Fracture and Fatigue Analysis of Composites 227 classical and modified Palmgren-Miner approach,for instance.This variable is most frequently treated as a random variable or a random process in stochastic modelling.Another group consists of mechanical models,where stress (A5.50)- (A5.53)or strain (A5.66)-(A5.67)limits are used instead of global life cycle number.Such models reflect the actual state of a composite during the fatigue process better and are more appropriate for the needs of computational probabilistic structural analysis.The combination of both approaches is proposed by Morrow in (A5.66)for constant stress amplitude and for different cycles by (A5.67).The overall fatigue analysis is then more complicated.However the most realistic model is obtained.Accidentally,Fong model is used,where damage function is represented by an exponential function of damage trend k,which is a compromise between counting fatigue cycles and mechanical tensor measurements. The very important problem is to distinguish the scale of application of the proposed model,especially in the context of determination of a fatigue crack length.The models valid for long cracks do not account for the phenomena appearing at the microscale of the composite specimen.On the contrary,cf. (A5.33),the microstructural parameter d is introduced,which makes it possible to include material parameters in the microscale in the equation describing the fatigue crack growth. All the models for the damage function can be extended on random variables theoretically,by perturbation methodology,or computationally,using the relevant MCS approach.The essential minor point observed in most of the formulae described above is a general lack of microstructural analysis.The two approaches analysed above can model cracks in real laminates,while other types of composites must be analysed using fatigue laws for homogeneous materials.This approach is not a very realistic one,since fatigue resistance of fibres,matrices,interfaces and interphases is essentially different.Considering the delamination phenomena during periodic stress changes,an analogous fatigue approach for fibre-matrix interface decohesion should be worked out.The probabilistic structural analysis of such a model can be made using SFEM computations or by a homogenisation. However a closed-form fatigue law should be completed first. As is known,there exist a whole variety of effective probabilistic methods in engineering.The usage of any of these approaches depends on the following factors:(a)type of random variables(normal,lognormal or Weibull,for instance), (b)probabilistic information on the input random variables,fields or processes(in the form of moments or probability density function (PDF)),(c)interrelations between particular probabilistic characteristics of the input (of higher to the first order,especially),(d)method of solution of corresponding deterministic problem and (e)available computational time as well as(f)applied reliability criteria. If the closed form solution is available or can be derived symbolically using computational algebra,then the probability density function(PDF)of the output can be found starting from analogous information about the input PDF.It can be done generally from definition-using integration methods,or,alternatively,by the characteristic function derivation.The following PDF are used in this case:
Fracture and Fatigue Analysis of Composites 227 classical and modified Palmgren-Miner approach, for instance. This variable is most frequently treated as a random variable or a random process in stochastic modelling. Another group consists of mechanical models, where stress (A5.50) - (A5.53) or strain (A5.66) - (A5.67) limits are used instead of global life cycle number. Such models reflect the actual state of a composite during the fatigue process better and are more appropriate for the needs of computational probabilistic structural analysis. The combination of both approaches is proposed by Morrow in (A5.66) for constant stress amplitude and for different cycles by (A5.67). The overall fatigue analysis is then more complicated. However the most realistic model is obtained. Accidentally, Fong model is used, where damage function is represented by an exponential function of damage trend k, which is a compromise between counting fatigue cycles and mechanical tensor measurements. The very important problem is to distinguish the scale of application of the proposed model, especially in the context of determination of a fatigue crack length. The models valid for long cracks do not account for the phenomena appearing at the microscale of the composite specimen. On the contrary, cf. (A5.33), the microstructural parameter d is introduced, which makes it possible to include material parameters in the microscale in the equation describing the fatigue crack growth. All the models for the damage function can be extended on random variables