For a univalent smooth mapping f of the unit disk D of complex plane onto the manifold f (D), let d f (z 0 ) be the radius of the largest univalent disk on the manifold f (D) centered at f (z 0 ) (|z 0 | < 1). The main aim of the present article is to investigate how the radius d h (z 0 ) varies when the analytic function h is replaced by a sense-preserving harmonic function f = h + g. The main result includes sharp upper and lower bounds for the quotient d f (z 0 )/d h (z 0 ), especially, for a family of locally univalent Q-quasiconformal harmonic mappings f = h + g on |z| < 1. In addition, estimate on the radius of the disk of convexity of functions belonging to certain linear invariant families of locally univalent Q-quasiconformal harmonic mappings of order α is obtained.2000 Mathematics Subject Classification. Primary: 30C62, 31A05; Secondary: 30C45,30C75. Key words and phrases. Locally univalent harmonic mappings, linear and affine invariant families, convex and close-to-convex functions, and covering theorems.