arXiv subjects

Lev Truskinovsky

Publications and source records attributed to Lev Truskinovsky.

At least 19 recordsLinked to original sources

arXiv2026

Elastodynamics from a variational standpoint: integral equalities and inequalities

As it is well known, solutions of equations of nonlinear elastodynamics, representing extremals of the action functional, can form shocks. We adapt the classical approach of Emmy Noether to such singular extremals and derive the appropriately generalized integral relations within Calculus of Variations. We apply them to elastodynamical extremals with shocks, obtaining new integral relations involving kinetic and elastic energies. For the extremals representing thermodynamically admissible (entropy) solutions of the corresponding hyperbolic Euler-Lagrange equations, the classical equalities, characterizing the variational approach of Noether, expectedly transform into ineqalities. We show that, rather remarkably, despite the crucial role of material velocity in the fully inertial energy redistribution processes, the corresponding kinetic energy can be completely eliminated from the expression for the dynamically stored elastic energy, even in the presence of shocks.

math-ph↗

arXiv2026

Clapeyron-type theorems in nonlinear elasticity

Clapeyron's Theorem of classical linear elasticity provides a way to explicitly express the energy stored in an equilibrium configuration in terms of the work of the forces applied on the boundary. We derive several new integral relations which can be viewed as nonlinear analogs of this classical result, reinterpreting them as rather general statements within Calculus of Variations. These relations reflect specific properties of Lagrangians, that we call ``partial variational symmetries'', since they are more general than classical variational symmetries. In the framework of nonlinear elasticity, partial variational symmetries, such as scale invariance, or scaling homogeneity, lead, via Noether's analysis to different nonlinear generalizations of Clapeyron's Theorem that combine naturally the work of physical and configurational forces. We present a series of illuminating examples showing the effectiveness of the obtained general results in different problems of nonlinear elasticity.

math-ph↗

arXiv2025

Self-induced marginality in plastically deformed crystals

Quasi-brittle plastic yielding is a salient feature of well-annealed glassy materials. Here we show that the same behavior is characteristic of perfect crystals after they experience mechanically driven elastic instability leading to massive nucleation of dislocations. We argue that such 'preparation' effectively converts an atomic configuration from crystalline to quasi-amorphous. To understand the nature of the subsequent intermittent mechanical response we study a model 2D crystal subjected to AQS driving and show that both pre- and post-yield dislocation avalanches exhibit power law statistics with similar exponents indicative of self-induced marginal stability.

cond-mat.mtrl-sci↗

arXiv2025

Micro-displacement tensor

We propose an extended kinematics of nominally elastic continuum solids allowing one to describe their mechanical interaction with micro-scale loading devices. The main new ingredient is the concept of a micro-displacement tensor which extends the conventional description of the deforming elastic solids in terms of macroscopic displacement vectors. We show that micro-displacement tensors are particularly useful in dealing with active incompatibility acquisition and its subsequent passive relaxation. We use the proposed approach to describe the energetics of surface deposition while accounting for the presence of micro-mechanical controls.To illustrate the effectiveness of the new conceptual scheme we present two case studies: crystallization from a melt resulting in pre-stress, and winding of a coil with controlled pre-stretch.

cond-mat.soft↗

arXiv2025

Generalized Clapeyron's theorem

Clapeyron's Theorem in classical linear elasticity provides a way to explicitly express the energy stored in an equilibrium configuration in terms of the work of the forces applied on the boundary. We derive several new integral relations which can be viewed as nonlinear analogs of this classical result, reinterpreting them as rather general statements within Calculus of Variations. In the framework of nonlinear elasticity these relations reflect various partial symmetries of the material response, for instance, scale-invariance or scaling homogeneity. In particular, when the energy functional is scale-free, the obtained result can be interpreted as the Generalized Clapeyron's Theorem (GCT). Remarkably, it combines rather naturally the work of physical and configurational forces. We present a series of illuminating case studies showing the variety of applications of various obtained relations in different seemingly unrelated problems of mechanics.

