# Research

** Few recent themes of research**

*Crystal plasticity and criticality*

- P. Zhang, O. U. Salman, 1, J.-Y. Zhang, G. Liu, J. Weiss, L. Truskinovsky, J. Sun, Taming intermittent plasticity at small scales,
*Acta Mat.,*128 , 351-364, 2017 - J. Weiss, S. Deschanel, W. Ben Rhouma and L. Truskinovsky, Plastic intermittency during cyclic loading: from dislocation patterning to micro-crack initiation,
*Phys. Rev. Materials,*3, 023603, 2019 - Baggio, R., E. Arbib, P. Biscari, S. Conti, L. Truskinovsky, G. Zanzotto, and O. U. Salman. “Landau-Type Theory of Planar Crystal Plasticity.”
*Physical Review Letters*123, n. 20, 205501, 2019 - Zhang, P., Salman, O. U., Weiss, J., and Truskinovsky, L, Variety of scaling behaviors in nanocrystalline plasticity,
*Phys. Rev. E*, (2020),

We quantified the coexistence of correlated and uncorrelated fluctuations in plastically deforming micropillars, demonstrated that the partition between the two is determined by sample size, and proposed quantitative strategies allowing one to temper plastic intermittency by artificially tailored disorder. We also analyzed collective dislocation dynamics during cycling loading and revealed a strong link between dislocation patterning, cyclic hardening/softening, and the intermittency of plasticity. We then developed a theory explaining why larger, high-symmetry crystals are mostly weak, ductile, and statistically subcritical, while smaller crystals with the same symmetry are strong, brittle and supercritical. Using a minimal automaton model we showed that with increasing size, submicron crystals undergo a crossover from spin-glass marginality to criticality characterizing the second order brittle-to-ductile transition. To show that nonlinear continuum elasticity can be effective in modeling plastic flows in crystals we reformulated it as a tensorial version of the Landau theory with an infinite number of equivalent energy wells whose configuration is dictated by the symmetry group GL(2; Z). As a proof of principle, we studied dislocation nucleation in a homogeneously sheared 2D square and hexagonal crystals and showed that the global tensorial invariance of the elastic energy foments the development of complexity in the configuration of collectively nucleating defects.

*Collective effects in fracture*

- I. Novak, Truskinovsky L., Segmentation in cohesive systems constrained by elastic environments,
*Phil. Trans. Roy. Soc. A*, 375: 20160160, 2017 - Borja da Rocha, H, and L. Truskinovsky. “Rigidity-controlled crossover: from spinodal to critical failure.”
*Physical Review Letters*124, no. 1 (2020): 015501.

The complexity of fracture-induced segmentation in elastically constrained cohesive systems originates from the presence of competing interactions. The role of discreteness in such phenomena is of interest in a variety of fields, from hierarchical self-assembly to developmental morphogenesis. We studied the analytically solvable example of segmentation in a breakable mass–spring chain elastically linked to a deformable lattice structure. We showed that, even in the continuum limit, the dependence of the segmentation topology on the stretching/pre-stress parameter in this problem is highly nontrivial taking the form of a devil’s type staircase. The failure in such a system is accompanied by intermittent fluctuations extending over a broad range of scales. Such scaling has been previously associated with either spinodal or critical points. We used an analytically transparent mean-field model to show that both analogies are relevant near the brittle-to-ductile transition. By interpreting rigidity as a time-like variable we revealed an intriguing parallel between earthquake-type critical failure and Burgers turbulence.

*Mechanics of phase transitions and elastic instabilities*

- Grabovsky, Y., Truskinovsky, L. When rank one convexity meets polyconvexity: an algebraic approach to elastic binodal,
*J. Nonlin. Sci.,*29, 1, 229–253, 2019 - Grabovsky, Y, L. Truskinovsky. “Explicit relaxation of a two-well Hadamard energy.”
*J.Elasticity*135, no. 1-2 (2019): 351-373. - Ciarletta, P., L. Truskinovsky. “Soft Nucleation of an Elastic Crease.”
*Physical Review Letters*122, no. 24 (2019): 248001. - Salman, O.U., Vitale, G. and Truskinovsky, L., 2019. Continuum theory of bending-to-stretching transition.
*Physical Review E*, 100(5), p.051001.

