Energy distribution in disordered elastic networks

Disordered networks are found in many natural and artificial materials, from gels or cytoskeletal structures to metallic foams or bones. Here, the energy distribution in this type of networks is modeled, taking into account the orientation of the struts. A correlation between the orientation and the energy per unit volume is found and described as a function of the connectivity in the network and the relative bending stiffness of the struts. If one or both parameters have relatively large values, the struts aligned in the loading direction present the highest values of energy. On the contrary, if these have relatively small values, the highest values of energy can be reached in the struts oriented transversally. This result allows explaining in a simple way remodeling processes in biological materials, for example, the remodeling of trabecular bone and the reorganization in the cytoskeleton. Additionally, the correlation between the orientation, the affinity, and the bending-stretching ratio in the network is discussed.

Figure 1

(Color online) Example of a network of connectivity $C=4$, with $N=200$ struts and $r/L_m=0.02$: (a) unloaded, (b) under a stretching strain $\epsilon =0.01$, and (c) under $\epsilon =0.10$. The calibration bar indicates the elastic energy of the struts per unit volume $w_{ij}$. (d) Sketch of a deformed strut between two generic nodes $i$ and $j$, indicating the displacement and the rotation angle of the nodes.

Figure 2

(Color online) Average elastic energy versus orientation of the struts at stretching strain $\epsilon =0.10$ for the three connectivities ($C=3$, 4, and 6) and two values of relative bending stiffness ($r/L_m=0.170$ and 0.006). The elastic energy is expressed as elastic energy of the struts per unit volume $w_{ij}$ divided by the mean value $w_m$. The struts are divided in ten groups between orientation $0°$ (struts parallel to the $x$ axis) and $90°$ (struts perpendicular to the $x$ axis, i.e., aligned to the loading direction). Error bars indicate standard deviation.

Figure 3

(Color online) General dependence between the dimensionless elastic energy per unit volume $w_{ij}/w_m$ and the orientation of the struts for the three cases of connectivity and the two cases of stretching strain. This is estimated by the slope $S$ of the linear regressions considering the orientation angle in radians. $S\ne 0$ means that the expected elastic energy for a strut depends on its orientation. Mean value and standard deviation computed for the four networks calculated at each condition.

Figure 4

(Color online) (a) Ratio of stretching or compression energy, $W_{str\text{−}com}/W_{tot}$, as a function of the relative bending stiffness, $r/L_m$. (b) Standard correlation coefficient between the computed relative displacement and the calculated affine relative displacement $\Delta r_{kl}$ of every two points as a function of $r/L_m$. It becomes equal to 1 when the deformation is affine. Mean value and standard deviation computed for the four networks calculated at each condition.