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