section method

To be published in Journal of Composite Materials, No 4, 2000

ESTIMATION OF THREE-DIMENSIONAL FIBRE-ORIENTATION DISTRIBUTION IN SHORT-FIBRE COMPOSITES BY A TWOSECTION METHOD G. Zak*

C. B. Park

B. Benhabib

Computer Integrated Manufacturing Laboratory Department of Mechanical and Industrial Engineering, University of Toronto 5 King's College Road, Toronto, Ontario, Canada M5S 3G8 *Department of Mechanical Engineering, Queen's University McLaughlin Hall, Kingston, Ontario, Canada K7L 3N6 e-mail: [email protected]

Abstract The overall fibre orientation in a composite can be statistically characterized by a fibreorientation distribution. Fibre-orientation studies have concentrated in two major areas: those dealing with continuous fibres and those dealing with short fibres. This paper presents a novel twosection based method for statistical characterization of the fibre-orientation distribution within short-fibre composites. This method produces unbiased distribution data for the near-zero misalignment angles and simultaneously resolves the orientation-duality problem. The method’s novelty lies in its ability to determine accurately (within 1 m) the relative positions of two sections. While taking advantage of the extra information provided by the two sections, the proposed method also makes use of the single-section method's main principles, namely: (1) intersection of a plane and a cylindrical fibre is an ellipse, and (2) the ellipse's shape is a function of fibre's orientation relative to the intersecting plane. Therefore, by sectioning a specimen, acquiring images of the cross-sectional surfaces, and collecting the fibre-ellipse data from these images, the orientation of every fibre intersecting the sectioning plane can be determined. The probability distribution for fibre orientations within the sectioning plane is then related to that within the specimen volume.

October 1999

To be published in Journal of Composite Materials, No 4, 2000

1 Introduction The mechanical properties of fibre-reinforced composites are largely determined by four primary factors: fibre content, aspect ratio, orientation, and the matrix itself. In particular, fibre orientation plays a very important role. This paper presents a novel method for statistical characterization of the 3-D fibre-orientation distribution within short-fibre composites. 1.1

Short-fibre-orientation studies The 3-D orientation of an individual fibre in a composite can be described by two angles, e.g.,

and

(Figure 1). The overall fibre orientation in a composite can then be statistically characterized

by a fibre-orientation distribution (FOD), p( , ), which is a joint probability density function describing the probability of finding fibres with orientation ( , ) in the specimen (Jain and Whetherhold, 1992). Frequently, it is desired to find the variation of probability as a function of either one of the two angles. This leads to the calculation of marginal probability density functions, p( ) and p( ). Fibre-orientation studies can be subdivided into two major areas: those dealing with continuous fibres and those dealing with short fibres. Continuous fibres tend to be highly aligned in their typical applications. Consequently, the FOD measurement methods applied to continuous fibres mainly focus on small angle misalignments from the nominal fibre direction (Davis, 1975, Yurgatis, 1987, Stecenko and Piggott, 1997). Most of the short-fibre orientation studies, on the other hand, concentrate on injection-moulded thermoplastic composites and characterize the local preferred fibre orientation due to the mould flow patterns. Short-fibre orientation studies can be further categorized by whether two- or three-dimensional fibre orientations are assumed. Assumption of 2-D fibre orientation implies that only in-plane fibre rotations are permitted, and they are naturally defined by a single angle. The 2-D nature of the problem considerably simplifies the data extraction process and makes it amenable to automation. Data can be derived either from polished sections by reflective microscopy or from microtomed samples by contact microradiography or transmission optical microscopy. Experimental results in this category are reported in Sirkis et al., 1994, Kamal, 1986, Vincent and Agassant, 1986, Ranganathan and Advani, 1990.

Page 2 of 26

To be published in Journal of Composite Materials, No 4, 2000 Z Section Plane

x0 y0 X

Y

Figure 1. Fibre orientation defined by two angles. On the other hand, to obtain three-dimensional FOD, Lian et al., 1994, use “optical sectioning” technique, whereby transparent samples with opaque tracer fibres are observed with an optical microscope. By focusing on a sequence of depths up to 1.5 mm into the sample, a series of sample “slices” are obtained. These 2-D slices are “stacked” to derive the 3-D fibre orientation. Although reliable results were obtained, the requirements for transparent matrix, opaque fibres, and depth restriction significantly limit the utility of this method. Fischer and Eyerer, 1988, Bay and Tucker, 1992, and Hine et al., 1993, all measure short-fibre orientations by examining polished cross-sections with an optical microscope. A similar method is suggested by Zhu et al., 1997. This single-section method considers the ellipse-shaped intersection of a cylinder (i.e., a fibre) with a plane to calculate the misalignment angle

between the plane’s

normal and the cylinder’s longitudinal axis based on the relationship: = acos( B A)

(1)

where A is the major radius and B is the minor radius of the ellipse, respectively. The rotation about the section-plane normal, or azimuth angle, , is defined by the direction of the major axis of the ellipse. The ellipse is located within the section plane by its centre coordinates (x0,y0).

