Chapter 1:Account for Random Microstructure in Multiscale Models Vadim V.Silberschmidt Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University,Ashby Road,Loughborough,Leics., LE11 3TU.UK 1.1 Introduction The accumulated in last decades knowledge of fibre-reinforced composite materials,their effective properties as well as deformation and damage processes in them confirms a random (probabilistic)character of their failure(see,e.g.[1-4]and references therein).Such a character is deter- mined by the specificity of microstructure of composites -a result of a manufacturing process of embedding of a huge number of reinforcing elements into a matrix.The resulting microscopic heterogeneity linked to randomness in positions of fibres,their bonding with the matrix,presence of microdefects,etc.causes a spatially and temporally non-uniform res- ponse to external loading even under macroscopically uniform loading conditions.The resulting pattern of deformation localisation and stress con- centrations is neither uniform nor periodic;it defines macroscopic non- uniformity in evolution of various damage mechanisms. At the current level of computational facilities,direct introduction of these stochastic microscopic features into computational models is pro- hibitive and counterproductive.A significantly better strategy is to employ multiscale models [5]that separate the levels of descriptions into (at least) local and global ones.The local level is used to incorporate details of a (real)microstructure of composites within a relatively small area(window) and to study the effect of its variability while the global one accounts for geometry of composite components/structures and loading/environmental conditions to study problems of their macroscopic behaviour,structural
Chapter 1: Account for Random Microstructure Vadim V. Silberschmidt Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Ashby Road, Loughborough, Leics., LE11 3TU, UK 1.1 Introduction The accumulated in last decades knowledge of fibre-reinforced composite materials, their effective properties as well as deformation and damage processes in them confirms a random (probabilistic) character of their failure (see, e.g. [1–4] and references therein). Such a character is determined by the specificity of microstructure of composites – a result of a manufacturing process of embedding of a huge number of reinforcing elements into a matrix. The resulting microscopic heterogeneity linked to randomness in positions of fibres, their bonding with the matrix, presence of microdefects, etc. causes a spatially and temporally non-uniform response to external loading even under macroscopically uniform loading conditions. The resulting pattern of deformation localisation and stress concentrations is neither uniform nor periodic; it defines macroscopic nonuniformity in evolution of various damage mechanisms. At the current level of computational facilities, direct introduction of these stochastic microscopic features into computational models is prohibitive and counterproductive. A significantly better strategy is to employ local and global ones. The local level is used to incorporate details of a (real) microstructure of composites within a relatively small area (window) and to study the effect of its variability while the global one accounts for geometry of composite components/structures and loading/environmental conditions to study problems of their macroscopic behaviour, structural in Multiscale Models multiscale models [5] that separate the levels of descriptions into (at least)
2 V.V.Silberschmidt integrity and/or durability.But such separation of scales presupposes a necessity to bridge them within a framework of a single computational approach.Generally,various schemes to account for material's random- ness can be employed both for various scales of modelling and bridging procedures. The diversity of composites (in terms of constituents,their mor- phology and a type of reinforcement)makes a general analysis of their behaviour,including damage accumulation,practically infeasible.Hence this chapter is limited to analysis of the effect of randomness in distri- butions of filaments in matrix on damage evolution in two-phase fibrous composites under external load.A vast literature on composites that assumes a periodic character of reinforcement is not considered here (though some of its results are employed as an obvious comparison basis).Since only plies of unidirectional (continuous)fibre-reinforced composites are con- sidered,the orientational randomness of inclusions is also not treated here. Though 3D studies and simulations are becoming a routine approach,and the respective experimental techniques,e.g.micro-X-ray computer tomo- graphy,can provide necessary volumetric data,for the sake of more transparency'a local modelling level in this chapter is limited to (pre- dominantly)2D analysis of unidirectional layers in the plane perpendicular to its fibres.This is due to the emphasis on transverse (matrix)cracking in cross-ply laminates,which is one of their main damage mechanisms under static and fatigue loading conditions [2,4].So,effectively,the (virginal) state of transverse cross-section of plies in such composites can be considered as a 2D distribution of circular inclusions in a matrix(Fig.1.1). Fig.1.1.Distribution of continuous graphite fibres in epoxy matrix in a transverse cross-section of a unidirectionally reinforced ply(digitalisation of a micrograph)
