Conewise linear elastic (CLE) materials are proposed as the proper generalization to two and three dimensions of one-dimensional bimodular models. The basic elements of classical smooth elasticity are extended to nonsmooth (or piecewise smooth) elasticity. Firstly, a necessary and sufficient condition for a stress-strain law to be continous across the interface of the tension and compression subdomains is established. Secondly, a sufficient condition for the strain energy function to be strictly convex is derived. Thirdly, the representations of the energy function, stress-strain law and elasticity tensor are obtained for orthotropic, transverse isotropic and isotropic CLE materials. Finally, the previous results are specialized to a piecewise linear stress-strain law and it is found out that the pieces must be polyhedral convex cones, thus the CLE name.