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Rock formations are very often characterized by the presence of fractures that have grown subcritically over geological time scales and under evolving stress fields. In mechanically layered systems, such fractures can either become layer-bound or penetrate into adjacent strata. The growth of fractures in brittle materials is generally dependent on the energy release rate, however no closed form analytical solutions exist for these when a crack tip is in the proximity of a material interface. In this study, we present new empirical formulas for calculating the energy release rate at the tip of a crack perpendicular to a material interface in a symmetric 3-layer system. In these formulas, the normalized energy release rate is expressed as the product of a base term that integrates the normalized stiffness modulus over the crack length, and a correction factor that accounts for the presence of a material interface. The latter is assumed to be dependent on two quantities: the ratio of the crack length to the inner layer thickness, and the contrast in material stiffness between the inner and outer layers. The correction factors are obtained by fitting the parameters of carefully chosen expressions to a set of finite element solutions in order to yield predictions that are accurate to within one percent of the numerical results., Comment: 16 pages, 15 figures, 2 tables

There is currently an increasing interest in developing efficient solvers for phase-field modeling of brittle fracture. The governing equations for this problem originate from a constrained minimization of a non-convex energy functional, and the most commonly used solver is a staggered solution scheme. This is known to be robust compared to the monolithic Newton method, however, the staggered scheme often requires many iterations to converge when cracks are evolving. The focus of our work is to accelerate the solver through a scheme that sequentially applies Anderson acceleration and over-relaxation, switching back and forth depending on the residual evolution, and thereby ensuring a decreasing tendency. The resulting scheme takes advantage of the complementary strengths of Anderson acceleration and over-relaxation to make a robust and accelerating method for this problem. The new method is applied as a post-processing technique to the increments of the solver, hence, the implementation can be done with minor modifications to already available software. Moreover, the cost of combining the two acceleration schemes is negligible. The robustness and efficiency of the method are demonstrated through numerical examples.

Many interesting phenomena are characterized by the complex interaction of different physical processes, each often best modeled numerically via a specific approach. In this paper, we present the design and implementation of an object-oriented framework for performing multiphysics simulations that allows for the monolithic coupling of different numerical schemes. In contrast, most of the currently available simulation tools are tailored towards a specific numerical model, so that one must resort to coupling different codes externally based on operator splitting. The current framework has been developed following the C++11 standard, and its main aim is to provide an environment that affords enough flexibility for developers to implement complex models while at the same time giving end users a maximum amount of control over finer details of the simulation without having to write additional code. The main challenges towards realizing these objectives are discussed in the paper, together with the manner in which they are addressed. Along with core objects representing the framework skeleton, we present the various polymorphic classes that may be utilized by developers to implement new formulations, material models and solution algorithms. The code architecture is designed to allow achievement of the aforementioned functionalities with a minimum level of inheritance in order to improve the learning curve for programmers who are not acquainted with the software. Key capabilities of the framework are demonstrated via the solution of numerical examples dealing on composite torsion, Biot poroelasticity (featuring a combined finite element-finite volume formulation), and brittle crack propagation using a phase-field approach.

Numerical simulations of brittle fracture using phase-field approaches often employ a discrete approximation framework that applies the same order of interpolation for the displacement and phase-field variables. Most common is to use linear finite elements to discretize the linear momentum and phase-field equations. However the use of $P_1$ Lagrange shape functions to model the phase-field is not optimal, since the latter develops cusps for fully developed cracks that in turn occur at locations correspoding to Gauss points of the associated FE model for the mechanics. Such feature is challenging to reproduce accurately with low order elements, and consequently element sizes must be made very small relative to the phase-field regularization parameter in order to achieve convergence of results with respect to the mesh. In this paper, we combine the standard $P_1$ FE discretization of stress equilibrium with a cell-centered finite volume approximation of the phase-field evolution equation based on the two-point flux approximation that is constructed on the same simplex mesh. Compared to a pure FE formulation utilizing linear elements, the proposed framework results in looser restrictions on mesh refinement with respect to the phase-field length scale. Furthermore, initialization of the history field is straightforward and accomplished through a local procedure. The ability to employ a coarser mesh relative to the traditional implementation is shown for several numerical examples, demonstrating savings in computational cost on the order of 50 to 80 percent for the studied cases.

Phase-field approaches to fracture based on energy minimization principles have been rapidly gaining popularity in recent years, and are particularly well-suited for simulating crack initiation and growth in complex fracture networks. In the phase-field framework, the surface energy associated with crack formation is calculated by evaluating a functional defined in terms of a scalar order parameter and its gradients, which in turn describe the fractures in a diffuse sense following a prescribed regularization length scale. Imposing stationarity of the total energy leads to a coupled system of partial differential equations, one enforcing stress equilibrium and another governing phase-field evolution. The two equations are coupled through an energy degradation function that models the loss of stiffness in the bulk material as it undergoes damage. In the present work, we introduce a new parametric family of degradation functions aimed at increasing the accuracy of phase-field models in predicting critical loads associated with crack nucleation as well as the propagation of existing fractures. An additional goal is the preservation of linear elastic response in the bulk material prior to fracture. Through the analysis of several numerical examples, we demonstrate the superiority of the proposed family of functions to the classical quadratic degradation function that is used most often in the literature., Comment: 33 pages, 30 figures