Analysis of residual stresses in a single fibre–matrix composite

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Abstract

Thermal stresses are induced in fibre/matrix composites due to the difference in coefficients of thermal expansion of the constituent materials. The present theoretical method is developed for the analysis of thermal stresses in fibres of finite length embedded in a matrix. Based on the complementary energy approach and minimization of the total complementary energy, the method yields closed-form solutions for the thermal residual stresses, which satisfy the requirements of a free surface around the fibre/matrix composite. The analysis shows that a radial-tensile stress concentration appears at both ends of the composite, subjecting the interface in the vicinity to tension. The interfacial shear stress peaks at some distance into the interface from the fibre ends. A parametric study further shows that thermal stresses are dependent upon the material and geometric properties of the constituents. Stress concentrations shift nearer to the ends of the composite with increase in fibre length while reduction in matrix radius will further accentuate the stress concentrations. A softer matrix tends to reduce the peak interfacial shear and radial-tensile stresses. A small mismatch in the coefficient of thermal expansion between fibre and matrix induces very low thermal stresses in the constituents. The stress concentrations, existing very close to the fibre ends tend to augment the fibre debonding stresses due to mechanical loading and will influence the analysis of interfacial properties in fibre pull- and push-out tests.

Introduction

A mismatch in the coefficients of thermal expansion (CTE) of the fibre and matrix results in a mismatch in strain which induces thermal stresses in the constituents. The CTE of the fibre in the longitudinal direction is usually much less than that of the matrix. Some carbon and aramid fibres have a small negative CTE. During the cooling process, both constituents shrink but the full contraction of the matrix is constrained as a consequence of being bonded to the fibre. On the other hand, the fibre is not only shortened by its own thermal shrinkage, but is also compressed by the constrained shrinkage of the matrix. As a result, residual axial compressive stresses, which decrease from a maximum at mid-length to zero at the fibre ends, are induced in the fibre. At the same time, the matrix is constrained by the fibre from fully shrinking, and as a result, is stressed in tension in a direction parallel to the fibre axis. Stress transfer from fibre to matrix takes place.

Several methods have been developed to determine thermal residual stresses in a fibre embedded in a matrix. These include the elasticity solution [1], finite element method [2] and cylinder theory [3], [4], [5] in which thermal stresses are determined for an infinitely long fibre surrounded by a matrix. The results from the latter are for generalized plane-strain conditions, which are not quite applicable to fibres of finite length such as those used in fibre pull- and push-out tests. A technique based on the complementary energy method is proposed. The method of complementary energy is preferred because the forces (or stresses in this case) rather than the displacements are treated as unknowns. The method provides a direct approach to obtain the residual stresses rather than the indirect approach of using displacements, which require the use of assumed higher-order displacement fields to obtain greater accuracy in the stresses. The complementary energy method reduces the complexity considerably. In the following sections, the equations used in the formulation of solutions for the thermal stresses are presented.

Section snippets

Stress function

Consider the single fibre/matrix composite depicted in Fig. 1 in which perfect interfacial bonding is assumed to exist between the fibre and matrix. The fibre of length L and radius a, is embedded concentrically in a matrix of radius b. For axial symmetric geometry, the cylindrical coordinates are r, θ and z. The boundaries are traction free and the composite is subjected to a decrease in temperature of 1°C. Actual thermal stresses are found by multiplying the thermal stress for a 1°C

Results and discussion

The method is applied to a fibre and matrix having these properties: Ef=230GPa, νf=0.2, αfL=−0.26×10−6/°C, αfT=26×10−6/°C, Em=3GPa, νm=0.4, αm=76×10−6/°C. The fibre has a radius of 3μm and a length of 1.0mm. The radius of the matrix is 1.0mm. A temperature difference of ΔT=−1°C is assumed. The actual residual stresses are found by multiplying the residual stress for a 1°C temperature decrease by the observed temperature decrease.

The residual stress fields along the fibre length, plotted in

Parametric study

The effects of fibre length, matrix radius, Young's modulus, CTE and Poisson's ratio of the matrix on the residual stress distributions were investigated. The stress variations along the z-axis are plotted against the fractional distance z/L.

Conclusions

Closed-form solutions for the thermal stresses in a fibre of finite length embedded in a matrix have been obtained with the total complementary energy principle approach. The complete axisymmetric state of stress is determined for a temperature difference of −1°C and the boundary conditions of a free surface around the composite are satisfied. In cases of large differences in temperature, the actual thermal stresses are obtained by multiplying the results for unit decrease in temperature by the

Acknowledgements

I would like to thank the reviewers for their helpful improvements to the manuscript.

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