Schedule for: 16w5090 - Variational Models of Fracture

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

The study of dynamic fracture is based on the dynamic energy-dissipation balance. It is easy to see that this condition is always satisfied by a stationary crack together with a displacement satisfying the system of elastodynamics. Therefore to predict crack growth a further principle is needed. In this talk we introduce a maximal dissipation condition that, together with elastodynamics and energy balance, provides a model for dynamic fracture, at least within a certain class of possible crack evolutions. In particular, we prove the existence of dynamic fracture evolutions satisfying this condition, subject to smoothness constraints, and exhibit an example to show that maximal dissipation can indeed rule out stationary cracks.

This talk will discuss phase-field modeling of dynamic instabilities of fast moving cracks in brittle solids. Experiments in thin gels have shown that cracks can attain extreme speeds approaching the shear wave speed when micro branching, which limits propagation to smaller speeds in thick samples, is suppressed. Furthermore, they have revealed the existence of an oscillatory instability with an intrinsic system-size-independent wavelength above a threshold speed. In apparent contradiction with experimental observations, the commonly used phase-field formulation of dynamic fracture yields crack that branch by tip splitting at roughly half the shear wave speed. A phenomenologically based phase-field formulation is proposed that can model crack propagation at extreme speeds by maintaining the wave speed constant inside the microscopic process zone. Simulations of this model for linear elasticity outside the process zone produce crack that tip split above a high threshold speed but no oscillations. In contrast, simulations for nonlinear neo-Hookean elasticity yield crack oscillations above a ultra-high speed threshold. Those oscillations have an intrinsic wavelength that scale with the size of the nonlinear zone surrounding the crack tip, which can be much larger than the process zone scale, and bear striking similarity with observed oscillations in thin gels.

This work was carried out in collaboration with Chih-Hung Chen and Eran Bouchbinder and his supported by a grant from the US-Israel Binational Science Foundation.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk a theoretical framework implementing the phase-field approach to fracture is used to couple the physics of flow through porous media and cracks with the mechanics of fracture. The main modeling challenge addressed in this work, which is a challenge for all diffuse crack representations, is on how to allow for the flow of fluid and the action of fluid pressure on the aggregate within the diffuse damage zone of the cracks. The theory is constructed by presenting the general physical balance laws, postulating a kinematic ansatz for an effective porosity, and conducting a consistent thermodynamic analysis to constrain the constitutive relationships. Constitutive equations that reproduce the desired responses at the various limits of the effective porosity are proposed in order to capture Darcy-type flow in the intact porous medium and Stokes-type flow within open cracks. A finite element formulation for the solution of the governing model equations is presented and discussed. Finally, the theoretical and numerical model is shown to compare favorably to several important analytical solutions. More complex and interesting calculations are also presented to illustrate some of the advantageous features of the approach.

In this talk I will review some recent results on the computation of energy release rates at the tip of 1D fractures with minimal smoothness assumption.
This is based on joint work with G. Francfort, J.-J. Marigo, A. Lemenant and J.-F. Babadjian.

In this talk, a variational model for gradient plasticity is proposed, which is based on an energy functional sum of a stored elastic bulk energy, a non-convex dissipative plastic energy, and a quadratic non-local term, depending on the gradient of the plastic strain. The basic modelling ingredients are presented in a simple one-dimensional setting, where the key physical aspects of the phenomena can easily be extracted. The evolution laws are deduced by using the mathematical tool of incremental energy minimization, and they are commented, highlighting the main differences and similarities with variational damage models. The typical assumptions of classical plasticity, such as yield condition, hardening rule, consistency condition, and elastic unloading, are obtained as necessary conditions for a minimum. Then, analytical solutions are determined, and attention is focused on the correlations between the convex-concave properties of the plastic energy and the distribution of the deformation field. The issue of solution stability is also addressed. Finally, some numerical results are discussed. First, tensile tests on steel bars and concrete samples are reproduced, and, then, a more complex two-dimensional crystal plasticity is proposed, and the process of microstructures evolution in metals is described by assuming a double-well plastic potential.

We consider a couple of evolutions for a phase field energy in brittle fracture. Both are obtained by time discretization using, as incremental problem, some alternate minimization scheme. We start from a time-discrete evolution, which resembles the alternate minimization scheme of Bourdin-Francfort-Marigo. Recasting the algorithm as a gradient flow, we provide a time-continuous limit, characterized in terms of a quasi-static evolution (more precisely a parametrized BV-evolution). Mechanically, the time-continuous evolution satisfies a suitable phase-field Griffith's criterion, at least in continuity points, while dissipation is thermodynamically consistent (with respect to the irreversibility constraint). Then, we consider a time-continuous system with an "irreversible Ginzburg-Landau" equation, for the phase field variable, paired with the elasto-static equilibrium equation. We provide existence by means of time-discrete alternate minimizing movement. Next, we study its quasi-static vanishing viscosity limit, again by means of a parametrized BV-evolution. Technically, characterizations are given both in terms of energy balance and by PDEs.

