A numerical study on the thermal expansion coefficients of fiber

A numerical study on the thermal expansion coefficients of fiber A numerical study on the thermal expansion coefficients of fiber

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3 composites with isotropic phases were derived using an extension of Hill’s method (1964). Exact relations were given from which composite thermal expansion coefficients may be determined if constituent properties and effective composite moduli are known. In a series of articles by Van Fo Fy (1965, 1966), exact thermal expansion coefficients for matrices reinforced by doubly periodic array of hollow or solid circular continuous fibers were developed by means of a detailed stress analysis. Another method for calculating upper and lower bounds on thermal expansion coefficients of isotropic and anisotropic composites with isotropic phases, and some explicit formulas for volumetric and linear coefficients of thermal expansion was given by Shapery (1968). In contrast to work by Levin and Van Fo Fy composites with an arbitrary number of different constituents and arbitrary phase geometry can be treated by Shapery’s formula. His method employed the complementary and potential energy principles of thermoelasticity theory in conjunction with a procedure for minimizing the difference between upper and lower bounds. It was shown that for some important cases the bounds coincide and therefore yield exact solutions. Shapery’s formula has been the most useful solution to determine the longitudinal thermal expansion coefficient afterwards. An alternative model for transverse thermal expansion of unidirectional composites was derived by Chamberlain (Rogers et al., 1977), using plane stress thick walled cylinder equations for the case of transversely isotropic fibers embedded in an isotropic cylindrical matrix. Rosen and Hashin (1970) developed relations to determine the upper and lower bounds on effective thermal expansion coefficients of anisotropic composites having any number of anisotropic phases using thermoelastic energy principles. The bounds they found are complicated algebraic expressions but reduction to simpler form is possible when symmetry of the phases and composite is taken into account. For isotropic phases, the thermal expansion results reduce to the bounds obtained by Shapery (1968). When the composite has only two phases, thermal expansion

4 coefficients coincide to give results in the form of unique functions of the elastic moduli. Another consideration was made by Sideridis (1994) who developed the model of the inhomogeneous boundary interphase. He studied the influence of the mode of variation of boundary’s thermal expansion coefficient, elastic modulus and Poisson’s ratio versus the polar radius between the fiber and the matrix in the representative volume element of a unidirectional fiber composite on the overall value of the composite thermal expansion coefficients. He also made experiments on E-glassepoxy composites and showed that for all volume fractions the thermal expansion values derived from his solutions are similar to both experiments and the respective values derived from the equations of Shapery and Van Fo Fy. An analytical and experimental investigation on thermal expansion coefficients of unidirectional composites was carried out by Ishikava, Koyama and Kobayashi (1978). The fiber anisotropy and temperature dependency of the constituent material properties were considered in the formulation of the problem. The solving technique they used was constructed by a slight modification of their previous investigation to calculate elastic moduli of unidirectional composites. The first point of the main purpose of the experiments was to reveal certain temperature dependent behavior of thermal expansion coefficient’s of the constituents of carbon-epoxy composites. The second was a comparison between the experimental and theoretical results and the third was to estimate unknown properties of carbon fibers. The influence of fiber type and orientation on thermal expansion coefficient of carbon-epoxy composites was discussed by Rogers et al. (1977). Interferometric measurements of the linear thermal expansion coefficients between approximately 90-400 K for a series of unidirectional and bidirectional specimens of epoxy resins reinforced with carbon fibers were made. The room temperature results shoved that linear thermal expansion coefficients of these composites are mostly influenced by the thermal and elastic properties of the constituents and the orientation of the fibers. At higher temperatures their results clearly showed significant changes in the

4<br />

<strong>coefficients</strong> coincide to give results in <strong>the</strong> form <strong>of</strong> unique functi<strong>on</strong>s <strong>of</strong> <strong>the</strong> elastic<br />

moduli.<br />

Ano<strong>the</strong>r c<strong>on</strong>siderati<strong>on</strong> was made by Sideridis (1994) who developed <strong>the</strong> model <strong>of</strong><br />

