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Mathematical morphology applied to fibre composite materials
MATHEMATICAL MORPHOLOGY APPLIED TO FIBRE COMPOSITE MATERIALS J. SERRA
Centre de Morphologie Math$matique de l'Ecole des Mines de Paris, 77305, Fontainebleau (France) and G. VERCHERY
Groupe Commun Ecole des M i n e s - - E N S T A , 75015, Paris (France) (Received: 21 N o v e m b e r , 1972)
SUMMARY
This paper is an introduction to a study of the application of the science of mathematical morphology and the texture analyser to fibre composite materials. The full study will include theoretical applications and experimental verification. In the first part the writers describe how the tensile and flexural rigidities of a composite strip can be measured, in conjunction with the morphological parameters of volume fraction of fibre or matrix. The second part presents the measurements of the number of connectivity of the closings of the fibres. This parameter is related to inter-fibre distances and, in particular, gives the minimum inter-fibre distance. The apparatus is briefly described and its original features pointed out. In addition, the basic notions of the hit-and-miss transformation are given. INTRODUCTION
The physical, and particularly mechanical, properties of heterogeneous materials (such as polycrystals, rocks and composite materials) are dependent on the distribution of the constitutive components. This distribution has both geometrical and physical characteristics. Geometrical characteristics: the shapes and position of the various particles. Physical characteristics: the nature of the components, their mechanical properties, crystal orientations, etc. These characteristics may be well-defined, as in the case of laminates, due to the ordered stacking of the material. However, generally speaking, the distribution is random, as a consequence of imperfect manufacture of a composite or by the very nature of the material, as in different polycrystals of the same element. Nevertheless, it is possible to define determinate parameters, for example the volume fraction of the various phases. 141
Fibre Science and Technology (6) (1973)--C Applied Science Publishers Ltd, England, 1973--Printed irt Great Britain
142
J. SERRA, G. VERCHERY
Two difficulties arise in determining the physical properties when theories are developed sufficiently to formulate the equation governing these properties. One of these problems is, with a known distribution, to actually solve the equations, which often have complicated boundary and interface conditions, and the other is in assessing the randomness of distribution. The principal methods of calculation for heterogeneous solids take into account, to a greater or lesser degree, both the above points. They are as follows: (i) Variational methods give boundsforthe elastic rigidities ofathree-dimensional body. ~' 2 (ii) By solving exact field equations (analytically or numerically) one derives values for the elastic constants of unidirectional composite materials with a perfect fibre distribution (hexagonal or square array). Off) One-fibre models of composite materials, followed by a mean on the whole body, give a mechanical behaviour. 3.4 To take account of the random distributions, it becomes necessary to measure the parameters of these distributions for the type of materials studied, for instance composite materials made by a given technology, or polycrystals solidified under given conditions. The above remarks point out the usefulness of experimental studies of the distributions: ~--to measure the parameters which are seen to be important when actual solution of the equations is possible. fl--to find the correlations between distribution parameters and the physical properties either when actual solution is not possible or the equations are unknown or give a faulty description of the behaviour. This paper illustrates such experimental studies with two cases taken for a composite material reinforced with unidirectional boron fibres. The first case (see above) uses simple concepts to calculate tensile and flexural rigidities of a composite rod from morphological measurements. The second attempts to define and quantify a ratio of regularity for the stacking of fibres in a transverse section. Such a ratio might be correlated to tensile or flexural strength, resilience, etc.
MATHEMATICAL MORPHOLOGY AND THE TEXTURE ANALYSER
An extensive description of mathematical morphology and the texture analyser can be found in references 5, 6 and 7. However, we summarise here the basic principles of this apparatus and the theory involved. The texture analyser is a new type of computer for geometrical data. The design and fabrication of its prototype were carried out by Serra et aL s, 9 Its development
MATHEMATICAL MORPHOLOGY APPLIED TO FIBRE COMPOSITE MATERIALS
143
and sale are now ensured by Leitz. Figure 1 shows the Leitz apparatus used in this study: it consists of a reflected light microscope, a TV camera and logical units for processing the camera signal. Several image analysers, based on this design, exist and possess similar optical and electronic (TV) characteristics. However, the Leitz analyser is quite different because of its logical units and belongs to a new generation of machines.
Fig. 1.
T h e Leitz texture analyser.
Formerly, the automatic devices were restricted to the automation of parameter measurements previously handmade: - - t h e surface ratio of phases - - t h e number of inclusions - - t h e statistical distribution of the length of the chords shut out by a given phase. These classical parameters may sometimes be shown to be inadequate, as in our two studies. Moreover, the two-dimensional images obtained by scanning (TV camera, microscope) need new concepts of analysis because of their twodimensional character. First it is necessary to take into account some theoretical requirements. For instance, parameters must be amenable to sampling analysis. A counter example is the value of the greatest chord of a three-dimensional element: it cannot be determined for measurements taken on a plane section, hence it is not a useful parameter.
