Micromechanical analysis of fiber-reinforced composites on account of influence of fiber coatings
Introduction
Thin coatings at the interfaces of the constituents of a composite material can make a substantial difference in the functional characteristics and reliability of composites. The optimum use of stiffness and strength properties of composites directly depends on the effectiveness of the transfer of load from the matrix to the inclusions, proceeding through the coatings. Furthermore, in the heterogeneous materials the greatest concentrations of local stresses occur, as a rule, on the interfaces between the constituents and, thus, the strength of coatings is one of the key factors, determining the load bearing capacity of composite as a whole. The fracture of coatings leads to the development of dislocations and cracks, which in the majority of the cases entails the rapid destruction of entire material.
The problems of analysis of the composites with the coatings were examined by many authors, see e.g., [1], [2], [3], [4], [5], [6], [7]. Introduction of a supplementary interphase layer between the constituents has been also used to model the effects of imperfect contact, see [8], [9], [10]. The analysis of the limiting cases of soft and rigid coatings is given in Ref. [11].
It should be noted, that interaction between the neighboring inclusions (fibers) can cause the significant variation of physical fields in the composite on the microlevel. Increase in the rigidity of fibers and their volume fraction (i.e., the decrease of distances between the neighboring fibers) leads to an increase in the local stresses on the interface of constituents. In this case the application of many known analysis methods can be limited by the difficulties of computational nature. Thus, analytical approaches based on representing stress fields in the form of expansions in various infinite series, can experience a deficiency in the convergence. Numerical methods require an increase in the mesh density and, accordingly, a significant increase in computing time.
In the present work we propose a new asymptotic method for calculating the effective moduli and local stresses in the periodically inhomogeneous composite materials on account of the micromechanical influence of fiber coatings. We use the asymptotic homogenization method, see e.g., [12], [13], [14], [15]. The asymptotic analytical solution of the unit cell problem is found with the aid of the method of perturbation of boundary shape [16], [17]. Local shear stresses along the fiber-coating and the matrix-coating interfaces are calculated. The influence of properties of coatings on the maximum local shear stresses on the interfaces of constituents is analysed. The obtained results are suitable for any values of stiffnesses and volume fractions of constituents, including the case of absolutely rigid fibers, converging to contact.
Section snippets
Original boundary-value problem
Consider the fiber-reinforced composites, consisting of the matrix Ωm and cylindrical fibers Ωf, divided by coating Ωc. Fibers are distributed periodically in the form of square lattice, see Fig. 1a, or in the form of hexagonal lattice, see Fig. 1b.
Let us examine the pure shear deformation in the longitudinal planes x1x2, x1x3, under the action of shear stresses τ21, τ31; and let us assume that the remaining stresses are equal to zero. In the literature this stress–strain state is referred as
Asymptotic homogenization method
The first phenomenological “averaged” models in the mechanics of inhomogeneous media appeared in the beginning of the 20th century in the classical works of Voigt [18] and Reuss [19]. Voigt [18] has proposed to calculate the effective properties of polycrystals via the simple arithmetical averaging of the properties of components (crystallites). The method of Reuss [19] was based on the averaging of the components of the tensor of reciprocal values. Only in the middle of 70s the asymptotic
Unit cell problem
As a rule, the solution of the local problem on the unit cell represents a basic difficulty in the practical applications of the asymptotic homogenization method. In the present paper, we find the asymptotic analytical solution of the unit cell problems (11), (12) using the method of perturbation of the boundary shape, see [16], [17], [20]. The basic idea of this method consists in representing the solution in the form of asymptotic expansion in powers of a certain small parameter, reflecting
Numerical examples
Let us first examine the limiting case, when coatings are absent (cc = 0). The obtained results in this limiting case for the effective longitudinal shear modulus 〈G〉 are shown in Table 1, and they are compared with the theoretical data from Ref. [21] in the case Gf/Gm = ∞. It should be noted that the obtained solution makes it possible to calculate 〈G〉 in the case of absolutely rigid fibers converging to contact, i.e., when Gf/Gm = ∞ and , which can be difficult to do using any other known
Conclusions
The new asymptotic method for the analysis of periodically inhomogeneous fiber-reinforced composites on account of the micromechanical influence of fiber coatings is proposed in the present work. The problem of longitudinal shear of the fiber-reinforced composites with square or hexagonal lattices of periodically distributed parallel cylindrical fibers is examined. The asymptotic homogenization method is used and the relevant local problem on the unit cell is solved with the aid of the method
Acknowledgements
This work is supported by the German Research Foundation (Deutsche Forschungsgemeinschaft), Grant No. WE 736/25-1 (I.V. Andrianov); by the NATO reintegration Grant No. NUKR.RIG.981491 (V.V. Danishevs’kyy); and by the Natural Sciences and Engineering Research Council of Canada, NSERC (A.L. Kalamkarov).
References (23)
- et al.
The effect of an inhomogeneous interphase on the elastic constants of transversely isotropic composites
Mech Mater
(1993) - et al.
Multiple-cell modelling of fiber-reinforced composites with the presence of interphases using the boundary element method
Comput Mater Sci
(2001) Thin interphase/imperfect interface in elasticity with application to coated fiber composites
J Mech Phys Solids
(2002)- et al.
A micromechanical model for effective elastic properties of particulate composites with imperfect interfacial bonds
Int J Solids Struct
(2005) - et al.
Effective properties and micro-mechanical response of filamentary composite wires under longitudinal shear
Eur J Mech A/Solids
(2005) - et al.
A rigorous analytical method for doubly periodic cylindrical inclusions under longitudinal shear and its application
Mech Mater
(2004) The theory of composites
(2002)Theory of reinforced materials with coatings
(1971)- et al.
Effect of interphases on micro and macromechanical behavior of hexagonal-array fiber composites
J Appl Mech
(1990) - et al.
Effect of fiber–matrix interphase defects on microlevel stress states at neighboring fibers
J Compos Mater
(1991)