math-ph↗

arXiv2025

Inertia-induced power-law scaling in martensites

Martensites subjected to quasistatic deformation are known to exhibit power law distributed acoustic emission in a broad range of scales, however, the origin of the observed scaling behavior and the mechanism of self-organization towards apparent criticality remains obscure. Here we argue that the power law structure of intermittent fluctuations can be at least partially interpreted as an effect of inertia. We build on the insight that inertial dynamics, evidenced by acoustic emission, can become an important factor if the underlying mechanical system is only marginally stable. We first illustrate the possibility of inertia-induced \textit{heavy-tailed} avalanche size distributions using a prototypical example of a discrete chain with bi-stable springs. We then explore the effects of inertia in fully realistic two- and three-dimensional continuum models of elastic phase transitions. In particular, we demonstrate that a three-dimensional model can produce not only qualitative but also quantitative agreement with experiment.

cond-mat.mtrl-sci↗

arXiv2025

Active chemo-mechanical solitons

In many biological systems localized mechanical information is transmitted by mechanically neutral chemical signals. Typical examples include contraction waves in acto-myosin cortex at cellular scale and peristaltic waves at tissue level. In such systems, chemical activity is transformed into mechanical deformation by distributed motor-type mechanisms represented by continuum degrees of freedom. To elucidate the underlying principles of chemo-mechanical coupling, we here present the simplest example, involving directional motion of a localized solitary wave in a distributed mechanical system, guided by a purely chemical cue. Our main result is that mechanical signals can be driven by chemical activity in a highly efficient manner.

cond-mat.soft↗

arXiv2024

Nonlinear stability in a free boundary model of active locomotion

Contraction-driven self-propulsion of a large class of living cells can be modeled by a Keller-Segel system with free boundaries. The ensuing "active" system, exhibiting both dissipation and anti-dissipation, features stationary and traveling wave solutions. While the former represent static cells, the latter describe propagating pulses (solitary waves) mimicking the autonomous locomotion of the same cells. In this paper we provide the first proof of the asymptotic nonlinear stability of both of such solutions, static and dynamic. 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 classically, 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 (GPG) theorem, which controls the entire spectrum via bounds on the resolvent operator. For both stationary and traveling wave solutions, nonlinear stability is then proved by showing how the nonlinear part of the problem may be dominated by the linear part and then employing a Gronwall inequality argument. The developed novel methodology can prove useful also in other problems involving non-self-adjoint (non-Hermitian or non-reciprocal) operators which are ubiquitous in the modeling of "active" matter.

math.AP↗

arXiv2024

Slip-dominated structural transitions

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 be also interpreted as microscopic shuffling, its precise crystallographic nature strongly suggests the plasticity-centered interpretation.

cond-mat.mtrl-sci↗

arXiv2024

Solitary waves and kinks in FPU lattices with soft-hard-soft trilinear interactions

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.

nlin.PS↗

arXiv2024

Optimal crawling: from mechanical to chemical actuation

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.

cond-mat.soft↗

arXiv2024

Active drive towards elastic spinodals

Active matter, exemplified by adaptive living materials such as the actomyosin cytoskeleton, can navigate material parameter space dynamically, leading to unconventional mechanical responses. In particular, it can self-drive toward elastic spinodal regimes, where inhomogeneous floppy modes induce elastic degeneracy and enable a controlled interplay between rigidity loss and recovery. Proximity to such marginal states leads to stress localization and the formation of force chains that can be actively assembled and disassembled. Here, we extend the classical notion of spinodal states to active solids and demonstrate how these extreme mechanical regimes can be actively accessed. Moreover, we show that in a nonlinear setting, crossing elastic spinodals generates new energy wells and makes force channeling an intrinsic feature of the emerging microstructure.

cond-mat.soft↗

arXiv2024

When discrete fronts and pulses form a single family: FPU chain with hardening-softening springs

We consider a version of the classical Hamiltonian FPU (Fermi-Pasta-Ulam) problem with nonlinear force-strain relation in which a hardening response is taken over by a softening regime above a critical strain value. We show that in addition to pulses (solitary waves) this discrete system also supports non-topological and dissipation-free fronts (kinks). Moreover, we demonstrate that these two types of supersonic traveling wave solutions belong to the same family. Within this family, solitary waves exist for continuous ranges of velocity that extend up to a limiting speed corresponding to kinks. As the kink velocity limit is approached from above or below, the solitary waves become progressively more broad and acquire the structure of a kink-antikink bundle. Direct numerical simulations and Floquet analysis of linear stability suggest that all of the obtained solutions are effectively stable. To motivate and support our study of the discrete problem we also analyze a quasicontinuum approximation with temporal dispersion. We show that this model captures the main effects observed in the discrete problem both qualitatively and quantitatively.