integrity and/or durability. But such separation of scales presupposes a necessity to bridge them within a framework of a single computational approach. Generally, various schemes to account for material’s randomness can be employed both for various scales of modelling and bridging procedures. The diversity of composites (in terms of constituents, their morphology and a type of reinforcement) makes a general analysis of their behaviour, including damage accumulation, practically infeasible. Hence this chapter is limited to analysis of the effect of randomness in distributions of filaments in matrix on damage evolution in two-phase fibrous composites under external load. A vast literature on composites that assumes a periodic character of reinforcement is not considered here (though some of its results are employed as an obvious comparison basis). Since only plies of unidirectional (continuous) fibre-reinforced composites are considered, the orientational randomness of inclusions is also not treated here. Though 3D studies and simulations are becoming a routine approach, and the respective experimental techniques, e.g. micro-X-ray computer tomography, can provide necessary volumetric data, for the sake of more ‘transparency’ a local modelling level in this chapter is limited to (predominantly) 2D analysis of unidirectional layers in the plane perpendicular to its fibres. This is due to the emphasis on transverse (matrix) cracking in cross-ply laminates, which is one of their main damage mechanisms under static and fatigue loading conditions [2, 4]. So, effectively, the (virginal) state of transverse cross-section of plies in such composites can be considered as a 2D distribution of circular inclusions in a matrix (Fig. 1.1). Fig. 1.1. Distribution of continuous graphite fibres in epoxy matrix in a transverse cross-section of a unidirectionally reinforced ply (digitalisation of a micrograph) 2 V.V. Silberschmidt
Chapter 1:Account for Random Microstructure 3 This chapter treats various aspects of randomness at various levels of modelling of fibre-reinforced cross-ply laminates-from the character of local distributions of fibres to non-uniformity of damage processes and cracking evolution and their influence on the composite's response to external loading. 1.2 Microstructures and Effective Properties Though microstructural randomness of composites was obvious to re- searchers from the very beginning of the studies of such materials,the main emphasis of research was on the determination of their overall properties that could allow the use of deterministic continuous descriptions.In other words,an inhomogeneous material (discrete medium)is substituted by an equivalent homogenous one(continuous medium).This can be implemented by means of homogenisation procedures,smearing'microscopic features at the macroscopic level of modelling.In many cases,an assumption of a coherent mixture or statistical homogeneity is employed:The spatial distribution of the phases is assumed to be macroscopically homogeneous [6-11].But even in this case,a full description of a composite with arbi- trary geometry of phases and their volume fractions is cumbersome,so the emphasis is shifted to estimates of the effects of structural and microscopic features(volume fractions,shape of filaments,the extent of randomness in their distributions,variations in dimensions,etc.).The implementation of all of the mentioned factors within the framework of a single model is a rather complicated task,so historically effects of a single feature (or of a few ones)were studied separately.The research started for cases with so- called 'dilute dispersions'[6],i.e.low-volume fractions of reinforcement in a matrix,to exclude the effects due to their interactions,but later on it was extended to arbitrary volume fractions. The main line of analysis was a use of periodic arrays of reinforcement in a matrix.Though micrographs of real microstructures vividly demon- strated deviations from regular patterns in distributions of inclusions (Fig.1.1),(relative)simplicity of the approach made it very attractive.The notion of representative volume,used to estimate the effective properties, is also introduced early in the study of composites.According to Hill [6],it means a sample with two main properties: 1.Its structure is 'entirely typical'for the composite. 2.It contains a 'sufficient number'of microstructural elements so that boundary conditions at the surface of the composite do not affect its effective properties
