In order to describe the ultra-slow mechanical behaviors of viscoelastic materials, a constitutive model based on the local structural derivatives is proposed in this paper, in which ln^α(1+t/τ₀)is the structural function. The new model has clear physical meaning and can accurately describe the very slow mechanical behaviors of viscoelastic materials, which is logarithmic-law dependent. Both of the traditional integer order and the fractal derivative constitutive models are not capable of quantifying the ultra-slow mechanical behaviors of viscoelastic materials. Currently, the Lomnitz model is commonly used to study the ultra-slow mechanical behaviors of viscoelastic materials, but it is an empirical model with unclear physical meaning. The creep experiment data of concrete under different experimental conditions were employed to compare the four models in analyzing the ultra-slow mechanical behaviors. The fitting results show that the local structural derivative model is feasible and effective in describing ultra-slow creep of the concrete. To give the applicable scope of different models, this paper also provides the local structural derivative Kelvin model to describe ultra-slow creep, which compared with the above mentioned local structural derivative Maxwell constitutive model.