Page 3 of 26

To be published in Journal of Composite Materials, No 4, 2000

1.2

Disadvantages of the single-section method The single-section method briefly addressed above has two shortcomings. The first one is the

significant bias and noise sensitivity in the estimation of the near-zero misalignment angles. This bias causes consistent errors in the misalignment angle estimates for fibres nearly aligned with the section normal. The errors result because noisy observations make it impossible to select correctly the major and minor radii of the nearly circular ellipses. Simply choosing the smaller measured radius as the minor leads to undercounting of fibres in the determination of FOD for very small angles. The maximum fibre misalignment angle that would still be affected by the bias,

bias,

is

naturally related to the actual measurement error of the ellipse radii. Yurgatis, 1987, addresses the above problem, while dealing with continuous fibres, by sectioning the specimen such that the average misalignment angle is about 85 , thus avoiding the biased estimation range. Hine et al., 1993, also note the existence of the above problem when dealing with aligned short fibres and recommend sectioning at a non-zero angle to the preferential fibre direction in order to improve the measurement accuracy. Although such an approach is suitable for well-aligned continuous or short fibres, it is not appropriate for short fibres with widely distributed orientations. Bay and Tucker, 1992, correctly note the true cause of the estimation bias. The authors’ proposed solution, however, relies on an operator’s discretion during image digitization. The second shortcoming of the single-section method is the existence of two equally possible alternative orientations for each elliptical cross-section (Figure 2), i.e., the orientation-duality problem. Both alternatives have the same misalignment angle , but the azimuth angles

differ by

180 . Z

Section Plane

X

Figure 2. Two possible fibre orientations based on the same elliptical cross-section. Page 4 of 26

To be published in Journal of Composite Materials, No 4, 2000

Zhu et al., 1997, propose a method which derives a complete three-dimensional FOD by combining data from two orthogonal plane sections, while Bay and Tucker, 1992, claim that data from three orthogonal section planes is required. The experimental implementation of such approaches may be practically difficult even though they are shown to be theoretically possible by such investigations. Additionally, the results would be valid only if the FOD remains unchanged between the two section locations. However, in short-fibre composites, the fibre orientation is frequently a strong function of location. 1.3

Proposed method In this paper, a novel method is proposed for obtaining a three-dimensional FOD by combining

data from two consecutive closely spaced cross-sections of a specimen. This two-section method produces unbiased distribution data for the near-zero misalignment angles and resolves the orientation duality problem. It consists of the following major steps: (i) Obtain two consecutive planar cross-sections of the specimen and fit ellipses to the fibre cross-sections observed within these sections; (ii) Using the two sets of data, identify the ellipses belonging to the same fibres and estimate the section-to-section transformation; and, (iii) Calculate ( , ) for each matched fibre and subsequently obtain the overall FOD.

2 3-D Fibre-Orientation Estimation – A Two-Section Method After giving a brief overview, this section derives the necessary theoretical foundation for the proposed two-section-based fibre-orientation-distribution estimation method.

Next, the section

discusses an important aspect of the proposed method, namely identification of fibre cross-sections belonging to the same fibres. Finally, the correction of the raw FOD data to account for variation in fibre observability with angle and length is addressed. 2.1

Overview The proposed method consists of the following steps: (1) Obtain a planar cross-section of the specimen, Section I.

Page 5 of 26

To be published in Journal of Composite Materials, No 4, 2000

(2) Fit ellipses to the fibre cross-sections observed within this Section I, obtaining for each fibre five ellipse parameters: centre coordinates, (x0, y0); Z-rotation angle, ; and major and minor radii, A and B, respectively. (3) Obtain a second planar cross-section of the same specimen, Section II, at a depth into the specimen roughly equal to one tenth1 of the fibre's expected length. (4) Fit ellipses to the fibre cross-sections of this Section II, obtaining a second set of the ellipse parameters. (5) Using the above two sets (i) identify the ellipses belonging to the same fibres in the two data sets, i.e., identify the matching ellipse pairs, and (ii) estimate transformation parameters between Section I and Section II data (i.e., the section-to-section alignment angle, the X-Y plane shift, and the depth distance). (6) Using the above transformation parameters, calculate for each fibre with a matching ellipse pair (i) the unbiased estimates of , and (ii) the estimates of

which span the full

360 range. (7) Obtain the FOD for the specimen at hand. The following two sections describe in detail the methodology for carrying out two critical tasks referred to in Step 5. 2.2

