Physics-Informed Neural Networks (PINNs) have been introduced as a promising method that can combine partial differential equations (PDEs) and measurement data in the loss function of the neural network. Collocation-based PINN methods incorporate the strong form of the governing PDE into the loss function, while variational PINNs (VPINNs) integrate residuals into the loss function using the weak formulation. Particularly for nonlinear problems, VPINNs demonstrate significantly higher accuracy compared to PINNs [1]. In this contribution, we extend the concept of VPINNs to solve problems in elasticity and contact mechanics. We deploy VPINNs in both the standard displacement-based and mixed formulation, which computes displacements and stresses as network output. Modified Legendre polynomials are chosen as test functions of the weak form. Two Gaussian quadrature approaches are used as numerical integration schemes, namely the Gauss-Legendre and Gauss-Labotto rules. Furthermore, a small hyperparameter optimization study is conducted to analyze the effect of different weak form residuals, the number of test functions, the number of Gauss points, and the number of elements. Boundary conditions of the investigated problems are enforced in two ways: as so-called soft constraints and hard constraints. Enforcing soft constraints is achieved by adding specific loss contributions to the overall loss function, while enforcing the hard constraints is accomplished by choosing an augmented network output that fulfills the boundary conditions directly. Since modified test functions are zero on boundaries, the traction term of the weak formulation vanishes. Thus, Neumann boundary conditions are enforced as both soft and hard constraints. Specific examples include forward and inverse problems of elasticity and contact mechanics, e.g. the Lam´e problem of elasticity, and the non-smooth contact between an elastic block and a rigid surface [2].