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The Elastic Fluctuation Tensor: Quantifying Stochastic Fluctuations of the Apparent Stiffness of Non-representative
Maximilian Krause1,2, Matti Schneider1,3,4
1Institute of Engineering Mathematics, University of Duisburg-Essen, Essen, Germany.
Journal of Elasticity
|June 8, 2026
Summary
We introduce the elastic fluctuation tensor to quantify stiffness variations in finite material volumes. This tensor captures essential fluctuations often missed, improving stochastic homogenization accuracy for materials with complex microstructures.
Area of Science:
- Materials Science
- Computational Mechanics
- Solid Mechanics
Background:
- Computational homogenization typically assumes infinite volume elements for effective material properties.
- Finite volume elements exhibit apparent stiffness fluctuations on the macroscale.
- Existing methods quantify thermal conductivity fluctuations but lack a similar tensor for elasticity.
Purpose of the Study:
- Introduce the eighth-order elastic fluctuation tensor to quantify apparent stiffness variations in finite microstructural volumes.
- Develop efficient tensor representations using group representation theory for various microstructure symmetries.
- Highlight the necessity of the full fluctuation tensor for accurate stochastic homogenization.
Main Methods:
- Defined the elastic fluctuation tensor as the infinite-volume limit of apparent stiffness covariance.
- Applied group representation theory to derive efficient tensor representations based on microstructure symmetry.
- Numerically computed the elastic fluctuation tensor for example materials like fiber-reinforced polypropylene and polycrystalline copper.
Main Results:
- The elastic fluctuation tensor inherits ensemble symmetries, leading to isotropy for rotationally symmetric ensembles.
- Efficient tensor representations were defined for different symmetry classes.
- Numerical computations confirmed theoretical convergence rates and symmetry properties.
- Fluctuations in anisotropic stiffness components are significant and often exceed those in isotropic components for isotropic microstructures.
Conclusions:
- The elastic fluctuation tensor is crucial for accurately quantifying uncertainties in stochastic homogenization.
- Considering the full tensor, not just isotropic components, is essential for microstructures with anisotropic features.
- Symmetry-based methods reduce computational cost and numerical errors in calculating the fluctuation tensor.
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