Analytical Approach to Quantifying the Non-Affine Behavior of Fiber Networks
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1060-LL09-03
Analytical Approach to Quantifying the Non-Affine Behavior of Fiber Networks Hamed Hatami-Marbini and Catalin Picu Rensselaer Polytechnic Institute, Troy, NY, 12180 Abstract Fiber networks deform nonaffinely due to their inhomogeneous structure. The degree of nonaffinity depends on many factors such as the spatial distribution and properties of fibers, the nature of the applied load, the type of joints and the length scale of observation. The “homogenized” response on given scale depends on the non-affine mechanics on all sub-scales. Here, a method is developed to map the non-affine mechanics of a regular network with a large density of defects into an equivalent continuum, which is then used to determine the homogenized elastic properties of the network. This semi-analytic method establishes a relationship between the structure of the network and its overall elastic properties. Furthermore, we develop a method to quantify the nonaffine strains in a random network and use it to study the dependence of the degree of non-affinity to the scale of observation as well as the dependence on the network architecture. 1. Introduction Fiber networks find many applications in diverse industries and products such as tissue templates, electrochemical substrates, paper products, thermal or sound insulation and cell structure among many others. Although the mechanical behavior of the fiber network may not be of primary interest in all of these applications, it needs to be well understood to predict their intended functionality. In order to study the mechanical behavior of such fiber networks, one has to start from the microstructure. If we assume that the structure behaves affinely under a uniform far field loading, a closed form relation can be obtained for the mechanical properties of the network [1-4]. However, it is broadly observed that this assumption is not accurate. The nonaffine deformation of a structure takes place at a lower energy level compared to the energy of the same structure with affine deformation. We begin analyzing this problem by considering a regular fiber network with a large number of randomly placed defects, i.e. missing links. The deformation is nonaffine due to the presence of defects. A unified analytical method that may address this type of problem does not exist. These issues have been studied mostly by means of numerical models [5-9]. Our goal here is to develop a formulation which enables us to map the deformation of the network to that of an equivalent continuum. The continuum has the same elastic constants as the perfect, non-defective network and is populated by point sources that reproduce the eigenfield of the actual network defects. To facilitate this representation, the eigenfield of each “elementary defect” is decomposed in terms of an independent basis. The component fields of this basis have physical meaning: each is the field produced by a force dipoles of various order [10]. Then, the decomposition reduces to finding the coefficients of this series. These coefficients are
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