3 This chapter treats various aspects of randomness at various levels of modelling of fibre-reinforced cross-ply laminates – from the character of local distributions of fibres to non-uniformity of damage processes and cracking evolution and their influence on the composite’s response to external loading. 1.2 Microstructures and Effective Properties Though microstructural randomness of composites was obvious to researchers from the very beginning of the studies of such materials, the main emphasis of research was on the determination of their overall properties that could allow the use of deterministic continuous descriptions. In other words, an inhomogeneous material (discrete medium) is substituted by an equivalent homogenous one (continuous medium). This can be implemented by means of homogenisation procedures, ‘smearing’ microscopic features at the macroscopic level of modelling. In many cases, an assumption of a coherent mixture or statistical homogeneity is employed: The spatial distribution of the phases is assumed to be macroscopically homogeneous [6–11]. But even in this case, a full description of a composite with arbitrary geometry of phases and their volume fractions is cumbersome, so the emphasis is shifted to estimates of the effects of structural and microscopic features (volume fractions, shape of filaments, the extent of randomness in their distributions, variations in dimensions, etc.). The implementation of all of the mentioned factors within the framework of a single model is a rather complicated task, so historically effects of a single feature (or of a few ones) were studied separately. The research started for cases with socalled ‘dilute dispersions’ [6], i.e. low-volume fractions of reinforcement in a matrix, to exclude the effects due to their interactions, but later on it was extended to arbitrary volume fractions. The main line of analysis was a use of periodic arrays of reinforcement in a matrix. Though micrographs of real microstructures vividly demonstrated deviations from regular patterns in distributions of inclusions (Fig. 1.1), (relative) simplicity of the approach made it very attractive. The notion of representative volume, used to estimate the effective properties, is also introduced early in the study of composites. According to Hill [6], it means a sample with two main properties: 1. Its structure is ‘entirely typical’ for the composite. 2. It contains a ‘sufficient number’ of microstructural elements so that boundary conditions at the surface of the composite do not affect its effective properties. Chapter 1: Account for Random Microstructure
4 V.V.Silberschmidt The main schemes used to determine the effective properties of composites are either the direct approaches,using,e.g.Voigt and Reuss estimates based on assumptions of uniform distributions of the stress and strain,respectively,or variational ones,employing,for instance,an elastic polarisation tensor [12].The latter scheme allows one to obtain much closer bounds for the effective moduli than the Voigt and Reuss estimates. The well-known Hashin-Shtrikman bounds are determined on the basis of the original variational approach;the classical extremum principles of mechanics are used in [13]to obtain bounds for the overall elastic properties of an inhomogeneous system composed of various solid phases at arbitrary concentrations with ideal bonding. The obtained results and bounds for elastic moduli explicitly depend on the volume fraction of constituents,or,for a two-phase composite,on the volume fraction of reinforcement due to an apparent relation V+Vm=1, (1.1) where Ir and Im are volume fractions of reinforcement (fibres)and matrix, respectively. For a case of fibre-reinforced composites with continuous fibres,one of the first results for bounds of the effective elastic moduli for a case of a transversely isotropic composite with fibres of the same diameter,arranged in a hexagonal array,was obtained in [9].More general results for a case of arbitrary geometry,restricted to the statistically transversal isotropy,are obtained in [14].At the same time,Hashin [14]noted that it was 'not known how to use statistical details of phase geometry in prediction of macroscopic elastic behaviour'.The solution was based on the analysis performed for a cylindrical sub-region,extending from base to base of the fibre-reinforced specimen (Hashin introduced there the well-known now abbreviation RVE for representative volume element)with its transverse cross-section being,on the one hand,considerably smaller than that of the entire specimen but,on the other hand,considerably larger than that of the filament. The Hashin's approach deals with a'cylinder assemblage'by contrast with the 'concentric composite circular cylinders'of Hill [7].Both ap- proaches provide the same bounds for the transverse plain-strain bulk modulus for a two-phase fibre-reinforced composite.Still,these approaches predicted a relatively broad interval of effective properties important for various application magnitudes of the volume fraction of fibres Vr~0.55. To improve the obtained bounds,approaches based on multi-point correlation functions were introduced.An example of such a function is the n-point probability function [15,16]
