The relationship between viscoelasticity and elasticity

royalsocietypublishing.org/journal/rspa Review Cite this article: Snoeijer JH, Pandey A, Herrada MA, Eggers J. 2020 The relationship between viscoelasticity and elasticity. Proc. R. Soc. A 476: 20200419. https://doi.org/10.1098/rspa.2020.0419 Received: 28 May 2020 Accepted: 14 October 2020 Subject Areas: physics Keywords: elasticity, viscoelasticity, capillarity, gels, instability Author for correspondence: J. H. Snoeijer e-mail: [email protected] The relationship between viscoelasticity and elasticity J. H. Snoeijer1,A.Pandey 1,M.A.Herrada 2and J. Eggers3 1Physics of Fluids Group, Faculty of Science and Technology, Mesa+ Institute, University of Twente, 7500 AE Enschede, The Netherlands 2Depto. de Mecánica de Fluidos e Ingeniería Aeroespacial, Universidad de Sevilla, 41092 Sevilla, Spain 3School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK JHS, 0000-0001-6842-3024;JE,0000-0002-0011-5575 Soft materials that are subjected to large deformations exhibit an extremely rich phenomenology, with properties lying in between those of simple fluids and those of elastic solids. In the continuum description of these systems, one typically follows either the route of solid mechanics (Lagrangian description) or the route of fluid mechanics (Eulerian description). The purpose of this review is to highlight the relationship between the theories of viscoelasticity and of elasticity, and to leverage this connection in contemporary soft matter problems. We review the principles governing models for viscoelastic liquids, for example solutions of flexible polymers. Such materials are characterized by a relaxation time λ, over which stresses relax. We recall the kinematics and elastic response of large deformations, and show which polymer models do (and which do not) correspond to a nonlinear elastic solid in the limit λ→∞. With this insight, we split the work done by elastic stresses into reversible and dissipative parts, and establish the general form of the conservation law for the total energy. The elastic correspondence can offer an insightful tool for a broad class of problems; as an illustration, we show how the presence or absence of an elastic limit determines the fate of an elastic thread during capillary instability. 2020 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited. 2 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... 1. Introduction The aim of this review is to expose systematically the relationship between the theories of viscoelasticity and of elasticity, and to leverage what can be learned from this connection. Given the very mature state of these fields, there exist many excellent review articles and monographs that cover all aspects of elastic liquids and elastic solids in great detail [1–16]. With this review, we therefore do not attempt a broad overview of these research areas, but very specifically focus on how elasticity and viscoelasticity are related. This relationship is much more difficult to find in the literature, but it can greatly contribute to the understanding of contemporary developments involving soft materials at large deformations. (a) Soft materials and large deformations Exceedingly soft solids, such as gels, elastomers and biological tissues, are extremely versatile and find numerous applications in nature and technology. Their mechanics is intricate: owing to their large deformability, soft solids can no longer be described within the framework of linear elasticity, but exhibit all the kinematic nonlinearities typical of the motion of fluids. This is the domain of large-deformation theory, which leads to nonlinear equations even if the elastic response of the material is perfectly linear. Figure 1 provides various contemporary illustrations of soft matter at very large deformation. Figure 1ashows an extremely tough hydrogel [17]. It is specifically designed to reversibly resist very large stretches, up to a factor of approximately 20, without fracture. A second example, given in figure 1b, consists of a liquid drop on a solid polydimethylsiloxane substrate [18]. The liquid–vapour interface creates a sharp elastic deformation, in the shape of a ridge around the droplet’s edge [22–24]. Interestingly, the dynamical spreading of drops on elastomers is dramatically slowed down by this deformation: during spreading, the ridge is transported along with the droplet edge and induces very large dissipation inside the substrate [25–28]—without any irreversible damage to the material. Indeed, highly deformable solids often exhibit strongly viscoelastic behaviour, where the dissipation occurs during transients of deformation. Such dissipation is actually exploited in the design of pressure-sensitive adhesives [29–31], and can also, for example, explain the delayed snap-through instability of jumping toy poppers [32]. So both fluids and soft solids can exhibit dissipation, and both share the same nonlinear kinematics under large deformations. The fundamental distinction between a (hyper)elastic solid and a (Newtonian) fluid is that the former maintains a permanent memory of its initial or ‘reference’ state to which it relaxes, whereas in a simple fluid all configurations are equivalent and only rates of deformation are important. Accordingly, the natural description of a solid is a ‘Lagrangian’ viewpoint, which follows the path of each element of the continuum as labelled by the reference state, and all forces are determined from this mapping from the reference to the ‘current’ state. Fluid motion can also be described using Lagrangian paths, which is a natural point of view when considering mixing and advection problems [33,34], but has also proved to be a fruitful way of looking at classical fluid mechanics problems [35,36]. However, fundamentally Lagrangian trajectories are extremely intricate, even for very simple time-independent flows [33], and thus it is much simpler to disregard the history of each particle. Instead, in the Eulerian point of view one considers snapshots of the velocity field only, which is sufficient to calculate rates. There are indeed numerous situations in which the material’s response exhibits both solidlike and fluid-like behaviour. Our focus is the connection that exists between fluid and solid mechanics in the limit of large relaxation times, and which persists in the case of large deformations. Here we remark that another relation exists that is different from the one that we address: it has been appreciated for a long time [37,38] that the linear equations for the velocity field of a viscous fluid, the so-called Stokes equations, and the equations of linear elasticity, in the incompressible limit of Poisson’s ratio being 1/2, are formally equivalent. For example, the theory of cracks in an elastic material [39] can be applied to the shape of free-surface cusps on the surface of a viscous fluid [40]. Taylor [41] noticed that the correspondence also applied to thin threads 3 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... (a) (d) (e) (b)(c) l = 17 2 cm water air solid Figure 1. Soft matter at large stretch. (a) Tough solid: a sheet of a soft but very tough hydrogel that exhibits a reversible deformationwhenstretchinguptoafactorof 17.Eventhepresenceofaholein thecentredidnotnucleateanyfracture. Adapted with permission from [17]. Copyright c Springer Nature. (b) Viscoelastic solid: liquid drops spreading over an elastomeric substrate cause largedeformationatthecontact line.Thetoppanelrevealsa‘wettingridge’arounda millimetricdrop(courtesy Mathijs Van Gorcum). The bottom panel is a magnified view of the wetting ridge at three phase contact lines. The scale bar is 2 µm. Adapted with permission from Park [18], under CC-BY 4.0 license. (c) Fracture in a viscoelastic liquid: the liquid is a functionalizedmicro-emulsion,formingtransientnetworks,andexhibitsbrittle fracture.Thethreadradiusatfractureisaround 0.3 mm. Adapted with permission from [19]. Copyright c Royal Society of Chemistry. (d) Elastic thread: when sufficiently soft, acylinderofcross-linkedagargelundergoesaRayleigh–Plateauinstability(cylinderradius0.24 mm).Adaptedwithpermission from [20]. Copyright c American Physical Society. (e) Viscoelastic thread: beads-on-a-string formation during the pinch-off of dilute(0.01 wt%)aqueouspolyacrylamidesolutionundergoingcapillarythinning(jetradius0.3 mm).Adaptedwithpermission from [21]. Copyright c Cambridge University Press. and sheets of viscous fluid, which on a short time scale are described by the nonlinear elastica equations and the equations for elastic sheets, respectively. This analogy has subsequently been derived more formally [42,43] and applied to many different physical situations. As a particularly instructive example for the relationship between elasticity and viscoelasticity made in the limit of large relaxation times, we consider the capillary instability of cylindrical jets [20,44–46]. Figure 1dshows a cylinder