INTRODUCTION
CONTINUUM MECHANICS V 0 A 0 u V A x 0 e 3 x e 2 O e 1 - volume / area V 0, V / A 0,A - base vectors { e 1, e 2, e 3 } - position vector x 0, x - displacement vector u - strains ε kl = 1 2 (u k,l + u l,k ) - compatibility relations - equilibrium equations σ ij,j + ρq i = 0 ; σ ij = σ ji - density ρ - load/mass q i - boundary conditions p i = σ ij n j - material model σ ij = N ij (ε kl )
MATERIAL BEHAVIOR σ ε t 1 t 2 t t 1 t 2 t σ ε t 1 t 2 t t 1 t 2 t σ ε t 1 t 2 t t 1 t 2 t
STRESS-STRAIN CURVES σ σ σ σ ε ε ε ε
FRACTURE
FRACTURE MECHANICS (Source: internet) questions : when crack growth? ( crack growth criteria) crack growth rate? residual strength? life time? inspection frequency? repair required? fields of science : material science and chemistry theoretical and numerical mathematics experimental and theoretical mechanics
OVERVIEW OF FRACTURE MECHANICS LEFM (Linear Elastic Fracture Mechanics) energy balance crack tip stresses SSY (Small Scale Yielding) DFM (Dynamic Fracture Mechanics) NLFM (Non-Linear Fracture Mechanics) EPFM (Elasto-Plastic Fracture Mechanics) Numerical methods : EEM / BEM Fatigue (HCF / LCF) CDM (Continuum Damage Mechanics) Micro mechanics micro-cracks (intra grain) voids (intra grain) cavities at grain boundaries rupture & disentangling of molecules rupture of atomic bonds dislocation slip
EXPERIMENTAL FRACTURE MECHANICS (Source: Internet)
LINEAR ELASTIC FRACTURE MECHANICS
DYNAMIC FRACTURE MECHANICS
NONLINEAR FRACTURE MECHANICS CTOD J-integral
NUMERICAL TECHNIQUES
FATIGUE
OBJECTIVES Insight in : crack growth mechanisms brittle / ductile energy balance crack tip stresses crack growth direction plastic crack tip zone crack growth speed nonlinear fracture mechanics numerical methods fatigue
FRACTURE MECHANICS
FRACTURE MECHANISMS shear fracture cleavage fracture fatigue fracture crazing de-adhesion
SHEARING dislocations voids crack dimples load direction
CLEAVAGE intra-granulair inter-granulair intra-granular HCP-, BCC-crystal T low ε high 3D-stress state inter-granular weak grain boundary environment (H 2 ) T high
FATIGUE clam shell pattern striations
CRAZING (Source: Internet) stress whitening crazing materials : PS, PMMA
DUCTILE - BRITTLE BEHAVIOR σ ABS, nylon, PC PE, PTFE 0 10 100 ε (%) surface energy : γ [Jm 2 ] solids : γ 1 [Jm 2 ] independent from cleavage/shearing ex.: alloyed steels; rubber
CHARPY V-NOTCH TEST
CHARPY CV-VALUE C v fcc (hcp) metals low strength bcc metals Be, Zn, ceramics high strength metals Al, Ti alloys T C v NDT FATT FTP T T t - Impact Toughness C v - Nil Ductility Temperature NDT - Nil Fracture Appearance Transition Temperature FATT(T t ) - Nil Fracture Transition Plastic FTP
THEORETICAL STRENGTH f r f f a 0 1 2 λ x r x σ S ( ) 2πx f(x) = f max sin ; x = r a 0 λ σ(x) = 1 ( ) 2πx f(x) = σmax sin S λ
ENERGY BALANCE available elastic energy per surface-unity [N m 1 ] U i = 1 S = required surface energy x=λ/2 x=0 x=λ/2 x=0 f(x) dx ( ) 2πx σ max sin dx λ =σ max λ π [Nm 1 ] U a = 2γ [Nm 1 ] energy balance at fracture U i = U a λ = 2πγ σ = σ max sin σ max ( ) x γ σ max
APPROXIMATIONS linearization σ = σ max sin ( ) x γ σ max x γ σ2 max linear strain of atomic bond elastic modulus ε = x a 0 x = εa 0 σ = εa 0 γ σ2 max E = ( dσ dε) x=0 = ( dσ dx a 0) x=0 = σ 2 max a 0 γ σ max = Eγ a 0 theoretical strength σ th = Eγ a 0
DISCREPANCY WITH EXPERIMENTAL OBSERVATIONS a 0 [m] E [GPa] σ th [GPa] σ b [MPa] σ th /σ b glass 3 10 10 60 14 170 82 steel 10 10 210 45 250 180 silica fibers 10 10 100 31 25000 1.3 iron whiskers 10 10 295 54 13000 4.2 silicon whiskers 10 10 165 41 6500 6.3 alumina whiskers 10 10 495 70 15000 4.7 ausformed steel 10 10 200 45 3000 15 piano wire 10 10 200 45 2750 16.4 discrepancy with experiments σ th σ b
GRIFFITH S EXPERIMENTS 11000 σ b [MPa] 170 10 20 d [µ] DEFECTS FRACTURE MECHANICS
CRACK LOADING MODES Mode I Mode II Mode III Mode I = opening mode Mode II = sliding mode Mode III = tearing mode
EXPERIMENTAL TECHNIQUES
SURFACE CRACKS (Source: Internet site www.venti-oelde.de (2009)) dye penetration small surface cracks fast and cheap on-site magnetic particles cracks disturbance of magnetic field surface cracks for magnetic materials only eddy currents impedance change of a coil penetration depth : a few mm s difficult interpretation
ELECTRICAL RESISTANCE
X-RAY (Source: Internet site University of Strathclyde (2006)) orientation dependency
ULTRASOUND piëzo-el. crystal wave sensor S in out t t
ACOUSTIC EMISSION registration intern sounds (hits)
ADHESION TESTS blade wedge test peel test (0 o and 90 o ) bending test scratch test indentation test laser blister test pressure blister test fatigue friction test
FRACTURE ENERGY
ENERGY BALANCE a B = thickness A = Ba Ḃ = 0 U e = U i + U a + U d + U k [Js 1 ] d dt ( ) = da d dt da ( ) = Ȧ d da ( ) = ȧ d da ( ) du e da = du i da + du a da + du d da + du k da [Jm 1 ] total energy balance du e da du i da = du a da + du d da + du k da [Jm 1 ]
GRIFFITH S ENERGY BALANCE no dissipation no kinetic energy energy balance energy release rate crack resistance force du e da du i da = du a da G = 1 ( due B da du ) i [Jm 2 ] da R = 1 ( ) dua = 2γ [Jm 2 ] B da Griffith s crack criterion G = R = 2γ [Jm 2 ]
GRIFFITH S ENERGY BALANCE du e = 0 a U a da needed G, R 2γ a c available a c U i du i da < du a da du i da > du a da du i da = du a da no crack growth unstable crack growth critical crack length
GRIFFITH STRESS σ y 2a a thickness B x σ U i = 2πa 2 B 1 σ 2 2 E ; U a = 4aB γ [Nm = J] Griffith s energy balance (du e = 0) G = 1 B ( ) dui da = 1 B ( ) dua = R 2πa σ2 da E = 4γ [Jm 2 ] Griffith stress σ gr = 2γE πa critical crack length a c = 2γE πσ 2
DISCREPANCY WITH EXPERIMENTAL OBSERVATIONS σ gr σ c reason remedy neglection of dissipation measure critical energy release rate G c glass G c = 6 [Jm 2 ] wood G c = 10 4 [Jm 2 ] steel G c = 10 5 [Jm 2 ] composite design problem / high alloyed steel / bone (elephant and mouse) energy balance G = 1 B ( due da du ) i da = R = G c critical crack length Griffith s crack criterion a c = G ce 2πσ 2 G = G c
COMPLIANCE CHANGE compliance : C = u/f F u P F u P a a + da F F a a + da a a + da du i du i du e u u fixed grips constant load
COMPLIANCE CHANGE : FIXED GRIPS fixed grips : du e = 0 du i =U i (a + da) U i (a) (< 0) = 1 2 (F + df)u 1 2 Fu = 1 2 udf Griffith s energy balance G= 1 2B udf da = 1 2B = 1 2B F2 dc da u 2 dc C 2 da
COMPLIANCE CHANGE : CONSTANT LOAD constant load du e = U e (a + da) U e (a) = Fdu du i =U i (a + da) U i (a) (> 0) = 1 2 F(u + du) 1 2 Fu = 1 2 Fdu Griffith s energy balance G= 1 2B F du da = 1 2B F2 dc da
COMPLIANCE CHANGE : EXPERIMENT F a 1 F u a 2 ap a 3 a 4 u G = shaded area a 4 a 3 1 B no fixed grips no constant load small deviation!!
EXAMPLE B F u 2h a u F u = Fa3 3EI = 4Fa3 EBh 3 C = u F = 2u F = 8a3 G = 1 B [ 1 2 F2dC da G c = 2γ F c = B a dc EBh 3 da = 24a2 EBh 3 ] = 12F2 a 2 [J m 2 ] EB 2 h 3 1 6 γeh3
EXAMPLE h a question : which h(a) makes dc da independent from a? C = u F = 2u F = 8a3 EBh 3 dc da = 24a2 EBh 3 choice : h = h 0 a n u = Fa 3 3(1 n)ei = 4Fa 3 (1 n)ebh 3 = 4Fa3(1 n) (1 n)ebh 3 0 C = 2u F = 8a3(1 n) (1 n)ebh 3 0 dc da = 24a(2 3n) EBh 3 0 dc da constant for n = 2 3 h = h 0 a 2 3
STRESS CONCENTRATIONS
DEFORMATION Q X + d X Q P u x + d x P X e 3 e 2 x e 1 x i =X i + u i (X i ) x i + dx i =X i + dx i + u i (X i + dx i ) =X i + dx i + u i (X i ) + u i,j dx j difference vector dx i = dx i +u i,j dx j = (δ ij +u i,j )dx j ds = d x = dx i dx i ds = d X = dx i dx i
