The investigation of discontinuous structures, including heterogeneities and crack growth with applications ranging from biological tissues over lightweight design to fiber-reinforced industrial structures with a focus on safety [1, 2], faces numerical challenges due to discontinuities and stress concentrations caused by substantial contrasts in stiffness. Physics-Informed Machine Learning [3] methods, such as the deep energy method (DEM) [4] have been shown to be highly versatile approaches with very promising results for nonlinear elastostatic problems [5, 6]. However, existing suggestions for the resolution of stress concentrations tend to require high computational cost and exhibit a behavior similar to overfitting. The current contribution addresses these gaps with mesh-based neural energy methods (M-NEM). Similar to the (DEM), a parametrized function approximator is trained to approximate the solution displacement field. For the analysis of cracks, additionally the damage field is trained as an output. The loss function is based on the minimum total potential energy principle and integrates Neumann boundary conditions. In contrast to most physics-informed neural network (PINN) setups, that represent collocation methods, the M-NEM discretizes the domain into finite elements applies algebraic derivatives of shape functions and Gauss quadrature. The current contribution compares multiple variants of function approximators, specifically multi-layer-perceptrons (MLP), Kolmogorov-Arnold-Networks (KAN) and radial basis function neural networks (RBFNN). In multiple case studies of standard crack tests, calcified hydrogels and the microstructure of fiber-reinforced polymers akin to those used in wind turbine blades, the M-NEM methods prove notable gains of accuracy and performance over the DEM, ranging up to one order of magnitude in computation time and three orders of accuracy. This leads to an accuracy comparable to the FEM while at the same time providing a significantly higher robustness against mesh density and time step width for crack propagation analysis. The best results are obtained with the RBFNN, suggesting the M-NEM with this combination as a suitable and robust technqiue for the analysis of discontinuous structures.