《纺织复合材料》课程参考文献（Computational Mechanics of Composite Materials）05 Fracture and Fatigue Models for Composites

Fracture and Fatigue Analysis of Composites 237 The entire procedure can be applied for the fatigue analysis by rewriting the material properties of the composite components in terms of the current fatigue cycle number.Then,homogenising the constitutive law for each cycle,the whole composite fatigue can be modelled in a global scale,without the necessity for a very precise microscale discretisation or computational substructuring;an analogous analysis can be carried out for composite materials with cracks [336,337],for instance.It should be underlined that the described homogenisation procedure is sensitive to the RVE determination from the entire composite and to the scale parameter relating this element,dimensions to the dimensions of the entire composite.The formula for effective elasticity tensor is rewritten under the assumption that this parameter tends to 0,which is a very unrealistic model. Furthermore,the homogenisation procedure can be established for random composites,too,only if the randomness does not influence the periodic character of the composite (especially during the fatigue process).Then,either MCS [191]as well as SFEM [192]can be utilised for this purpose.Therefore,starting from probabilistic characteristics of the composite properties,the expected values, variances (or standard deviations)as well as higher order moments (in the statistical estimation only)can be computed. A very important issue from the technological point of view is the presence of the interface defects (usually with stochastic nature)appearing and growing between the composite components.Various computational models are proposed in this case in terms of special purpose spring finite elements or,alternatively,using the interphase as a new,separate material between the original composite components.This new material can be constructed from the original semicircular defects with random parameters,smeared (averaged probabilistically)over the entire interphase region according to the stochastic model introduced in Chapter 2; the composite with such an introduced interphase is then homogenised.To utilise the model for fatigue life cycle analysis,the geometrical and physical properties of the composite should be described in terms of the fatigue cycle number and then homogenised cycle by cycle for the needs of computational simulation of the composite

Fracture and Fatigue Analysis of Composites 237 The entire procedure can be applied for the fatigue analysis by rewriting the material properties of the composite components in terms of the current fatigue cycle number. Then, homogenising the constitutive law for each cycle, the whole composite fatigue can be modelled in a global scale, without the necessity for a very precise microscale discretisation or computational substructuring; an analogous analysis can be carried out for composite materials with cracks [336,337], for instance. It should be underlined that the described homogenisation procedure is sensitive to the RVE determination from the entire composite and to the scale parameter relating this element, dimensions to the dimensions of the entire composite. The formula for effective elasticity tensor is rewritten under the assumption that this parameter tends to 0, which is a very unrealistic model. Furthermore, the homogenisation procedure can be established for random composites, too, only if the randomness does not influence the periodic character of the composite (especially during the fatigue process). Then, either MCS [191] as well as SFEM [192] can be utilised for this purpose. Therefore, starting from probabilistic characteristics of the composite properties, the expected values, variances (or standard deviations) as well as higher order moments (in the statistical estimation only) can be computed. A very important issue from the technological point of view is the presence of the interface defects (usually with stochastic nature) appearing and growing between the composite components. Various computational models are proposed in this case in terms of special purpose spring finite elements or, alternatively, using the interphase as a new, separate material between the original composite components. This new material can be constructed from the original semicircular defects with random parameters, smeared (averaged probabilistically) over the entire interphase region according to the stochastic model introduced in Chapter 2; the composite with such an introduced interphase is then homogenised. To utilise the model for fatigue life cycle analysis, the geometrical and physical properties of the composite should be described in terms of the fatigue cycle number and then homogenised cycle by cycle for the needs of computational simulation of the composite

