Elastic moduli and vibrational modes in jammed particulate packings
H Mizuno and K Saitoh and LE Silbert, PHYSICAL REVIEW E, 93, 062905 (2016).
When we elastically impose a homogeneous, affine deformation on amorphous solids, they also undergo an inhomogeneous, nonaffine deformation, which can have a crucial impact on the overall elastic response. To correctly understand the elastic modulus M, it is therefore necessary to take into account not only the affine modulus MA, but also the nonaffine modulus M-N that arises from the nonaffine deformation. In the present work, we study the bulk (M = K) and shear (M = G) moduli in static jammed particulate packings over a range of packing fractions phi. The affine M-A is determined essentially by the static structural arrangement of particles, whereas the nonaffine M-N is related to the vibrational eigenmodes. We elucidate the contribution of each vibrational mode to the nonaffine M-N through a modal decomposition of the displacement and force fields. In the vicinity of the (un)jamming transition phi(c), the vibrational density of states g(omega) shows a plateau in the intermediate-frequency regime above a characteristic frequency omega*. We illustrate that this unusual feature apparent in g(omega) is reflected in the behavior of M-N: As phi -> phi(c), where omega* -> 0, those modes for omega < omega* contribute less and less, while contributions from those for omega > omega* approach a constant value which results in MN to approach a critical value M-Nc, as M-N - M-Nc similar to omega*. At phi(c) itself, the bulk modulus attains a finite value K-c = K-Ac - K-Nc > 0, such that K-Nc has a value that remains below K-Ac. In contrast, for the critical shear modulus G(c), G(Nc) and G(Ac) approach the same value so that the total value becomes exactly zero, G(c) = G(Ac) - G(Nc) = 0. We explore what features of the configurational and vibrational properties cause such a distinction between K and G, allowing us to validate analytical expressions for their critical values.
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