This paper proposes a constitutive model for polycrystalline shape memory alloy (SMA) wires arising from micromechanical arguments. The texture of the polycrystal is captured through the volume fractions and the maximal transformation strain in each crystalline orientation. As a result, the model is able to reproduce texture effects such as nonlinear hardening during phase transformation. An attractive feature of the proposed model is that closed-form expressions of the material response can be obtained for typical thermomechanical loadings of interest in SMA, such as cyclic traction at high temperature or thermal cycling at a fixed stress. Those analytical solutions are notably useful for identifying the constitutive parameters of the model. A temperature-controlled testing apparatus for SMA wires was developed for performing a reliable characterization of Nickel-Titanium wires. All model parameters have been identified by means of three tests: differential scanning calorimetry, isothermal traction test and thermal cycling at constant stress.