Phase-field models for the quasi-static simulation of brittle fracture where the crack is approximated by a damage phase-field are limited by the necessary memory and computation time. In this contribution, we study the applicability of low-rank methods to phase-field fracture models, specifically the tensor train (TT) format. To this end, we investigate the low-rank structure of the crack phase-field. Additionally, we present an implementation of an alternating minimization scheme to solve the coupled displacement and damage problem in the TT format. We show the evolution of the TT ranks of the displacement and damage fields for a specific example