<strong>the</strong> inhomogeneous boundary interphase. He studied <strong>the</strong> influence <strong>of</strong> <strong>the</strong> mode <strong>of</strong><br />

variati<strong>on</strong> <strong>of</strong> boundary’s <strong>the</strong>rmal expansi<strong>on</strong> coefficient, elastic modulus and Poiss<strong>on</strong>’s<br />

ratio versus <strong>the</strong> polar radius between <strong>the</strong> <strong>fiber</strong> and <strong>the</strong> matrix in <strong>the</strong> representative<br />

volume element <strong>of</strong> a unidirecti<strong>on</strong>al <strong>fiber</strong> composite <strong>on</strong> <strong>the</strong> overall value <strong>of</strong> <strong>the</strong><br />

composite <strong>the</strong>rmal expansi<strong>on</strong> <strong>coefficients</strong>. He also made experiments <strong>on</strong> E-glassepoxy<br />

composites and showed that for all volume fracti<strong>on</strong>s <strong>the</strong> <strong>the</strong>rmal expansi<strong>on</strong><br />

values derived from his soluti<strong>on</strong>s are similar to both experiments and <strong>the</strong> respective<br />

values derived from <strong>the</strong> equati<strong>on</strong>s <strong>of</strong> Shapery and Van Fo Fy.<br />

An analytical and experimental investigati<strong>on</strong> <strong>on</strong> <strong>the</strong>rmal expansi<strong>on</strong> <strong>coefficients</strong> <strong>of</strong><br />

unidirecti<strong>on</strong>al composites was carried out by Ishikava, Koyama and Kobayashi<br />

(1978). The <strong>fiber</strong> anisotropy and temperature dependency <strong>of</strong> <strong>the</strong> c<strong>on</strong>stituent material<br />

properties were c<strong>on</strong>sidered in <strong>the</strong> formulati<strong>on</strong> <strong>of</strong> <strong>the</strong> problem. The solving technique<br />

<strong>the</strong>y used was c<strong>on</strong>structed by a slight modificati<strong>on</strong> <strong>of</strong> <strong>the</strong>ir previous investigati<strong>on</strong> to<br />

calculate elastic moduli <strong>of</strong> unidirecti<strong>on</strong>al composites. The first point <strong>of</strong> <strong>the</strong> main<br />

purpose <strong>of</strong> <strong>the</strong> experiments was to reveal certain temperature dependent behavior <strong>of</strong><br />

<strong>the</strong>rmal expansi<strong>on</strong> coefficient’s <strong>of</strong> <strong>the</strong> c<strong>on</strong>stituents <strong>of</strong> carb<strong>on</strong>-epoxy composites. The<br />

sec<strong>on</strong>d was a comparis<strong>on</strong> between <strong>the</strong> experimental and <strong>the</strong>oretical results and <strong>the</strong><br />

third was to estimate unknown properties <strong>of</strong> carb<strong>on</strong> <strong>fiber</strong>s.<br />

The influence <strong>of</strong> <strong>fiber</strong> type and orientati<strong>on</strong> <strong>on</strong> <strong>the</strong>rmal expansi<strong>on</strong> coefficient <strong>of</strong><br />

carb<strong>on</strong>-epoxy composites was discussed by Rogers et al. (1977). Interferometric<br />

measurements <strong>of</strong> <strong>the</strong> linear <strong>the</strong>rmal expansi<strong>on</strong> <strong>coefficients</strong> between approximately<br />

90-400 K for a series <strong>of</strong> unidirecti<strong>on</strong>al and bidirecti<strong>on</strong>al specimens <strong>of</strong> epoxy resins<br />

reinforced with carb<strong>on</strong> <strong>fiber</strong>s were made. The room temperature results shoved that<br />

linear <strong>the</strong>rmal expansi<strong>on</strong> <strong>coefficients</strong> <strong>of</strong> <strong>the</strong>se composites are mostly influenced by<br />

<strong>the</strong> <strong>the</strong>rmal and elastic properties <strong>of</strong> <strong>the</strong> c<strong>on</strong>stituents and <strong>the</strong> orientati<strong>on</strong> <strong>of</strong> <strong>the</strong> <strong>fiber</strong>s.<br />

At higher temperatures <strong>the</strong>ir results clearly showed significant changes in <strong>the</strong>

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