in anisotropic materials: Stroh formalism;• Riemann-Hilbert formulation;• Symmetric and skew-symmetric weight functions;• Stress intensity factors evaluation;• Application: point forces applied at crack faces;• Conclusions;Interfacial cracks: Stroh formalism• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References3 / 19• Quasi-static semi-infinite plane interfacial crack with generalloading acting on the faces;• Displacements and tractions in terms of functions of complexvariable zj = x1 + µjx2:u,1(x1, x2) = 2Re[Ag(z)], T (x1, x2) = 2Re[Bg(z)],Assuming Stroh representation:[Qik + (Rik +Rki)µj + Tikµ2j ]Akj = 0Bij = (Rki + µjTik)AkjRiemann-Hilbert formulation• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References4 / 19Traction-free crack problem:-Free traction condition at x1 < 0 ;-Tractions and displacements continuity at x1 > 0 ;-Boundary conditions at the interface yields to a R-H problem:h+(x1) +H−1Hh−(x1) = T (x1) for x1 > 0h+(x1) +H−1Hh−(x1) = 0 for x1 < 0WhereH = Y(1) +Y(2) andY = iAB−1;Mirror traction-free problem• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References5 / 19Mirror traction-free crack problem:-Free traction at x1 > 0 ;-Tractions and displacements continuity at x1 < 0 ;-At the interface:w+(x1) +H−1Hw−(x1) = 0 for x1 > 0w+(x1) +H−1Hw−(x1) = Σ(x1) for x1 < 0-U singular solutions of the mirror problem;Weight functions• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References6 / 19• Symmetric weight functions:[U](x1) = U(x1, x2 = 0+)−U(x1, x2 = 0−)• Skew-symmetric weight functions:〈U〉(x1) = 12(U(x1, x2 = 0+) +U(x1, x2 = 0−))• Mirror traction-free problem is solved in Fourier space;• A Wiener-Hopf-like equation is derived:[Uˆ]+(ξ) = − 1|ξ|{ReH− i sign(ξ) ImH}Σˆ−(ξ),• The skew-symmetric weight function become:〈Uˆ〉(ξ) = − 12|ξ|{ReW−i sign(ξ) ImW}Σˆ−(ξ), ξ ∈ R.WhereH = Y(1) +Y(2) andW = Y(1) −Y(2);Decoupling plane and antiplane strain and stress• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References7 / 19• Materials whereA,B andY and thenH andW have thefollowing structure are considered: ∗ ∗ 0∗ ∗ 00 0 ∗• Uncoupled inplane and antiplane strain and stresses;• Monoclinic and orthotropic materials have this property;• Physical tractions:T (x1) =1√2pix1Re(Kxiε1 w)⇒ Mode I and IIT3(x1) = K3√2pix1 ⇒ Mode III• K = KI + iKII , andw = (w1, w2) is a complex vector;• K3 is a real scalar;• Same behaviour for the singular solution (Σ,U);Stress intensity factors evaluation• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References8 / 19Betti integral’s theorem relates (u,T (+)) to (U,Σ(−)):• For plane strain:[Uˆ]+TRTˆ+ − Σˆ−TR[uˆ]− = −[Uˆ]+TR〈pˆ〉 − 〈Uˆ〉TR[pˆ]Where R is the rotation matrix:R =(−1 00 1)• For antiplane strain:[Uˆ3]Tˆ +3 − Σˆ3[uˆ3]− = −[Uˆ3] 〈pˆ3〉 − 〈Uˆ3〉[pˆ3]• Integral formulas for stress intensity factors:K =M−112pii∫ ∞−∞{[Uˆ]+T (τ)R〈pˆ〉(τ) + 〈Uˆ〉T (τ)R[pˆ](τ)}dτK3 =12piiK33∫ ∞−∞{[Uˆ3]+(τ)〈pˆ3〉(τ) + 〈Uˆ3〉(τ)[pˆ3](τ)}dτWhereK = (K,K)T ;Plane strain in orthotropic bimaterials I• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References9 / 19• 2D vector problem in orthotropic bimaterials;• Symmetric bimaterial matrix:H =(H11 −iβ√H11H22iβ√H11H22 H22)• Bimaterial parameters:H11 = [2nλ14 (s˜11s˜22)12 ](1) + [2nλ14 (s˜11s˜22)12 ](2),H22 = [2nλ− 14 (s˜11s˜22)12 ](1) + [2nλ−14 (s˜11s˜22)12 ](2),β√H11H22 = [((s˜11s˜22)12 + s˜12)](2) − [((s˜11s˜22)12 + s˜12)](1),Where: λ = s˜11s˜22, ρ = 122s˜12+s˜66√s˜11s˜22, n =(12(1 + ρ)) 12 ,• Generalized Dundurs parameter connected to oscillatory index:ε =12piln(1− β1 + β)• Homogeneous material ⇒ β, ε = 0, no oscillations;Plane strain in orthotropic bimaterials II• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References10 / 19• Skew-symmetric bimaterial matrix:W = Y(1) −Y(2) =(δ1H11 iγ√H11H22−iγ√H11H22 δ2H22)• Bimaterial parameters:δ1 =[2nλ14 (s˜11s˜22)12 ](1) − [2nλ 14 (s˜11s˜22) 12 ](2)H11,δ2 =[2nλ−14 (s˜11s˜22)12 ](1) − [2nλ− 14 (s˜11s˜22) 12 ](2)H22,γ =[((s˜11s˜22)12 + s˜12)](1) + [((s˜11s˜22)12 + s˜12)](2)√H11H22,• Homogeneous material ⇒ δ1, δ2 = 0, but γ 6= 0 then even inhomogeneous case we have non-zero skew-symmetric weightfunctions;Plane strain in orthotropic bimaterials: weight functions• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References11 / 19• Fourier transform symmetric and skew-symmetric weightfunctions:[Uˆ]+ = −√H11H22|ξ|√H11H22iβsign(ξ)−iβsign(ξ)√H22H11 Σˆ−(ξ);〈Uˆ〉 = −√H11H222|ξ| δ1√H11H22−iγsign(ξ)+iγsign(ξ) δ2√H22H11 Σˆ−(ξ);• Inverting these expressions we get [U] and 〈U〉.• Since Mode I and II are coupled:U =(U11 U21U12 U22),Σ =(Σ11 Σ21Σ12 Σ22)• [Uˆ]⇒ Wiener-Hopf equation;Asymmetric loading• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References12 / 19Asymmetric point forces acting on the crack faces:〈p2〉(x1) = −F2 δ(x1 + a)− F4 δ(x1 + a+ b)− F4 δ(x1 + a− b)[p2] (x1) = −Fδ(x1 + a) + F2 δ(x1 + a+ b) + F2 δ(x1 + a− b)Symmetric and skew-symmetric components ⇒ K = KS +KA;Plane strain: SIF• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References13 / 19 !"b/a0 "!# 0.4 1$!"0.6 0.8 !""!"%a KIS/F#!"& = '1/2& = '1/4& = 0ab/a0 "!# 0.4 10.6 0.8b#!(& = )1/2#!"%a KIA/F& = )1/4 !( & = 0& = 1/4  !" & = 1/2 "!("!"b/a0 "!# 0.4 1"!*0.6 0.8)"!+)"!#)"!*%a KIIS/F"!""!+"!#& = )1/2 & = )1/4 & = 0& = 1/4 & = 1/2 cb/a0"!*"!# 0.4 1 0.6 0.8d"!+"!#%a KIIA/F"!"=& )1/2& = )1/4& = 0& = 1/4 & = 1/2 )"!#)"!+)"!*Symmetric vs skew-symmetric SIF• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References14 / 19Β = 1  2Β = 1  4Β = -1  2Β = -1  40.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.5baKiAKiS• As b/a→ 1 increase, KAI ≈ 40%− 50% of KSI ;• Skew-symmetric part of the loading is not negligible and needs tobe taken into account;Antiplane strain: weight functions• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References15 / 19• Anisotropic materials with symmetry plane at x3 = 0 areconsidered;• Fourier transform of weight functions:[Uˆ3](ξ) = −H33|ξ| Σˆ3(ξ); 〈Uˆ3〉(ξ) =η2[Uˆ3](ξ);• Inverting we obtain:[U3](x1) =H33√2pix121 ; 〈U3〉(x1) =η2√2piH33x121 ;• H33 =[√s˜44s˜55 − s˜245](1)+[√s˜44s˜55 − s˜245](2);η =([√s˜44s˜55 − s˜245](1)−[√s˜44s˜55 − s˜245](2))/H33• Homogeneous material ⇒ η = 0, Mode III is symmetric;Antiplane strain: SIF• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References16 / 190.