## Elastic networks

Various elastic systems can be understood as networks of interconnected struts which deform by a combination of bending, stretching, twisting and shearing mechanisms. Examples include paper sheets, polymer gels, protein networks and cytoskeletal structures , crystal atomic lattices, granular materials, foams, wood, bones... Moreover, the pairwise interaction potentials used in standard elastic percolation models can also be identified with the strain energy of elastic struts. Despite extensive research, the connection between the mechanical properties of such networks on a macroscopic level and the description of their structures on a microscopic level has not been completely elucidated yet.

Interestingly, under identical loading conditions, some structures appear to deform primarily through the local stretching of the struts, while in other structures the elastic energy is stored via local bending (twisting and shearing contributions are usually neglected). For instance, "foam-like" cellular architectures tend to be bending-dominated, while fibrous architectures exhibit a rich mechanical behavior: computational studies of the two-dimensional elastic deformation of a network of cross-linked fibers have shown a transition from a nonaffine, bending-dominated regime to an affine, stretch-dominated regime with increasing density of fibers. Recent experimental studies, have confirmed this transition.

We performed an analytical study of the macroscopic properties of elastic networks of beams. Our analysis provides a convenient set of geometrical criteria to identify the networks which deform exclusively by extension or compression of their members. Values of the effective elastic moduli of such networks are also derived. These results shed light on the relationship between microscopic geometry and macroscopic mechanics of elastic networks. In particular, they explain the difference in mechanical behavior of cellular and fiber networks.

Interestingly, under identical loading conditions, some structures appear to deform primarily through the local stretching of the struts, while in other structures the elastic energy is stored via local bending (twisting and shearing contributions are usually neglected). For instance, "foam-like" cellular architectures tend to be bending-dominated, while fibrous architectures exhibit a rich mechanical behavior: computational studies of the two-dimensional elastic deformation of a network of cross-linked fibers have shown a transition from a nonaffine, bending-dominated regime to an affine, stretch-dominated regime with increasing density of fibers. Recent experimental studies, have confirmed this transition.

We performed an analytical study of the macroscopic properties of elastic networks of beams. Our analysis provides a convenient set of geometrical criteria to identify the networks which deform exclusively by extension or compression of their members. Values of the effective elastic moduli of such networks are also derived. These results shed light on the relationship between microscopic geometry and macroscopic mechanics of elastic networks. In particular, they explain the difference in mechanical behavior of cellular and fiber networks.

## Transport networks

Finding the most efficient transport network is an issue arising in a wide variety of contexts. One can cite, among others, the water, natural gas and power supply of a city, telecommunication networks, rail and road traffic, and more recently the design of labs-on-chips or microfluidic devices. Moreover, this problem also appears in theoretical works intending to describe the architecture of the vascular systems of living organisms. Generally speaking, consider a set of sources and sinks embedded in a two- or three-dimensional space, their respective number and locations being fixed. The flow rates into the network from each source, and out of the network through each sink, are also given. The problem consists in interconnecting the sources and sinks via possible intermediate junctions, referred to as additional nodes, in the most efficient way. That is, to minimize a cost function of general form $\sum_{k} w_k f(i_k)$ , where the summation is over all the links that constitute the network. $w_k$ is the "weight" associated with the $k^{th}$ link, and is some function of the flow rate $i_k$ carried by this link. Minimization of the cost function can be done over different optimization parameters and with different constraints.

We have studied the structure of pipe networks that minimize the total dissipation rate $U=\sum_{k}r_k i_k^2$ is studied, where the weight $r_k$ is the "flow resistance" of pipe $k$, defined as: $r_k=\rho ~ l_k/ s_k^m$ , $\rho$ being some positive constant, $l_k$ and $s_k$ the length and cross-sectional area of each pipe respectively, and $m$ a positive constant characterizing the flow profile. For most flows encountered in physics, $m \geq 1$ . We have characterized the geometrical and topological features of such networks.

We have studied the structure of pipe networks that minimize the total dissipation rate $U=\sum_{k}r_k i_k^2$ is studied, where the weight $r_k$ is the "flow resistance" of pipe $k$, defined as: $r_k=\rho ~ l_k/ s_k^m$ , $\rho$ being some positive constant, $l_k$ and $s_k$ the length and cross-sectional area of each pipe respectively, and $m$ a positive constant characterizing the flow profile. For most flows encountered in physics, $m \geq 1$ . We have characterized the geometrical and topological features of such networks.