Accurate estimation of the section-to-section transformation parameters Considering the fibre's centre axis as a line in 3-D space, the fibre's orientation can be

unambiguously determined given the X-Y coordinates of the line's intersection with two off-set parallel planes and given the plane-to-plane separation distance. Such information, however, is not directly available as a result of examining the two sections. First, the X-Y coordinates observed by examining each section plane are likely to be translated and rotated with respect to each other, since the specimen must be removed from the setup for repolishing, and, second, the section-to-section separation distance is unknown. Therefore, to calculate the fibre's three-dimensional orientation, one must first find a precise spatial relationship between the two specimen sections.

1

The recommended distance of 1/10th of the fibre’s length is an empirically determined number which was found to be a reasonable compromise between two conflicting requirements: if the distance is too small then the accuracy of the misalignment angle estimate will suffer; if it is too large there will be too few fibres crossing both section planes. Page 6 of 26

To be published in Journal of Composite Materials, No 4, 2000

Defining transformation parameters For parallel section planes, four transformation parameters are required: xt, yt, zt, and

t,

Figure

3, to relate the two section coordinate frames. It will be assumed that the alignment angle

t

is

relatively small and that it can be readily determined by matching the orientation of linear features on both sections of the specimen. The remaining three parameters need to be determined through the following derivation using data for fibres with

>

bias,

where the ellipse-based

estimates are

reliable. Estimating the parameters zt and xt After examining the two sections and matching the ellipses belonging to the same fibres, a set of ellipse parameters belonging to Section I, ( x o(1) , y o(1) , ( x o( 2 ) , y o( 2 ) ,

(2)

,

( 2)

(1)

,

(1)

) , and that belonging to Section II,

) , are obtained for each fibre, Figure 3. However, one must note that the ellipse

centre coordinates, ( x o(1) , y o(1) ) and ( xo( 2 ) , yo( 2 ) ) , in the above sets are expressed in their own respective coordinate frames. Furthermore, the misalignment angles ( angles (

(1)

and

(1)

(2)

and

) and the azimuth

(2)

) are calculated by the single-section method, and, therefore, the

values only

span a range of 180 . The alignment angle

t

is assumed to be known at this point, and, therefore, the Section II

ellipse centre coordinates ( x0( 2 ) , y 0( 2 ) ) can be rotated by

t

about Z2 axis to align the Section II and

Section I frames. As a result, the rotated ellipse-centre coordinates are obtained as follows: ~ x0( 2 ) = x0( 2 ) sin

t

y 0( 2 ) cos

t

and ~ y 0( 2 ) = x 0( 2 ) sin

t

+ y 0( 2 ) cos

t

.

(2)

Then, Section II ellipse-centre coordinates can be expressed in Section I frame, Figure 3, as: xo( 21) = ~ xo( 2) + xt

and

y o( 21) = ~ y o( 2) + y t .

(3)

Let us also define the following variables: x = xo( 21)

xo(1)

and

y = yo( 21)

yo(1) ,

(4)

~ x =~ xo( 2 )

x o(1)

and

~ y=~ y o( 2)

y o(1) .

(5)

Page 7 of 26

To be published in Journal of Composite Materials, No 4, 2000

xt

Z2

yt

x0(2)

y0(2)

Y2

X2 t

Z1

x0(1) y0(21)

y0(1) Y1

X1

y x x0(21)

zt

Figure 3. Geometry of the fibre intersection by two parallel section planes (four section-to-section transformation parameters are circled). where ( x , y ) is the actual shift of the fibre ellipse centre from Section I to II, when both are projected onto the X1-Y1 plane, and ( ~ x, ~ y ) are the X and Y differences between the ellipse centre positions as observed in their respective frames after the frame alignment. The ( ~ x, ~ y ) values would be available directly, but the ( x , y ) can only be found once (xt, yt) parameters have been determined. From Figure 3: tan = tan =

y , x

and

(6)

x2 + y2 . zt

(7)

Substituting y from (6) into (7) and rearranging: Page 8 of 26

To be published in Journal of Composite Materials, No 4, 2000

tan 2 1 + tan 2

2

x

=

zt

.

(8)

.

(9)

Let us define a variable c , such that c

tan 2 1 + tan 2

=

Also, from (3), (4), and (5), x= ~ x + xt

y= ~ y + yt .

(10)

Substituting x from (10) into (8) and rearranging: c

=

~ x + xt . zt

(11)

Equation (11) can be rewritten as: ~ x = zt c

xt .

Equation (12) describes a linear relationship between c

(12) and

x~ can be obtained from the ellipse centre coordinates, and c

x~ . For each matched ellipse pair, can be obtained from the single-

section-based orientation estimates. Subsequently, zt and xt can be accurately estimated via a regression analysis for the above linear relationship. Estimating the parameter yt The following relationship, from (10), is used to determine the last of the three section-tosection transformation parameters, yt: yt = y

~ y.

(13)

Examining Figure 3, y = r sin where

and

r = z t tan ,

(14)

r is the distance between the matched ellipse centres when expressed in Section I

coordinates. Substituting for y in (13): y t = z t tan sin The

and

~ y.

are known from the single-section ellipse observations, and

(15) ~ y is known from

matched ellipse centres. Thus, if there are n fibres found with matched ellipse pairs, yt can be estimated using (15) by: Page 9 of 26

To be published in Journal of Composite Materials, No 4, 2000

yˆ t =

1 n

n i =1

(z t tan

i

sin

i

~ yi ) .

(16)

Determining the fibre orientation At this point, the correct fibre orientations,

and , can be determined by applying Equations

(6) and (7) to all fibres. Since the correct signs of x and y are now known, the ellipse-based orientation estimation duality problem is eliminated, yielding 2.3

values over the full 360 range.

Matching the fibre cross-sections An essential step for the implementation of the proposed two-section method is the

identification of the fibre cross-sections belonging to the same fibre (in other words, matching fibres between the two sections) (Step 5 in the Section 2.1). Our experience of analyzing cross-sectional images of short-fibre specimens has shown that this is not a trivial task. Several specific problems contributing to the task's difficulty have been identified. The first problem arises because, after examining the first section and prior to examining the second section, the specimen must be removed from the setup for repolishing. Therefore, unless special care is taken, there will usually be a relative translation and rotation between the two sections. The second problem is caused by the variability of fibre orientations within the short-fibre composite. When a fibre is intersected by two offset parallel planes, the location of the elliptical cross-section within the second section plane shifts relative to the coordinate frame fixed to the first section by ( x, y ) , Figure 3. This shift is naturally a function of the fibre’s orientation, ( , ) , and, therefore, varies from fibre to fibre. The third problem is caused by the short length of the fibres considered herein. For such short fibres, a significant fraction of fibres will not extend from one section to another, and therefore there will not be two matching cross-sections. Due to the above problems, the pattern of fibre cross-sections changes significantly from one section to the other, which makes it impossible to reliably identify the matching fibres based on the raw cross-sectional images. Therefore, a method has been developed to assist with fibre matching process by predicting the locations of the Section II fibre ellipses based on the fibre ellipse data from Section I. The steps involved in fibre matching are given below: Page 10 of 26

To be published in Journal of Composite Materials, No 4, 2000

(1) Identify the fibre ellipse boundaries in Section I and Section II. (2) Obtain fibre orientations based on the fibre ellipses in Section I using the single-section method. (3) Predict location of fibre ellipses in Section II based on (a) the fibre orientations derived from Section I ellipses and (b) an approximate estimate of the depth separation, ~ z . t

(4) Translate and rotate fibre ellipses observed in Section II in order to express them in Section I coordinates. Use approximate estimates of parameters ( ~ xt , ~ y t ) for translation and

t

for rotation;

(5) Superimpose the predicted locations of Section I fibre ellipses over the Section II ellipses (all expressed with respect to the Section I frame); (6) Switch each fibre’s orientation as needed between the two possible alternatives of the single-section-based orientation estimate, with the consequent difference in the shift direction. (7) Identify the ellipses closest in terms of location, shape, and orientation as “matching” (i.e., belonging to the same fibre). Estimating section-to-section transformation parameters The forgoing procedure indicates that to accomplish fibre matching, the values of the four section-to-section transformation parameters need to be found. As noted in Section 2.2, three of these parameters (X-Y translation (xt, yt) and section-plane separation zt) are also determined by the proposed FOD estimation method. It appears that, in order to estimate the section-to-section transformation parameters, fibre matching is required and, in order to perform fibre matching, the parameter values are required. The circularity of the above definition is resolved by starting with approximate parameter values: since the matching process only requires making the binary "match" vs. "no-match" decisions, the parameter values do not need to be precisely known at this stage. Additionally, an approximate value for zt should be available since the thickness of material removed would be monitored during the repolishing.