of a fully cross-linked agar gel, which possesses a well-defined reference state [20]. Despite its elasticity, the cylinder exhibits a Rayleigh–Plateau instability that one usually associates with liquid jets [47]. The cross-linked network ultimately prohibits break-up and leads to the formation of thin elastic threads. For comparison, figure 1e shows a jet consisting of a dilute polymer suspension, a viscoelastic liquid; here break-up does occur, and the tenuous liquid filaments become thinner over time [47,48]. Conversely, complex liquids whose microstructure develops transient elastic networks can exhibit solid-like brittle fracture [19,49], as shown in figure 1c. In this case, the deformation is initially liquid-like but at some point breaks as if it were a solid. It can be argued, as we will do, that viscoelastic liquids can be used as a universal modelling paradigm for a broad class of soft matter systems such as in figure 1. In contrast to elastic solids, 4 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... viscoelastic liquids such as polymer solutions or emulsions do not possess a permanent reference state. Instead, they exhibit a fading memory of any prior deformation, which is characterized by a relaxation time λ, which is the time scale over which elastic stresses relax during flow. Such complex fluids are extremely common and important [2–4,50], as they occur whenever large and flexible molecules or other similar structural elements are present in the flow, as is the case in a vast range of biological and industrial contexts. In practice, soft materials with a complex internal structure possess a broad distribution of time scales, and can exhibit a power-law response [51,52] rather than the conventional exponential response. Whenever a polymer is transported by a flow, it leaves its preferred reference state and exerts a force back on the liquid. To make the extremely complicated interactions between liquid and microstructure tractable, the polymer is often modelled as two beads, convected by the flow and connected by an elastic spring [2]. If the spring is soft, the polymer experiences large deformations, as the beads follow the complicated Lagrangian trajectories of the flow. As a result, the response becomes very nonlinear even if the spring is Hookean. A fluid in which the stress consists of a contribution of damped Hookean springs (damping due to friction with the solvent complemented by the Newtonian stress of the solvent) is known as an Oldroyd-B fluid. Owing to its conceptual simplicity it has become one of the most popular models of elastic liquids, although it neglects any nonlinear response of the spring as well as interactions between constituents. The relaxation time λof the model polymer derives from the ratio of the frictional force between a bead and the surrounding liquid divided by the spring constant, ensuring return to an equilibrium state. In weak flows, such that the relaxation time multiplied by a typical rate of deformation of the flow is small, the polymer remains close to its equilibrium shape, and only makes a linear, Newtonian contribution to the stress. Even if the flow is strong, on a time scale that is much larger than λ, the polymer will have ‘forgotten’ the deformations it experienced in the past. Only in the limit λ→∞will each bead follow its Lagrangian path as a passive tracer and produce an elastic response associated with large-deformation elasticity. In other words, upon varying the time scale λ, viscoelasticity continuously bridges the gap between a Newtonian liquid and a perfectly elastic solid. Various pioneering works are actually based on this idea, using a continuum formulation of viscoelasticity that is based on the theory of elasticity with an additional relaxation process [12,53–55]. We further illustrate the correspondence between elasticity and viscoelasticity using the thinning of a viscoelastic cylinder under capillary action (cf. figure 1d,e). Figure 2 shows the capillary thinning as modelled by the Oldroyd-B fluid. For λ=0, the liquid has no memory and the thread thickness hthr tends to zero like a power law [47,57], following a perfectly Newtonian pinch-off. For finite λ, polymers become increasingly stretched by the elongational flow near any potential pinch point, and the effective elongational viscosity increases exponentially. As a result, a very uniform thread is formed whose radius decreases exponentially on a time scale set by λ. At λ=∞ one recovers a purely elastic behaviour, in this case that of a neo-Hookean solid. As the solid becomes increasingly deformed by surface tension, elastic stresses build up until they balance surface tension and a stationary thread of constant radius is formed. (b) This review As highlighted above, this review focuses on the specific issue of how the theories of elasticity and of viscoelasticity are related, and to leverage what can be learned from this connection in the context of recent research. This explicit relation is not frequently explored, but, in fact, can be a very insightful and powerful tool for challenging problems, such as those in figure 1. As an example, we recently solved a long-standing problem in the break-up of dilute polymer suspensions by exploiting the elastic correspondence in the limit λ=∞[48]. Conversely, viscoelastic liquids probed at high rates have been used as a model soft elastic solid at large deformation [58]. The elastic correspondence offers a way to bridge the gap between different communities working in various areas of soft matter (biophysics, chemistry, engineering, fluid physics), 5 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... 0 100 200 t/t l = 0 l = • elastic 300 400 500 10–2 10–1 1 Newtonian hthr ˜ e–t/3l hthr/R0 Figure2. Thinningdynamicsofaviscoelasticliquidthread.Theinsetshowsaliquidthreadbetweentwodropsofdilutepolymer suspension (courtesy A. Deblais and D. Bonn; see also [56]). The main panel shows the thinning dynamics of the thread as described by an Oldroyd-B fluid, for different values of the relaxation time λ. The thread thickness hthr is scaled by the initial jet radius R0, while time is scaled by the capillary time τ=ρR3 0/γ . The model continuously bridges between Newtonian liquids (λ=0) and elastic solids (λ=∞). (Online version in colour.) which often use different modelling approaches to mechanics. These approaches are very well documented in established reviews and monographs [1–16]. However, there is a barrier to crossing the disciplines because of differences in mathematical formulation; the purpose of this review is to offer a unified exposition of elastic liquids and elastic solids. For example, while fluid mechanicians are mostly familiar with an Eulerian description, solid mechanics is most naturally expressed using the Lagrangian description. In addition, constitutive relations for liquids are usually formulated in terms of the stress tensor, while hyperelastic solids are defined by a (strain-dependent) free energy density [8]. We remark that a generalization of the concept of hyperelasticity is known as implicit elasticity [13,15], in which the elastic internal energy is dependent on both stress and strain, a situation we will not consider here. We will use the elastic strain energy density of the polymers to present a systematic way of deriving an energy balance equation for viscoelastic fluids. While a mainstay of Newtonian fluid mechanics, energetic arguments are not much used for elastic fluids. Yet they reveal important characteristics of such fluids, as energy can now be stored temporarily in its elastic form, transported to other parts of the flow and eventually injected back as kinetic energy of the flow. The review is organized as follows. In §2, we briefly summarize classical continuum theory, presenting side by side the formulations of viscoelasticity and of large-deformation elasticity. Section 3 explores what we call the ‘elastic correspondence’, by investigating viscoelastic liquid models in the limit λ→∞. In particular, we show which models converge (or do not converge) to elastic solids when taking this limit. The elastic correspondence is then exploited in detail by the example of capillary thinning in §4, highlighting the importance of whether or not the elastic correspondence exists. This also offers a new modelling paradigm to viscoelastic solids, based on models of viscoelastic liquids. Section 5 discusses a thermodynamic approach to viscoelastic liquids, and we discuss the relation between stress, energy and dissipation. We close with a discussion in §6. 