In the variational problems involving non-convex integral functionals, finding the binodal, the boundary of validity of the quasiconvexity of the energy density, is of central importance. We developed a systematic methodology for identifying a part of the binodal corresponding to simple laminates by showing that in this case the supporting null-Lagrangians, establishing polyconvexity, can be constructed explicitly. We used these ideas to compute an explicit quasiconvex envelope for a subclass of double-well Hadamard energies which model materials undergoing isotropic-to-isotropic elastic phase transitions. Creasing is another ubiquitous instability in solids; however, its inception has remained enigmatic as it cannot be captured by the standard linearization techniques. We showed that despite its fundamentally nonlinear nature, creasing has its origin in marginal stability which is, however, obscured by the dominance of long-range elastic interactions. We obtained an analytic instability criterion showing an excellent agreement with both physical experiments and direct numerical simulations. Yet another highly interesting elastic instability, is associated with the transition from bending-dominated to stretching-dominated elastic response in semiflexible fibrous networks. We proposed a simple continuum model of this transition with macroscopic strain playing the role of order parameter. The theory predicts that bending-to-stretching transition should proceed through propagation of the fronts separating domains with affine and non-affine elastic response and this prediction is confirmed by experiment.

*Dispersive dynamics of moving defects*

- L. Truskinovsky, A.Vainchtein, Strictly supersonic solitary waves in lattices with second-neighbor interactions,
*Physica D,*389 , 24–50, 2019 - Gorbushin, N., Truskinovsky, L. Supersonic kinks and solitons in active solids.
*Philosophical Transactions of the Royal Society A*, 378 (2162), 20190115 (2020). - Gorbushin, N., Mishuris G. Truskinovsky, L. Frictionless motion of lattice defects,
*Physical Review Letters,*(2020) in print. - Gavrilyuk, S, Nkonga, B.,S. Keh-Ming, Truskinovsky , L “Stationary shock-like transition fronts in dispersive systems”,
*Nonlinearity*(2020) in print.

Energy dissipation by fast crystalline defects takes place mainly through the resonant interaction of their cores with periodic lattice. We showed that the resultant effective friction can be reduced to zero by appropriately tuned acoustic sources located on the boundary of the body. To illustrate the general idea, we considered three prototypical models describing the main types of strongly discrete defects: dislocations, cracks and domain walls. In a related project we considered a nonlinear mass–spring chain with first and second-neighbor interactions and showed that, in contrast to the case of simple chains, solitary waves in such systems are strictly supersonic. To capture this effect we developed a new and unconventional, higher-order quasicontinuum approximation carrying more than one length scale. We also showed that steadily propagating nonlinear waves in discrete matter can be driven internally. In contrast to subsonic kinks in passive bi-stable chains that are necessarily dissipative, supersonic kinks in such active systems were shown to be purely anti-dissipative. In a different but related project we showed that, contrary to popular belief, dispersive regularization of hyperbolic systems does not exclude the development of the localized shock-like transition fronts. To guide the numerical search of such solutions, we generalized Rankine-Hugoniot relations to the case of higher order dispersive discontinuities and study their properties in an idealized case of a transition between two periodic wave trains with different wave lengths.

*Mechanics of incompatible growth*

- G. Zurlo, Truskinovsky L., Printing Non-Eucledian Solids, 2017,
*Phys. Rev. Lett.,*119, 048001, 2017 - G. Zurlo, Truskinovsky L., Inelastic Surface Growth,
*Mech. Res. Com.,*93, 174-179, 2018 - Truskinovsky, L , and G. Zurlo. “Nonlinear elasticity of incompatible surface growth.”
*Physical Review E*99, no. 5 (2019): 053001.