144
J. SERRA~ G. VERCHERY
The theory of 'hit-or-miss transformation' by structuring elementsS.6-7 enables us to express clearly all the theoretical requirements and the morphological concepts needed to produce more useful parameters from experimentation. The basic idea is as follows. The specimen to be analysed is divided into two sets of points by a discriminator, namely F and M, where M is the complement of F. Let kr(x) be the indicator of the set F, i.e. : Ikr(x) = 1 if x EF
l kr(x) = 0
if
x EM
(1)
The structuring element, H, is a given geometrical shape whose implantation will vary in space by translation. Let us fix Hx at some point, x, and examine two possible relationships between H x and F. The first defines erosion and occurs when Hx is included in F(H~ c F), thus giving the set of the centres of the structuring elements included in the set, F. The set is called the 'eroded of F according to H ' and is denoted by F G / t . The second defines dilatation and occurs when H~ hits F(Hx c~ F # q~) thus giving the set of the centres of the structuring elements which hit F. This is called the 'dilated of F according to H ' and is denoted by F ~9/~/. These two transformations are called 'hit-or-miss transformations' (HMT). It can be shown that all the classical measurements appear as special cases of H M T (the volume ratio is given by an H reduced to a point, the chord distribution by choosing a segment for H and the number of particles by choosing a quadruplet (~ ~) for H). Two important formulae associated with H M T are:
F(~ [l~" l U M21 = [ F ~ ~/11 N IF e ~r21
(2)
Their importance comes from the fact that they permit a series of elementary geometric operations to be processed by our texture analyser. The first (eqn. (2)) is the law of shunt connections for logics. To erode F by M 1 u M2, it is sufficient to erode, at the same time, F by M1 and M2 in two elementary moduli and to combine the results. Hence, from a set of basic structuring elements it will be possible to constitute, as a mosaic, a considerable number of new structuring elements of various shapes and sizes. The second formula (eqn. (3)) is the law of iterative logics, or series-connected logics. It can be shown that this law gives easy access to the second dimension by combining one-dimensional logics, working in various directions (hexagonal erosions).
MATHEMATICAL MORPHOLOGY APPLIED TO FIBRE COMPOSITE MATERIALS I
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Fig. 2.
Fig. 3.
A sandwich composite.
Scanning electron micrograph of the specimen.
146
J. SERRA, G. VERCHERY THE SPECIMENS AND THEIR PREPARATION
Four types of composite material strips with rectangular cross-section (about 1 mm x 10 mm) were used. All were reinforced with unidirectional boron fibres (100 pm diameter) and had the following compositions: (i) Two aluminium-matrix composites of 10 and 12 layers with quasi-hexagonal stacking. (ii) A 12-layer epoxy-matrix composite. (iii) A sandwich epoxy-matrix composite with facings of three and four layers and an epoxy core (Fig. 2).
Fig. 4.
Light micrograph of fibres.
For the aluminium matrix we found that the best results were obtained on the texture analyser with a somewhat unusual preparation of the sections: after a first polish, the matrix is etched with hydrofluoric acid, then repolished. Figures 3 and 4 are micrographs of the surface after preparation (scanning electron and photomicrographs). The epoxy-matrix composites gave a good contrast (light fibres and dark matrix) after a single polishing.
M A T H E M A T I C A L M O R P H O L O G Y A P P L I E D T O FIBRE C O M P O S I T E MATERIALS
147
TENSILE A N D F L E X U R A L STIFFNESSES
Calculation of stiffnesses We need to make a number of assumptions before applying the theory, namely that the components of the body are isotropic, linearly elastic, with different Young's moduli (E = EM for the matrix and E = Ep for the fibres) but with the same Poisson's ratio. In addition, we assume that plane sections remain plane when the composite bar is subjected to a longitudinal force, N, or a couple, C x (Fig. 5).
z
Fig. 5. Schematic view of the composite strip. In any cross-section, S, the matrix has a volume ratio, VM, and the fibres have the ratio Vr. C o m p o u n d bars with such assumptions were first studied by Muskhelishvili, l o who introduced the concepts of reduced moments of inertia and principle axes. In our case, the width, L, of the cross-section includes a great number of fibres, hence we may assume that any irregularities of pitch along Ox are unimportant and that, in consequence, the principal axis of inertia about Oy is also the reduced axis and is at x = L/2. However, the reduced principal axis of inertia about Ox, A, is not at the mid-point of the depth, h, due to irregularities in the stacking, or, in some sandwich bars, to an unbalanced sequence. It lies at the ordinate 6 = (h/2) + fl (Fig. 6) given by:
f E(y s
- 6) d x d y = 0
(4)
148
J. SERRA, G. VERCHERY
This fact is significant of the coupling effect between tensile and bending properties. The tensile stiffness of the bar is: A = fsEdxdy
(5)
whereas the reduced bending stiffness (the 'reduced flexural rigidity', IE, of reference I0) is:
D* = Is E(y - 6) 1 dx dy
(6)
The 'coupling stiffness' is defined as:
and is consequently related to fl: B = fiA
(8)
Definitions of mean moduli In order to compare the composite with homogeneous materials we shall introduce quantities having the dimensions of a pressure, the mean Young's
,I_
L
I Z
o
Fig. 6.