2. Classical continuum theory (a) Viscoelastic fluids The equations of motion for viscoelastic fluids are most commonly expressed in the Eulerian description, using a velocity field v(x,t) that is a function of space xand time t.Hereweconsider 6 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... the fluid to be incompressible, ∇·v=0, and described by the momentum balance ρ∂v ∂t+v·∇v=∇·σ, (2.1) where σis the stress tensor. The stress tensor is split into a Newtonian contribution (coming, for example, from a solvent) and a polymeric (viscoelastic) contribution σp, σ=−pI+ηs˙ γ+σp≡−pI+τ, (2.2) where the deviatoric stress τis the contribution excluding the pressure. We defined the rate-of- deformation tensor to be ˙ γ=(∇v)+(∇v)T(2.3) and ηsis the solvent viscosity. Any isotropic contribution to the stress can be written as part of the pressure p. The non-Newtonian contribution σporiginates from the presence of the microstructure inside the fluid. Though we will refer to σpas the ‘polymeric stress’, having in mind dilute polymer suspensions, the concept equally applies to emulsions whose microstructure is described by droplet deformations [59,60]. The stress σpis governed by a separate evolution equation, the constitutive equation, which encodes the non-Newtonian properties of the fluid. Specifically, for viscoelastic liquids, the constitutive equation describes the relaxation of stress over a time scale λ. Many different approaches to modelling σpcan be found in the literature [2–7,9,50]; these can be roughly divided into two categories. The first approach is that the polymer stress σpshould be a functional of the history of deformation [5,9]. These constitutive relations are expressed either as a differential equation or in an integral form. In the latter case, one explicitly performs an average over past deformations with a weight factor that encodes the fading memory. Many of these constitutive equations are based on systematic expansions, where at each order one gradually adds more information of the deformation history and/or more nonlinearity [5]. The advantage of such an approach is that the calculation of stress is direct, though care must be taken that the formulated model is thermodynamically consistent [6]. In the second approach, the polymer stress is not expressed directly in terms of the deformation, but rather in terms of an order parameter field, A(x,t), that characterizes the state of the polymer (or, more generally, of the microstructure). In this approach, the polymeric stress is written as σp=σp(A). The coupling to the deformation history is then achieved by a separate relaxation equation for A(x,t), which describes how the microstructure evolves over time. The advantage of an order parameter description is that it can be embedded in a thermodynamically consistent framework, as done, for example, in the bracket [6] and the GENERIC [7] formalisms. For polymer solutions and emulsions, the natural order parameter Ais the so-called conformation tensor: it is a second-rank tensor that characterizes the amount of stretch of the polymer. The tensorial nature arises since stretching will in general be different along different directions. It has the property that the polymer stress vanishes when A=I,whereIis the identity tensor, and its principal values represent the stretches along principal directions. Another merit of the conformation tensor description is that it can be derived by coarse-graining microscopic models based on suspended bead-and-spring dumbbells [2]. As we will see below, the formulation in terms of a conformation tensor offers a natural connection to the theory of elasticity—which is the central purpose of this review. Therefore, in the remainder of the paper we focus primarily on constitutive modelling based on a conformation tensor. In §3e, and in a few other places, we briefly discuss how the conformation tensor models are connected to descriptions based on the history of deformation. Though the relation σp(A) and the evolution equation for Acan often be derived from microscopic models, we here follow a purely continuum approach, without referring to any microscopic model. Once again, the continuum description has the advantage of exposing the correspondence to the theory of elasticity—the ‘elastic limit’ is obtained by considering viscoelastic fluid models in the limit λ→∞. We will see that this elastic correspondence completely determines the dependence σp(A). 7 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... (i) The subtlety of the time derivative Let us start by considering the simplest type of model based on the conformation tensor. In this case, the relation σp(A) is linear σp=μ(A−I), (2.4) where μis the analogue of the elastic shear modulus. The state variable Amust have the property that it evolves towards its relaxed state A=Iin the limit of long times. Once more, the simplest way of doing this is through a linear relaxation law ˙ A=−1 λ(A−I), which is known as a Maxwell model. In such a model, an initial condition Arelaxes exponentially towards A=Ion a time scale λ. However, it has been known for a long time [2,61] that, for the dynamics of a second-rank tensor to be frame invariant [50], i.e. to be independent of the frame of reference, the ordinary time derivative needs to be replaced by one of two frame-invariant derivatives or a linear combination of the two. The so-called upper convected derivative  A=∂A ∂t+v·∇A−(∇v)T·A−A·(∇v)(2.5) is derived from the requirement that its components transform consistently as the components of a contravariant tensor. The first two terms on the right are the convected derivative of a material point, ensuring Galilean invariance; the last two terms make sure that Atransforms correctly under deformations by the flow. However, a covariant formulation does equally well from the point of view of frame invariance, but yields a different derivative, known as the lower convected derivative,  A=∂A ∂t+v·∇A+A·(∇v)T+(∇v)·A. (2.6) As the names suggest, these derivatives have natural geometric interpretation in curvilinear coordinates that are convected with the flow [50,61,62]. For completeness, the curvilinear description is given in appendix A, where we discuss in detail the geometric interpretation. From the point of view of frame invariance, one is thus left with a somewhat unpleasant ambiguity. Namely, the derivatives  Aand  A, and linear combinations of the two, are equally admissible when building a theory for viscoelastic fluids. The resulting mechanical behaviour, however, is manifestly different depending on the choice of the derivative. In particular, when using the upper convected derivative, the stress will grow exponentially in a strong extensional flow, as expected from a bead-and-spring model [2], where the two beads will be separated by the flow. However, the ambiguity of the time derivative can be lifted more generally, without relying on any microscopic model. This becomes particularly clear when working out the correspondence to the theory of elasticity. Using the upper convected derivative, a linear relaxation law thus takes the form  A=−1 λ(A−I). (2.7) It is instructive to combine (2.4) and (2.7), such that we have a single equation of motion for the polymeric stress. Introducing the polymeric viscocity ηp=μλ, this gives σp+λ σp=ηp˙ γ. (2.8) This is a tensorial form of the upper convected Maxwell model. The stress tensor (2.2) with σp given by (2.8) is known as the Oldroyd-B model [2]; in the limit of vanishing rates of deformation, it describes a Newtonian fluid of total viscosity η0=ηs+ηp.Figure 3 provides a schematic of the Oldroyd-B fluid, whose mechanical response consists of a Newtonian solvent in parallel with a so-called upper convected Maxwell fluid. When omitting the solvent, one recovers the upper convected Maxwell model. 8 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... mhp hs hs g (solvent) sp (polymer) Figure 3. Qualitative representation of the Oldroyd-B fluid. The deviatoric stress is given by the sum of the Newtonian stress of the solvent (viscosity ηs) and the viscoelastic stress of the upper convected Maxwell fluid. The latter is characterized by a polymer viscosity ηpand an elastic modulus μ.Theratioofηp/μ gives the polymer relaxation time λ. Note that the springdashpot analogy is not to be taken quantitatively, certainly not at large deformations where one encounters strongly nonlinear responses. Equation (2.8) is an example of a constitutive relation where σpis expressed in terms of the history of deformation. Here this is in the form of a differential equation, where deformation is encoded in ˙ γand in ∇vin the upper convected derivative. In fact, (2.8) is a special case of the more general expansion by Oldroyd [5], known as the eight-constant model, in which all terms to second order in stress and strain rate are retained, while respecting frame invariance. We return to Oldroyd’s eight-constant model in §3e. For now, let us remark that, in such an expansion, there is no mathematical ground to anticipate whether taking  σpor  σpin (2.8) would do a better job in describing polymeric fluids. Although the Oldroyd-B model is very popular owing to its simplicity, there are many relevant physical effects which are not captured. For example, it incorrectly describes the shear-thinning behaviour of polymeric