Originating in the field of biomechanics, Fung's model of quasi-linear viscoelasticity (QLV) is one of the most popular constitutive theories employed to compute the time-dependent relationship between stress and deformation in soft solids. It is one of the simplest models of nonlinear viscoelasticity, based on a time-domain integral formulation. In the present study, we consider the QLV model incorporating a single scalar relaxation function. We provide natural internal variables of state, as well as a consistent expression of the free energy to illustrate the thermodynamic consistency of this version of the QLV model. The thermodynamic formulation highlights striking similarities between QLV and the internal-variable models introduced by Holzapfel and Simo. Finally, the dissipative features of compressible QLV materials are illustrated in simple tension.
We show that the Persson-Brener theory of crack propagation in viscoelastic solids gives a viscoelastic fracture energy factor $G/G_0 = 1+f$ which is nearly the same as the viscoelastic factor obtained using the cohesive-zone model. We also discuss finite-size effects and comment on the use of crack propagation theories for "solids" with a viscoelastic modulus that vanishes at zero frequency.
Many manmade and naturally occurring materials form viscoelastic solids. The increasing use of biologically inspired elastomeric material and the extreme mechanical response these elastomers offer drive the need for a better, more intuitive and quantitative understanding of the mechanical response of such continua. This is in particular important in determining the stability of viscoelastic structure 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 characterized by temporall…
We calculate the friction force acting on a hard cylinder or spherical ball rolling on a flat surface of a viscoelastic solid. The rolling friction coefficient depends non-linearly on the normal load and the rolling velocity. For a cylinder rolling on a viscoelastic solid characterized by a single relaxation time Hunter has obtained an exact result for the rolling friction, and our result is in very good agreement with his result for this limiting case. The theoretical results are also in good agreement with experiments of Greenwood and Tabor. We suggest that measurements of rolling friction o…
The effective dynamic properties of specific periodic structures involving rubber-like materials can be adjusted by pre-strain, thus facilitating the design of custom acoustic filters. While nonlinear viscoelastic behaviour is one of the main features of soft solids, it has been rarely incorporated in the study of such phononic media. Here, we study the dynamic response of nonlinear viscoelastic solids within a 'small-on-large' acoustoelasticity framework, that is we consider the propagation of small amplitude waves superimposed on a large static deformation. Incompressible soft solids whose b…
Using the finite element method we quantitatively analyse temperature field evolution in a viscoelastic solid undergoing a loading--unloading process. In particular we investigate the temperature field inside a Kelvin--Voigt type viscoelastic body with a thin cutout. We find that the viscosity significantly contributes to the temperature field changes, and that the temperature field changes initiated by the loading--unloading process are strongly concentrated at the tip of the thin cutout. The predicted temperature field qualitatively corresponds to the temperature field observed in experiment…
When quickly detaching an elastomer from a counterface, viscoelasticity dramatically increases the perceived adhesion relative to its adiabatic or equilibrium value. Here, we report simulations on the sticking contact between a rigid cylinder and a viscoelastic half space revealing a maximum in the work of adhesion at intermediate pull-off velocities. Maximum tensile forces yet increase monotonically with the pull-off speed and the crack-tip speed in accordance with the Persson-Brener approach. As predicted theoretically, the fracture mode transitions from interfacial crack propagation to quas…
We study pattern formation during tensile deformation of confined viscoelastic layers. The use of a model system (PDMS with different degrees of crosslinking) allows us to go continuously from a viscous liquid to an elastic solid. We observe two distinct regimes of fingering instabilities: a regime called "elastic" with interfacial crack propagation where the fingering wavelength only scales with the film thickness, and a bulk regime called "viscoelastic" where the fingering instability shows a Saffman-Taylor-like behavior. We find good quantitative agreement with theory in both cases and pres…