,It
The composite strip section.
moduli. By comparison with the tensile rigidity of an homogeneous body of the same cross-section, S, we can define a mean Young's modulus in tension, equal to: E =
A J'E dx dy - - S S dx dy
(9)
MATHEMATICAL MORPHOLOGY APPLIED TO FIBRE COMPOSITE MATERIALS
149
which quantity proves to be equal to the modulus given by the usual rule of mixtures: ff~ = EMVM + E v V v (10) The bending stiffness case is more complex. Comparison can be made with an homogeneous bar of the same cross-section, S, of principal bending rigidity, I: i ~ - - D* _
I
I E ( y - 6) 2 d x d y I[Y - (h/2)] 2 dx dy
(11)
or to an homogeneous bar having the same neutral axis, A, and the same surface strains as the compound bar: D*
/~
=
1"-%- =
I E ( y - 6) z dx dy I(Y
-- fi)2
(12)
dx dy
In general, /~ _> E and the equality holds for a balanced composite. A pure bending test (Fig. 7) gives D*, the ratio of the couple to the curvature. If such a test is interpreted with the usual strength of material expressions, it gives /~. If
/
//////
Fig. 7. Pure bending test. strains are measured on the outer and inner surfaces of the bent bar, it is possible to determine the neutral axis and to calculate J~. In this paper we use only the latter definition of J~ for the mean Young's modulus in bending. M e a s u r e m e n t and computation
The field observed is a fraction, l, of the width, L, over the full depth of the bar. The texture analyser uses a mask which selects for scanning a line, i, parallel to Ox. Two consecutive lines, i and i + 1, are separated by a distance, t. An automatic device ensures the translation of the microscope stage along Oy. At each step, the analyser can measure the length percentage, v(i), of the fibres found in the line and compute the summations, Y~v(i), E (i - 1)v(i) and E (i - 1)2v(i), which are used in the approximate expressions of the integrals defining stiffnesses and mean moduli. One line is scanned in 0.04 sec and measurement and computation are completed within a few minutes.
150
J. S E R R A , G . V E R C H E R Y
Discussion of the method The precision of the results must be considered from both the technical and statistical points of view. (a) Discussion of the measurement technology: The first problem lies in the choice of the light threshold so that the real geometry of fibres is not distorted. A curve of the percentage of the apparent fibre area versus threshold is plotted. The theoretical curve has a point of inflexion at the optimal threshold. On the experimental curve (Fig. 8) we first exclude, by visual observation, the threshold positions
80 70
% length
/J.............J
6O 5O #0. %0,
1
I
I
I
i
~o
20
0o
hO
~0
!
60
IL
]'h reshoPd
Fig. 8. Light function of the threshold. where the distortion is obvious. We then take as a threshold the middle of the remaining interval. The precision improves as the slope of the curve diminishes and the error caused by one interval of light threshold is very small, of the order of 10 -3 Another cause of error results from the fluctuations due to the resolution of the camera and the signal sampling necessary for logical treatment. The corresponding error has been estimated by repeating the measurement of the same field and the variance of the results is less than 0.5 per cent. (b) Statistics: A primary source of error is the variation (up to 10 per cent) in the mean Young's moduli measured in each field across the composite bar. To limit the variation, a mean value, based on four to six readings, was retained. Error also depends on the size of the inter-line distance, t. A fixed value of 19.6 #m provided measurement with almost no variance.
151
MATHEMATICAL MORPHOLOGY APPLIED TO FIBRE COMPOSITE MATERIALS
Presentation of the results (Table 1) The following values were taken for the Young's moduli of the components: Boron 450 x Aluminium 75 x Epoxy 5 × (1 Pa = 10 -7 h bar
109 Pa 109 Pa 109 Pa ~ 1.45 x 10-4psi) TABLE 1
Specimens
n
Vv
1 2 3 4
12 10 12 7
0.502 0.524 0.448 0.164
D*/Lm.N 43"6 24"8 71"5 148
~S 109 Pa 263 271 204 77"9
/~ I09 P a 222 215 180 151
R1
R2
0.84 0-79 0"88 1.94
0"89 0'87 0.85 0.77
n: number of layers. R 1: ratio/~'//~ as measured in the real case R2: ratio/~//~ as predicted for a regular hexagonal array with the volume ratio VF. 11
ANALYSIS OF THE FIBRE CLUSTERS
Purpose and method This second part of our study tries to give a new tool for a better understanding of the fracture problem of ductile matrix composites. During a tensile test, when a fibre has broken, the load which was carried by that fibre is mainly transferred to the nearest fibres. In a regular hexagonal array, the six neighbouring fibres carry an equal excess loading, but, for a real (irregular) array, it is thought that the nearest single fibre carries a great deal of the load. Therefore the probability of a failure process by a chain reaction is greatly increased by the irregularities and the fracture properties (ultimate tensile strength, fatigue crack propagation, resilience, etc . . . . ) are related to the 'regularity ratio' (if defined) of the cross-section pattern of the fibres, for a given volume ratio, VF. This 'regularity ratio' is experimentally reached by means of the following geometrical transforms. First, a dilatation changes the 0t-state into the fl-state (Fig. 9). The structuring element, H, is an hexagon of side a, which may be taken as a circle in the analysis. The radius of the fibre being b, two fibres with the distance between centres 2d, overlap after the dilatation if the inequality holds: a _> d - b
(13)
Hence we obtain a parameter fulfilling our expectations in the number of connectivity of the dilated set F ~ / - ) , measured as a function of the radius, a, of the structuring element, H.
152
J. SERRA, G. VERCHERY
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© ©
©
U ~
B ~)H Fig. 9.
(B~) H) O H
Dilatation a n d closing of the fibres.
