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# Shear shocks in fragile networks

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved November 1, 2013 (received for review July 31, 2013)

## Significance

In ordinary solids, shocks are extreme mechanical phenomena: they occur when rigid materials are subjected to violent impacts. However, weakly connected amorphous solids, such as glasses and polymer networks, can be prepared in a state of vanishing rigidity in which even the tiniest (shear) perturbations elicit an extreme mechanical response. When that happens, these materials are not just soft but they become fragile. In our theoretical and computational study, we elucidate the exotic mechanisms whereby nonlinear sound propagates in fragile materials composed of a disordered network of beads connected by springs. Furthermore, we propose a unique approach to infer the fascinating properties of these fragile materials from acoustic experiments that measure the speed and width of strongly nonlinear shock waves.

## Abstract

A minimal model for studying the mechanical properties of amorphous solids is a disordered network of point masses connected by unbreakable springs. At a critical value of its mean connectivity, such a network becomes fragile: it undergoes a rigidity transition signaled by a vanishing shear modulus and transverse sound speed. We investigate analytically and numerically the linear and nonlinear visco-elastic response of these fragile solids by probing how shear fronts propagate through them. Our approach, which we tentatively label shear front rheology, provides an alternative route to standard oscillatory rheology. In the linear regime, we observe at late times a diffusive broadening of the fronts controlled by an effective shear viscosity that diverges at the critical point. No matter how small the microscopic coefficient of dissipation, strongly disordered networks behave as if they were overdamped because energy is irreversibly leaked into diverging nonaffine fluctuations. Close to the transition, the regime of linear response becomes vanishingly small: the tiniest shear strains generate strongly nonlinear shear shock waves qualitatively different from their compressional counterparts in granular media. The inherent nonlinearities trigger an energy cascade from low to high frequency components that keep the network away from attaining the quasi-static limit. This mechanism, reminiscent of acoustic turbulence, causes a superdiffusive broadening of the shock width.

Many natural and manmade amorphous structures ranging from glasses to gels can be modeled as disordered viscoelastic networks of point masses (nodes) connected by springs. Despite its simplicity, the spring network model uncovers the remarkable property that the rigidity of an amorphous structure depends crucially on its mean coordination number *z*, i.e., the average number of nodes that each node is connected to (1). For an unstressed spring network in *D* dimensions, the critical coordination number separates two disordered states of matter: above the system is rigid, below it is floppy. Therefore, can be identified as a critical point in the theory of rigidity phase transitions (2⇓–4).

Various elastic properties are seen to scale with the control parameter , close to the critical point (1). For example, the shear modulus vanishes as a power law of (5⇓–7). At the critical point, the disordered network becomes mechanically fragile in the sense that shear deformations cost no energy within linear elasticity. This property is shared with packings and even chains of Hertzian grains just in contact, which display also a zero bulk modulus (3, 8). There is, however, an important difference. In the disordered spring networks, local particle interactions are harmonic. Fragility and the incipient nonlinear behavior is a collective phenomenon triggered by the mean global topology of the network that is composed of unbreakable Hookean springs (no smooth deformation can change *z*; springs must be added or removed). By contrast, in granular media, additional and crucial nonlinearities are typically introduced also at the local level, e.g., by the Hertzian interaction between grains or particle rearrangements that leads to a distinct shock phenomenology (9⇓⇓⇓⇓–14).

In the present work, we adopt a blend of theoretical and numerical analysis to study the out of equilibrium response of viscoelastic random spring networks subjected to a constant influx of energy. Our approach suggests an alternative route to standard oscillatory rheology that consists in constantly shearing one of the boundaries while monitoring how the ensuing shear front propagates throughout the sample (Fig. 1). This method, which we tentatively label shear front rheology (SFR), is of general applicability, but it may be particularly appropriate to investigate critical systems whose dynamics is undergoing a dramatic slow-down, such as biopolymer networks or gels with a sufficiently small shear modulus and sound speed, so that transients become the main observables. Our findings demonstrate that the mechanical response near the critical point is inherently nonaffine, non–quasi-static, and nonlinear.

