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Within this work, we develop a phase-field description for simulating fractures in nearly incompressible materials. It is well-known that low-order approximations generally lead to volume-locking behaviors. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor–Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on three numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson’s ratio approximating the incompressible limit, are presented.

In this work, we examine a numerical phase-field fracture framework in which the crack irreversibility constraint is treated with a primal-dual active set method and a linearization is used in the degradation function to enhance the numerical stability. The first goal is to carefully derive from a complementarity system our primal-dual active set formulation, which has been used in the literature in numerous studies, but for phase-field fracture without its detailed mathematical derivation yet. Based on the latter, we formulate a modified combined active-set Newton approach that significantly reduces the computational cost in comparison to comparable prior algorithms for quasi-monolithic settings. For many practical problems, Newton converges fast, but active set needs many iterations, for which three different efficiency improvements are suggested in this paper. Afterwards, we design an iteration on the linearization in order to iterate the problem to the monolithic limit. Our new algorithms are implemented in the programming framework pfm-cracks [T. Heister, T. Wick; pfm-cracks: A parallel-adaptive framework for phase-field fracture propagation, Software Impacts, Vol. 6 (2020), 100045]. In the numerical examples, we conduct performance studies and investigate efficiency enhancements. The main emphasis is on the cost complexity by keeping the accuracy of numerical solutions and goal functionals. Our algorithmic suggestions are substantiated with the help of several benchmarks in two and three spatial dimensions. Therein, predictor-corrector adaptivity and parallel performance studies are explored as well.

Within this work, we develop a phase-field description for simulating fractures in incompressible materials. Standard formulations are subject to volume-locking when the solid is (nearly) incompressible. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor-Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on two numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson's ratio approaching the incompressible limit, are presented.

This work presents a new adaptive approach for the numerical simulation of a phase-field model for fractures in nearly incompressible solids. In order to cope with locking effects, we use a recently proposed mixed form where we have a hydro-static pressure as additional unknown besides the displacement field and the phase-field variable. To fulfill the fracture irreversibility constraint, we consider a formulation as a variational inequality in the phase-field variable. For adaptive mesh refinement, we use a recently developed residual-type a posteriori error estimator for the phase-field variational inequality which is efficient and reliable, and robust with respect to the phase-field regularization parameter. The proposed model and the adaptive error-based refinement strategy are demonstrated by means of numerical tests derived from the L-shaped panel test, originally developed for concrete. Here, the Poisson's ratio is changed from the standard setting towards the incompressible limit \\(\\nu \\to 0.5\\).

In this work, we consider fracture propagation in nearly incompressible and (fully) incompressible materials using a phase-field formulation. We use a mixed form of the elasticity equation to overcome volume locking effects and develop a robust, nonlinear and linear solver scheme and preconditioner for the resulting system. The coupled variational inequality system, which is solved monolithically, consists of three unknowns: displacements, pressure, and phase-field. Nonlinearities due to coupling, constitutive laws, and crack irreversibility are solved using a combined Newton algorithm for the nonlinearities in the partial differential equation and employing a primal-dual active set strategy for the crack irreverrsibility constraint. The linear system in each Newton step is solved iteratively with a flexible generalized minimal residual method (GMRES). The key contribution of this work is the development of a problem-specific preconditioner that leverages the saddle-point structure of the displacement and pressure variable. Four numerical examples in pure solids and pressure-driven fractures are conducted on uniformly and locally refined meshes to investigate the robustness of the solver concerning the Poisson ratio as well as the discretization and regularization parameters.

In this work, we consider adaptive mesh refinement for a monolithic phase-field description for fractures in brittle materials. Our approach is based on an a posteriori error estimator for the phase-field variational inequality realizing the fracture irreversibility constraint. The key goal is the development of a reliable and efficient residual-type error estimator for the phase-field fracture model in each time-step. Based on this error estimator, error indicators for local mesh adaptivity are extracted. The proposed estimator is based on a technique known for singularly perturbed equations in combination with estimators for variational inequalities. These theoretical developments are used to formulate an adaptive mesh refinement algorithm. For the numerical solution, the fracture irreversibility is imposed using a Lagrange multiplier. The resulting saddle-point system has three unknowns: displacements, phase-field, and a Lagrange multiplier for the crack irreversibility. Several numerical experiments demonstrate our theoretical findings with the newly developed estimators and the corresponding refinement strategy.