STRAINS ds 2 =dx i dx i = [(δ ij + u i,j )dx j ][(δ ik + u i,k )dx k ] =(δ ij δ ik + δ ij u i,k + u i,j δ ik + u i,j u i,k )dx j dx k =(δ jk + u j,k + u k,j + u i,j u i,k )dx j dx k =(δ ij + u i,j + u j,i + u k,i u k,j )dx i dx j =dx i dx i + (u i,j + u j,i + u k,i u k,j )dx i dx j =ds 2 + (u i,j + u j,i + u k,i u k,j )dx i dx j ds 2 ds 2 =(u i,j + u j,i + u k,i u k,j )dx i dx j =2γ ij dx i dx j Green-Lagrange strains γ ij = 1 2 (u i,j + u j,i + u k,i u k,j ) linear strains ε ij = 1 2 (u i,j + u j,i )
COMPATIBILITY 3 displacement components 9 strain components 6 dependencies 6 compatibility equations 2ε 12,12 ε 11,22 ε 22,11 = 0 2ε 23,23 ε 22,33 ε 33,22 = 0 2ε 31,31 ε 33,11 ε 11,33 = 0 ε 11,23 + ε 23,11 ε 31,12 ε 12,13 = 0 ε 22,31 + ε 31,22 ε 12,23 ε 23,21 = 0 ε 33,12 + ε 12,33 ε 23,31 ε 31,32 = 0
STRESS unity normal vector stress vector Cauchy stress components n = n i e i p = p i e i p i = σ ij n j stress cube σ 33 σ 23 σ 13 σ 32 3 2 σ 21 σ 31 σ 12 σ 22 1 σ 11
LINEAR ELASTIC MATERIAL BEHAVIOR σ ij = C ijkl ε lk material symmetry isotropic material 2 mat.pars
HOOKE S LAW FOR ISOTROPIC MATERIALS σ ij = E 1 + ν ε ij = 1 + ν E ( ε ij + ( σ ij ν ) 1 2ν δ ijε kk ν 1 + ν δ ijσ kk ) i = 1,2, 3 i = 1, 2,3 σ 11 σ 22 σ 33 σ 12 σ 23 σ 31 = α 1 ν ν ν 0 0 0 ν 1 ν ν 0 0 0 ν ν 1 ν 0 0 0 0 0 0 1 2ν 0 0 0 0 0 0 1 2ν 0 0 0 0 0 0 1 2ν ε 11 ε 22 ε 33 ε 12 ε 23 ε 31 α = E/[(1 + ν)(1 2ν)] ε 11 ε 22 ε 33 ε 12 ε 23 ε 31 = 1 E 1 ν ν 0 0 0 ν 1 ν 0 0 0 ν ν 1 0 0 0 0 0 0 1 + ν 0 0 0 0 0 0 1 + ν 0 0 0 0 0 0 1 + ν σ 11 σ 22 σ 33 σ 12 σ 23 σ 31
EQUILIBRIUM EQUATIONS stress components on stress cube σ 12 σ 13 + σ 13,3 dx 3 σ 23 + σ 23,3 dx 3 σ 33 + σ 33,3 dx 3 σ 22 σ 32 σ 11 σ 21 1 σ 11 + σ 11,1 dx 1 σ 21 + σ 21,1 dx 1 σ 31 + σ 31,1 dx 1 3 2 σ 13 σ 23 σ 33 σ 31 σ 12 + σ 12,2 dx 2 σ 22 + σ 22,2 dx 2 σ 32 + σ 32,2 dx 2 volume load ρq i force equilibrium σ ij,j + ρq i = 0 i = 1,2, 3 moment equilibrium σ ij = σ ji
PLANE STRESS σ 33 = σ 13 = σ 23 = 0 equilibrium (q i = 0) σ 11,1 + σ 12,2 = 0 ; σ 21,1 + σ 22,2 = 0 compatibility 2ε 12,12 ε 11,22 ε 22,11 = 0 Hooke s law σ ij = E 1 + ν ε ij = 1 + ν E ( ε ij + ( σ ij ν ) 1 ν δ ijε kk ν 1 + ν δ ijσ kk ) i = 1,2 i = 1, 2 Hooke s law in matrix notation ε 11 ε 22 ε 12 σ 11 σ 22 σ 12 = 1 E = E 1 ν 2 1 ν 0 ν 1 0 0 0 1 + ν 1 ν 0 ν 1 0 0 0 1 ν σ 11 σ 22 σ 12 ε 11 ε 22 ε 12 ε 33 = ν E (σ 11 + σ 22 ) = ν 1 ν (ε 11 + ε 22 ) ε 13 = ε 23 = 0
PLANE STRAIN ε 33 = ε 13 = ε 23 = 0 equilibrium (q i = 0) σ 11,1 + σ 12,2 = 0 ; σ 21,1 + σ 22,2 = 0 compatibility 2ε 12,12 ε 11,22 ε 22,11 = 0 Hooke s law ε ij = 1 + ν E (σ ij νδ ij σ kk ) i = 1, 2 ( ) σ ij = ε ij + E 1 + ν Hooke s law in matrix notation σ 11 σ 22 σ 12 ε 11 ε 22 ε 12 = E (1 + ν)(1 2ν) = 1 + ν E ν 1 2ν δ ijε kk 1 ν ν 0 ν 1 ν 0 0 0 1 i = 1, 2 1 ν ν 0 ν 1 ν 0 0 0 1 2ν σ 11 σ 22 σ 12 ε 11 ε 22 ε 12 σ 33 = Eν (1 + ν)(1 2ν) (ε 11 + ε 22 ) = ν (σ 11 + σ 22 ) σ 13 = σ 23 = 0
DISPLACEMENT METHOD σ ij,j = 0 σ ij = E 1 + ν E 1 + ν ( ε ij,j + ( ε ij + ε ij = 1 2 (u i,j + u j,i ) ν ) 1 2ν δ ijε kk,j ν ) 1 2ν δ ijε kk,j = 0 E 1 + ν BC s 1 2 (u i,jj + u j,ij ) + Eν (1 + ν)(1 2ν) δ iju k,kj = 0 u i ε ij σ ij
STRESS FUNCTION METHOD ψ(x 1,x 2 ) σ ij = ψ,ij + δ ij ψ,kk σ ij,j = 0 ε ij = 1 + ν E (σ ij νδ ij σ kk ) ε ij = 1 + ν E { ψ,ij + (1 ν)δ ij ψ,kk } 2ε 12,12 ε 11,22 ε 22,11 = 0 2ψ,1122 + ψ,2222 + ψ,1111 = 0 (ψ,11 + ψ,22 ),11 + (ψ,11 + ψ,22 ),22 = 0 Laplace operator : 2 = 2 x 2 1 + 2 x 2 2 = ( ) 11 + ( ) 22 bi-harmonic equation 2 ( 2 ψ) = 4 ψ = 0 BC s } ψ σ ij ε ij u i
CYLINDRICAL COORDINATES z e z e t e r x e 3 e 2 e 1 θ r y vector bases { e 1, e 2, e 3 } { e r, e t, e z } e r = e r (θ) = e 1 cosθ + e 2 sinθ e t = e t (θ) = e 1 sin θ + e 2 cosθ θ { e r(θ)} = e t (θ) ; θ { e t(θ)} = e r (θ)
LAPLACE OPERATOR z e z e t e r x e 3 e 2 e 1 θ r y gradient operator = e r r + e t 1 r θ + e z z Laplace operator two-dimensional 2 = = 2 r 2 + 1 r 2 = 2 r 2 + 1 r r + 1 r 2 2 θ 2 + 2 z 2 r + 1 r 2 2 θ 2
BI-HARMONIC EQUATION bi-harmonic equation ( 2 r 2 + 1 r stress components r + 1 2 ) ( 2 ψ r 2 θ 2 r + 1 ψ 2 r r + 1 2 ) ψ r 2 θ 2 σ rr = 1 r σ tt = 2 ψ r 2 ψ r + 1 2 ψ r 2 θ 2 σ rt = 1 ψ r 2 θ 1 2 ψ r r θ = r ( 1 r ) ψ θ = 0
CIRCULAR HOLE IN INFINITE PLATE y σ σ r θ x 2a ( 2 r 2 + 1 r stress components r + 1 2 ) ( 2 ψ r 2 θ 2 r + 1 ψ 2 r r + 1 2 ) ψ r 2 θ 2 σ rr = 1 r σ tt = 2 ψ r 2 ψ r + 1 2 ψ r 2 θ 2 σ rt = 1 ψ r 2 θ 1 2 ψ r r θ = r ( 1 r ) ψ θ = 0
LOAD TRANSFORMATION σ σ θ σ rr σ rt σ rr σ rt 2a 2b equilibrium σ rr (r = b,θ) = 1 2 σ + 1 2 σ cos(2θ) σ rt (r = b, θ) = 1 2 σ sin(2θ) two load cases I. σ rr (r = a) = σ rt (r = a) = 0 σ rr (r = b) = 1 2 σ ; σ rt(r = b) = 0 II. σ rr (r = a) = σ rt (r = a) = 0 σ rr (r = b) = 1 2 σ cos(2θ) ; σ rt(r = b) = 1 2 σ sin(2θ)
LOAD CASE I σ rr (r = a) = σ rt (r = a) = 0 σ rr (r = b) = 1 2 σ ; σ rt(r = b) = 0 Airy function ψ = f(r) stress components bi-harmonic equation σ rr = 1 r ψ r + 1 2 ψ r 2 θ = 1 df 2 rdr σ tt = 2 ψ r = d2 f 2 dr 2 σ rt = r ( 1 r ψ θ ) = 0 ( d 2 dr + 1 )( d d 2 f 2 rdr dr + 1 ) df = 0 2 rdr
SOLUTION general solution ψ(r) = A lnr + Br 2 ln r + Cr 2 + D stresses σ rr = A + B(1 + 2 ln r) + 2C r2 σ tt = A + B(3 + 2 lnr) + 2C r2 strains (from Hooke s law for plane stress) ε rr = 1 [ ] A E r2(1 + ν) + B{(1 3ν) + 2(1 ν) lnr} + 2C(1 ν) ε tt = 1 1 [ Ar ] E r (1 + ν) + B{(3 ν)r + 2(1 ν)r lnr} + 2C(1 ν)r compatibility ε rr = du dr = d(rε tt) dr B = 0 2 BC s and b a A and C σ rr = 1 2 σ(1 a2 r 2 ) ; σ tt = 1 2 σ(1 + a2 r 2 ) ; σ rt = 0
LOAD CASE II σ rr (r = a) = σ rt (r = a) = 0 σ rr (r = b) = 1 2 σ cos(2θ) ; σ rt(r = b) = 1 2 σ sin(2θ) Airy function ψ(r, θ) = g(r) cos(2θ) stress components σ rr = 1 r bi-harmonic equation ( 2 r 2 + 1 r ( d 2 dr 2 + 1 r ψ r + 1 2 ψ ; σ r 2 θ 2 tt = 2 ψ r 2 σ rt = 1 ψ r 2 θ 1 2 ψ r r θ = r r + 1 2 ) {( d 2 g r 2 θ 2 d dr 4 r 2 ( 1 r ψ θ ) dr + 1 dg 2 r dr 4 ) r 2g ) cos(2θ) = 0 )( d 2 g dr + 1 dg 2 r dr 4 r 2g } cos(2θ) = 0
SOLUTION general solution g = Ar 2 + Br 4 + C 1 r 2 + D ψ = ( Ar 2 + Br 4 + C 1 r 2 + D ) cos(2θ) stresses ( σ rr = 2A + 6C r + 4D ) cos(2θ) 4 r 2 ( σ tt = 2A + 12Br 2 + 6C ) cos(2θ) r 4 ( σ rt = 2A + 6Br 2 6C r 2D ) sin(2θ) 4 r 2 4 BC s and b a A,B,C and D ( ) σ rr = 1 2 σ 1 + 3a4 r 4a2 cos(2θ) 4 r 2 σ tt = 1 2 σ ( 1 + 3a4 r 4 ) cos(2θ) ( σ rt = 1 2 σ 1 3a4 r + 2a2 4 r 2 ) sin(2θ)
STRESSES FOR TOTAL LOAD σ rr = σ ) ) ] [(1 a2 + (1 + 3a4 2 r 2 r 4a2 cos(2θ) 4 r 2 σ tt = σ ) ) ] [(1 + a2 (1 + 3a4 cos(2θ) 2 r 2 r 4 σ rt = σ ] [1 3a4 2 r + 2a2 sin(2θ) 4 r 2
SPECIAL POINTS σ rr (r = a, θ) = σ rt (r = a,θ) = σ rt (r, θ = 0) = 0 σ tt (r = a,θ = π 2 ) = 3σ σ tt (r = a,θ = 0) = σ stress concentration factor K t = σ max σ = 3 [-] K t is independent of hole diameter!
STRESS GRADIENTS large hole : smaller stress gradient larger area with higher stress higher chance for critical defect in high stress area
ELLIPTICAL HOLE y σ σ yy σ a b radius ρ x σ yy (x = a,y = 0) = σ ( 1 + 2 a ) b 2σ a/ρ = σ ( 1 + 2 ) a/ρ stress concentration factor K t = 2 a/ρ [-]
CRACK TIP STRESSES
COMPLEX PLANE x 2 θ r x 1 crack tip = singular point complex function theory complex Airy function (Westergaard, 1939)
COMPLEX VARIABLES x 2 e i r θ z e r x 1 z z = x 1 + ix 2 = re iθ z = x 1 ix 2 = re iθ x 1 = 1 (z + z) 2 x 2 = 1 2i (z z) = 1 i(z z) 2 z = x 1 e r + x 2 e i = x 1 e r + x 2 i e r = (x 1 + ix 2 ) e r
COMPLEX FUNCTIONS complex function f(z) = φ + iζ = φ(x 1, x 2 ) + iζ(x 1, x 2 ) = f f( z) = φ(x 1,x 2 ) iζ(x 1, x 2 ) = f φ = 1 2 {f + f } ; ζ = 1 2 i{f f } 2 φ = 2 ζ = 0 appendix!!
