Stress chain analysis

Stress chain analysis is a powerful tool to reveal the non-continuum features of granular dynamics. Stress chains are seen in laboratory experiments with photo-elastic granular materials and are generally thought to linkage the most compressive principal stress vectors of each particle that have magnitudes larger than the average value. Although stress chains are seen in the large principal stress vectors, such images are too complex to analyze because most of the large stress vectors are not part of the stress chains. Thus, identifying stress chains is important for analyzing the complex 3D dynamics of granular materials. We quantitatively identify the stress chains via the method proposed by¹⁹. Thus, we first calculate the most compressive principal stress vector (σ₃) from the particle stress tensor as follows:

where 1 < c < n_c denotes all particle pairs in contact, V is the volume of a particle, f_I is the I-th component of the force between two contacting particles, and r_J is the J-th component of the radius vector from the center of the particle to the point of contact with the paired particle. Tension is positive value in Eq. (¹³); thus, the most compressive vector is the minor principal stress σ₃. To form the stress chains, i-th particles should have |σ3i| greater than the average and should contact j-th particles. In addition, the stress vectors of the i-th and j-th particles (i.e. σ3i and σ3j) should be aligned in the relative position vector δⁿ_ij within a 45° angle. Finally, we count the length of the strongly connected particles and identify the stress chains that are longer than a given number of particles, which is four in this case.