Elasticity bounds from effective field theory

Lasma Alberte*

Abdus Salam International Centre for Theoretical Physics (ICTP),Strada Costiera 11, 34151, Trieste, Italy

Matteo Baggioli†

Crete Center for Theoretical Physics, Institute for Theoretical and Computational PhysicsDepartment of Physics, University of Crete, 71003 Heraklion, Greece

Víctor Cáncer Castillo‡ and Oriol Pujolas §

Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology (BIST)Campus UAB, 08193 Bellaterra, Barcelona, Spain

(Received 12 September 2018; published 23 September 2019)

Phonons in solid materials can be understood as the Goldstone bosons of the spontaneously brokenspacetime symmetries. As such, their low energy dynamics are greatly constrained and can be captured bystandard effective field theory methods. In particular, knowledge of the nonlinear stress-strain curvescompletely fixes the full effective Lagrangian at leading order in derivatives. We attempt to illustrate thepotential of effective methods focusing on the so-called hyperelastic materials, which allow large elasticdeformations. We find that the self-consistency of the effective field theory imposes a number of bounds onphysical quantities, mainly on the maximum strain and maximum stress that can be supported by themedium. In particular, for stress-strain relations that at large deformations are characterized by a power-lawbehavior σðεÞ ∼ εν, the maximum strain exhibits a sharp correlation with the exponent ν.

DOI: 10.1103/PhysRevD.100.065015

I. INTRODUCTION

A prominent and early example of an effective fieldtheory (EFT) is the theory of elasticity: the continuum-limitdescription of a solid’s mechanical response, including itssound wave excitations—the phonons [1,2]. As in hydro-dynamics, elasticity theory can be phrased as a derivativeexpansion for an effective degree of freedom (d.o.f.)—thedisplacement vector of the solid elements with respect toequilibrium. Importantly, the classic elasticity theory can bepromoted to the nonlinear regime, addressing the responseto finite deformations [3–5]. Operationally, this is done byfinding the stress-strain relations for both the finite shear orbulk strain applied to the material. These diagrams encodeseveral response parameters (such as the proportional limitor the failure point; see Ref. [5] for definitions), which are

well-defined material properties that go deep into thenonlinear response regime. Typically, these parametersare difficult to compute from the microscopic constituents,so there is a chance that EFT methods may help in theunderstanding of some nonlinear elasticity phenomena.From the viewpoint of quantum field theory (QFT), it is

clear that elasticity theory can be treated as a nontrivial (i.e.,interacting) EFT. The way this theory works as an EFT,however, is quite different from other well-known exam-ples, mostly because the underlying symmetry breakingpattern involves spacetime symmetries. The purpose of thiswork is to revisit finite elasticity theory from the viewpointof QFT. We aim at clarifying how the EFT methodologyworks for broken spacetime symmetries and find novelrelations between (and bounds on) various nonlinear elasti-city parameters.

II. FROM GOLDSTONES TOSTRESS-STRAIN CURVES

We start by stating the precise QFT sense in whichelasticity theory can be treated as an EFT. The firstrequirement is that the material must have a separationof scales; we shall consider only low frequency (acoustic)phonons; any other mode is considered as much heavierand integrated-out. (Materials displaying scale invariance

*lalberte@ictp.it†mbaggioli@physics.uoc.gr; www.thegrumpyscientist.com.‡vcancer@ifae.es§pujolas@ifae.es

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Funded by SCOAP3.

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violate this assumption and deserve a separate treatment.)Under this condition, we can exploit the fact that thephonons can be viewed as the Goldstone bosons of trans-lational symmetry breaking [6–8]. As such, we obtain theirfully nonlinear effective action by the means of the standardcoset construction [9]. For simplicity, we shall work in2þ 1 spacetime dimensions, where the dynamical d.o.f. arecontained in two scalar fields ϕIðxÞ. The internal symmetrygroup is assumed to be the two-dimensional Euclideangroup, ISOð2Þ, acting like translations and rotations in thescalar fields space. The theory then must be shift invariantin the ϕI’s, implying that any field configuration that islinear in the spacetime coordinates will satisfy the equa-tions of motion. The equilibrium configuration of anisotropic material is given by

ϕIeq ¼ δIJx

J: ð1Þ

This vacuum expectation value spontaneously breaks thesymmetry group ISOð2Þ × ISOð2; 1Þ down to the diagonalsubgroup.Following the coset construction method, one concludes

that the effective action at lowest order in derivatives takesthe form

S ¼ −Z

d3xffiffiffiffiffiffi−g

pVðX; ZÞ; ð2Þ

with X and Z defined in terms of the scalar fields matrix1

I IJ ¼ gμν∂μϕI∂νϕ

J as X ¼ trðI IJÞ, Z ¼ detðI IJÞ. Thefunction VðX; ZÞ is “free,” and its form depends on thesolid. In this language, the phonons πI are identified asthe small excitations around the equilibrium configurationdefined through ϕI ¼ ϕI

eq þ πI . Plugging this decomposi-tion into (2), one can find the phonon kinetic terms andtheir self-interactions ð∂πÞn. The leading phonon effectiveoperators are determined by a few Wilson coefficientsthat are related to the lowest derivatives of V evaluatedon the equilibrium configuration; see Ref. [10] for details.(Analogous results can be found in Ref. [11] for super-conductors.) The effective action (2) also encodes theresponse to finite (large) deformations, and for that, theglobal form of VðX; ZÞ is needed.By symmetry considerations, one cannot restrict the

action (2) any further. To identify what is the functionVðX; ZÞ for a given material, one needs more information,some kind of constitutive relation. According to the finiteelasticity literature (see, e.g., Ref. [5]), the function VðX; ZÞis naturally identified with the so-called strain-energyfunction. This is a function of the principal invariants

characterizing the materials state of deformation. It encodesthe full nonlinear response for the so-called Cauchyhyperelastic solids, for which plastic and dissipative effectscan be ignored [4].The form of V can then be found from the stress-strain

relations measured in both the shear and the bulk channelsof real solids (see, e.g., Refs. [3,4,12]). More specifically,the response of the material to constant and homogeneousdeformations can be deduced from configurations of theform. These can be reduced to configurations of the form

