Project Sponsor
Procter&Gamble
www.pg.com
Faculty Participants
Catalin R. Picu
Professor
Current and Former Students
Hamed Hatami-Marbini
Monica Soare
Ali Shahsavari
Introduction
Random fiber networks are the building blocks of many biological and non-biological
systems. They generally play a structural role, but may also contribute to transport. The
major advantages provided by designs using fibers are: lightweight (with a small volume
of material one can span and provide stiffness over large embedding volumes), adjustable
stiffness obtained by anisotropic/preferential fiber distributions, large deformations
without loss of carrying capacity.
In the non-living world, paper and various cellulose structures used for fluid
absorption and retention or as insulators are made from cellulose fibers. The fibers are
randomly distributed and may be bonded to each other or not. Rubber is a material owing
its exceptional properties to the molecular network forming its structure. Various
consumer products such as baby diapers and wipes are made from non-woven random
fiber networks of polymeric fibers.
Fiber networks are broadly encountered in the living world as well. The
cytoskeleton of eukaryotic cells is composed from a set of interpenetrating semi-flexible
fiber networks: F-actin fibers, microtubules and intermediate filaments. The cytoskeleton
performs multiple functions providing a “scaffold” structure to the cell. The microtubules
act also as guides for transport across the cytoplasm to the nucleus, while the entire
structure is thought to contribute in an essential way to chemo-mechanical transduction.
Figure 1 shows a clump of polypropylene fibers which are not bonded to each
other, but long enough to be entangled, and a fibrous tissue scaffold (from Chandran and
Barocas, J. Biomech. Eng. 128, 259, (2006)).
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Elongation [mm]
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Figure 1. a) Clump of entangled polypropylene fiber network, b) tissue scaffold.
Most tissues have a fiber network-type microstructure. For example, the extracellular
matrix has a collagen network component, and cartilage is a network infiltrated with
water, proteoglycans and other macromolecules, with essential role in controlling wear
and friction.
The objective of this program is to develop methodologies to predict the mechanics of
large, stochastic fiber networks starting from the behavior of individual fibers and the
multiscale structure. Such understanding and modeling and simulation capability are
desirable in the design of many consumer products, the design of tissue scaffolds and
understanding the mechanics of complex networks, such as the cytoskeleton.
Cross-Linked Fiber Networks
The results to date include:
a) the discovery of long range correlations in the mechanical fields defined over
network domains [1],
b) the development of a method to address mechanics problems over domains with
heterogeneous microstructure exhibiting long range correlations of the elastic
moduli [3,4],
c) the characterization of the non-affine deformation of networks of semi-flexible
fibers [4,5].
Figure 2 shows a schematic of a system constructed by dropping fibers of same length, at
random, in a square domain and bonding them at all crossing points. Consider a very
large system composed from millions of fibers for which the solution of a boundary value
problem is sought. For example, the model may represent a gel or a tissue scaffold
subjected to some generic boundary conditions. One seeks the stress/strain state at some
point inside the domain produced by this boundary excitation.
In principle, one may solve the problem constructing a large model in which all
fibers are explicitly represented. However, this option is quickly ruled out as the size of
the system increases. In addition, because the problem is stochastic, many realizations
need to be considered. The problem becomes intractable very quickly. Recently we
developed the physical insight and the computational methodology that makes possible
approaching the solution at a fraction of the cost of this “brute force” procedure.
Fig. 2 Large network subjected to boundary conditions. When zooming in, the
structure appears self-similar. This is seen by overlapping a regular grid and
computing the fiber density and effective stiffness over each square.
The density of fibers computed over each square patch in Fig. 2 has fluctuations
that scale as a power law of the probing length scale and is long-range, power law
correlated.
a)
b)
Fig. 3 (a) Distribution of effective elastic moduli over a network domain with  = 0.02. The network
contains ~4000 fibers of length L0 = 0.1 (lengths normalized with the size of the entire domain). (b)
Auto-correlation function (ACF) of Young’s modulus (computed from maps of type 3(a)) showing
power law scaling over a range of scales. Data for networks of various density lc (represents the
mean segment length) and probed at various length scales, , are shown. The range of the power
law scaling can be increased by considering longer fibers and considerably larger models.
Since the stiffness scales with the density, the same properties are expected for the
elastic constants as well. To demonstrate this property, the stiffness is computed
independently of the density for each square patch in Fig. 2, without “removing” the
patch from the network. Figure 3(a) shows a map of Young’s modulus for such network.
Further, this information is used to evaluate the auto-correlation function (ACF) of the
local stiffness (Fig. 3(b)). The range of scales over which this scaling property is
observed is bounded below by the mean segment length (lc) and above by the entire fiber
length L0, in systems of chopped fibers, or by the fiber persistence length in systems with
“infinite” fibers.
Solving boundary value problems defined over domains having stochastic microstructure
and long-range correlated elastic constants is not trivial. We developed a method to
address such cases. The method was developed for materials with fractal microstructure
(see also the project on “Mechanics of Materials with Fractal Microstructure”) and
adapted to address the network problem discussed here. It is based on the stochastic finite
element method. An example is shown in Fig. 4: a rectangular domain is subjected to
uniaxial tension with the tensile force being non-uniform across the model width. Fig. 4b
shows the mean of the resulting displacement across the width.
a)
b)
Fig. 4. a) fiber network domain subjected to uniaxial loading, and b) distribution of
(mean) displacements across the width of the model.
This method requires only one solution and applies for any probing length scale.
Obtaining the solution takes less than an hour on a single processor, while using the
“brute force” method takes weeks.
The deformation of random fiber networks is non-affine, i.e. the local strain is
different from the global strain. This is due to the extreme heterogeneity of the network.
The implication is that the global response is more compliant than predicted based on the
affine deformation assumption. We investigated the variation of the degree of nonaffinity with the scale of observation and determined that as one zooms in (decreasing
delta), the non-affinity increases as a power law of delta. The variation of the strain-based
measure of non-affinity Hij (d) = < (eij - eijaffine )2 > for a dense network is shown in Fig.
5.
Fig. 5. Power law scaling of the strain non-affinity measure H.
[1] R.C. Picu and H. Hatami-Marbini, Heterogeneous long range correlated deformation
of semi-flexible random fiber networks, Phys. Rev. E, 80, 046703 (2009).
[2] M.A. Soare, and R.C. Picu, Boundary value problems defined on stochastic selfsimilar multiscale geometries, Int. J. Num. Meth. Eng. 74, 668-696 (2007).
[3] M.A. Soare, and R.C. Picu, Spectral decomposition of random fields defined over the
generalized Cantor set, Chaos, Solit. Fract. 37, 566-573 (2008).
[4] H. Hatami-Marbini, and R. C. Picu, Scaling of non-affine deformation in random
semi-flexible fiber networks Phys. Rev. E 77, 062103 (2008).
[5] H. Hatami-Marbini, R.C. Picu, Effect of fiber orientation on the non-affine
deformation of random fiber networks, Acta Mechanica, 205, 77-84 (2009).
Entangled Fiber Networks
Entangled fiber networks preserve all interesting features of the bonded (cross-linked)
networks, with some additional challenges. We are currently studying the statistics of the
deformation in fiber masses subjected to compression and/or shear. An example of such
system (a model) is shown in Fig. 6.
Fig. 6. a) Ensemble of entangled non-bonded fibers, and b) energy per fiber under
compression of the system in a). The effect of friction between fibers is shown.