While determining the stability of an unconstrained elastic structure is a straightforward task, this is not the case for viscoelastic structures. Seemingly elastically stable conformations of viscoelastic structures may gradually creep until stability is lost, and conversely, creeping does not necessarily imply that a structure will eventually become unstable. Understanding instabilities in viscoelastic structures requires a more intuitive description of viscoelasticity to allow analytical results and quantitative predictions. In this work we put forward a metric description of viscoelasticit…
Understanding surface mechanics of soft solids, such as soft polymeric gels, is crucial in many engineering processes, such as dynamic wetting and adhesive failure. In these situations, a combination of capillary and elastic forces drives the motion, which is balanced by dissipative mechanisms to determine the rate. While shear rheology (i.e. viscoelasticity) has long been assumed to dominate the dissipation, recent works have suggested that compressibility effects (i.e. poroelasticity) could play roles in swollen networks. We use fast interferometric imaging to quantify the relaxation of surf…
Viscoelastic liquids are characterized by a finite static viscosity and a zero yield stress, whereas soft solids have an infinite viscosity and a non-zero yield stress. The rheological nature of viscoelastic materials has long been a challenge, and it is still a matter of debate. Here, we provide for the first time the constitutive equations of linear viscoelasticity for magnetic wires in yield stress materials, together with experimental measurements using Magnetic Rotational Spectroscopy (MRS). With MRS, the wires are submitted to a rotational magnetic field as a function of frequency and th…
New exact solutions are exhibited within the framework of finite viscoelasticity. More precisely, the solutions correspond to finite-amplitude, transverse, linearly-polarized, inhomogeneous motions superposed upon a finite homogeneous static deformation. The viscoelastic body is composed of a Mooney-Rivlin viscoelastic solid, whose constitutive equation consists in the sum of an elastic part (Mooney-Rivlin hyperelastic model) and a viscous part (Newtonian viscous fluid model). The analysis shows that the results are similar to those obtained for the purely elastic case; inter alia, the normals…
We have derived the general solution of a wave equation describing the dynamics of two-layere viscoelastic polymer materials of arbitrary thickness deposited on solid (quartz) surfaces in a fluid environment. Within the Voight model of viscoelastic element, we calculate the acoustic response of the system to an applied shear stress, i.e. we find the shift of the quartz generator resonance frequency and of the dissipation factor, and show that it strongly depends on the viscous loading of the adsorbed layers and on the shear storage and loss moduli of the overlayers. These results can readily b…
We introduce a model for nonlinear viscoelastic solids where traveling shear waves with compact support are possible. We obtain an exact compact solution. We also derive a new Burger's type evolution equation associated with the introduced constitutive equation.
The one-dimensional impact problem in the case of Fung's quasi-linear viscoelastic model is studied for the relaxation function of the standard solid model (or Zener model). At that, quasi-linear viscoelastic Maxwell and Kelvin-Voigt models are recovered as limit cases. The results of numerical simulations for some illustrative values of the dimensionless problem parameters are presented.
A paradigm for internally driven matter is the active nematic liquid crystal, whereby the equations of a conventional nematic are supplemented by a minimal active stress that violates time reversal symmetry. In practice, active fluids may have not only liquid crystalline but also viscoelastic polymer degrees of freedom. Here we explore the resulting interplay by coupling an active nematic to a minimal model of polymer rheology. We find that adding polymer can greatly increase the complexity of spontaneous flow, but can also have calming effects, thereby increasing the net throughput of spontan…
In this paper, we present a qualitative analysis of the dissipative processes during the failure of the interface between a viscoelastic polymer and a solid surface. We reassess the "viscoelastic trumpet" model [P.-G. de Gennes, C. R. Acad. Sci. Paris, 307, 1949 (1988)], and show that, for a crosslinked polymer, the interface toughness G(V) starts from a relatively low value, G_0, due to local processes near the fracture tip, and rises up to a maximum of order $G_0 (μ_{\infty}/μ_0)$ (where $μ_0$ and $μ_{\infty}$ stand for the elastic modulus of the material, respectively at low and high strain…
