Figure 4:

Simulations on matrix models containing two contracting inclusions. (a) A representative model containing two inclusions of diameter D = 2L_f, separated by inter-inclusion spacing d. The central dipole of length 2L_f is transverse to the imaginary line joining the centers of the inclusions. The inclusions were radially contracted by 40%. (b) Enlarged view of the central dipole which samples fibers passing through the circular regions of radius 0.25L_f centered at the dipole’s end points. (c) For four independent matrices, the ratio of the values of matrix stiffness sensed by the dipole after and before the contraction of inclusions (k_s/k_s,ref) were recorded, with k_s/k_s,ref < 1 indicating softening. Each marker color indicates an independent model. (d) A representative deformed state of the matrix with d = 8.5D, where the force chains consisting of top 5% most stretched fibers are shown in red. (e) The deformed matrix of panel d, where the top 5% most buckled fibers are shown blue. (f) The dipole of the system described in panels d and e. The matrix fibers in its microenvironment are colored based on the values of their excess lengths. (g, h, i) A representative deformed state of the matrix with d = 4.5D, where the force chains, buckled fibers, and dipole are shown, similar to panels d–f. (j, k, l) A representative deformed state of the matrix with d = 0.5D, where the force chains, buckled fibers, and dipole are shown, similar to panels d–f. (m) Plot showing the dependence of the average excess length of fibers in the microenvironment of the dipole to the model parameter d/D. (n) The normalized matrix stiffness in the vicinity of the transverse dipole (k_s/k_s,ref < 1) was negatively correlated to the average excess length of fibers in the microenvironment of the dipole. (o) The normalized matrix stiffness (k_s/k_s,ref) against the model parameter d/D. Panels a–o considered the transversely aligned dipole of panel a. (p) The matrix stiffness in the direction longitudinal to the imaginary center-line of the inclusions (inset) was plotted against the model parameter d/D. More than half (18 out of 28 instances, 95% confidence interval [13, 23]) of the data points exhibited k_s/k_s,ref < 1, indicating softening. In panels m–p, each marker color indicates an independent model. In panels c & n–p, the black dashed line at k_s/k_s,ref = 1 indicates no matrix softening.