First, the approach toward the critical point is accompanied microscopically by an increasingly heterogeneous displacement field that is nonaffine with respect to the uniform shear applied at the boundary (6, 15⇓⇓⇓–19). We show that the increasingly nonaffine response, as is lowered toward the rigidity transition, is reflected in SFR by the diverging widths of the shear fronts. This effect is a dynamical analog of the diverging width of magnetic domain walls near the Curie point: it is captured in our elastic models by a divergent disorder-induced effective viscosity. Thus, even in the limit of vanishing microscopic coefficient of viscous dissipation, a disordered spring network behaves as if it were overdamped. A large fraction of the energy injected at the boundary is leaked into nonaffine fluctuations and unless a constant influx of energy is maintained at the boundaries, coherent wave propagation cannot be observed. This effect is reminiscent of the strong ultrasound attenuation in smectic liquid crystals, where the role of the soft nonaffine fluctuations triggered by disorder is played by soft layer-bending fluctuations at a finite temperature (20).

Second, we find that, no matter how slowly you shear, the process can never be considered quasi-static at the critical point. In oscillatory rheology, this means that the frequency range , below which the network must be sheared to observe a frequency-independent (dc) response, becomes vanishingly small (e.g., ) as the critical point is approached (21). The progressive breakdown of the quasi-static approximation is signaled in SFR by the front width broadening superdiffusively with time. We show that, from the superdiffusion exponent in SFR, you can read off the power law exponent of the loss modulus as a function of frequency in oscillatory rheology. In this regime, the network has no time to respond elastically, so there is no front propagation only the superdiffusive broadening that would be also observed for . Away from the critical point, simple diffusive broadening of the front is recovered after a divergent time scale that is inversely proportional to .

Third, close to the critical point, the shear modulus and transverse sound speed become vanishingly small as the regime of linear response progressively vanishes (22). Thus, at the rigidity transition, shear deformations can no longer propagate as linear shear waves because the structure is mechanically fragile. Instead, we demonstrate that even for the tiniest strains, nonuniform strongly nonlinear shock waves arise. Away from the critical point, linear shear waves are restored but only below a critical strain, , that vanishes at the transition. For strains larger than , the same shear shocks that emanate from the critical point are responsible for the transmission of mechanical energy. We show that the speed at which these nonlinear fronts move can serve as a dynamical probe of the nonlinear exponent in the stress-strain relation. Unlike linear fronts, the shock width broadens superdiffusively (even at late times) as a result of a nonlinear energy cascade reminiscent of acoustic turbulence. This mechanism acts as a source that generates higher and higher harmonics that keep the network away from attaining the quasi-static limit.

Experimental and theoretical investigations of compressional shocks and solitons have been carried out extensively in granular media prepared in a state of zero external pressure (termed sonic vacuum) in which the bulk modulus vanishes as a result of nonlinear grain interactions (9, 11, 14). By contrast, shear shocks triggered by global nonlinearities are less explored experimentally with the notable exception of biogels that share with our models the property of having a very small shear modulus typically of the order of kilopascals (23), but they are typically stabilized by osmotic pressure or activity (1, 24). Nonetheless, it is much easier to break the sound barrier (and observe genuine acoustic Mach cones) in human tissues, where the speed of linear transverse sound is of the order of meters per second, than in metals where it is thousands times higher. Indeed, supersonic shear imaging has recently developed into a powerful diagnostic tool for medical applications (25, 26).

## Divergent Shear Front Width

We construct computer models of weakly connected 2D random viscoelastic networks from highly compressed jammed packings of frictionless poly-disperse disks (3, 27). One first identifies the disk centers as point particles (network nodes) and then models the interactions between overlapping disks using two sided harmonic springs of varying rest length to eliminate any prestress existing in the original jammed packings. The result is a highly coordinated spring network that serves as the seed from which families of networks with a wide range of *z* are generated by removing springs. We have investigated two cutting protocols that result in two distinct ensembles of random spring networks.

The first protocol generates nearly homogeneous isotropic networks by progressively removing bonds with the highest *z*, such that spatial *z* fluctuations are reduced by construction, a scenario assumed in several mean field theories of the jamming transition (18, 28). The second protocol consists in removing springs randomly (subject to the no-local-floppiness constraint ) and results in less homogeneous but still isotropic structures, whose critical exponents are close to the ones obtained in rigidity percolation (7). These two ensembles of networks display different critical exponents but share the crucial qualitative features that arise from the presence of a rigidity transition at . For simplicity, we present most of our results in the context of the homogeneous networks while noting how the exponents are modified if random cutting is used.