LAPLACE OPERATOR complex function g(x 1, x 2 ) = g(z, z) Laplacian 2 g = 2 g x 2 1 + 2 g x 2 2 derivatives (see App. A) g = g z + g z = g x 1 z x 1 z x 1 z + g z 2 g x 2 1 = 2 g z + 2 2 g 2 z z + 2 g z 2 g = g z + g z = i g x 2 z x 2 z x 2 z i g z 2 g x 2 2 = 2 g z + 2 2 g 2 z z 2 g z 2 Laplacian 2 g = 2 g x 2 1 + 2 g x 2 2 = 4 2 g z z 2 = 4 2 z z
BI-HARMONIC EQUATION Airy function ψ(z, z) bi-harmonic equation 2 ( 2 ψ(z, z) ) = 0
SOLUTION OF BI-HARMONIC EQUATION real part of complex function f satisfies Laplace eqn. 2 ( 2 ψ(z, z) ) = 2 (φ(z, z)) = 0 φ = f + f choice Airy function 2 ψ = 4 2 ψ z z = φ = f + f integration ψ = 1 2 [ zω + z Ω + ω + ω ] unknown functions : Ω ; Ω ; ω ; ω
STRESSES Airy function ψ = 1 2 [ zω + z Ω + ω + ω ] stress components σ ij =σ ij (z, z) = ψ,ij + δ ij ψ,kk σ 11 = ψ,11 + ψ,γγ = ψ,22 =Ω + Ω 1 2 { zω + ω + z Ω + ω } σ 22 = ψ,22 + ψ,γγ = ψ,11 =Ω + Ω + 1 2 { zω + ω + z Ω + ω } σ 12 = ψ,12 = 1 2 i { zω + ω z Ω ω }
DISPLACEMENT u 2 u e i e 2 x 2 θ r u 1 e r e 1 x 1 definition of complex displacement u=u 1 e 1 + u 2 e 2 =u 1 e r + u 2 e i =u 1 e r + u 2 i e r =(u 1 + iu 2 ) e r =u e r u=u 1 + iu 2 =u 1 (x 1, x 2 ) + iu 2 (x 1,x 2 ) = u(z, z) ū=u 1 iu 2 = ū(z, z)
SCHEMATIC u z u z ū z u(z, z) with int. const. M(z) u z + ū z M(z), M( z) u z + ū z = ε 11 + ε 22 σ 11, σ 22 M(z) u(z, z)
DISPLACEMENT DERIVATIVES u z = u x 1 x 1 z + u x 2 x 2 = 1 2 z = 1 2 { u + i u x 1 x 2 } { u1 + i u 2 + i u 1 u 2 x 1 x 1 x 2 x 2 u z = u x 1 x 1 z + u x 2 x 2 z = 1 2 { u1 = 1 2 = 1 2 { u i u x 1 x 2 } + i u 2 i u 1 + u 2 x 1 x 1 x 2 x 2 { ε 11 + ε 22 + i ū z = ū x 1 x 1 z + ū x 2 x 2 = 1 2 ( u2 u 1 x 1 x 2 z = 1 2 )} { ū i ū x 1 x 2 } { u1 i u 2 i u 1 u 2 x 1 x 1 x 2 x 2 ū z = ū x 1 x 1 z + ū x 2 x 2 z = 1 2 { u1 = 1 2 = 1 2 { ū + i ū x 1 x 2 } i u 2 + i u 1 + u 2 x 1 x 1 x 2 x 2 { ε 11 + ε 22 i ( u2 x 1 u 1 x 2 )} } = 1 2 (ε 11 ε 22 + 2iε 12 ) } } = 1 2 (ε 11 ε 22 2iε 12 ) }
GENERAL SOLUTION u z = 1 2 (ε 11 ε 22 + 2iε 12 ) Hooke s law (pl.strain) u z = 1 2 1 + ν E = 1 + ν E [σ 11 σ 22 + 2iσ 12 ] [z Ω + ω ] Integration u = 1 + ν E [ ] z Ω + ω + M
INTEGRATION FUNCTION u= 1 + ν E [ ] z Ω + ω + M u z = 1 + ν E ū= 1 + ν E ū z = 1 + ν E [ Ω + M ] [ zω + ω + M ] [ Ω + M ] u z + ū z = 1 + ν E [ Ω + Ω + M + M ] u z + ū z =ε 11 + ε 22 = 1 + ν E [(1 2ν)(σ 11 + σ 22 )] = (1 + ν)(1 2ν) E 2 [ Ω + Ω ] M + M = (3 4ν) [ Ω + Ω ] M= (3 4ν)Ω = κω u= 1 + ν E [ ] z Ω + ω κω
CHOICE OF COMPLEX FUNCTIONS Ω = (α + iβ)z λ+1 = (α + iβ)r λ+1 e iθ(λ+1) ω = (γ + iδ)z λ+1 = (γ + iδ)r λ+1 e iθ(λ+1) Ω = (α iβ) z λ+1 = (α iβ)r λ+1 e iθ(λ+1) Ω = (α iβ)(λ + 1) z λ = (α iβ)(λ + 1)r λ e iθλ ω = (γ iδ) z λ+1 = (γ iδ)r λ+1 e iθ(λ+1) displacement u = 1 2µ rλ+1 [ κ(α + iβ)e iθ(λ+1) (α iβ)(λ + 1)e iθ(1 λ) (γ iδ)e iθ(λ+1) ] with µ = E 2(1 + ν) displacement finite λ > 1
DISPLACEMENT COMPONENTS u = 1 2µ rλ+1 [ κ(α + iβ)e iθ(λ+1) e iθ = cos(θ) + i sin(θ) (α iβ)(λ + 1)e iθ(1 λ) (γ iδ)e iθ(λ+1) ] u= 1 2µ rλ+1 [ { κα cos(θ(λ + 1)) κβ sin(θ(λ + 1)) i α(λ + 1) cos(θ(1 λ)) β(λ + 1) sin(θ(1 λ)) } γ cos(θ(λ + 1)) + δ sin(θ(λ + 1)) + { κα sin(θ(λ + 1)) + κβ cos(θ(λ + 1)) α(λ + 1) sin(θ(1 λ)) + β(λ + 1) cos(θ(1 λ)) + } ] γ sin(θ(λ + 1)) + δ cos(θ(λ + 1)) =u 1 + iu 2
MODE I
MODE I : DISPLACEMENT x 2 θ r x 1 displacement for Mode I u 1 (θ > 0) = u 1 (θ < 0) u 2 (θ > 0) = u 2 (θ < 0) } β = δ = 0 Ω = αz λ+1 = αr λ+1 e i(λ+1)θ ω = γz λ+1 = γr λ+1 e i(λ+1)θ
MODE I : STRESS COMPONENTS σ 11 =(λ + 1) [ αz λ + α z λ 1 2 σ 22 =(λ + 1) [ αz λ + α z λ + 1 2 { αλ zz λ 1 + γz λ + αλz z λ 1 + γ z λ}] { αλ zz λ 1 + γz λ + αλz z λ 1 + γ z λ}] σ 12 = 1 2 i(λ + 1) [ αλ zz λ 1 + γz λ αλz z λ 1 γ z λ] with z = re iθ ; z = re iθ σ 11 =(λ + 1)r [ λ αe iλθ + αe iλθ }] {αλe i(λ 2)θ + γe iλθ + αλe i(λ 2)θ + γe iλθ 1 2 σ 22 =(λ + 1)r [ λ αe iλθ + αe iλθ + }] {αλe i(λ 2)θ + γe iλθ + αλe i(λ 2)θ + γe iλθ 1 2 σ 12 = 1 2 i(λ + 1)rλ [ αλe i(λ 2)θ + γe iλθ αλe i(λ 2)θ γe iλθ ] with e iθ + e iθ = 2 cos(θ) ; e iθ e iθ = 2i sin(θ) σ 11 =2(λ + 1)r λ [ α cos(λθ) + 1 2 σ 22 =2(λ + 1)r λ [ α cos(λθ) 1 2 {αλ cos((λ 2)θ) + γ cos(λθ)}] {αλ cos((λ 2)θ) + γ cos(λθ)}] σ 12 =(λ + 1)r λ [αλ sin((λ 2)θ) + γ sin(λθ)]
STRESS BOUNDARY CONDITIONS σ 11 = 2(λ + 1)r λ [ α cos(λθ) + 1 2 σ 22 = 2(λ + 1)r λ [ α cos(λθ) 1 2 {αλ cos((λ 2)θ) + γ cos(λθ)}] {αλ cos((λ 2)θ) + γ cos(λθ)}] σ 12 = (λ + 1)r λ [αλ sin((λ 2)θ) + γ sin(λθ)] crack surfaces are stress free σ 22 (θ = ±π) = σ 12 (θ = ±π) = 0 [ det (λ 2) cos(λπ) cos(λπ) λ sin(λπ) sin(λπ) [ ][ (λ 2) cos(λπ) cos(λπ) λ sin(λπ) sin(λπ) α γ ] ] = [ 0 0 ] = sin(2λπ) = 0 2πλ = nπ λ = 1 2, n, with n = 0, 1,2,.. 2
STRESS FIELD λ = 1 2 α = 2γ ; λ = 0 α = 1 2 γ λ = 1 2 α = 2γ ; λ = 1 α = γ σ 11 =2γr 1 2 cos( 1 2 θ) [ 1 sin( 3 2 θ) sin(1 2 θ)] + σ 22 =2γr 1 2 cos( 1 2 θ) [ 1 + sin( 3 2 θ) sin(1 2 θ)] + σ 12 =2γr 1 2 [ cos( 1 2 θ) cos(3 2 θ) sin(1 2 θ)] +
MODE I : STRESS INTENSITY FACTOR definition stress intensity factor K ( Kies ) ( ) K I = lim 2πr σ22 θ=0 r 0 = 2γ 2π [ m 1 2 N m 2 ]
MODE I : CRACK TIP SOLUTION stress components σ 11 = σ 22 = σ 12 = K I 2πr [ cos( 1 2 θ) { 1 sin( 1 2 θ) sin(3 2 θ)}] K I 2πr [ cos( 1 2 θ) { 1 + sin( 1 2 θ) sin(3 2 θ)}] K I 2πr [ cos( 1 2 θ) sin(1 2 θ) cos(3 2 θ)] displacement components u 1 = K I r 2µ 2π [ cos( 1 2 θ) { κ 1 + 2 sin 2 ( 1 2 θ)}] u 2 = K I r 2µ 2π [ sin( 1 2 θ) { κ + 1 2 cos 2 ( 1 2 θ)}] plane stress plane strain κ = 3 ν 1 + ν κ = 3 4ν
MODE II
MODE II : DISPLACEMENT x 2 θ r x 1 displacements for Mode II u 1 (θ > 0) = u 1 (θ < 0) u 2 (θ > 0) = u 2 (θ < 0) } α = γ = 0 Ω = iβz λ+1 = iβr λ+1 e i(λ+1)θ ω = iδz λ+1 = iδr λ+1 e i(λ+1)θ
MODE II : STRESS INTENSITY FACTOR definition stress intensity factor K ( Kies ) ( ) K II = lim 2πr σ12 θ=0 r 0 [ m 1 2 N m 2 ]
MODE II : CRACK TIP SOLUTION stress components σ 11 = K II 2πr [ sin( 1 2 θ) { 2 + cos( 1 2 θ) cos(3 2 θ)}] σ 22 = K II 2πr [ sin( 1 2 θ) cos(1 2 θ) cos(3 2 θ)] σ 12 = K II 2πr [ cos( 1 2 θ) { 1 sin( 1 2 θ) sin(3 2 θ)}] displacement components u 1 = K II r 2µ 2π [ sin( 1 2 θ) { κ + 1 + 2 cos 2 ( 1 2 θ)}] u 2 = K II r 2µ 2π [ cos( 1 2 θ) { κ 1 2 sin 2 ( 1 2 θ)}] plane stress plane strain κ = 3 ν 1 + ν κ = 3 4ν
MODE III
LAPLACE EQUATION ε 31 = 1 2 u 3,1 ; ε 32 = 1 2 u 3,2 Hooke s law σ 31 = 2µε 31 = µu 3,1 ; σ 32 = 2µε 32 = µu 3,2 equilibrium σ 31,1 + σ 32,2 = µu 3,11 + µu 3,22 = 0 2 u 3 = 0
MODE III : DISPLACEMENT general solution u 3 = f + f specific choice f = (A + ib)z λ+1 f = (A ib) z λ+1
MODE III : STRESS COMPONENTS σ 31 = 2(λ + 1)r λ {A cos(λθ) B sin(λθ)} σ 32 = 2(λ + 1)r λ {A sin(λθ) + B cos(λθ)} σ 32 (θ = ±π) = 0 [ sin(λπ) cos(λπ) sin(λπ) cos(λπ) ][ A B ] = [ 0 0 ] det [ sin(λπ) cos(λπ) sin(λπ) cos(λπ) ] = sin(2πλ) = 0 2πλ = nπ λ = 1 2, n,.. with n = 0, 1,2,.. 2 crack tip solution λ = 1 2 A = 0 σ 31 = Br 1 2{sin( 1 2 θ)} ; σ 32 = Br 1 2{cos( 1 2 θ)}
MODE III : STRESS INTENSITY FACTOR definition stress intensity factor ( ) K III = lim 2πr σ32 θ=0 r 0
MODE III : CRACK TIP SOLUTION stress components σ 31 = K III 2πr [ sin( 1 2 θ)] σ 32 = K III 2πr [ cos( 1 2 θ)] displacement u 3 = 2K III r µ 2π [ sin( 1 2 θ)]
CRACK TIP STRESS (MODE I, II, III) σ τ τ σ τ τ Mode I Mode II Mode III σ ij = K I 2πr f Iij (θ) ; σ ij = K II 2πr f IIij (θ) ; σ ij = K III 2πr f IIIij (θ) crack intensity factors (SIF) K I = β I σ πa ; K II = β II τ πa ; K III = β III τ πa
K-ZONE D K-zone : D I II D II D I
SIF FOR SPECIFIED CASES W 2a σ W τ 2a K I = σ ( πa sec πa W K II = τ πa ) 1/2 small a W σ K I = σ [ a 1.12 π 0.41 a W + a W ( a ) 2 ( a ) 3 18.7 38.48 + W W ( ] a 4 53.85 W) 1.12σ πa small a W σ K I = σ [ a 1.12 π + 0.76 a W a W a ( a ) 2 ( ] a 3 8.48 + 27.36 W W) 1.12σ πa
W 2a σ K I /σ 2.5 2 1.5 1 full 1st term 0.5 0 0 0.1 0.2 0.3 0.4 a/w a W σ K I /σ 2.5 2 1.5 1 full 1st term 0.5 0 0 0.1 0.2 0.3 0.4 a/w plots are made with Kfac.m.
P K I = PS BW 3/2 [ 2.9 ( a W)1 2 W a P/2 S P/2 4.6 37.6 ( a ( W)3 2 a + 21.8 W)5 2 ( a W)7 2 + 37.7 ] ( a 2 W)9 P a W P K I = P BW 1/2 185.5 1017 [ 29.6 ( a W)1 2 ( a ( W)3 2 a + 655.7 W)5 2 ( a W)7 2 + 638.9 ] ( a 2 W)9 2a p K I = p πa W p per unit thickness
P 100 80 full 1st term W a P/2 S P/2 K I /P 60 40 20 0 0 0.1 0.2 0.3 0.4 a/w a P W K I /P 200 150 100 full 1st term P 50 0 0 0.1 0.2 0.3 0.4 a/w plots are made with Kfac.m.