We describe our approach to simulating curvilinear brittle fractures in two-dimensions based on the use of Universal Meshes. A Universal Mesh is one that can be used to mesh a class of geometries by slightly perturbing some nodes in the mesh, and hence the name universal. In this way, as the crack evolves, the Universal Mesh is always deformed so as to exactly mesh the crack surface. The advantages of such an approach are: (a) no elements are cut by the crack, (b) new meshes are automatically obtained as the crack evolves, (c) the crack faces are exactly meshed with a conforming mesh at all times, and the quality of the surface mesh is guaranteed to be good, and (d) apart from duplicating degrees of freedom when the crack grows, the connectivity of the mesh and the sparsity of the associated stiffness matrix remains unaltered. In addition to the mesh, we are now able to compute stress intensity factors with any order of convergence, which gives us unprecedented accuracy in computing the crack evolution. As a result, we observe first order convergence of the crack path as well as the tangent to the crack path in a number of different examples. In the presentation I will introduce the notion of a Universal Mesh, illustrate the progress we have made so far with some examples, and then focus on the simulation of curvilinear fractures, and on the tools we created to compute stress intensity factors. In particular, show examples in which the computed crack path converge to the exact crack path, regardless of the mesh. If time permits, simulation of thermally induced fracture and hydraulic fractures will be discussed.
[1] R. Rangarajan and A.J. Lew, Universal Meshes: A method for triangulating planar curved domains immersed in nonconforming triangulations, International Journal for Numerical Methods in Engineering, 98(4), 236–264, 2014.
[2] R. Rangarajan and A.J. Lew, Analysis of a method to parameterize planar curves immersed in triangulations, SIAM Journal for Numerical Analysis,51(3), 1392-1420, 2013.
[3] Maurizio M. Chiaramonte, Yongxing Shen, Leon M. Keer, and Adrian J. Lew, Computing stress intensity factors for curvilinear cracks,International Journal for Numerical Methods in Engineering, (2015).
[4] M. Hunsweck, Y. Shen and A.J. Lew, A finite element approach to the simulation of hydraulic fractures with lag, International Journal for Numerical and Analytical Methods in Geomechanics, 37(9), 993-1015, 2013.
[5] R. Rangarajan, M. M. Chiaramonte, M. J. Hunsweck, Y. Shen, and A. J. Lew, Simulat- ing curvilinear crack propagation in two dimensions with universal meshes, International Journal for Numerical Methods in Engineering, 2014.

This work presents proppant and fluid-filled fracture with quasi-Newtonian fluid in a poroelastic medium. Lower-dimensional fracture surface is approximated by using the phase field function. The two-field displacement phase-field system solves fully-coupled constrained minimization problem due to the crack irreversibility. This constrained optimization problem is handled by using active set strategy. The pressure is obtained by using a diffraction equation where the phase-field variable serves as an indicator function that distinguishes between the fracture and the reservoir. Then the above system is coupled via a fixed-stress iteration. The transport of the proppant in the fracture is modeled by using a power-law fluid system. The numerical discretization in space is based on Galerkin finite elements for displacements and phase-field, and an Enriched Galerkin method is applied for the pressure equation in order to obtain local mass conservation. The concentration is solved with cell-centered finite elements. Nonlinear equations are treated with Newton's method. Predictor-corrector dynamic mesh refinement allows to capture more accurate interface of the fractures with reasonable number for degree of freedoms.
[1] Lee, S. and Wheeler, M. and Wick, T.; Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model
[2] Lee, S. and Mikelic, A. and Wheeler, M. and Wick, T.; Phase-field modeling of proppant-filled fractures in a poroelastic medium

