RVE Problem: Mathematical aspects and related stochastic mechanics

Karimi, Pouyan; Malyarenko, Anatoliy; Ostoja–Starzewski, Martin; Zhang, Xian

doi:10.1016/j.ijengsci.2019.103169

Cited by 14 publications

(8 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In 2014-2020 the authors of the survey published a series of papers and books [52,65,67,70,68,69,71,73,98] in collaboration with colleagues, where they extended the results described above. We describe the above extensions in Sections 4 and 5.…”

Section: A Short History Of the Topicmentioning

confidence: 84%

“…Solutions in both cases have been obtained via a cellular automata method which (i) allows assignment of heterogeneous material properties from cell to cell and (ii) in the limit of infinitesimal cells, is analytically equivalent to the continuum elastodynamics equations [86]. Reprinted from [52] by permission of Elsevier.…”

Section: From a Random Microstructure To Mesoscale Responsementioning

confidence: 99%

See 1 more Smart Citation

Tensor- and spinor-valued random fields with applications to continuum physics and cosmology

Malyarenko¹,

Ostoja–Starzewski²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp. rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp. its tensor square), making the field isotropic.Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic.For convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix.

show abstract

Section: A Short History Of the Topicmentioning

confidence: 84%

Section: From a Random Microstructure To Mesoscale Responsementioning

confidence: 99%

Tensor- and spinor-valued random fields with applications to continuum physics and cosmology

Malyarenko¹,

Ostoja–Starzewski²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In addition, more focus concentrated to the relationship between fiber VF and numerical results are in RVE scale, [ 62–65 ] while in EUC scale, people pay more attention to the influence of interface. [ 66–69 ] Therefore, in ilka scales, we also discussed the strain energy, mean normal stress and shear stress, while d i / d CF = 2 with

V^{RVE}

is 3%, 10%, 30% respectively in RVE, and

V^{EUC} = 10 %

with d i / d CF is 1, 1.5, 2.0 and 2.5 respectively in EUC.…”

Section: Resultsmentioning

confidence: 99%

A dual‐scale homogenization method to study the properties of chopped carbon fiber reinforced epoxy resin matrix composites

Zheng

Qiu

et al. 2022

Polymer Composites

View full text Add to dashboard Cite

In this work, a dual‐scale homogenization (DSH) procedures including the representative volume element (RVE) and Unit cell (UC) with Halpin–Tsai models in the UC scale and space direction‐sensitive Hotelling expansion (SDSHE) models in RVE scale separately. These two methods are introduced and implemented to predict the response of effective elastic properties in chopped fiber reinforced epoxy matrix structures. A post‐processing scheme of subdomain in UC scale based on Gauss theorem is proposed in SDSHE method, which is used to extract the average strain and stress in UC three‐dimensional structure. In addition, the X‐ray computed tomography was used to determine the fourth order tensor orientation angle and the probability distribution information of chopped fiber length in testing samples. Homogenization properties in RVE scale, and inclusion volume fraction (VF), thickness of interface and orientation tensors are designed in UC scale and compared. The excellent correlations with experiments prove the effectiveness of the proposed models. Furthermore, the algorithm proposed in this paper realizes the homogenization of the two scales in the process of numerical analysis, and it will effectively take into account many problems that cannot be achieved by other algorithms, such as the path strain of the model, the thickness of the interface, and so on.

show abstract

“…To facilitate comparison, we correspond the hypothetical scale model of the present approach to the fine-scale model of the AM&CM-based approach, and somewhat more loosely, component hypomodels and relation models of the present approach to coarse-grained models and transfer operators of the AM&CM-based approach respectively. Credibility assessment of an AM&CM-based model involves verification of mathematical and thermodynamic consistency of each transfer operator [ 10 , 13 ], which highlights the unique role the laws of conservation of energy and entropy play in AM&CM-based approaches. The fine-scale model and the transfer operators carry within themselves empirical knowledge of the system being modelled (e.g.…”

Section: Discussionmentioning

confidence: 99%

“…In systems with periodicity or macrohomogeneity, the size of the representative volume element (RVE) defines the characteristic finer scale. It should be noted that identifying the RVE size is not always straightforward, especially in the presence of material randomness [ 10 , 13 ]. In practical uses of this approach, the requirement for asymptotic scale separation (e.g.…”

Section: Introductionmentioning

confidence: 99%

A systematic approach to the scale separation problem in the development of multiscale models

Bhattacharya

Lacroix

et al. 2021

PLoS ONE

View full text Add to dashboard Cite

Throughout engineering there are problems where it is required to predict a quantity based on the measurement of another, but where the two quantities possess characteristic variations over vastly different ranges of time and space. Among the many challenges posed by such ‘multiscale’ problems, that of defining a ‘scale’ remains poorly addressed. This fundamental problem has led to much confusion in the field of biomedical engineering in particular. The present study proposes a definition of scale based on measurement limitations of existing instruments, available computational power, and on the ranges of time and space over which quantities of interest vary characteristically. The definition is used to construct a multiscale modelling methodology from start to finish, beginning with a description of the system (portion of reality of interest) and ending with an algorithmic orchestration of mathematical models at different scales within the system. The methodology is illustrated for a specific but well-researched problem. The concept of scale and the multiscale modelling approach introduced are shown to be easily adaptable to other closely related problems. Although out of the scope of this paper, we believe that the proposed methodology can be applied widely throughout engineering.

show abstract

RVE Problem: Mathematical aspects and related stochastic mechanics

Cited by 14 publications

References 32 publications

Tensor- and spinor-valued random fields with applications to continuum physics and cosmology

Tensor- and spinor-valued random fields with applications to continuum physics and cosmology

A dual‐scale homogenization method to study the properties of chopped carbon fiber reinforced epoxy resin matrix composites

A systematic approach to the scale separation problem in the development of multiscale models

Contact Info

Product

Resources

About