fluids, and in a strong extensional flow the stress will grow indefinitely. This can be avoided by incorporating the fact that the spring can only reach a finite extension, by making the spring constant increase as full extension is reached. There exist numerous extensions of the Oldroyd-B equations in that spirit; for example, taking into account nonlinearity in both (2.4) and (2.7), or in the solvent contribution in (2.2). In §5c, we supply a list of various models. Apart from the question of frame invariance, models have to be consistent with the requirements of thermodynamics [6,7,59,60,63]. (b) Elasticity We now turn to a brief exposition of the theory of elasticity [8,64]. While fluid mechanics is usually expressed in the Eulerian formulation, using the spatial coordinates xto describe the system, nonlinear (finite deformation) elasticity is written in a Lagrangian formulation based on material coordinates. This is because elastic solids exhibit a well-defined (undeformed) reference state, in which elastic energy is minimal and the elastic stress vanishes. The material coordinates in this reference state are denoted by X. Deformations are described by a mapping x=χ(X,t), where x denotes the position of a material point after the deformation, which used to be at Xbefore the deformation. In fluid mechanics, the mapping is known as a Lagrangian path of a particle with label X. Flow corresponds to the case where the mapping χis time dependent, though even static deformations are of interest in the context of solid mechanics. The theory of (hyper)elastic solids is based on the idea that deformations are perfectly reversible, without any dissipation, so that their constitutive behaviour can be formulated in terms of an elastic free energy. Figure 1aoffers a spectacular example of a solid with a perfectly reversible response. If the medium is isotropic, the density of elastic energy Wcan only depend on the change in distance between material points generated by a deformation [10]. To evaluate this change of distance, we introduce the deformation gradient tensor F=∂x/∂X.Namely,ifds is the distance between two points which used to be a distance dSapart, we obtain [8], using 9 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... dx=F·dX, that ds2−dS2=dXT·FT·F−I·dX. (2.9) The deformation is thus encoded in Green’s deformation tensor C=FT·F, which is a symmetric second-rank tensor that is defined on the reference configuration (the tensor (FT·F−I)/2 is called the finite-strain tensor). The stored density of elastic energy must be a function of C,or,more specifically, of the invariants of C. The energy function W=W(C) must also have the property that it assumes a minimum for C=I, when the material is undeformed from its reference state. This means that any deformation costs energy, which is a necessary condition for the unstressed statetobestable. Given that we are interested in the connection to Eulerian theory for viscoelastic liquids, we will not pursue further the Lagrangian formulation of elasticity—for that we refer to [8,64]. All we need for the present discussion is that Cshares the same eigenvalues as those of the Finger tensor [8], defined as B=F·FT. This can be understood when expressing Falong principal directions. In that case Fis diagonal and the eigenvalues {Λi}are the principal stretches, expressing the corresponding ratio of dsand dS; the corresponding Band Care then also diagonal, with identical components {Λ2 i}. So, why do we wish to use the Finger tensor? In contrast to C, the Finger tensor Bis an Eulerian tensor, defined on the current configuration,1and is therefore more appropriate when connecting to the Eulerian description of viscoelastic liquids. Given that the invariants of C are the same as those of B, we can thus write W=W(B) for the elastic free energy density. Though not necessary, from now we on focus on incompressible deformations (as is the case for most polymeric solids), for which det(F)=1. Once the free energy is specified, the (Cauchy) stress tensor for incompressible media follows as [8,64] σp=∂W ∂F·FT=2∂W ∂B·B, (2.10) where in the second step we exploited the symmetry of B. This expression is a consequence of the virtual work principle [9], which requires that any change in the elastic energy density satisfies [10]dW dt=σp:(∇v)T=1 2σp:˙ γ, (2.11) where in the second step we used the symmetry of σp. The derivation of (2.10) and (2.11) will be spelled out in §5. As we argued before, Wcan only be a function of one of the invariants of B, which can be written as I1=Bkk,I2=1 2B2 kk −BijBijand I3=det(B), (2.12) where we recall that I3=det(F)2=1 for incompressible media. Hence, we can write the free energy as a function of the first two invariants only: W(I1,I2). The constraint I3=1 will be ensured by an isotropic pressure acting as a Lagrange multiplier. Using the relation between energy and stress (2.10), and the definitions of the invariants (2.12), we obtain σp=2W1B+2W2(tr(B)B−B·B), (2.13) where W1≡∂W/∂I1and W2≡∂W/∂I2. This can be simplified using the Cayley–Hamilton theorem, which reads det(B)B−1=B2−tr(B)B+1 2tr(B)2−tr(B2)I. (2.14) Using det(B)=det(F)2=1, the stress can now be written as σp=2W1(B−I)+2W2I−B−1, (2.15) where for convenience we have absorbed an isotropic contribution into the pressure. 1More precisely, Bis a tensor in the vector space defined by x, while Cis a tensor in the vector space defined by X. These aspects are made explicit in the curvilinear description given in appendix A. 16 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... substrates [27,28,78], which was shown in figure 1b. The spreading dynamics of such drops is known to be extremely slow because of dissipation inside the solid. This phenomenon is called viscoelastic braking [25,26], and much research is currently dedicated to finding appropriate models for this coupled fluid–structure interaction problem [28,79,80]. The present analysis demonstrates that (reference state-preserving) viscoelastic solids can indeed be captured by a fully Eulerian approach, through viscoelastic fluids (like Oldroyd-B) in the limit λ→∞. By now, it should be clear that this route only works for affine models, with an upper convected derivative for the conformation tensor. (e) A brief note on constitutive models without the conformation tensor The discussion so far has been restricted to constitutive relations based on the conformation tensor A, the reason being the elegant connection between Aand the Finger tensor Bin the theory of elasticity. A natural question is whether the above considerations carry over to other viscoelastic theories, which do not explicitly involve the conformation tensor. Specifically, many models are based on the idea of expressing stress directly as a functional of the history of deformation [5,9]. Here we briefly touch upon such models, by expanding on the discussion of the upper convected Maxwell model. (i) Integral forms We first discuss the integral formulation of viscoelasticity, which is based on a memory kernel acting on past deformations. The approach is illustrated through a direct integration of the upper convected Maxwell model, as given by (2.7). The integration was already achieved for λ=∞ (see the discussion around (3.1)), but the integral exists also at finite λ[3]. For a stress-free initial condition at t=0, the solution reads3 A(t)=e−(t/λ)B(t)+1 λt 0dte−(t−t)/λBt(t), (3.13) whereweintroduced Bt(t)=F(t)·F−1(t)·F−T(t)·FT(t). (3.14) The interested reader can find the derivation in appendix B. In this review, we focus on materials with a single time scale only. This can easily be generalized to multiple modes [2], with separate time constants λi. In the limit of a continuous, broad distribution of time scales, one arrives at power-law materials [13,51,52] for which the exponential kernel of (3.13) is replaced by a power law. The object Bt(t) can be seen as a generalization of the Finger tensor: while B(t)≡B0(t)measures the stretches compared with a ‘reference’ state at t=0, the tensor Bt(t) measures the stretches at time tcompared with the state at another time t. The elastic correspondence is easily recovered in the integral formalism. Taking λ→∞at finite t, (3.13) reduces to A(t)=B(t)atalltimest≥0. For viscoelastic fluids with a finite λ, the initial condition plays no specific role and it is more natural to express (3.13) as A(t)=1 λt −∞ dte−(t−t)/λBt(t). (3.15) Once multiplied by the shear modulus, to obtain σp, this form goes by the name of the Lodge equation [3]. This form nicely reveals that the stress can indeed be considered as an integral over the entire history of deformation. The associated kernel exp(−(t−t)/λ) accounts for the fading memory over a time λ. A more general integral formulation of viscoelasticity goes back to the Kaye–Bernstein– Kearsly–Zapas model (KBKZ), which is directly inspired by the theory of elasticity [53,55]. In 3We define the reference state at t=0, which implies that B(0) =I. A stress-free initial condition implies A(0) =I, which is the case in (3.13). 