Effective Elastic Moduli in Solids with High Crack Density

We investigate the weakening of elastic materials through randomly distributed circles and cracks numerically and compare the results to predictions from homogenization theories. We find a good agreement for the case of randomly oriented cracks of equal length in an isotropic plane-strain medium for lower crack densities; for higher densities the material is weaker than predicted due to precursors of percolation. For a parallel alignment of cracks, where percolation does not occur, we analytically predict a power law decay of the effective elastic constants for high crack densities, and confirm this result numerically.Comment: 8 page

Comparison of phase-field models for surface diffusion

The description of surface-diffusion controlled dynamics via the phase-field method is less trivial than it appears at first sight. A seemingly straightforward approach from the literature is shown to fail to produce the correct asymptotics, albeit in a subtle manner. Two models are constructed that approximate known sharp-interface equations without adding undesired constraints. Linear stability of a planar interface is investigated for the resulting phase-field equations and shown to reduce to the desired limit. Finally, numerical simulations of the standard and a more sophisticated model from the literature as well as of our two new models are performed to assess the relative merits of each approach. The results suggest superior performance of the new models in at least some situations.Comment: 23 pages, 16 figures, submitted to PRE, this paper is closely related to cond-mat/0607823, in which some of the analytical derivations are already given and the 3D case is treate

Free-boundary problem of crack dynamics: phase field modeling

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

Free-boundary problem of crack dynamics: phase field modeling

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

Phase-field modeling of surface diffusion

In the classical description of surface diffusion, transport on a curved interface is associated with the Laplace – Beltrami operator acting on a chemical potential (difference). An early attempt to model surface diffusion via the phase-field approach goes back to Cahn, Elliott and Novick-Cohen; they use a scalar mobility approaching zero in the bulk. Similar models have been proposed first on the basis of heuristic ideas and then underpinned by asymptotic analysis. As it turns out, most of these analyses suffer from a subtle flaw, due not to a miscalculation but rather to early termination of the calculation. The asymptotic analysis provides all the equations desired for the correct sharp-interface limit. Unfortunately, it provides an additional equation, which is one restriction too many. Consequences for dynamical simulations of this kind of model are explored numerically. We construct two models based on the introduction of a tensorial mobility that approximate known sharp-interface equations without adding undesired constraints. Numerical simulations suggest superior performance of the new models in at least some situations.Read More: http://www.hanser-elibrary.com/doi/abs/10.3139/146.11029

vorgelegt von Diplom-Physiker

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