We study elastic rectifying strips constructed from timelike curves constituting one of the causal characters of a curve in Minkowski 3-space. Even if elastic surfaces correspond to critical points of Willmore functional, we instead find extremals of the Sadowsky functional, because Willmore functional is proportional to the Sadowsky functional for rectifying strips. We then provide a characterization of timelike critical points of Sadowsky functional with two Euler-Lagrange equations. When we choose different variations, we derive two conservation laws, and by using these rules, we introduce two new kinds of elastic strips with timelike directrix (the base curve). We next establish a relation between elastic strips with timelike directrix and spacelike elastic curves on de Sitter 2-space and pseudohyperbolic 2-space. Finally, we verify that the semi-Riemannian Hopf cylinder associated to the tangent imagine of the timelike curve defining a force-free strip with timelike directrix which is one of the new types elastic strips gives rise to a Willmore surface in anti de Sitter 3-space.