While most of fundamental physical questions in dynamic fracture mechanics are settled science, there remains a significant gap between this fundamental understanding and our ability to apply it in computational models of failure in the complex systems and materials that arise in geophysics, biology, and contemporary engineering design. This talk describes recent progress at the University of Illinois and the University of Tennessee Space Institute in developing new numerical methods and models intended to close at least some aspects of this gap. A spacetime discontinuous Galerkin (SDG) method is the numerical foundation for our fracture model. This particular SDG method [1] is tailored to the requirements of hyperbolic systems, and differs from most others in that it is asynchronous, locally implicit, embarrassingly parallel, and supports fine-grain adaptive meshing. It enforces jump conditions with respect to Riemann solutions on element boundaries to preserve the characteristic structure of the underlying system. As with other DG methods, conservation fields balance to within machine precision on every (spacetime) element. We model crack opening and closure with specialized Riemann solutions for the various modes of frictional dynamic contact [2]. We weakly enforce these Riemann solutions using the same framework that enforces jump conditions and boundary conditions at inter-element and domain boundaries elsewhere in the SDG formulation. This approach produces contact conditions that are distinct from those that arise from simple constraints against inter-element penetration. We can implement cohesive fracture models in this SDG framework by incorporating traction– separation laws in the Riemann solutions [3]. However, in this presentation we focus on interfacial damage as an alternative means to model fracture along sharp interfaces. Time-delay evolution equations determine the damage rate as functions of the tractions and velocity jumps across fracture surfaces. A probabilistic model for microscopic flaws provides a mechanism for nucleating new fracture surfaces and is sufficient to captures crack branching. Adaptive refinement ensures that the solution fields are well resolved at multiple crack tips and along wave fronts. The same adaptive procedures continuously reconfigure the mesh so that it follows the crack paths determined by our physical model. We discuss some of the open challenges in modeling fracture with interfacial damage and present several numerical examples to demonstrate existing capabilities.

References
[1] R. Abedi, R.B. Haber, B. Petracovici. A spacetime discontinuous Galerkin method for elastodynamics with element-level balance of linear momentum, Comput. Methods Appl. Mech. Eng. 195 (2006) 3247–3273.
[2] R. Abedi, R.B. Haber. Riemann solutions and spacetime discontinuous Galerkin method for linear elastodynamic contact. Comput. Methods Appl. Mech. Engrg. 270 (2014) 150–177.
[3] R. Abedi, M.A. Hawker, R.B. Haber, K. Matouš. An adaptive spacetime discontinuous Galerkin method for cohesive models of elastodynamic fracture, Int. J. Numer. Methods Eng. 1 (2009) 1–42.

Work in collaboration with Reza Abedi, Mechanical, Aerospace & Biomedical Engineering, University of Tennessee Space Institute (UTSI) / Knoxville (UTK), 411 B. H. Goethert Parkway, Tullahoma, TN 37388

My presentation introduces the peridynamic model for predicting the initiation and evolution of complex fracture patterns. The model, a continuum variant of Newton's second law, uses integral rather than partial differential operators where the region of integration is over a domain. The force interaction is derived from a novel nonconvex strain energy density function, resulting in a nonmonotonic material model. The resulting equation of motion is proved to be mathematically well-posed. The model has the capacity to simulate nucleation and growth of multiple, mutually interacting dynamic fractures. In the limit of zero region of integration, the model reproduces the classic Griffith model of brittle fracture. The simplicity of the formulation avoids the need for supplemental kinetic relations that dictate crack growth or the need for an explicit damage evolution law.

We consider a nonlinear diffusion equation with irreversible property and construct a unique strong solution by using implicit time discretization. A new regularity estimate for the classical obstacle problem is established and is used in the construction of the strong solution. As an application, we consider a quasi-static fracture model of brittle material using the idea of the phase field model. The Francfort-Marigo energy which is based on the classical Griffith theory is introduced, where the sharp crack profile is approximated by a smooth damage function using the idea of the Ambrosio-Tortorelli regularization. The crack propagation model is derived as a gradient flow of the energy of the damage variable with an irreversible constraint. Some numerical examples in various settings computed by finite element method are also presented in the talk.
The contents is based on the joint works with Goro Akagi (Kobe Univ.) and with Takeshi Takaishi (Hiroshima Kokusai Gakuin Univ.).

This work is devoted to a numerical implementation of the Francfort-Marigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith-type dissipated energy. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well-separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasi-static damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack propagation, including kinking and branching. Several numerical examples in 2d and 3d are discussed.
This is a joint work with F. Jouve and N. Van Goethem.