We may go further in the analysis by means of an hexagonal closing. 5,7 The dilated set F @/~ is then eroded by the structuring element, H, and gives the set (F t~ H) @ H (Fig. 9(T)). After this transform, the fibres which overlapped by dilatation may, or may not, be linked. More exactly, the link between two fibres remains if it is possible to place element H in the thinnest part of their overlapping area, i.e., if the inequality holds: d2 _ b2
a _> . - 2b
Fig. 10(a)
(14)
C~
0
0
©
©
II
It
~°
v
~o
~°
il
°o
~D
MATHEMATICALMORPHOLOGYAPPLIED TO FIBRE COMPOSITEMATERIALS
155
The main advantage of the closing compared with the dilatation is a more striking view of the results. Figure 10 (a to e) shows the development of the closing process, when side a of the structuring hexagon increases clusters of fibres appear more and more linked.
Experimental results We have measured the hexagonal closings of the fibres in five areas which approximately represent a whole section of the bar. The measurement interval is the side of the hexagon which was used for closing, equal to 3.4/~m. We shall not discuss here in detail the relationship between the measurement area chosen and the objects to be measured within that area: obviously when two fibres are on
75 ~ N(o)
\
50
x
g,4 /~
-2ft.
Fig. 11. Real array. Number of connectivity, N, of the closings.
~q
156
J. SERRA, G. VERCHERY
both sides of the area boundary they are never linked, whatever the radius of the structuring element. However, these measurement errors, sometimes large, can be completely corrected by the analyser 9 which is at present the only apparatus capable of such a correction. The results shown in Fig. 11 represent the number of connectivity after the closing, given by the number of fibre chains minus the number of enclaves in the chains. The slight increase at the beginning of the curve results from small internal defects of the fibres and must not be taken into account. LS(o)
-g-- d-b dLbt
"TE"
-2N
Fig. 12.
Hexagonal array. Number of cormectivity, N, of the closings.
After the second closing (6.8 pm) these defects no longer have an effect and fibre contacts begin to appear. Every new contact reduces by one the number of connectivity. The meaning of the curve is shown by comparison with the case of a regular hexagonal array (Fig. 12), where 2d is the distance between centres, b is the fibre radius, and N is the number of fibres in the area. In this case all the contacts occur at the same time, when a reaches the value ( d 2 - b2)/2b and forms links containing
MATHEMATICALMORPHOLOGYAPPLIED TO FIBRE COMPOSITEMATERIALS
157
2N pores, which disappear simultaneously when the dilatation of the fibres covers all the field, i.e., when a reaches the value 2d/(3)½ - b. In a real random array the irregularity of stacking causes a less abrupt decrease and negative values are smaller, which is explained by the small number of links formed. The slope of the curve can be interpreted as the opposite of the number of new neighbours formed for a considered distance, a, a useful parameter to correlate with fracture properties. A second useful parameter is amin = ( d 2 m i n - b2)/2b, when 2dmin is the smallest distance between centres of any two fibres. The point ami, on the curve (Fig. 11) is taken as the intercept of the maximum value of the number of connectivity 5 with the mean negative slope.
CONCLUSIONS The texture analyser has shown itself to be useful for rapid morphological measurements on composite fibre materials, the specimens requiring a minimum amount of surface preparation. The calculation of flexural and tensile stiffnesses is practically impossible by any other means yet the analyser gives a precise result since it takes into account both the position and the size of each element. The same can be said when we need to measure the volume fraction of fibres in a composite. The analyser permits a precise value of the local and global volume fractions. The preliminary results presented in Table 1 agree well with known unidirectional fibre composite behaviour. The plot of 'number of connectivity' v. the structuring hexagon radius, gives a qualitative assessment of the regularity of the fibre distribution, a regular distribution has a large negative slope and an irregular distribution a smaller one. In addition, the curve provides the value of the smallest inter-fibre distance. This parameter is thought to be of importance in explaining real composite fracture properties, however extensive experimental work is required to verify this.
REFERENCES 1. W. VOIGT, Lehrbuch der Kristallphysik. Teubner, Berlin, 1910. 2. A. REUSS, Bereclmung der Fliessgreuze yon Mischkristallen auf Grund der Plastizit~itsbedingung fflr Einkristalle, Z A M M , 9 (1929) p. 49 3. A. KELLYand G. J. DAVmS,The principles of the fibre reinforcement of metals. Metallurgical Review, 10 (1965) pp. 1-77. 4. R. HILL,Theory of mechanical properties of fibre-strengthened materials: III, Self-consistent model, J. Mech. Phys. Solids, 13 (1965) p. 189. 5. J. SERRA,Introduction ~t la morphologie math6matique, Fasc. N ° 3 des Cahiers du C M M , Ecole des Mines de Paris, 1969. 6. G. MATrmRON,Elements pour une th~orie des Milieux Poreux, Masson, Paris, 1967.
158
J. SERRA, G. VERCHERY
J. SERRA, Stereology and structuring elements, J. of Microscopy, 95, Pt. 1 (1972) pp. 93-103. J. SERRA, IRSID--Brevet de l'Analyseur de Textures No. 1449.059. J. C. KLEIN and J. SERRA,The texture analyser, J. of Microscopy, 95, Pt. 2 (1972) pp. 349-56. N.I. MUSKHELISHVILI,Some basic problems of the mathematical theory of elasticity. Chapter 24, translated from the fourth Russian edition by J. R. M. Radok, Noordhoff, 1963. 11. J. HUNEAU, Tension et flexion des composites; Modules moyens. Rapport interne du Centre des Mat6riaux, Ecole des Mines de Paris, 1971.
7. 8. 9. 10.