The main schemes used to determine the effective properties of composites are either the direct approaches, using, e.g. Voigt and Reuss estimates based on assumptions of uniform distributions of the stress and strain, respectively, or variational ones, employing, for instance, an elastic polarisation tensor [12]. The latter scheme allows one to obtain much closer bounds for the effective moduli than the Voigt and Reuss estimates. The well-known Hashin–Shtrikman bounds are determined on the basis of the original variational approach; the classical extremum principles of mechanics are used in [13] to obtain bounds for the overall elastic properties of an inhomogeneous system composed of various solid phases at arbitrary concentrations with ideal bonding. The obtained results and bounds for elastic moduli explicitly depend on the volume fraction of constituents, or, for a two-phase composite, on the volume fraction of reinforcement due to an apparent relation f m V V+ =1, (1.1) where Vf and Vm are volume fractions of reinforcement (fibres) and matrix, respectively. For a case of fibre-reinforced composites with continuous fibres, one of the first results for bounds of the effective elastic moduli for a case of a transversely isotropic composite with fibres of the same diameter, arranged in a hexagonal array, was obtained in [9]. More general results for a case of arbitrary geometry, restricted to the statistically transversal isotropy, are obtained in [14]. At the same time, Hashin [14] noted that it was ‘not known how to use statistical details of phase geometry in prediction of macroscopic elastic behaviour’. The solution was based on the analysis performed for a cylindrical sub-region, extending from base to base of the fibre-reinforced specimen (Hashin introduced there the well-known now abbreviation RVE for representative volume element) with its transverse cross-section being, on the one hand, considerably smaller than that of the entire specimen but, on the other hand, considerably larger than that of the filament. The Hashin’s approach deals with a ‘cylinder assemblage’ by contrast with the ‘concentric composite circular cylinders’ of Hill [7]. Both approaches provide the same bounds for the transverse plain-strain bulk modulus for a two-phase fibre-reinforced composite. Still, these approaches predicted a relatively broad interval of effective properties important for various application magnitudes of the volume fraction of fibres Vf ≈ 0.55. To improve the obtained bounds, approaches based on multi-point correlation functions were introduced. An example of such a function is the n-point probability function [15, 16] 4 V.V. Silberschmidt
Chapter 1:Account for Random Microstructure 5 S（2X) (1.2) where I(x)is the characteristic function (known also as indicator function [171)of the phase 1 (e.g.inclusions) I(x)= 1,if x belongs to phasel, (1.3) 10,otherwise; angular brackets denote an ensemble average. The volume fraction Vr is a one-point probability function.The two- point probability function (for a phase iof a composite can be interpreted as a probability that two points at positions xi and x2 belong to this phase [18].For statistically isotropic media,the two-point probability function depends only on the distancer=x-2between the points, and the simplified notationS(can be used.For a statistically isotropic fibrous composite,two estimates hold S(0)=V (1.4) and lim (r)=V2 r (1.5) Such correlation functions are normally referred to as microstructural descriptors,a thorough review of various types of which is given in [17]. To introduce the extent of connectedness of microstructural elements into consideration (that the two-point probability function lacks),another statistical measure -lineal-path function is introduced in [19].This parameter denoted 1(x)is linked to the probability that a line seg- ment spanning from xi and x2 is situated entirely in the phase i. Three-point correlation functions are employed in [20]to obtain the bounds for elastic properties of composites.One disadvantage of the approach is the use of different correlation functions to define the upper and lower bounds of properties.So,Milton [21,22]introduced 'simplified bounds'for two-component composites that depend on the volume fraction of two 'fundamental geometric parameters'=1-52 and m=1-n2 (5,mE[0,1]).These bounds are more restrictive than the Hashin- Shtrikman bounds(up to five times narrower according to Milton [21]);the latter correspond to cases =m=0 and 5=m=1.The self-consistent
5 1 2 1 ( , , , ) ( ), n n ni i S I = xx x x … = ∏ (1.2) where I(x) is the characteristic function (known also as indicator function [17]) of the phase 1 (e.g. inclusions) 1, if belongs to phase1, ( ) 0, otherwise; I = ⎧ ⎨ ⎩ x x (1.3) angular brackets denote an ensemble average. The volume fraction Vf is a one-point probability function. The twopoint probability function ( ) 2 12 (, ) i S x x for a phase i of a composite can be interpreted as a probability that two points at positions x1 and x2 belong to this phase [18]. For statistically isotropic media, the two-point probability function depends only on the distance 1 2 r = x x − between the points, and the simplified notation ( ) 2 ( ) i S r can be used. For a statistically isotropic fibrous composite, two estimates hold f 2 f S V (0) = (1.4) and f 2 2 f lim ( ) . r Sr V →∞ = (1.5) Such correlation functions are normally referred to as microstructural descriptors, a thorough review of various types of which is given in [17]. To introduce the extent of connectedness of microstructural elements into consideration (that the two-point