5 Fracture and Fatigue Models for Composites 5.1 Introduction The effective fatigue model for engineering composites analysis is decisive for a precise estimation of the overall life of this structure and satisfactory reliability analysis of such materials.Various theoretical,experimental and computational criteria must be satisfied in the same time to obtain such a model [37,172,246,2981. These criteria may include material properties of composite constituents [226,258], composite type [229](ductile or brittle components),spatial distribution,length (continuity)as well as size effect of the reinforcing fibres [219,220,335],frequency effects [350],load amplitude type [48](constant or not),micromechanical phenomena [110,217,279],etc.First of all,a very precise,experimentally based deterministic idea of fatigue life cycle estimation has to be proposed.It should be adequate for the composite components,the technology applied and numerical methodology implemented.Monitoring of most engineering composites and preventing the fatigue failure is very complicated and usually demands very modern technology [360].It is widely known that the interface conditions and phenomena can be decisive factors for both static fracture and fatigue resistance of laminates,fibre-and particle-reinforced composites.Analytical models even in the case of linear elasticity models are complicated [369],therefore numerical analysis is very popular in this area.Engineering FEM software makes it possible to simulate delamination processes [362]and fatigue damage [62,277]in fibre-reinforced composites as well as time-dependent interlaminar debonding processes [69],for instance. The application of the well-known Palmgren-Miner or Paris-Erdogan laws is not always recommended as the most effective method in spite of their simplicity or wide technological usage.The choice of fatigue theory should be accompanied with a corresponding sensitivity analysis,where physical and material input parameters included into the fatigue life cycle equation are treated as design variables.Due to the sensitivity gradients determination,the most decisive parameters should be considered,while the remaining ones,considering further stochastic analysis complexity,may be omitted.The sensitivity gradients can be determined analytically using symbolic computation packages (MAPLE, MATLAB,MATHEMATICA,etc.)or may result from discrete FEM computations,for instance.A related problem is to decide if the local concept of composite fatigue is to be applied (critical element concept,for instance),where local fatigue damage causes global structural changes of the composite reliability. This results in computational FEM or Boundary Element Method (BEM)based
5 Fracture and Fatigue Models for Composites 5.1 Introduction The effective fatigue model for engineering composites analysis is decisive for a precise estimation of the overall life of this structure and satisfactory reliability analysis of such materials. Various theoretical, experimental and computational criteria must be satisfied in the same time to obtain such a model [37,172,246,298]. These criteria may include material properties of composite constituents [226,258], composite type [229] (ductile or brittle components), spatial distribution, length (continuity) as well as size effect of the reinforcing fibres [219,220,335], frequency effects [350], load amplitude type [48] (constant or not), micromechanical phenomena [110,217,279], etc. First of all, a very precise, experimentally based deterministic idea of fatigue life cycle estimation has to be proposed. It should be adequate for the composite components, the technology applied and numerical methodology implemented. Monitoring of most engineering composites and preventing the fatigue failure is very complicated and usually demands very modern technology [360]. It is widely known that the interface conditions and phenomena can be decisive factors for both static fracture and fatigue resistance of laminates, fibre- and particle-reinforced composites. Analytical models even in the case of linear elasticity models are complicated [369], therefore numerical analysis is very popular in this area. Engineering FEM software makes it possible to simulate delamination processes [362] and fatigue damage [62,277] in fibre-reinforced composites as well as time-dependent interlaminar debonding processes [69], for instance. The application of the well-known Palmgren-Miner or Paris-Erdogan laws is not always recommended as the most effective method in spite of their simplicity or wide technological usage. The choice of fatigue theory should be accompanied with a corresponding sensitivity analysis, where physical and material input parameters included into the fatigue life cycle equation are treated as design variables. Due to the sensitivity gradients determination, the most decisive parameters should be considered, while the remaining ones, considering further stochastic analysis complexity, may be omitted. The sensitivity gradients can be determined analytically using symbolic computation packages (MAPLE, MATLAB, MATHEMATICA, etc.) or may result from discrete FEM computations, for instance. A related problem is to decide if the local concept of composite fatigue is to be applied (critical element concept, for instance), where local fatigue damage causes global structural changes of the composite reliability. This results in computational FEM or Boundary Element Method (BEM) based
