Stereological measurement and error estimates for three‐dimensional fiber orientation

Abstract

A method is presented for measuring three‐dimensional fiber orientation in fiber‐reinforced polymers and placing confidence limits on the results. The orientations of individual fibers are determined from the elliptical intersections between the cylindrical fibers and a polished section. This can be done using either manual digitization or automated image analysis. Volume averages for the sample are computed using an orientation‐dependent weighting function that corrects for the bias of an area‐based sample. Equations are developed for nonuniform fiber lengths, using both number‐average and weight‐average measures of orientation. Sources of systematic, measurement, and sampling error are discussed and equations for sampling error and the propagation of measurement error are derived. The results use a second‐rank tensor to characterize fiber orientation, but the error analysis can be applied to any type of orientation parameter. We implement the technique using manual digitization of optical micrographs. Our implementation accurately measures samples with known orientation, and produces identical results from two perpendicular sections of a glass fiber/nylon injection‐molded sample.

Second‐order stereology of spatial fibre systems

“…So their investigation is an important theme in stereology and spatial statistics. There are many publications on the subject, the majority of which are devoted to diameter(s), axial ratio and other size estimation (Hennig & Elias, 1964;Elias & Hyde, 1980;Bishop et al, 1996;Krasnoperov et al, 2001a,b;Krasnoperov & Gerasimov, 2003a,b), determination of the length density or total length (Gokhale, 1990;Adair et al, 1994;Benes et al, 1997;Gundersen, 2002;Kubínová et al, 2002;Mouton et al, 2002) and defining an orientation distribution (Stoyan, 1985;Bay & Tucker, 1992;Vogel et al, 1993;Mattfeldt et al, 1994;Redon et al, 1998;Eberhardt & Clarke, 2001) of the fibres. In the present paper, methods of second-order stereology of fibres developed in Stoyan (1984Stoyan ( , 1985 and Stoyan et al (1995) are reconsidered, with the existing estimator of the K-function for spatial fibres being improved.…”

“…It is . The intersection angles β i can be measured on the fibre sectioning profiles using previously published methods (Stoyan, 1985;Bay & Tucker, 1992;Mattfeldt et al, 1994;Eberhardt & Clarke, 2001;Krasnoperov et al, 2001b). Then the estimator Est{L V K 3D (r)} defined in Eqs (9)-(11) can be used for L V K 3D (r).…”

Complex fibers orientation distribution evaluation in short glass fiber-reinforced thermoplastic (PA66 GF50)

“…Target is to illustrate and quantify the skincore effect of such samples. The calculation of the orientation tensor and its components are well described in [1][2][3], [7][8][9][10]. For simplicity purpose, only the two components of the second order orientation tensor (a11 and a22) are plotted in Fig.…”

“…To reconstruct the global tensor orientation, it's also necessary to take into account that the probability of cutting a fiber perpendicular to the surface which depends of the length-'l' of the considered fiber, its diameter-'d' and its orientation. This function is defined in [2], [7][8][9][10].…”

Local anisotropy analysis of injection moulded fibre reinforced polymer composites

“…describing techniques based on this principle have been published and this method is still used, due to its simplicity and low cost. However, it has been soon highlighted that these measurements are affected by big uncertainties, particularly because of difficulties in deriving volume properties from area-based measurements [8] and in obtaining high angular resolution because of the low eccentricity of the elliptical footprints. Moreover, the same elliptical section may correspond to two symmetrical angles θ and 180° -θ [9].…”