Journal of Computational Engineering

Volume 2016 (2016), Article ID 2753187, 9 pages

http://dx.doi.org/10.1155/2016/2753187

## A Monte-Carlo Algorithm for 3D Fibre Detection from Microcomputer Tomography

^{1}Justus Liebig University, Institute for Theoretical Physics, Heinrich-Buff-Ring 16, 35392 Giessen, Germany^{2}Technische Hochschule Mittelhessen, Institute of Mechanics and Materials, Wiesenstr. 14, 35390 Giessen, Germany

Received 14 March 2016; Revised 15 July 2016; Accepted 11 August 2016

Academic Editor: Amine Ammar

Copyright © 2016 Robert Gloeckner et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A model-based approach to analyze fibre distributions in polymer composites applicable for high fibre content is suggested. The algorithm is a four-step iterative method using Monte-Carlo techniques in order to increase speed and robustness for fibre detection. Samples with up to 20% volume fraction of glass fibres and different matrix polymers (PP, PBT) have been analyzed regarding distributions of orientation and length and thickness of the fibres.

#### 1. Introduction

Short fibre reinforced polymer composites are appealing to mid-sized industry, as their use promises low cost continuous production of parts with enhanced properties. Local fibre length distribution (FLD) and fibre orientation distribution (FOD) [1] determine mechanical properties [2–9] but are strongly affected by production processes [10]. The most commonly used manufacturing process for components made of short fibre reinforced plastics is injection moulding. Hereby, molten plastic material and glass fibres are filled into a mould cavity under high pressure. Particularly in thin walled structural parts, the fibres are oriented in tree layers: two boundary layers and a core layer whose thickness depends on the viscosity of the polymer matrix. The fibres in the boundary layers are mostly oriented in filling direction while the fibres in the core layer are oriented perpendicular to the filling direction. This influences strongly the anisotropy of the composite. Within the development process of moulded plastic parts, flow simulations are performed, for example, in order to avoid weld lines in highly stressed areas by optimization of the positions of the gates. Moreover, the local FOD can also be approximated within such flow simulations. This local FOD information can then be used in anisotropic finite element analysis for the structural part under mechanical loading. However, flow simulations are based on idealized fibre models and lead to uncertainty about FLD as well as FOD [11–13]. A comparison of the FOD data computed by flow simulation with real FOD test data is indispensable. Once the test data is available the material parameters, for example, the fibre interaction coefficient in [12], can be calibrated and flow simulation becomes much more predictive. Thus, the usage of short fibre reinforced polymers in safety related components demands effective and accurate quality testing.

While other methods either lack a complete characterization [14, 15] or require too much preparation and labour time [16–18], microcomputer tomography as a nondestructive testing method is a suitable alternative for those needs [19–22]. Common fibre contents of commercially available composites vary between 10% and 70% weight. Hence, due to clusters and accumulations, reliable methods will be needed that are able to differentiate and locate single fibres therein.

The system used in this work is a X-ray desktop microcomputer tomography (CT). Measurements were performed using a SkyScan 1172. Its microfocus sealed X-ray tube operated at 20–100 kV/0–250 A below a maximum power of 10 W. Typical volume picture dimensions used in this work are 1024 1024 900 cubic voxels of 1.7 m length. This CT together with the algorithm presented in this paper has recently been used in [23] to compute the elastic properties of a short-fibre plastic based on real measured microstructure data by homogenization. In the present work we set our focus on the numerical methodology of fibre detection and discuss the Monte-Carlo algorithm in detail.

#### 2. Model-Based Algorithm

A common approach of model-based image analysis is to extract image primitives and to establish a correspondence between these and the model primitives. Probabilistic object pattern matching speeds up the scanning process, provides good scaling capabilities if done in parallel, and increases tolerance against noise and distortion. For an overview of image analysis of materials structures, the reader is referred to [24] and the references therein.

The fibre model consists of a chord of cylinders of constant radii and various lengths. The radius of scannable fibres is limited by the minimum cylinder length.

All Monte-Carlo pattern-matching processes run independent of each other. Their results are merged in a control-process which detects doublets and optimizes the vector data. Subsequently, this process modifies the voxel data as well as the parameters of the pattern-matching processes and initiates their restart.

##### 2.1. Pattern Matching

Core algorithm steps are Monte-Carlo scanning for separate fibre parts and cylinder integral extrusion for fibre length detection. Scanning for separate fibre parts (index ) is done by calculating radial density functions in randomly chosen spherical volumes (centres ):whereand is a chosen tolerance.

To approximate radial density, the spherical volume is divided into spherical shell lists of random point-symmetric coordinate pairs within nonequidistant radius-intervals:where denotes the fibre radius and is the pattern radius of the final sphere. Using correlating point-symmetric coordinates according to (2) removes voxels from neighbour fibres as well as noise. Figure 1 shows the radial density as a function of the integral sphere radius and the cylinder radius , which represents the fibre. The density, that is, the probability that a random point within the sphere is also located within the fibre, is for . For an increasing sphere radius we have a decreasing probability and thus a decreasing density distribution which finally tends to zero.