Once the networks are generated, we shear the left most edge at a constant speed , as in Fig. 1, and follow the evolution of the resulting shear velocity profile by averaging out the longitudinal particle speeds over bins along the *x* direction, effectively creating a 1D front profile propagating in the *x* direction. The dynamics is obtained by numerically integrating Newton’s equations of motion using the velocity Verlet method subject to periodic boundary conditions in the *y* direction and hard walls (to which the first and last columns of particles are attached) in the *x* direction (a hard wall can only move up and down as a whole, but no relative motion of the particles is allowed). The samples are composed of up to identical particles with mass *m*. In addition to the Hookean interaction (with a spring constant *k*) between connected particles, *i* and *j*, we include the effects of viscous dissipation: , where *b* is the microscopic damping constant, and are the velocities of a pair of particles connected by a spring. Time is measured in units of and the damping constant in units of , which is equivalent to setting and . Lengths are measured in units of *d* the mean spring length at rest.

In Fig. 2, we illustrate the late-times dynamics of the transverse velocity profiles , normalized by the shearing speed , for a small shear strain , where is the critical strain needed to elicit a nonlinear response, discussed in detail later. Inspection of the schematics in Fig. 1 shows how the shear strain, , is implicitly controlled by via the relation . The bottom left panel shows successive snapshots of the (transverse) velocity profiles as a function of *x* (after averaging the longitudinal velocity fluctuations) in a sample at . After an initial transient, a linear shear front propagates at a constant transverse sound speed, , where *ρ* is the mass density and *G* the shear modulus. In the right inset of Fig. 2, the numerical value of *G* extracted from the speeds of shear fronts propagating in homogeneously cut networks is plotted against (red dots). We find that it is consistent with the expected linear scaling (3, 5, 8) (the best fit solid red line has a slope close to 1.1). By contrast, a power law exponent numerically closer to 1.4 is found for randomly cut networks.

In the main panel of Fig. 2, we show the velocity profiles obtained from molecular dynamics simulations after the fronts have traveled half the length of homogeneous random networks with different (the numerical data are represented by symbols with different colors). To unambiguously extract the dependence of the width on , we first fit the data with an analytical solution for the velocity profile derived in the *SI Appendix*. Next, we define the rescaled width that normalizes out the diffusive spreading in time and plot as a function of , as red dots in the main panel of Fig. 3. The solid line that fits the data for has slope on a log plot, hence for homogeneously cut networks. The divergent width of the shear fronts is a striking signature (in real space) of the important diverging length scale proportional to that accompanies the jamming/unjamming transition (29⇓⇓–32).

Note, however, that the scaling exponent of the width (as for the shear modulus one) is not universal; instead, it depends on the spring cutting protocol adopted to generate the network. If springs are cut randomly, we find that *α* is numerically close to (Fig. 3, red data in the lower inset), but the basic phenomenology is essentially unaltered. The divergence of the front width as a function of occurs irrespective of the choice of the microscopic coefficient of dissipation even when , provided that disorder is strong (Fig. 3, green data in the lower inset). As we shall see in the next section, the divergent front width results from the divergence of nonaffine fluctuations at the critical point: it can be modeled analytically on introducing a divergent effective viscosity.

## Shear Front Rheology and Effective Viscosity

Consider first the general linear visco-elastic relation between the stress , the strain , and the strain rate in the frequency independent (quasi-static) regimewhere the loss modulus (viscosity) and shear modulus *G* are constants, independent of frequency. For random networks, however, these moduli can be scaling functions of that we denote by and , respectively. On expressing the shear strain as (Fig. 1), we write down Newton’s equation asThis linear second-order partial differential equation can be readily cast in terms of the velocity field . Inspection of Eq. **2** reveals that the resulting velocity profile propagates as a linear wave with speed proportional to (from the second-order terms), whereas its width spreads diffusively as (from the third-order term). Hence, the scaling observed in the fluctuation-averaged profiles of Fig. 2 implies that with .