Fracture and Fatigue Analysis of Composites 223 analyses of the whole composite in its real configuration,including the microgeometry and all interface phenomena into it.Alternatively,the homogenisation method can be applied,where the complementary energy or potential energy of the entire system is the only measure of composite fatigue. Then,the global discretisation of the original structure is used instead and the equivalent,homogeneous medium is simulated numerically. Next,an appropriate analytical or computational stochastic analysis method corresponding to the level of randomness of input parameters is considered.The Monte Carlo simulation based analysis,stochastic second or third order perturbation method or,alternatively,stochastic spectral analysis can be taken into account.The first method does not have any restrictions on input random variable probabilistic moment interrelations.However,time consuming computations can be expected.Numerical analysis using the second approach implementation is very fast,but not sufficiently effective for larger than 10%variations of input random parameters,while the last approach has some limitations on convergence of the output parameters and fields.The choice between the methods proposed is implied by the availability of the experimental techniques,considering the input randomness level.On the other hand this choice is determined by relevant reliability criteria for composites.Furthermore,having collected most of the deterministic fatigue concepts for composites,corresponding stochastic equations can be obtained automatically using analytical derivation or computer simulation techniques. Combination of deterministic models and stochastic methods requires another engineering decision about the choice of the randomness type to be analysed.It is known from recent references in this area that (i)random variables,(ii)random fields as well as (iii)stochastic processes can be considered as the input of the entire fatigue analysis.According to the state-of-the-art research,the first two types of randomness can be considered together with FEM or BEM based computational simulation,while the stochastic processes can be used in terms of direct simulation of the fatigue process when the analytical solution is known. Some approximate methods of combining discrete modelling with stochastic degradation of homogeneous materials are available in reliability modeling; however without any application in engineering composites area until now. Various fatigue models worked out for composites can be classified in different ways:using the scale of the model application (local or global)or considering the main goal of the analysis (fatigue cycle number,its stiffness reduction,its crack growth or damage function determination),the analysis type (deterministic, probabilistic or stochastic)as well as the composite material type (ceramic, polymer-based,metal matrix and so forth). Considering various scales of engineering composites and fatigue phenomena related to them,the local and,alternatively,global approaches are considered. Local and microlocal models represented by the critical element concept [299], assume that there exists so-called critical element in the entire composite structure that controls the total fatigue damage (as well as subcrtitical elements,too),and then the local damage is governing the reliability of the whole composite structure
Fracture and Fatigue Analysis of Composites 223 analyses of the whole composite in its real configuration, including the microgeometry and all interface phenomena into it. Alternatively, the homogenisation method can be applied, where the complementary energy or potential energy of the entire system is the only measure of composite fatigue. Then, the global discretisation of the original structure is used instead and the equivalent, homogeneous medium is simulated numerically. Next, an appropriate analytical or computational stochastic analysis method corresponding to the level of randomness of input parameters is considered. The Monte Carlo simulation based analysis, stochastic second or third order perturbation method or, alternatively, stochastic spectral analysis can be taken into account. The first method does not have any restrictions on input random variable probabilistic moment interrelations. However, time consuming computations can be expected. Numerical analysis using the second approach implementation is very fast, but not sufficiently effective for larger than 10% variations of input random parameters, while the last approach has some limitations on convergence of the output parameters and fields. The choice between the methods proposed is implied by the availability of the experimental techniques, considering the input randomness level. On the other hand this choice is determined by relevant reliability criteria for composites. Furthermore, having collected most of the deterministic fatigue concepts for composites, corresponding stochastic equations can be obtained automatically