This contribution presents a data-driven modelling framework with a physics-inspired structure for representing the nonlinear behaviour of path-dependent materials in structural components subjected to seismic loading. Gated recurrent units (GRUs) are employed to implicitly capture the material's internal state variables through a hidden state, while a fully connected (FC) layer maps this state to the material response in terms of stress. The incremental formulation and appropriate dimensional consistency are explicitly considered in both the training and compiling frameworks. The proposed material model is trained using a limited experimental dataset obtained from earthquake inspired cyclic tests on high-strength steel reinforcing bars. To enhance robustness and extrapolation capability, a few physics-based constraints are incorporated into the loss function during training. Lastly, the trained model is compiled within the open-source Open System for Earthquake Engineering Simulation (OpenSees) framework. The extrapolation performance of the proposed data-driven material model is evaluated and compared with that of conventional phenomenological material models under cyclic loading paths not included in the training data. Its performance is also evaluated in a numerical simulation reproducing a full-scale shake-table test of a reinforced concrete bridge column under seismic loading. The results demonstrate that the physics-inspired GRU-FC model can predict and extrapolate under unseen cyclic loading conditions at the material-scale simulation level. However, the model demonstrated limited extrapolation capability and less accurate tangent stiffness when integrated into full-scale structural simulations under complex seismic loading, an unaddressed challenge.