Hashin

Symposium on Mechanics of Composite Materiais, Zvi Hashin and Cad T. Herakovich (Editors). Biacksburg, Virginia, 16-19 August 1982, Pergamon Press, New York, 1983, pp. 323-337. 

Hashin and Rosen [3-8] extended Hashin s work to fiber-reinforced composite materiais. The fibers have a circular cross section and can be hoilow or solid. Two cases were treated (1) identical fibers in a he gotiaLarray and (2) fibers of various diameters (but same ratio of insldeto outside diameter, if hollow) in a random array. The two types of arrays are depicted in Figure 3-21. In both cases, the basic anaiysis... 

Figure 3-21 Hashin and Rosen s Fiber-Reinforcement Geometries...

A variation on the exact soiutions is the so-caiied seif-consistent modei that is explained in simpiest engineering terms by Whitney and Riiey [3-12]. Their modei has a singie hollow fiber embedded in a concentric cylinder of matrix material as in Figure 3-26. That is, only one inclusion is considered. The volume fraction of the inclusion in the composite cylinder is the same as that of the entire body of fibers in the composite material. Such an assumption is not entirely valid because the matrix material might tend to coat the fibers imperfectiy and hence ieave voids. Note that there is no association of this model with any particular array of fibers. Also recognize the similarity between this model and the concentric-cylinder model of Hashin and Rosen [3-8]. Other more complex self-consistent models include those by Hill [3-13] and Hermans [3-14] which are discussed by Chamis and Sendeckyj [3-5]. Whitney extended his model to transversely isotropic fibers [3-15] and to twisted fibers [3-16]. 

The experimental results for 6 2 ai" also shown in Figure 3-44, along with theoretical results from Equation (3.67) for C = 0,. 2,. 4, and 1. As with the previous moduli, the experimental data are bounded by curves for C = 0 and C =. 4. The upper (parallel-connected phases) and lower (series-connected phases) bounds due to Paul (see Section 3.3) are shown to demonstrate the accuracy of the bounds in the present case where E is much greater than E. The lower bound results of Hashin and Rosen [3-8] correspond to C = 0, but their upper bound is below some of the experimental data in Figure 3-44. 

Figure 3-52 Cumulative Number of Fiber Fractures versus Percentage of Ultimate Composite Load (After Rosen, Dow, and Hashin [3-29])...

Zvi Hashin, The Elastic Moduli of Heterogeneous Materials, Journal of Applied Mechanics, March 1962, pp. 143-150. 

Zvi Hashin and S. Shtrikman, A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials, Journal of the Mechanics and Physics of Solids, March-April 1963, pp. 127-140. 

B. Walter Rosen, Norris F. Dow, and ZvI Hashin, Mechanical Properties of Fibrous Composites, NASA CR-31, April 1964. 

Analysis of the relationships between the moduli and bond strength between particles [222] has shown that for Vf = 0.1 — 0.15 the concentration dependence of the modulus corresponds to the lower curve in the Hashin-Shtrikman equation [223] (hard inclusion in elastic matrix), and for Vf — 0.34 to the upper boundary (elastic inclusion in a hard matrix). The 0.1 to 0.34 range is the phase inversion region. 

A study of the effect of the mesophase layer on the thermomechanical behaviour and the transfer mechanism of loads between phases of composites will be presented in this study. Suitable theoretical models shall be presented, where the mesophase is taken into consideration as an additional intermediate phase. To a first approximation the mesophase material is considered as a homogeneous isotropic one, while, in further approximations, more sophisticated models have been developed, in which the mesophase material is considered as an inhomogeneous material with progressively varying properties between inclusions and matrix. Thus, improvements of the basic Hashin-Rosen models have been incorporated, making the new models more flexible and suitable to describe the real behaviour of composites. 

A satisfactory model for particulates is a modification of the well-known model proposed by Hashin 1 According to this model the composite consists of three phases the matrix, the inclusion, and a third phase, called the mesophase, which corresponds to the zone of imperfections, surrounding the inclusions2,3). 

Fig. la and b. Principal sections of the Hashin two-phase model and its respective three-layer unfolding model for a typical particulate composite... 

A better approach for the Rosen-Hashin models is to adopt models, whose representative volume element consists of three phases, which are either concentric spheres for the particulates, or co-axial cylinders for the fiber-composites, with each phase maintaining its constant volume fraction 4). 

The novel element in these models is the introduction of a third phase in the Hashin-Rosen model, which lies between the two main phases (inclusions and matrix) and contributes to the progressive unfolding of the properties of the inclusions to those of the matrix, without discontinuities. Then, these models incoporate all transition properties of a thin boundary-layer of the matrix near the inclusions. Thus, this pseudo-phase characterizes the effectiveness of the bonding between phases and defines a adhesion factor of the composite. 

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

The explicit formulae given by Rosen55 are also of value. They are derived from a model consisting of a random assemblage of composite cylinders (Hashin and Rosen56 ) and expressed in terms of the axial Young modulus E, the Poisson ratio for uniaxial stress in the fibre direction v, the transverse plane strain bulk modulus k, the axial shear modulus G and the transverse shear modulus G. ... 

These results of Walpole61 include as special cases those of Hill47 and of Hashin and Shtrikman48. For anisotropic phases Walpole58 gives bounds on the five elastic moduli of an aligned array of transversely isotropic elements and for randomly oriented fibrous inclusions in an isotropic matrix. For the former case (alignment) the bounds are expressed in terms of phase concentration q and the quantities k, 1, m, n, p defined as follows k - 1/2 (Cjj + C, m — 1/2 (Cn — C22), = C13, n = C33, p = C44 = C55. 

Gray and McCrum735 used the Hashin-Shtrikman theory to explain the origin of the y relaxation in PE and PTFE, Maeda et al.745 have given exact analyses of several two phase models for semi-crystalline polymers and Buckley755 represented a biaxially oriented sheet of linear polyethylene by a two phase composite model. 

The foregoing summary of applications of composites theory to polymers does not claim to be complete. There are many instances in the literature of the use of bounds, either the Voigt and Reuss or the Hashin-Shtrikman, of simplified schemes such as the Halpin-Tsai formulation84, of simple models such as the shear lag or the two phase block and of the well-known Takayanagi models. The points we wish to emphasize are as follows. 

M. Arcan, Z. Hashin, and A. Voloshin, A Method to Produce Uniform Plane-Stress States with Applications to Fiber-Reinforced Materials, Exp. Mech., 18[4], 141-146 (1978). 

The Kerner equation, a three phase model, is applicable to more than one type of inclusion, Honig (14,15) has extended the Hashin composite spheres model to include more than one inclusion type. Starting with a dynamic theory and going to the quasi-static limit, Chaban ( 6) obtains for elastic inclusions in an elastic material... 

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