[911 Operator Full Crack [Crack Serial Key
Facunda Ganesh
The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. Following our recent work [C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically-consistent phase field models of fracture: Variational principles and multi-field fe implementations, International Journal for Numerical Methods in Engineering DOI:10.1002/nme.2861] on phase-field-type fracture, we propose in this paper a new variational framework for rate-independent diffusive fracture that bases on the introduction of a local history field. It contains a maximum reference energy obtained in the deformation history, which may be considered as a measure for the maximum tensile strain obtained in history. It is shown that this local variable drives the evolution of the crack phase field. The introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a new algorithmic treatment of diffusive fracture. Here, we propose an extremely robust operator split scheme that successively updates in a typical time step the history field, the crack phase field and finally the displacement field. A regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added. The proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples.
The prediction of failure mechanisms due to crack initiation and propagation in solids is of great importance for engineering applications. Following the classical treatments of Griffith [2] and Irwin [3], cracks propagate if the energy release rate reaches a critical value. The Griffith theory provides a criterion for crack propagation but is insufficient to determine curvilinear crack paths, crack kinking and branching angles. In particular, such a theory is unable to predict crack initiation. These defects of the classical Griffith-type theory of brittle fracture can be overcome by variational methods based on energy minimization as suggested by Francfort & Marigo [4], see also Bourdin, Francfort & Marigo [5], Dal Maso & Toader [6] and Buliga [7]. The regularized setting of their framework, considered in Bourdin, Francfort & Marigo [8], [5], is obtained by Γ-convergence inspired by the work on image segmentation by Mumford & Shah [9]. We refer to Ambrosio & Tortorelli [10] and the reviews of Dal Maso [11] and Braides [12], [13] for details on Γ-convergent approximations of free discontinuity problems. The approximation regularizes a sharp crack surface topology in the solid by diffusive crack zones governed by a scalar auxiliary variable. This variable can be considered as a phase field which interpolates between the unbroken and the broken state of the material. Conceptually similar are recently outlined phase field approaches to brittle fracture based on the classical Ginzburg-Landau type evolution equation as reviewed in Hakim & Karma [14], see also Karma, Kessler & Levine [15] and Eastgate et al. [16]. In contrast to the above mentioned rate-independent approach, these models are fully viscous in nature and mostly applied to dynamic fracture. The phase field approaches to fracture offer important new perspectives towards the theoretical and computational modeling of complex crack topologies. Recall that finite-element-based numerical implementations of sharp crack discontinuities, such as interface element formulations, element and nodal enrichment strategies suffer in the case of three-dimensional applications with crack branching. Advanced XFEM-based methods for sharp crack propagation are outlined in Belytschko et al. [17], [18] We also refer to the adaptive interface methods by Grses & Miehe [19] and Miehe & Grses [21], [24], [25] for the modeling of configurational-force-driven sharp crack propagation. In contrast, phase-field-type diffusive crack approaches avoid the modeling of discontinuities and can be implemented in a straightforward manner by coupled multi-field finite element solvers. In the recent work Miehe, Welschinger & Hofacker [1], we outlined a general thermodynamically-consistent framework of phase field modeling in fracture mechanics and considered aspects of its numerical implementation. It overcomes some of the difficulties inherent in the above mentioned phase field approaches, by definition of fracture only in tension, precise characterization of the dissipation and introduction of the viscosity as a regularization of the rate-independent formulation. The numerical treatment was based on a monolithic three-field saddle point principle, that includes beside the displacement and fracture phase field a particular crack driving force field.
In this work, we develop a new model of phase-field-type fracture that substantially enhances our formulations in Miehe, Welschinger & Hofacker [1]. The key novel aspects are the formulation of the balance equation for the evolution of the phase field in terms of a strain-history source term, which allows an extremely simple and robust staggered solution of the two-field problem in terms of the algorithm summarized in Box 1. On the theoretical side, we outline an incremental variational formulation for a rate-independent evolution of the crack phase model. The important characteristic is the introduction of a crack surface density function that governs the dissipation in the diffusive fracture theory. The key observation then is that the possible discontinuous evolution of the phase field in time is driven by a local history field. It contains a maximum reference energy obtained in the deformation history, which may be considered as a measure for the maximum tensile strain obtained in history. It is shown that this local variable drives the evolution of the crack phase field. The introduction of the history field provides a very transparent representation of the balance equation that governs the evolution of the diffusive crack topology. It represents a source term in the balance equation for the phase-field. In order to stabilize post-critical solution paths, we may add a viscous regularization. This consists of the introduction of a viscous term that supplements the crack surface resistance. A particular ingredient is that the rate-independent limit is recovered by simply setting the viscosity to zero. This feature of the proposed theory is a very important element of a robust numerical implementation. The proposed approach is embedded in the theory of gradient-type materials with a characteristic length scale, such as outlined in the general context by Capriz [22], Mariano [23] and Fremond [24]. On the numerical side, we develop a robust scheme for the incremental update of the fracture phase field and the displacement field. It represents an operator split algorithm within a typical time step that allows a staggered update of the phase field and the displacement field. The central idea for the algorithmic decoupling of the coupled equations is an approximation of the current history field that provides the crack source term in the diffusive crack topology equation. The formulation results in two linear problems for the successive update of the phase field and the displacement field within a typical time step. The proposed staggered scheme is extremely robust. The formulation may be considered as a canonical framework for the implementation of phase-field-type fracture in the rate-independent setting.
We start our investigation in Section 2 with a descriptive motivation of a regularized crack topology based on a phase field. This treatment results in the definition of a crack surface functional, depending on the crack phase field, that Γ-converges for vanishing length-scale parameter to a sharp crack topology. We consider this crack functional as the crack surface itself and postulate that it should stay constant or grow for arbitrary loading processes. The next modeling step, discussed in Section 3, is concerned with the definition of energy storage and dissipation functionals which depend on the fracture phase field. Here, we consider stored energy functions where the degradation due to the growing phase field acts only on a properly defined positive (tensile) part of the stored energy. The dissipation functional is directly related to the evolution of the crack surface functional via a material parameter that represents the critical energy release. With the dissipation and energy functionals at hand, we derive the governing balance equations from an incremental variational principle. For the quasi-static processes under consideration, the coupled system of equations consists of the static equilibrium condition for the degraded stresses and the balance equation for the phase field. It characterizes a two-field problem with the displacement field and the phase-field as the primary variables. For this coupled field problem, we develop in Section 4 the new staggered solution scheme which contains minimizers for each partial problem. The discrete counterparts of the continuous variational principles results in two linear algebraic Euler equations for the algorithmically decoupled incremental problem. Finally, Section 5 outlines representative numerical examples which demonstrate the features and algorithmic robustness of the proposed phase field models of fracture.
With the idea of a diffusive crack topology at hand, we develop in this section a constitutive framework of phase-field-type fracture for the rate-independent setting. The subsequent formulation in terms of a history field is motivated in Appendix A by a simple one-dimensional structure of local damage mechanics. Fig. 4 provides a visual guide to the subsequent developments.
795a8134c1