A numerical convergence study regarding homogenization assumptions in phase field modeling

Kiefer, Björn; Furlan, T.; Mosler, Jörn

doi:10.1002/nme.5547

Cited by 20 publications

(40 citation statements)

References 28 publications

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“…The second kind of models considers all jump conditions in the stresses and strains across the interface and contains the Reuss and Voigt type homogenization methods as limiting cases . It has been shown that the quality of the solution and the rate of convergence to the sharp interface solution depend on the chosen homogenization methods and that the homogenization method of the second kind outperforms the other ones . For that reason, in the presented contribution a homogenization method is derived that fulfills the kinematic compatibility across the interface and static equilibrium at the interface.…”

Section: Introductionmentioning

confidence: 99%

A diffuse modeling approach for embedded interfaces in linear elasticity

et al. 2019

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In this contribution, we present a diffuse modeling approach to embed material interfaces into nonconforming meshes with a focus on linear elasticity. For this purpose, a regularized indicator function is employed that describes the distribution of the different materials by a scalar value. The material in the resulting diffuse interface region is redefined in terms of this indicator function and recomputed by a homogenization of the adjacent material parameters. The applied homogenization method fulfills the kinematic compatibility across the interface and the static equilibrium at the interface. In addition, an hℓ‐adaptive refinement strategy based on truncated hierarchical B‐spline is applied to provide an appropriate and efficient approximation of the diffuse interface region. We justify mathematically and demonstrate numerically that the applied approach leads to optimal convergence rates in the far field for one‐dimensional problems. A two‐dimensional example illustrates that the application of the hℓ‐adaptive refinement strategy allows for a clear reduction of the error in the near and far field and a good resolution of the local stress and strain fields at the interface. The use of a higher continuous B‐spline basis leads to efficient computations due to the higher continuity of the diffuse interface model.

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Section: Introductionmentioning

confidence: 99%

A diffuse modeling approach for embedded interfaces in linear elasticity

et al. 2019

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“…However, better approaches exist, which account for the stress and strain discontinuity across the interface simultaneously [17,21,22,23]. The improvement manifests itself in better convergence rates and the fact, that the kinematics of an energy driven, moving interface are captured correctly [24]. The improved homogenization scheme is often referred to as partial rank-I relaxation, which will become apparent in the remainder of the paper, and yields a pointwise fulfillment of the equilibrium and compatibility within the diffuse region.…”

Section: Introductionmentioning

confidence: 99%

Phase-field modeling of fracture in heterogeneous materials: jump conditions, convergence and crack propagation

2020

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In this contribution, a variational diffuse modeling framework for cracks in heterogeneous media is presented. A static order parameter smoothly bridges the discontinuity at material interfaces, while an evolving phase-field captures the regularized crack. The key novelty is the combination of a strain energy split with a partial rank-I relaxation in the vicinity of the diffuse interface. The former is necessary to account for physically meaningful crack kinematics like crack closure, the latter ensures the mechanical jump conditions throughout the diffuse region. The model is verified by a convergence study, where a circular bi-material disc with and without a crack is subjected to radial loads. For the uncracked case, analytical solutions are taken as reference. In a second step, the model is applied to crack propagation, where a meaningful influence on crack branching is observed, that underlines the necessity of a reasonable homogenization scheme. The presented model is particularly relevant for the combination of any variational strain energy split in the fracture phase-field model with a diffuse modeling approach for material heterogeneities.

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“…To obtain the mechanical energy and the mechanical driving force, a homogenization scheme is applied to calculate the individual phase-field strain distribution, which is presented in [53][54][55] and applied to OpenPhase in [56]. Therefore, a partial rank-one homogenization is used, which satisfies the static equilibrium and the strains compatibility condition at the interfaces.…”

Section: Homogenization Of Eigenstrains On Microscalementioning

confidence: 99%

“…cf. [54], where n is the normal vector of the interface and a characterizes the jump of the displacement gradient All other parameters are explained in Table 4. The effective (homogenized) eigenstrain tensor representing the internal laminate structure of one martensite grain can be computed by volume averaging, whereas the laminate consists of equal amounts of the two variants.…”

Section: Homogenization Of Eigenstrains On Microscalementioning

confidence: 99%

Experimental and Numerical Investigations of the Development of Residual Stresses in Thermo-Mechanically Processed Cr-Alloyed Steel 1.3505

Behrens

Schröder

Brands

et al. 2019

Metals

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Residual stresses in components are a central issue in almost every manufacturing process, as they influence the performance of the final part. Regarding hot forming processes, there is a great potential for defining a targeted residual stress state, as many adjustment parameters, such as deformation state or temperature profile, are available that influence residual stresses. To ensure appropriate numerical modeling of residual stresses in hot forming processes, comprehensive material characterization and suitable multiscale Finite Element (FE) simulations are required. In this paper, experimental and numerical investigations of thermo-mechanically processed steel alloy 1.3505 (DIN 100Cr6) are presented that serve as a basis for further optimization of numerically modeled residual stresses. For this purpose, cylindrical upsetting tests at high temperature with subsequently cooling of the parts in the media air or water are carried out. Additionally, the process is simulated on the macroscale and compared to the results based on the experimental investigations. Therefore, the experimentally processed specimens are examined regarding the resulting microstructure, distortions, and residual stresses. For the investigation on a smaller scale, a numerical model is set up based on the state-data of the macroscopic simulation and experiments, simulating the transformation of the microstructure using phase-field theory and FE analysis on micro- and meso-scopic level.

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A numerical convergence study regarding homogenization assumptions in phase field modeling

Cited by 20 publications

References 28 publications

A diffuse modeling approach for embedded interfaces in linear elasticity

A diffuse modeling approach for embedded interfaces in linear elasticity

Phase-field modeling of fracture in heterogeneous materials: jump conditions, convergence and crack propagation

Experimental and Numerical Investigations of the Development of Residual Stresses in Thermo-Mechanically Processed Cr-Alloyed Steel 1.3505

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