17 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... this model, the non-Newtonian contribution to the stress is written as σp=t −∞ dtW1(t−t)Bt(t)−W2(t−t)Bt(t)−1. (3.16) Here the connection to the elasticity theory is very explicit: the stress is of the same form as (2.15), with time-dependent elastic moduli W1,W2that serve as memory kernels. The Lodge equation (3.15) is recovered by λW1=μexp(−(t−t)/λ)andW2=0. Let us remark that, in the general case where both kernels W1(t−t)andW2(t−t) are non-zero, the KBKZ model cannot be reduced to a simple conformation tensor description. We can interpret the integral over Bt(t) as the conformation tensor Ausing (3.15). However, the integral over [Bt(t)]−1does not lead to A−1, not even when both kernels have the exact same time dependence. Hence, the deformation history in (3.16) gives rise to two independent ‘state variables’ (or two conformation tensors) that determine the polymer stress. (ii) Expansions The upper convected Maxwell model can thus be expressed in various equivalent forms: the conformation tensor description, (2.4) and (2.7), the integral (3.16), but also of course in the more common differential from (2.8). This, however, is not the case for all viscoelastic models, in particular when the models result from expansions. For example, the Oldroyd-B model can be seen as a special case of Oldroyd’s eight-constant model [2,50], σ+λ1  σ+λ2˙ γ·σ+σ·˙ γ+λ3tr(σ)˙ γ+λ4(σ:˙ γ)I=η˙ γ+λ5  ˙ γ+λ6˙ γ·˙ γ+λ7(˙ γ:˙ γ)I. (3.17) Equation (3.17) is an expansion that contains all quadratic terms in stress and strain rate, provided they satisfy frame invariance. The Oldroyd-B and Johnson–Segalman fluids are particular versions of it, with ηp,ηs,μand aas the only non-zero constants. As far as we are aware, however, the general eight-constant model cannot be reduced to a description in terms of a conformation tensor, given by some expression σp(A) and a relaxation equation for A. In its most general form, the eight-constant model therefore does not converge to an elastic solid when λ→∞. Another special case of the Oldroyd eight-constant model is the so-called second-order fluid [5,50], defined by the constitutive equation σp=b2  ˙ γ+b11 ˙ γ·˙ γ. (3.18) This results from the so-called Rivlin–Ericksen expansion, which gradually builds in memory of past deformations. The quadratic terms in (3.18) are the lowest order to give non-Newtonian effects. Indeed, the second-order fluid cannot be represented in terms of a conformation tensor. This can be seen by considering the case where the flow is suddenly stopped at some time t0,so that ˙ γ=0for t>t0. Evaluating the polymer stress in (3.18), we find that σp=0for t>t0.Hence any stress present at t0is instantaneously relaxed, which is incompatible with a gradually relaxing conformation tensor. By the same argument it is also clear that the second-order fluid has no limit in which it can converge to solid-like behaviour. Let us emphasize that the absence of an elastic correspondence should not be seen as a shortcoming of a model. In the case of the second-order fluid, the perturbative expansion was not designed to capture strongly unsteady effects, but rather to capture nearly steady flows. For example, in spite of its simplicity, (3.18) exhibits normal stress differences in shear flow, and successfully captures various viscoelastic phenomena [81–83]. Similarly, the Johnson–Segalman model was never intended to describe elastic solids, but rather to capture non-monotonic stress relaxation. 18 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... 4. Collapse of a cylinder under surface tension We now illustrate the importance of the elastic correspondence through the collapse of a (visco)elastic cylinder under surface tension. As was discussed in the Introduction, figure 1d shows the capillary instability for an elastic solid that consists of a cross-linked agar gel [20]. It is cross-linked to such a degree that the gel possesses (and maintains) a reference state, which ultimately prevents the break-up of the thin threads. The structures that appear, in particular the thin threads, strongly resemble those seen during the capillary break-up of viscoelastic liquids that do not possess a reference state [21,45,84]. Figure 1eshows the break-up of a water jet containing a low concentration of a high molecular flexible polymer, with a relaxation time of about 0.01s. The break-up process repeats itself periodically in space but with a time delay in between, so one can see an almost cylindrical thread at different stages of thinning. An alternative geometry is that of a liquid bridge between two plates, which leads to a single thread. In each case, the thread radius is observed to thin exponentially in time [21,84,85]. As another amusing example of the interaction between elasticity and capillarity, albeit in a different context, we mention the interaction of an elastic beam with a liquid drop [86]. Below we will first re-derive the classical result of exponential thinning of viscoelastic fluids, and show how the elastic correspondence is actually a key element in solving the problem. Subsequently, the usefulness of the elastic correspondence is underlined by numerical solutions of viscoelastic solids, by means of an Oldroyd-B fluid in the limit λ→∞. The result is compared with a purely elastic simulation of a neo-Hookean solid. Finally, it is shown that the collapse changes dramatically when the model does not exhibit an elastic correspondence, as is exemplified by the Johnson–Segalman fluid. (a) Viscoelastic fluid The capillary thinning of an infinitely long liquid cylinder is due to an elongational flow, defined by the Eulerian velocity field (3.4). From a Lagrangian perspective, this can be seen as a stretching of the cylinder by a rate ˙, and a lateral contraction dictated by volume conservation. Denoting the cylinder radius by h(t), the radially inward flow implies ˙ h=−(1/2)h, so that h=h0e−(1/2)˙t. (4.1) The goal is to determine the value of ˙and the constant h0(which is not equal to the initial thread radius R0) for the process of capillary thinning. (i) Stretch and relax We consider the conformation tensor Ain the Oldroyd-B fluid during the thinning, determined by (2.7) with elongational flow. For a rubber band (λ→∞) the conformation tensor would simply follow the Finger tensor, which is given by (3.5). The general solution with finite relaxation time was already given in integral form in (3.13). One verifies that Bt zz(t)=exp(2˙(t−t)), representing the exponential separation over a time lag t−t. With this, the integral (3.13) gives the axial component of the conformation tensor Azz(t)=2˙λ 2˙λ −1e(2˙−(1/λ))t−1 2˙λ −1, (4.2) and similarly one finds for the radial component Arr(t)=˙λ ˙λ +1e−(˙+(1/λ))t+1 ˙λ +1. (4.3) The solution (4.2) very nicely brings out the competition between stretching and relaxation. The term exp(2˙t) reflects the stretching of the polymer by the flow; at the same time, the polymer relaxation reduces the stretch as exp(−t/λ). When ˙λ > 1/2, as will be the case in capillary 19 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... thinning, the exponential stretching dominates over the relaxation. When ˙λ < 1/2, the relaxation is strong enough that Azz saturates at some finite value. The value of ˙can be found when demanding that the polymer stress σp∼μAzz has the same time dependence as the capillary pressure γ/h(t)∼exp(˙t/2). Equating this exponential to that in (4.2), one finds ˙λ =2/3. One thus concludes that the thinning of the thread scales as h∼exp(−t/3λ), a result that goes back to Entov [87]. (ii) The elastic correspondence Figure 2 shows numerical simulations of the break-up in the Oldroyd-B fluid [48]. For finite λ,we indeed observe the expected exponential thinning dynamics (blue line). However, the initial stage of the break-up (on the scale of the capillary time τ) is not exponential; in fact it closely follows the Newtonian break-up, λ=0, plotted as the purple line. In this early regime, the polymer is not yet sufficiently stretched to compete with capillary forces, and we observe a near independence of λ. In the elastic limit λ=∞(red line), the thread does not go to zero at all, but saturates at a finite thickness. This corresponds to a neo-Hookean solid, and, for example, describes the cross-linked agar gel of figure 2a. So how can we compute the prefactor h0of the thinning law (4.1)? For this, we make use of the elastic correspondence. Upon inspection of figure 2, one can infer that h0is essentially given by the final thickness of the purely elastic thread (this becomes exact when τλ). This final thickness follows from an elasto-capillary stress balance. The elastic stress is simply that of a neo-Hookean rubber band, with elongation stretch given by (R0/h)2,whereR0is the initial cylinder radius. When sufficiently soft, the elastic stress scales as ∼μ(R0/h)4. Balancing this with the capillary stress γ/h, one obtains the elasto-capillary length [45] e=μR4 0 γ1/3 . (4.4) A detailed analysis matching the cylinder to a large drop shows that the exact prefactor in (4.1) reads h0=e/21/3[48]. The elastic correspondence goes much beyond computing h0. Using a lubrication description, it was conjectured by Entov & Yarin [44] and confirmed in [21,84] that the entire shape of the thinning thread—at finite λ—could be described by that of the corresponding elastic solid. This is a scheme that we recently confirmed to be true in general, beyond the lubrication description [48]. (b) Viscoelastic solid We argued in §3d that the elastic correspondence allows us to model viscoelastic solids as an Oldroyd-B fluid, taking the limit λ→∞. This idea is tested by two distinct numerical schemes to compute the capillary collapse. We first consider an Eulerian simulation for the Oldroyd-B fluid with infinite relaxation time, followed by a Lagrangian simulation of a neo-Hookean elastic cylinder. The final state should be the same because of the elastic correspondence, but there is an important difference: because of the solvent viscosity, the Oldroyd-B model is able to capture the dynamics of the solid in the presence of viscous damping. (i) The Oldroyd-B fluid as a viscoelastic solid We simulate the Oldroyd-B equations (2.1), (2.2), (2.8) in the limit of λ→∞, taken such that μ= ηp/λ remains finite. Then the polymeric stress is governed by  σp=0. The collapse is driven by surface tension, and the stress boundary condition at the free surface is n·σ=−γκn, (4.5) where κ=1 h(1 +h2 z)1/2−hzz (1 +h2 z)3/2,n=er−ezhz (1 +h2 z)1/2(4.6) 20 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... 0.2 0.4 0.6 0.8 1.0 0 50 100 20 1 2 46 z/R0 r/R0 20 1 2 46 z/R0 r/R0 20 1 2 46 z/R0 r/R0 20 1 2 46 z/R0 r/R0 rR0 3/g hmin/R0 (a)( b)( c) (d)( e) t/ Figure 5. (a) Time evolution of the minimum thread radius for the viscoelastic solid, simulated by an Oldroyd-B fluid with λ=∞. Parameters are =0.05, ηs/ργR3 0=0.79 and μR0/γ =0.0119. (b–e) Polymeric stresses in the final state of the fluid simulation: (b)σp,zz;(c)σp,zr;(d)σp,rr;(e)σp,θθ. The purple to yellow (dark to light) colour gradient represents the change from minimum to maximum stress, respectively. (Online version in colour.) are (twice) the mean curvature and the surface normal, respectively. If h(z,t) is the thread profile, the kinematic boundary condition becomes ∂h ∂t+uz(z,h)∂h ∂z=ur(z,h), (4.7) where v=urer+uzezin cylindrical coordinates. As an initial condition, we take the free surface shape h(z,0)≡h0(z)=R01−cos z 2R0, (4.8) and the velocity field vanishes initially; boundary conditions are periodic. First, we will consider the case that stresses vanish initially. To illustrate the predictions of (3.3), we will then consider an initial uniform axial stress. We have carried out a simulation for a fluid cylinder of radius R0, which is slightly perturbed according to (4.8) with =0.05. Material parameters are fixed by dimensionless numbers ηs/ργR3 0=0.79 and μR0/γ =0.0119. In order to calculate the interface evolution accurately, we apply the boundary fitted coordinate method, where the liquid domain is mapped onto a rectangular domain through a coordinate transformation. The hydrodynamic equations are discretized in this domain using fourth-order finite differences, with 22 equally spaced points in the radial direction and 1000 equally spaced points in the axial direction. An implicit time advancement is performed using the second-order backward finite differences with a fixed time step 0.05ρR3 0/γ ; details of the numerical procedure can be found elsewhere [88]. We begin with the case where there is no stress in the initial condition. In figure 5a, we show the minimum thread radius hmin as a function of time. As the bridge collapses, elastic stress builds up until it is balanced by surface tension and hmin approaches a constant value, as shown in figure 5. At this point, the solution becomes stationary, time derivatives vanish and the velocity goes to zero. As a result, the solvent viscosity does not affect the final state, which should be identical to that of a neo-Hookean solid. However, the Oldroyd-B simulation also captures the transient dynamics of the viscoelastic solid. For completeness, figure 5b–dshows the different components of the stress tensor in the static final state. The axial stress σp,zz is highest inside the thread, where fluid elements are stretched the most in the axial direction. Radial stresses σp,rr, on the other hand, are most pronounced inside the drop, where fluid elements are stretched in the radial direction. 21 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... 00123456 1.0 1.5 0.5 r/R0 00123456 1.0 1.5 0.5 r/R0 00123456 1.0 1.5 0.5 r/R0 z/R0 (a) (c) (b) Figure 6. The elastic simulation: (a) the reference state, μR0/γ =∞;(b)μR0/γ =0.2; (c)μR0/γ =0.1. Red (cyan) lines describe constant values of η(ξ). The arrows indicate the direction of the mesh deformation. (Online version in colour.) (ii) The neo-Hookean solid We now calculate the steady state of an elastic neo-Hookean material using nonlinear elasticity, as described by σ=μ(B−I)−pI, (4.9) and subject to the incompressibility constraint J=det(F)=1. First we discuss the case without pre-stress. The pressure pis adjusted such that J=1 is satisfied. Instead of a dynamical equation, the condition for static equilibrium reads ∇·σ=0, i.e. (2.1) with v=0, with the elasto-capillary boundary condition (4.5). To determine the final state of the collapsed cylinder, we solve a nonlinear set of equations corresponding to the above conditions, based on a (stationary) mapping x=x(X), as illustrated in figure 6. To this end, we write the mapping in cylindrical coordinates: r=r(R,Z), z=z(R,Z). The coordinates Rand Zare the radial and axial coordinates of the cylinder in the reference state. Using general formulae for Fin cylindrical coordinates [89], incompressibility amounts to detF=r R∂r ∂R ∂z ∂Z−∂r ∂Z ∂z ∂R=1, (4.10) while the stress can be computed from the Finger tensor B=F·FT=⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ∂r ∂R2+∂r ∂Z20∂r ∂R ∂z ∂R+∂r ∂Z ∂z ∂Z 0r R20 ∂r ∂R ∂z ∂R+∂r ∂Z ∂z ∂Z0∂z ∂Z2+∂z ∂R2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . (4.11) The solution depends on the dimensionless number R0μ/γ . In the ‘soft’ limit where the thread becomes very thin, rR0, the thickness of the thread scales as the elasto-capillary length scale e given by (4.4). 22 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... 0123456 z = R0 r = R0 0 0.5 1.0 1.5 2.0 fluid simulation: s0R0/g = 0 elastic simulation: s0R0/g = 0 fluid simulation: s0R0/g = 0.1 elastic simulation: s0R0/g = 0.1 Figure 7. A comparison between the final shape of an elastic bridge driven by surface tension, using an Oldroyd-B fluid with λ=∞, and a neo-Hookean elastic material. The elastic shear modulus is μ=ηp/λ =0.0119γ/R0. The green and red symbols (large deformation) represent vanishing initial stress, while to obtain the cyan and blue symbols (small deformation) the bridge contained a uniform initial axial stress σ0. In each case, the results are virtually identical. (Online version in colour.) To solve the problem numerically, we define the reference state by R=h0(ξ)ηand Z=ξ, with the elastic domain defined by η∈[0,1] and ξ∈[0,2πR0]; h0is once again defined by (4.8) and =0.05. We are looking for two unknown functions fand g,wherer=r(R,Z)=f(η,ξ)and z=z(R,Z)=g(η,ξ), as well as the pressure p(η,ξ). These three unknowns are found from solving the three equations (4.10), (2.1) at steady state and with boundary conditions (4.5). The free surface h(z) then is given by the parametric representation h(g(1,ξ)) =f1(1, ξ), from which the curvature κcan be evaluated. In figure 6, the domain is discretized using fourth-order finite differences with 301 equally spaced points in the ξdirection and 11 Chebyshev collocation points in the η direction. For the results presented in figure 7, a finer mesh was used with 2001 equally spaced points in the ξdirection. The resulting system of nonlinear equations is solved using a Newton– Raphson technique [88]. We solve the problem by starting with the reference state as the initial guess and μR0/γ sufficiently large (μR0/γ =100) to ensure the convergence of the Newton– Raphson iterations. Once we get a solution, we use this solution in a new run with a smaller value of μR0/γ . The result is shown in figure 6, which describes the deformation of the mesh as well as of the free surface, for various values of μR0/γ . The resulting shapes closely match those in figure 2 at moderate stiffness, and agree with simulations in [46]. In figure 7, we compare the elastic equilibrium state (red crosses) with the stationary state reached in the simulation of the Oldroyd-B model for μR0/γ =0.0119 (green circles). The agreement is perfect—illustrating that, for this problem where a stationary state is