A probabilistic virtual process chain to quantify process-induced uncertainties in Sheet Molding Compounds

Meyer, Nils; Gajek, Sebastian; Görthofer, Johannes; Hrymak, Andrew N.; Kärger, Luise; Henning, Frank; Schneider, Matti; Böhlke, Thomas

doi:10.1016/j.compositesb.2022.110380

Cited by 9 publications

(3 citation statements)

References 114 publications

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“…Fiber-orientation tensors [1] date back to the far-reaching works [2,3] and describe the relevant features of the fiber-orientation distribution of discontinuous fiber-reinforced composites. Within the virtual development and design process of such composites [1,[4][5][6], fiberorientation tensors appear in material modeling [7][8][9][10][11][12], microstructure generation [13][14][15][16][17][18][19], mold filling or flow simulations [20][21][22][23][24][25] and the experimental computer tomography analysis [26][27][28]. This wide field of application motivates a detailed understanding of the mathematical properties of fiber-orientation tensors.…”

Section: State Of the Artmentioning

confidence: 99%

On the Phase Space of Fourth-Order Fiber-Orientation Tensors

Bauer

Schneider

Böhlke

2023

J Elast

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Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation tensor is obtained via a closure approximation. Unfortunately, such a description limits the predictive capabilities of the modeling process significantly, because the wealth of possible fourth-order fiber-orientation tensors is not exploited by such closures, and the restriction to second-order fiber-orientation tensors implies artifacts. Closures based on the second-order fiber-orientation tensor face a fundamental problem – which fourth-order fiber-orientation tensors can be realized? In the literature, only necessary conditions for a fiber-orientation tensor to be connected to a fiber-orientation distribution are found. In this article, we show that the typically considered necessary conditions, positive semidefiniteness and a trace condition, are also sufficient for being a fourth-order fiber-orientation tensor in the physically relevant case of two and three spatial dimensions. Moreover, we show that these conditions are not sufficient in higher dimensions. The argument is based on convex duality and a celebrated theorem of D. Hilbert (1888) on the decomposability of positive and homogeneous polynomials of degree four. The result has numerous implications for modeling the flow and the resulting microstructures of fiber-reinforced composites, in particular for the effective elastic constants of such materials. Based on our findings, we show how to connect optimization problems on fourth-order fiber-orientation tensors to semi-definite programming. The proposed formulation permits to encode symmetries of the fiber-orientation tensor naturally. As an application, we look at the differences between orthotropic and general, i.e., triclinic, fiber-orientation tensors of fourth order in two and three spatial dimensions, revealing the severe limitations inherent to orthotropic closure approximations.

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Section: State Of the Artmentioning

confidence: 99%

On the Phase Space of Fourth-Order Fiber-Orientation Tensors

Bauer

Schneider

Böhlke

2023

J Elast

Self Cite

View full text Add to dashboard Cite

show abstract

“…Fiber-orientation tensors [1] date back to the far-reaching works [2,3] and describe the relevant features of the fiber-orientation distribution of discontinuous fiber-reinforced composites. Within the virtual development and design process of such composites [1,[4][5][6], fiber-orientation tensors appear in material modeling [7][8][9][10][11][12], microstructure generation [13][14][15][16][17][18][19], mold filling or flow simulations [20][21][22][23][24][25] and the experimental computer tomography analysis [26][27][28]. This wide field of application motivates a detailed understanding of the mathematical properties of fiber-orientation tensors.…”

Section: State Of the Artmentioning

confidence: 99%

On the phase space of fourth-order fiber-orientation tensors

Bauer¹,

Schneider²,

Böhlke³

2022

Preprint

View full text Add to dashboard Cite

Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation tensor is obtained via a closure approximation. Unfortunately, such a description limits the predictive capabilities of the modeling process significantly, because the wealth of possible fourth-order fiber-orientation tensors is not exploited by such closures, and the restriction to second-order fiber-orientation tensors implies artifacts. Closures based on the second-order fiber-orientation tensor face a fundamental problem -which fourth-order fiberorientation tensors can be realized? In the literature, only necessary conditions for a fiber-orientation tensor to be connected to a fiber-orientation distribution are found. In this article, we show that the typically considered necessary conditions, positive semidefiniteness and a trace condition, are also sufficient for being a fourth-order fiber-orientation tensor in the physically relevant case of two and three spatial dimensions. Moreover, we show that these conditions are not sufficient in higher dimensions. The argument is based on convex duality and a celebrated theorem of D. Hilbert (1888) on the decomposability of positive and homogeneous polynomials of degree four. The result has numerous implications for modeling the flow and the resulting microstructures of fiber-reinforced composites, in particular for the effective elastic constants of such materials.

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“…Most notably, DMNs predict the effective properties for arbitrarily nonlinear constituents and complex three-dimensional microstructures with high accuracy and provide speed-ups of up to five orders of magnitude [8,9], allowing for conducting DMN-accelerated large-scale component simulations on commodity hardware. Extensions to treat thermomechanically coupled problems [1] and problems with fluctuating microstructure characteristics [8,10] are possible.…”

Section: Introductionmentioning

confidence: 99%

Material‐informed training of viscoelastic deep material networks

Gajek

Schneider

Böhlke

2023

Proc Appl Math and Mech

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Deep material networks (DMN) are a data‐driven homogenization approach that show great promise for accelerating concurrent two‐scale simulations. As a salient feature, DMNs are solely identified by linear elastic precomputations on representative volume elements. After parameter identification, DMNs act as surrogates for full‐field simulations of such volume elements with inelastic constituents. In this work, we investigate how the training on linear elastic data, i.e., how the choice of the loss function and the sampling of the training data, affects the accuracy of DMNs for inelastic constituents. We investigate linear viscoelasticity and derive a material‐informed sampling procedure for generating the training data and a loss function tailored to the problem at hand. These ideas improve the accuracy of an identified DMN and allow for significantly reducing the number of samples to be generated and labeled.

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A probabilistic virtual process chain to quantify process-induced uncertainties in Sheet Molding Compounds

Cited by 9 publications

References 114 publications

On the Phase Space of Fourth-Order Fiber-Orientation Tensors

On the Phase Space of Fourth-Order Fiber-Orientation Tensors

On the phase space of fourth-order fiber-orientation tensors

Material‐informed training of viscoelastic deep material networks

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