Geometrically frustrated solids with a non-Euclidean reference metric are ubiquitous in biology and are becoming increasingly relevant in technological applications. Often they acquire a targeted configuration of incompatibility through the surface accretion of mass as in tree growth or dam construction. Surface growth is a crucial component of many natural and artificial processes, from cell proliferation to additive manufacturing. To study of incompatible surface growth we developed a new continuum approach and showed that geometrical frustration developing during deposition can be fine-tuned to ensure a particular behavior of the system. We obtained in this way an explicit 3D printing protocol for arteries, which guarantees stress uniformity under inhomogeneous loading, and for explosive plants, allowing a complete release of residual elastic energy with a single cut. Our new theory accounts for both physical and geometrical nonlinearities of an elastic solid.

*Mechanics of muscle contraction and collective folding*

- Caruel, M., Truskinovsky, L., Bi-stability resistant to fluctuations,
*J. Mech. Phys. Solids,*109 ,117–141, 2017 - Caruel, M., Truskinovsky, L., Physics of muscle contraction,
*Rep. Prog. Phys*., 81 036602124, 2018 - H. Borja da Rocha, Truskinovsky, L., Functionality of disorder in muscle mechanics,
*Phys. Rev. Lett.*122, 088103, 2019 - F. Manca, F. Pincet, Truskinovsky, L J. E. Rothman, L. Foret and M. Caruel, SNARE machinery is optimized for ultra-fast fusion,
*PNAS*, 116 (7) 2435-2442, 2019 - H. Borja da Rocha, Truskinovsky L, Equilibrium unzipping at finite temperature,
*Arch. Appl. Mech*., 89, no. 3 (2019): 535-544. - Sheshka, R., Truskinovsky, L. (2020). Power-Stroke-Driven Muscle Contraction. In:
*The Mathematics of Mechanobiology*, pp. 117-207. Springer, - H. Borja da Rocha, Truskinovsky L, Equilibrium unzipping at finite temperature,
*Arch. Appl. Mech*., 89, no. 3 (2019): 535-544

This project is aimed at complementing the conventional chemistry-centered models of force generation in skeletal muscles by mechanics-centered models. The main passive effect is the fast force recovery which does not require the detachment of myosin cross-bridges from actin filaments and can operate without a specialized supply of metabolic fuel. In mechanical terms, it can be viewed as a collective folding-unfolding phenomenon in the system of interacting bi-stable units and modeled by near equilibrium Langevin dynamics. Since the elastic interaction in this system has a long-range character, the behavior of the system in force and length controlled ensembles is different; in particular, it can have two distinct order-disorder–type critical points. We showed that the account of the disregistry between myosin and actin filaments places the elementary force-producing units of skeletal muscles close to both such critical points. Using this idea as a bio-mimetic analogy we proposed a simple micro-mechanical device that does not lose its snap-through behavior in an environment dominated by fluctuations. It has several degrees of freedom that can cooperatively resist the de-synchronizing effect of random perturbations. We then used similar ideas to study thermally activated unzipping, which can be also modeled as a debonding process. A related approach was also used in the modeling of SNARE proteins which zipper to form complexes that power vesicle fusion with target membranes in a variety of biological processes. We proposed that, similar to the case of muscle myosins, the ultrafast fusion results from cooperative action of many SNAREpins. Incorporating this idea in a simple coarse-grained model results in the prediction that there should be an optimum number of SNAREpins for submillisecond fusion: three to six over a wide range of parameters. This prediction was fully supported by more recent experiment. To show that acto-myosin contraction can be propelled directly through a conformational change, we then developed a new interpretation of muscle contraction machinery with myosin power-stroke being the main active mechanism.

*Cell motility and active particles*

- T. Putelat, P. Recho, Truskinovsky L, Mechanical stress as a regulator of cell motility,
*Phys. Rev. E*, 97, 1, 012410, 2018 - Recho, P , T. Putelat, and Truskinovsky L. “Force-induced repolarization of an active crawler.”
*New Journal of Physics*21, no. 3 (2019): 033015. - García-García, R., P. Collet, and Truskinovsky L. “Guided active particles.”
*Physical Review E*100, no. 4 (2019): 042608. - Recho, P., Putelat, T. and Truskinovsky, L., 2019. Active gel segment behaving as an active particle.
*Physical Review E*, 100 (6), p.062403.