theoretically, by perturbation methodology, or computationally, using the relevant MCS approach. The essential minor point observed in most of the formulae described above is a general lack of microstructural analysis. The two approaches analysed above can model cracks in real laminates, while other types of composites must be analysed using fatigue laws for homogeneous materials. This approach is not a very realistic one, since fatigue resistance of fibres, matrices, interfaces and interphases is essentially different. Considering the delamination phenomena during periodic stress changes, an analogous fatigue approach for fibre-matrix interface decohesion should be worked out. The probabilistic structural analysis of such a model can be made using SFEM computations or by a homogenisation. However a closed-form fatigue law should be completed first. As is known, there exist a whole variety of effective probabilistic methods in engineering. The usage of any of these approaches depends on the following factors: (a) type of random variables (normal, lognormal or Weibull, for instance), (b) probabilistic information on the input random variables, fields or processes (in the form of moments or probability density function (PDF)), (c) interrelations between particular probabilistic characteristics of the input (of higher to the first order, especially), (d) method of solution of corresponding deterministic problem and (e) available computational time as well as (f) applied reliability criteria. If the closed form solution is available or can be derived symbolically using computational algebra, then the probability density function (PDF) of the output can be found starting from analogous information about the input PDF. It can be done generally from definition – using integration methods, or, alternatively, by the characteristic function derivation. The following PDF are used in this case:
228 Computational Mechanics of Composite Materials lognormal for stress and strain tensors,lognormal and Gaussian distributions for elastic properties as well as for the geometry of fatigue specimen.Weibull density function is used to simulate external loads(shifted Rayleigh PDF,alternatively), yield strength as well as the fracture toughness,while the initial crack length is analysed using a shifted exponential probability density function. As is known [313],one of the following computational methods can be used in probabilistic fatigue modelling:Monte Carlo simulation technique,stochastic (second or higher order)perturbation analysis as well as some spectral techniques (Karhunen-Loeve or polynomial chaos decompositions).Alternatively,Hermitte- Gauss quadratures (HGQ)or various sampling methods (Latin Hypercube Sampling-LHS,for instance)in conjunction with one of the latters may be used. Computational experience shows that simulation and sampling techniques are or can be implemented as exact methods.However their time cost is very high. Perturbation-based approaches have their limitations on higher order probabilistic moments,but they are very fast.The efficiency of spectral methods depends on the order of decomposition being used,but computational time is close to that offered by the perturbation approach.Unfortunately,there is no available full comparison of all these techniques-comparison of MCS and SFEM can be found in [208]. HGQ with SFEM in [237]and stochastic spectral FEM with MCS in [113,114].A lot of numerical experiments have been conducted in this area,including cumulative damage analysis of composites by the MCS approach (Ma et al.[243]) and simulation of stochastic processes given by (A5.30)-(A5.38).However,the problem of an appropriate conjunction of stochastic processes and structural analysis using FEM or BEM techniques has not been solved yet. Let us analyse the application of the perturbation technique to damage function D extension,where it is a function of random parameter vector b.Using a stochastic Taylor expansion it is obtained that Db)=Db°)+b'Db°)+E2b'bDrb°) (5.2) Then,according to the classical definition,the expected value of this function can be derived as E[D(b)]=jD(b)p(b)db (5.3) =jb6+b'D-6+e2山山D严6》pbd =D6+D严6)cob,b) while variance is va()pVar(b (5.4)