probability function lacks), another statistical measure – lineal-path function – is introduced in [19]. This parameter denoted ( ) 2 12 (, ) i L x x is linked to the probability that a line segment spanning from x1 and x2 is situated entirely in the phase i. Three-point correlation functions are employed in [20] to obtain the bounds for elastic properties of composites. One disadvantage of the approach is the use of different correlation functions to define the upper and lower bounds of properties. So, Milton [21, 22] introduced ‘simplified bounds’ for two-component composites that depend on the volume fraction of two ‘fundamental geometric parameters’ ξ1 = 1 − ξ2 and η1 = 1 − η2 (ξ1, η1 ∈ [0,1]). These bounds are more restrictive than the Hashin– Shtrikman bounds (up to five times narrower according to Milton [21]); the latter correspond to cases ξ1 = η1 = 0 and ξ1 = η1 = 1. The self-consistent Chapter 1: Account for Random Microstructure
6 V.V.Silberschmidt approximations of [11,23]correspond-to the same order of approximation -to s=m=Ve.The fourth-order correlation functions for composites are suggested in [24]. An alternative approach to the self-consistent scheme is introduced in [25,26]and coined differential effective medium theory in [261.According to Norris [27],the suggested approach is rooted in the idea of Roscoe [28] that extended the famous Einstein's results on suspensions [29,30].One of the advantages of the differential scheme-as compared to the self- consistent one-is that it distinguishes between the two phases.One phase is taken as a matrix while the second-filament-is incrementally added to it from zero concentration to the final value [25,271.At each stage of the process,the added inclusions are considered to be embedded in a homo- geneous material,corresponding to the composite formed by the matrix and all the previously added inclusions.This process is described by the tensorial differential equation of the following structure: dL I dV,1-V (L,-L)E, (1.6) with an obvious condition L(V=0)=L2 (1.7) Here L is the (fourth-order)tensor of effective moduli of the two-phase composite;L and L2 are moduli of inclusions and matrix,respectively; E =[I+P(L-L)]is a strain concentration tensor;I is a unit tensor and tensor P was introduced by Hill [23].A more generalised scheme is suggested in [27],where particles'of both matrix and inclusions can be added simultaneously to the initial material. 1.3 Microstructures and Their Descriptors Since transversal arrangements of fibres in unidirectional layers of real composites are vividly random(Fig.1.1),researchers trying to adequately describe them are confronted with several problems: 1.Characterisation of random microstructures 2.Comparison of random and periodic microstructures 3.Introduction of real microstructures into models The first problem is traditionally solved with the help of the automatic image analysis (AIA)and various tessellation schemes.An attempt to
approximations of [11, 23] correspond – to the same order of approximation – to ξ1 = η1 = Vf. The fourth-order correlation functions for composites are suggested in [24]. An alternative approach to the self-consistent scheme is introduced in [25, 26] and coined differential effective medium theory in [26]. According to Norris [27], the suggested approach is rooted in the idea of Roscoe [28] that extended the famous Einstein’s results on suspensions [29, 30]. One of the advantages of the differential scheme – as compared to the selfconsistent one – is that it distinguishes between the two phases. One phase is taken as a matrix while the second – filament – is incrementally added to it from zero concentration to the final value [25, 27]. At each stage of the process, the added inclusions are considered to be embedded in a homogeneous material, corresponding to the composite formed by the matrix and all the previously added inclusions. This process is described by the tensorial differential equation of the following structure: 1 1 f f d 1 ( ), d 1 V V = − − L L LE (1.6) with an obvious condition f 2 L L ( 0) . V = = (1.7) Here L is the (fourth-order) tensor of effective moduli of the two-phase composite; L1 and L2 are moduli of inclusions and matrix, respectively; 1 1 E I PL L =+ − [ ( )] is a strain concentration tensor; I is a unit tensor and tensor P was introduced by Hill [23]. A more generalised scheme is suggested in [27], where ‘particles’ of both matrix and inclusions can be added simultaneously to the initial material. 1.3 Microstructures and Their Descriptors Since transversal arrangements of fibres in unidirectional layers of real composites are vividly random (Fig. 1.1), researchers trying to adequately describe them are confronted with several problems: 1. Characterisation of random microstructures 2. Comparison of random and periodic microstructures 3. Introduction of real microstructures into models The first problem is traditionally solved with the help of the automatic image analysis (AIA) and various tessellation schemes. An attempt to 6 V.V. Silberschmidt
