Isolating the Effect of Pre-strain in Disordered Solids

The energy landscapes of disordered solids yield many different equilibrium particle configurations that are at a non-zero energy minimum. In those cases, the individual inter-particle forces need not be zero, but they are still required to be in force balance. The presence of such forces is due to the pre-strain on each bond. These forces have often been neglected when analyzing the elastic behavior of solids formed from disordered networks. Here, we show that, independent of configuration, changing the magnitude of the pre-strain changes the mechanics of the system in a characteristic manner: the modulus remains constant as a function of pre-strain out to a crossover value, determined by the coordination number, Z, and then varies approximately as the square root of the pre-strain. We present an argument for the value of the crossover pre-strain in terms of Z. The variation with the square root of the pre-strain can be traced to the fact that the disorder causes the average over all the network bonds to sample a wide range of bond angles. Given that disordered solids are used to study material trainability, we have used our analysis to understand the limits of training in materials with pre-strain.