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