K-BASED CRACK GROWTH CRITERIA K I = K Ic ; K II = K IIc ; K III = K IIIc K Ic = Fracture Toughness calculate K I,K II,K III - analytically - literature - relation K G - numerically (EEM, BEM) experimental determination of K Ic,K IIc, K IIIc - normalized experiments (exmpl. ASTM E399) - correlation with C v ( KAN p. 18 : K 2 Ic E = mcn v )
RELATION G K I y σ yy x a a crack length a σ yy (θ = 0, r = x a) = σ a 2(x a) ; u y = 0 crack length a + a σ yy (θ = π,r = a + a x) = 0 u y = (1 + ν)(κ + 1) E σ a + a 2 a + a x plane stress : κ = 3 ν 1 + ν plane strain : κ = 3 4ν
RELATION G K I (CONTINUED) accumulation of elastic energy U=2B a+ a a =Bf( a) a 1 2 σ yy dx u y = B a+ a a σ yy u y dx energy release rate G= 1 B lim a 0 ( ) U a = lim a 0 f( a) = (1 + ν)(κ + 1) 4E σ 2 aπ = (1 + ν)(κ + 1) 4E K 2 I plane stress G = K2 I E plane strain G = (1 ν 2 ) K2 I E
MULTI MODE LOAD G = 1 ( ) c1 K 2 I + c 2 K 2 II + c 3 K 2 III E plane stress G = 1 E (K2 I + K 2 II) plane strain G = (1 ν2 ) (K 2 I + K 2 E II) + (1 + ν) K 2 III E
THE CRITICAL SIF VALUE σ K I K Ic 2a σ B B c B K Ic = σ c πa B c = 2.5 ( ) 2 KIc σ y
K IC VALUES Material σ v [MPa] K Ic [MPa m ] steel, 300 maraging 1669 93.4 steel, 350 maraging 2241 38.5 steel, D6AC 1496 66.0 steel, AISI 4340 1827 47.3 steel, A533B reactor 345 197.8 steel, carbon 241 219.8 Al 2014-T4 448 28.6 Al 2024-T3 393 34.1 Al 7075-T651 545 29.7 Al 7079-T651 469 33.0 Ti 6Al-4V 1103 38.5 Ti 6Al-6V-2Sn 1083 37.4 Ti 4Al-4Mo-2Sn-0.5Si 945 70.3
MULTI-MODE CRACK LOADING
MULTI-MODE CRACK LOADING Mode I Mode II Mode I + II Mode I + II crack tip stresses s ij Mode I s ij = K I 2πr f Iij (θ) Mode II s ij = K II 2πr f IIij (θ) Mode I + II s ij = K I 2πr f Iij (θ)+ K II 2πr f IIij (θ)
STRESS COMPONENT TRANSFORMATION (b) e 2 p n e 2 e 1 θ e 1 e 1 = cos(θ) e 1 + sin(θ) e 2 = c e 1 + s e 2 e 2 = sin(θ) e 1 + cos(θ) e 2 = s e 1 + c e 2 stress vector and normal unity vector p = p 1 e 1 + p 2 e 2 = p 1 e 1 + p 2 e 2 [ ] [ ][ ] [ p 1 c s = p 2 s c p 1 p 2 p 1 p 2 ] = [ c s s c ][ p 1 p 2 ] = T p p p = T T p idem : ñ = T T ñ
TRANSFORMATION STRESS MATRIX p = σñ T p = σ Tñ p = T T σ T ñ = σ ñ σ = T T σ T σ = T σ T T [ σ 11 σ 12 σ 21 σ 22 ] = = [ [ = c s s c c s s c ] [ ] [ σ 11 σ 12 σ 21 σ 22 ][ c s s c cσ 11 + sσ 12 sσ 11 + cσ 12 cσ 21 + sσ 22 sσ 21 + cσ 22 c 2 σ 11 + 2csσ 12 + s 2 σ 22 csσ 11 + (c 2 s 2 )σ 12 + csσ 22 csσ 11 + (c 2 s 2 )σ 12 + csσ 22 s 2 σ 11 2csσ 12 + c 2 σ 22 ] ]
CARTESIAN TO CYLINDRICAL TRANSFORMATION σ yy σ xy e t e r σ xx e 2 r θ e 1 σ rt σ rr σ tt e r = c e 1 + s e 2 ; e t = s e 1 + c e 2 [ σ rr σ rt σ tr σ tt ] = = [ c s s c ][ σ xx σ xy σ xy σ yy ][ c s s c c 2 σ xx + 2csσ xy + s 2 σ yy csσ xx + (c 2 s 2 )σ xy + csσ yy csσ xx + (c 2 s 2 )σ xy + csσ yy s 2 σ xx 2csσ xy + c 2 σ yy ]
CRACK TIP STRESSES : CARTESIAN σ yy σxy σ xx σ xx = K I 2πr f Ixx (θ) + K II 2πr f IIxx (θ) σ yy = K I 2πr f Iyy (θ) + K II 2πr f IIyy (θ) σ xy = K I 2πr f Ixy (θ) + K II 2πr f IIxy (θ) f Ixx (θ) = cos( θ 2 )[ 1 sin( θ 2 ) sin(3θ 2 )] f IIxx (θ) = sin( θ 2 )[ 2 + cos( θ 2 ) cos(3θ 2 )] f Iyy (θ) = cos( θ 2 ) [ 1 + sin( θ 2 ) sin(3θ 2 )] f IIyy (θ) = sin( θ 2 ) cos(θ 2 ) cos(3θ 2 ) f Ixy (θ) = sin( θ 2 ) cos(θ 2 ) cos(3θ 2 ) f IIxy (θ) = cos( θ 2 ) [ 1 sin( θ 2 ) sin(3θ 2 )]
CRACK TIP STRESSES : CYLINDRICAL σ rr σ rt θ σ tt σ rr = K I 2πr f Irr (θ) + K II 2πr f IIrr (θ) σ tt = K I 2πr f Itt (θ) + K II 2πr f IItt (θ) σ rt = K I 2πr f Irt (θ) + K II 2πr f IIrt (θ) f Irr (θ) = [ 5 4 cos(θ 2 ) 1 4 cos(3θ 2 )] f IIrr (θ) = [ 5 4 sin(θ 2 ) + 3 4 sin(3θ 2 )] f Itt (θ) = [ 3 4 cos(θ 2 ) + 1 4 cos(3θ 2 )] f IItt (θ) = [ 3 4 sin(θ 2 ) 3 4 sin(3θ 2 )] f Irt (θ) = [ 1 4 sin(θ 2 ) + 1 4 sin(3θ 2 )] f IIrt (θ) = [ 1 4 cos(θ 2 ) + 3 4 cos(3θ 2 )]
MULTI-MODE LOAD σ 22σ12 σ 22 σ 12 σ 12 2a σ 11 2a σ 11 e 2 e 2 e 1 θ σ 12 σ 22 e 1 load [ σ 11 σ 12 σ 21 σ 22 ] = c 2 σ 11 + 2csσ 12 + s 2 σ 22 csσ 11 + (c 2 s 2 )σ 12 + csσ 22 csσ 11 + (c 2 s 2 )σ 12 + csσ 22 s 2 σ 11 2csσ 12 + c 2 σ 22 crack tip stresses s ij = K I 2πr f Iij (θ) + K II 2πr f IIij (θ) with K I = β σ 22 πa ; KII = γ σ 12 πa σ 11 does not do anything
EXAMPLE MULTI-MODE LOAD 2a σ kσ σ 12 σ 22 2a σ 11 σ 12 θ load σ 11 = c 2 σ 11 + 2csσ 12 + s 2 σ 22 = c 2 kσ + s 2 σ σ 22 = s 2 σ 11 2csσ 12 + c 2 σ 22 = s 2 kσ + c 2 σ σ 12 = csσ 11 + (c 2 s 2 )σ 12 + csσ 22 = cs(1 k)σ crack tip stresses s ij = K I 2πr f Iij (θ) + K II 2πr f IIij (θ) stress intensity factors K I = β I σ 22 πa = βi (s 2 k + c 2 )σ πa K II = β II σ 12 πa = βii cs(1 k)σ πa
EXAMPLE MULTI-MODE LOAD σ t 2a σ a p R t θ σ 22 σ 12 σ 11 σ t = pr t = σ ; σ a = pr 2t = 1 2 σ k = 1 2 σ 22 = s 2 1 2 σ + c2 σ ; σ 12 = cs(1 1 2 )σ = 1 2 csσ K I = σ 22 πa = ( 1 2 s2 + c 2 )σ πa = ( 1 2 s2 + c 2 ) pr t πa K II = σ 12 πa = 1 2 csσ = 1 2 cs pr t πa
CRACK GROWTH DIRECTION
CRACK GROWTH DIRECTION criteria for crack growth direction : maximum tangential stress (MTS) criterion strain energy density (SED) criterion requirement : crack tip stresses in cylindrical coordinates
MAXIMUM TANGENTIAL STRESS CRITERION Erdogan & Sih (1963) σ rr σ rt θ σ tt Hypothesis : crack growth towards local maximum of σ tt σ tt θ = 0 and 2 σ tt θ 2 < 0 θ c σ tt (θ = θ c ) = σ tt (θ = 0) = K Ic 2πr crack growth
3 2 σ tt θ = 0 K I [ 1 2πr 4 sin(θ 2 ) 1 4 sin(3θ 2 )] + 3 2 K II [ 1 2πr 4 cos(θ 2 ) 3 4 cos(3θ 2 )] = 0 K I sin(θ) + K II {3 cos(θ) 1} = 0 3 4 2 σ tt θ 2 < 0 K I [ 1 2πr 4 cos(θ 2 ) 3 4 cos(3θ 2 )] + 3 4 K II 2πr [ 1 4 sin(θ 2 ) + 9 4 sin(3θ 2 )] < 0 σ tt (θ = θ c ) = K Ic 2πr 1 K I 4 [ 3 cos( θ c K 2 ) + cos( 3θ c 2 )] + Ic 1K II 4 [ 3 sin( θ c K 2 ) 3 sin( 3θ c 2 )] = 1 Ic
MODE I LOAD K II = 0 σ tt θ = K I sin(θ) = 0 θ c = 0 2 σ tt θ 2 < 0 θc σ tt (θ c ) = K Ic 2πr K I = K Ic
MODE II LOAD K I = 0 σ tt θ = K II(3 cos(θ c ) 1) = 0 2 σ tt θ 2 θ c = ± arccos( 1 3 ) = ±70.6o < 0 θc θ c = 70.6 o σ tt (θ c ) = K Ic 2πr K IIc = 3 4 K Ic τ θ c τ
MULTI-MODE LOAD K I [ sin( θ 2 ) sin(3θ 2 )] + K II[ cos( θ 2 ) 3 cos(3θ 2 )] = 0 K I [ cos( θ 2 ) 3 cos(3θ 2 )] + K II[sin( θ 2 ) + 9 sin(3θ 2 )] < 0 K I [3 cos( θ 2 ) + cos(3θ 2 )] + K II[ 3 sin( θ 2 ) 3 sin(3θ 2 )] = 4K Ic K I f 1 K II f 2 = 0 K I f 2 + K II f 3 < 0 K I f 4 3K II f 1 = 4K Ic ( KI K Ic ( KI ( KI K Ic ) f 1 ) f 2 + ) f 4 3 K Ic ( KII K Ic ( KII K Ic ( KII K Ic ) f 2 = 0 ) f 3 < 0 ) f 1 = 4
MULTI-MODE LOAD 0 10 20 30 θ c 40 50 60 70 0 0.2 0.4 0.6 0.8 1 K I /K Ic 0.9 0.8 0.7 0.6 K II /K Ic 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 K I /K Ic Plot made with
STRAIN ENERGY DENSITY (SED) CRITERION Sih (1973) σ rr σ rt θ σ tt U i = Strain Energy Density (Function) = εij 0 σ ij dε ij S = Strain Energy Density Factor = ru i = S(K I,K II,θ) Hypothesis : crack growth towards local minimum of SED S θ = 0 and 2 S θ > 0 θ 2 c S(θ = θ c ) = S(θ = 0, pl.strain) = S c crack growth