using analytical derivation or computer simulation techniques. Combination of deterministic models and stochastic methods requires another engineering decision about the choice of the randomness type to be analysed. It is known from recent references in this area that (i) random variables, (ii) random fields as well as (iii) stochastic processes can be considered as the input of the entire fatigue analysis. According to the state-of-the-art research, the first two types of randomness can be considered together with FEM or BEM based computational simulation, while the stochastic processes can be used in terms of direct simulation of the fatigue process when the analytical solution is known. Some approximate methods of combining discrete modelling with stochastic degradation of homogeneous materials are available in reliability modeling; however without any application in engineering composites area until now. Various fatigue models worked out for composites can be classified in different ways: using the scale of the model application (local or global) or considering the main goal of the analysis (fatigue cycle number, its stiffness reduction, its crack growth or damage function determination), the analysis type (deterministic, probabilistic or stochastic) as well as the composite material type (ceramic, polymer-based, metal matrix and so forth). Considering various scales of engineering composites and fatigue phenomena related to them, the local and, alternatively, global approaches are considered. Local and microlocal models represented by the critical element concept [299], assume that there exists so-called critical element in the entire composite structure that controls the total fatigue damage (as well as subcrtitical elements, too), and then the local damage is governing the reliability of the whole composite structure
224 Computational Mechanics of Composite Materials This assumption results in the fact that the whole composite,together with microstructural defects increasing during fatigue processes,should be discretised for the FEM or BEM simulation.Taking into account the application of the probabilistic analysis,the model implies the randomness in microgeometry of the composite,which is extremely difficult in computational simulation,as is shown below.Some special purpose algorithms are introduced to replace the randomness in composite interface geometry with the stochasticity of material thermoelastic properties. Alternatively,a homogenisation method is proposed for more efficient fracture and fatigue phenomena analysis [223]that originated from analysis of linear periodic elastic composites without defects.The main idea is to find the medium equivalent to the original composite in terms of complementary energy,or potential energy,equal for both media.The final goal of the homogenisation procedure is to find the effective material characteristics defining the equivalent homogeneous medium.The effective constitutive relations can be found for the composite with elastic,elastoplastic or even viscoelastoplastic components with and/or without microstructural defects.The general assumption of the model means,however,that every local phenomenon can be averaged in some sense in the entire composite volume and that the global,not local,phenomena result in the overall composite fatigue. 5.2 Existing Techniques Overview Taking into account the results of fatigue analysis,four essentially different approaches can be observed:(i)direct determination of the fatigue cycle number N, (ii)fatigue stiffness reduction where mechanical properties of the composite are decreased in the function of N,(iii)observation of the crack length growth a as a function of fatigue cycle number(as da/dN,taking into account the physical nature of fatigue phenomenon)or,alternatively,(iv)estimation of the damage function in terms of dD/dN.A damage function is usually proposed as follows: (1)D=0 with cycle number n=0; (2)D=1,where failure occurs; (3)D=AD,where AD is the amount of damage accumulation during fatigue at stress level ri.Generally,the function D can be represented as D=D(n,r,f,T,M....) (5.1) where n indexes a number of the current fatigue cycle,r is the applied stress level,f denotes applied stress frequency,T is temperature,while M denotes the moisture content.Then,contrary to the crack length growth analysis,the damage function can be proposed each time in a different form as a function of various structural parameters
