Modelling heterogeneities in the cross-linked bacterial sacculus

Examining the design principles of biological materials, in particular the presence of inhomogeneities in their ultrastructure is the key to understanding the often remarkable mechanical properties possessed by them. In this work, motivated by the question of understanding the effect of variability in the material properties of the peptide cross-linkers on the bulk mechanical properties of the cell wall structure of bacteria, we study a spring system in which variability is encoded by assigning values of spring constants and rupture strengths of the constituent springs from appropriate probability distribution. Using analytical methods and computer simulations, we study the response of the spring system to shear loading and observe how heterogeneities inherent in the system can heighten the resistance to failure. We derive the force extension relation of the system and explore the effect that the disorder in values of spring constant and rupture strength has on load carrying capacity of the system and failure displacement. We also study a discrete step shear loading of the system, exhibiting a transition from quasi-brittle to brittle response controlled by the step size, providing possible framework to experimentally quantify the disorder in analogous structures. The model studied here will also be useful in general to understand fiber bundles exhibiting disorder in the elasticity and rupture strengths of constituent fibers.