Centre de Morphologie Math$matique de l'Ecole des Mines de Paris, 77305, Fontainebleau (France) and G. VERCHERY
Groupe Commun Ecole des M i n e s - - E N S T A , 75015, Paris (France) (Received: 21 N o v e m b e r , 1972)
SUMMARY
This paper is an introduction to a study of the application of the science of mathematical morphology and the texture analyser to fibre composite materials. The full study will include theoretical applications and experimental verification. In the first part the writers describe how the tensile and flexural rigidities of a composite strip can be measured, in conjunction with the morphological parameters of volume fraction of fibre or matrix. The second part presents the measurements of the number of connectivity of the closings of the fibres. This parameter is related to inter-fibre distances and, in particular, gives the minimum inter-fibre distance. The apparatus is briefly described and its original features pointed out. In addition, the basic notions of the hit-and-miss transformation are given. INTRODUCTION
The physical, and particularly mechanical, properties of heterogeneous materials (such as polycrystals, rocks and composite materials) are dependent on the distribution of the constitutive components. This distribution has both geometrical and physical characteristics. Geometrical characteristics: the shapes and position of the various particles. Physical characteristics: the nature of the components, their mechanical properties, crystal orientations, etc. These characteristics may be well-defined, as in the case of laminates, due to the ordered stacking of the material. However, generally speaking, the distribution is random, as a consequence of imperfect manufacture of a composite or by the very nature of the material, as in different polycrystals of the same element. Nevertheless, it is possible to define determinate parameters, for example the volume fraction of the various phases. 141
Fibre Science and Technology (6) (1973)--C Applied Science Publishers Ltd, England, 1973--Printed irt Great Britain
142
J. SERRA, G. VERCHERY
Two difficulties arise in determining the physical properties when theories are developed sufficiently to formulate the equation governing these properties. One of these problems is, with a known distribution, to actually solve the equations, which often have complicated boundary and interface conditions, and the other is in assessing the randomness of distribution. The principal methods of calculation for heterogeneous solids take into account, to a greater or lesser degree, both the above points. They are as follows: (i) Variational methods give boundsforthe elastic rigidities ofathree-dimensional body. ~' 2 (ii) By solving exact field equations (analytically or numerically) one derives values for the elastic constants of unidirectional composite materials with a perfect fibre distribution (hexagonal or square array). Off) One-fibre models of composite materials, followed by a mean on the whole body, give a mechanical behaviour. 3.4 To take account of the random distributions, it becomes necessary to measure the parameters of these distributions for the type of materials studied, for instance composite materials made by a given technology, or polycrystals solidified under given conditions. The above remarks point out the usefulness of experimental studies of the distributions: ~--to measure the parameters which are seen to be important when actual solution of the equations is possible. fl--to find the correlations between distribution parameters and the physical properties either when actual solution is not possible or the equations are unknown or give a faulty description of the behaviour. This paper illustrates such experimental studies with two cases taken for a composite material reinforced with unidirectional boron fibres. The first case (see above) uses simple concepts to calculate tensile and flexural rigidities of a composite rod from morphological measurements. The second attempts to define and quantify a ratio of regularity for the stacking of fibres in a transverse section. Such a ratio might be correlated to tensile or flexural strength, resilience, etc.
MATHEMATICAL MORPHOLOGY AND THE TEXTURE ANALYSER
An extensive description of mathematical morphology and the texture analyser can be found in references 5, 6 and 7. However, we summarise here the basic principles of this apparatus and the theory involved. The texture analyser is a new type of computer for geometrical data. The design and fabrication of its prototype were carried out by Serra et aL s, 9 Its development
MATHEMATICAL MORPHOLOGY APPLIED TO FIBRE COMPOSITE MATERIALS
143
and sale are now ensured by Leitz. Figure 1 shows the Leitz apparatus used in this study: it consists of a reflected light microscope, a TV camera and logical units for processing the camera signal. Several image analysers, based on this design, exist and possess similar optical and electronic (TV) characteristics. However, the Leitz analyser is quite different because of its logical units and belongs to a new generation of machines.
Fig. 1.
T h e Leitz texture analyser.
Formerly, the automatic devices were restricted to the automation of parameter measurements previously handmade: - - t h e surface ratio of phases - - t h e number of inclusions - - t h e statistical distribution of the length of the chords shut out by a given phase. These classical parameters may sometimes be shown to be inadequate, as in our two studies. Moreover, the two-dimensional images obtained by scanning (TV camera, microscope) need new concepts of analysis because of their twodimensional character. First it is necessary to take into account some theoretical requirements. For instance, parameters must be amenable to sampling analysis. A counter example is the value of the greatest chord of a three-dimensional element: it cannot be determined for measurements taken on a plane section, hence it is not a useful parameter.