To uncover the physical origin of the diverging viscosity , note that the displacement field in Eq. **2** describes only the dynamics of the affine deformations that the network undergoes. However, the relative magnitude of the nonaffine fluctuations , which have been averaged out in Fig. 2, diverges while approaching the critical point as for a homegeneously cut network (5, 18), as verified in Fig. 3 (*Upper Inset*). In our treatment, we implicitly capture the presence of the large nonaffine fluctuations by a renormalization of the effective viscosity that diverges as the critical point is approached (even if ). The scaling exponent *χ* (that is equal to 1 for homogenous network) is suitably chosen to match the amount of kinetic energy leaked into the nonaffine fluctuations that have been integrated out.

By assuming that *G* and are frequency independent, we restrict the domain of validity of our predictions to the so-called quasistatic regime. The shear fronts enter this regime only after a characteristic time scale that diverges at the transition. The dependence of , derived in the *SI Appendix*, can be guessed from the ratio , which for homogeneously cut networks gives , whereas the randomly cut networks are found numerically to have a scaling exponent close to −3. For , the network has “no time” to respond elastically, so we are effectively probing the properties of a floppy structure, even if . The experimental signature of this regime is that the shear deformation continuously applied at the edge penetrates the sample mainly by broadening its width instead of propagating, as illustrated in the inset of Fig. 4.

In oscillatory rheology, the regime corresponds to the nonquasistatic regime (with ), for which the viscosity is no longer frequency independent (21, 33, 34). Instead, it exhibits power law scaling (21). In the random networks we investigate, the frequency scale is distinct from the frequency scale that appears in the density of vibrational modes (35, 36) or in the phonon diffusivity (29, 30, 37).

Heuristic considerations (formalized in the *SI Appendix*) suggest that the front width broadens superdiffusively for . The superdiffusive spreading is corroborated in the main panel of Fig. 4, where the width is shown as a function of for a range of , indicated by different colors. From the measured superdiffusion exponent , we infer that (independently of cutting protocol): this scaling is often encountered in rheological studies of grains and emulsions (21, 38). For the fronts propagate ballistically, and their widths broaden diffusively with an *ω* independent effective viscosity , for homogeneously cut networks. Consistently, all data can be collapsed onto a single master curve when *w* is rescaled by a power of close to 3 (note implies ).

The more general response function treatment, detailed in *SI Appendix*, formalizes how to extract the full *ω* dependence of the moduli from the velocity profiles. Consider the frequency dependent constitutive relation for a linear visco-elastic materialwhere is the (Laplace transformed) complex modulus, whose real and imaginary parts correspond (aside from an *ω* factor) to the shear (storage) and viscosity (loss modulus), respectively. Assuming that the random spring network is homogeneous and isotropic, we solve the equation of motion for the velocity field to obtain (*SI Appendix*)where is the strain speed imposed at the boundary (Fig. 1), and *x* is a coarse-grained variable along the longitudinal direction.

We can readily invert the Laplace transform to obtain the velocity profile, , as a function of time, once an appropriate choice for the complex modulus is made or vice versa. The quantitative match between the predictions of Eq. **4** (see the continuous lines in the main panel of Fig. 2) and our numerical data validates the concept of an effective viscosity. It is a useful tool to capture the divergent nonaffine fluctuations responsible for its breakdown within ordinary continuum elastic theory.

## Shear Shocks and Energy Cascade

As the critical point is approached, the linear approximation, adopted in the previous sections, progressively breaks down because the shear modulus and transverse sound speed become vanishingly small. This raises the question of how energy is transferred in such a fragile material. A simple guess for the elastic part of the nonlinear stress readswhere is a nonlinear elastic coefficient that does not vanish at , and *n* is a nonlinear exponent to be inferred from shear front rheology experiments ( is the most common result that ensures mirror symmetry). Note that the viscous contribution to the stress-strain relation discussed in Eq. **3** is still present in the nonlinear regime. However, it is only the elastic contribution in Eq. **2** that determines the front speed; dissipation controls only the front width. Hence, we shall address the effect of nonlinearity and dissipation separately.