238 Computational Mechanics of Composite Materials 5.3.1 Delamination of Two-Component Curved Laminates Let us consider a two-component elastic transversely isotropic material in two-dimensional space defined by the polar coordinate system y=fR,}(cf. Figures 5.40-5.43).It is necessary to introduce the following relations: (a)the gap between two surfaces g(R,o)=2(R,Θ)-(R,Θ) (5.40) (b)the relative tangential slip of two surfaces s(R,O)=哈2(R,O)-(R,O) (5.41) (c)the normal traction oR(R,Θ)=o(R,Θ)-oW(R,Θ) (5.42) (d)the shear traction oa(R,O)=o8(R,Θ)-o(R,o),「e={Tc:R=R:oe0,∞} (5.43) where Ro is the radius of the interface curvature.Since(5.40)-(5.43)are referred to the composite interface (cracked or joined)Te(R=Ro=const)only,then their radial dependence is neglected.The equilibrium problem of linear elasticity is given by the following equations system [95]: equilibrium equations 3dx10+(Ox-Go)+bx=0 (5.44) OR R OR R +10Go+2Gme+be=0 d0e腿 2 (5.45) aR R a R where bg and be denote the body force components; strain-displacement relations -0治+货6 1(1 dug due ue (5.46 ER=

238 Computational Mechanics of Composite Materials 5.3.1 Delamination of Two-Component Curved Laminates Let us consider a two-component elastic transversely isotropic material in two-dimensional space Ω defined by the polar coordinate system y={R,Θ} (cf. Figures 5.40-5.43). It is necessary to introduce the following relations: (a) the gap between two surfaces () () () ,Θ = ,Θ − ,Θ (2) (1) g R uR R uR R (5.40) (b) the relative tangential slip of two surfaces () () () ,Θ = Θ ,Θ − Θ ,Θ (2) (1) s R u R u R (5.41) (c) the normal traction () () () ,Θ = ,Θ − ,Θ (2) (1) σ R R σ R R σ R R (5.42) (d) the shear traction Θ () () () ,Θ = Θ ,Θ − Θ ,Θ (2) (1) σ R R σ R R σ R R , Γc = { } Γc : R = R0 ;Θ ∈ 0,∞ (5.43) where R0 is the radius of the interface curvature. Since (5.40) - (5.43) are referred to the composite interface (cracked or joined) Γc (R=R0=const) only, then their radial dependence is neglected. The equilibrium problem of linear elasticity is given by the following equations system [95]: • equilibrium equations ( ) 0 1 1 + − + = ∂ ∂ + ∂ ∂ Θ Θ R R R R b R R R R σ σ σ σ (5.44) 0 1 2 + + = ∂Θ ∂ + ∂ ∂ Θ Θ Θ Θ b R R R R R σ σ σ (5.45) where bR and bΘ denote the body force components; • strain-displacement relations R uR R ∂ ∂ ε = , R u u R R + ∂Θ ∂ = Θ Θ 1 ε , ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ∂ ∂ + ∂Θ ∂ = Θ Θ Θ R u R u u R R R 1 2 1 ε (5.46)

Fracture and Fatigue Analysis of Composites 239 constitutive relations C Cun Cu3 ER C C C (5.47) Ee C3331 C3332 C333 The following boundary conditions are employed: ug =ig and ue =ie on T (5.48) tR=ig and te-ioon「a (5.49) g(Θ)=0；s(Θ)=0→cR(Θ)=0；Oa(Θ)=0on「 (5.50) g(Θ)=0；s(Θ)≠0→cR(Θ)<0：lba(Θ=cR(Θ)omTe (5.51) g(Θ)>0；s(Θ)=00rs(Θ)≠0oR(Θ)=0,09（Θ）=0onTc (5.52) g(Θ)<0；s(⊙)≠0R(Θ)<0：ba(Θ=oR(Θ)on「。 (5.53) signΘ=sign(s日）on「e (5.54) where u denotes the constant friction coefficient.Then,the near-tip stress field is described in the polar coordinate system as x)=fr,0}(cf.Figure 5.6). a 票 R Iu Figure 5.5.Two-component curved laminate structure

