This thesis describes the behavior of cracks and pores under the influence of elastic and curvature effects. In a continuum theory approach, these structure deformations are treated as free moving boundaries. Our investigation start with well established sharp interface equations for which no fully dynamical solutions exist so far. The equations include only linear dynamical elasticity, surface energy and non-equilibrium transport theory. By proper use of the phase-field concept, we are now able to tackle the fully time-dependent free moving boundary problem to describe crack propagation in a fully self-consistent way. We concentrate on two material transport processes, namely surface diffusion and phase transition dynamics. We show analytically that the intuitive and widely used approach for constructing a phase-field model for surface diffusion fails, since it does not reduce to the desired sharp interface equations, providing an uncontrolled approximation to the dynamics. We then develop two completely new models that ensure the correct asymptotic behavior and support our analytical findings by numerical simulations, which are are computationally very demanding due to the high order equations that have to be solved. We therefore derive another phase-field model based on a phase transition process. Incorporating elastodynamic effects into the theory makes the simultaneous self-consistent selection of a tip radius scale and the propagation velocity possible. Our simulations show that it describes the complicated tip behavior and the elastic far-field behavior correctly, also allowing the numerical extraction of quantities like the stress intensity factor. Our results agree with those found in the literature for the case of steadily propagating cracks and extend them into the previously unaccessible parameter regime of large elastic driving forces, Here, we are able to resolve a dynamical tip-splitting instability, in agreement with experimental observations. Structures that are subjected to external loading often contain many small cracks already, which can weaken the structure substantially, depending on the initial crack density. We performed simulations of static inclusions and compared the results with the predictions we obtained with analytic approximation schemes. The use of our scheme reveals that the complicated three-dimensional behavior of the elastic modulus as a function of the crack density for randomly oriented cracks reduces to a simple exponential decay and exhibits the inability of the often used differential homogenization method to predict percolation, i.e. breaking of the system. The parallel arrangement of slit-like cracks, where percolation does not occur, is not easily accessible to the standard analytical techniques. We could show by use of thin-plate theory, scaling arguments and numerical calculations that for this geometrical setup, the relevant effective elastic constant decays not exponentially as for randomly oriented cracks, but as a power-law instead. Our method can thus describe morphological surface instabilities, fast crack propagation and even the collective behavior of multi-cracked materials with high quantitative precision
