Lev Truskinovsky’s research while affiliated with French National Centre for Scientific Research and other places

The simplest model of contraction-driven self-propulsion of keratocytes reduces to a one dimensional Keller-Segel system with free boundaries. This "active" model involving both dissipation and anti-dissipation features stationary and traveling wave solutions representing static and moving cells, respectively. The goal of this paper is to provide the first rigorous proof of the asymptotic nonlinear stability of both of such solutions. In the case of stationary solutions, the linear stability is established using the spectral theorem for compact, self-adjoint operators, and thus linear stability is determined solely by eigenvalues. For traveling waves the picture is more complex because the linearized problem is non-self-adjoint, opening the possibility of a "dark" area in the phase space which is not "visible" in the purely eigenvalue/eigenvector approach. To establish linear stability in this case we employ spectral methods together with the Gearhart-Pruss-Greiner Theorem, which requires a uniform bound on the resolvent of the linear operator. For both stationary and traveling wave solutions, nonlinear stability is then proved by showing how the nonlinear part of the problem may be controlled by the linear part and then employing a Gr\"onwall inequality argument. The developed methodology can prove useful also in other problems involving non-Hermitian and non-reciprocal operators which are ubiquitous in the description of active matter.

We consider a version of the classical Hamiltonian Fermi–Pasta–Ulam (FPU) problem with a trilinear force–strain relation of soft–hard–soft type that is in general non-symmetric. In addition to the classical spatially localized solitary waves, such hardening–softening model also exhibits supersonic kinks and finite-amplitude, spatially delocalized flat-top solitary waves that acquire the structure of a kink–antikink bundle when their velocity approaches the kink limit. Exploiting the fact that traveling waves are periodic modulo shift by a lattice spacing, we compute these solutions as fixed points of the corresponding nonlinear map and investigate how their properties depend on the parameter measuring the asymmetry of the problem. In a particularly interesting case when one of the soft regimes has zero elastic modulus, we obtain explicit solutions for sufficiently slow solitary waves. In contrast to conventional delocalization in the sonic limit, the corresponding compact structures mounted on a constant background become localized at the lattice scale as their velocity tends to zero. Numerical simulations of Riemann-type initial value problem in this degenerate limit show the emergence of Whitham shocks that involve periodic trains of solitary waves. We investigate stability of the obtained solutions using direct numerical simulations and Floquet analysis. We also obtain explicit solutions for a quasicontinuum model that captures the most important features of the discrete problem.

We use molecular dynamics to show that plastic slip is a crucial component of the transformation mechanism of a square-to-triangular structural transition. The latter is a stylized analog of many other reconstructive phase transitions. To justify our conclusions we use a novel atomistically-informed mesoscopic representation of the field of lattice distortions in molecular dynamics simulations. Our approach reveals a hidden alternating slip distribution behind the seemingly homogeneous product phase which points to the fact that lattice invariant shears play a central role in this class of phase transformations. While the underlying pattern of anti-parallel displacements may also be interpreted as microscopic shuffling, its precise crystallographic nature strongly suggests the plasticity-centered interpretation.

We argue that nucleation of brittle cracks in initially flawless soft elastic solids is preceded by a nonlinear elastic instability, which cannot be captured without accounting for geometrical precise description of finite elastic deformation. As a prototypical problem we consider a homogeneous elastic body subjected to tension and assume that it is weakened by the presence of a free surface which then serves as a site of crack nucleation. We show that in this maximally simplified setting, brittle fracture emerges from a symmetry breaking elastic instability activated by softening and involving large elastic rotations. The implied bifurcation of the homogeneous elastic equilibrium is highly unconventional for nonlinear elasticity as it exhibits an extraordinary sensitivity to geometry, reminiscent of the transition to turbulence in fluids. We trace the post-bifurcational development of this instability beyond the limits of applicability of scale free continuum elasticity and use a phase-field approach to capture the scale dependent sub-continuum strain localization, signaling the formation of actual cracks.

We consider a version of the classical Hamiltonian Fermi-Pasta-Ulam (FPU) problem with a trilinear force-strain relation of soft-hard-soft type that is in general non-symmetric. In addition to the classical spatially localized solitary waves, such hardening-softening model also exhibits supersonic kinks and finite-amplitude, spatially delocalized flat-top solitary waves that acquire the structure of a kink-antikink bundle when their velocity approaches the kink limit. Exploiting the fact that traveling waves are periodic modulo shift by a lattice spacing, we compute these solutions as fixed points of the corresponding nonlinear map and investigate how their properties depend on the parameter measuring the asymmetry of the problem. In a particularly interesting case when one of the soft regimes has zero elastic modulus, we obtain explicit solutions for sufficiently slow solitary waves. In contrast to conventional delocalization in the sonic limit, these compact structures mounted on a constant background become localized at the lattice scale as their velocity tends to zero. Numerical simulations of Riemann-type initial value problem in this degenerate model show the emergence of Whitham shocks that involve periodic trains of solitary waves. We investigate stability of the obtained solutions using direct numerical simulations and Floquet analysis. We also obtain explicit solutions for a quasicontinuum model that captures some important features of the discrete problem.

Taking inspiration from the crawling motion of biological cells on a substrate, we consider a physical model of self-propulsion where the spatio-temporal driving can involve both, a mechanical actuation by active force couples, and a chemical actuation through controlled mass turnover. We show that the competition and cooperation between these two modalities of active driving can drastically broaden the performance repertoire of the crawler. When the material turnover is slow and the mechanical driving dominates, we find that the highest velocity at a given energetic cost is reached when actuation takes the form of an active force configuration propagating as a traveling wave. As the rate of material turnover increases, and the chemical driving starts to dominate the mechanical one, such a peristalsis-type control progressively loses its efficacy, yielding to a standing wave type driving which involves an interplay between the mechanical and chemical actuation. Our analysis suggests a new paradigm for the optimal design of crawling biomimetic robots where the conventional purely mechanical driving through distributed force actuators is complemented by a distributed chemical control of the material remodeling inside the force-transmitting machinery.