Geometry and the onset of rigidity in a disordered network

Disordered spring networks that are undercoordinated may abruptly rigidify when sufficient strain is applied. Since the deformation in response to applied strain does not change the generic quantifiers of network architecture - the number of nodes and the number of bonds between them - this rigidity transition must have a geometric origin. Naive, degree-of-freedom based mechanical analyses such as the Maxwell-Calladine count or the pebble game algorithm overlook such geometric rigidity transitions and offer no means of predicting or characterizing them. We apply tools that were developed for the topological analysis of zero modes and states of self-stress on regular lattices to two-dimensional random spring networks, and demonstrate that the onset of rigidity, at a finite simple shear strain $\gamma^\star$, coincides with the appearance of a single state of self stress, accompanied by a single floppy mode. The process conserves the topologically invariant difference between the number of zero modes and the number of states of self stress, but imparts a finite shear modulus to the spring network. Beyond the critical shear, we confirm previously reported critical scaling of the modulus. In the sub-critical regime, a singular value decomposition of the network's compatibility matrix foreshadows the onset of rigidity by way of a continuously vanishing singular value corresponding to nascent state of self stress.Comment: 6 pages, 6 figue

Self-stresses control stiffness and stability in overconstrained disordered networks

We investigate the interplay between pre-stress and mechanical properties in random elastic networks. To do this in a controlled fashion, we introduce an algorithm for creating random freestanding frames that support exactly one state of self stress. By multiplying all the bond tensions in this state of self stress by the same number---which with the appropriate normalization corresponds to the physical pre-stress inside the frame---we systematically evaluate the linear mechanical response of the frame as a function of pre-stress. After proving that the mechanical moduli of affinely deforming frames are rigourously independent of pre-stress, we turn to non-affinely deforming frames. In such frames, pre-stress has a profound effect on linear response: not only can it change the values of the linear modulus---an effect we demonstrate to be related to a suppressive effect of pre-stress on non-affinity---but pre-stresses also generically trigger bistable mechanical response. Thus, pre-stress can be leveraged to both augment the mechanical response of network architectures on the fly, and to actuate finite deformations. These control modalities may be of use in the design of both novel responsive materials and soft actuators.Comment: 13 pages, 9 figure