approached, the Oldroyd-B fluid in the limit of λ→∞ converges to a neo-Hookean solid. A detailed similarity analysis of this problem is provided in [48]. To further test (3.3) in the case of a pre-stressed material, we repeat the same analysis for both the fluid and the elastic case, but assuming an initial purely axial stress. This means that the fluid equations are solved with the initial condition σp,zz =σ0, and are solved until a new stationary state is reached. As for the elastic simulation, (3.3) is used with σ(0) p,zz =σ0. As seen in figure 7 (cyan diamonds, fluid simulation; blue crosses, elastic simulation), the equilibrium is now reached at a much larger value of hmin, because less of a build-up of elastic stresses is needed to counter surface 23 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... a = 0.1 a = 0.9 a = 0.1 a = 0.9 a = 1 r r z hmin / R0 01020304050 0.5 1.0 –2 0 2 –5 0 5 z –5 0 5 –2 0 2 rR0 3/g t/ (a)(b)(i) (ii) Figure 8. (a) The minimum thread radius hmin for a Johnson–Segalman fluid, as a function of time for different values of a. Only in the case of non-affine motion (a=1) does the thread attain a static elastic solution. Break-up is observed for all values a= 1. (b) Typical interface shapes for the collapse of a liquid bridge of Johnson–Segalman fluid close to break-up, for two values of the parameter a. The case a=0.1 corresponds to t=29.7747 and hmin =0.000886, while for a=0.9, t=37.271 and hmin =0.0124. (Online version in colour.) tension. Once more, there is perfect agreement with the Oldroyd-B fluid simulation, illustrating the general formula (3.3) for arbitrary initial conditions. Thus, we have demonstrated two very different numerical solutions to the same problem of finding the equilibrium shape of an elastic thread constricted by surface tension, one being Eulerian and the other Lagrangian. This represents a satisfying confirmation of the equivalence of fluid flow and nonlinear elasticity in the limit of infinite relaxation time, and presents a valuable check on the stability and reliability of the underlying numerical methods. (c) The absence of an elastic limit: the Johnson–Segalman fluid The above comparison was an illustration of our result, which assigns a unique elastic limit to the Oldroyd-B model for large relaxation times. The scenario changes dramatically in the presence of non-affine motion, for which there is no elastic correspondence. This will be explored for a Johnson–Segalman fluid with infinite relaxation time, characterized by  (A)a=0. The other equations remain the same. Previous analysis of the long-wavelength limit has shown [45,90] that there can be no static solution for a<1/2. Owing to the presence of non-affine slip, the amount of polymer stretch is not enough for the elastic stress to balance the capillary stress in a thinning cylinder. This means that the motion will not be arrested by surface tension, and the thread breaks up in finite time [45,90]. By contrast, the slender analysis predicts that for a>1/2 the thread will not break up. Johnson–Segalman fluid simulations are carried out by integrating (3.10) with λ=∞.We consider different values of awith the same numerical technique as described at the beginning of the section, using the same parameters =0.05, ηs/ργR3 0=0.79 and μR0/γ =0.0119 as before. Figure 8ashows hmin as a function of time for three different values of the slip parameter a.Ascan be seen in the figure, the solution reaches a steady state only for the affine case a=1 (black dotted line), as seen before in figure 5. In line with the lubrication analysis of [90], for a=0.1 the thread thickness approaches zero linearly, and at a finite time (red dashed line). Remarkably, however, the case a=0.9 also leads to break-up (blue solid line), a feature that was not predicted in the lubrication framework. Figure 8bhighlights further the behaviour of the pinching threads. Figure 8b(i) shows that for a=0.1 the thread profile remains slender, and break-up is described by the similarity theory of Fontelos [90]. Figure 8b(ii) shows the case a=0.9, which exhibits break-up despite 24 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... the lubrication prediction that it should not. Indeed, the corresponding interface now develops a more complicated structure, with pinching near the drops, which violates the assumption of slenderness. We thus conclude that non-affine motion, for which the elastic limit does not exist, has a dramatic effect on the dynamics, in this case on the capillary collapse of a cylinder. 5. Energy and dissipation There are striking dissimilarities in the approach to viscoelastic liquids and to elastic solids. We have seen in §2 that the constitutive relations in elasticity theory are defined by a free energy functional W, while the constitutive relations for viscoelastic liquids are most commonly expressed in terms of the stress tensor σp. A fully conservative formulation of viscoelasticity is not possible, of course, since the process of relaxation involves dissipation. Still, a thermodynamic approach is very feasible, as described in Beris & Edwards [6]. This thermodynamic route was previously proposed by Leonov [54], and was formalized using the bracket [6] and the GENERIC formalisms [7]. We refer to Pasquali & Scriven [65] for a detailed discussion on how these theories are related. The idea is again to describe the microstructure of complex fluids by a thermodynamic state variable, or order parameter, say A(x,t) in the case of the conformation tensor. The key element of the thermodynamic route is to associate an elastic free energy density with this field, W(A), which represents the free energy stored in stretched polymers. The formulation is then very similar to the theory of elasticity, which was based on W(B), except that one allows for dissipative processes via relaxation, expressed by  A= 0. An important advantage of the thermodynamic formulation of viscoelasticity is that, by construction, the constitutive relations are consistent with the laws of thermodynamics. Here we further elaborate these ideas, and derive the explicit form of the energy equation for viscoelastic fluids. This serves two purposes. (i) It is surprisingly difficult to find general expressions for dissipation in common models of viscoelastic fluids [91]. In particular, we did not succeed in finding an energy equation for viscoelastic fluids that would generalize the usual energy conservation law for Newtonian flows. (ii) The thermodynamic formulation offers a very natural connection between elasticity and viscoelasticity, with the viscoelastic free energy W(A) playing the exact same role as the elastic free energy W(B). Rather than following the formalisms of statistical physics [6,7], we here offer a direct mechanistic route by separating the reversible and dissipative parts of the energy equation. (a) Flow versus relaxation The order parameter formulation of viscoelasticity has the following ingredients: (i) A symmetric rank-2 tensor-order parameter field A(x,t), which quantifies the stretched state of the polymer. (ii) An elastic free energy density W(A), which is minimal for A=I. (iii) A relaxation equation towards A=I, governing dissipation. Since the conformation tensor now plays the role of a thermodynamic state variable, it is important to give it a proper definition. For microscopic bead–spring models, the conformation tensor can be expressed in terms of the (averaged) end-to-end vector of the dumbbell. To mimic the theory of elasticity, however, we here look for a purely continuum definition. The quantification of the amount by which the polymer is stretched requires a comparison of the current state with a relaxed (isotropic) state at which the system is stress free. In a purely continuum description, the definition of Atherefore requires a concept that can be characterized as an ‘instantaneous reference state’—this describes at each instant in time the state in which the polymer would be stress-free [5]. In a purely elastic material, devoid of relaxation, the reference state is the same at all times and can, for example, be chosen as the initial condition. For 25 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... viscoelastic liquids, however, the memory of the initial condition is gradually fading. For instance, let us consider a case where, by applying appropriate constraint forces, we instantaneously stop the flow. The final state of the liquid will ultimately become stress-free, but this takes some time. In this situation, however, the current state of the liquid no longer evolves since the flow has stopped: it is the instantaneous reference state that will relax towards the liquid’s final state. These ideas can be formalized upon introducing curvilinear material coordinates that describe the dynamics of material points following the flow. The curvilinear description allows us to properly define the conformation tensor and to distinguish its time evolution