Chapter 1:Account for Random Microstructure 7 quantify the random distribution of filaments(second phase)in a matrix by means of AIA and Dirichlet cell tessellation procedures was undertaken in [31,321.Voronoi tessellation,based on discretisation of a domain into multi-sided convex polygons(known as Voronoi)each containing no more than a single filament,is also used to estimate the character of distribution of distances between filaments [33,34].The distribution of cells is sup- posed to be of the Poisson type with the cumulative probability distribution function accounting for non-overlapping assemblage of filaments (known as Gibbs hard-core process) 可1 -l P心，>)=1-e0-7月 (1.8) It describes the cumulative probability that the local volume fraction of fibres V exceeds a value V.V denotes a mean volume fraction.In the case of the unidirectional 2D composite with random fibre spacing Vr=h/c, where h and c are a fibre radius and a half-spacing between (centres of) neighbouring fibres,respectively(Fig.1.2).The corresponding probability density function has the following form [34]: o-点小 (1.9) The exact relation for the probability density function for inter-fibre spacing x in the case of random impenetrable fibres of unit diameter is obtained in [35,36] (1.10) 2h Fig.1.2.Longitudinal cross-section of unidirectional fibre-reinforced composite
7 quantify the random distribution of filaments (second phase) in a matrix by means of AIA and Dirichlet cell tessellation procedures was undertaken in [31, 32]. Voronoi tessellation, based on discretisation of a domain into multi-sided convex polygons (known as Voronoi) each containing no more than a single filament, is also used to estimate the character of distribution of distances between filaments [33, 34]. The distribution of cells is supposed to be of the Poisson type with the cumulative probability distribution function accounting for non-overlapping assemblage of filaments (known as Gibbs hard-core process) f f f f f 1 ˆ ( ) 1 exp 1 . 1 V PV V V V ⎡ ⎛ ⎞⎤ > =− − − ⎢ ⎜ ⎟⎥ ⎣ − ⎝ ⎠⎦ (1.8) It describes the cumulative probability that the local volume fraction of fibres f Vˆ exceeds a value Vf; Vf denotes a mean volume fraction. In the case of the unidirectional 2D composite with random fibre spacing Vf = h/c, where h and c are a fibre radius and a half-spacing between (centres of) neighbouring fibres, respectively (Fig. 1.2). The corresponding probability density function has the following form [34]: f f f 2 f f f f 1 1 ( ) exp 1 . 1 1 V V p V VV V V ⎡ ⎛ ⎞⎤ = −− ⎢ ⎜ ⎟⎥ − − ⎣ ⎝ ⎠⎦ (1.9) The exact relation for the probability density function for inter-fibre spacing x in the case of random impenetrable fibres of unit diameter is obtained in [35, 36] f f f f ( ) exp ( 1) . 1 1 V V p x x V V ⎡ ⎤ = −− ⎢ ⎥ − − ⎣ ⎦ (1.10) Fig. 1.2. Longitudinal cross-section of unidirectional fibre-reinforced composite Chapter 1: Account for Random Microstructure
8 V.V.Silberschmidt A study of micrographs of a carbon fibre-reinforced PEEK prepreg, containing about 2,000 fibres with the volume fraction close to 50%,has shown that the distribution of Voronoi distances-distances in an arbitrary direction from the centroid of a fibre to the Voronoi cell boundary-can be assumed as a random one [37].The Voronoi distance is also used as a random variable of the statistical description suggested in [38]. 1.3.1 Parameters of Microstructure Various parameters are introduced to quantify the extent of non-uniformity in distributions of filaments in composites.Several such parameters are suggested in [39].The first one-homogeneity distribution parameter g- characterises the closeness of N particles (e.g.fibres in a transversal cross- section)within the window with area 4 5= (1.11) √A/N This parameter is a ratio of two magnitudes of an inter-particle distance, one,dp,corresponding to the peak of probability density diagram for this parameter and another being an effective average of it.Obviously,for a square latticeg=1;its value diminishes with the increase in clusterisation. Another parameter-an anisotropy parameter of the first kind n-can also be applied to a distribution of cylindrical fibres in a transversal cross- section.It is introduced as [39] 7=N∑cos22, (1.12) i=l where is an orientation angle for the direction from the centre of the window to the centroid of particle i.For a statistically isotropic distribution, this parameter should vanish Several parameters are suggested to characterise the extent of clustering and the properties of clusters (see,e.g.[40]).Still,in traditional carbon fibre-reinforced composites with Vr>0.5,the clusters are less obvious (if at all)than in metal matrix composites(MMCs). As it is shown in [41],real distributions of fibres in unidirectional composites are neither periodic nor fully random,thus presupposing employ- ment of measures that provide additional quantitative characteristics of the exact type of microstructures.So,based on the works of Ripley [42,43],a second-order intensity function K(r)was introduced to describe dis- tributions of points in the following form [41]:
