Mechanical response of packings of nonspherical particles: A case study of two-dimensional packings of circulo-lines

Designing the pressure-dependent shear modulus using tessellated granular metamaterials

Jammed packings of granular materials display complex mechanical response. For example, the ensemble-averaged shear modulus 〈G〉 increases as a power law in pressure p for static packings of soft spherical particles that can rearrange during compression. We seek to design granular materials with shear moduli that can either increase or decrease with pressure without particle rearrangements even in the large-system limit. To do this, we construct tessellated granular metamaterials by joining multiple particle-filled cells together. We focus on cells that contain a small number of bidisperse disks in two dimensions. We first study the mechanical properties of individual disk-filled cells with three types of boundaries: periodic boundary conditions (PBC), fixed-length walls (FXW), and flexible walls (FLW). Hypostatic jammed packings are found for cells with FLW, but not in cells with PBC and FXW, and they are stabilized by quartic modes of the dynamical matrix. The shear modulus of a single cell depends linearly on p. We find that the slope of the shear modulus with pressure λ_{c}<0 for all packings in single cells with PBC where the number of particles per cell N≥6. In contrast, single cells with FXW and FLW can possess λ_{c}>0, as well as λ_{c}<0, for N≤16. We show that we can force the mechanical properties of multicell granular metamaterials to possess those of single cells by constraining the end points of the outer walls and enforcing an affine shear response. These studies demonstrate that tessellated granular metamaterials provide a platform for the design of soft materials with specified mechanical properties.

The structural, vibrational, and mechanical properties of jammed packings of deformable particles in three dimensions

We investigate the structural, vibrational, and mechanical properties of jammed packings of deformable particles with shape degrees of freedom in three dimensions (3D). Each 3D deformable particle is modeled as a surface-triangulated polyhedron, with spherical vertices whose positions are determined by a shape-energy function with terms that constrain the particle surface area, volume, and curvature, and prevent interparticle overlap. We show that jammed packings of deformable particles without bending energy possess low-frequency, quartic vibrational modes, whose number decreases with increasing asphericity and matches the number of missing contacts relative to the isostatic value. In contrast, jammed packings of deformable particles with non-zero bending energy are isostatic in 3D, with no quartic modes. We find that the contributions to the eigenmodes of the dynamical matrix from the shape degrees of freedom are significant over the full range of frequency and shape parameters for particles with zero bending energy. We further show that the ensemble-averaged shear modulus 〈G〉 scales with pressure P as 〈G〉 ∼ Pβ, with β ≈ 0.75 for jammed packings of deformable particles with zero bending energy. In contrast, β ≈ 0.5 for packings of deformable particles with non-zero bending energy, which matches the value for jammed packings of soft, spherical particles with fixed shape. These studies underscore the importance of incorporating particle deformability and shape change when modeling the properties of jammed soft materials.