An eigenstrain formulation for the prediction of elastic moduli of defective fiber networks

A method for predicting the elastic moduli of a regular network populated by a large number of randomly located defects is presented. The prediction is based exclusively on the stiffness of individual fibers and the location of defects. The method requires a preliminary calibration step in which the eigenstrains associated with “elementary defects” of the regular network are fully characterized. Each type of defect is represented by a superposition of singular point sources in 2D elastostatics producing a field identical to the eigenstrain of the respective defect. The amplitude of the point sources is determined by probing the eigenstrain with a series of path independent integrals. This “spectral decomposition” represents the generalization that allows applying methods developed to account for crack–crack interaction in fracture mechanics to situations in which the interacting sources have eigenstrains obtained by the superposition of multiple types of singularities. Once the representation of each elementary defect is determined, any distribution of defects in the network can be mapped into a distribution of point sources in an equivalent continuum. This allows inferring the elastic behavior of a defective network of any distribution and concentration of defects. The method discussed here provides an efficient way to treat the non-affine deformation of defective regular fiber networks.