Machine-learning models have shown their promising potential in the field of computational mechanics, but still face several shortcomings that prevent them from becoming standalone solvers. In view of the above challenges we are proposing I-FENN, a hybrid framework where Neural Networks (NNs) are directly deployed within the Finite Element Method (FEM) as approximators of the solution of governing PDEs. The key idea of I-FENN is to decompose the problem such that a pre-trained NN predicts one physical state variable, and then the prediction is integrated within a generic FEM solver to compute the remaining state variables like a typical user-defined material model. This iterative process is repeated until the system residuals have converged, abiding therefore with the long-standing practices in the field and ensuring the accuracy of the numerical solution. To implement the framework, a) we first adopt a Temporal Convolutional Network (TCN) to capture the history dependence of nonlocal strain in a coarsely meshed domain and b) we integrate the trained TCN within the nonlinear solver using either a full or a modified Newton-Raphson scheme to analyze a fine mesh idealization of the investigated topology. We note that very strict convergence criteria are satisfied across all the load increments, and we successfully simulate the entire load history analysis. Our results demonstrate computational savings in the order of 30% - 60% compared to conventional FEM (monolithic or staggered schemes), depending on the problem under consideration.