Fracture and Fatigue Analysis of Composites 239 • constitutive relations ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ Θ Θ Θ Θ R R R R C C C C C C C C C ε ε ε σ σ σ 3331 3332 3333 2221 2222 2223 1111 1112 1113 (5.47) The following boundary conditions are employed: R R u = uˆ and Θ = Θ u uˆ on Γu (5.48) R R t t = ˆ and Θ = Θ t t ˆ on Γσ (5.49) g(Θ) = 0 ; s(Θ) = 0 ⇒σR (Θ) = 0 ; 0 σ RΘ (Θ) = on Γc (5.50) g(Θ) = 0 ; s(Θ) ≠ 0 ⇒σR (Θ) < 0 ; (Θ) = µ σ (Θ) σ RΘ R on Γc (5.51) g(Θ) > 0 ; s(Θ) = 0 or s(Θ) ≠ 0 σR (Θ) = 0 , 0 σ RΘ (Θ) = on Γc (5.52) g(Θ) < 0 ; s(Θ) ≠ 0 σR (Θ) < 0 ; (Θ) = µ σ (Θ) σ RΘ R on Γc (5.53) sign( ( )) sign( ) s( ) R σ Θ = Θ on Γc (5.54) where µ denotes the constant friction coefficient. Then, the near-tip stress field is described in the polar coordinate system as {x}={r,θ} (cf. Figure 5.6). Figure 5.5. Two-component curved laminate structure Γσ Ω1 Ω2 R0 Γu x1 x2 a Θ g(Θ) Γc R

240 Computational Mechanics of Composite Materials 21 a xj=rcos0 x2=rsine 02 Figure 5.6.Near-tip field It is assumed that both crack surfaces are modelled as perfectly smooth-there are no neither meso-nor micro-asperities on this surface in the context of the FEM contact model presented by [49,371,382],however application of the Boundary Element Method is also known,see [374].Considering future particle- reinforced composites delamination simulations,the 3D contact algorithms must be employed [284,322].The asymptotic nature of the elastic fields near a transition in the boundary conditions (crack tip)is expressed by the analytic functions and therefore,the description of a near-tip stress for an interface crack between two different transversely isotropic in a plane stress problem and the traction-free crack surfaces is given as follows [301]: oi=Re Krie(2m5z伯，e)+ImKrie](2mro5z(e,e), (5.55) where i.j=1.2,(0),()are the angular functions derived using the Muskhelishvili potentials;describes here the oscillatory stress singularity given as r=cos(e Inr)+isin(Inr) (5.56) The angular functions correspond to the normal and in-plane shear tractions, respectively,on interface ahead crack tip (x>0;0=0)at a distance r given by [140,222: tio=K(2nr)as or (5.57) ox=Relkr"2r)as and Imlkrxr)os Moreover,the functions ),(0,)are related to the elastic properties of the bimaterial specimen using the oscillatory index e given by

240 Computational Mechanics of Composite Materials x1=rcosθ x2=rsinθ Figure 5.6. Near-tip field It is assumed that both crack surfaces are modelled as perfectly smooth – there are no neither meso- nor micro-asperities on this surface in the context of the FEM contact model presented by [49,371,382], however application of the Boundary Element Method is also known, see [374]. Considering future particle￾reinforced composites delamination simulations, the 3D contact algorithms must be employed [284,322]. The asymptotic nature of the elastic fields near a transition in the boundary conditions (crack tip) is expressed by the analytic functions and therefore, the description of a near-tip stress for an interface crack between two different transversely isotropic in a plane stress problem and the traction-free crack surfaces is given as follows [301]: Re[ ] r () () 2 , Im[ ]( ) 2 ( , ) i 0.5 0.5 = Σ ∈ + Σ ∈ ∈ − ∈ − σ π θ π θ II ij I i ij ij K r Kr r , (5.55) where i,j=1,2, ) Σ (θ,∈ I ij , ) Σ (θ,∈ II ij are the angular functions derived using the Muskhelishvili potentials; r i∈ describes here the oscillatory stress singularity given as r () () r i r i = cos ∈ln + sin ∈ ln ∈ (5.56) The angular functions correspond to the normal and in-plane shear tractions, respectively, on interface ahead crack tip (x1>0; θ=0) at a distance r given by [140,222]: ( )− ∈ σ + σ = π . i K r r 0 5 22 12 i 2 or [ ]( ) 0.5 22 Re 2 ∈ − = Kr r i σ π and [ ]( ) 0.5 12 Im 2 ∈ − = Kr r i σ π (5.57) Moreover, the functions ) Σ (θ,∈ I ij , ) Σ (θ,∈ II ij are related to the elastic properties of the bimaterial specimen using the oscillatory index ∈ given by Ω2 x1 x2 r θ a Ω1