Deterioration of mechanical and hydraulic properties of rock masses and subsequent problems are closely related to changes in the stress state, formation of new cracks, and increase of permeability in porous media saturated with freely moving fluids. In fully saturated rocks, fluid and solid phases are interconnected and the interaction between fluid and rock is characterized by coupled diffusion-deformation mechanisms that convey an apparent time-dependent character to the mechanical properties of the rock. The two governing equations of the coupled problem are the linear momentum balance and the continuity equation (mass conservation). The kinematic quantities that characterize this picture are the porous solid displacement and the rate of fluid volume per unit area. Hydro-mechanical coupling arises from the influence of the mechanical variables (stress, strain and displacement) on the continuity equation, where the primary variable is the fluid pressure, and from the influence of the hydraulic variables (pore pressure and seepage velocity) on the equilibrium equations, where the primary variables are the displacements. We describe a coupled approach to model damage induced by hydro-mechanical processes in low permeability solids. We describe the solid as an anisotropic brittle continuum where the damage is characterized by the formation of nested micro-structures in the form of equi-distant parallel faults, characterized by distinct orientation and spacing. The particular geometry of the faults allows for the analytical derivation of the porosity and of the anisotropic permeability of the solid. The fractured medium can be regarded as an anisotropic porous material. Classic methods can be applied to describe the porous-mechanical behavior of the solid to estimate the flow of fluids across the medium according to the presence of a fluid pressure gradient. The approach can be used for a wide range of engineering problems, ranging from the prevention of water or gas outburst into underground mines to the prediction of the integrity of reservoirs for underground CO2 sequestration or hazardous waste storage.
The work is done in collaboration with M.L. De Bellis, G. Della Vecchia and M. Ortiz.

This talk is devoted to highlighting the interplay between fracture and delamination in thin films. The usual scaling law on the elasticity parameters and the toughness of the medium with respect to its thickness gives rise to traditional cracks which are invariant in the transverse direction. We will show that, upon playing on this scaling law, it is also possible to observe debonding effects (delamination as well as decohesion) through the appearance of cracks which are orthogonal to the thin direction. Starting from a three-dimensional brittle elastic thin film, we will first present how both phenomena can be recovered independently through a Gamma-convergence analysis as the thickness tends to zero. Then, working on a “toy model" for scalar anti-plane displacements, we will show how both phenomena can be obtained at the same time. Some partial results in the full three-dimensional case will be presented.
These are joint works with Blaise Bourdin, Duvan Henao, Andrès Leon Baldelli. and Corrado Maurini.

Despite many applications of hydraulic fracturing, coupling of reservoir fluid flow, heat transfer, and fracture(s) propagation, especially in the presence of other pre-existing fractures, remains a major challenge in predicting the well stimulation and the evolution of well injectivity in the petroleum industry. To date simulation has focused upon the problem of a single planar in mode-I driven by a pressurized fluid using classical fracture mechanics models, or superposition of single planar fractures whilst neglecting the interaction between fractures. In contrast, realistic applications involve multiple fractures propagating along complex and unknown paths. Recently, the variational approach to fracture, which was originally proposed by Francfort and Marigo (1998) and numerically implemented by Bourdin et al. (2001, 2008), has been increasingly applied to simulation of hydraulic fracturing because of its ability to track any number of arbitrary fractures in an efficient manner. In this talk, we present a variational fracture model extended to hydraulic fracturing by accounting for pressure force within the fracture and in-situ stresses. We then show illustrative examples to demonstrate that the model responses are closely matched with existing analytical solutions of fluid-driven fracture propagation. Finally we will present several applications to practical problems.

This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity, in the case of metals, and classical rubber elasticity for polymers. When hardening exponents for metals are given values consistent with observation, or when chain failure is accounted for in polymers, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type for metals and fractional strain-gradient elasticity for polymers. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets in metals and to crazes in polymers, and that it requires a well-defined energy per unit fracture area. The optimal scaling laws show that ductile fracture is cohesive in nature, i.e., it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that microplasticity and surface energy play as contributors to the specific fracture energy of the material.

We define two finite element methods for elliptic problems with possibly discontinuous diffusion coefficients and divergence constraints where the meshes are not aligned with the interface. The first method is based on Immersed Interface Methods while the second one on Nitsche's methods. In order to obtain apriori error estimates totally independent of the contrast between diffusion coefficients and independent on how the interface crosses the mesh, we consider stabilizations based on the jump of the flux as well as the jump of solution across elements.

Hybrid atomistic to continuum coupling methods have been developed to obtain the accuracy of atomistic modeling in the neighborhood of crack tips, while using continuum modeling to include long range elastic effects. We will present a survey of recent work to analyze the lattice stability error introduced by atomistic to continuum coupling methods. These error estimates are then utilized to develop accurate blended coupling methods with controllable error.

Joint work with Christoph Ortner, Mathew Dobson, Xingjie Helen Li, Derek Olson, and Brian Van Koten.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.

In this talk I will talk about the stability of homogeneous states for gradient damage models. We will show how to exploit the second order derivative of the total energy to discriminate stable and unstable homogeneous states depending on the hardening properties, the loading and the size of the sample.

This is based on joint work with J-J Marigo and Corrado Maurini.