nlin.PS↗

arXiv2023

Rigidity-induced critical points

While classical theory of phase transitions deals with systems where shape variation is energetically neutral, the account of rigidity can lead to the emergence of new thermodynamic features. One of them is a special type of critical points that are characteristic of phase transitions specifically in solids. We develop a general theory of such rigidity-induced critical points and illustrate the results by analyzing in detail the case of an isotropic, geometrically nonlinear solid undergoing a volumetric phase transition at zero temperature.

cond-mat.mtrl-sci↗

arXiv2023

A class of nonlinear elasticity problems with no local but many global minimizers

We present a class of models of elastic phase transitions with incompatible energy wells in any space dimension, where an abundance of Lipschitz global minimizers in a hard device coexists with a complete lack of strong local minimizers. The analysis hinges on the proof that every strong local minimizer in a hard device is also a global minimizer which is applicable much beyond the chosen class of models. Along the way we show that a new proof of sufficiency for a subclass of affine boundary conditions can be built around a novel nonlinear generalization of the classical Clapeyron theorem, whose subtle relation to dynamics was studied extensively by R. Fosdick.

math.AP↗

arXiv2023

Solid phase transitions in the liquid limit

We address the fundamental difference between solid-solid and liquid-liquid phase transitions within the Ericksen's nonlinear elasticity paradigm. To highlight ideas, we consider the simplest nontrivial 2D problem and work with a prototypical two-phase Hadamard material which allows one to weaken the rigidity and explore the nature of solid-solid phase transitions in a ``near-liquid'' limit. In the language of calculus of variations we probe limits of quasiconvexity in an ``almost liquid'' solid by comparing the thresholds for cooperative (laminate based) and non-cooperative (inclusion based) nucleation. Using these two types of nucleation tests we obtain for our model material surprisingly tight two-sided bounds on the elastic binodal without directly computing the quasiconvex envelope.

math-ph↗

arXiv2022

Beyond the classical Cauchy-Born rule

Physically motivated variational problems involving non-convex energies are often formulated in a discrete setting and contain boundary conditions. The long-range interactions in such problems, combined with constraints imposed by lattice discreteness, can give rise to the phenomenon of geometric frustration even in a one-dimensional setting. While non-convexity entails the formation of microstructures, incompatibility between interactions operating at different scales can produce nontrivial mixing effects which are exacerbated in the case of incommensuration between the optimal microstructures and the scale of the underlying lattice. Unraveling the intricacies of the underlying interplay between non-convexity, non-locality and discreteness, represents the main goal of this study. While in general one cannot expect that ground states in such problems possess global properties, such as periodicity, in some cases the appropriately defined global solutions exist, and are sufficient to describe the corresponding continuum (homogenized) limits. We interpret those cases as complying with a Generalized Cauchy-Born (GCB) rule, and present a new class of problems with geometrical frustration which comply with GCB rule in one range of (loading) parameters while being strictly outside this class in a complementary range. A general approach to problems with such mixed behavior is developed.

math.AP↗

arXiv2021

Mean field fracture in disordered solids: statistics of fluctuations

Power law distributed fluctuations are known to accompany \emph{terminal} failure in disordered brittle solids. The associated intermittent scale-free behavior is of interest from the fundamental point of view as it emerges universally from an intricate interplay of threshold-type nonlinearity, quenched disorder, and long-range interactions. We use the simplest mean-field description of such systems to show that they can be expected to undergo a transition between brittle and quasi-brittle (ductile) responses. While the former is characterized by a power law distribution of avalanches, in the latter, the statistics of avalanches is predominantly Gaussian. The realization of a particular regime depends on the variance of disorder and the effective rigidity represented by a combination of elastic moduli. We argue that the robust criticality, as in the cases of earthquakes and collapsing porous materials, indicates the self-tuning of the system towards the boundary separating brittle and ductile regimes.

cond-mat.dis-nn↗