We developed an analytically transparent one-dimensional model of cell motility accounting for active myosin contraction and induced actin turnover. We showed that stretching can polarize static cells and initiate cell motility while squeezing can symmetrize and arrest moving cells. We then performed formal model reduction transforming this continuum model into a zero-dimensional model of an active particle. Both models give rise to hysteretic force-velocity relations showing that an active agent can support two opposite polarities under the same external force and that it can maintain the same polarity while being dragged by external forces with opposite orientations. This double bi-stability results in a rich dynamic repertoire which we illustrated by studying static, stalled, and motile regimes. The model also predicts steady oscillations of cells attached to an elastic environment and offers a self-consistent mechanical explanation for all experimentally observed outcomes of cell collision tests. To account for the possibility of an externally driven taxis in active systems, we also developed a model of a guided active drift which relies on the presence of an external guiding field and a vectorial coupling between the mechanical degrees of freedom and a chemical reaction.

**Ongoing Projects**

**Theme 1 (crystal plasticity and criticality).** Despite the success of our automata-based 2D description of plastic criticality, we do still not know how to deal with engineering problems of practical interest involving 3D crystal samples with complex geometries and generic loadings. It is therefore important to bring criticality down from the level of prototypical modeling to real engineering practice where, ultimately, information on the statistical structure of fluctuations can be used as a way of predicting the life-time of a structure. A crucial step is then to put forward a first comprehensive 3D tensorial model of criticality in crystal plasticity. The first challenge is to construct the appropriate strain energy functions with GL(3,Z) symmetry allowing one to differentiate between FCC, BCC and HCP lattices. While the statistical structure of plastic fluctuations in each of these lattices is known to be quantitatively different, the power law distributed fluctuations emerge ubiquitously when the system size reaches a sub-micron scale. Understanding this intriguing size effect poses major challenge for the theory and we are addressing it in our current work. )

**Theme 2 (precursors of fatique).** Our recent results suggest that a micro-cracking regime, preceding the formation of a terminal fatigue macro-crack must saturate in a (spinodal) steady state characterized by a power law acoustic emission. It can be then used as an indicator of a system being close to failure. We propose to couple our tensorial model of critical shakedown in plasticity with a tensorial model describing criticality induced by damage. We are currently working on development of a comprehensive meso-scopic model accounting for the crossover from the dislocational mechanism of inelastic deformation to the macro-cracking mechanism. The model is based on the energy potential with GL(3,Z) symmetry, showing simultaneously a softening behavior in tension.

**Theme 3 (elastic instabilities and fragmentation). **Our ability to model various fragmentation patterns produced by an instability of a softening elastic body under stretch opens the way to understand patterns formation in biological systems, most importantly, those associated with vertebral segmentation. We have recently shown that a simple model of a softening finite slab in tension generates a rich morphological diagram with a complex crossover between various diffuse and localized modes including necking, shear banding and wrinkling. In particular, we have showed that infinitesimal variations of the aspect ratio can produce fundamental restructuring of the critical modes which suggests statistical approach to critical patterns resembling the approach adopted in the theory of fluid turbulence. We plan to advance with weakly non-linear analytical study and further develop a finite element code capable of capturing complex fracturing patterns away from the initial bifurcation threshold. Our preliminary results show the formation of intriguing hierarchical chessboard patterns pointing towards the idea of scale free fragmentation. The corresponding bifurcated branch emerges as a result of a loss of strong ellipticity. This analysis should be complemented by the study elastic instabilities due to the loss of complementing condition which can lead to other types of complex fragmentation patterns.

