Properties for Design of Composite Structures. Neil McCartney

that are consistent with isotropic properties.

      3.4 Shear Modulus

      3.4.1 Spherical Particle Embedded in Infinite Matrix Material Subject to Pure Shear Loading

      For a state of pure shear, and in the absence of thermal effects, the displacement field of a homogeneous sample of material referred to a set of Cartesian coordinates (x1,x2,x3) has the form

      (3.31)

      and the corresponding strain and stress components are given by

      (3.32)

      (3.33)

      The parameters γ and τ are the shear strain (half the engineering shear strain) and shear stress, respectively, such that τ=2μγ where μ is the shear modulus of an isotropic material. The principal values of the stress field are along (tension) and perpendicular to (compression) the line x2=x1.

      A single spherical particle of radius a is now placed in, and perfectly bonded to, an infinite matrix, where the origin of spherical polar coordinates (r, θ, ϕ) is taken at the centre of the particle. The system is then subject only to a shear stress applied at infinity. At the particle/matrix interface the following perfect bonding boundary conditions must be satisfied:

(3.34)

      A displacement field equivalent to that used by Hashin [5], based on the analysis of Love [8, Equations (5)–(7)] that leads to a stress field satisfying the equilibrium equations and the stress-strain relations (3.15) with ΔT=0, can be used to solve the embedded isolated sphere problem (see Appendix A). The displacement and stress fields in the particle are bounded at r = 0 so that

(3.35)

(3.36)

      In the matrix the displacement field and stress field (stresses bounded as r→∞) have the form

(3.37)