An Analytic Definition of the Border Polymerization Line for Axisymmetric Composite Rods
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An Analytic Definition of the Border Polymerization Line for Axisymmetric Composite Rods S. N. Grigoriev & A. N. Krasnovskii & I. A. Kazakov & K. V. Kvachev
Received: 27 February 2013 / Accepted: 12 March 2013 / Published online: 2 April 2013 # Springer Science+Business Media Dordrecht 2013
Abstract The paper describes a theoretical study of the border polymerization line for rods of circular cross-section during the pultrusion process. The influence of the pull speed, the type of filler and fiber volume fraction on the position of the border polymerization line is investigated. It is shown that pre-heating of the fiber/resin system helps to speed up the polymerization process. Keywords Composite material . Pultrusion . Border line . Die . Specific heat . Degree of polymerization
1 Introduction Pultrusion is one of the most common ways to manufacture long-length composite profiles with longitudinal arrangement of the reinforcing filaments. The capabilities to improve performance of pultruded products depending on the reinforcement scheme already studied [1–4]. Currently, the primary task is to increase the productivity of the process and reduce the production cost without a loss of quality of manufactured products. Mathematical modeling of the pultrusion process [5–9] is one of the ways to reduce the cost of production
S. N. Grigoriev University Administration, Moscow State Technological University (STANKIN), Vadkovsky lane, 3а, 127055 Moscow, Russia e-mail: [email protected] A. N. Krasnovskii (*) : I. A. Kazakov : K. V. Kvachev Department of Composite Materials, Moscow State Technological University (STANKIN), Vadkovsky lane, 3а, 127055 Moscow, Russia e-mail: [email protected] I. A. Kazakov e-mail: [email protected] K. V. Kvachev e-mail: [email protected]
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Appl Compos Mater (2013) 20:1055–1064
output adjustment. The authors obtained an analytical expression allowing to define the boundary between the polymerized and uncured material for composite rods of circular cross-section.
2 Statement of the Problem and Derivation of a Formula for Border Line Figure 1 shows a longitudinal section of the die with a composite material located inside. The gray color marked polymerized part of the rod. We assume that the uncured material is a Newtonian viscous fluid. Because the die has a cylindrical shape and temperature of the die has axisymmetric distribution, then the problem is axisymmetric. Then we can assume that we have polymerized ring with outer radius R (the radius of the die) and the inner radius r(x) (Fig. 1) in each section of the rod inside a die during the polymerization. Clear that the border line is determined by the distribution of temperature. Therefore, to find the border line between polymerized and uncured phases need to use the governing equation of heat flow [10]. In general, this equation for the cylindrical coordinate system is given by @ @T da 1 @ @T þ rl ρcv dT 1 dt ¼ ρH dt r @r @r 2 2 þ @x l2 @x þ 2 ð1Þ @vx 2 x r r þ2μ @v þ @v þ 12 @v þ vrr @r @x @x þ @r W
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