ϕIstr ¼ OI

JxJ; OI

J ¼ α

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ε2=4

pε=2

ε=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ε2=4

p!;

ð3Þwhere ε and α − 1 are the shear and the bulk strainsrespectively, and they induce constant but nontrivial valuesof Xjstr ¼ α2ð2þ ε2Þ and Zjstr ¼ α4. The amount of stressin the material generated by (or needed to support) such aconfiguration depends only on the strains ε and α and onthe shape of VðX; ZÞ; see, e.g., Eq. (9). The upshot is that itis possible to reconstruct the full form of the effectiveLagrangian (up to an irrelevant overall constant) by justmeasuring the stress-strain relations, that is, from theresponse to time-independent and homogeneous deforma-tions. This already illustrates how the solid EFTs retainpredictive power.The next apparent challenge from the QFT viewpoint is

that the real-world stress-strain curves typically exhibit adramatic feature: they terminate at some point, correspond-ing to the breaking (or elastic failure) of the material. It isthen natural to ask how exactly is the breaking seen in theEFT. Must the function VðX; ZÞ be singular? Or does thebreaking correspond to a dynamical process (e.g., aninstability) that can be captured within the EFT witha regular VðX; ZÞ? We argue below that the latter possibil-ity can certainly arise, allowing one to extract relationsbetween the parameters that control the large deformations.The main task then is to analyze the stability properties

of the strained configuration (3). This can be done bysetting ϕI ¼ ϕI

str þ πI in (2) and expanding for “small” πI .In doing so, one easily finds that the phonon soundspeeds depend on the applied strain OI

J. This is a long-known phenomenon, the acoustoelastic effect; see, e.g.,Refs. [13–18]. Still, we argue here that this can have a greatimpact on the stress-strain relations, eventually limiting themaximal stress that a material can withstand. The reason isthat generically, increasing the strain results in increasing/decreasing the various sound speeds—typically in anunbounded fashion. In particular, in most cases, past somelarge enough strain value, εmax, one of the sound speedsbecomes either i) imaginary or ii) superluminal. Case iimplies that the material develops an instability and it mustevolve to a different ground state. Case ii prevents theexistence of a Lorentz invariant ultraviolet completion.

1We retain the curved spacetime metric gμν only to make itclear how the energy-momentum tensor arises from this action. Inpractice, we shall always work on the Minkowski backgroundημν ¼ diagð−1;þ1;þ1Þ.

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Therefore, the effective low energy description (2) must bephysically invalid at least for such a large deformation.In any case, one can translate the constraints i and ii asupper bounds on the maximum allowed strain that iscompatible with the given choice of VðZ; XÞ. We remarkthat these bounds arise even for smooth choices of theeffective Lagrangian VðZ; XÞ, and yet they naturally lead tostress-strain curves that terminate at some point ε ¼ εmax;see Fig. 1 for some illustrative examples.Additionally, demanding that none of these pathologies

occurs for materials that we know admit large deformations(elastomers) significantly constrains the stress-strain curvesand therefore the possible nonlinear response of materialson quite general grounds. We illustrate the point byfocusing on materials/EFTs which allow for large defor-mations and which realize stress-strain curves with apower-law scaling,

σ ∼ εν for ε ≫ 1: ð4Þ

Henceforth, we shall refer to ν as the strain exponent. Aswe show below, both the maximum strain and the exponentν are bounded from above, and there is a general relationbetween the two. It is unclear to us to what extent theseresults were already known before. Nonetheless, our maingoal is to show how the EFT perspective presented herebrings some additional layer of understanding to thesephenomena.First, let us obtain the corresponding stress-energy tensor

by varying the action with respect to the curved spacetimemetric gμν and evaluating it on the Minkowski background,gμν ¼ ημν:

Tμν ¼ −2ffiffiffiffiffiffi−gp δS

δgμν

����g¼η

¼ −ημνV þ 2∂μϕI∂νϕIVX

þ 2ð∂μϕI∂νϕIX − ∂μϕ

I∂νϕJI IJÞVZ: ð5Þ

For any time-independent scalar field configurations, thestress-energy tensor components are

Ttt ≡ ρ ¼ V; ð6Þ

Txx ≡ p ¼ −V þ XVX þ 2ZVZ; ð7Þ

Txy ¼ 2∂xϕ

I∂yϕIVX; ð8Þ

where VX ≡ ∂V=∂X, etc. Henceforth, we shall work withthe deformed field configuration (3), which introducesboth the shear and bulk deformation. In particular, whensetting α ¼ 1, it describes a pure shear strain (i.e., volumepreserving) in the ðx; yÞ directions induced by ε ≠ 0. Forε ¼ 0 and α ≠ 1, the same setup encodes a pure bulk strain.In the considered scalar field background configuration,X and Z take the values Xjstr ¼ α2ð2þ ε2Þ, Zjstr ¼ α4.In particular, the full nonlinear stress-strain curve for

pure shear deformations as a function of ε reads

σðεÞ≡ Txy ¼ 2ε

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ε2

4

rVXð2þ ε2; 1Þ: ð9Þ

The analogous stress-strain curve for pure bulk defor-mations can also be found by expressing ΔTx

x ¼ Txx − Tx

xjeqas a function of the bulk strain, α − 1. It is thus clear thatfrom the knowledge (measurement) of both shear and bulkdiagrams one can extract the shape of VðX; ZÞ—the fulleffective Lagrangian. For instance, under the assumptionthat the Z-dependence is negligible, then from a given σðεÞshear stress-strain curve, one can extract

