In a concurrent multiscale (FE2) modeling approach the complex microstructure of composite materials is explicitly modeled on a finer scale and nested to each integration point of the macroscale. However, such generality is often associated with exceedingly high computational costs in real-scale applications. In this work, a novel Neural Network (NN) is used as the constitutive model for the microscale to tackle that issue. Unlike conventional NNs, the proposed network employs the actual material models used in the full-order micromodel as the activation function of one of the layers. The NN's capabilities are assessed (i) for a single micromodel level, where its performance is compared to that of a Recurrent Neural Network (RNN), and (ii) for an FE2 example. A highlight of the proposed network is the ability to predict unloading/reloading behavior without ever seeing it during training, a stark contrast with highly popular but data-hungry models such as RNNs.