Origins of the Poynting effect in elastic networks

Poynting observed that some elastic networks expand when sheared, while others contract. We study the origins of this nonlinear effect in random spring networks, a minimal model for biopolymer gels, foams, and other mechanical networks. We show theoretically and verify numerically that the sign of the Poynting effect is predicted by the microscopic Grüneisen parameter, which measures how stretching the system shifts its eigenfrequencies. Spring networks contract under shear (a negative Poynting effect) because their eigenfrequencies shift upwards. We find that the amplitude of the Poynting effect is sensitive to the network's preparation protocol, and it diverges at the isostatic point. Finally, we illustrate correlations between the Poynting effect and the manner in which a network stiffens under strain.