224 Computational Mechanics of Composite Materials This assumption results in the fact that the whole composite, together with microstructural defects increasing during fatigue processes, should be discretised for the FEM or BEM simulation. Taking into account the application of the probabilistic analysis, the model implies the randomness in microgeometry of the composite, which is extremely difficult in computational simulation, as is shown below. Some special purpose algorithms are introduced to replace the randomness in composite interface geometry with the stochasticity of material thermoelastic properties. Alternatively, a homogenisation method is proposed for more efficient fracture and fatigue phenomena analysis [223] that originated from analysis of linear periodic elastic composites without defects. The main idea is to find the medium equivalent to the original composite in terms of complementary energy, or potential energy, equal for both media. The final goal of the homogenisation procedure is to find the effective material characteristics defining the equivalent homogeneous medium. The effective constitutive relations can be found for the composite with elastic, elastoplastic or even viscoelastoplastic components with and/or without microstructural defects. The general assumption of the model means, however, that every local phenomenon can be averaged in some sense in the entire composite volume and that the global, not local, phenomena result in the overall composite fatigue. 5.2 Existing Techniques Overview Taking into account the results of fatigue analysis, four essentially different approaches can be observed: (i) direct determination of the fatigue cycle number N, (ii) fatigue stiffness reduction where mechanical properties of the composite are decreased in the function of N, (iii) observation of the crack length growth a as a function of fatigue cycle number (as da/dN, taking into account the physical nature of fatigue phenomenon) or, alternatively, (iv) estimation of the damage function in terms of dD/dN. A damage function is usually proposed as follows: (1) D=0 with cycle number n=0; (2) D=1, where failure occurs; (3) ∑ = = ∆ n i D Di 1 , where ∆Di is the amount of damage accumulation during fatigue at stress level ri. Generally, the function D can be represented as D = D( ) n,r, f ,T, M ,... (5.1) where n indexes a number of the current fatigue cycle, r is the applied stress level, f denotes applied stress frequency, T is temperature, while M denotes the moisture content. Then, contrary to the crack length growth analysis, the damage function can be proposed each time in a different form as a function of various structural parameters
Fracture and Fatigue Analysis of Composites 225 Let us note that direct determination of fatigue cycle number makes it possible to derive,without any further computational simulations,the life of the structure till the failure,while the stiffness reduction approach is frequently used together with the FEM or BEM structural analyses.The crack length growth and damage function approach are used together with the structural analysis FEM programs, usually to compute the stress intensity factors.However final direct or symbolic integration of crack length or damage function is necessary to complete the entire fatigue life computations. Considering the mathematical nature of the fatigue life cycle estimation,the deterministic approach can be applied,where all input parameters are defined uniquely by their mean values.Otherwise,the whole variety of probabilistic approaches can be introduced where fatigue structural life is described as a simple random variable with structural parameters defined deterministically and random external loads.The cumulative fatigue damage can be treated as a random process, where all design parameters are modelled as stochastic parameters.However,in all probabilistic approaches sufficient statistical information about all input parameters is necessary,which is especially complicated in the last approach where random processes are considered due to the statistical input in some constant periods of time(using the same technology to assure the same randomness level). The analysis of fatigue life cycle number begins with direct estimation of this parameter by a simple power function(A5.1)consisting of stress amplitude as well as some material constant(s).Alternatively,an exponential-logarithmic equation can be proposed (A5.2),where temperature,strength and residual stresses are inserted.Both of them have a deterministic form and can be randomised using any of the methods described below.The weak point is the homogeneous character of the material being analysed;to use these criteria for composites,the effective parameters should be calculated first.In contrary to theoretical models,the experimentally based probabilistic law can be proposed where parameters of the Weibull distribution of static strength are inserted (A5.3);it is important to underline that this law does not have its deterministic origin. More complicated from the viewpoint of engineering practice are the stiffness reduction models (cf.A5.4-A5.7),where structural material characteristics are reduced together with a successive fatigue cycle number increase.The stiffness reduction model is used in FEM or BEM dynamical modelling to recalculate the component stiffness in each cycle.It is done using a linear model for stiffness reduction,cf.(A5.5),as well as some power laws (see (A5.4),for example) determined on the basis of mechanical properties reduction rewritten for homogeneous media only.An alternative power law presented as(A5.7)consists of the time of rupture,creep and fatigue,measured in hours.Considering the random analysis aspects,a probabilistic treatment of material properties seems to be much more justified. Deterministic fatigue crack growth analysis presented by(A5.8)-(A5.29)can be classified taking into account the physical basis of this law formation,such as energy approaches (A5.8)-(A5.11),crack opening displacement (COD)based approaches (A5.12),(A5.15)-(A5.17),(A5.19)and (A5.20),continuous
