An algorithm for generating microstructures of fiber‐reinforced composites with long fibers

Schneider, Matti

doi:10.1002/nme.7110

Cited by 10 publications

(3 citation statements)

References 91 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For a start, it permits assessing the realizability of a given fourth-order tensor as a fiber-orientation tensor in a simple and straightforward way. Consider, for instance, fiber-microstructure generators [18,19,78], which serve as the basis for full-field simulations and where the fourth-order fiberorientation tensor serves as the input. Then, if the prescribed tensor does not belong to the set Cand(d), it cannot be realized by a fiber microstructure at all, independent of the size of the microstructure and the number of considered fibers.…”

Section: Discussionmentioning

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

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.

show abstract

Section: Discussionmentioning

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

“…For a start, it permits assessing the realizability of a given fourth-order tensor as a fiber-orientation tensor in a simple and straightforward way. Consider, for instance, fiber-microstructure generators [18,19,73], which serve as the basis for full-field simulations and where the fourth-order fiber-orientation tensor serves as the input. Then, if the prescribed tensor does not belong to the set Cand(d), it cannot be realized by a fiber microstructure at all, independent of the size of the microstructure and the number of considered fibers.…”

Section: Discussionmentioning

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.

show abstract

“…With the development of fiber art in handicraft design, fiber art attracts people's attention with its unique practicality, artistry and ecology and gradually becomes the emotional bond that connects people with handicraft design [4]. The development of traditional Chinese folk crafts is a long-term and arduous project that requires the joint efforts of the state, society and individuals.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Consumer Perceived Needs of Fiber Arts and Crafts in the Internet Era with Perceptual Engineering as an Orientation

Wang,

Abidin,

Shaari

et al. 2023

Applied Mathematics and Nonlinear Sciences

View full text Add to dashboard Cite

Starting from the construction of the perceptual engineering model, this paper gives the research objective of consumer perceived demand for fiber art handicrafts in the Internet era and designs the corresponding perceived SD scale according to consumer perceived demand. Factor analysis, the results show that for embroidery handicrafts, more attention is placed on grad in consumer perceived needs, with a factor loading eigenvalue of 4.647, and the grade variability of embroidery handicrafts is 27.336%. The consumer perception under the cluster 1 category of handmade blankets is mainly the grade, and its percentage is the highest at 26.698%, and the influence factor of style in bamboo and grass woven products is the highest at 23.078%. The design of fiber art handicrafts oriented to perceptual engineering needs to fully explore the consumer’s perception needs in order to promote innovation and development of fiber art handicrafts.

show abstract

An algorithm for generating microstructures of fiber‐reinforced composites with long fibers

Cited by 10 publications

References 91 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

Analysis of Consumer Perceived Needs of Fiber Arts and Crafts in the Internet Era with Perceptual Engineering as an Orientation

Contact Info

Product

Resources

About