The spreading of liquid drops on soft substrates is extremely slow, owing to strong viscoelastic dissipation inside the solid. A detailed understanding of the spreading dynamics has remained elusive, partly owing to the difficulty in quantifying the strong viscoelastic deformations below the contact line that determine the shape of moving wetting ridges. Here we present direct experimental visualisations of the dynamic wetting ridge, complemented with measurements of the liquid contact angle. It is observed that the wetting ridge exhibits a rotation that follows exactly the dynamic liquid cont…
The progressive onset of slip at the wall, which corresponds to a slip length increasing with the solicitation time before reaching a plateau, has been investigated for model viscoelastic polymer solutions, allowing one to vary the longest relaxation time while keeping constant solid - fluid interactions. A hydrodynamic model based on a Maxwell fluid and the classical Navier's hypothesis of a linear response for the friction stress at the interface fully accounts for the data, without any adjustable parameter. No viscoelastic response needs be postulated for the friction, reflecting the local …
We present a constitutive model capturing some of the experimentally observed features of soft biological tissues: nonlinear viscoelasticity, nonlinear elastic anisotropy, and nonlinear viscous anisotropy. For this model we derive the equation governing rectilinear shear motion in the plane of the fiber reinforcement; it is a nonlinear partial differential equation for the shear strain. Specializing the equation to the quasi-static processes of creep and recovery, we find that usual (exponential-like) time growth and decay exist in general, but that for certain ranges of values for the materia…
Lubrication flows appear in many applications in engineering, biophysics, and in nature. Separation of surfaces and minimisation of friction and wear is achieved when the lubrication fluid builds up a lift force. In this paper we analyse soft lubricated contacts by treating the solid walls as viscoelastic: soft materials are typically not purely elastic, but dissipate energy under dynamical loading conditions. We present a method for viscoelastic lubrication and focus on three canonical examples, namely Kelvin-Voigt-, Standard Linear-, and Power Law-rheology. It is shown how the solid viscoela…
Borosilicones (e.g., Silly Putty) have been known for 70 years, but their peculiar behaviors have remain unexplained. In this work, experiment and theory are used to show that they are network liquids---dynamic macromolecules that appear elastic on short timescales but exhibit flow on longer timescales. Each borosilicone is a vast covalent network of silicone polymer chains joined by trifunctional boron crosslinks. At any instant, the borosilicone is a highly-crosslinked elastic material. Because the boron crosslinks are temporary, however, the network evolves with time and the borosilicone ex…
A theory of reciprocating contacts for linear viscoelastic materials is presented. Results are discussed for the case of a rigid sphere sinusoidally driven in sliding contact with a viscoelastic half-space. Depending on the size of the contact, the frequency and amplitude of the reciprocating motion, and on the relaxation time of the viscoelastic body, we establish that the contact behavior may range from the steady-state viscoelastic solution, in which traction forces always oppose the direction of the sliding rigid punch, to a more elaborate trend, never observed before, which is due to the …
We study the propagation of small amplitude waves superimposed on a large static deformation in a nonlinear viscoelastic material of differential type. We use bulk waves and surface waves to address the questions of dissipation and of material and geometric stability. In particular, the analysis provides bounds on the constitutive parameters and on the pre-deformation that ensure linearized stability in the neighbourhood of a large pre-stretch. This type of result is relevant to the imaging of biological soft tissues using acoustical techniques, where pre-deformation is known to increase contr…
Soft tissues are complex media, they display a wide range of mechanical properties such as anisotropy and non-linear stress-strain behaviour. They undergo large deformations and they exhibit a time-dependent mechanical behaviour, i.e. they are viscoelastic. In this chapter we review the foundations of the linear viscoelastic theory and the theory of Quasi-Linear Viscoelasticity (QLV) in view of developing new methods to estimate the viscoelastic properties of soft tissues through model fitting. To this aim, we consider the simple torsion of a viscoelastic Mooney-Rivlin material in two differen…
We review a theory of crack propagation in viscoelastic solids. We consider both cracks in infinite systems and in finite sized systems. As applications of the theory we consider two adhesion problems, namely pressure sensitive adhesives and the ball-flat adhesion problem. We also study crack propagation in the pig skin dermis, which is of medical relevance, and rubber wear in the context of tires.