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.02.53.0baaK3SFΒΗ = -1  4Η = -1  2Η = -1Η = 1Η = 1  2Η = 1  40.0 0.2 0.4 0.6 0.8 1.0-1.0-0.50.00.51.0baaK3AF• Same loading configuration directed along x3:〈p3〉(x1) = −F2 δ(x1+a)− F4 δ(x1+a+ b)− F4 δ(x1+a− b)[p3] (x1) = −Fδ(x1+a)+ F2 δ(x1+a+ b)+ F2 δ(x1+a− b)• Homogeneous material ⇒ η = 0, K3 is symmetric;Symmetric vs skew-symmetric SIF• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References17 / 19Η = -1Η = -1  2Η = -1  4Η = 1Η = 1  4Η = 1  20.0 0.2 0.4 0.6 0.8 1.0-1.0-0.50.00.51.0baK3AK3S• As for plane strain, KA3 increase with b/a, expecially for|η| > 1/2;• For b/a > 0.5, skew-symmetric part of the loading is notnegligible;Conclusions• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References18 / 19• A new general approach for deriving the weight functions for 2Dinterfacial cracks in anisotropic bimaterials has been developed;• For perfect interface conditions, the new method avoid the use ofWiener-Hopf technique and the challenging factorization problemconnected;• Both symmetric and skew-symmetric weight functions can bederived by means of the new approach;• Weight functions can be used for deriving singular integralformulation of interfacial cracks in anisotropic media;• The proposed method can be applied for studying interfacialcracks problems in many materials:monoclinic, orthotropic, cubic,piezoelectrics, poroelastics, quasicrystals;Furter developments:-Applications to steady state moving cracks and wavy cracks;-Analysis of inclusions effects of interface cracks propagation;-Extension to 3D case;References• Outline• Interfacial cracks:Stroh formalism• Riemann-Hilbertformulation• Mirror traction-freeproblem• Weight functions• Decoupling plane andantiplane strain andstress• Stress intensityfactors evaluation• Plane strain inorthotropic bimaterials I• Plane strain inorthotropic bimaterials II• Plane strain inorthotropic bimaterials:weight functions• Asymmetric loading• Plane strain: SIF• Symmetric vsskew-symmetric SIF• Antiplane strain:weight functions• Antiplane strain: SIF• Symmetric vsskew-symmetric SIF• Conclusions• References19 / 19-[1] Y. Antipov, J. Mech. Phys. Solids, 57, 1051, (1999).-[2] J.P. Bercial-Velez et Al., J. Mech. Phys. Solids, 53, 1128, (2005).-[3] H.F. Bueckner, Int. J. Solids Struct., 23, 57, (1985).-[4] H.F. Bueckner, Eng. Anal. Bound. Elem., 6, 3, (1989).-[5] H. Gao Int. J. Fract., 56, 139, (1992).-[6] L. Ma and Y. Chen, Acta Mec. Sin., 20, 82, (2004).-[7] K.P. Meade and L.M. Keer, J. Elasticity, 14, 3, (1984).-[8] L. Morini et Al., ArXiv:1202.5418v1, (2012).-[9] L.Morini et Al., ArXiv:1205.1321v1, (2012).-[10] A. Piccolroaz et Al., J. Mech. Phys. Solids, 55, 1575, (2007).-[11] A. Piccolroaz et Al., J. Mech. Phys. Solids, 57, 1657, (2009).-[12] A. Piccolroaz et Al. Int. J. Fract., 166, 41, (2010).-[13] J.R. Willis and A.B. Movchan, J. Mech. Phys. Solids, 43, 319,(1995).-[14] Z. Suo Proc. R. Soc. Lond. A, 427, 331, (1990).

Interfacial cracks in bi-material solids: Stroh formalism and skew-symmetric weight functions

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Archivio istituzionale della ricerca - Università di Modena e Reggio Emilia