Fracture and Fatigue Analysis of Composites 241 K1/G+1/G2 E= 2π K2/G2+/G (5.58) where the Kolosov constant K is given as [158,259] 3-V (5.59) K = "for the plane stress a 1+V K =3-4v for the plane strain;n=1,2. (5.60) where va and G denote the Poisson ratio and shear modulus of the nth component, respectively.Next,the elastic Dundur mismatch parameters are defined by a=Ck,+小-G,k+) (5.61) G(K2+1)+G2(K1+1) and B=G(2-1)-G:(k,-1) GK2+1)+G2(K1+1) Then,it is possible to rewrite (5.58)in the following way: (5.62) 2 1+B The fracture modes I and II [54]of the SIF in the case of an interface crack between dissimilar isotropic materials are now coupled together into the single complex SIF K=Ki+iK2 uniquely characterising the singular stress field;Ki and K2 are the functions of a distance r from the tip and may be denoted as follows: K(r)=Re(Kr)and K2(r)=Im(Kr) (5.63) The associated relative crack surfaces displacements (Au=ui(r,0=)-u(r,0=-))at a distance r behind the tip (x<0;0=t)are described in the following way: △41+i4 (5.64) Finally,the ERR for the crack propagation along the interface may be given as ERR- KK (5.65) where K=K1-iK2 is the conjugate complex SIF.It finally gives

Fracture and Fatigue Analysis of Composites 241 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + ∈= 2 2 1 1 1 2 1 1 ln 2 1 G G G G κ κ π (5.58) where the Kolosov constant κn is given as [158,259] n n n + ν − ν κ = 1 3 for the plane stress (5.59) n n κ = 3 for the plane strain; − 4ν n=1,2. (5.60) where νn and Gn denote the Poisson ratio and shear modulus of the nth component, respectively. Next, the elastic Dundur mismatch parameters are defined by ( )( ) ( )( ) 1 1 1 1 1 2 2 1 1 2 2 1 + + + + − + = κ κ κ κ α G G G G and ( )( ) ( )( ) 1 1 1 1 1 2 2 1 1 2 2 1 + + + − − − = κ κ κ κ β G G G G (5.61) Then, it is possible to rewrite (5.58) in the following way: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − ∈= β β π 1 1 ln 2 1 (5.62) The fracture modes I and II [54] of the SIF in the case of an interface crack between dissimilar isotropic materials are now coupled together into the single complex SIF K=K1+iK2 uniquely characterising the singular stress field; K1 and K2 are the functions of a distance r from the tip and may be denoted as follows: ( ) ∈ = i K (r) Re Kr 1 and ( ) ∈ = i K (r) Im Kr 2 (5.63) The associated relative crack surfaces displacements (∆ui=ui(r,θ=π)-ui(r, θ=-π)) at a distance r behind the tip (x1<0; θ=±π) are described in the following way: ( ) () 0.5 2 2 1 1 1 2 1 2 cosh 2 1 1 ∈ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ∈ ∈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − ∆ + ∆ = π π ν ν r i Kr G G u i u i (5.64) Finally, the ERR for the crack propagation along the interface may be given as ( ) ∈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − = π ν ν 2 2 2 1 1 4cosh 1 1 KK G G ERR , (5.65) where K =K1-iK2 is the conjugate complex SIF. It finally gives