144
J. SERRA~ G. VERCHERY
The theory of 'hit-or-miss transformation' by structuring elementsS.6-7 enables us to express clearly all the theoretical requirements and the morphological concepts needed to produce more useful parameters from experimentation. The basic idea is as follows. The specimen to be analysed is divided into two sets of points by a discriminator, namely F and M, where M is the complement of F. Let kr(x) be the indicator of the set F, i.e. : Ikr(x) = 1 if x EF
l kr(x) = 0
if
x EM
(1)
The structuring element, H, is a given geometrical shape whose implantation will vary in space by translation. Let us fix Hx at some point, x, and examine two possible relationships between H x and F. The first defines erosion and occurs when Hx is included in F(H~ c F), thus giving the set of the centres of the structuring elements included in the set, F. The set is called the 'eroded of F according to H ' and is denoted by F G / t . The second defines dilatation and occurs when H~ hits F(Hx c~ F # q~) thus giving the set of the centres of the structuring elements which hit F. This is called the 'dilated of F according to H ' and is denoted by F ~9/~/. These two transformations are called 'hit-or-miss transformations' (HMT). It can be shown that all the classical measurements appear as special cases of H M T (the volume ratio is given by an H reduced to a point, the chord distribution by choosing a segment for H and the number of particles by choosing a quadruplet (~ ~) for H). Two important formulae associated with H M T are:
F(~ [l~" l U M21 = [ F ~ ~/11 N IF e ~r21
(2)
Their importance comes from the fact that they permit a series of elementary geometric operations to be processed by our texture analyser. The first (eqn. (2)) is the law of shunt connections for logics. To erode F by M 1 u M2, it is sufficient to erode, at the same time, F by M1 and M2 in two elementary moduli and to combine the results. Hence, from a set of basic structuring elements it will be possible to constitute, as a mosaic, a considerable number of new structuring elements of various shapes and sizes. The second formula (eqn. (3)) is the law of iterative logics, or series-connected logics. It can be shown that this law gives easy access to the second dimension by combining one-dimensional logics, working in various directions (hexagonal erosions).
MATHEMATICAL MORPHOLOGY APPLIED TO FIBRE COMPOSITE MATERIALS I
IIIli
III
145
III
QQOOOOOQO
%%0000000o0600
0 0 0 0 0 0 0 0
00000000
oOoOoOoOoOoOoOoOo I
I
Fig. 2.
Fig. 3.
A sandwich composite.
Scanning electron micrograph of the specimen.
146
J. SERRA, G. VERCHERY THE SPECIMENS AND THEIR PREPARATION
Four types of composite material strips with rectangular cross-section (about 1 mm x 10 mm) were used. All were reinforced with unidirectional boron fibres (100 pm diameter) and had the following compositions: (i) Two aluminium-matrix composites of 10 and 12 layers with quasi-hexagonal stacking. (ii) A 12-layer epoxy-matrix composite. (iii) A sandwich epoxy-matrix composite with facings of three and four layers and an epoxy core (Fig. 2).
Fig. 4.
Light micrograph of fibres.
For the aluminium matrix we found that the best results were obtained on the texture analyser with a somewhat unusual preparation of the sections: after a first polish, the matrix is etched with hydrofluoric acid, then repolished. Figures 3 and 4 are micrographs of the surface after preparation (scanning electron and photomicrographs). The epoxy-matrix composites gave a good contrast (light fibres and dark matrix) after a single polishing.
M A T H E M A T I C A L M O R P H O L O G Y A P P L I E D T O FIBRE C O M P O S I T E MATERIALS
147
TENSILE A N D F L E X U R A L STIFFNESSES
Calculation of stiffnesses We need to make a number of assumptions before applying the theory, namely that the components of the body are isotropic, linearly elastic, with different Young's moduli (E = EM for the matrix and E = Ep for the fibres) but with the same Poisson's ratio. In addition, we assume that plane sections remain plane when the composite bar is subjected to a longitudinal force, N, or a couple, C x (Fig. 5).
z
Fig. 5. Schematic view of the composite strip. In any cross-section, S, the matrix has a volume ratio, VM, and the fibres have the ratio Vr. C o m p o u n d bars with such assumptions were first studied by Muskhelishvili, l o who introduced the concepts of reduced moments of inertia and principle axes. In our case, the width, L, of the cross-section includes a great number of fibres, hence we may assume that any irregularities of pitch along Ox are unimportant and that, in consequence, the principal axis of inertia about Oy is also the reduced axis and is at x = L/2. However, the reduced principal axis of inertia about Ox, A, is not at the mid-point of the depth, h, due to irregularities in the stacking, or, in some sandwich bars, to an unbalanced sequence. It lies at the ordinate 6 = (h/2) + fl (Fig. 6) given by:
f E(y s
- 6) d x d y = 0
(4)
148
J. SERRA, G. VERCHERY
This fact is significant of the coupling effect between tensile and bending properties. The tensile stiffness of the bar is: A = fsEdxdy
(5)
whereas the reduced bending stiffness (the 'reduced flexural rigidity', IE, of reference I0) is:
D* = Is E(y - 6) 1 dx dy
(6)
The 'coupling stiffness' is defined as:
and is consequently related to fl: B = fiA
(8)
Definitions of mean moduli In order to compare the composite with homogeneous materials we shall introduce quantities having the dimensions of a pressure, the mean Young's
,I_
L
I Z
o
Fig. 6.