228 Computational Mechanics of Composite Materials lognormal for stress and strain tensors, lognormal and Gaussian distributions for elastic properties as well as for the geometry of fatigue specimen. Weibull density function is used to simulate external loads (shifted Rayleigh PDF, alternatively), yield strength as well as the fracture toughness, while the initial crack length is analysed using a shifted exponential probability density function. As is known [313], one of the following computational methods can be used in probabilistic fatigue modelling: Monte Carlo simulation technique, stochastic (second or higher order) perturbation analysis as well as some spectral techniques (Karhunen-Loeve or polynomial chaos decompositions). Alternatively, HermitteGauss quadratures (HGQ) or various sampling methods (Latin Hypercube Sampling – LHS, for instance) in conjunction with one of the latters may be used. Computational experience shows that simulation and sampling techniques are or can be implemented as exact methods. However their time cost is very high. Perturbation-based approaches have their limitations on higher order probabilistic moments, but they are very fast. The efficiency of spectral methods depends on the order of decomposition being used, but computational time is close to that offered by the perturbation approach. Unfortunately, there is no available full comparison of all these techniques – comparison of MCS and SFEM can be found in [208], HGQ with SFEM in [237] and stochastic spectral FEM with MCS in [113,114]. A lot of numerical experiments have been conducted in this area, including cumulative damage analysis of composites by the MCS approach (Ma et al. [243]) and simulation of stochastic processes given by (A5.30) - (A5.38). However, the problem of an appropriate conjunction of stochastic processes and structural analysis using FEM or BEM techniques has not been solved yet. Let us analyse the application of the perturbation technique to damage function D extension, where it is a function of random parameter vector b. Using a stochastic Taylor expansion it is obtained that () () ( ) 2 , 0 2 0 0 , 0 1 (b) b b b r r r s rs D = D + ε∆b D + ε ∆b ∆b D (5.2) Then, according to the classical definition, the expected value of this function can be derived as [ ] ( ) () () ( ) ( ) () ( ) rs r s r r r s rs D D Cov b b D b D b b D p d E D D p d , ( ) ( ) ( ) ( ) , 0 2 0 0 1 2 , 0 2 0 0 , 0 1 b b b b b b b b b b b = + = + ∆ + ∆ ∆ = ∫ ∫ +∞ −∞ +∞ −∞ ε ε (5.3) while variance is ( ) ( ) 2 b b Var D Var D ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = (5.4)
Fracture and Fatigue Analysis of Composites 229 Since this function is usually used for damage control,which in the deterministic case is written as Ds1,an analogous stochastic formulation should be proposed.It can be done using some deterministic function being a combination of damage function probabilistic moments as follows: Db)≤guDb)Ds1 (5.5) where u(D(b))denote some function of up to kth order probabilistic moments. Usually,it is carried out using a stochastic 'envelope'function being the upper bound for the entire probability density function as,for instance 3(ux [D(b)])=E[D(b)]-3D(b) (5.6 This formula holds true for Gaussian random deviates only.It should be underlined that this approximation should be modified in the case of other random variables,using the definition that the value of damage function should be smaller than 1 with probability almost equal to 1;the lower bound can be found or proposed analogously.In the case of classical Palmgren-Miner rule (A5.45),with fatigue life cycle number N treated as an input random variable, D=” (5.7) N≡b the expected value is derived as follows [215]: ED]=D°+D-NVar(W)=” +a入) (5.8) and the variance in the form of ((N) (5.9) Var(D)=(DNVar(N)= It is observed that the methodology can also be applied to randomise all of the functions D listed in the appendix to this chapter with respect to any single or any vector of composite input random parameters.In contrast to the classical derivation of the probabilistic moments from their definitions,there is no need to make detailed assumptions on input PDF to calculate expected values and variances for the inversed random variables in this approach. Let us determine for illustration the number of fatigue cycles of cumulative damage of a crack at the weld subjected to cyclic random loading with the specified expected value and standard deviation (or another second order
Fracture and Fatigue Analysis of Composites 229 Since this function is usually used for damage control, which in the deterministic case is written as 1 D ≤ , an analogous stochastic formulation should be proposed. It can be done using some deterministic function being a combination of damage function probabilistic moments as follows: D(b) ≤ g( ) [ ] D(b) ≤1 µ k (5.5) where ( ) D(b) µk denote some function of up to kth order probabilistic moments. Usually, it is carried out using a stochastic ‘envelope’ function being the upper bound for the entire probability density function as, for instance g( ) [ ] D(b) E[ ] D(b) 3 D(b) µk = − (5.6) This formula holds true for Gaussian random deviates only. It should be underlined that this approximation should be modified in the case of other random variables, using the definition that the value of damage function should be smaller than 1 with probability almost equal to 1; the lower bound can be found or proposed analogously. In the case of classical Palmgren-Miner rule (A5.45), with fatigue life cycle number N treated as an input random variable, N n D = , N ≡ b (5.7) the expected value is derived as follows [215]: ( ) [ ] ( ) ( ) 