SED U i = 1 2E (σ2 xx + σ 2 yy + σ 2 zz) ν E (σ xxσ yy + σ yy σ zz + σ zz σ xx ) + 1 2G (σ2 xy + σ 2 yz + σ 2 zx) σ xx = K I 2πr cos( θ 2 ) [ 1 sin( θ 2 ) sin(3θ 2 )] K II 2πr sin( θ 2 )[ 2 + cos( θ 2 ) cos(3θ 2 )] σ yy = K I 2πr cos( θ 2 ) [ 1 + sin( θ 2 ) sin(3θ 2 )] + K II 2πr sin( θ 2 ) cos(θ 2 ) cos(3θ 2 ) σ xy = K I 2πr sin( θ 2 ) cos(θ 2 ) cos(3θ 2 ) + K II 2πr cos( θ 2 )[ 1 sin( θ 2 ) sin(3θ 2 )]
SED FACTOR S = ru i = S(K I, K II, θ) = a 11 k 2 I + 2a 12 k I k II + a 22 k 2 II with a 11 = 1 16G (1 + cos(θ))(κ cos(θ)) a 12 = 1 16G sin(θ){2 cos(θ) (κ 1)} a 22 = 1 16G {(κ + 1)(1 cos(θ))+ (1 + cos(θ))(3 cos(θ) 1)} k i = K i / π S θ = 0 k 2 I 16G {2 sin(θ) cos(θ) (κ 1) sin(θ)} + k Ik II 16G {2 4 sin2 (θ) (κ 1) cos(θ)} + k 2 II { 6 sin(θ) cos(θ) + (κ 1) sin(θ)} = 0 16G 2 S θ 2 > 0 k 2 I 16G {2 4 sin2 (θ) (κ 1) cos(θ)} + k Ik II { 8 sin(θ) cos(θ) + (κ 1) sin(θ)} + 16G k 2 II 16G { 6 + 12 sin2 (θ) + (κ 1) cos(θ)} > 0
MODE I LOAD S = a 11 k 2 I = σ2 a {1 + cos(θ)}{κ cos(θ)} 16G S θ = sin(θ){2 cos(θ) (κ 1)} = 0 θ c = 0 or arccos ( 1 2 (κ 1)) 2 S θ 2 = 2 cos(2θ) (κ 1) cos(θ) > 0 θ c = 0 S(θ c ) = σ2 a 16G {2}{κ 1} = σ2 a 8G (κ 1) S c = S(θ c, pl.strain) = (1 + ν)(1 2ν) 2πE K 2 Ic
MODE II LOAD S = a 22 k 2 II = τ2 a 16G [(κ + 1){1 cos(θ)} + {1 + cos(θ)}{3 cos(θ) 1}] S θ = sin(θ) [ 6 cos(θ) + (κ 1)] = 0 2 S θ 2 = 6 cos2 (θ) + (κ 1) cos(θ) > 0 θ c = ± arccos ( 1(κ 1)) 6 S(θ c ) = τ2 a 16G { 1 12 ( κ2 + 14κ 1)} S(θ c ) = S c τ c = 1 192GSc a κ 2 + 14κ 1
MULTI-MODE LOAD; PLANE STRAIN 10 0 10 ν=0.0.75 ν=0.1 ν=0.2 ν=0.3 ν=0.4 ν=0.495 20 30 θ c 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K /K I Ic 1 0.9 0.8 0.7 K II /K Ic 0.6 0.5 0.4 ν=0.0.75 ν=0.1 ν=0.2 ν=0.3 ν=0.4 ν=0.495 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K /K I Ic Plots made with sedmm2.m
MULTI-MODE LOAD; PLANE STRESS 10 0 10 ν=0.0.75 ν=0.1 ν=0.2 ν=0.3 ν=0.4 ν=0.495 20 30 θ c 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K /K I Ic 1 0.9 0.8 0.7 0.6 K II /K Ic 0.5 0.4 0.3 0.2 ν=0.0.75 ν=0.1 ν=0.2 ν=0.3 ν=0.4 ν=0.495 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K /K I Ic Plots made with sedmm2.m
MULTI-MODE LOAD; PLANE STRAIN k I = σ a sin 2 (β) ; k II = σ a sin(β) cos(β) S = σ 2 a sin 2 (β) { a 11 sin 2 (β) + 2a 12 sin(β) cos(β) + a 22 cos 2 (β) } S θ = (κ 1) sin(θ c 2β) 2 sin{2(θ c β)} sin(2θ c ) = 0 2 S θ 2 = (κ 1) cos(θ c 2β) 4 cos{2(θ c β)} 2 cos(2θ c ) > 0 σ β 2a σ θ c 90 θ c 80 70 ν = 0.5 60 50 40 30 ν = 0 20 ν = 0.1 10 0 0 10 20 30 40 50 60 70 80 90 β From Gdoutos
DYNAMIC FRACTURE MECHANICS
DYNAMIC FRACTURE MECHANICS impact load (quasi)static load fast fracture - kinetic approach - static approach
CRACK GROWTH RATE Mott (1948) energy balance du e da du i da = du a da + du d da + du k da σ y 2a a thickness B x σ du e da = 0 ; du d da = 0 U a = 4aBγ U i = 2πa 2 B 1 σ 2 2 E du a da = 4γB du i da = 2πaBσ2 E
KINETIC ENERGY U k = 1 2 ρb Ω material velocity ( u 2 x + u 2 y) dxdy u x u y = du y dt = du y da da dt = du y da s U k = 1 2 ρs2 B assumption Ω ( ) 2 duy dxdy da ds da = 0 du k da = 1 2 ρs2 B Ω u y = 2 2 σ E ( ) 2 duy dxdy da d da a2 ax du y da = 2 σ E 2a x a2 ax du k σ B( da = ρs2 E ( σ = ρs 2 B E ) 2 a Ω ) 2 a k(a) 1 a 3 x 2 (x 2a) (a x) 2 dxdy
ENERGY BALANCE 2πaσ 2 E = 4γ + ρs 2 ( σ E) 2 ak s = ( )1 E ρ 2 ( 2π k )1 2 ( 1 2γE πaσ 2 )1 2 2π k 0.38 ; a c = 2γE πσ 2 ; c = ( ds ) da 0!! E ρ ( s = 0.38 c 1 a c a )1 2 s 0.38 c a a c
EXPERIMENTAL CRACK GROWTH RATES steel copper aluminum glass rubber E [GPa] 210 120 70 70 20 ρ [kg/m 2 ] 7800 8900 2700 2500 900 ν 0.29 0.34 0.34 0.25 0.5 c [m/sec] 5190 3670 5090 5300 46 s [m/sec] 1500 2000 s/c 0.29 0.38 0.2 < s c < 0.4
ELASTIC WAVE SPEEDS C 0 = elongational wave speed = C 1 = dilatational wave speed = C 2 = shear wave speed = E ρ κ + 1 κ 1 µ ρ µ ρ C R = Rayleigh velocity = 0.54 C 0 á 0.62 C 0
CORRECTIONS ( Dulancy & Brace (1960) s = 0.38 C 0 1 a ) c a ( Freund (1972) s = C R 1 a ) c a
CRACK TIP STRESS Yoffe (1951) : σ Dij = K D 2πr f ij (θ,r, s, E, ν)
CRACK BRANCHING Yoffe (1951) σ Dij = K ID 2πr f ij (θ, r, s, E,ν) σ Dtt (θ) σ Dtt (θ = 0) 0.9 max 1 0.6 0 s c R 0.87 σ tt π π 2 θ crack branching volgens MTS Source: Gdoutos (1993) p.245
FAST FRACTURE AND CRACK ARREST K D K Dc (s, T) K D < min 0<s<C R K Dc (s, T) = K A crack growth crack arrest
EXPERIMENTS Source: KAN1985 p.210 High Speed Photography : 10 6 frames/sec Robertson : CA Temperature (CAT) test (KAN1985 p.258)
PLASTIC CRACK TIP ZONE
VON MISES AND TRESCA YIELD CRITERIA Von Mises Tresca W d = W d c (σ 1 σ 2 ) 2 +(σ 2 σ 3 ) 2 +(σ 3 σ 1 ) 2 = 2σ 2 y τ max = τ maxc σ max σ min = σ y yield surface in principal stress space
PRINCIPAL STRESSES AT THE CRACK TIP plane stress state σ zz = σ zx = σ zy = 0 σ xx σ xy 0 σ = σ xy σ yy 0 det(σ σi) = 0 0 0 0 characteristic equation σ [ σ 2 σ(σ xx + σ yy ) + (σ xx σ yy σ 2 xy) ] = 0 σ 1 = 1 2 (σ xx + σ yy ) + { 1 4 (σ } xx σ yy ) 2 + σ 2 1/2 xy σ 2 = 1 2 (σ xx + σ yy ) { 1 4 (σ } xx σ yy ) 2 + σ 2 1/2 xy σ 3 = 0 plane strain state σ 3 = ν(σ 1 + σ 2 )
PRINCIPAL STRESSES AT CRACK TIP σ 1 = 1 2 (σ xx + σ yy ) + { 1 4 (σ xx σ yy ) 2 + σ 2 xy} 1/2 σ 2 = 1 2 (σ xx + σ yy ) { 1 4 (σ xx σ yy ) 2 + σ 2 xy} 1/2 σ 3 = 0 or σ 3 = ν(σ 1 + σ 2 ) crack tip stresses σ ij = K I 2πr f Iij (θ) σ 1(+),2( ) = 1 4 K I 2πr [ cos( θ 2 )± ] { 2 cos( θ 2 ) sin(θ 2 ) sin(3θ 2 )} 2 { + sin( θ 2 ) cos(θ 2 ) cos(3θ 2 )} 2 σ 1 = K I 2πr cos( θ 2 ){1 + sin(θ 2 )} σ 2 = K I 2πr cos( θ 2 ){1 sin(θ 2 )} σ 3 = 0 or σ 3 = 2νK I 2πr cos( θ 2 ) plane stress σ 1 > σ 2 > σ 3 plane strain σ 1 > σ 2 > σ 3 or σ 1 > σ 3 > σ 2
1000 ν = 0.25 1000 ν = 0.35 800 800 600 600 σ σ 400 400 200 200 0 0 20 40 60 80 100 θ 0 0 20 40 60 80 100 θ 1000 ν = 0.45 1000 ν = 0.5 800 800 600 600 σ σ 400 400 200 200 0 0 20 40 60 80 100 θ 0 0 20 40 60 80 100 θ Plots are made with plstr.m.
VON MISES PLASTIC ZONE Von Mises yield criterion (σ 1 σ 2 ) 2 + (σ 2 σ 3 ) 2 + (σ 3 σ 1 ) 2 = 2σ 2 y plane stress σ 3 = 0 (σ 1 σ 2 ) 2 + σ 2 2 + σ 2 1 = 2σ 2 y K 2 I 2πr y cos 2 ( θ 2 )[ 6 sin 2 ( θ 2 ) + 2] = 2σ 2 y r y = K2 I 2πσ 2 y cos 2 ( θ 2 ) [ 1 + 3 sin 2 ( θ 2 )] = K2 I 4πσ 2 y [ 1 + cos(θ) + 3 2 sin2 (θ) ] plane strain σ 3 = ν(σ 1 + σ 2 ) (ν 2 ν + 1)(σ 2 1 + σ 2 2) + (2ν 2 2ν 1)σ 1 σ 2 = σ 2 y K 2 I 2πr y cos 2 ( θ 2 )[ 6 sin 2 ( θ 2 ) + 2(1 2ν)2] = 2σ 2 y r y = K2 I 4πσ 2 y [ (1 2ν) 2 {1 + cos(θ)} + 3 2 sin2 (θ) ]
VON MISES PLASTIC ZONE 1 Von Mises plastic zones pl.stress pl.strain 0.5 0 0.5 1 0.5 0 0.5 1 1.5 Plot made with plazone.m.
TRESCA PLASTIC ZONE σ max σ min = σ y plane stress {σ max,σ min } = {σ 1,σ 3 } K I 2πry [ cos( θ 2 ) + cos( θ 2 ) sin(θ 2 ) ] = σ y r y = K2 I 2πσ 2 y [ cos( θ 2 ) + cos( θ 2 ) sin(θ 2 ) ] 2 plane strain I σ 1 > σ 2 > σ 3 {σ max, σ min } = {σ 1, σ 3 } r y = K2 I 2πσ 2 y [ (1 2ν) cos( θ 2 ) + cos( θ 2 ) sin(θ 2 ) ] 2 plane strain II σ 1 > σ 3 > σ 2 {σ max, σ min } = {σ 1, σ 2 } r y = K2 I 2πσ 2 y sin 2 (θ)
TRESCA PLASTIC ZONE 1 0.5 Tresca plastic zones pl.stress pl.strain sig3 = min pl.strain sig2 = min 0 0.5 1 0.5 0 0.5 1 1.5 Plot made with plazone.m.