In this work, we consider pressurized phase-field fracture problems in nearly and fully incompressible materials. To this end, a mixed form for the solid equations is proposed. To enhance the accuracy of the spatial discretization, a residual-type error estimator is developed. Our algorithmic advancements are substantiated with several numerical tests that are inspired from benchmark configurations. Therein, a primal-based formulation is compared to our newly developed mixed phase-field fracture method for Poisson ratios approaching \\(\\nu \\to 0.5\\). Finally, for \\(\\nu = 0.5\\), we compare the numerical results of the mixed formulation with a pressure robust modification.

In this work, we present crack propagation experiments evaluated by digital image correlation (DIC) for a carbon black filled ethylene propylene diene monomer rubber (EPDM) and numerical modeling with the help of variational phase-field fracture. Our main focus is the evolution of cracks in one-sided notched EPDM strips containing a circular hole. The crack propagation experiments are complemented with investigations identifying the mechanical material properties as well as the critical strain energy release rate. For simulating the evolution of cracks with a given notch, phase-field fracture modeling is a popular approach. To avoid volume-locking effects considering fractures in nearly incompressible materials, a quasi-static phase-field fracture model in its classical formulation is reformulated with the help of a mixed form of the solid-displacement equation. The new established mixed phase-field fracture model is applied to simulate crack propagation in punctured EPDM strips by using the experimentally identified material parameters with mixed finite elements. To discuss agreements and point out challenges and differences, the crack paths, the maximal force response, the traverse displacement at the crack start, as well as force-displacement curves of the experimental and numerical results are compared.

Arabidopsis thaliana CRT1 (compromised for recognition of Turnip Crinkle Virus) was previously shown to be required for effector-triggered immunity. Sequence analyses previously revealed that CRT1 contains the ATPase and S5 domains characteristic of Microchidia (MORC) proteins; these proteins are associated with DNA modification and repair. Here we show that CRT1 and its closest homologue, CRH1, are also required for pathogen-associated molecular pattern (PAMP)-triggered immunity, basal resistance, non-host resistance and systemic acquired resistance. Consistent with its role in PAMP-triggered immunity, CRT1 interacted with the PAMP recognition receptor FLS2. Subcellular fractionation and transmission electron microscopy detected a subpopulation of CRT1 in the nucleus, whose levels increased following PAMP treatment or infection with an avirulent pathogen. These results, combined with the demonstration that CRT1 binds DNA, exhibits endonuclease activity, and affects tolerance to the DNA-damaging agent mitomycin C, argue that this prototypic eukaryotic member of the MORC superfamily has important nuclear functions during immune response activation. The CRT1 gene in Arabidopsis confers effector-triggered immunity. Here Kang et al. show that CRT1 has a broader endonuclease role in plant innate immunity, including basal, non-host and systemic acquired resistance, and becomes partially localized to the nucleus upon immune receptor activation.

MORC1 and MORC2, two of the seven members of the Arabidopsis (Arabidopsis thaliana) Compromised Recognition of Turnip Crinkle Virusl subfamily of microrchidia Gyrase, Heat Shock Protein90, Histidine Kinase, MutL (GHKL) ATPases, were previously shown to be required in multiple layers of plant immunity. Here, we show that the barley (Hordeum vulgare) MORCs also are involved in disease resistance. Genome-wide analyses identified five MORCs that are 37% to 48% identical on the protein level to AtMORCl. Unexpectedly, and in clear contrast to Arabidopsis, RNA interference-mediated knockdown of MORC in barley resulted in enhanced basal resistance and effector-triggered, powdery mildew resistance locus A12-mediated resistance against the biotrophic powdery mildew fungus (Blumeria graminis f. sp. hordei), while MORC overexpression decreased resistance. Moreover, barley knockdown mutants also showed higher resistance to Fusarium graminearum. Barley MORCs, like their Arabidopsis homologs, contain the highly conserved GHKL ATPase and S5 domains, which identify them as members of the MORC superfamily. Like AtMORCl, barley MORC1 (HvMORC1) binds DNA and has Mn ²⁺-dependent endonuclease activities, suggesting that the contrasting function of MORC1 homologs in barley versus Arabidopsis is not due to differences in their enzyme activities. In contrast to AtMORCs, which are involved in silencing of transposons that are largely restricted to pericentromeric regions, barley MORC mutants did not show a loss-of-transposon silencing regardless of their genomic location. Reciprocal overexpression of MORC1 homologs in barley and Arabidopsis snowed that AtMORC1 and HvMORC1 could not restore each other's function. Together, these results suggest that MORC proteins function as modulators of immunity, which can act negatively (barley) or positively (Arabidopsis) dependent on the species.