In most elastic media, the first term in Eq. **5** dominates as long as . Thus, the fundamental mechanical excitations are (weakly interacting) transverse phonons. By contrast, in fragile matter, the usual approach needs to be turned around because is the small parameter. Hence, the basic excitations are now obtained from the nonlinear term; they are shock-like solutions of the equations of motion derived from Eq. **5**. By equating the two terms in Eq. **5**, we find the threshold , above which the nonlinear response kicks in, which is indeed much smaller than close to the transition.

On defining a nonlinear shear modulus , we can obtain the characteristic nonlinear front speed asNotice that for , approaches the transverse linear speed of sound . For , instead, becomes independent of and controlled only by the applied strain: the hallmark of a strongly nonlinear wave.

In the main panel of Fig. 5*A*, we plot vs. for homogeneously cut networks of various and achieve a very good data collapse. For , we find (i.e., ), which implies that (recall ), consistently with the findings in ref. 18. Hence, the linear sound regime , where , progressively shrinks to zero as the critical point is approached. Furthermore, the (cutting-protocol independent) finding and mirror symmetry imply that the stress attains the surprising nonanalytic form at the critical point (18). We find that the simple analytic expression , plotted as a continuous line in the inset of Fig. 5*A*, fits our data very well for all and *γ*.

We emphasize that, in fragile elastic networks, strongly nonlinear waves can be generated without necessarily breaking the springs, because the critical strain for the onset of shock waves is vanishingly small. By contrast, in noncohesive granular media, where the grain interactions are purely repulsive, the propagation of a strongly nonlinear wave typically induces bond breaking and rearrangements (39⇓–41).

The dynamics of random spring networks is inherently overdamped. Hence, we do not observe a stationary shock solution. Instead, the shock width broadens as it propagates, as shown in Fig. 5*B*. However, there is a striking difference with the linear case, which is illustrated in Fig. 5*C*. In the highly nonlinear regime, the shock width (closed symbols) broadens superdiffusively as even at late-times , when the linear fronts (open symbols) have clearly crossed over to diffusive broadening. This means that the quasi-static approximation, from which the late-times diffusive broadening was derived, ceases to be valid in the nonlinear regime.

The resilience of the superdiffuive broadening in the nonlinear regime is one of the central results of our work: it can be explained qualitatively by an energy cascade mechanism from low to high *ω*, reminiscent of acoustic turbulence. When large oscillatory strains with frequency components are applied to the network, nonlinearities act as an additional source that generates higher and higher harmonics, past the characteristic threshold . These higher harmonics keep the network away from attaining the quasi-static limit. Consequently, one of the tangible experimental signatures of this energy cascade is the persistence of the regime of superdiffusive broadening even at long times when diffusive broadening is observed for linear fronts (Fig. 5*C*) (the constant shear we apply excites simultaneously all *ω* components).

To sum up, we find that at (or near) the isostatic point of a random network of masses permanently connected by Hookean springs (*i*) shear fronts propagate as supersonic shocks with speed and (*ii*) their widths broaden superdiffusively as . We propose that the inherent nonlinearities at the critical point trigger an energy cascade from low- to high-frequency components that causes the breakdown of the quasi-static approximation and gives rise to superdiffusion. In the linear regime, we observe at late times a diffusive broadening controlled by an effective shear viscosity that diverges at the critical point. Even in the limit of vanishing microscopic coefficient of dissipation, (strongly) disordered networks behave as if they were overdamped because energy is irreversibly leaked into diverging nonaffine fluctuations. The shear front rheology approach illustrated here can be used more broadly to infer the viscoelastic properties of any material undergoing very slow dynamics from (nonlinear) acoustic experiments.

## Acknowledgments

We wish to acknowledge helpful conversations with L. Gomez, B. Tighe, A. Zaccone, M. van Hecke, M. Wyart, P. Hebraud, and F. Lequeux. This work was supported by the Stichting voor Fundamenteel Onderzoek der Materie and the Nederlandse Organisatie voor Wetenschappelijk Onderzoek.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: vitelli{at}lorentz.leidenuniv.nl.

Author contributions: V.V. designed research; S.U., N.U., B.v.O., and V.V. performed research; S.U., N.U., and B.v.O. analyzed data; and S.U., N.U., and V.V. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1314468110/-/DCSupplemental.

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