In this paper we study swimming of a model organism, the so-called Taylor's swimming sheet, in a viscoelastic fluid close to a solid boundary. This situation comprises natural habitats of many swimming microorganisms, and while previous investigations have considered the effects of both swimming next to a boundary and swimming in a viscoelastic fluid, seldom have both effects been considered simultaneously. We re-visit the small wave amplitude result obtained by Elfring and Lauga (Gwynn J. Elfring and Eric Lauga, in Saverio E. Spagnolie, editor, Complex Fluids in Biological Systems, Springer N…
We introduce a comprehensive scheme to physically quantify both viscous and elastic rheological nonlinearities simultaneously, using an imposed large amplitude oscillatory shear (LAOS) strain. The new framework naturally lends a physical interpretation to commonly reported Fourier coefficients of the nonlinear stress response. Additionally, we address the ambiguities inherent in the standard definitions of viscoelastic moduli when extended into the nonlinear regime, and define new measures which reveal behavior that is obscured by conventional techniques.
Polymer gels behave as soft viscoelastic solids and exhibit a generic nonlinear mechanical response characterized by pronounced stiffening prior to irreversible failure, most often through macroscopic fractures. Here, we aim at capturing the latter scenario for a protein gel using a nonlinear integral constitutive equation built upon ($i$) the linear viscoelastic response of the gel, here well described by a power-law relaxation modulus, and ($ii$) the nonlinear viscoelastic properties of the gel, encoded into a "damping function". Such formalism predicts quantitatively the gel mechanical resp…
Viscoelastic fluids are non-Newtonian fluids that exhibit both "viscous" and "elastic" characteristics in virtue of mechanisms to store energy and produce entropy. Usually the energy storage properties of such fluids are modelled using the same concepts as in the classical theory of nonlinear solids. Recently new models for elastic solids have been successfully developed by appealing to implicit constitutive relations, and these new models offer a different perspective to the old topic of elastic response of materials. In particular, a sub-class of implicit constitutive relations, namely relat…
Stress Softening Damage in Strongly Nonlinear Viscoelastic Soft Materials A Physics Informed Data Driven Constitutive Model with Time Temperature Coupling
Characterizing the softness of deformable materials having partial elastic and partial viscous behaviour via soft lubrication experiments has emerged as a versatile and robust methodology in recent times. However, a straightforward extension of the classical elastohydrodynamic lubrication theory that is commonly employed for characterizing elastic materials turns out to be rather inadequate in explaining the response of such viscoelastic materials subjected to dynamic loading conditions, despite adhering to a mathematically acceptable framework via the complex Young's modulus as a material pro…
Experiments have shown that shear waves induced in brain tissue can develop into shock waves, thus providing a possible explanation of deep traumatic brain injuries. Here, we study the formation of shock waves in soft viscoelastic solids subject to an imposed velocity at their boundary. We consider the plane shearing motion of a semi-infinite half-space, which corresponds to a spatially one-dimensional problem. Incompressible soft solids whose behaviour is described by the Fung-Simo quasi-linear viscoelasticy theory (QLV) are considered, where the elastic response is either exponential or poly…
We study the effect of viscoelastic dynamics on the frictional properties of a (mean field) spring-block system pulled on a rough surface by an external drive. When the drive moves at constant velocity V, two dynamical regimes are observed: at fast driving, above a critical threshold Vc, the system slides at the drive velocity and displays a friction force with velocity weakening. Below Vc the steady sliding becomes unstable and a stick-slip regime sets in. In the slide-hold-slide driving protocol, a peak of the friction force appears after the hold time and its amplitude increases with the ho…
Minimally invasive experimental methods that can measure local rate dependent mechanical properties are essential in understanding the behaviour of soft and biological materials in a wide range of applications. Needle based measurement techniques such as Cavitation Rheology and Volume Controlled Cavity Expansion (VCCE), allow for minimally invasive local mechanical testing, but have been limited to measuring the elastic material properties. Here, we propose several enhancements to the VCCE technique to adapt it for characterization of viscoelastic response at low to medium stretch rates ($10^{…
