Failure trajectories, probable failure zones, and damage indices are some of the key quantities of relevance in brittle fracture mechanics . High-fidelity numerical solvers that reliably estimate these relevant quantities exist but they are computationally demanding requiring a high resolution of the crack. Moreover, independent simulations need to be carried out even for a small change in domain parameters and/or material properties. Therefore, fast and generalizable surrogate models are needed to alleviate the computational burden but the discontinuous and complex nature of fracture mechanics presents a major challenge to developing such models. We propose a physics-informed variational formulation of DeepONet (V-DeepONet) for brittle fracture analysis. V-DeepONet is trained to map the initial configuration of the defect to the relevant fields of interests (e.g., damage and displacements). Once the network is trained, the entire global solution can be rapidly obtained for any initial crack configuration and loading steps on that domain. While the original DeepONet is solely data-driven, we take a different path to train the V-DeepONet by imposing the governing equations in a variational form with some labeled data. We demonstrate the effectiveness of V-DeepOnet through two benchmarks of brittle fracture and verify its accuracy using results from high-fidelity solvers. Encoding the physical laws to the model with data enhancement in training renders the surrogate model capable of accurately performing both interpolation and extrapolation tasks. Considering that fracture modeling is very sensitive to fluctuations, the proposed V-DeepONet with a hybrid training strategy is able to predict the quantities of interests with good accuracy, which can be easily extended to a wide array of dynamical systems with complex responses.