,It
The composite strip section.
moduli. By comparison with the tensile rigidity of an homogeneous body of the same cross-section, S, we can define a mean Young's modulus in tension, equal to: E =
A J'E dx dy - - S S dx dy
(9)
MATHEMATICAL MORPHOLOGY APPLIED TO FIBRE COMPOSITE MATERIALS
149
which quantity proves to be equal to the modulus given by the usual rule of mixtures: ff~ = EMVM + E v V v (10) The bending stiffness case is more complex. Comparison can be made with an homogeneous bar of the same cross-section, S, of principal bending rigidity, I: i ~ - - D* _
I
I E ( y - 6) 2 d x d y I[Y - (h/2)] 2 dx dy
(11)
or to an homogeneous bar having the same neutral axis, A, and the same surface strains as the compound bar: D*
/~
=
1"-%- =
I E ( y - 6) z dx dy I(Y
-- fi)2
(12)
dx dy
In general, /~ _> E and the equality holds for a balanced composite. A pure bending test (Fig. 7) gives D*, the ratio of the couple to the curvature. If such a test is interpreted with the usual strength of material expressions, it gives /~. If
/
//////
Fig. 7. Pure bending test. strains are measured on the outer and inner surfaces of the bent bar, it is possible to determine the neutral axis and to calculate J~. In this paper we use only the latter definition of J~ for the mean Young's modulus in bending. M e a s u r e m e n t and computation
The field observed is a fraction, l, of the width, L, over the full depth of the bar. The texture analyser uses a mask which selects for scanning a line, i, parallel to Ox. Two consecutive lines, i and i + 1, are separated by a distance, t. An automatic device ensures the translation of the microscope stage along Oy. At each step, the analyser can measure the length percentage, v(i), of the fibres found in the line and compute the summations, Y~v(i), E (i - 1)v(i) and E (i - 1)2v(i), which are used in the approximate expressions of the integrals defining stiffnesses and mean moduli. One line is scanned in 0.04 sec and measurement and computation are completed within a few minutes.
150
J. S E R R A , G . V E R C H E R Y
Discussion of the method The precision of the results must be considered from both the technical and statistical points of view. (a) Discussion of the measurement technology: The first problem lies in the choice of the light threshold so that the real geometry of fibres is not distorted. A curve of the percentage of the apparent fibre area versus threshold is plotted. The theoretical curve has a point of inflexion at the optimal threshold. On the experimental curve (Fig. 8) we first exclude, by visual observation, the threshold positions
80 70
% length
/J.............J
6O 5O #0. %0,
1
I
I
I
i
~o
20
0o
hO
~0
!
60
IL
]'h reshoPd
Fig. 8. Light function of the threshold. where the distortion is obvious. We then take as a threshold the middle of the remaining interval. The precision improves as the slope of the curve diminishes and the error caused by one interval of light threshold is very small, of the order of 10 -3 Another cause of error results from the fluctuations due to the resolution of the camera and the signal sampling necessary for logical treatment. The corresponding error has been estimated by repeating the measurement of the same field and the variance of the results is less than 0.5 per cent. (b) Statistics: A primary source of error is the variation (up to 10 per cent) in the mean Young's moduli measured in each field across the composite bar. To limit the variation, a mean value, based on four to six readings, was retained. Error also depends on the size of the inter-line distance, t. A fixed value of 19.6 #m provided measurement with almost no variance.
151
MATHEMATICAL MORPHOLOGY APPLIED TO FIBRE COMPOSITE MATERIALS
Presentation of the results (Table 1) The following values were taken for the Young's moduli of the components: Boron 450 x Aluminium 75 x Epoxy 5 × (1 Pa = 10 -7 h bar
109 Pa 109 Pa 109 Pa ~ 1.45 x 10-4psi) TABLE 1
Specimens
n
Vv
1 2 3 4
12 10 12 7
0.502 0.524 0.448 0.164
D*/Lm.N 43"6 24"8 71"5 148
~S 109 Pa 263 271 204 77"9
/~ I09 P a 222 215 180 151
R1
R2
0.84 0-79 0"88 1.94
0"89 0'87 0.85 0.77
n: number of layers. R 1: ratio/~'//~ as measured in the real case R2: ratio/~//~ as predicted for a regular hexagonal array with the volume ratio VF. 11
ANALYSIS OF THE FIBRE CLUSTERS
Purpose and method This second part of our study tries to give a new tool for a better understanding of the fracture problem of ductile matrix composites. During a tensile test, when a fibre has broken, the load which was carried by that fibre is mainly transferred to the nearest fibres. In a regular hexagonal array, the six neighbouring fibres carry an equal excess loading, but, for a real (irregular) array, it is thought that the nearest single fibre carries a great deal of the load. Therefore the probability of a failure process by a chain reaction is greatly increased by the irregularities and the fracture properties (ultimate tensile strength, fatigue crack propagation, resilience, etc . . . . ) are related to the 'regularity ratio' (if defined) of the cross-section pattern of the fibres, for a given volume ratio, VF. This 'regularity ratio' is experimentally reached by means of the following geometrical transforms. First, a dilatation changes the 0t-state into the fl-state (Fig. 9). The structuring element, H, is an hexagon of side a, which may be taken as a circle in the analysis. The radius of the fibre being b, two fibres with the distance between centres 2d, overlap after the dilatation if the inequality holds: a _> d - b
(13)
Hence we obtain a parameter fulfilling our expectations in the number of connectivity of the dilated set F ~ / - ) , measured as a function of the radius, a, of the structuring element, H.
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J. SERRA, G. VERCHERY
© ©
© ©
©
U ~
B ~)H Fig. 9.
(B~) H) O H
Dilatation a n d closing of the fibres.
