Statistical physics of interacting dislocation loops and their effect on the elastic moduli of isotropic solids

We present a field-theoretical study of interacting dislocation loops and their effect on the elastic moduli of isotropic three-dimensional solids. We find that the shear modulus decreases to a finite limit with increasing density and size of dislocation loops and vanishes only with the appearance of open dislocation lines that terminate on the boundaries of the sample. Using the random phase approximation, we analyze the correlations of dislocation “charges” and “currents” and show that interaction between dislocations leads to screening of long-range correlations. Variational and perturbative methods are used to show that fluctuation-induced attraction between segments of dislocation loops leads to the shrinking of the radii of gyration of loops compared to their Gaussian dimensions. The applicability of our model assumptions to real solids is discussed.