INFLUENCE OF THE PLATE THICKNESS CRITICAL PLATE THICKNESS B c > 25 3π ( KIc σ y ) 2 > 2.5 ( ) 2 KIc σ y
SHEAR PLANES Source: Gdoutos p.60/61/62; Kanninen p.176
IRWIN PLASTIC ZONE CORRECTION σ xx σ yy σ xx σ yy σ y σ y a r y r a r y r p r θ = 0 σ xx = σ yy = K I 2πr yield σ xx = σ yy = σ y r y = 1 2π equilibrium no longer satisfied correction required shaded area equal ( ) 2 KI σ y σ y r p = ry 0 σ yy (r) dr = K I 2π ry 0 r 1 2 dr = 2K I 2π ry r p = 2K I 2π ry σ y r p = 1 π ( KI σ y ) 2 = 2 r y
DUGDALE-BARENBLATT PLASTIC ZONE CORRECTION y σ y σ x a r p load σ load σ y K I (σ) = σ π(a + r p ) K I (σ y ) = 2σ y a + rp π σ ( ) a arccos a + r p singular term = 0 K I (σ) = K I (σ y ) ( ) a πσ = cos a + r p 2σ y r p = πk2 I 8σ 2 y
PLASTIC CONSTRAINT FACTOR 1 2 {(σ 1 σ 2 ) 2 + (σ 2 σ 3 ) 2 + (σ 3 σ 1 ) 2 } = [ ] 1 n m + n2 + m 2 mn σ max = σ y pcf = σ max σ y = 1 1 n m + n2 + m 2 mn PCF AT THE CRACK TIP pl.sts n = [ 1 sin( θ 2 )] / [ 1 + sin( θ 2 )] ; m = 0 pl.stn n = [ 1 sin( θ 2 )] / [ 1 + sin( θ 2 )] ; m = 2ν/ [ 1 + sin( θ 2 )] PCF AT THE CRACK TIP IN THE CRACK PLANE pl.sts n = 1 ; m = 0 pcf = 1 pl.stn n = 1 ; m = 2ν pcf = 1 1 4ν + 4ν 2
PLASTIC ZONES IN THE CRACK PLANE r y r p criterion state r y or r p (K I /σ y ) 2 Von Mises plane stress Von Mises plane strain Tresca plane stress 1 2π 1 18π 1 2π Tresca plane strain σ 1 > σ 2 > σ 3 1 18π ( KI σ y ( KI σ y ( KI σ y ( KI σ y ) 2 0.1592 ) 2 0.0177 ) 2 0.1592 ) 2 0.0177 Tresca plane strain σ 1 > σ 3 > σ 2 0 0 Irwin plane stress Irwin plane strain (pcf = 3) Dugdale plane stress Dugdale plane strain (pcf = 3) 1 π 1 π π 8 π 8 ( KI σ y ( KI ) 2 0.3183 ) 2 0.0354 3σ y ) 2 0.3927 ( KI σ y ( KI 3σ y ) 2 0.0436
SMALL SCALE YIELDING LEFM & SSY correction effective crack length a eff Irwin / Dugdale-Barenblatt correction SSY : outside plastic zone : K I (a eff )-stress a eff = a + (r y r p ) K I = β I (a eff )σ πa eff
NONLINEAR FRACTION MECHANICS
CRACK-TIP OPENING DISPLACEMENT crack tip displacement u y = σ πa r 2µ 2π [ sin( 1 2 θ) { κ + 1 2 cos 2 ( 1 2 θ)}] displacement in crack plane θ = π; r = a x u y = (1 + ν)(κ + 1) E σ 2 2a(a x) Crack Opening Displacement (COD) δ(x) = 2u y (x) = (1 + ν)(κ + 1) E σ 2a(a x) Crack Tip Opening Displacement (CTOD) δ t = δ(x = a) = 0
CTOD BY IRWIN σ xx σ xx σ yy σ yy σ y σ y a r y r a r y r p r effective crack length a eff = a + r y = a + 1 2π ( ) 2 KI σ y
CTOD BY IRWIN δ(x) = = (1 + ν)(κ + 1) E (1 + ν)(κ + 1) E σ 2a eff (a eff x) σ 2(a + r y )(a + r y x) (1 + ν)(κ + 1) δ t = δ(x = a) = σ 2(a + r y )r y E (1 + ν)(κ + 1) = σ 2ar y + 2r E 2 y (1 + ν)(κ + 1) E σ 2ar y plane stress : δ t = 4 K 2 I = 4 G π Eσ y πσ y [ ] 1 4(1 ν 2 ) plane strain : δ t = 3 π K 2 I Eσ y
CTOD BY DUGDALE y σ y σ x a r p σ effective crack length ( ) 2 KI a eff = a + r p = a + π 8 σ y
CTOD BY DUGDALE displacement from requirement singular term = 0 : ū y (x) ū y (x) = (a + r p)σ y πe [ { x sin 2 (^γ γ) ln a + r p sin 2 (^γ + γ) cos(^γ) ln } + { } ] 2 sin(^γ) + sin(γ) sin(^γ) sin(γ) ( ) x γ = arccos a + r p ; ^γ = π 2 σ σ y Crack Tip Opening Displacement δ t = lim 2ū x a y(x) = 8σ { ( va π πe ln sec 2 series expansion & σ σ y σ σ y )} plane stress : δ t = K2 I = G Eσ y σ y [ ] 1 plane strain : δ t = (1 ν 2 ) K2 I 2 Eσ y
CTOD CRACK GROWTH CRITERION δ t (G,K I ) at LEFM δ t = measure for deformation at crack tip (LEFM) δ t = measure for (large) plastic deformation at crack tip (NLFM) criterion δ t = δ tc ( ε, T) δ t calculate or measure δ tc experimental determination (ex. BS 5762)
J-INTEGRAL n t x 2 Ω V S e 2 Γ positive e 1 x 1 definition J k -integral (vector!!) J k = Γ ( ) u i Wn k t i x k dγ W = specific energy = J = J 1 = Γ Epq ( ) u i Wn 1 t i x 1 0 σ ij dε ij dγ [ ] N m
INTEGRAL ALONG CLOSED CURVE J k = Γ ) (Wδ jk σ ij u i,k n j dγ inside Γ no singularities Stokes (Gauss in 3D) Ω ( dw dε mn ε mn x j δ jk σ ij,j u i,k σ ij u i,kj ) dω homogeneous hyper-elastic σ mn = W ε mn linear strain ε mn = 1 2 (u m,n+u n,m ) equilibrium equations σ ij,j = 0 Ω Ω { 1 2 σ mn(u m,nk + u n,mk ) σ ij u i,kj } (σ mn u m,nk σ ij u i,kj ) dω = 0 dω =
PATH INDEPENDENCE x 2 Ω n e 2 e 1 Γ+ Γ n Γ B x 1 Γ A f k dγ + f k dγ + f k dγ + f k dγ = 0 Γ A Γ B Γ Γ + f 1 dγ + f 1 dγ + f 1 dγ + f 1 dγ = 0 Γ A Γ B Γ Γ + no loading of crack faces : n 1 = 0 ; t i = 0 on Γ + and Γ f 1 dγ + f 1 dγ = 0 Γ A Γ B f 1 dγ = J 1A ; f 1 dγ = J 1B Γ A Γ B J 1A + J 1B = 0 J 1A = J 1B
RELATION J K lin. elast. material : W = 1 2 σ mnε mn = 1 4 σ mn(u m,n + u n,m ) J k = Γ = Mode I + II + III Γ ( 1 4 σ mn(u m.n + u n,m )δ jk σ ij u i,k ) n j dγ ( 1 2 σ mnu m,n δ jk σ ij u i,k )n j dγ σ ij = 1 2πr [K I f Iij + K II f IIij + K III f IIIij ] u i = u Ii + u IIi + u IIIi substitution and integration over Γ = circle J 1 = J 2 = (κ + 1)(1 + ν) 4E (κ + 1)(1 + ν) K I K II 2E ( K 2 I + KII) 2 (1 + ν) + K 2 III E
RELATION J G Mode I J 1 = J = (κ + 1)(1 + ν) 4E K 2 I = G plane stress plane strain κ + 1 = 3 ν 1 + ν + 1 + ν 1 + ν = 4 1 + ν J = 1 E K2 I κ + 1 = 4 4ν J = (1 ν2 ) E K 2 I
RELATION J δ T plane stress Irwin J = π 4 σ yδ t Dugbale J = σ y δ t plane strain Irwin J = π 4 3 σy δ t Dugbale J = 2σ y δ t PLASTIC CONSTRAINT FACTOR J = mσ y δ t m = 0.111 + 0.817 a W + 1.36 σ u σ y
RAMBERG-OSGOOD MATERIAL LAW ε = σ ( ) n σ + α ε y0 σ y0 σ y0 n strain hardening parameter (n 1) n = 1 linear elastic n ideal plastic 5 Ramberg Osgood for α = 0.01 n = 1 4 n = 3 3 σ/σ y0 2 n = 5 n = 7 1 n = 13 0 0 1 2 3 4 5 6 ε/ε y0
HRR-SOLUTION HRR - solution σ ij = σ y0 β r 1 n+1 σ ij (θ) u i = αε y0 β n r 1 n+1 ũ i (θ) with : β = [ J ασ y0 ε y0 I n ] 1 n+1 (I n from num. anal.) I n 5 2.5 plane strain plane stress 0 5 10 15 n
J-INTEGRAL CRACK GROWTH CRITERION LEFM : J k G (K I, K II, K III ) NLFM : Ramberg-Osgood : J determines crack tip stress criterion J = J c calculate J J Ic from experiments e.g. ASTM E813
NUMERICAL FRACTURE MECHANICS
NUMERICAL FRACTURE MECHANICS Methods EEM ; BEM Calculations G K δ t J Simulation crack growth
QUADRATIC ELEMENTS ξ 2 ξ 1 4 7 ξ 2 3 ξ 1 8 6 1 5 3 2 4 7 3 4 7 6 8 1 5 6 2 1 8 5 2 isoparametric coordinates : 1 ξ i 1 shape functions for each node n ψ n (ξ 1,ξ 2 ) = quadratic in ξ 1 and ξ 2
CRACK TIP MESH bad approximation stress field results are mesh-dependent 1/ r
SPECIAL ELEMENTS enriched elements crack tip field added to element displacement field structure K and changes f transition elements for compatibility hybrid elements modified variational principle
QUARTER POINT ELEMENTS 8 4 7 3 6 1 5 2 p 3p 3 5 6 1 4 2 4 8 1 7 p 3 6 5 2 3p 4 1 3 2 Distorted Quadratic Quadrilateral (1/ r) Distorted Quadratic Triangle (1/ r) Collapsed Quadratic Quadrilateral (1/ r) Collapsed Distorted Linear Quadrilateral (1/r) good approximation stress field (1/ r or 1/r) bad approximation non-singular stress field standard FEM-programs can be used
CRACK TIP ROZET Quarter Point Elements : 8x Transition Elements : number is problem dependent Buffer Elements
ONE-DIMENSIONAL CASE ξ = 1 ξ = 0 ξ = 1 1 x 3 2 position x= 1 2 ξ(ξ 1)x 1 + 1 2 ξ(ξ + 1)x 2 (ξ 2 1)x 3 = 1 2 ξ(ξ + 1)L (ξ2 1)x 3 displacement and strain u = 1 2 ξ(ξ 1)u 1 + 1 2 ξ(ξ + 1)u 2 (ξ 2 1)u 3 du dξ = (ξ 1 2 )u 1 + (ξ + 1 2 )u 2 2ξu 3 du dx = du dξ dξ dx = du dξ /dx dξ
MID POINT ELEMENT mid-point element : x 3 = 1 2 L ξ = 1 ξ = 0 ξ = 1 1 x 3 2 x = 1 2 ξ(ξ + 1)L (ξ2 1) 1 2 L = 1 2 (ξ + 1)L dx dξ = 1 2 L du du dx = dξ 1 2 L du dx x=0 ξ= 1 = ( 2 L ) {( 3 2 ) u1 + ( ) } 1 2 u2 + 2u 3
QUARTER POINT ELEMENT quarter-point element : x 3 = 1 4 L ξ = 1ξ = 0 ξ = 1 1 x 3 2 x = 1 2 ξ(ξ + 1)L (ξ2 1) 1 4 L = 1 4 (ξ + 1)2 L ξ + 1 = dx dξ = 1 2 (ξ + 1)L = xl 4x L du du dx = dξ du xl dx x=0 ξ= 1 = singularity 1 x
VIRTUAL CRACK EXTENSION METHOD (VCEM) u u a a + a I II fixed grips G = 1 B du e da = 0 du i da 1 B U i (a + a) U i (a) a
VCEM : STIFFNESS MATRIX VARIATION BG = du i da = C 1 2ũT a ũ with C = C(a+ a) C(a) G from analysis crack tip mesh only nodal point displacement : ± 0.001 element size not possible with crack tip in interface unloaded crack plane no thermal stresses
STRESS INTENSITY FACTOR calculate G I and G II with VCEM calculate K I and K II from K 2 I = E G I ; K 2 II = E G II plane stress E = E plane strain E = E/(1 ν 2 ) difficult for crack propagation study
SIF : STRESS FIELD extrapolation to crack tip ( ) K I = lim 2πr σ22 θ=0 r 0 ( ) K II = lim 2πr σ12 θ=0 r 0 p p 4 p 3 p 2 1 θ r K K p1kp2 K p3 Kp4 r 1 r 2 r 3 r 4 r questions : which elements? how much elements? which integration points?
SIF : DISPLACEMENT FIELD crack tip displacement y-component u y = 4(1 ν2 ) E [ K I = lim r 0 E 4(1 ν 2 ) r 2π K I g ij (θ) ] 2π r u y(θ = 0) more accurate than SIF from stress field
J-INTEGRAL J = Γ ( ) u i Wn 1 t i x 1 with W = dγ ε σ ij dε ij 0
J-INTEGRAL : DIRECT CALCULATION J = 2 y [ W 2 x ( σ xx u x x + σ yx [( σ xy u x x + σ yy )] u y x )] u y x dy dx W = E 2(1 ν 2 ) (ε2 xx + 4νε xx ε yy + 2(1 ν)ε 2 xy + ε 2 yy) path through integration points no need for quarter point elements
J-INTEGRAL : DOMAIN INTEGRATION x 2 Ω n q = 0 e 2 e1 Γ + Γ n Γ B Γ A x 1 Ω q = 1 J = Ω q x j ( ) u i σ ij Wδ 1j x 1 dω interpolation q e = Ñ T (ξ ) q e
DE LORENZI J-INTEGRAL : VCE TECHNIQUE J= Ω q x j ( ) u i σ i1 Wδ 1j x 1 u i qp i dγ x 1 Γ s Ω dω q(ρq i ρü i ) u i x 1 dω + Ω qσ ij ε o ij x 1 dω rigid region elongation a of crack translation δx 1 of internal nodes fixed position of boundary q = δx 1 a = shift function (0 < q < 1)
CRACK GROWTH SIMULATION Node release Moving Crack Tip Mesh Element splitting Smeared crack approach
NODE RELEASE node collocation technique
MOVING CRACK TIP MESH
ELEMENT SPLITTING
SMEARED CRACK APPROACH e 2 e 1 e 2 n 2 e 1 σ 2 n 1 σ1 n 2 n 1
TELETEKST WO 3 OKTOBER 2007 Van de 274 stalen bruggen in ons land kampen er 25 met metaalmoeheid. Dat is de uitkomst van een groot onderzoek van het ministerie van Verkeer. Bij twaalf bruggen zijn de problemen zo groot dat noodmaatregelen nodig zijn. Ook de meer dan 2000 betonnen bruggen en viaducten zijn onderzocht. De helft daarvan moet nog nader worden bekeken. Ze gaan mogelijk minder lang mee dan was berekend, maar de veiligheid komt volgens het ministerie niet in gevaar. Verkeersbeperkende maatregelen zijn dan ook niet nodig. Die werden in april wel getroffen voor het vrachtverkeer over de Hollandse Brug bij Almere.