We use molecular dynamics simulations of a model three-dimensional particulate gel, to investigate the linear viscoelastic response. The numerical simulations are combined with a novel test protocol (the optimally- windowed chirp or OWCh), in which a continuous exponentially-varying frequency sweep windowed by a tapered cosine function is applied. The mechanical response of the gel is then analyzed in the Fourier domain. We show that i) OWCh leads to an accurate computation of the full frequency spectrum at a rate significantly faster than with the traditional discrete frequency sweeps, and wi…
Viscoelasticity has been described since the time of Maxwell as an interpolation of purely viscous and purely elastic response, but its microscopic atomic-level mechanism in solids has remained elusive. We studied three model disordered solids: a random lattice, the bond-depleted fcc lattice, and the fcc lattice with vacancies. Within the harmonic approximation for central-force lattices, we applied sum-rules for viscoelastic response derived on the basis of non-affine atomic motions. The latter motions are a direct result of local structural disorder, and in particular, of the lack of inversi…
Epithelial tissues act as barriers and, therefore, must repair themselves, respond to environmental changes and grow without compromising their integrity. Consequently, they exhibit complex viscoelastic rheological behavior where constituent cells actively tune their mechanical properties to change the overall response of the tissue, e.g., from solid-like to fluid-like. Mesoscopic mechanical properties of epithelia are commonly modeled with the vertex model. While previous studies have predominantly focused on the rheological properties of the vertex model at long time scales, we systematicall…
We study the sliding friction for viscoelastic solids, e.g., rubber, on hard flat substrate surfaces. We consider first the fluctuating shear stress inside a viscoelastic solid which results from the thermal motion of the atoms or molecules in the solid. At the nanoscale the thermal fluctuations are very strong and give rise to stress fluctuations in the MPa-range, which is similar to the depinning stresses which typically occur at solid-rubber interfaces, indicating the crucial importance of thermal fluctuations for rubber friction on smooth surfaces. We develop a detailed model which takes i…
The functionality of adhesives relies on their response under the application of a load. Yet, it has remained a challenge to quantitatively relate the macroscopic dynamics of peeling to the dissipative processes inside the adhesive layer. Here we investigate the peeling of a reversible adhesive made of a polymer gel, measuring the relationship between the peeling force, the peeling velocity, and the geometry of the interface at small-scale. Experiments are compared to a theory based on the linear viscoelastic response of the adhesive, augmented with an elastocapillary regularization approach. …
We determine the linear viscoelastic response of jammed packings of athermal repulsive viscous spheres, a model for emulsions, wet foams, and soft colloidal suspensions. We numerically measure the complex shear modulus, a fundamental characterization of the response, and demonstrate that low frequency response displays dynamic critical scaling near unjamming. Viscoelastic shear response is governed by the relaxational eigenmodes of a packing. We use scaling arguments to explain the distribution of eigenrates, which develops a divergence at unjamming. We then derive the critical exponents chara…
A predicting model for soft Dielectric Elastomer Generators (DEGs) must consider a realistic model of the electromechanical behaviour of the elastomer filling, the variable capacitor and of the electrical circuit connecting all elements of the device. In this paper such an objective is achieved by proposing a complete framework for reliable simulations of soft energy harvesters. In particular, a simple electrical circuit is realised by connecting the capacitor, stretched periodically by a source of mechanical work, in parallel with a battery through a diode and with an electrical load consumin…
The steady flow of three viscoelastic fluids (Oldroyd-B, FENE-P, and Owens model for blood) in a two-dimensional channel, partly bound by a deformable, finite thickness neo-Hookean solid, is computed. The limiting Weissenberg number beyond which computations fail to converge is found to increase with increasing dimensionless solid elasticity parameter Γ, following the trend Owens > FENE- P > Oldroyd-B. The highly shear thinning nature of Owens model leads to the elastic solid always collapsing into the channel, for the wide range of values of Γ considered here. In the case of the FENE-P and Ol…
Needs to impart appropriate elasticity and high toughness to viscoelastic polymer materials are ubiquitous in industries such as concerning automobiles and medical devices. One of the major problems to overcome for toughening is catastrophic failure linked to a velocity jump, i.e., a sharp transition in the velocity of crack propagation occurred in a narrow range of the applied load. However, its physical origin has remained an enigma despite previous studies over 35 years. Here, we propose an exactly solvable model that exhibits the velocity jump incorporating linear viscoelasticity with a cu…