VðXÞ ≃Z

X

2

dxσð ffiffiffiffiffiffiffiffiffiffiffi

x − 2p Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2=4 − 1

p :

To make the connection to the linear elasticity theoryexplicit, one considers small shear and bulk deformations,i.e., small values of ε and α − 1. Then, as usual, elasticdeformations at the linear level are described in terms of thedisplacement tensor

εij ¼1

2ð∂iδϕj þ ∂jδϕiÞ; ð10Þ

where δϕI ≡ ϕI − ϕIeq is the displacement away from the

equilibrium state, ϕIeq ¼ xI . A deformation of the body that

changes its volume is given by the compression or bulkstrain as εii ¼ ∂iδϕ

i. In turn, a deformation that only affectsits shape—pure shear—is given by εik − 1

2δikεjj.

Expanding both the stress-energy tensor components (7),(8) and the displacement tensor (10) up to linear order in εand α − 1, one recovers the usual expression in 2þ 1dimensions,

FIG. 1. The nonlinear shear stress-strain curve σðεÞ for thebenchmark model (15) for B ¼ 1.6 and A ¼ 0.05, 0.2, 0.35, 0.5,0.61 (from bottom to top). The black stars represent the “break-ing” points of the material arising due to the onset of gradientinstability; the red dot indicates the onset of superluminality.

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T linij ¼ ðpþ KεkkÞδij þ 2G

�εij −

1

2δijεkk

�; ð11Þ

where p is the equilibrium pressure and G and K are theshear and bulk elastic moduli. In the case of a pure sheardeformation, this gives Txy ¼ 2Gεxy þ � � �, and we can readoff the shear modulus G as

G ¼ 2VXð2; 1Þ: ð12ÞSimilarly for the case of the pure bulk strain (ε ¼ 0), we

first note that Eq. (7) holds at nonlinear level, i.e., forarbitrarily large values of α. In order to find the linear bulkmodulus, we expand both the bulk strain and the bulk stressΔTx

x around the equilibrium value α ¼ 1. For the stress, thisgives ΔTii ¼ 2Kεii þ � � � with the equilibrium pressuregiven in (7) and εii ¼ 2ðα − 1Þ. The bulk modulus is then

K ¼ 2ZVZ þ 4Z2VZZ þ 4XZVXZ þ X2VXX; ð13Þwhere all the quantities are evaluated at X ¼ 2, Z ¼ 1.All the details concerning the consistency and stability of

perturbations around the strained background configurationare given in the Appendix. There, we find that the spectrumof perturbations contains two gapless phonon modes,

ω� ¼ c�ðα; εÞk; ð14Þwith the sound speeds bearing a nonlinear dependence onthe strain parameters α, ε. For the consistency and stabilityof a given VðX; ZÞ around the background (3), we requirethe absence of i) modes with negative kinetic energy, i.e.,ghosts; ii) negative sound speeds squared, i.e., gradientinstability; and iii) superluminal propagation. In each case,this leads to a certain value of maximal strain, εmax, beyondwhich one of these consistency conditions is violated.A typical stress-strain curve exhibiting this behavior,obtained for a given choice of VðX; ZÞ, is shown in Fig. 1.It is important to remark that our expressions for εmax

derived in Appendix should be interpreted as giving anupper bound on the maximum strain that the material cansupport, since other effects not included here can enterbefore, thus lowering the actual maximum ε. For instance,one expects plastic/dissipative effects to enter at some pointin real materials. However, this alters our analysis only forεplastic < εmax; thus, we still obtain an upper bound on themaximum reversible deformability.It is interesting to consider the possibility that it really is

the εmax found here (or a value very close to it) thatcorresponds to the physical limitation to the materialdeformation. In this case, the EFT gives partial informationon how the material might “break.” As was mentionedearlier, there are twomain options: that the breakdown is dueto gradient instability or due to reaching superluminality.In the case of gradient instability, one expects that, like

any instability, this is physically resolved by a transitionto another ground state, most likely described by a

different EFT. The specific nature of this transition remainshidden in the leading order low energy EFT presented inthis work. For instance, whether the gradient instabilitydevelops as a soft (slow) or hard (fast) process depends onthe nature of the next-to-leading order corrections toVðX; ZÞ. One may speculate that the hard case correspondsto a breaking of the material and the soft case correspondsto the necking phenomenon—a decrease in the cross sec-tional areaof amaterial sample that is often seenunder tensilestress. This would resemble the so-called soft phononinstability observed in some materials; see Refs. [19–26].Concerning superluminality, let us emphasize that in

contrast to ghost and gradient instabilities the issue ofsuperluminal propagation relates to the possibility of aLorentz invariant UV completion, not to the stability ofpropagation [27]. In order to apprehend the physicalpicture, it is instructive to recall a classic in field theory:the example given by high spin fields where the problem ofsuperluminality is known to arise [28]. As discussed inRefs. [29,30], there are two ways to resolve the problem,which require augmenting the EFT either by higher orderoperators or with additional light d.o.f. Any of the tworesolutions makes it manifest that the naive EFT truncation[akin to the one that we are doing in Eq. (2)] breaks down.Moreover, it also gives an idea of how—what that trunca-tion might be missing. In our case, this means that in thevicinity of violating the no-superluminality conditioncorrections to the particular shape of VðX; ZÞ that weconsider must become important either by the presence ofadditional operators or light fields. The possibility thathigher order operators (with more derivatives) can fix thesuperluminality problem while keeping the rest of theelastic response properties is nontrivial, and we leave itfor future research. On the other hand, the possibility thatone needs to supplement the benchmark model with otherlight d.o.f. seems quite reasonable—after all, in real-worldmaterials, phonons do couple to many other modes. If thisis the resolution, then the physical interpretation of thebound given by superluminality is that εmax can be under-stood as an upper limit on when these light d.o.f. have to betaken into account.