We may go further in the analysis by means of an hexagonal closing. 5,7 The dilated set F @/~ is then eroded by the structuring element, H, and gives the set (F t~ H) @ H (Fig. 9(T)). After this transform, the fibres which overlapped by dilatation may, or may not, be linked. More exactly, the link between two fibres remains if it is possible to place element H in the thinnest part of their overlapping area, i.e., if the inequality holds: d2 _ b2
a _> . - 2b
Fig. 10(a)
(14)
C~
0
0
©
©
II
It
~°
v
~o
~°
il
°o
~D
MATHEMATICALMORPHOLOGYAPPLIED TO FIBRE COMPOSITEMATERIALS
155
The main advantage of the closing compared with the dilatation is a more striking view of the results. Figure 10 (a to e) shows the development of the closing process, when side a of the structuring hexagon increases clusters of fibres appear more and more linked.
Experimental results We have measured the hexagonal closings of the fibres in five areas which approximately represent a whole section of the bar. The measurement interval is the side of the hexagon which was used for closing, equal to 3.4/~m. We shall not discuss here in detail the relationship between the measurement area chosen and the objects to be measured within that area: obviously when two fibres are on
75 ~ N(o)
\
50
x
g,4 /~
-2ft.
Fig. 11. Real array. Number of connectivity, N, of the closings.
~q
156
J. SERRA, G. VERCHERY
both sides of the area boundary they are never linked, whatever the radius of the structuring element. However, these measurement errors, sometimes large, can be completely corrected by the analyser 9 which is at present the only apparatus capable of such a correction. The results shown in Fig. 11 represent the number of connectivity after the closing, given by the number of fibre chains minus the number of enclaves in the chains. The slight increase at the beginning of the curve results from small internal defects of the fibres and must not be taken into account. LS(o)
-g-- d-b dLbt
"TE"
-2N
Fig. 12.
Hexagonal array. Number of cormectivity, N, of the closings.
After the second closing (6.8 pm) these defects no longer have an effect and fibre contacts begin to appear. Every new contact reduces by one the number of connectivity. The meaning of the curve is shown by comparison with the case of a regular hexagonal array (Fig. 12), where 2d is the distance between centres, b is the fibre radius, and N is the number of fibres in the area. In this case all the contacts occur at the same time, when a reaches the value ( d 2 - b2)/2b and forms links containing
MATHEMATICALMORPHOLOGYAPPLIED TO FIBRE COMPOSITEMATERIALS
157
2N pores, which disappear simultaneously when the dilatation of the fibres covers all the field, i.e., when a reaches the value 2d/(3)½ - b. In a real random array the irregularity of stacking causes a less abrupt decrease and negative values are smaller, which is explained by the small number of links formed. The slope of the curve can be interpreted as the opposite of the number of new neighbours formed for a considered distance, a, a useful parameter to correlate with fracture properties. A second useful parameter is amin = ( d 2 m i n - b2)/2b, when 2dmin is the smallest distance between centres of any two fibres. The point ami, on the curve (Fig. 11) is taken as the intercept of the maximum value of the number of connectivity 5 with the mean negative slope.
CONCLUSIONS The texture analyser has shown itself to be useful for rapid morphological measurements on composite fibre materials, the specimens requiring a minimum amount of surface preparation. The calculation of flexural and tensile stiffnesses is practically impossible by any other means yet the analyser gives a precise result since it takes into account both the position and the size of each element. The same can be said when we need to measure the volume fraction of fibres in a composite. The analyser permits a precise value of the local and global volume fractions. The preliminary results presented in Table 1 agree well with known unidirectional fibre composite behaviour. The plot of 'number of connectivity' v. the structuring hexagon radius, gives a qualitative assessment of the regularity of the fibre distribution, a regular distribution has a large negative slope and an irregular distribution a smaller one. In addition, the curve provides the value of the smallest inter-fibre distance. This parameter is thought to be of importance in explaining real composite fracture properties, however extensive experimental work is required to verify this.
REFERENCES 1. W. VOIGT, Lehrbuch der Kristallphysik. Teubner, Berlin, 1910. 2. A. REUSS, Bereclmung der Fliessgreuze yon Mischkristallen auf Grund der Plastizit~itsbedingung fflr Einkristalle, Z A M M , 9 (1929) p. 49 3. A. KELLYand G. J. DAVmS,The principles of the fibre reinforcement of metals. Metallurgical Review, 10 (1965) pp. 1-77. 4. R. HILL,Theory of mechanical properties of fibre-strengthened materials: III, Self-consistent model, J. Mech. Phys. Solids, 13 (1965) p. 189. 5. J. SERRA,Introduction ~t la morphologie math6matique, Fasc. N ° 3 des Cahiers du C M M , Ecole des Mines de Paris, 1969. 6. G. MATrmRON,Elements pour une th~orie des Milieux Poreux, Masson, Paris, 1967.
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J. SERRA, G. VERCHERY
J. SERRA, Stereology and structuring elements, J. of Microscopy, 95, Pt. 1 (1972) pp. 93-103. J. SERRA, IRSID--Brevet de l'Analyseur de Textures No. 1449.059. J. C. KLEIN and J. SERRA,The texture analyser, J. of Microscopy, 95, Pt. 2 (1972) pp. 349-56. N.I. MUSKHELISHVILI,Some basic problems of the mathematical theory of elasticity. Chapter 24, translated from the fourth Russian edition by J. R. M. Radok, Noordhoff, 1963. 11. J. HUNEAU, Tension et flexion des composites; Modules moyens. Rapport interne du Centre des Mat6riaux, Ecole des Mines de Paris, 1971.
7. 8. 9. 10.