A study of micrographs of a carbon fibre-reinforced PEEK prepreg, containing about 2,000 fibres with the volume fraction close to 50%, has shown that the distribution of Voronoi distances – distances in an arbitrary direction from the centroid of a fibre to the Voronoi cell boundary – can be assumed as a random one [37]. The Voronoi distance is also used as a random variable of the statistical description suggested in [38]. 1.3.1 Parameters of Microstructure Various parameters are introduced to quantify the extent of non-uniformity in distributions of filaments in composites. Several such parameters are suggested in [39]. The first one – homogeneity distribution parameter ξ – characterises the closeness of N particles (e.g. fibres in a transversal crosssection) within the window with area A p . / d A N ξ = (1.11) This parameter is a ratio of two magnitudes of an inter-particle distance, one, dp, corresponding to the peak of probability density diagram for this parameter and another being an effective average of it. Obviously, for a square lattice ξ = 1; its value diminishes with the increase in clusterisation. Another parameter – an anisotropy parameter of the first kind η – can also be applied to a distribution of cylindrical fibres in a transversal crosssection. It is introduced as [39] 1 1 cos 2 , N i N i η θ = = ∑ (1.12) where θi is an orientation angle for the direction from the centre of the window to the centroid of particle i. For a statistically isotropic distribution, this parameter should vanish. Several parameters are suggested to characterise the extent of clustering and the properties of clusters (see, e.g. [40]). Still, in traditional carbon fibre-reinforced composites with Vf ≥ 0.5, the clusters are less obvious (if at all) than in metal matrix composites (MMCs). As it is shown in [41], real distributions of fibres in unidirectional composites are neither periodic nor fully random, thus presupposing employment of measures that provide additional quantitative characteristics of the exact type of microstructures. So, based on the works of Ripley [42, 43], a second-order intensity function K(r) was introduced to describe distributions of points in the following form [41]: 8 V.V. Silberschmidt
Chapter 1:Account for Random Microstructure 9 K()=4( N2台w (1.13) This function characterises the expected number of further points (e.g. centres of fibres)within the distance r from an arbitrary point,normalised by their intensity (i.e.the number of points per unit area).Here,A is an area of the sampling window,containing N points,and (r)is the number of points situated within the distance r from the point k.The weighting factor w is introduced to account for the edge effects;it is equal to the ratio of the circumference of the circle situated within the window.If the entire circle with radius r is situated within the window,w&=1 and it is smaller than unity otherwise.The second-order function was applied to specimens of unidirectional fibre-reinforced composites exposed to different levels of external pressure during curing;also statistics for orientations and distances between fibres were used in terms of cumulative distribution functions.It was shown that these parameters,obtained with the use of image analysis from micrographs of real specimens,significantly differ from those of artificial microstructures with the same number of fibres,obtained by the Poisson process [41].Unfortunately,second-order functions are not able to determine sub-patterns in distributions,so either parameters of a higher order or combinations of second-order functions with some other parameters should be used [44]. The second-order intensity function K(r)can also be used to derive another quantitative parameter,characterising randomness in distribution of fibres (their centroids).It can be introduced in the following way [41, 44,45].The average number of fibre centroids located within a circular ring of radius r and thickness dr with a centre at a given fibre centroid is dK(r)=K(r+dr)-K(r). (1.14) Dividing (1.14)by the area of the ring 2dr,one can obtain the local spatial density of fibres.The ratio of the latter and the average spatial density NIA forms the radial distribution function [41,45] g)=、AdK) (1.15) 2πrNdr Obviously,for a random Poisson process g(r)=1.The value ro,for which g(ro)=1,is a characteristic scale of the local disorder in an ensemble. In parallel with statistical characterisation of distributions of micro- scopic features (e.g.filaments in a matrix)in composites,various topological characteristics are introduced.An obvious development in this direction is application of fractals [39,46,47]
9 2 1 ( ) ( ) . N k k k A I r K r N w = = ∑ (1.13) This function characterises the expected number of further points (e.g. centres of fibres) within the distance r from an arbitrary point, normalised by their intensity (i.e. the number of points per unit area). Here, A is an area of the sampling window, containing N points, and Ik(r) is the number of points situated within the distance r from the point k. The weighting factor wk is introduced to account for the edge effects; it is equal to the ratio of the circumference of the circle situated within the window. If the entire circle with radius r is situated within the window, wk = 1 and it is smaller than unity otherwise. The second-order function was applied to specimens of unidirectional fibre-reinforced composites exposed to different levels of external pressure during curing; also statistics for