In this talk, we study the rigidity properties of a differential inclusion in linearized elasticity. We will introduce and discuss a set K of three diagonal strains that are pairwise incompatible but with non-trivial (symmetrized) rank-1-convex hull. This is called a T3 structure. We first prove that K is rigid at the level of exact solutions, namely that Lipschitz maps whose gradient is locally in K must be affine. After this, we study the scaling behaviour of the corresponding singularly-perturbed elastic energy, giving quantitative information on the flexibility of K in the sense of approximate solutions. This is based on a joint work with R. Indergand, D. Kochmann, A. Rüland, and C. Zillinger.