into a part due to ‘flow’ and a part that represents ‘relaxation’. We refer the reader to appendix A for details. Below we exploit this separation of flow and relaxation, which enables us to separate the storage of energy (stretching by flow) and its dissipation (via relaxation). (b) Stress, energy and dissipation (i) Splitting the work: energy storage versus dissipation With a continuum definition of Ain place, we can find the correct structure of W(A) by borrowing the elastic energy W(B), as found on nonlinear elasticity. The energy must be a function of the invariants I1=Akk,I2=1 2A2 kk −AijAijand I3=det(A), (5.1) with Ataking the role of Bin (2.12). There should be no confusion from using the same notation for the invariants of A. The choice of W(A) naturally determines the elastic limit, while the relaxation equation for Aaccounts for irreversible dissipation. We now proceed to derive the expression for the stress and the dissipation, focusing first on affine polymer models. Given that we chose the conformation tensor Ato express the stretching of the polymer, it must relax according to λ  A=f(A). The idea is that the reversible part of the deformation has the same form as the reversible change in free energy (2.11), so the remainder corresponds to dissipation. Writing the work in symmetric form (1/2)σp:˙ γ, and introducing the volumetric dissipation rate p, energy conservation requires that 1 2σp:˙ γ=dW dt+p, (5.2) where we have performed a split into reversible and irreversible parts: any work done during the deformation must be either stored in elastic energy or dissipated. With this convention, pmust be positive in order to be consistent with thermodynamics. The time derivative dW/dtcan be calculated using the definition of  A, yielding dW dt=∂W ∂A:dA dt=∂W ∂A:(∇v)T·A+A·(∇v)+  A =∂W ∂A·A:˙ γ+∂W ∂A:  A, (5.3) where in the last line we made use of the symmetry of A. As anticipated in (5.2), this nicely separates into a term due to flow, proportional to ˙ γ, and a term associated with the relaxation law, proportional to  A. Comparing (5.3) and (5.2), we obtain the expression for the stress σp=2∂W ∂A·A. (5.4) The same expression for stress is obtained in the GENERIC formalism based on continuum considerations [7] or for specific cases of microscopic bead–spring models [3]. As expected, this is exactly the form of the elastic stress (2.10), with Areplacing B. The second term in (5.3) can be 32 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... (b) Flow Now, we investigate the effect of flow on the metric. First, we define the velocity as v=dx dtqi =viei, (A 11) expressed using the basis defined by (A 4), where from now on d/dtmeans the time derivative at constant material points qi. Using (A 5) and (A 4), the time derivative of the metric tensor is dgij dt=⎛ ⎝ ∂vkek ∂qi·ej⎞ ⎠+ei·∂vmem ∂qj≡vk ;iek·ej+ei·(em)vm ;j =gkjvk ;j+gmivm ;j=vj;i+vi;j≡˙γij, (A 12) where we used the definition of the covariant derivative, denoted by (..);j. Hence, the rate of strain tensor ˙ γdirectly gives the change of the metric of material coordinates by the flow. Remembering that Bij =Gij (cf. (A 8)), we see that the Finger tensor evolves according to dBij dt=d dt∂qi ∂X·∂qj ∂X=0 (A 13) during flow. This time derivative vanishes because the reference state X(qi) is independent of time. Note, however, that the covariant components Bij are not constant in time, since dBij dt=d dtgikgjmBkm=2˙γikBk j. (A 14) For a general tensor A, the derivatives dAij/dtand dAij/dt, respectively, correspond to the components of the upper and lower convected derivatives [61,96], i.e.  A=dAij dtei⊗ej,  A=dAij dtei⊗ej. (A 15) The equivalence with the definitions (2.17) and (2.18) follows from transforming Aij and Aij, respectively, from the Lagrangian (co-moving) material coordinates qito an Eulerian coordinate system ¯ qithat is fixed in space. In this fixed coordinate system, the tensor components denoted by ¯ Aij can be obtained using the transformation Fk i=∂¯ qk/∂qi. Transforming dAij/dtto the fixed frame then gives Fp idAij dtFq j=Fp i d dt(F−1)i k¯ Akm(F−1)j mFq j. (A 16) On the right-hand side, we recognize the definition (2.17) for  A, now in the form of the components of the fixed Eulerian system ¯ qi. In a similar fashion, one derives (A 15) for the lower convected derivative. Hence, dBij/dt=0 implies that the upper convected derivative of the Finger tensor vanishes, i.e.  B=0. (c) Elasticity In the theory of elasticity, the energy density Wis a function of the invariants of B. Assuming incompressibility I3=det(B)=1, the energy is of the form W(I1,I2), where the first and second 33 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... invariants are defined as I1=Bi i=gijBij,I2=1 2gijBij2−BijBij=1 2gijBij2−gimgjnBmnBij. (A 17) The expression for stress is obtained from time derivatives, according to the virtual work principle 1 2σij ˙γij =dW dt=W1dI1 dt+W2dI2 dt. (A 18) Using dBij/dt=0, we find dI1 dt=dgij dtBij =˙γijBij,dI2 dt=dgij dt ∂I2 ∂gij =˙γij I1Bij −Bi nBnj, (A 19) both of which are proportional to ˙γij. Hence, from (A 18) we can read off the stress as σij =2∂W ∂gij =2W1Bij +2W2I1Bij −Bi nBnj. (A 20) (d) Viscoelasticity When an elastic liquid is suddenly arrested, the polymer will relax towards an isotropic equilibrium confirmation. The system slowly forgets about the history of deformation prior to the arrest, and all stress and polymer stretches are ultimately relaxed. When expressing the polymer deformation in terms of the elementary lengths between qiand qi+dqi, we can still write ds2−dS2=gij −Gijdqidqj, (A 21) and we associate an elastic energy with the polymer strain. Owing to the fading memory of the initial state x(qi,t=0), however, the object Gij can no longer be identified with the timeindependent metric of this initial condition. Instead, Gij reflects the metric of the ‘instantaneous reference state’ X(qi,t) that progressively tends to evolve towards the current state. From the above, we have a very clear definition of flow and relaxation: at fixed material point qi, flow refers to the time dependence of x, while relaxation implies the time dependence of X.We now exploit this further using (A 7) and (A 6), and obtain Gij =∂X ∂qi·∂X ∂qi,Gij =∂qi ∂X·∂qj ∂X. (A 22) We can then define the conformation tensor A, whose eigenvalues give the stretches of the polymer. This is in direct analogy to the Finger tensor B, the only difference being that the stretches need to be measured with respect to the instantaneous reference state X(qi,t), rather than x(qi,t=0). Hence, A=∂qi ∂X·∂qj ∂Xei⊗ej=Gijei⊗ej. (A 23) At the start of §5, we referred to X(qi,t)=X∗, so that we there defined A=F∗·F∗T.The components Aij in (A 23) do not involve any flow, but only relaxation, as relaxation includes timedependent effects only of X(qi,t). Hence, if we wish to express a relaxation law directly in terms of the conformation tensor A, we necessarily arrive at dAij dt=1 λfij(A), (A 24) which in the fixed frame corresponds to the upper convected derivative. By contrast, the covariant components Aij =gikgjmAkm do exhibit a time dependence due to the flow, via dgij/dt.Hence,to quantify the relaxation in terms of the conformation tensor A, one automatically singles out the upper convected derivatives as the appropriate time derivative. 34 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: 20200419 ........................................................... In analogy with elasticity theory, we introduce an elastic energy W(I1,I2) associated with the first and second invariants of A. Once again we employ the virtual work principle, but now including dissipation p, 1 2σij ˙γij =dW dt+p. (A 25) The dissipation is necessary since the elastic energy exhibits an extra time dependence associated with relaxation, dW dt=∂W ∂gij ˙γij +∂W ∂Aij dAij dt. (A 26) Again, the terms proportional to ˙γij provide the stress, so that stress and dissipation can be separated as σij =2∂W ∂gij ,p=−∂W ∂Aij dAij dt. (A 27) Appendix B. Integrating the upper convected Maxwell model In order to integrate the upper convected Maxwell model, we write (2.7) using (2.17) as F·d dtF−1·A·F−T·FT=−1 λ(A−I),(B1) so that, upon defining D=F−1·A·F−T, we can write dD dt=−1 λD−F−1·F−T.(B2) Note that Ais defined on the Eulerian domain, while Dis defined on the reference domain. Therefore, its time derivative does not contain any convective terms, and one can treat (B2) as an ordinary differential equation. The term g(t)=F−1(t)·F−T(t) can be treated as some function of time that is imposed externally by the flow. One verifies that the solution to (B 2) reads D(t)=e−t/λD0+1 λt 0dte−(t−t)/λg(t), (B 3) where D0=D(0) is a constant of integration. Transforming the result back to the Eulerian domain, by the operation F(t)·(···)·FT(t), we obtain A(t)=e−(t/λ)F(t)·D0·FT(t)+1 λt 0dte−(t−t)/λF(t)·g(t)·FT(t). (B 4) For a stress-free initial condition, D0=I, this gives (3.13). References 1. 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