FATIGUE ± 1850 (before Griffith!) : cracks at diameter-jumps in axles carriages / trains failure due to cyclic loading with small amplitude Wöhler : systematic experimental examination cyclic loading : variable mechanical loads vibrations pressurization / depressurization thermal loads (heating / cooling) random external loads
CRACK SURFACE clam shell markings (beach marks) - irregular crack growth - crack growth under changing conditions striations - sliding of slip planes - plastic blunting / sharpening of crack tip - regular crack growth
EXPERIMENTS full-scale testing a.o. train axles airplanes laboratory testing harmonic loading constant force/moment strain/deflection SIF
TRAIN AXLE D = 0.75 [m] 1 rev = πd = π 0.75 2.25 [m] 1 km = 1000 m = 1000 2.25 = 4000 9 445 [c(ycles)] 1 day Maastricht - Groningen = 2.5 333 [km] = 1000 [km] 1 day Maastricht - Groningen = 445 10 3 [c] 1 year = 300 445 10 3 [c] = 1335 10 5 [c] 1.5 10 8 [c] frequency : 100 [km/h] = 445 10 2 [c/h] = 44500 3600 = 12.5 [c/sec] = 12.5 [Hz]
FATIGUE LOAD stress controlled σ σ max σ m σ min 0 0 i i + 1 t N σ = σ max σ min ; σ a = 1 2 σ σ m = 1 2 (σ max + σ min ) ; R σ = σ min /σ max ; σ a = 1 R σ m 1 + R - frequency bending 30-80 Hz tensile electric 50-300 Hz mechanic < 50 Hz hydraulic 1-50 Hz - no influence frequency for ± 5000 [c/min] (metals) strain controlled ε ; ε a ; ε m ; R ε
FATIGUE LIMIT σ σ th N σ < σ th : no increase of damage materials with fatigue limit mild steel low strength steels Ti / Al / Mg -alloys materials without fatigue limit some austenitic steels high strength steels most non-ferro alloys Al / Mg-alloys
(S-N)-CURVE B.S. 3518 part I 1984 : S = σ max S σ th 0 0 log(n f ) reference : R = 1 and σ m = 0 σ max = 1 2 σ fatigue life : N f at σ max (= S) fatigue limit : σ th (= σ fat ) N f = (±10 9 ) fatigue strength : σ e = σ max when N f 50 10 6 steels : σ th 1 2 σ b
(S A -N)-CURVE B.S. 3518 part I 1984 : S a = 1 2 σ = σ a S a σ th 0 0 log(n f ) reference : R = 1 and σ m = 0 σ a = σ max (S a N) curve = (S N) curve
EXAMPLES 600 550 500 450 steelt1 400 σ max [MPa] 350 300 250 steel1020 200 150 100 Mgalloy Al2024T4 10 4 10 5 10 6 10 7 10 8 10 9 N f
INFLUENCE OF AVERAGE STRESS σ a σ m σ th 0 0 log(n f )
CORRECTION FOR AVERAGE STRESS Gerber (1874) σ a σ a = 1 ( ) 2 σm σ u Goodman (1899) σ a σ a = 1 σ m σ u Soderberg (1939) σ a σ a = 1 σ m σ y0 σ u σ y0 : tensile strength : initial yield stress
(P-S-N)-CURVE 550 500 450 400 σ max [MPa] 350 300 90% prob.failure 250 50% prob.failure 200 150 10% prob.failure 100 10 4 10 5 10 6 10 7 10 8 N f
HIGH/LOW CYCLE FATIGUE S a σ m = 0 0 0 LCF 4 5 HCF log(n f ) high cycle fatigue N f > ±50000 low stresses LEFM + SSY stress-life curve Basquin relation K max = βσ max πa ; Kmin = βσ min πa ; K = β σ πa low cycle fatigue N f < ±50000 high stresses EPFM strain-life curve Manson-Coffin relation
BASQUIN RELATION 1 2 σ = σ a = σ f(2n f ) b σn b f = constant σ f = fatigue strength coefficient σ b (monotonic tension) b = fatigue strength exponent (Basquin exponent) log ( ) σ 2 log(2n f )
MANSON-COFFIN RELATION 1 2 εp = ε f(2n f ) c ε p N c f = constant ε f = fatigue ductility coefficient ε b (monotonic tension) c = fatigue ductility exponent ( 0.5 < c < 0.7) log ( ) ε p 2 log(2n f )
TOTAL STRAIN-LIFE CURVE log( ε 2 ) log(n f ) ε 2 = εe 2 + εp 2 = 1 E σ f(2n f ) b + ε f(2n f ) c
INFLUENCE FACTORS load spectrum stress concentrations stress gradients material properties surface quality environment
LOAD SPECTRUM sign / magnitude / rate / history multi-axial lower f.limit than uni-axial
STRESS CONCENTRATIONS ρ σ max = K t σ nom σ th (notched) = 1 K t σ th (unnotched) σ th (notched) = 1 K f σ th (unnotched) ; 1 < K f < K t K f : fatigue strength reduction factor (effective stress concentration factor) notch sensitivity factor q(ρ) K f = 1 + q(k t 1) Peterson : q = 1 1 + a ρ with a = material parameter Neuber : q = 1 + 1 b ρ with b = grain size parameter
STRESS GRADIENTS full-scale experiments necessary
MATERIAL PROPERTIES grain size/structure : small grains higher f.limit at low temp. large grains higher f.limit at high temp. (less grain boundaries less creep) texture inhomogeneities and flaws residual stresses fibers and particles
10µm surface extrusions & intrusions notch + inclusion of O 2 etc. bulk defect internal surfaces internal grain boundaries / triple points (high T) voids manufacturing minimize residual tensile stresses surface finish minimize defects (roughness) surface treatment (mech/temp) residual pressure stresses coating environmental protection high σ y0 more resistance to slip band formation
ENVIRONMENT temperature creep - fatigue low temperature : ships / liquefied gas storage elevated temperature (T > 0.5T m ) : turbine blades creep mechanism : diffusion / dislocation movement / migration of vacancies / grain boundary sliding grain boundary voids / wedge cracks chemical influence corrosion-fatigue
CRACK GROWTH a I II III a c a c σ a f a 1 a i N i N f N I : N < N i - N i = fatigue crack initiation life - a i = initial fatigue crack II : N i < N < N f - slow stable crack propagation - a 1 = non-destr. inspection detection limit III : N f < N - global instability - towards catastrophic failure - a = a c : failure rest-life N r N f = 1 N N f
CRACK GROWTH MODELS da dn striation spacing 6 ( ) 2 K (Bates, Clark (1969)) E da dn f(σ, a) σm a n ; m 2 7 ; n 1 2 da dn δ t ( K)2 Eσ y (BRO263) da dn da K dn K E Source: HER1976a p515 Paris law : da dn = C( K)m
PARIS LAW log ( ) da dn [mm/c] -1-2 -3-4 -5-6 -7-8 -9 0 1 2 3 4 log C = 8.7 log( K) [MPa m] ( ) da da dn = C( K)m log = log(c) + m log( K) dn ( ) da log( K) = 0 log(c) = log = 8.7 dn C = 2 10 9 [mm] [MPa m] m m = ( 2) ( 4) (2) (1.5) = 4
LIMITS OF PARIS LAW log( da dn ) power law growth R rapid crack growth slow crack growth K th K c log( K) Paris law : da dn = C( K)m da dn = C(K max) m R 0 K K th roughness induced crack closure K < K th growth very short cracks (10 8 mm/cycle) dangerous overestimation of fatigue life σ m R ( 7 9 10 12 100 102 1)
PARIS LAW PARAMETERS material K th [MNm 3/2 ] m[-] C 10 11 [!] mild steel 3.2-6.6 3.3 0.24 structural steel 2.0-5.0 3.85-4.2 0.07-0.11 idem in sea water 1.0-1.5 3.3 1.6 aluminium 1.0-2.0 2.9 4.56 aluminium alloy 1.0-2.0 2.6-3.9 3-19 copper 1.8-2.8 3.9 0.34 titanium 2.0-3.0 4.4 68.8
CONVERSION da dn = C ( σ πa) m C = da dn ( σ πa) m [in] and [ksi] [m] and [MPa] 1 [ in ] [ ksi in ] = 0.0254[m] m {6.86 [ MPa] 0.0254[m]} m ( ) 0.0254 [ m] = (1.09) m [ MPa m ] m [m] and [MPa] [mm] and [MPa] 1 [ m ] [ MPa m ] = 10 3 [ mm ] m {[ MPa] 10 3 [ mm ]} m ( ) 10 3 = { [ mm ] 10 3 } m [ MPa mm ] m
FATIGUE LIFE : ANALYTICAL INTEGRATION integration Paris law fatigue life N f N f N i = [ ( σ) m β m C( 2 ) π) m (1 m 2 ) a(1 m f 1 ( ai a f ) (1 m 2 ) ]
FATIGUE LIFE : NUMERICAL INTEGRATION set σ, N, a c initialize N = 0, a = a 0 while a < a c end K = β σ π a da dn = C ( K)m a = da dn N a = a + a N = N + N
INITIAL CRACK LENGTH 120 C = 4.56e 11 ; m = 2.9 ; DN = 100 100 80 a0 = 1 [mm] a0 = 0.1 [mm] a [mm] 60 40 20 0 0 1 2 3 4 5 N [c] x 10 6 aluminum ; σ = 50 [MPa] 120 C = 0.25e 11 ; m = 3.3 ; DN = 1000 a0 = 1 [mm] 100 a0 = 1.1 [mm] 80 a [mm] 60 40 20 0 0 2 4 6 8 10 12 14 N [c] x 10 6 mild steel ; σ = 50 [MPa]
FATIGUE LOAD integration Paris law for different load levels fatigue life at a f = a c = 2γ π E σ 2 N f aluminum C = 4.56e 11 ; m = 2.9 E = 70 [GPa] ; γ = 1 [J/m 2 ] σ [MPa] 25 50 75 100 a 0 [mm] 0.1 0.1 0.1 0.1 a c [mm] 56 28 12.5 7 N f [c] 35070000 4610000 1366000 572000 60 50 a [mm] 40 30 20 10 0 0 1 2 3 4 N [c] x 10 7
OTHER CRACK GROW LAWS Erdogan (1963) ( general empirical law ) da dn = C(1 + β)m ( K K th ) n K Ic (1 + β) K with β = K max + K min K max K min Broek & Schijve (1963) da dn = CK2 max K Forman (1967) ( K max K c ) da dn = C( K) n (1 R)K c K with R = K min K max Donahue (1972) ( K K th ) da dn = C( K K th) m with K th = (1 R) γ K th (R = 0) Walker (1970) ( influence R ) { } m da K dn = C with m = 0.4 ; n = 0.5 (1 R) n
Priddle (1976) ( K K th & K max K c ) da dn = C ( K Kth K Ic K max ) m with K th = A(1 R) γ and 1 2 γ 1 [Schijve (1979)] McEvily & Gröger (1977) ( theoretical ) da dn = A ( ( K K th ) 2 1 + Eσ v with K th = A, K 0 1 R 1 + R K 0 K K Ic K max influence environment ) NASA / FLAGRO program (1989) da dn = C(1 R)m K n ( K K th ) p [(1 R)K Ic K] q m = p = q = 0 Paris m = p = 0,q = 1 Forman p = q = 0, m = (m w 1)n Walker
CRACK GROWTH AT LOW CYCLE FATIGUE σ slipline θ λσ θ Kanninen & Atkinson (1980) da dn = 3 sin 2 (θ) cos 2 ( θ 2 ) 9 sin(θ) ( ) K 1 βγ 1 2 Eσ v K 2 max {1 (1 λ) σ max σ v } θ = cos ( ) 1 1 3 β = 0.5 + 0.1R + 0.4R 2 γ da dn = 7 64 2 K Eσ v (1 0.2R 0.8R 2 ) K 2 max {1 (1 λ) σ max σ v }
CRACK GROWTH AT LOW CYCLE FATIGUE J-integral based Paris law da dn = C ( J) m with J = Γ { } W u i n 1 t i x 1 dγ ; W = ε pq max ε pqmin σ ij dε ij
LOAD SPECTRUM σ 0 N n 1 n 2 n 3 n 4 Palmgren-Miner (1945) L i=1 n i N if = 1 life time by piecewise integration da dn f( K,K max) no interaction interaction Palmgren-Miner no longer valid : L i=1 n i N if = 0.6 2.0
MINER S RULE 1 2 3 4 n 1 n 2 N 1f N 2f n 3 N 3f n 4 N 4f rest life N r N f : 1 1 n 1 2 3 4 N 1f ( 1 n 1 N 1f ) n 2 N 2f ( 1 n 1 n ) 2 n 3 N 1f N 2f N 3f ( 1 n 1 n 2 n ) 3 N 1f N 2f N 3f n 4 N 4f = 0
RANDOM LOAD σ 0 t cyclic counting procedure - (mean crossing) peak count - range pair (mean) count - rain flow count statistical representation load spectrum
MEASURED LOAD HISTORIES instrumentation with strain gages at critical locations measure load history continuous monitoring during service update spectrum standard spectra
TENSILE OVERLOAD (K max ) OL K max a a b 2 b 1 N
CRACK RETARDATION Al 2024-T3 (Hertzberg, 1976) K % P max nr. P max delay [MPa m] [-] [-] [10 3 cycles] 15 53 1 6 15 82 1 16 15 109 1 59 16.5 50 1 4 16.5 50 10 5 16.5 50 100 9.9 16.5 50 450 10.5 16.5 50 2000 22 16.5 50 9000 44 23.1 50 1 9 23.1 75 1 55 23.1 100 1 245
PLASTIC ZONE RESIDUAL STRESS σ σ yy σ σ 1 σ v 0 t A B σ σ yy B 1 A 1 σ yy σ = 0 B 2 ε yy A 2
CRACK RETARDATION MODELS Willenborg (1971) K R = φ [(K max ) OL [ 1 a r y ] K max ] ; a < r y K R = residual SIF ; K R = 0 delay distance φ = [1 (K th /K max )](S 1) 1 ; S = shut-off ratio (K max ) OL K max a a r y Johnson (1981) R eff = K min K R K max K R ; r y = 1 βπ β = plastic constraint factor ( (Kmax ) OL σ v ) 2
CRACK RETARDATION Elber (1971) [?] K eff = U K ; U = 0.5 + 0.4R with 0.1 R 0.7 Schijve (1981) [?] U = 0.55 + 0.33R + 0.12R 2 with 1.0 < R < 0.54
DESIGN AGAINST FATIGUE - infinite life design - safe life design - damage tolerant design - fail safe design
INFINITE LIFE DESIGN σ < σ th (σ < σ e ) no fatigue damage sometimes economically undesirable
SAFE LIFE DESIGN determine load spectra empirical rules / numerical analysis / laboratory tests fatigue life : (S N)-curves apply safety factors sometimes safety factors are undesirable (weight) stress-life design or strain-life design
STRESS/STRAIN LIFE DESIGN Basquin 1 2 σ = σ f(2n f ) b 1 2 εe = 1 E σ f(2n f ) b Manson-Coffin 1 2 εp = ε f(2n f ) c combination ε = ε e + ε p 1 2 ε = 1 2 σ f(2n f ) b + ε f(2n f ) c log ( ) ε 2 log(2n f )
DAMAGE TOLERANT DESIGN dangerous situations not acceptable safety factors undesirable determine load spectra periodic inspection (insp. schedules) monitor cracks NDT important calculate safe rest life ( integrate appropriate da -growth law ) dn repair when necessary
FAIL SAFE DESIGN design for safety : crack arrest / etc.