III. RESULTS IN A SCALING MODEL

For concreteness, we shall focus on the simple potential

VðX; ZÞ ¼ ρeqXAZðB−AÞ=2; ð15Þ

where ρeq is the dimensionful energy density set by theequilibrium configuration. The reason for choosing thisform is that it realizes a power-law scaling like (4) at largedeformations, ε ≫ 1. This behavior is observed in hypere-lastic rubberlike materials, and there are many phenom-enological models [4,12,31–36] that reduce to (15) at largestrains with various strain exponents ν. Here, we areinterested in characterizing how the stress-strain curves

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(and mainly the maximum stress and strain) depend on theparameters A and B. Let us also note that there are twospecial “corners” in parameter space: for A ¼ 0, the bench-mark potential describes a perfect fluid [7,9]; for A ¼ 1,B ¼ 1, the model reduces to two free scalar fields.We first find that the linear elastic moduli for the

potential (15) take the simple form

G ¼ ρeq2AA; K ¼ ρeq2

ABðB − 1Þ: ð16Þ

They are both positive for A > 0, B > 1. Moreover, thePoisson’s ratio—the negative ratio of transverse to axialstrain—for our models is readily obtained as (see Ref. [37])

r≡ K −GK þ G

¼ BðB − 1Þ − ABðB − 1Þ þ A

: ð17Þ

The result is shown in Fig. 2. At large B, the ratio is close toits upper bound, meaning that the models are close toperfect incompressible elastic materials. At small values ofB and large A, the ratio tends to its lower negative bound.A negative Poisson’s ratio is typical of more exotic (the so-called auxetic) materials like some foams and metamate-rials. Interestingly, the limit of free canonical scalars is inthat regime. Finally, for most of the models described,−0.5 < r < 0.5, as is common for steels and rigidpolymers.For the full nonlinear response to pure shear, Eq. (9)

gives

σðεÞ ¼ ρeqAεffiffiffiffiffiffiffiffiffiffiffiffiffiε2 þ 4

pðε2 þ 2ÞA−1: ð18Þ

This is shown in Fig. 1 for various values of A and B.Notably, the stress-strain curves obtained from thebenchmark models mimic a large variety of materialsincluding fibers, glasses, and elastomers [12]. More pre-cisely, Eq. (18) describes Neo-Hookean systems whichfollow Hooke’s law at small strain but exhibit nonlinearpower-law scalings at large deformations [5]. Similarly, thenonlinear response to a pure compression, that we define asκ ≡ α − 1, reads

ΔTiiðκÞ ¼ ρeq2Aþ1ðB − 1Þ½ðκ þ 1Þ2B − 1�: ð19Þ

We show the full nonlinear response to pure bulk defor-mation for various values of B in Fig. 3. As per con-struction, at large strains, ε, κ ≫ 1, the nonlinear stressesdisplay power-law scalings of the form

σðεÞ ∼ Aε2A; ΔTiiðκÞ ∼ ðB − 1Þκ2B; ð20Þ

from where we read off the shear and bulk strain exponentsas νshear ¼ 2A and νbulk ¼ 2B. Note that, as can be seenfrom Eq. (16), A and B also control the linear shear andbulk moduli.Combining the requirements of the absence of ghosts,

gradient instabilities, and superluminal propagation withthe positivity of the elastic moduli, K and G, constrains theallowed range of parameters. In the simple case of lineardeformations, we obtain the following allowed region forthe exponents A and B:

0 ≤ A ≤ 1 and 1 ≤ B ≤ffiffiffiffiffiffiffiffiffiffiffi1 − A

pþ 1: ð21Þ

The analysis can be extended to a finite strain and asmentioned above leads us to another important result: theexistence of a maximum strain εmax that can be supportedby the system before the onset of one of the aforementionedpathologies. How εmax depends on the strain exponents isshown in Fig. 4; the exact analytic expressions can be foundin the Appendix.2 We must emphasize that the εmaxobtained in this way is not meant to be the actual maximumdeformation that a material with the aforementioned scalingproperties can withstand but rather an upper bound on it.Still, this already provides quite a lot of information. Forinstance, in the large (yellowish) area of Fig. 4 where εmaxonly reaches values of approximately 1, one can alreadydiscard the existence of very elastic materials that exhibitscaling as in (4) with those scaling exponents.We note that the regions in the A − B parameter space

where large strains can be supported are near the specialpoints A ¼ 1, B ¼ 1 (free scalars) or A ¼ 0 (fluid limit).

FIG. 2. Poisson’s ratio r in the allowed parameter region givenin Eq. (21).

2Let us remark that, as can be inferred from Eq. (18), thepower-law scaling can really be reached only for ε ≳ 2. There-fore, the limits shown in Fig. 4 can only be extended to a materialfollowing (20) at large strains in the bluish part of the diagram.

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Therefore, for the model (15), we expect the real-world(nonrelativistic) solids to lie near the A ¼ 0 axis. In thislimit, the maximum strain is set from the absence ofgradient instability for almost all values of B.

Intriguingly, for A ≪ 1, a number of “universal” corre-lations appear. First, we find a universal scaling of themaximum strain

εmax ≃ffiffiffi2

p �B − 1

A

�1=4

: ð22Þ

Inserting this in the expression (9) for the nonlinear shearstress, we further obtain

σmax ≡ σðεmaxÞ ¼ ρeqA: ð23Þ

This shows a linear dependence of the maximal stresssupported by a material on the strain exponent A, which inour simple model controls also the linear elastic modulus.Similar linear correlations are observed experimentally invarious materials [38–42]. Additionally, we also find a clearrelation between the hardness and the maximum strain,σmax ∼ ε−4max. Let us emphasize, however, that whether thecorrelations that we find can be extrapolated to real-worldmaterials strongly depends on i)whether their stress-energyfunction V behaves as a power law at large strain and ii)whether they can support large deformations.Finally, let us note that within the benchmark model (15)

there are no constraints on the bulk strain κ arising from theconsistency and stability requirements. This is a conse-quence of (15) being a monomial. For more generalchoices, additional bounds can arise. Let us also mentionthat for B ∈ ð0; 1Þ it is possible to achieve a negative bulkmodulus, K < 0, in a way that is perfectly consistent fromthe EFT perspective. In particular, as long as K > −G, thestability constraint c2þ > 0 is still satisfied. This has alsobeen studied in four dimensions [43,44] and observedexperimentally [45].

IV. NONRELATIVISTIC SOLIDS

The benchmark model, Eq. (15), considered above hasbeen useful for exhibiting the constraining power of theEFT methods; however, it has one disadvantage: the regionin parameter space giving small sound speeds as in real-world elastic materials is very small. To be more specific,the typical sound speeds are at most of order approximately10−4 in the units of the speed of light. In the parameterspace A, B, this corresponds to the corner where both A andB − 1 are of order 10−8, or less. The problem with this isthat in the benchmark model (15) A and B also control theexponents in the stress-strain relation at large strain, σ ∼ εν

with νshear ¼ 2A for pure shear and νbulk ¼ 2B for pure bulkdeformations respectively. It follows that the benchmarkmodels can only cover realistic materials with very specificexponents, basically νshear ∼ 10−8 and νbulk ≃ 2. Clearly,there has to be a way around this limitation because elasticmaterials with more generic values for νbulk=shear do existand one expects that a similar EFT construction should

FIG. 3. The nonlinear bulk stress-strain curve for the bench-mark model and the parameter values A ¼ 0.5 and B ¼ 1.1, 1.3,1.5, 1.7, 1.9. The large strain scaling is set by ΔTii ∝ κ2B.

FIG. 4. The allowed parameter region (21) for the benchmarkmodel (15). The left, bottom, and right edges are respectivelygiven by the gradient instability, positivity of the bulk modulus,and superluminality. The red line separates the region where themaximum strain is due to the gradient instability (left) and theregion where it is due to superluminality (right). Large strains[and therefore the power-law behavior (4)] are realized in thebluish area.

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describe them. The obvious guess is that the benchmarkchoice, Eq. (15), is too restrictive. In this section, we showhow to deform the model in order to have small speeds ofsound while keeping large deformability and genericexponents.Fortunately, there is a well-motivated and unique way to

ensure that the sound speeds become as small as neededwhile keeping the stress-strain relations untouched. This isachieved by adding an extra term to the potential δV ∝

ffiffiffiffiZ

pwith a large coefficient in front. This term is special formany reasons. Physically, it is proportional to the massdensity of the material [6]. This immediately explains whythe coefficient in front of it must be large in the non-relativistic materials. The mass density contributes to theLagrangian (an energy density) weighted by c2 [6] and ismuch larger than the typical stresses in solids. Related tothis, in the fluctuations around any background, this termonly produces temporal kinetic terms, as can be easily seenin Eqs. (A4)–(A9) in Appendix, noting that this termsatisfies δVZ þ 2ZδVZZ ¼ 0. Therefore, this new termonly contributes to the denominators in the formulas forthe speeds, and so enhancing it decreases the speeds.Moreover, an important feature of this term is that it does

not affect the bulk stress Tii nor the shear stress Tij, so itdoes not alter the stress-strain relations [this is clear fromEqs. (7) and (9)]. This term only appears in the energydensity T00, as it must be, since it only accounts for theinertial mass and thus it contributes like “dust” (pressure-less fluid). This is crucial to retain the predictive/constraining power of the EFT framework because in orderto go to the nonrelativistic regime it suffices to add onesingle parameter in the full nonlinear Lagrangian.For these reasons, it suffices to switch to the following

model,

VðX; ZÞ ¼ ρ0

� ffiffiffiffiZ

pþ v2

�X2

�AZ

B−A2

�; ð24Þ

with v a small parameter (which is a measure of the typicalspeeds in the units of the speed of light). This guaranteesthat the material is nonrelativistic while keeping the non-linear static elastic response the same as in the benchmarkmodel (15).The discussion about the stability and consistency of this

model is also mentioned in the Appendix. In summary, wefind that for v ≪ 1 there are two new regions in the A − Bparameter space that allow i) small velocities and ii) ε ≫ 1(i.e., a very elastic material), as can be seen in Figs. 5 and 6.The first region is close to the line A ¼ B but with (A < B).The other one extends for A > B relatively close to A ¼ 1.These two regions are conceptually on a different levelfrom the EFT standpoint because the constraint on ε arisesfrom gradient instability or superluminality in either case.The separatrix between the two regions is given by the redline in the plots. For v ≪ 1, this line is very close to the line

A ¼ B at small A, B. Importantly, both regions containsizeable values for A, B. The first conclusion, then, is that,indeed, adding a large mass-density term

ffiffiffiffiZ

pto the

Lagrangian opens up the possibility to model nonrelativ-istic materials with sizeable shear and bulk exponents,νshear ¼ 2A and νbulk ¼ 2B.

FIG. 5. Expanded parameter space for v2 ¼ 0.2. The red linesplits the regions where the limit on the maximal strain comesfrom superluminality (on the right) and from gradient instability(on the left). The green dashed line is A ¼ B. In the region A ≥ B,the maximum strain is only dictated by subluminality.

FIG. 6. Expanded parameter space for v2 ¼ 10−8. The sub-luminal constraint in A < B is now located at larger values of Aand B. In the region A ≥ B, the maximum strain is only dictatedby subluminality.

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In the new region with A < B, the subluminality con-dition does not play any role (for v ≪ 1 and moderatevalues of A, B), so these are reasonable candidate EFTs tomodel realistic materials. In this region, the EFT again“predicts” that the maximum strain εmax and the bulk/shearexponents are related in a simple way. One can see that εmaxscales with the exponents as

εmax ∼ffiffiffi2

p �A

B − A

�1=4

; A < B: ð25Þ

Interestingly enough, even though this differs from Eq. (22)[valid for the benchmark model (15) at A ≪ 1], we stillhave some relation εmaxðA;BÞ.The second new region (for A > B) instead is only

constrained by the subluminality condition, and so thebounds are less powerful. Specifically, for v ≪ 1, we find

ε2max ∼ 21

v2=AðAðAþ 1Þ þ BðB − 3ÞÞ−1=A: ð26Þ

Notice that the maximum strain scales as ενSmax ∼ 1=v2,where νS ≡ 2A. This scaling can be understood because forlarge shear deformation the phonon speed cþ grows asc2þ ∼ v2ε2A. Since the constraint obtained within the EFT isreally only an upper limit on the strain, one obtains only avery large upper bound—a very loose bound.As is clear from Figs. 5 and 6, the two regions actually

touch each other; therefore, at some point, one of the speedsmust increase also in the A < B region. Since in this regionεmax comes from the gradient instability of one of the twomodes, i.e., by setting one of the speeds c−ðεmaxÞ ¼ 0, agood notion of how nonrelativistic the material is at thebreaking point is given by the other phonon speed, i.e.,cþðεmaxÞ. At ε ¼ 0, all the speeds are granted to be of orderv, while, by definition, cþðεmaxÞ ¼ 1 on the separatrixbetween the two types of the new regions. However, in trulynonrelativistic materials, one does not expect cþðεmaxÞ toraise to such large values. Therefore, in order to be morerealistic, we can also impose that the speeds do not varymuch from their values at equilibrium (ε ¼ 0) to thebreaking point. To this end, we introduce the ratio

r≡ cþðεmaxÞcþðε ¼ 0Þ ð27Þ

and demand that in the region A < B it is allowed to grow atmost by a factor 10 − 102. In the limit v ≪ 1, r is onlya function of A and B, and in the region B ∼ A, we find,using (25),

r ∼ ðB − AÞ−A=4: ð28Þ

This new constraint is shown in Fig. 7. This figure showsthat there is indeed an overlap between the regions

corresponding to large deformation (εmax significantlygreater than 1), moderate r, and large exponents. The sizeof this region in parameter space depends on the criteria forr and εmax, but one can say that it extends to next to theA ¼ B line within a few percent. In this region, nontrivialrelations such as (27) or (25) should hold.As a final remark, let us emphasize the most basic

property of this new region: it is close to A ¼ B. In otherwords, this corresponds to very elastic realistic materialsthat display a power-law stress-strain curve both for shearand bulk deformations, with nearly equal bulk and shearexponents, νshear ≃ νbulk.

V. DISCUSSION

In conclusion, let us highlight that EFT methods for solidmaterials allow us to extract nontrivial information andbounds on their nonlinear elastic response. The list ofobservables that are fixed (to the leading order in the EFT)once the strain-energy function VðX; ZÞ is known includesall the n-point phonon correlation functions; the phonon-phonon self-interactions; and, most remarkably, how thesedepend on the applied stresses—the first example of thisbeing the acoustoelastic effect. The correlations obtained inthis way are most directly relevant for materials that admitlarge deformations and where dissipative effects areunimportant.3

For a specific application, we have studied how themaximal strain supported by a given material is constrainedby the consistency of the EFT. Focusing on the class ofmaterials with power-law stress-strain relations at largestrains, σ ∼ εν, we find several universal relations between

FIG. 7. Speed ratio constraints for r ¼ 10, 102. Dashed linesshow where εmax ¼ 3, 5, 8 (blue, orange, green).

3For recent EFT-like efforts to include dissipation in fluids andviscoelastic materials; see Refs. [46–59].

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intrinsically nonlinear response parameters, such as themaximum stress and the strain exponent.An interesting case is represented by the conformal

solids limit, realized by potentials of the form VðX; ZÞ ¼X3=2FðX=Z1=2Þ, which preserve scale invariance and implyTμμ ¼ 0 [60] (see also Refs. [61,62]). In this case, the bulk

modulus is directly proportional to the energy densityK ¼ 3=4ρ, as observed in earlier holographic models[62]. Concerning the strain exponents, scale invariancefixes νbulk ¼ 3 and bounds νshear ≤ 3=2. Let us emphasize,however, that the notion of a conformal solid, understood asan EFT with a Lagrangian of the above form, should bedistinguished from a system of which the low energydynamics is controlled by a strongly coupled infrared fixedpoint. In that case, the standard EFT methods are notgranted to apply. A study of the nonlinear elasticity for thatcase using holographic techniques is deferred to a separatework [63].We have also shown how to extend the analysis to

nonrelativistic materials, with realistically small soundspeeds. Our main conclusion—that the EFT methodprovides nontrivial relations between nonlinear responseparameters—remains true also in this regime. Moreover, letus make a remark about the region close to A ¼ B of thesenonrelativistic solids. In this region, the EFT method is themost informative, so it is worth trying to compare itspredictions to data. A proper analysis of the experimentaldata on real-world elastomers is well beyond the scope ofthis work, but we would like to make one comment. It isknown [3] that a very successful way to fit the nonlinearresponse of some rubbers consists of writing VðX; ZÞ as asum of a few powers of the matrix XIJ ¼ ∂μϕ

I∂μϕJ asV ¼ ΣnμnTr½ðXIJÞpn � with some constants μn, pn. At largedeformations, these models are dominated by the term withthe highest power; call it p. It is easy to see that takingV ¼ Tr½ðXIJÞp� does not strictly coincide with our bench-mark models for any A, B; however, it does lead to aresponse at large strains very similar to that in our bench-mark model with A ¼ B ¼ p (for instance in the stress-strain relations). This is encouraging because it wouldsuggest that the models in the region near A ¼ B couldcorrespond to these rubbers. It would be interesting to seewhether (27) or (25) holds for them. We leave thesequestions for the future.Furthermore, it would be desirable to introduce dis-

sipative and thermal effects within the EFT picture ofcondensed matter systems [54,64]. In this regard, the holo-graphic description could provide valuable supplementaryinsight [61–63,65–69]. We hope to return to some of thesepoints eventually.

ACKNOWLEDGMENTS

We thank Alex Buchel, Carlos Hoyos, Karl Landsteiner,Mikael Normann, Giuliano Panico, Napat Poovuttikul,

Kostya Trachenko, and Alessio Zaccone for useful dis-cussions and comments about this work and the topicsconsidered. M. B. is supported in part by the AdvancedERC grant SM-grav, Grant No. 669288. V. C. C. and O. P.acknowledge support by the Spanish Ministry of Education& Science under Grants No. FPA2014-55613-P andNo. FPA2017-88915-P and the Severo Ochoa excellenceprogram of MINECO (Grants No. SO-2012- 0234 andNo. SEV-2016- 0588), as well as by the Generalitat deCatalunya under Grant No. 2014-SGR-1450. M. B. wouldlike to thank Iceland University and Queen MaryUniversity for the warm hospitality during the completionof this work.

APPENDIX: FLUCTUATIONS ANDCONSISTENCY

In order to study the stability of perturbations around thestrained background configuration, we expand the scalarfields as ϕI ¼ ϕI

str þ πI. To identify the propagating d.o.f.,we perform the decomposition into longitudinal and trans-verse fluctuations by splitting πI ¼ πIL þ πIT , with πL=Tsatisfying

OIK∂Iπ

KL ¼ 0; εIJOK

I ∂KπTJ ¼ 0: ðA1Þ

This gives two dynamical scalar modes that can be definedthrough

πIL ¼ OIK∂KπL; πIT ¼ εIJOKJ ∂Kπ

T: ðA2ÞConstraining the spatial dependence to πL=T ¼ πL=Tðt; xÞand redefining πL=T → πL=T=

ffiffiffiffiffiffiffiffi−∂2

x

p, we obtain the follow-

ing quadratic action for the fluctuations,

δS2 ¼Z

d3x½NT _π2T þ NL _π

2L þ 2NTL _πT _πL − c2Tð∂xπTÞ2

− c2Lð∂xπLÞ2 − 2c2TL∂xπT∂xπL�; ðA3Þ

where the parameters NT , NL, NTL and c2T , c2L, c

2TL depend

on both the shear and bulk strains; i.e., they are functions ofε and α. The explicit expressions in terms of the derivativesof the function VðX; ZÞ, defined in Eq. (15), are found to be

NT ¼ 1

2ððX2 − 2ZÞVZ þ XVXÞ; ðA4Þ

NL ¼ ZVZ þ X2VX; ðA5Þ

NTL ¼ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZðX2 − 4ZÞ

qVZ; ðA6Þ

c2L ¼ ZðVZ þ 2ZVZZÞ þ1

2XðVX þ 4ZVXZ þ XVXXÞ;

ðA7Þ

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c2T ¼ 1

4ððX2 − 4ZÞðVZ þ 2ZVZZÞ þ 2XVXÞ ðA8Þ

c2TL ¼ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZðX2 − 4ZÞ

qðVZ þ 2ZVZZ þ XVXZÞ; ðA9Þ

with all the quantities evaluated on the scalar field back-ground solution ϕI

str.Let us emphasize that for a nondiagonal matrix OI

J thetransverse and longitudinal modes remain mixed both withrespect to time and spatial derivatives. In order to study thestability of fluctuations, we therefore first introduce thekinetic matrix as

N ¼�

NT NTL

NTL NL

�: ðA10Þ

The absence of ghostlike excitations then requires that theeigenvalues of the kinetic matrix, λ�, are positive. Thisgives the first condition for stable propagation of themodes: λ� > 0.It is straightforward to determine the true dynamical

modes described by the action (A3) by working at the levelof the equations of motion of the mixed fields πL=T . AfterFourier transforming as πL=T ¼ aL=Teiωt−ikx, we can solvefor the spectrum of perturbations to obtain

ω2� ¼ c2�ðα; εÞk2: ðA11Þ

The other conditions for consistency that we are going toimpose are thus:

(i) c2� ≥ 0, i.e., the absence of gradient instabilities;(ii) c2� ≤ 1, i.e., the absence of superluminal modes.The exact expressions of the kinetic eigenvalues can be

put in the form

λ� ¼ c2

�1�

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

4dc2

r �; ðA12Þ

with

c ¼ NL þ NT; ðA13Þ

d ¼ NTNL − N2TL ¼ detN : ðA14Þ

Similarly, the sound speeds can be expressed as

c2� ¼ a2d

�1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

4bda2

r �ðA15Þ

with

a ¼ c2TNL þ c2LNT − 2c2TLNTL; ðA16Þ

b ¼ c2Tc2L − c4TL: ðA17Þ

Let us point out that evaluating the sound speeds c� atα ¼ 1 and ε ¼ 0 we find that the result coincides with thestandard relationships obeyed by the transverse and longi-tudinal phonons of the equilibrium state ϕI

eq ¼ xI ,

cT ¼ffiffiffiffiffiffiffiffiffiffiffiffiG

ρþ p

s; cL ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiK þ Gρþ p

s; ðA18Þ

where ρ and p are the equilibrium energy density andpressure, as in Eqs. (6) and (7). The K and G refer to thelinearized bulk and shear moduli, defined in Eqs. (13)and (12).The conditions necessary to ensure the positivity of λ�

then read

c > 0; d ≥ 0; 1 −4dc2

≥ 0: ðA19Þ

The first two constraints above can be expressed asinequalities for quadratic polynomials in ε2. For the bench-mark model, we find that upon setting

A − B < 0; A > 0 ðA20Þ

these are satisfied for any choice of ε, while the lastcondition is fulfilled automatically for arbitrary choiceof A, B, ε.The conditions necessary for avoiding the gradient

instability are in turn

a > 0; b ≥ 0; 1 −4bda2

≥ 0 ðA21Þ

and are slightly harder to satisfy. It is easy to see that bysetting

Aþ B > 1 ðA22Þ

and assuming that (A20) holds the condition a > 0 can besatisfied for arbitrary values of ε. However, for these valuesof A, B, the equation b ¼ 0 defines an inverse parabola inthe ε2 space with two real roots ε2� only when

B − 1 > 0: ðA23Þ

Hence, the condition b ≥ 0 is only satisfied forε2 ∈ ½ε2−; ε2þ�. Since we are only interested in positivevalues of ε2, then we conclude that the condition b ≥ 0imposes a constraint on the maximal allowed strain appliedto our system given by

ε2max ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAðB − AÞ þ Aþ BðB − 1Þ

AðB − AÞ

s− 2: ðA24Þ

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Analyzing the last condition in (A21) analytically becomesmore involved. We find, however, that in the parameterregion

B ≤1

2ð2 − Aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4 − 3A2

pÞ ðA25Þ

the maximal strain is determined by the onset of thegradient instability and is thus given by (A24). Only inthe region complementary to (A25) is the maximal strainfixed by requiring the absence of superluminal propagation,finding

ε2max ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAðAþ B − 2Þ

A2 þ AðB − 1Þ þ ðB − 2ÞB

s− 2: ðA26Þ

We present the full constraints on the parameter spaceobtained numerically in the main text.Finally, let us quote our results for the simple case of

linear deformations, i.e., of zero background shear strain,i.e., ε ¼ 0. We obtain the following allowed region for theexponents A and B:

0 ≤ A ≤ 1 and 1 ≤ B ≤ffiffiffiffiffiffiffiffiffiffiffi1 − A

pþ 1: ðA27Þ

More specifically, the two kinetic eigenvalues in this caseare equal and given by λ� ¼ 2−1þA−2BB imposing theconstraint B > 0. The sound speeds are in turn given byc2− ¼ A

B and c2þ ¼ B − 1þ AB. The absence of gradient

instabilities is thus setting A ≥ 0 and B − 1 ≥ −A=B.The latter constraint can be made stronger by requiringthe positivity of the bulk modulus, leading to B ≥ 1; thepositivity of the shear modulus gives again A ≥ 0.We can now repeat the exercise of finding the speeds and

all the constraints for the nonrelativistic solid modelpresented in Sec. IV. In the limit of infinitesimal strain,the transverse and longitudinal modes, as defined in (A2),

decouple at the level of the quadratic action (A3). Indeed,for ε ¼ 0, we find that NTL ¼ c2TL ¼ 0. From the positivityof the remaining quantities NT , NL, c2T , c

2L, we arrive at the

following set of conditions on the parameters of our model:

A > 0; Aþ BðB − 1Þ > 0; 1þ Bv2 > 0: ðA28ÞThe propagation speeds of the canonically normalizedmodes are then given as

c2TNT

¼ v2A1þ Bv2

;c2LNL

¼ v2ðAþ BðB − 1ÞÞ1þ Bv2

: ðA29Þ

We thus see that with this choice of potential bothpropagation speeds in the infinitesimal strain limit scalewith v. Hence, in order to go to nonrelativistic speeds, wejust need to set v ≪ 1. Let us also point out that the twospeeds are related as c2L ¼ c2T þ v2BðB − 1Þ. The secondterm in this relation comes from the linear bulk modulus,defined in Eq. (13). For the new choice of potential, itequals K ¼ ρ0v2ðB − 1ÞB, and thus for a negative B − 1,the bulk modulus becomes negative. Henceforth, we shallonly consider B ≥ 1.The additional term in the potential also enables us to

expand the allowed parameter space for A, B. In particular,by analyzing the stability conditions (A21), we find that themaximal strain is only set by the requirement of the absenceof gradient instability for the parameter values A < B. Itsvalue remains unaffected by the new term, i.e., it does notdepend on v, and is still given by (A24), with the additionalrequirement (coming from ε ¼ 0) that Aþ BðB − 1Þ > 0.In the remaining the parameter space, the maximal strain isdetermined by the superluminality constraint. The newterm in the potential pushes the superluminality constraintfarther away, thus expanding the allowed region for A, B.This is shown in Fig. 5.

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