orientations and distances between fibres were used in terms of cumulative distribution functions. It was shown that these parameters, obtained with the use of image analysis from micrographs of real specimens, significantly differ from those of artificial microstructures with the same number of fibres, obtained by the Poisson process [41]. Unfortunately, second-order functions are not able to determine sub-patterns in distributions, so either parameters of a higher order or combinations of second-order functions with some other parameters should be used [44]. The second-order intensity function K(r) can also be used to derive another quantitative parameter, characterising randomness in distribution of fibres (their centroids). It can be introduced in the following way [41, 44, 45]. The average number of fibre centroids located within a circular ring of radius r and thickness dr with a centre at a given fibre centroid is dK r K r dr K r ( ) ( ) ( ). = + − (1.14) Dividing (1.14) by the area of the ring 2πrdr, one can obtain the local spatial density of fibres. The ratio of the latter and the average spatial density N/A forms the radial distribution function [41, 45] ( ) ( ) . 2 A dK r g r π rN dr = (1.15) Obviously, for a random Poisson process g(r) = 1. The value r0, for which g(r0) = 1, is a characteristic scale of the local disorder in an ensemble. In parallel with statistical characterisation of distributions of microscopic features (e.g. filaments in a matrix) in composites, various topological characteristics are introduced. An obvious development in this direction is application of fractals [39, 46, 47]. Chapter 1: Account for Random Microstructure
10 V.V.Silberschmidt A multifractal formalism can provide useful information on the type of the random distribution of fibres in the matrix [48].It characterises the spatial scaling of non-uniform distributions:A local probability (number of fibres)P;in the ith box(element)from a set of boxes,compactly covering the area of interest,scales with the box size as P(0c1, (1.16) where the scaling exponent a;is known as singularity strength.According to the multifractal theory [49,50],the number of elements with probability characterised by the same singularity strength is linked to the box size by the fractal(Hausdorff)dimension fa) N(a)al-r(@) (1.17) The function fa),known as multifractal spectrum,describes the con- tinuous (but finite)spectrum of scaling exponents for a random distribution. As it was shown in [48],the distribution of carbon fibres in epoxy matrix is multifractal;the respective multifractal spectrum was calculated. 1.3.2 Local Volume Fraction Analysing the effects of microstructural randomness,an obvious idea is to consider the volume fraction of reinforcement not only in terms of a global description,i.e.as a parameter characterising the entire composite,but also as a field function,introducing the idea of a local volume fraction.A direct comparison of various parts of the composite (Fig.1.3)vividly demonstrates that the volume fraction of fibres depends on a location in a composite.In Torquato [17],it is introduced as an average over a volume element (observation window)Po of the composite with the centroid at x 4倒-s0-2 (1.18) where /(x)is the characteristic function (see (1.3)),z characterises any point in Vo and e(x-z)is the indicator function z-x∈V0: (1.19) 10, otherwise
A multifractal formalism can provide useful information on the type of the random distribution of fibres in the matrix [48]. It characterises the spatial scaling of non-uniform distributions: A local probability (number of fibres) Pi in the ith box (element) from a set of boxes, compactly covering the area of interest, scales with the box size l as () ,i Pl l i α ∝ (1.16) where the scaling exponent αi is known as singularity strength. According to the multifractal theory [49, 50], the number of elements with probability characterised by the same singularity strength is linked to the box size by the fractal (Hausdorff) dimension f(α) ( ) () . f N l α α − ∝ (1.17) The function f(α), known as multifractal spectrum, describes the continuous (but finite) spectrum of scaling exponents for a random distribution. As it was shown in [48], the distribution of carbon fibres in epoxy matrix is multifractal; the respective multifractal spectrum was calculated. 1.3.2 Local Volume Fraction Analysing the effects of microstructural randomness, an obvious idea is to consider the volume fraction of reinforcement not only in terms of a global description, i.e. as a parameter characterising the entire composite, but also as a field function, introducing the idea of a local volume fraction. A direct comparison of various parts of the composite (Fig. 1.3) vividly demonstrates that the volume fraction of fibres depends on a location in a composite. In Torquato [17], it is introduced as an average over a volume element (observation window) V0 of the composite with the centroid at x 0 f 0 1 ( ) ( ) ( )d , V V I V = − θ ∫ x x xzz (1.18) where I(x) is the characteristic function (see (1.3)), z characterises any point in V0 and θ (x − z) is the indicator function 1, , 0 ( ) 0, otherwise. V θ − ∈ − = ⎧ ⎨ ⎩ z x x z (1.19) 10 V.V. Silberschmidt