Using the observation data of SNeIa, CMB and BAO, we establish two concrete f (T ) models with nonminimal torsion-matter coupling extension. We study in detail the cosmological implication of our models and find they are successful in describing the observation of the Universe, its large scale structure and evolution. In other words, these models do not change the successful aspects of ΛCDM scenario under the error band of fitting values as describing the evolution history of the Universe including radiation-dominated era, matter-dominated era and the present accelerating expansion. Meanwhile, the significant advantage of these models is that they could avoid the cosmological constant problem of ΛCDM. A joint analysis is performed by using the data of CMB+BAO+JLA, which leads to Ωm0 = 0.255 ± 0.010, Ω b0 h 2 = 0.0221 ± 0.0003 and H0 = 68.54 ± 1.27 for model I and Ωm0 = 0.306 ± 0.010, Ω b0 h 2 = 0.0225 ± 0.0003 and H0 = 60.97 ± 0.44 for model II at 1σ confidence level. The evolution of the decelaration parameter q(a) and the effective equation of state wDE(a) are displayed. Furthermore, The resulted age of the Universe from our models is consistent with the ages of the oldest globular clusters. As for the fate of the Universe, model I results in a de Sitter accelerating phase while model II appears a power-law one, even though wDE0 < −1 makes model I look like a phantom at present time.PACS numbers: 04.50. Kd, 95.36.+x arXiv:1511.07935v1 [gr-qc]