Machine learning and deep learning approaches usually require a large amount of simulation or experimental data, which might be hard to obtain due to the complexity of simulations and the high costs of experiments. On the other hand, data scarcity often causes data-driven approaches to fail to deliver adequate accuracy. Thus, data-driven methods with noisy real-world observations (sensor or experimental data) or wrongly labeled datasets are prone to mostly making wrong predictions as there is no physics-based control mechanism to validate the predictions. To compensate for the lack of adequate data and unknown black-box behavior of the data-driven approaches, physics-informed neural networks (PINNs) have been proposed. PINNs combine differential equations and measurement data in the loss function of the neural network. Based on sensor data of a physical object, PINNs can be used in hybrid digital twins of civil engineering structures. In this contribution, we extend the concept of physics-informed neural networks to solve problems in solid and contact mechanics. We deploy PINNs in the mixed variable formulation which computes both displacements and stresses as network output so that only first-order derivatives of the network input are required. 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. The mixed variable formulation enables also enforcing stress boundary conditions as hard constraints. In comparison to pure solid mechanics, contact problems involve additional constraints. Concretely, a set of Karush-Kuhn-Tucker inequality conditions can be enforced using the sign function to distinguish points that are in the active or inactive contact regions. As an alternative, we explore an adopted Sigmoid function with finite gradients. Moreover, a so-called complimentary function that reduces inequality constraints into single equality is investigated. Specific examples include the four-point bending test of a reinforced concrete beam with measurement data, contact between two elastic blocks, and the Hertzian contact problem.