Elastic fields in a random pack of rigid bonded spheres

A solution technique is presented for determining the small-deformation elastic fields (displacement vector, strain tensor, and stress tensor) at all points in a matrix material filled with an arbitrary dispersion of rigid spherical particles bonded to the matrix. The matrix is treated as a linear, homogeneous, isotropic material whose two material constants are arbitrary. With knowledge of the detailed fields among the particles, we can deduce composite mechanical properties and estimate failure for arbitrary particle-size distributions. The three-dimensional particle packs containing arbitrary size distributions are created using the reduced-dimension packing algorithms. Typically, packs containing several thousand particles are used although much larger packs are readily created. Interactions of the particles with each other and with an externally applied uniform stress field are described with a multipole expansion of the Navier equation centered on each particle. For a given region of composite material, the local elastic fields, as well as particle displacements and rotations, are obtained by summing over the multipoles of all spheres near that region. In a densely packed composite, particles strongly interact with each other. To obtain the myriads of multipole moments describing these interactions, we iterate among the multipole expansions about all spheres. For a given sphere, the multipole moments can be obtained one by one due to the orthogonality of the Navier multipoles, thereby greatly reducing the computational intensity. We perform simulations on various random and ordered packs and compare with experimental data and other theories. Lastly, we discuss extensions of this solution technique that should enhance its speed and applicability.