Page 11 of 26

To be published in Journal of Composite Materials, No 4, 2000

Once a minimum number of matching fibres have been identified, an initial estimate of the parameters can be made using the equations given in Section 2.2, and these estimates can in turn be used to enhance the accuracy of the overlapping ellipse display. As more fibres are matched, the accuracy of the parameter estimates would be further improved. After all the matching fibres are identified, the translation and depth parameters are accurately estimated and used subsequently to obtain accurate fibre orientations. Superimposing the fibre ellipses For the ellipses to be matched, they must be superimposed on the same display. This task is accomplished in two steps. First, the locations of Section II ellipses are predicted based on the fibre orientation data extracted from the Section I ellipses. In other words, the fibre orientation is calculated by applying the single-section method to the Section I ellipses and the location of each fibre's intersection with the Section II plane is predicted. Second, the ellipse locations from both sections are expressed with respect to a common frame. Herein, Section I frame was selected arbitrarily as the common reference frame. Examining Figure 3, we can see that, for each Section I ellipse, the location of Section II ellipse centre with respect to the Section I frame can be predicted by: xˆ o( 21) = xo(1) + ~ z t tan cos * and yˆ o( 21) = y o(1) + ~ z t tan sin * , where the azimuth angle

depending on which of the two alternative is estimated using Equation (1); and ~ z is the initial approximate estimate

orientations is selected;

*=

or

*=

(17)

+ ,

t

of the parameter zt. To express them in Section I frame, the centres of Section II ellipses are first rotated by angle and then translated by ( ~ x ,~ y ): t

x0( 21) = x0( 2) sin

t

t

t

y 0( 2) cos

t

xt and y 0( 21) = x 0( 2 ) sin +~

t

+ y 0( 2 ) cos

t

yt . +~

(18)

where ( ~ xt , ~ y t ) are the initial estimates of the parameters (xt, yt). Selecting correct orientation alternative Projecting Section I ellipses at the Section II depth requires selection of the correct alternative from the two possible orientations ( * =

or * =

+ ). The correct orientation is found by

examining each of the two alternatives and selecting the one which produces a closer match. In the Page 12 of 26

To be published in Journal of Composite Materials, No 4, 2000

current implementation, this step is performed manually, whereby a Section I ellipse is "toggled" between the two alternatives through operator's action and the closer match is visually identified. Then, the ellipses are matched, and the fibre's orientation alternative is recorded together with the identifications of the matching ellipses. The above process is expected to be well-suited for automation, since the matching decisions can be made by automatically toggling between the alternative orientations and seeking the closest matches in terms of ellipse orientation, eccentricity, and minor diameter. 2.4

Correcting for fibre observability For any fibre, the probability of its intersection with a section plane decreases with the

increased misalignment angle and reduced fibre length, leading to progressive undercounting of fibres as the misalignment angle increases (DeHoff and Rhines, 1968). To correct for this bias, a weighting function, Wobs, which will be referred to as the observability-correction function, must be applied to the sectional distribution data. Alternative forms of this function were presented by Zhu et al., 1997, Bay and Tucker, 1992, and M ginger and Eyerer, 1991, Table 1, where s = l d is the fibre aspect ratio, l is the fibre length, and d its diameter. Table 1. Alternative forms of the observability-correction function. Zhu et al., 1997

Bay and Tucker, 1992

M ginger and Eyerer, 1991

Wobs =

Wobs

1 s cos + sin

1 , for = s cos 1, for

>

= acos

1 s

c

1

Wobs =

1

2

s cos where

c

4

+

s+ 1+

2

s (1 + ) + s+

is found by solving

f

is the fibre volume fraction

Page 13 of 26

sin

(1 + 2 +

s= 4

and

2

1 2 f

2

cos 2

) 2

2

+ sin

4

2

To be published in Journal of Composite Materials, No 4, 2000

In this paper, the function pobs represents the probability of observing a particular fibre type (i.e., of particular orientation, length, or both) when sectioning a sample. It is related to the abovementioned weighting functions as: pobs = ( K obs Wobs ) 1 ,

(19)

where Kobs is an unknown proportionality constant and Wobs can be in any one of the forms given in Table 1. Three cases of aspect ratios are considered below: (1) constant aspect ratio for all fibres, (2) variable aspect ratio, and (3) variable aspect ratio constrained by layer height. The first case applies to all composites with relatively narrow fibre aspect ratio distributions; the second applies when the aspect ratio is widely distributed and the distribution function is known; and the third applies to layered composites with widely distributed fibre aspect ratios, where the layer height restricts certain combinations of fibre length and orientation. Although examples of the first two cases have been noted in the literature, the third case represents a novel contribution by this paper. Case 1: Constant aspect ratio The probability of finding a fibre with the misalignment angle of

i,

for 0