Cells exploit a phase transition to mechanically remodel the fibrous extracellular matrix

Through mechanical forces, biological cells remodel the surrounding collagen network, generating striking deformation patterns. Tethers—tracts of high densification and fibre alignment—form between cells, thinner bands emanate from cell clusters. While tethers facilitate cell migration and communication, how they form is unclear. Combining modelling, simulation and experiment, we show that tether formation is a densification phase transition of the extracellular matrix, caused by buckling instability of network fibres under cell-induced compression, featuring unexpected similarities with martensitic microstructures. Multiscale averaging yields a two-phase, bistable continuum energy landscape for fibrous collagen, with a densified/aligned second phase. Simulations predict strain discontinuities between the undensified and densified phase, which localizes within tethers as experimentally observed. In our experiments, active particles induce similar localized patterns as cells. This shows how cells exploit an instability to mechanically remodel the extracellular matrix simply by contracting, thereby facilitating mechanosensing, invasion and metastasis

On the critical nature of plastic flow: one and two dimensional models

Steady state plastic flows have been compared to developed turbulence because the two phenomena share the inherent complexity of particle trajectories, the scale free spatial patterns and the power law statistics of fluctuations. The origin of the apparently chaotic and at the same time highly correlated microscopic response in plasticity remains hidden behind conventional engineering models which are based on smooth fitting functions. To regain access to fluctuations, we study in this paper a minimal mesoscopic model whose goal is to elucidate the origin of scale free behavior in plasticity. We limit our description to fcc type crystals and leave out both temperature and rate effects. We provide simple illustrations of the fact that complexity in rate independent athermal plastic flows is due to marginal stability of the underlying elastic system. Our conclusions are based on a reduction of an over-damped visco-elasticity problem for a system with a rugged elastic energy landscape to an integer valued automaton. We start with an overdamped one dimensional model and show that it reproduces the main macroscopic phenomenology of rate independent plastic behavior but falls short of generating self similar structure of fluctuations. We then provide evidence that a two dimensional model is already adequate for describing power law statistics of avalanches and fractal character of dislocation patterning. In addition to capturing experimentally measured critical exponents, the proposed minimal model shows finite size scaling collapse and generates realistic shape functions in the scaling laws.Comment: 72 pages, 40 Figures, International Journal of Engineering Science for the special issue in honor of Victor Berdichevsky, 201

On the Energy Scaling Behaviour of Singular Perturbation Models with Prescribed Dirichlet Data Involving Higher Order Laminates

Motivated by complex microstructures in the modelling of shape-memory alloys and by rigidity and flexibility considerations for the associated differential inclusions, in this article we study the energy scaling behaviour of a simplified $m$-well problem without gauge invariances. Considering wells for which the lamination convex hull consists of one-dimensional line segments of increasing order of lamination, we prove that for prescribed Dirichlet data the energy scaling is determined by the \emph{order of lamination of the Dirichlet data}. This follows by deducing (essentially) matching upper and lower scaling bounds. For the \emph{upper} bound we argue by providing iterated branching constructions, and complement this with ansatz-free \emph{lower} bounds. These are deduced by a careful analysis of the Fourier multipliers of the associated energies and iterated "bootstrap arguments: based on the ideas from \cite{RT21}. Relying on these observations, we study models involving laminates of arbitrary order.Comment: 51 pages, 10 figures; improved lower bounds to avoid losses in scaling; periodic extension instead of whole space extension (Lemma 4.2); added perimeter contribution in Lemma 4.5; added Corollary 4.9; corrected some typo

On Scaling Properties for Two-State Problems and for a Singularly Perturbed T3T_3 Structure

In this article we study quantitative rigidity properties for the compatible and incompatible two-state problems for suitable classes of $\mathcal{A}$-free operators. In particular, in the compatible setting we prove that all first order, constant-rank, linear operators with affine boundary data which enforce oscillations yield the typical $\epsilon^{\frac{2}{3}}$-lower scaling bounds. As observed in \cite{CC15} for higher order operators this may no longer be the case. Revisiting the example from \cite{CC15}, we show that this is reflected in the structure of the associated symbols and that this can be exploited for a new Fourier based proof of the lower scaling bound. Moreover, building on \cite{RT22, GN04, PP04}, we discuss the scaling behaviour of a $T_3$ structure for the divergence operator. We prove that as in \cite{RT22} this yields a non-algebraic scaling law.Comment: 42 pages, comments welcom

On the error estimate of gradient inclusions

The numerical analysis of gradient inclusions in a compact subset of $2\times 2$ diagonal matrices is studied. Assuming that the boundary conditions are reached after a finite number of laminations and using piecewise linear finite elements, we give a general error estimate in terms of the number of laminations and the mesh size. This is achieved by reduction results from compact to finite case.Comment: 21 pages, 4 figure

Partition of unity-based discontinuous finite elements: GFEM, PUFEM, XFEM

International audienceIn this paper we review some basic notions of partition of unity-based discontinuous finite elements showing their relation to the Generalized Finite Element Method. A minimal one-dimensional example illustrates some of the issues related to the computer implementation of the method and highlights the relative simplicity of the approach. The ability of the approach in describing displacement discontinuities independently of the finite element mesh is shown in a classical crack propagation problem in an elastic medium. We also illustrate some limitations of this method when used in conjunction with the dummy stiffness approach.Dans cet article, nous présentons les principes élémentaires des éléments finis discontinus basés sur la méthode de la partition de l'unité en montrant leur relation avec la méthode des éléments finis généralisée. Les aspects liés à l'implantation dans un code ainsi que la simplicité de cette méthode sont illustrés sur un exemple unidimensionnel. La capacité de cette approche à représenter des discontinuités en déplacement indépendamment du maillage élément fini est montrée sur la propagation d'une fissure dans un milieu élastique. Nous montrons aussi certaines limitations de cette méthode quand elle est utilisée avec l'approche "dummy stiffness"

Multiscale resolution in the computation of crystalline microstructure

Summary.: This paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy. It is shown that branching of different variants near interfaces of twinned martensite and austenite phases leads to reduced energies in finite element approximations. Such behavior of minimizing deformations is understood for an extended model that involves surface energies. Moreover, the closely related question of the role of different growth conditions of the employed bulk energy is discussed. By explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders. For first order laminates the estimates are optima