A phase field model for fracture

The variational formulation of brittle fracture as formulated for example by Francfort and Marigo in [1], where the total energy is minimized with respect to any admissible crack set and displacement field, allows the identification of crack paths, branching of preexisting cracks and even crack initiation without additional criteria. For its numerical treatment a continuous approximation of the model in the sense of Γ-convergence has been presented by Bourdin in [2]. In the regularized Francfort–Marigo model cracks are represented by an additional field variable (secondary variable) s∈[0,1] which is 0 if the material is cracked and 1 if it is undamaged. In this work, we reinterpret the crack variable as a phase field order parameter and address cracking as a phase transition problem. The crack growth is governed by the evolution equation of the order parameter which resembles the Ginzburg–Landau equation. The numerical treatment is done by finite elements combined with an implicit Euler scheme for the time integration. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interpretation of parameters in phase field models for fracture

Phase field fracture models typically feature a length parameter, which controls the width of the diffuse transition zone between broken and undamaged material. In the limit case of a vanishing length parameter, these models converge to a sharp crack formulation. From this point of view, the length scale parameter is a purely auxiliary numerical quantity. However, the study of the stability of homogeneous solutions in a one dimensional setting permits a different interpretation. Since the length parameter is directly related to the critical stress at which the homogeneous solution becomes unstable and crack nucleation occurs, it can be related to the strength of the material. In this regard, the length parameter itself may be seen as a material parameter. These analytical findings are approved by finite element simulations. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

From phase field to sharp cracks: Convergence of quasi-static evolutions in a special setting

We consider the quasi-static evolution of a straight crack within the recently developed phase-field approach and the classical sharp crack approach, and we show a strong correlation between the outcomes from the two approaches: the corresponding energies, minimizers, energy release rates and quasi-static evolutions converge as the internal length parameter of the phase-field model tends to zero. A crucial point in the proof is a novel representation of the energy release rate, which allows one to pass to the limit under weak convergence of the strains.     

An LMI description for the cone of Lorentz-positive maps II

Let L _n be the n-dimensional second-order cone. A linear map from ? ^m to ? ⁿ is called positive if the image of L _m under this map is contained in L _n . For any pair (n,?m) of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality of size (n???1)(m???1) that describes this cone.     

The dual description of a cone integer-valuedly-generated by a finite set of vectors

A discrete analog is obtained in the paper for certain classical results of the theory of linear inequalities.     

Optimal boundary controls for a phase field model

The phase field model is a nonlinear system of parabolic equationswhich describes the phase transitions between two differentphases, e.g. solid and liquid. In this paper, we consider ageneral optimal boundary control problem which is governed bythis model. The existence of the solutions of the phase fieldmodel is established by a rigorous analysis of the method oflines. The existence of the optimal solutions and the necessaryconditions for optimality are proved. For a special unconstrainedboundary control problem, we also prove some results concerningthe uniqueness of the optimal solutions. For a special constrainedboundary control problem, we obtain a result concerning thebang-bang principle.     

A novel treatment of crack boundary conditions in phase field models of fracture

In the phase field approach for fracture an additional scalar field is introduced in order to describe the state of the material between intact and fully broken. So far, for the loading dependent degradation of stiffness (damage) either the volumetric-deviatoric split of strain [1, 2] or the spectral decomposition [3, 4] is used. In contrast to such an isotropic degradation of stiffness, the fully broken state represents a crack with a particular orientation. Both aforementioned approaches do not take the crack orientation into account. This may lead to the violation of the crack boundary conditions. In order to satisfy these conditions the phase field approach is modified here by taking the orientation of the crack into account. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of turbulent dispersion using a simple random model of the flow field

A method is devised to simulate the movement and spreading of a patch of contaminant in two-dimensional turbulent flow. The turbulent motion is exponentially divided into components of differing wave number, adjacent components being made to have correlation times differing by a factor of two. The turbulent motion is then reconstructed by replacing each component with a sinusoidal advection field having a randomly directed wave number. Contaminant particles are advected by each of the reconstructed components, the smallest scale components being applied first. A computer simulation was performed, using a Kolmogorov k^{-$\text{5}\text{3}$} turbulent energy spectrum. Batchelor's σ ∝ t^{$\text{3}\text{2}$} law for the spreading of a contaminant patch was reproduced, approximately, as was Richardson's non-Gaussian asymptotic form of the distance-neighbour function.     

An error estimate for a finite-element scheme for a phase field model

In this paper we propose a fully discrete finite-element schemeto solve a non-linear system of parabolic equations for a phasefield model and demonstrate an error estimate of optimal orderin L² for this scheme. This error estimate conforms with thenumerical results presented at the end of this paper.     

Krasnosel'skii-type characterizations for a cone

Three combinatorial criteria for cones involving the concepts of R-visibility and clear R-visibility introduced for sets in Euclidean space are established. They are direct analogues of Krasnosel'skii-type theorems existing for starshaped sets.The author is with the Department of Mathematics, University of Notre Dame, Indiana, on leave from the Mathematical Institute of the Polish Academy of Sciences.