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 MN 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 φ. The affine MA is determined essentially by the static structural arrangement of particles, whereas the nonaffine MN is related to the vibrational eigenmodes. We elucidate the contribution of each vibrational mode to the nonaffine MN through a modal decomposition of the displacement and force fields. In the vicinity of the (un)jamming transition φc, the vibrational density of states g(ω) shows a plateau in the intermediate-frequency regime above a characteristic frequency ω∗. We illustrate that this unusual feature apparent in g(ω) is reflected in the behavior of MN: As φ→φc, where ω∗→0, those modes for ω<ω∗ contribute less and less, while contributions from those for ω>ω∗ approach a constant value which results in MN to approach a critical value MNc, as MN-MNc∼ω∗. At φc itself, the bulk modulus attains a finite value Kc=KAc-KNc>0, such that KNc has a value that remains below KAc. In contrast, for the critical shear modulus Gc, GNc and GAc approach the same value so that the total value becomes exactly zero, Gc=GAc-GNc=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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics