The metric description of viscoelasticity and instabilities in viscoelastic solids
TThe metric description of viscoelasticity and instabilities in viscoelastic solids
Erez Y. Urbach and Efi Efrati ∗ Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel (Dated: June 5, 2018)Many manmade and naturally occurring materials form viscoelastic solids. The increasing useof biologically inspired elastomeric material and the extreme mechanical response these elastomersoffer drive the need for a better, more intuitive and quantitative understanding of the mechanicalresponse of such continua. This is in particular important in determining the stability of viscoelasticstructure over time where in lieu of robust rules we often must resort to simulations.In this work we put forward a metric description of viscoelasticity in which the continua is char-acterized by temporally evolving reference lengths quantified by a rest reference metric. This restreference metric serves as a state variable describing the result of the viscoelastic flow in the system,and allows us to provide robust claims regarding stability of incompressible isotropic viscoelasticmedia. We demonstrate these claims for a simple bistable systems of three standard-linear-solidspring dashpot assemblies where the predicted rules can be verified by explicit calculations, andalso show quantitative agreement with recent experiments in viscoelastic silicone rubber shells thatdisplay delayed stability loss.
I. INTRODUCTION
Elastomeric solids are ubiquitous materials composedof cross-linked long molecular chains and display extrememechanical response properties. Examples include latexrubber, silicone rubbers [1] as well as the crosslinked pro-teins resilin and elastin responsible for the mechanicalproperties of tissues such as ligaments and arteries [2–4]. Elastomeric solids are characterized by low elasticmoduli, and reversible deformations even at strains ex-ceeding one hundred percent. However, they also showstress relaxation when held at constant displacement andcreep under a constant load [5]. These creep and stressrelaxation phenomena are dissipative yet are reversible.Unlike viscoelastic fluids, elastomeric solids retain thetopology of material elements in the body indefinitely.Moreover, they display only fractional stress relaxation,supporting a finite fraction of the stress even after arbi-trarily long relaxation times.One of the most fascinating phenomena displayed byelastomeric materials is delayed stability loss, in which afast instability releasing the elastically stored energy ina system is preceded by a slow creep. Such phenomenahave been observed in the rapid snapping of the Venusfly-trap leaf [6] activated by the plant to capture prey, inthe passive snap through of thin elastomeric shells knownas jumping poppers [7, 8], and for non-elastomeric mate-rials in the slow crustal dynamics leading to some earth-quakes [9, 10].The equations of state for viscoelastic materials com-monly relate the stress to the full history of the strainin the body through a material dependent memory ker-nel [5]. Such equations of state accurately capture thematerial response, and may be used to numerically studythe response of viscoelastic structures given their geome-try and loading conditions [11, 12]. However, they rarely ∗ efi[email protected] allow explicit solutions and provide very little insight tothe state of the viscoelastic material and its general re-sponse properties. The lack of intuition for viscoelasticdynamics is exceptionally evident when considering vis-coelastic instabilities; as linear stress relaxation acts toreduce local stresses it is expected to have a stabilizingeffect and in particular never cause a meta-stable stateto lose stability. More elaborated variations of this handwaving argument, as we show in section IV B, only proveit stronger rather than refute it, thus elucidating the sub-tle nature of viscoelastic instabilities.In this works we describe a metric approach to vis-coelasticity and viscoelastic stability. We begin by de-scribing one dimensional viscoelastic systems throughtemporally evolving rest lengths and specifically formu-late the dynamics of the standard linear solid (SLS)model as a spring with temporally evolving rest length.We then consider an assembly of three SLS springs inthe form of a Von-Misses truss as a first example of tran-sient elastic stability in viscoelastic systems. We con-clude the section describing this one dimensional motionby using the temporally evolving rest lengths to provegeneral claims regarding the elastic stability and station-arity of a general 1D, SLS systems.The intuitive results we obtain for one dimensional SLSsystems can be generalized to full three dimensional sys-tems. We next do so by constructing a covariant metricdescription of viscoelastic solids. The theory describesthe material response as elastic with respect to a timedependent three dimensional reference metric. We eluci-date the notions of quasi static approximation, transientelastic stability and isotropicity in light of this new de-scription. We then discuss results regarding the quasi-static dynamics when considering isotropic and incom-pressible materials. Several claims regarding the elasticstability and stationary states of such bodies are shown. a r X i v : . [ c ond - m a t . s o f t ] J un II. VISCOELASTICITY AND DELAYEDSTABILITY LOSS IN 1D (1 − β ) kβk η L L d FIG. 1. Standard linear solid with total stiffness k , β andviscosity η . A. Standard linear solid
The simplest intuitive model capable of displayingboth finite stress relaxation and a finite response at highrates is called the standard linear solid model (SLS) [13].The SLS is a generalization of two spring-dashpot-models- the Kelvin-Voigt model, accounting for the creep re-sponse, and the Maxwell model capturing the stress re-laxation and high loading rate response [14]. The SLSconsist of spring of stiffness βk and dashpot of viscos-ity η connected in series, both connected in parallel toanother spring of stiffness (1 − β ) k (see FIG. 1). The di-mensionless constant, 0 ≤ β ≤
1, accounts for differencebetween the spring stiffness. The total length of the sys-tem is denoted L , the length on the dashpot is L d , andthe rest length of the long and short springs read ¯ L and˜ L , respectively. The total force is then F = − (1 − β ) k ( L − ¯ L ) − βk ( L − L d − ˜ L ) , (1)while force balance between the spring and dashpot inseries yields the closure relation η ˙ L d = βk ( L − L d − ˜ L ) . (2)Rapid variation of the total length L → L + ∆ L willelongate the springs while leaving the dashpot length, L d unchanged. The force under such conditions will increaseby ∆ F = k ∆ L , similar to a simple spring of stiffness k .Motivated by the temporal scale separation between theelastic response of viscoelastic solids and their typicalcreep rate we seek to rephrase the force equations as asimple elastic spring. By setting the reference length ¯ L = (1 − β ) ¯ L + β ˜ L + βL d , and substituting in eq.(1) we obtain F = − k ( L − ¯ L ) . (3) k k FIG. 2. The SLS von-Mises truss model. Point mass con-nected to two diagonal SLS with k each, and a vertical SLSwith total k . Both with dashpot η and compliance β . This is supplemented by the closure relation describingthe temporal evolution of the reference length˙¯ L = − τ (cid:0) β ( ¯ L − L ) + (1 − β )( ¯ L − ¯ L ) (cid:1) , (4)where τ = ηβk . The reference length evolves simultane-ously towards L , the present state of the system, andtowards ¯ L , the rest length to which it will asymptoti-cally approach if left unconstrained. The two simulta-neous evolution terms are weighted by the dimensionlessfactor 0 ≤ β ≤ β = 0 corresponds to the elastic casein which ¯ L starts and stays at ¯ L . β = 1 corresponds toa Maxwell material where the material reference length¯ L has no preferred rest value and approaches L , relaxingthe force it supports to zero. One can easily generalizethe SLS model to account for multiple relaxation timescales by adding more spring dashpot pairs to the sys-tem in parallel, and even account for a distribution oftime scales that could give rise to non-exponential relax-ation; e.g. power-law or logarithmic. Such generaliza-tions will not change the notion of the reference lengthor its interpretation through eq.(3), but only change theclosure relation describing its temporal evolution:¯ L ( t ) = (1 − β ) ¯ L − β (cid:90) t −∞ ˙ φ ( t − s ) L ( s ) ds. (5)Note that β retains its meaning as the fraction of forcethat is relaxed asymptotically in a constant displacementsetting. As we will see later this quantity dominatesmost of the questions of stability in viscoelasitc systems,while the actual functional form of the memory kernel, φ ,only influences the temporal approach to instability. Forthis reason it suffices to examine the question of stabil-ity loss in 1D considering only the behavior of a simpleSLS model. While equations (3) and (5) are a mere re-formulation of the familiar viscoelastic dynamics, statingthe problem in these new variables allows an intuitiveinterpretation of the dynamics and deeper insight to thebehavior such systems exhibit. B. SLS von-Mises truss
One of the most striking and subtle viscoelastic phe-nomena is delayed instability. To capture the essence ofthe phenomena we consider three SLS’s of similar mate-rial parameters β and τ , arranged to form a von-Misestruss [15] as illustrated in figure 2. The truss is composedof a point mass that moves along the z axis. It is con-nected to two diagonal SLS’s of stiffness k that create abi-stable elastic potential. The symmetry of this poten-tial is broken by a vertical SLS of stiffness k . The restlengths are chosen such that the system is relaxed andstationary at position z = 1 at rest lengths ¯ L = √ L = 0. The lengths of each spring as function of z are L ( z ) = √ z and L ( z ) = | z − | respectively. Asdisplayed on the previous section, each SLS is associatedwith a reference length ¯ L i that slowly evolves accordingto Eq. (4). The force on the mass is minus the derivativewith respect to z of the instanteneous elastic energy E ( z ) = 2 k L ( z ) − ¯ L ) + k L ( z ) − ¯ L ) (6)We set α = (cid:113) k k and note that for the purely elastictruss bi-stability is obtained for small values of α whereasmono-stability is displayed at higher α , where only the z > dEdz = 0. For these purposes it is instructiveto employ a normalized energy function e ( z ) = E ( z ) k = ( L ( z ) − ¯ L ) + ( αL ( z ) − α ¯ L ) (7)This energy function may be interpreted as the Euclidean“distance” between the normalized reference lengths ¯L = ( ¯ L , α ¯ L ) and the normalized configuration lengths L ( z ) = ( L ( z ) , αL ( z )). We can thus use this 2D ’phasespace’ of normalized lengths to understand the behaviorof the system. The admissible (normalized) lengths ofthe springs L ( z ) form a one dimensional curve parame-terized by z , whereas the reference lengths ¯L could beanywhere in the two dimensional space. The Euclideandistance between the configuration L and the referencelengths ¯L is exactly the elastic energy of the configura-tion. Specifically, as L ( z ) = ( √ z , α | z − | ), z < , α ) (cor-responding to z = 0) and asymptotic line ( L , αL ) as z → −∞ . The rest lengths ¯L = ( √ ,
0) is a point onthe admissible lengths with z = 1, or ¯L = L ( z = 1) (seeFIG. 3).To understand the dynamics of the system, we choosethe following protocol; the mass is taken from rest, where L i = ¯ L i = ¯ L i , abruptly to z = − t = 0 the reference lengths ¯L are atthe rest length values ¯L = ( √ ,
0) whereas the springsare held at the configuration L hold = ( √ , α ). By Eq. 4 during the holding the reference lengths ¯L evolve alongthe line connecting ¯L and L hold up to a stationary po-sition between them (See FIG. 3). We note that due tothe relaxation the reference lengths assume values noton the curve of admissible lengths, L ( z ), and thereforedo not correspond to any realizable configuration of thesystem. Nevertheless ¯L might have stable realizable con-figurations. An elastic equilibrium corresponds to a pointof minimal “distance” according to (7) between L ( z ) and ¯L ( t ). Geometrically, this minimization condition is sat-isfied if the line between the reference lengths ¯L and theconfiguration L is normal to the admissible lengths curve ¯L ( z ) (see FIG. 3). After the release of the mass from L hold it will elastically snap to the closest stable point.Using this geometrical interpretation of the viscoelasticquasi-static evolution, we can explain how different val-ues of α create different dynamical phases.We first consider cases with large α → ∞ , illustratedin FIG. 3a. At rest, ¯L = ¯L , only the rest lengths’spoint ¯L is stable. That is, no other line connecting thepoint ¯L with the curve L ( z ) meets the curve L ( z ) per-pendicularly. During the holding, the reference lengths ¯L approache L hold . However, still there is only one stabledistance minimizing solution, which in turn is obtainedby slightly perturbing the rest solution ¯L = ¯L . In otherwords, the elastic potential remain mono-stable along theentire relaxation. Upon release from L hold , the configu-ration will snap to L s and evolve back with ¯L to ¯L . Putin more conventional terms the relaxation of the forcesduring the holding was not enough to stabilize the massnear the position of relaxation and when released it snapsback to a position near its original rest position.The second limit to consider is that of small α → α diminish the value of the ec-centricity towards unity, shrink the hyperbola verticallyand transform the two asymptotic lines to become closerto parallel. Thus, as expected, the rest lengths ¯L is nowassociated with two stable points; the original rest posi-tion and another stable point, L i , situated closer to theinverted configuration L hold , and associated with z < L s for some another z > L i for different z <
0. The system remains bi-stable,and the relaxation only further lowers the instantaneouselastic energy of the inverted states (as their distancedecreases).If for this case we follow a different protocol in whichthe systems starts at rest and then brought abruptly tothe state L = L which is a locally stable equilibriumwith respect to ¯L = ¯L , then while ¯L will evolve towards L , the actual configuration will remain unchanged at L = L . We term such states stationary states. Thisstationarity property can be easily understood geomet-rically. In the protocol described ¯L evolves along theline connecting L and ¯L for every point on this linethe assumed configuration L is locally the closest to it,and thus the position of the elastic equilibrium does notchange. FIG. 3. The normalized lengths 2D space for (a) α = 1, (b) α = 0 . α = 0 .
1. The curve of admissible lengths L ( z ) is drawn in thick (black) line. The initial lengths ¯L =( √ ,
0) is drawn in (red) filled circle. During the relaxationthe reference lengths ¯L - drawn in gray filled circle - evolvein the direction of the holding lengths L hold = L ( z = −
1) =( √ , α ). The stable positions of a specific reference lengthsis geometrically the normals from its point to the curve. (a)Unstable - even along the relaxation the system remain mono-stable. (b) Finitely stable - bistability emerge because of therelaxation, and dissolve finite time after releasing the system.(c) Stable - the system remains bistable during the holding. Last we consider the case of intermediate values of α that display delayed instability. One could find values of α large enough such that for ¯L = ¯L only the state L = ¯L is stable, yet small enough such that after relaxation at L hold for time t there will be two distinct locally stableequilibria with respect to ¯L ( t ), as depicted in FIG. 3b. When released from relaxation the system will assumethe state L = L i , where z <
0. However, the referencelength ¯L continuously evolves towards collinearity with L and ¯L . However, as no state other than the trivial oneis stable with respect to ¯L = ¯L such a process cannotconverge to a stable state and thus must at some timeloose stability.The Von-Mises truss SLS models displays non-trivialphenomenology of instabilities of viscoelastic systems, allof which are intuitively explained through the graphicalrepresentation. If the system at rest can be brought toanother locally stable equilibrium, then the equilibriumpoint will remain unchanged despite the visco-elastic evo-lution. In particular it will never become unstable. Con-versely, state that lose stability at a finite time must havenot been stable with respect to their rest state and theirstability must have been acquired through relaxation.Last, every acquired stable point cannot remain stableindefinitely and must become unstable. The results ob-tained here were obtained for the specific geometry andconstitutive relations of the Von-Mises SLS system. How-ever, they remain valid for rather general linear visco-elastic systems. In appendix A we show that they holdfor an arbitrary collection of SLS’s provided the massthey are connected to is constrained to move along a lineand they all share the same β and τ . We next come togeneralize these result for the general isotropic and in-compressible three dimensional linearly viscoelastic con-tinua. III. METRIC LINEAR VISCOELASTICITY
We now turn to discuss linear viscoelasticity of contin-uous bodies. Such systems are amenable to a geometricreformulation similar to what we did for the SLS model.Here we need to exploit an infinitesimal analog of theevolving reference lengths, which we achive by resortingto an elastic description using metric tensors.
A. Linear viscoelastic continuum
We start by equipping the body with a material (orLagrangian) coordinate system x , x , x . r ( x i , t ) is theconfiguration of the body at time t in R . The config-uration manifest itself on the coordinate system (up totime dependent rigid motions) by the induced Euclideanmetric g ij = ∂ i r · ∂ j r , which measure the infinitesimallengths in the solid [16]. In analogy with ¯ L we definethe rest reference metric ¯ g ij as the metric on which thebody is locally stress-free and stationary. That is, if thebody was to be cut to infinitesimal pieces and each ofthe pieces allowed to freely relax indefinitely, then thelengths of each piece will approach those represented bythe rest reference metric ¯ g ij . The strain tensor of thebody is ε ij ( t ) = (cid:0) g ij ( t ) − ¯ g ij (cid:1) . (8)Different measures of stress correspond to different mea-sures of volume. The second Piola-Kirchoff stress tensor S ij is the virtual work conjugate of the metric, per unitvolume (cid:112) ¯ g [16, 17] δW = − (cid:90) S ij δε ij (cid:112) ¯ g d x = − (cid:90) S ij δg ij (cid:112) ¯ g d x (9)Intuitively, the energy difference obtained by varyingonly the distance between two points is exactly the forcebetween them. The general viscoelastic model assumesthe stress at time t is a functional of the entire historyof the strain S ij ( t ) = S ij [ ε ij ( s )] of times s ≤ t . ¯ g ij wasdescribed above as the only stationary stress-free config-uration. By Eq. (8), S ij ( t ) = 0 for all t if and only if thestrain ε ij ( t ) = 0 for all t . Following [18] we can approx-imate the functional S ij [ ε ij ] around the zero function,and obtain a linear functional S ij ( t ) = C ijkl ε kl + (cid:90) t −∞ Φ ijkl ( t − s ) ε kl ( s ) ds. (10) C ijkl is the stiffness tensor, Φ ijkl ( s ) is the memory kernel accounting for the viscoelastic dissipation. Φ( s ) shouldfade over time, thus we take ˙Φ( s ) < s → ∞ ) =0 [18].Similar to the SLS, Eq. (10) predicts that instanta-neous incremental deformations ∆ g ij lead to linear stressincrements ∆ S ij = C ijkl ∆ g kl . Following the above no-tion of temporal reference length ¯ L , we define the time-dependent reference metric ¯ g ij of the body to satisfy S ij ( t ) = C ijkl ( g kl ( t ) − ¯ g kl ( t )) . (11)The temporal evolution of the reference metric can bededuced from equations (8), (11) and (20), and reads¯ g ij ( t ) = ¯ g ij − C − ijkl (cid:90) t −∞ ˙Φ klmn ( t − s ) (cid:0) g mn ( s ) − ¯ g mn (cid:1) ds (12)= ¯ g ij − (cid:90) t −∞ ˙ˆΦ klij ( t − s ) (cid:0) g kl ( s ) − ¯ g kl (cid:1) ds. (13)ˆΦ klij ( s ) = C − ijkl Φ klmn ( s ) is the metric memory kernel . Aswe can see, ¯ g ij is defined only on cases where the stiff-ness tensor is invertible, a reasonable assumptions formost solids. As expected, ¯ g ij ( t ) remains unchanged byinstantaneous variations of g ij . We may thus consider itas the slow state variable describing the viscoelastic evo-lution of the material. At each moment in time we mayconsider the system as an elastic system with respect tothe reference metric ¯ g ij [16, 19]. B. The quasi static approximation
Viscoelastic systems are dissipative, thus the notionof an elastic free energy is ill defined. Nonetheless, the virtual work of spatial variations performed by the bodyover a short period ∆ t →
0, coincides with the instanta-neous elastic energy functional [16] E (cid:2) ε el ij (cid:3) = (cid:90) C ijkl ε el ij ε el kl (cid:112) | ¯ g | d x, (14)where the elastic strain is again ε el ij = ( g ij − ¯ g ij ). Typi-cally the elastic response time scale in elastomers is muchsmaller than the viscoelastic relaxation time. In suchcases we can eliminate inertia from the system and ap-proximate the motion of material as quasi static evo-lution between elastic equilibrium states. That is, theconfiguration at every moment in time, given by themetric g ij ( t ), minimizes the instantaneous elastic en-ergy functional (14). If we can characterize the possi-ble Euclidean metrics by finite set of variables λ α , i.e. g ij ( x ) = g ij [ λ α ]( x ) then the minimization condition isfor all α , reads δEδλ α = 0 or F α [ g ij ] = − (cid:90) δg ij δλ α S ij (cid:112) | ¯ g | d x = 0 , (15)which is the vanishing of the generalized forces. Forthis extremal point to be a minima, the Hessian, δ Eδλ α δλ β ,needs to be positive definite.Condition (15) gives g ij ( t ) as function of ¯ g ij ( t ), whichin turn evolve according to Eq. (13). The quasi staticevolution gives an intuitive interpretation of the slow evo-lution of viscoelastic bodies. Over time, the material’sconfiguration quickly minimizes its ’distance’ to the ref-erence metric. The idea that in viscoelasticity bodiesadapts to their configuration is implemented by the slowdynamics of the reference metric from the rest referencemetric toward the recent configurations. This coupledevolution of g ij and ¯ g ij is also simpler to analyze bothnumerically and theoretically compared with the full dy-namics. C. Acquired stability and stability loss
In section II B we describe how the evolution of the ref-erence lengths in discrete, 1D systems can result in nontrivial evolution of the configuration (characterized by z ).When the parameter α is appropriately set the von-Misestruss could acquire new type of stable states; namelystates that are unstable initially, but acquire stabilitythrough stress relaxation at a strained state. Such statesalso exist for continuous viscoelastic systems, and simi-larly to the discrete case described above can be shownin some cases to not be able to remain stable indefinitely.A given reference metric ¯ g ij can yield multiple elas-tically stable configurations of the instantaneous elasticenergy functional (14). Under external loading, as thereference metric slowly evolves viscoelastically accordingto Eq. (13) it could acquire new stable configurations,merge existing stable points or cause stable elastic con-figurations to lose their stability. Acquiring stability ismost easily understood by examining loading at constantdisplacements. Consider a body taken from rest to aninitially unstable configuration g hold ij . The body is heldfixed at this position and the reference metric ¯ g ij startsfrom ¯ g ij and slowly advances toward the metric g hold ij ,and the forces pushing the body away from its presentstate gradually relax. When releasing the body an ini-tially unstable metrics near g hold ij might display transientstability. This phenomenon is called acquired stability .State with acquired stability lose their stabilitythrough delayed stability loss . Seemingly stable stategradually creep until at some point elastic stability islost, and the body rapidly snaps to a near stable configu-ration. Both acquired stability and delayed stability lossare expected for linear viscoelastic bodies, close to multi-stability. Described quantitatively in section IV below,as the memory kernel become comparable to the stiffness,more configurations are able to acquire stability. D. Isotropicity and homogeneity
Considering isotropic materials the stiffness and mem-ory kernel tensors C ijkl and Φ ijkl ( s ) further simplify.Isotropicity is the infinitesimal invariance to rotation insome configuration of the body. More generally, it is theinfinitesimal invariance to orthogonal transformations in-duced by some metric on the body. This metric describeshow each piece of the body should be deformed in orderto have the same value under rotations.By Eq. (14), the metric associated with the stiffness isthe metric used to raise the indices of the elastic strainto create the scalar energy density. In linear elastic-ity, in which ¯ g ij = ¯ g ij , the stiffness is usually taken asisotropic with respect to the reference metric ¯ g ij [16].This amounts to assuming the body is isotropic in theundeformed configuration. The natural generalization ofthis assumption to viscoelasticity is to take the stiffnessas isotropic with respect to the instantaneous referencemetric ¯ g ij . This will yield isotropic response proper-ties about the instantaneous stress free state. However,an isotropic elastic response is often intimately relatedto isotropic material composition and internal structurewhich in our case is set by the rest reference metric¯ g ij . Therefore, we assume here the stiffness tensor tobe isotropic with respect to the rest reference metric C ijkl = λ ¯ g ij ¯ g kl + 2 µ ¯ g ik ¯ g jl . (16)We note that in principle, some materials may displaya different constitutive behaviors, and in particular beisotropic with respect the instantaneous reference metric¯ g ij in which case the elasticity tensor should be adaptedaccordingly. Moreover, while it is reasonable to as-sume that the memory kernel Φ ijkl ( s ) accounting for theviscoelastic response will also be isotropic with respectto ¯ g ij , there is no restriction preventing it from beingisotropic with respect to any linear combination of ¯ g ij and ¯ g ij . The assumption of homogeneity is usually obtained bytaking the tensor at hand as constant in space. However,because both C ijkl and Φ ijkl ( s ) are densities of proper-ties, homogeneity is a matter of measure. Directly fromEq. (14), constant stiffness gives equal energy densityper ’initial’ volume element (cid:112) ¯ g d x (with the same de-formation). Here too, in principle, one can account forother measures induced by other metrics, and thus fordifferent types of homogeneity. Thus even when a ma-terial is considered isotropic and homogeneous, there aresome choices as to how to express these symmetries inthe constitutive relations.We now focus on the reasonable and simplest op-tion, where the stiffness and the memory kernel are bothisotropic and homogeneous with respect to ¯ g ij . The gen-eral form of the metric memory kernel ˆΦ klij ( s ) isˆΦ klij ( s ) = βφ ( s ) δ ki δ lj + ψ ( s ) ¯ g ij ¯ g kl (17)where 0 < β < φ (0) = 1, andboth φ ( s ) , ψ ( s ) decrease monotonically to zero as s → ∞ .Substitution in Eq. (13) gives the isotropic evolution ofthe reference metric¯ g ( t ) = (1 − β − α ( x, t )) ¯ g ij − β (cid:90) t −∞ ˙ φ ( t − s ) g ij ( s ) ds, (18)where α ( x, t ) = (cid:90) t −∞ ˙ ψ ( t − s ) (cid:0) ¯ g kl g kl ( x, s ) − (cid:1) ds. (19)Note that α ( x, t ) is the only term in Eq. (18) that mixesthe indices of the metrics. IV. VISCOELASTICITY OF INCOMPRESSIBLEMATERIALSA. Constitutive relations
Incompressible isotropic materials are described byPoisson ratio ν = . The incompressible Poissonian con-traction is expected to hold also during the viscoelasticrelaxation. Thus, we expect ψ ( s ) = 0. Put in Eq. (17)and Eq (10), we obtain the following constitutive relationbetween stress and strain S ij ( t ) = C ijkl (cid:18) ε kl ( t ) + β (cid:90) t −∞ ˙ φ ( t − s ) ε kl ( s ) ds (cid:19) . (20)The incompressible evolution of the reference metric (byequations (13) and (17)) also gives¯ g ij ( t ) = (1 − β ) ¯ g ij − β (cid:90) t −∞ ˙ φ ( t − s ) g ij ( s ) ds. (21)The dimensionless factor β quantifies the degree of vis-coelasticity in the system. It corresponds to the asymp-totic value of the fraction of stress relaxed in a constant g ¯ g stat ¯ g ¯ g ( t ) Admissiblemetrics
FIG. 4. A schematic representation of the metrics collinear-ity. The minimization of the metric g is constrained and per-formed with respect to a subset of metrics that correspondto realizable configurations. Such metrics are in particularorientation preserving and Euclidean. This set is representedby the dashed line above. Given a reference metric ¯ g the re-alized metric will correspond to the closest point from the setof admissible metrics to ¯ g according to the distance functiongiven by the instantaneous elastic energy (14). Starting fromrest, ¯ g evolves from ¯ g towards g . Conversely, if g is the clos-est admissible metric to ¯ g s tat it is also such for ¯ g which iscollinear with the two. displacement experiment starting from rest. Imposing astationary solution for ¯ g ij and g ij in Eq. (21), we obtain¯ g stat ij = βg ij + (1 − β )¯ g ij . (22)Here β is shown to control the degree to which the refer-ence metric of a viscoelastic body will approach the cur-rent configuration metric; β = 0 describes a purely elas-tic material with a constant reference metric, and β = 1describes a viscoelastic fluid in which no information isconserved indefinitely, asymptotically all stresses are re-laxed, and the notion of the rest reference metric ¯ g ij hasno meaning.As seen from Eq. (22), the long time stability of abody depends only on β , and not on the specific memorykernel. It is thus instructive to discuss the SLS model.The memory kernel for the SLS model is given by φ ( s ) = e − s / τ . In analogy to Eq. (4), the differential equation forthe reference metric reads˙¯ g ij ( t ) = − τ (cid:0) β (¯ g ij ( t ) − g ij ( t )) + (1 − β ) (cid:0) ¯ g ij ( t ) − ¯ g ij (cid:1)(cid:1) = − τ (cid:0) ¯ g ij ( t ) − ¯ g stat ij ( t ) (cid:1) . (23) B. Stationary states and the rest system
We now turn to examine the stationary solutions ofthe incompressible evolution rules Eq. (21) and(15). Asdescribed on section II B and in appendix A, the sta-tionary states of one dimensional SLS structures are ex-tremal solutions of the rest elastic system ¯ L i = ¯ L i . Sim-ilar claim can be made for general linear viscoelasticand incompressible bodies. By Eq (22) at stationarity¯ g stat ij = βg ij + (1 − β )¯ g ij . Configuration g ij is stationary if it is in elastic equilibrium with respect to its stationaryreference metric ¯ g stat ij . Thus, the generalized forces withrespect to ¯ g stat ij must vanish0 = F stat α [ g ij ]= − (cid:90) C ijkl δg ij δλ α ( g kl − ¯ g stat kl ) (cid:112) | ¯ g | d x = − (1 − β ) (cid:90) C ijkl δg ij δλ α ( g kl − ¯ g kl ) (cid:112) | ¯ g | d x = (1 − β ) F α [ g ij ] .F α [ g ij ] is the generalized force calculated in the rest elas-tic system ¯ g ij = ¯ g ij . Therefore the configuration is anextremal of the elastic energy functional with respect tothe rest reference metric F α [ g ij ] = 0. We thus obtainedthat all the stationary configurations of viscoelastic bodyare extremals of its the rest elastic system.Following Eqs. (14) and (22), we arrive to the followingrelation between the energy Hessians of the stationaryand rest systems δ E stat δλ α δλ β = (1 − β ) δ E δλ α δλ β + β (cid:90) C ijkl δg ij ( x ) δλ α δg kl ( x ) δλ β (24)Drucker stability criterion states that C ijkl is positivedefinite, thus also (cid:82) C ijkl δg ij ( x ) δλ α δg kl ( x ) δλ β is a positivedefinite matrix [14]. If the configuration was initiallystable and δ E δλ α δλ β is positive definite, then the sta-tionary Hessian is also positive definite (sum of twopositive definite matrices). Thus initially stable (notonly extremal) configuration is also stationary. ButEq. (24) also open the possibility for non-stable butextremal configuration of the rest system, for examplea saddle point, to stabilize indefinitely. Eq. (24) alsoelucidate the basic intuition that when a system is heldat constant displacement, stress relaxation stabilizesthe configuration; as we can see from the equation,originally non positive Hessian become ’more positive’along the relaxation. Both features depend on the valueof β . As β → β → t < g ij ( t ) = ¯ g ij ( t ) = ¯ g ij . At t = 0 the system isbrought abruptly to an elastically stable conifguration g ij (0) (cid:54) = ¯ g ij , such that g ij (0) satisfies (15). We wish toshow that ˙ g ij ( t ) = 0 satisfies the evolution rules for all t >
0. First, substitution in Eq. (21) gives¯ g ij ( t ) = ¯ g + β (1 − φ ( t )) (cid:0) g ij − ¯ g ij (cid:1) , (25)where g ij = g ij ( t ) is constant in time. More impor-tantly, we have the evolution of the elastic strain ε el ij ( t ) = ( g ij − ¯ g ij ( t )) ε el ij ( t ) = (1 − β (1 − φ ( t ))) (cid:0) g ij − ¯ g ij (cid:1) = (1 − β + βφ ( t )) ε el ij ( t = 0) (26)It is left to prove that the constant g ij remains at localminimum of the instantaneous energy functional with re-spect to Eq. (25). First, the generalized force F α ( t ) = − (cid:90) C ijkl δg ij δλ α ε el kl ( t ) (cid:112) | ¯ g | d x = (1 − β + βφ ( t )) F α ( t = 0)= 0The last step uses the stability of g ij at t = 0 (withrespect to ¯ g ij ). Second, the Hessian δ E ( t ) δλ α δλ β = (1 − β + βφ ( t )) δ E ( t = 0) δλ α δλ β + β − φ ( t )) (cid:90) C ijkl δg ij ( x ) δλ α δg kl ( x ) δλ β By assumption, δ E ( t =0) δλ α δλ β is a positive definite matrix. Aswe note above, the right matrix is also positive definite. δ E ( t ) δλ α δλ β is thus a sum of two positive definite matrices,and therefore also positive definite for all t >
0. Thus,we have shown that ˙ g ij ( t ) = 0 for any t > ijkl ( s ) but only underthe assumption that the metric kernel is scalar ˆΦ klij = βφ ( s ) δ ki δ lj . Still, the result are valid to any incompress-ible solid in a continuous manner. That is, the actualvariation from these result diminish as the material athand is closer to incompressibility.Intuitively, both claims are due to collinearity of g ij ,¯ g ij and ¯ g ij under the appropriate initial condition (seeFIG. 4). If the evolution of the reference metric startsat rest ¯ g ij = ¯ g ij then evolving toward g stat ij also evolve inthe direction of g ij . If the metric was originally stable -with ’minimal length’ to ¯ g ij = ¯ g ij , than it will also beminimal to any reference metric ¯ g ij ( t ) on the line betweenthe rest reference metric to g ij . C. Transient elastic stability and snap-through
We have so far discussed the behavior of viscoelasticsystems in states that are stable when arrived to from rest. We next come to discuss the complementary statesthat are unstable. In particular we would like to addresshow could they acquire stability and also in turn howthis stability is lost. Jumping poppers are thin rubbershells. Very thin poppers display almost isometric bi-stability, while thick poppers have only one elasticallystable shape. However, when an appropriately cut pop-per is flipped inside out and laid on the table [Fig. (5)]it creeps for a few seconds and snaps back to its originalshape, much in the same manner systems with interme-diate values of α in the Von-Mises truss model of sectionII B acquired and lost stability.One general corollary of the previous section is thatstability loss can occur only from configurations thatwere “originally” unstable - not a stable configurationof the rest elastic system ¯ g ij = ¯ g ij . If the configurationwas originally stable, than by the above claim it is alsostationary, and its stability can never be lost. Thus asystem exhibiting temporary stability followed by a snapthough, was necessarily away of its rest system ¯ g ij (cid:54) = ¯ g ij before the free evolution. Therefore if a popper snap afinite time after being inverted, its stability (prior to thesnap) was necessarily acquired, presumably by holding itinverted for a long enough duration.Second, acquired stability from configurations thatwere not extremal initially (e.g. saddle points) cannotpersist indefinitely and must be eventually lost. If a con-figuration is stationary after a relaxation that startedfrom rest then it must also be extremal with respect tothe rest reference metric. In the case of the popper itmeans that if it was originally unstable, there is no way(by holdings the popper inverted or by any other de-formations) to stabilize the popper indefinitely after theexternal loads are removed. For such cases the invertedshapes cannot be stationary states of the system, and thepopper will always eventually snap back to an rest state. V. SUMMARY AND DISCUSSION
Naively one expects stress relaxation to only diminishinternal forces and thus act to stabilize a system ren-dering the notion of viscoelastic instabilities elusive. Wehave shown this intuition to indeed be correct; if a systemis brought abruptly from its rest state into an elasticallymeta-stable state then stress relaxation will never causeit to lose stability. We moreover proved that in such asetting the configuration of the system will show no evo-lution despite the continuous vicoelastic attenuation ofinternal stresses.Having formulated the viscoelastic evolution in termsof metric tensors we observed that the instantaneous ref-erence metric evolves not only toward g , the assumedconfiguration, but also toward ¯ g , its rest metric. Whenstarting from rest only the former is observed. In thiscase the reference metric evolves along a straight lineand as a result the elastic energy minimizer at time t with respect to ¯ g ( t ) remains the minimizer for subse- (a) I n v e rt e dS tr a i g h t t hold t flip (b) .
50 0 .
52 0 .
54 0 .
56 0 .
58 0 . h/r min ∞ F li p t i m e [ s ec ] Stable Acquired stability UnstableLong holding timeNo holding (c)
FIG. 5. (a) Straight and inverted poppers. (b) The ex-periment protocol: The cone is held in the inverted shapefor a time t hold and then released; it flips after a time t flip .(c) The flipping time as function of the cone thickness forthe two limiting cases of immediate release and long holdingtime. The elastic cone was simulated with geometrical proper-ties r min = 10[mm] , r max = 25[mm]. The material propertiestaken were β = 0 .
3, Young’s modulus E = 2 . ν = 0 .
47. A similar phase plot presented in [12] andin [8]. quent times, as explained graphically in FIG. (4). Forevery other case, in which ¯ g does not evolve solely to-ward g , we expect a visible creep and in particular suchsystems may display viscoelastic instabilities. We haveshown that the only states that can display viscoelasticinstabilities are states that are elastically unstable withrespect to ¯ g but have acquired stability through stressrelaxation. We have also shown that every instance of such an acquired stability cannot persist indefinitely andmust be lost in time.These results are expected to hold for every isotropicand incompressible linearly viscoelastic material regard-less of the form of history dependence of the stress relax-ation kernel. However, considering a single SLS systemor any other purely one dimensional viscoelastic systemwe do not observe this rich phenomenology of instabili-ties due to the trivial structure of the set of admissiblestates. The ability of a viscoelastic system to display de-layed instabilities relies on the non-trivial shape of the setof admissible states. As observed in FIG. (3) for the SLSVon-Misses truss, only when the set of admissible statesis sufficiently curved can it support multi-stability. Forcontinuous media the structure of admissible (Euclidean)states is non-convex. As a result we may bring a systemsthat is initially at rest in an Euclidean configuration, toanother Euclidean state by the application of externalforces, and the viscoelastic evolution will result in a non-Euclidean instantaneous reference metric, giving rise toa geometrically frustrated state. In such cases residualstress is like to appear even in lieu of multistability.While in the theory presented here the rest metric, ¯ g ij ,is assumed constant one could consider growing tissue orplastically deforming material by allowing the rest metricto vary in time. The variation could be provided by ex-ternal conditions or obey some additional constitutive re-lations. This will result in a covariant elaso-visco-plastictheory. Such a description may provide crucial insightinto growing and deforming biological polymers wherea rapid growth or shrinkage events may be masked bythe slow evolution of the instantaneous reference metric.The applicability of such a theory, reminiscent of the ad-ditive decomposition of strains [19, 21], to growing anddeforming tissue remains to shown. ACKNOWLEDGMENTS
The authors would like to thank D. Biron, G. Cohen,A. Grosberg, S.M. Rubinstein, E. Sharon, Y. Bar-Sinaiand D. Vella for helpful discussions, as well as acknowl-edge the Aspen Center for Physics, which is supportedby National Science Foundation grant PHY-1607761, forhosting many of these discussions. [1] L. R. G. Treloar,
The physics of rubber elasticity (OxfordUniversity Press, USA, 1975).[2] S. O. Andersen, Biochimica et biophysica acta , 249(1963).[3] R. E. Neuman and M. A. Logan, Journal of BiologicalChemistry , 549 (1950).[4] J. Gosline, M. Lillie, E. Carrington, P. Guerette, C. Or-tlepp, and K. Savage, Philosophical Transactions of theRoyal Society of London B: Biological Sciences , 121 (2002).[5] R. Christensen, Theory of viscoelasticity: an introduction (Elsevier, 2012).[6] Y. Forterre, J. M. Skotheim, J. Dumais, and L. Mahade-van, Nature , 421 (2005).[7] A. Pandey, D. E. Moulton, D. Vella, and D. P. Holmes,EPL (Europhysics Letters) , 24001 (2014).[8] E. Y. Urbach and E. Efrati, “Delayed instabilities in vis-coelastic solids through a metric description,” (2018). [9] A. M. Freed and J. Lin, Nature , 180 (2001).[10] Z. Reches, G. Schubert, and C. Anderson, Journal ofGeophysical Research: Solid Earth , 21983 (1994).[11] M. Santer, International Journal of Solids and Structures , 3263 (2010).[12] A. Brinkmeyer, M. Santer, A. Pirrera, and P. Weaver,International Journal of Solids and Structures , 1077(2012).[13] C. Zener, Elasticity and Anelasticity of Metals (Univer-sity of Chicago Press, 1948) google-Books-ID: FKcZA-AAAIAAJ.[14] A. F. Bower,
Applied mechanics of solids (CRC press,2009).[15] Z. Baˇzant and L. Cedolin,
Stability of Structures: Elastic,Inelastic, Fracture, and Damage Theories , Dover bookson engineering (Dover Publications, 1991).[16] E. Efrati, E. Sharon, and R. Kupferman, Journal of theMechanics and Physics of Solids , 762 (2009).[17] L. D. Landau and E. Lifshitz, Course of TheoreticalPhysics , 109 (1986).[18] B. D. Coleman and W. Noll, Reviews of Modern Physics , 239 (1961).[19] E. Efrati, E. Sharon, and R. Kupferman, Soft Matter ,8187 (2013).[20] For the case of elasto-plasticity, a similar idea was firstproved but never published by H. Aharony. This type ofbehavior was also conjecture [22].[21] E. H. Lee (ASME, 1969).[22] B. Hayman, Proceedings of the Royal Society of Lon-don A: Mathematical, Physical and Engineering Sciences , 393 (1978).[23] N. J. Hoff, The Aeronautical Quarterly , 1 (1956).[24] Technically, another possibility is that the point will con-tinuously progress to one of the rest elastic system ex-tremum points. * Appendix A: 1D SLS truss and rest statestationarity