**Theme 4 (scale free patterns as energy minimizers). **Regularization of integral variational problems in solid mechanics involving nonconvex energies is often achieved through the transition from continuum to discrete setting. The presence of a rigid atomic scale in such problems can give rise to the phenomenon of geometrical frustration when the scale of the defect microstructure becomes incommensurate with the underlying lattice. We propose to study analytically tractable examples of nonconvex relaxation in lattice problems with and without incommensuration which originate from fracture mechanics and the theory of elastic phase transitions. Our main focus will be on the ‘devilish’ features of the relaxed energies and the remarkably complex nature of their dependence on the parameters of the problem in the case of relaxation with incommensuration. In addition to explicit computation of global minimizers in the problems of interest we plan to develop Gamma – equivalent continuum approximations of the incommensurate lattice problems which carry an irrefutable signature of the original discreetness.

**Theme 5 (optimization of shape memory alloys).** Criticality can be used not only to detect instabilities but also to optimize material properties as we have shown in our recent study of the training-induced critical behavior of martensitic materials. The fact that commercial shape memory alloys found through trial and error are very close to the self organized criticality domain suggests that scale free response may have been an implicit design criterion explaining their exceptional performance. We plan to study whether self-organization of cyclically loaded (trained) crystals towards criticality is sensitive to structural parameters and pose the problem of optimal design (through alloying) for criticality. Our preliminary study shows that the desirable behavior will require a particular combination of elastic stiffness and the transformation strain and the obtained empirical relations need to be rationalized.

**Theme 6 (skeletal muscles and active contraction).** There is strong experimental evidence that active cytoskeletal networks are only marginally stable. Motor activity is fundamental for the ability of such systems to reach and sustain the critical regime. The goal of this project is to develop a continuum theory of skeletal muscles viewed as an active cytoskeletal network with internal driving. Despite the significant progress in the understanding of the microscopic and mesoscopic aspects of muscle mechanics, achieved in the last years, the problem of self-tuning mechanism, bringing sarcomeric systems towards criticality, remains open. The proximity to the critical point allows the muscle system to amplify interactions, ensure strong feedback, and achieve considerable robustness in front of random perturbations. Most importantly, it is a way to quickly and robustly switch back and forth between highly efficient synchronized stroke and stiff behavior in the desynchronized state which currently presents a mystery.

**Theme 7 (mechanics of cytoskeleton).** The objective of this project is to develop a continuum theory of the active cytoskeleton of eucaryotic cells. Cytoskeleton can transmit mechanical loads by either stretching or bending of its micro constituent fibers. Its mechanical response is governed by the degree of active cross-linking and can shift between almost solid-like and almost fluid-like. A continuum model of such trans-rheological mechanical response involving marginally rigid materials will be developed, which is capable of displaying solid to fluid transition as a result of the tuning of its microstructure. The model will be used to explains and predict the wide spectrum of stress responses observed in cellular cytoskeleton, including the transition from diffuse to force-chain-like stress propagation. The resulting continuum theory can be expected to display stress response of elliptic, parabolic, and hyperbolic types. Our first results show that self-organizing fibrous micro-texture of the active continuum (encoded in a hierarchy of fabric tensors) can be used to drive the material from one of these response to another. For instance, in the elastically degenerate hyperbolic regime the material exhibits focused, or channeled, transmission of forces.

**Theme 8 (solitons in active media). **Active matter is a continuum description of a collection of nonlinearly interacting agents which are driven by internally fueled engines. Collective activity in such systems can propagate with the formation of activity fronts separating passive and active phases of the matter. In various systems of self-propelled objects, ranging from microorganisms to swarming robots, the activity fronts were found to bound the characteristic activity pulses. The unique properties of active pulses or solitons have been recently exploited to enable a wide range of applications, including impact mitigation, asymmetric transmission, switching, and focusing. Propagation of active waves waves can be as well harnessed to make flexible structures crawl as in the case of the retrograde peristaltic waves observed in earthworms. Such pulse-driven locomotion offers a universal general scheme for flexible machines to move. We propose to develop a systematic model of autonomous pulses of activity and determine the rules of their velocity selection. Our preliminary study the simplest mass-spring active chain will be extended towards more complex models of activity and more realistic higher dimensional setting.