Fracture and Fatigue Analysis of Composites 225 Let us note that direct determination of fatigue cycle number makes it possible to derive, without any further computational simulations, the life of the structure till the failure, while the stiffness reduction approach is frequently used together with the FEM or BEM structural analyses. The crack length growth and damage function approach are used together with the structural analysis FEM programs, usually to compute the stress intensity factors. However final direct or symbolic integration of crack length or damage function is necessary to complete the entire fatigue life computations. Considering the mathematical nature of the fatigue life cycle estimation, the deterministic approach can be applied, where all input parameters are defined uniquely by their mean values. Otherwise, the whole variety of probabilistic approaches can be introduced where fatigue structural life is described as a simple random variable with structural parameters defined deterministically and random external loads. The cumulative fatigue damage can be treated as a random process, where all design parameters are modelled as stochastic parameters. However, in all probabilistic approaches sufficient statistical information about all input parameters is necessary, which is especially complicated in the last approach where random processes are considered due to the statistical input in some constant periods of time (using the same technology to assure the same randomness level). The analysis of fatigue life cycle number begins with direct estimation of this parameter by a simple power function (A5.1) consisting of stress amplitude as well as some material constant(s). Alternatively, an exponential-logarithmic equation can be proposed (A5.2), where temperature, strength and residual stresses are inserted. Both of them have a deterministic form and can be randomised using any of the methods described below. The weak point is the homogeneous character of the material being analysed; to use these criteria for composites, the effective parameters should be calculated first. In contrary to theoretical models, the experimentally based probabilistic law can be proposed where parameters of the Weibull distribution of static strength are inserted (A5.3); it is important to underline that this law does not have its deterministic origin. More complicated from the viewpoint of engineering practice are the stiffness reduction models (cf. A5.4-A5.7), where structural material characteristics are reduced together with a successive fatigue cycle number increase. The stiffness reduction model is used in FEM or BEM dynamical modelling to recalculate the component stiffness in each cycle. It is done using a linear model for stiffness reduction, cf. (A5.5), as well as some power laws (see (A5.4), for example) determined on the basis of mechanical properties reduction rewritten for homogeneous media only. An alternative power law presented as (A5.7) consists of the time of rupture, creep and fatigue, measured in hours. Considering the random analysis aspects, a probabilistic treatment of material properties seems to be much more justified. Deterministic fatigue crack growth analysis presented by (A5.8) - (A5.29) can be classified taking into account the physical basis of this law formation, such as energy approaches (A5.8) - (A5.11), crack opening displacement (COD) based approaches (A5.12), (A5.15) - (A5.17), (A5.19) and (A5.20), continuous
226 Computational Mechanics of Composite Materials dislocation formalism (A5.13),skipband decohesion (A5.18),nucleation rate process models (A5.14)and(A5.15),dislocation approaches (A5.23)and(A5.24), monotonic yield strength dependence (A5.25)and (A5.31)as well as another mixed laws (A5.26)-(A5.30)and (A5.32)-(A5.35).Description of the derivative da/dN enables further integration and determination of the critical crack length.The second classification method is based on a verification of the validity of a particular theory in terms of elastic (A5.8)-(A5.20),(A5.26)-(A5.30), (A5.32)-(A5.34)or elastoplastic (A5.22)-(A5.25)and (A5.31)mechanism of material fracture.Most of them are used for composites,even though they are defined for homogeneous media,except for the Ratwani-Kan and Wang- Crossman models (A5.21)and (A5.22),where composite material characteristics are inserted.All of the homogeneous models contain stress intensity factor AK in various powers (from 2 to n),while composite-oriented theories are based on delamination length parameter.The structure of these equations enables one to include statistical information about any material or geometrical parameters and, next,to use a simulation or perturbation technique to determine expected values and variances of the critical crack length,which are very useful in stochastic reliability analysis. An essentially different methodology is proposed for the statistical analysis [9,35,130,288,333,349,359]and in the stochastic case [241,244,373],where the crack size and/or components material parameters,their spatial distribution may be treated as random processes (cf.egns (A5.36)-(A5.44)).Then,various representations