## Statistical physics of cellular systems

Cellular materials are interesting both as disordered media with well-defined structural elements, and as models for more complex systems such as biological tissues. Among them, foams are ubiquitous in our daily lives and in many industries. Bubble monolayers are much easier to observe and to study. Such quasi two-dimensional foams are characterised by their number of bubbles, \(N\), area distribution, $p(A)$, and number-of-sides distribution, $p(n)$. When the liquid fraction is very low (“dry” foams), their bubbles are polygonal, with shapes that are locally governed by Plateau’s laws. Each side is a thin liquid film with a uniform surface tension; its curvature is determined by the difference of pressure between the two bubbles it separates. Bubble size distribution and packing (or “topology”) are crucial in determining e.g. rheological properties or coarsening rate. When a foam is shuffled (either mechanically or thermally), $N$ and $p(A)$ remain fixed, but bubbles undergo “T1” neighbour changes, which induce a random exploration of the foam configurations.

In two-dimensional foams, we explore the relation between the distributions of bubble number-of-sides (topology) and bubble areas (geometry). We develop a statistical model which takes into account physical ingredients based on Plateau laws. The model predicts that the mean number of sides of a bubble with area $A$ within a foam sample with moderate size dispersity is given by:

\[\bar{n}(A) = 3\left(1+\dfrac{\sqrt{A}}{\langle \sqrt{A} \rangle} \right),\]

where $\langle . \rangle$ denotes the average over all bubbles in the foam. The model also relates the \textit{topological disorder} $ \Delta n / \langle n \rangle =\sqrt{\langle n^2 \rangle - \langle n \rangle^2}/\langle n \rangle$ to the (known) moments of the size distribution:

\[\left(\dfrac{\Delta n}{\langle n \rangle}\right)^2=\frac{ 1 }{4}\left(\langle A^{1/2} \rangle \langle A^{-1/2} \rangle+\langle A \rangle \langle A^{1/2} \rangle^{-2} -2 \right).\]

Extensive data sets arising from experiments and simulations all collapse surprisingly well on a straight line, even at extremely high values of geometrical disorder. At the other extreme, when approaching the perfectly regular honeycomb pattern, we identify and quantitatively discuss a crystallisation mechanism whereby topological disorder vanishes.

In two-dimensional foams, we explore the relation between the distributions of bubble number-of-sides (topology) and bubble areas (geometry). We develop a statistical model which takes into account physical ingredients based on Plateau laws. The model predicts that the mean number of sides of a bubble with area $A$ within a foam sample with moderate size dispersity is given by:

\[\bar{n}(A) = 3\left(1+\dfrac{\sqrt{A}}{\langle \sqrt{A} \rangle} \right),\]

where $\langle . \rangle$ denotes the average over all bubbles in the foam. The model also relates the \textit{topological disorder} $ \Delta n / \langle n \rangle =\sqrt{\langle n^2 \rangle - \langle n \rangle^2}/\langle n \rangle$ to the (known) moments of the size distribution:

\[\left(\dfrac{\Delta n}{\langle n \rangle}\right)^2=\frac{ 1 }{4}\left(\langle A^{1/2} \rangle \langle A^{-1/2} \rangle+\langle A \rangle \langle A^{1/2} \rangle^{-2} -2 \right).\]

Extensive data sets arising from experiments and simulations all collapse surprisingly well on a straight line, even at extremely high values of geometrical disorder. At the other extreme, when approaching the perfectly regular honeycomb pattern, we identify and quantitatively discuss a crystallisation mechanism whereby topological disorder vanishes.

## Morphogenesis of the tubular network of *Physarum Polycephalum*

Throughout life, adequate adaptation and growth of vascular networks are essential for our well being. In many diseases,

We study experimentally the vascular morphogenesis of model biological systems.

*i.e.*cancer, hypertension, macular degeneration, diabetic retinopathy, etc., vascular adaptation is impaired. Understanding how vascular networks form might help to develop better treatment strategies or maybe even ways to prevent these diseases. Vascular morphogenesis is partly determined by self organized actions of shear stresses generated by the circulating blood and mechanical stresses generated by the growing, deforming surrounding tissue. The core of this work is to study the role of mechanical self-organized processes in growing tissue for vascular morphogenesis and to identify the basic physical laws that shape the vascular networks.We study experimentally the vascular morphogenesis of model biological systems.

To watch a time lapse movie of Physarum Polycephalum growing on a map of France, click here.