We present new nonlinear, time-dependent convective hydrodynamical models of RR Lyrae stars computed assuming a constant helium-to-metal enrichment ratio and a broad range in metal abundances (Z=0.0001--0.02). The stellar masses and luminosities adopted to construct the pulsation models were fixed according to detailed central He burning Horizontal Branch evolutionary models. The pulsation models cover a broad range in stellar luminosity and effective temperatures and the modal stability is investigated for both fundamental and first overtones. We predict the topology of the instability strip as a function of the metal content and new analytical relations for the edges of the instability strip in the observational plane. Moreover, a new analytical relation to constrain the pulsation mass of double pulsators as a function of the period ratio and the metal content is provided. We derive new Period-Radius-Metallicity relations for fundamental and first-overtone pulsators. They agree quite well with similar empirical and theoretical relations in the literature. From the predicted bolometric light curves, transformed into optical (UBVRI) and near-infrared (JHK) bands, we compute the intensity-averaged mean magnitudes along the entire pulsation cycle and, in turn, new and homogenous metal-dependent (RIJHK) Period-Luminosity relations. Moreover, we compute new dual and triple band optical, optical--NIR and NIR Period-Wesenheit-Metallicity relations. Interestingly, we find that the optical Period-W(V,B-V) is independent of the metal content and that the accuracy of individual distances is a balance between the adopted diagnostics and the precision of photometric and spectroscopic datasets.
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