3 0 0 , 2 0 1 Var N N n N n E D D D Var N NN = + = + (5.8) and the variance in the form of ( ) ( ) ( ) ( ) ( ) 4 0 2 2 , Var N N n Var D D Var N N = = (5.9) It is observed that the methodology can also be applied to randomise all of the functions D listed in the appendix to this chapter with respect to any single or any vector of composite input random parameters. In contrast to the classical derivation of the probabilistic moments from their definitions, there is no need to make detailed assumptions on input PDF to calculate expected values and variances for the inversed random variables in this approach. Let us determine for illustration the number of fatigue cycles of cumulative damage of a crack at the weld subjected to cyclic random loading with the specified expected value and standard deviation (or another second order
230 Computational Mechanics of Composite Materials probabilistic characteristics)of Ao.Let us assume that the crack in a weld is growing according to the Paris-Erdogan law,cf.(A5.26),described by the equation =cyo元"a是 (5.10) dN and that y≠Y(a).Then a-jclYso)"dN (5.11) Q which gives by integration that 1 (5.12) -罗+ a+1=Cyo元)N+D,Der Taking for N=0 the initial condition a=a,it is obtained that (5.13) a for K=學-1，B=CYo√元m (5.14) Therefore,the number of cycles to failure is given by (5.15) The following equation is used to determine the probabilistic moments of the number of cycles for a crack to grow from the initial length a;to its final length af AN=j- 1 -da 4C(△K (5.16) Substituting for AK one obtains △W=19 1da Corπ号y"a (5.17)
230 Computational Mechanics of Composite Materials probabilistic characteristics) of ∆σ. Let us assume that the crack in a weld is growing according to the Paris-Erdogan law, cf. (A5.26), described by the equation ( ) 2 m C Y a dN da m = ∆σ π (5.10) and that Y≠Y(a). Then ( ) ∫ = ∫C Y∆ dN a da m m σ π 2 (5.11) which gives by integration that a C( ) Y N D m m m = ∆ + − + − + σ π 1 2 2 1 1 , D ∈ℜ (5.12) Taking for N=0 the initial condition a=ai, it is obtained that a N a k i − β = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 1 1 (5.13) for 1 2 κ = − m , ( ) m β κai C Y σ π κ = ∆ (5.14) Therefore, the number of cycles to failure is given by β 1 N f = (5.15) The following equation is used to determine the probabilistic moments of the number of cycles for a crack to grow from the initial length ai to its final length af: ( ) ∫ ∆ ∆ = f i a a m da C K N 1 (5.16) Substituting for ∆K one obtains ( ) ∫ ∆ ∆ = f i m m a m a m da C Y a N 2 2 1 1 σ π (5.17)
Fracture and Fatigue Analysis of Composites 231 By the use of a stochastic second order perturbation technique we determine the expected value of△Was Eilw]=Awho+;a。arao) 204σ月 (5.18) and the variance of number of cycles as w)- Var(Ao) (5.19) Adopting m=2 it is calculated using(5.17)and(5.18)that gatraaloa (5.20) and a:E△o】 (5.21) The following data are adopted in probabilistic symbolic computations: EAo]=6 -6 =10.0 MPa,a;=25 mm and obtained experimentally C=1.64x1010,Y=1.15.The visualisation of the first two probabilistic moments of fatigue cycle number is done using the symbolic computation program MAPLE as functions of the coefficient of variation a(Ao)and the final crack length as.The results of the analysis in the form of deterministic values,corresponding expected values and standard deviations are presented below with the design parameters marked on the horizontal axes. 1e+007 8e+006 6e+006 4e+006 2e+006 02502d1501050i250500ias20i4000450.0 Figure 5.1.Deterministic values of fatigue cycles (dN)
Fracture and Fatigue Analysis of Composites 231 By the use of a stochastic second order perturbation technique we determine the expected value of ∆N as [ ] ( ) ( ) ( ) ( ) ( ) σ σ σ σ ∆ ∂ ∆ ∂ ∆ ∆ ∆ = ∆ ∆ + Var N E N N 2 0 2 0 0 2 1 (5.18) and the variance of number of cycles as ( ) ( ) ( ) ( ) ( ) σ σ σ ∆ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∆ ∂ ∆ ∆ ∆ = Var N Var N 2 0 0 (5.19) Adopting m=2 it is calculated using (5.17) and (5.18) that [ ] [ ] [ ] ( )⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∆ ∆ + ∆ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ = σ π σ σ Var a E E a CY E N i f 2 2 4 1 6 ln 1 (5.20) and ( ) ( ) [ ] σ α σ π ∆ ∆ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∆ = 4 2 2 4 2 ln 4 a E a C Y Var N i f (5.21) The following data are adopted in probabilistic symbolic computations: E[ ] . MPa max min ∆σ = σ − σ = 10 0 , ai=25 mm and obtained experimentally C=1.64x10-10, Y=1.15. The visualisation of the first two probabilistic moments of fatigue cycle number is done using the symbolic computation program MAPLE as functions of the coefficient of variation α(∆σ) and the final crack length af. The results of the analysis in the form of deterministic values, corresponding expected values and standard deviations are presented below with the design parameters marked on the horizontal axes. Figure 5.1. Deterministic values of fatigue cycles (dN)