Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales

Flodén, Liselott; Persson, Jens

doi:10.3934/nhm.2016012

Cited by 4 publications

(2 citation statements)

References 25 publications

(60 reference statements)

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“…All efforts regarding consistent homogenization assumptions within phase field modeling thus propose a staggered structure decomposed into a transient and a stationary problem, i.e., _ A nþ1 ¼ GðA 0 Þ and A 0 ¼ FðA 1 ; A 2 ; :::; A n Þ. Consequently, even if the stationary part could be derived by appropriate homogenization methods, the transient part represents an additional modeling step on the macroscale. A recently investigated, alternative procedure, that directly results in a macroscopic relation with associated rates of homogenized quantities, could eventually be derived by prescribing evolution equations on the microscale itself and applying the concept of multiscale convergence, as outlined in [27,28] for partial differential equations including the Landau-Ginzburg relations as a special case. Such an approach involves multiple spatial and temporal scales so that not only effective interpolation rules between singlephase properties but also long-term evolution and stationary microstructure prediction could potentially be obtained in a mathematically rigorous fashion.…”

Section: Discussionmentioning

confidence: 99%

Unequally and Non-linearly Weighted Averaging Operators as a General Homogenization Approach for Phase Field Modeling of Phase Transforming Materials

Oertzen

Kiefer

2022

Shap. Mem. Superelasticity

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The phase field method has been shown to have tremendous potential to serve as a continuum modeling approach of microstructural evolution mechanisms in many contexts, such as alloy solidification, fracture, and chemo-mechanics. By replacing sharp interfaces between phases with a diffuse representation, additional degrees of freedom, namely order parameters, enter the continuum model, in order to describe the current phase state at each material point. Single-phase properties thus need to be interpolated carefully within diffuse interface regions by applying mixture rules subject to specific, microscopic constraints in an underlying homogenization framework. However, there exists a variety of well-established nonlinear interpolation schemes—especially incorporating symmetric or hyperspherical order parameters—for which it turns out that they cannot consistently be described within conventional homogenization theories. To overcome this problem, an extension toward unequally, non-linearly weighted averaging operators is presented, in which conventional, unweighted homogenization represents a special case. The embedding of Reuss–Sachs, Taylor–Voigt, and rank-one convexification models—extended by nonlinear interpolation—within the proposed framework is demonstrated by identifying necessary constraints on corresponding weighting functions. Since this concept establishes a generalization of conventional homogenization, the following question arises: Could any effective property interpolation within the diffuse interface fit into the proposed framework by choosing appropriate weighting functions, and if so, under which microscopic constraints? To this end, the concepts of macroscopic links and domain relations are introduced and applied for conventional homogenization schemes in phase field modeling. Important, yet often subtle, implications of such theoretical considerations on the prediction of microstructure formation and evolution by means of phase field modeling are the focus of discussion in this contribution.

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Section: Discussionmentioning

confidence: 99%

Unequally and Non-linearly Weighted Averaging Operators as a General Homogenization Approach for Phase Field Modeling of Phase Transforming Materials

Oertzen

Kiefer

2022

Shap. Mem. Superelasticity

View full text Add to dashboard Cite

show abstract

“…A recently investigated alternative procedure, which directly results in a macroscopic relation with associated rates of homogenized quantities, could eventually be derived by prescribing evolution equations on the microscale itself and applying the concept of multiscale convergence. This notion was outlined in [9,10], for partial differential equations that include the Landau-Ginzburg relations as a special case. Such an approach involves multiple spatial and temporal scales, so that not only effective interpolation rules between single-phase properties, but also long-term evolution and stationary microstructure prediction could potentially be obtained in a mathematically rigorous fashion.…”

Section: Discussionmentioning

confidence: 99%

The concept of unequally and nonlinearly weighted averaging operators as a fundamental homogenization framework in phase‐field modeling

Oertzen

Kiefer

2023

Proc Appl Math and Mech

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In recent years, the phase-field method has gained tremendous potential to serve as a continuum modeling approach of micro structural evolution mechanisms for multiple applications such as alloy solidification, phase transitions, and fracture. By replacing sharp interfaces between phases with a diffusive formulation, additional degrees of freedom, namely order parameters, enter the continuum model to locally describe the current phase composition at each material point. Single phase properties thus need to be interpolated carefully within diffuse interface regions by applying mixture rules subject to specific constraints in an underlying homogenization framework, c.f. [1,2]. However, there exists a variety of well established nonlinear interpolation schemes that can only partially, or not at all, comply with conventional homogenization approaches.To this end, an extension towards unequally and nonlinearly weighted averaging operators is presented, in which conventional unweighted homogenization represents a special case. The embedding of nonlinearly interpolated Reuss-Sachs, Taylor-Voigt and rank-one homogenization models within the given framework is demonstrated by means of energy relaxation. Furthermore, upper and lower energetic bounds are compared between unweighted and nonlinearly weighted averages of order distributions, where the weighted case is shown to yield an improved effective energy description.

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Homogenization of phase transforming materials: The concept of phase-morphology and variable scale separations

von Oertzen,

Kiefer

2025

Journal of the Mechanics and Physics of Solids

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Homogenization of nonlinear dissipative hyperbolic problems exhibiting arbitrarily many spatial and temporal scales

Cited by 4 publications

References 25 publications

Unequally and Non-linearly Weighted Averaging Operators as a General Homogenization Approach for Phase Field Modeling of Phase Transforming Materials

Unequally and Non-linearly Weighted Averaging Operators as a General Homogenization Approach for Phase Field Modeling of Phase Transforming Materials

The concept of unequally and nonlinearly weighted averaging operators as a fundamental homogenization framework in phase‐field modeling

Homogenization of phase transforming materials: The concept of phase-morphology and variable scale separations

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