Here we examine an explicitly solvable discrete systemthat is more complex than the three spring truss. Thissystem is composed of a point mass m , free to move alongthe ˆ z axis whose position we denote z ( t ). The mass isconnected through SLS’s to N pinned positions in the 2Dspace ( x i , z i ). All the SLS’s share the same β, τ but maydiffer in their stiffnesses k i . This choice is well justifiedwhen considering large structures, or composite mate-rials, made of many identical SLS’s. The compositionsof identical SLS’s (in parallel or in series) leave β and τ unchanged. Thus the effective description of an SLSstructure can assign different stiffnesses to distinct modesof deformations, but take β and τ as intrinsic materialproperties. The length on each of the SLS’s is denoted L i = (cid:112) x i + ( z i − z ) , and the reference length and restlength of each SLS is denoted ¯ L i and ¯ L i respectively.We assume a temporal scale separation between thefast elastic time-scale and the slow viscoelastic timeshared by all the springs, and that no external time scaleenters the problem. We may thus reduce the problem to studying the quasi static evolution of the system inwhich every configuration the system assumes minimizesthe instantaneous elastic energy that in turn depends onthe slowly evolving reference lengths ¯ L i , E ( z ) = N (cid:88) i =1 k i (cid:0) L i − ¯ L i (cid:1) . (A1)In this quasi-static limit the force on the point mass van-ishes0 = F ( z ) = − dEdz = − N (cid:88) i =1 k i dL i dz (cid:0) L i − ¯ L i (cid:1) . (A2)The rest length ¯ L i change in time and, as in the caseof the Von-Mises truss, could change the shape of theinstantaneous elastic energy. For example they couldcause an unstable state to become stable through vis-coelastic relaxation. Such states with acquired stabilityin viscoelastic systems were first identified in the contextof creep buckling [22, 23], and revisited recently wherethis phenomenon was termed temporary bistability [11]or pseudo bistability [12], as these states were observednumerically to always lose their stability.In order to analyze the dynamic of the system, we dif-ferentiate the force Eq. (A2) in time, and use Eq. (4): d Edz ˙ z = N (cid:88) i =1 k i dL i dz ˙¯ L i = 1 τ N (cid:88) i =1 k i dL i dz (cid:0) L i − ¯ L i − (1 − β ) (cid:0) L i − ¯ L i (cid:1)(cid:1) Substitution of Eq. (A2) gives d Edz ˙ z = 1 − βτ F ( z ) , (A3)where F ( z ) = − (cid:80) Ni =1 k i dL i dz (cid:0) L i − ¯ L i (cid:1) is the force onthe mass at z , with respect to reference lengths thatequal the rest lengths, ¯ L i = ¯ L i . We thus obtained thatwhen the point mass is in a convex region of the poten-tial ( d Edz > z changes continuously and the sign of ˙ z remains the same as the sign of the force in the rest elas-tic system F ( z ). When the second derivative vanishes d Edz = 0 the system loses its transient elastic stability,and snaps to another stable point.Immediate result of Eq. (A3) is that stationary statesof Eqs. (A3) and (4) are all extremal points of the restelastic system’s potential. That is assuming both ˙ z =0 and ˙¯ L i = 0 implies F ( z ) = 0. This result agreeswith the phenomenology of the von-Mises truss describedabove. Both the unstable and the finitely stable caseseventually result in the original position - stable point ofthe rest elastic system. The stable phase satisfy the claimtrivially, by staying stable from the very rest. We havethus also shown that in general, any acquired stability1(initially non extremal point that has become stable),will eventually result in a snap [24].The sign of ˙ z in Eq. (A3) is independent of the ref-erence lengths. Thus, mass left on a stable point of therest elastic system F ( z ) = 0 will remain static, althoughall the reference length ¯ L i will continue to change. In-tuitively, this phenomenon occurs because the relaxationtakes the same force fraction of each SLS over time, pre-serving the vanishing total force on the point mass. Oneonly need to check whether d E ( t ) dz > d E ( t ) dz = N (cid:88) i =1 k i d L i dz (cid:0) L i − ¯ L i ( t ) (cid:1) + N (cid:88) i =1 k i (cid:18) dL i dz (cid:19) (A4)Assuming the point to be at rest ¯ L i = ¯ L i at t = 0, andtaking constant z and L i ’s, Eq. (4) gives¯ L i ( t ) = (1 − B ( t )) ¯ L i + B ( t ) L i , where B ( t ) = β (cid:0) − e − t / τ (cid:1) . Thus d E ( t ) dz = (1 − B ( t )) d E ( t = 0) dz + B ( t ) N (cid:88) i =1 k i (cid:18) dL i dz (cid:19) (A5)Because 0 < B ( t ) < t > d E ( t =0) dz > t >
0. We have therefore obtainedthat stable points of the rest elastic system are also static,when arrived to directly from rest.By considering the SLS as a spring with dynamic ref-erence length we were able to show explicitly that theresults obtained for the Von-Mises truss SLS system arein fact rather general and in particular apply even whenthe system display multiple metastable states and is com-prised of many springs of different stiffnesses providedtheir β and ττ