A continuum model for two-dimensional fiber networks
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A continuum model for two-dimensional fiber networks Eveline Baesu1, Minrong Zheng and Doina Beljic Department of Engineering Mechanics University of Nebraska-Lincoln Lincoln, NE 68588 ABSTRACT Elastic-plastic deformation of a continuum formed by continuously distributed fibers is described. Applications to the mechanical characterization of nanofibers, and to biological materials such as cellular cytoskeleton and tissue scaffolds are indicated.
INTRODUCTION A brief account of the mechanics of ideally flexible elastic-plastic fibers is given. The model is based on the idea that the fibers are continuously distributed, as in some existing theories of networks. Although many aspects of this formulation translate easily to 3D, only 2D aspects of it are considered here. Moreover, for illustrative purposes only, the case of two fiber families is considered in more detail. In this formulation, the properties of the continuum can be inferred from those of each fiber family. The converse is also true provided that the number of fiber families is three or less. These features make the theory very amenable to experimental treatment. The present theory has many applications including fibrous materials without a significant matrix, such as paper, and certain textiles, as well as to biological materials such as cellular cytoskeleton and tissue scaffolds. The theory presented here is for regular orientations of fiber families with no sliding between the fibers, but it is easily extended to the case of a certain distribution function of random fiber orientation, and also to frictional sliding. These results will appear in a upcoming publication. However, the theory presented here is sufficient to illustrate the essential qualitative features of the theory.
BACKGROUND The mathematical modeling of such a continuum dates back to Tchebychev [1], who studied an orthogonal network of inextensible fibers in the context of the “clothing problem” – mapping an orthogonal network to a given shape. Over the past forty years, the literature has grown considerably. Examples are the work of Stoker [2], Rivlin [3], and Adkins [4], Pipkin [5-7] for inextensible fibers. For extensible fiber systems we cite the work of Steigmann [8], for elastic fibers and McLaughlin [9] and Baesu [10], for the elastic-plastic case, and the references cited therein.
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Corresponding Author
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KINEMATICS At this stage, we consider a 2D continuum consisting of n distinct fiber families. Let I = {1,…,n} be the index set of fibers, and let Li(X), i∈I be the field of unit vectors tangential to the i-th family of fibers in the reference plane Ω, and let li(X), i∈I, be the field of unit tangents to the same fibers after deformation. Here X, and x are the position vector of a material point in the reference and present configuration, respectively. We have X = Xαeα, α = 1,2, and x = xjej, j = 1,2,3, in which {ej} is a fixed orthonormal basis. The deformation of such a continuum would map a 2D surface into a 3D one as we allow for the possibility of a curved
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