ENGINEERING PLASTICS (POLYMERS) ABS acrylonitrilbutadieenstyreen EM HIPS high-impact polystyreen TP LDPE low-density polyetheen TP Nylon PC polycarbonaat TP PMMA polymethylmethacrylaat (plexiglas) TP PP polypropeen TP PPO polyfenyleenoxyde TP PS polystyreen TP PSF PolySulFon TP PTFE polytetrafluoretheen (teflon) TP PVC polyvinylchloride TP PVF polyvinylfluoride TP PVF2 polyvinylidieenfluoride TP
MECHANICAL PROPERTIES - (nonlinear) elastic - visco-elastic - thermal influences - anisotropy
DAMAGE shearing (shear yielding) no change in density crazing (normal yielding) change in density : 40-60 % decrease shear bands crazes
PROPERTIES OF ENGINEERING PLASTICS AC CZ K Ic PMMA a + 13.2 PS a + 17.6 PSF a - low PC a - high main chain segmental motions energy dissipation Nylon 66 cr - main chain segmental motions energy dissipation crystalline regions crack retardation PVF2 sc PET sc amorphous strain induced crystallization at crack tip CPLS - cross-linked suppressed crazing HIPS + µ-sized rubber spheres enhanced crazing ABS blending AC : a = amorphous AC : c = crystalline AC : sc = semicrystalline AC : cr = crystalline regions CZ = crazing K Ic = fracture toughness in MPa m
FATIGUE amorphous crystalline high low molecular weight main chain motions toughening
FCP FOR POLYMERS 10 2 da dn [mm/cycle] 3 10 3 1 2 3 4 5 10 3 3 10 4 10 4 0.5 1 K 2 5 [MPa m] 1 : PMMA 5 Hz crazing 2 : LDPE 1 Hz 3 : ABS 10 Hz 4 : PC 10 Hz no crazing 5 : Nylon 10 Hz cristalline regions
FCP FOR POLYMERS : CRYSTALLINE VERSUS AMORPHOUS da dn [mm/cycle] 10 1 10 2 2 3 4 6 5 7 10 3 8 10 4 1 9 10 5 0.5 1 : PS 5 : PC 2 : PMMA 6 : Nylon 6.6 3 : PSF 7 : PVF2 4 : PPO 8 : PET 9 : PVC 1.0 2.0 5.0 10.0 K [MPa m]
FCP POLYMERS : TOUGHENING da dn [mm/cycle] 10 2 1 2 3 4 10 3 10 4 0.5 1.0 2.0 K [MPa m] 5.0 1 : CLPS 2 : PS 3 : HIPS 4 : ABS
COHESIVE ZONE MODELS
COHESIVE ZONE MODELS n t traction = f(opening) T = f( ) normal / tangential T n = f n ( n ) ; T t = f t ( t ) coupling T n = f n ( n, t ) ; T t = f t ( t, n ) MODELS polynomial piece-wise linear rigid-linear exponential
EXPONENTIAL COHESIVE ZONE LAW φ( n, t ) = φ n t [1 ( 1 + ) ( n exp ) ( )] n exp 2 t δ n δ n δ 2 t T n = φ n = φ n φn δ n T t = φ t = 2 φ t φt δ t ( n δ n ( t δ t ) exp ( ) ( ) n exp 2 t δ n δ 2 t ) ( 1 + ) ( ) ( n exp 2 t exp ) n δ n δ 2 t δ n δ n = φ n T n,max exp(1) ; δ t = φ t T t,max 1 2 exp(1) 1 1 0.5 0.5 Tn/Tn,max 0 Tt/Tt,max 0 0.5 0.5 1 1 0 1 2 3 4 1 n /δ n t /δ t 2 1 0 1 2 parameters : φ n,φ t, δ n,δ t
COUPLING : TRACTION 1 1 0.5 0.5 Tn/Tn,max 0 Tt/Tt,max 0 0.5 0.5 1 1 0 1 2 3 4 1 n /δ n t /δ t 2 1 0 1 2
T t = max{t t ( t, n,max )} T n = max{t n ( n, t,max )} 1 1 0.8 0.8 T t /T t,max 0.6 0.4 T n/tn,max 0.6 0.4 0.2 0.2 0 1 0 1 2 3 4 n,max /δ n 0 2 1 0 1 2 t,max /δ t
COUPLING : DISSIPATED ENERGY φ n = 100 Jm 2 ; φ t = 80 Jm 2 100 100 80 80 W[Jm 2 ] 60 40 W[Jm 2 ] 60 40 W n W n 20 W t 20 W t W tot W tot 0 0 2 4 6 8 0 n,max /δ n t,max /δ t 0 2 4 6 8 MODE-MIXITY 100 80 W[Jm 2 ] 60 40 W n W t W tot 20 0 0 20 40 60 80 100 α
WEIGHTED RESIDUAL FORMULATION FOR CZ
EQUILIBRIUM EQUATION σ + ρ q = 0 x V with : gradient operator σ : Cauchy stress tensor ρ : density q : volume load v : material velocity
WEIGHTED RESIDUAL FORMULATION σ + ρ q = 0 x V [ ] w σ + ρ q dv = 0 w( x) V V ( w) c : σ dv = V w ρ q dv + A w t da w( x) f i ( w, σ) = f e ( w, t, q) w( x)
TRANSFORMATION TO INITIAL STATE V ( w) c : σdv = f e ( w, t, q) w( x) = F c 0 ( w) c = ( 0 w) c F 1 dv = det(f) dv 0 = J dv 0 ( 0 w) c F 1 : σj dv 0 = f e0 w( x) V 0 ( 0 w) c : (F 1 σj)dv 0 = f e0 w( x) V 0 ( 0 w) c : T dv 0 = f e0 w( x) V 0 f i0 ( w,t) = f e0 ( w, t 0, q 0 ) w( x)
INTERNAL LOAD : BULK AND VOLUME f i0 = ( 0 w) c : T dv 0 V 0 = ( 0 w) c : T dv 0 + ( 0 w) c : T dv 0 = f i0b + f i0cz V 0b V 0 cz
f i0cz = ( 0 w) c : T dv 0 V 0 cz f i0cz = V 0 cz f i0cz =d 0 =d 0 = d 0 h 0 w = δ u ( 0 w) c : T = ( 0 δ u) c : T = δε l : T δε l : T dv 0 = d 0 δε l : T da 0 A 0 cz δε l = δε l e 0 e 0 = δ ε l e 0 T = T e 0 e 0 = T e 0 A 0 cz A 0 cz A 0 cz δ ε l T da 0 = d 0 (δ u) h 0 T da 0 w T da 0 A 0 cz } δε l : T = δ ε l T δ u h 0 T da 0
FINITE ELEMENTS FOR CZ
TWO-DIMENSIONAL CZ ELEMENT 3 A 1 4-1 Q T P 0 ξ η 1 B 2 f i0cz = d 0 h 0 A 0 cz w T da 0 = d 0 h 0 1 ξ= 1 1 η= 1 w T h 0 2 l 0 2 dηdξ = d 0 2 1 ξ= 1 1 η= 1 w T l 0 2 dηdξ = d 0l 0 2 1 η= 1 w T dη
LOCAL VECTOR BASE 3 4 T e n T B 1 A e t 2 w= w t e t + w n e n = [ w t w n ] T =Tt e t + T n e n = [ T t T n ] [ e t e n ] [ ] e t e n = T T ẽ = w T ẽ f i0cz = d 0l 0 2 1 η= 1 w T (η)t (η) dη
INTERPOLATION w T (η)=[ w t w n ] = = [ ] w A t w A n w B t w B n = w AB T N T (η) 1 2 1 2 (1 η) 0 1 0 2 (1 η) (1 + η) 0 (1 + η) 0 1 2 w A t = w 4 t w 1 t ; w A n = w 4 n w 1 n w B t = w 3 t w 2 t ; w B n = w 3 n w 2 n w AB = w A t w A n w B t w B n = 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 w 1 t w 1 n w 2 t w 2 n w 3 t w 3 n w 4 t w 4 n =P l w w T T = l P w T N T
LOCAL GLOBAL w l = w 1 t w 1 n w 2 t w 2 n w 3 t w 3 n w 4 t w 4 n = c s 0 0 0 0 0 0 s c 0 0 0 0 0 0 0 0 c s 0 0 0 0 0 0 s c 0 0 0 0 0 0 0 0 c s 0 0 0 0 0 0 s c 0 0 0 0 0 0 0 0 c s 0 0 0 0 0 0 s c w 1 1 w 1 2 w 2 1 w 2 2 w 3 1 w 3 2 w 4 1 w 4 2 = R w w T T = l P w T N T (η) = w T R T P T N T (η) f i0cz = w T d 0l 0 2 R T P T 1 η= 1 N T (η)t (η) dη = w T f i0 cz (R, T )
ASSEMBLING + WRF w T f i0 b + w T f i0 cz = w T f e0 w i0 f (σ, F) + b f i0 (R, T ) = cz f e0 f i0 = f e0
ITERATIVE PROCEDURE T = T + δt ; R = R + δr R f = i + δf i i f δf i0 cz = d 0l 0 2 R T P T 1 η= 1 N T (η) δt (η) dη δt (η) = T δ = T t T t [ ] t n δ t T n T n δ n t n = M(η) δ (η) δ (η)=n(η)δ AB = N(η)P δũ l = N(η)P R δũ δf i0 cz = [ d 0 l 0 2 R T P T =K cz δũ { 1 η= 1 N T (η)m(η) N(η) dη }P R ] δũ
EXAMPLES : SINGLE MODE 7 8 5 6 1 0.5 np7 np8 fy/tn,max 0 3 4 0.5 1 2 1 1 0 1 2 3 4 u y /δ n 1 np7 np8 0.5 7 8 3 4 5 6 1 2 fx/tt,max 0 0.5 1 3 2 1 0 1 2 3 u x /δ t
MULTIMODE WITHOUT ROTATION 7 8 5 6 3 4 1 2 2 1.5 np7 np8 1 0.8 np7 np8 fy/tn,max 1 0.5 fx/tt,max 0.6 0.4 0 0.2 0.5 0 1 2 3 4 5 u y /δ n 0 0 0.5 1 1.5 2 2.5 u x /δ t 1 0.8 ip1 ip2 0.7 0.6 ip1 ip2 Tn/Tn,max 0.6 0.4 Tt/Tt,max 0.5 0.4 0.3 0.2 0.2 0.1 0 0 1 2 3 4 5 0 n /δ n t /δ t 0 0.5 1 1.5 2 2.5
ROTATION 8 7 6 5 3 4 1 2 1.4 1.2 np7 np8 1 np7 np8 1 0.5 fy/tn,max 0.8 0.6 0.4 fx/tt,max 0 0.2 0.5 0 0.2 0 0.5 1 1.5 2 2.5 u y /δ n 1 0.5 0.4 0.3 0.2 0.1 0 u x /δ t 1 0.8 ip1 ip2 0.25 0.2 ip1 ip2 Tn/Tn,max 0.6 0.4 Tt/Tt,max 0.15 0.1 0.2 0.05 0 0 0.5 1 1.5 2 2.5 0 n /δ n t /δ t 0 0.05 0.1 0.15 0.2
CHOICE OF REFERENCE LINE (M T B) 8 8 8 7 7 7 6 6 6 5 5 5 3 4 3 4 3 4 1 2 1 2 1 2 0.25 0.2 ip1 ip2 0.4 0.35 0.3 ip1 ip2 Tt/Tt,max 0.15 0.1 Tt/Tt,max 0.25 0.2 0.15 0.05 0.1 0.05 0 0 0.05 0.1 0.15 0.2 0 t /δ t t /δ t 0 0.1 0.2 0.3 0.4 0.04 0.03 ip1 ip2 0.02 Tt/Tt,max 0.01 0 0.01 0.02 0.03 0.04 0.06 0.04 0.02 0 0.02 0.04 t /δ t
CZ FOR LARGE DEFORMATIONS 40 30 T [MPa] 20 10 0 0 1 2 3 4 5 6 [µm] = e ; T = T( ) e ; T = f( ) φ ; T = δ ( ) ( exp ) δ δ φ = =0 T( ) d ; T max = φ δ exp(1)
MODE-MIXITY d = d 1 d 2 T = φ ( ) exp δ δ ( ) ( exp α d ) δ 2 120 110 α = 0.2 α = 0.0 α = 0.2 50 40 α = 0.2 α = 0.0 α = 0.2 W [Jm 2 ] 100 90 T [MPa] 30 20 80 0 30 60 90 β [ o ] 10 0 0 2 4 6 8 10 [µm] parameters : φ, δ, α
POLYMER COATED STEEL FOR PACKAGING
EASY PEEL-OFF LID
SOLDER JOINT FATIGUE