and types of random fields and stochastic processes are used,such as stationary and nonstationary Gaussian white noise,homogeneous Poisson counting process [204]as well as Markovian [304],birth and death or renewal processes.However all of them are formulated for a globally homogeneous material.These methods are intuitively more efficient in real fatigue process modelling than deterministic ones,but they require definitely a more advanced mathematical apparatus.Further,randomised versions of deterministic models can be applied together with structural analysis programs,while stochastic characters of a random process cannot be included without any modification in the FEM or the BEM computer routines.An alternative option for stochastic models of fatigue is experimentally based formulation of fatigue law,where measurements of various material parameters are taken in constant time periods.Then,statistical information about expected values and higher order probabilistic characteristics histories is obtained,which allows approximation of the entire fatigue process.Such a method, used previously for homogeneous structural elements,is very efficient in stochastic reliability prognosis and then random fatigue process can be included in SFEM computations.Let us observe that formulations analogous to the ones presented above can be used for ductile fracture of composites where initiation,coalescence and closing of microvoids are observed under periodic or quasiperiodic external loads. A wide variety of fatigue damage function models is collected at the end of the appendix.The basic rules are based on the numbers of cycles to failure ((A5.45)- (A5.48),(A5.54)-(A5.57),(A5.63)-(A5.65)and(A5.67)illustrated with
226 Computational Mechanics of Composite Materials dislocation formalism (A5.13), skipband decohesion (A5.18), nucleation rate process models (A5.14) and (A5.15), dislocation approaches (A5.23) and (A5.24), monotonic yield strength dependence (A5.25) and (A5.31) as well as another mixed laws (A5.26) - (A5.30) and (A5.32) - (A5.35). Description of the derivative da/dN enables further integration and determination of the critical crack length. The second classification method is based on a verification of the validity of a particular theory in terms of elastic (A5.8) - (A5.20), (A5.26) - (A5.30), (A5.32) - (A5.34) or elastoplastic (A5.22) - (A5.25) and (A5.31) mechanism of material fracture. Most of them are used for composites, even though they are defined for homogeneous media, except for the Ratwani-Kan and WangCrossman models (A5.21) and (A5.22), where composite material characteristics are inserted. All of the homogeneous models contain stress intensity factor ∆K in various powers (from 2 to n), while composite-oriented theories are based on delamination length parameter. The structure of these equations enables one to include statistical information about any material or geometrical parameters and, next, to use a simulation or perturbation technique to determine expected values and variances of the critical crack length, which are very useful in stochastic reliability analysis. An essentially different methodology is proposed for the statistical analysis [9,35,130,288,333,349,359] and in the stochastic case [241,244,373], where the crack size and/or components material parameters, their spatial distribution may be treated as random processes (cf. eqns (A5.36) - (A5.44)). Then, various representations and types of random fields and stochastic processes are used, such as stationary and nonstationary Gaussian white noise, homogeneous Poisson counting process [204] as well as Markovian [304], birth and death or renewal processes. However all of them are formulated for a globally homogeneous material. These methods are intuitively more efficient in real fatigue process modelling than deterministic ones, but they require definitely a more advanced mathematical apparatus. Further, randomised versions of deterministic models can be applied together with structural analysis programs, while stochastic characters of a random process cannot be included without any modification in the FEM or the BEM computer routines. An alternative option for stochastic models of fatigue is experimentally based formulation of fatigue law, where measurements of various material parameters are taken in constant time periods. Then, statistical information about expected values and higher order probabilistic characteristics histories is obtained, which allows approximation of the entire fatigue process. Such a method, used previously for homogeneous structural elements, is very efficient in stochastic reliability prognosis and then random fatigue process can be included in SFEM computations. Let us observe that formulations analogous to the ones presented above can be used for ductile fracture of composites where initiation, coalescence and closing of microvoids are observed under periodic or quasiperiodic external loads. A wide variety of fatigue damage function models is collected at the end of the appendix. The basic rules are based on the numbers of cycles to failure ((A5.45) - (A5.48), (A5.54) - (A5.57), (A5.63) - (A5.65) and (A5.67)) illustrated with
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)