# Viscoelasticity **Viscoelasticity** is the property of a material that responds to deformation both as an elastic solid and as a viscous liquid. A perfect spring stores every joule put into it and gives it back; a perfect dashpot dissipates every joule as [[Heat|heat]]. A viscoelastic material does both at once, so its stress depends on the whole history of strain rather than on the strain alone: hold it at fixed strain and the stress decays, hold it at fixed stress and the strain grows. [[Polymer|Polymers]] above their [[Glass_transition|glass transition]], molten glass, wood, asphalt, [[Bone|bone]] and living tissue are all viscoelastic, and so is any metal held long enough and hot enough to [[Creep_(deformation)|creep]]. In the microsim below the reader runs a [[Maxwell_model|Maxwell element]] (a spring and a dashpot in series) and a [[Kelvin–Voigt_material|Kelvin–Voigt element]] (the same two in parallel) side by side under one load. A toggle chooses the experiment: under a step of strain the Maxwell element relaxes as `sigma(t) = sigma_0·exp(-t/tau)`, while under a step of stress the Kelvin–Voigt element creeps as `eps(t) = (sigma_0/E)·(1 - exp(-t/tau))`. The control is the relaxation time `tau = eta/E`, which also moves when the reader slides [[Temperature|temperature]] through an ILLUSTRATIVE [[Time–temperature_superposition|time–temperature]] shift, and the loss tangent is read out at 1 Hz. On the [[Materials_science|Materials science]] flagship this article serves the *Viscoelasticity* section of Part VIII, Industry, as the time-dependent half of the pair whose equilibrium half is [[Rubber_elasticity|rubber elasticity]]. ## Background The two limiting idealizations are old. Hooke's spring, `sigma = E·eps`, and Newton's viscous liquid, `sigma = eta·(d eps/dt)`, describe materials whose stress depends on strain alone or on strain rate alone. The combination came from the kinetic theory of gases: in 1867 James Clerk Maxwell, treating a gas as a medium with an elasticity that decays, wrote the series spring-and-dashpot equation that still carries his name.[^maxwell1867][^ferry-models] Ludwig Boltzmann generalized it in 1874 into the superposition principle underlying all of linear viscoelasticity — the stress now is the sum of the decaying contributions of every past increment of strain.[^boltzmann1874][^ferry-linear] The subject became an engineering discipline with the synthetic polymers of the twentieth century, whose stiffness can change by three or four orders of magnitude across a few tens of degrees.[^callister-visco] John Ferry's *Viscoelastic Properties of Polymers* organized the measurements and the shifting rules that read decades of time out of hours of experiment.[^ferry-tts] ## Elastic versus viscoelastic behavior The distinction is a bookkeeping question about energy. Load a purely elastic bar and unload it: the two paths on the [[Stress–strain_curve|stress–strain curve]] lie on top of one another, and the work done is recovered. Load and unload a viscoelastic bar and the paths separate into a hysteresis loop whose enclosed area is the [[Energy|energy]] lost per cycle as heat. Bounce a steel ball and a rubber ball of the same stiffness and the rubber ball returns less of its kinetic energy, because part of the strain energy went into internal motion instead of back into the bounce. The same difference shows in time. An elastic solid reaches its final strain the instant the load arrives; a [[Viscosity|viscous]] liquid never reaches one at all. A viscoelastic solid does both in sequence — a fast, almost elastic response, then a slower approach to a finite equilibrium strain — while a viscoelastic liquid does the fast part and then flows without limit. The sim's two elements are that pair. The Maxwell element is the liquid: its dashpot passes strain along forever and the stress falls to zero. The Kelvin–Voigt element is the solid: its spring eventually carries the whole load, and creep stops at `eps_inf = sigma_0/E`. ## Linear viscoelasticity and nonlinear viscoelasticity A material is linearly viscoelastic when doubling the load doubles the response at every time, and when responses to loads applied at different times simply add. Both statements are Boltzmann's superposition principle, and together they mean one measured function — the relaxation modulus `E(t)`, or equivalently the creep compliance `J(t)` — predicts the response to any loading history.[^ferry-linear] Every model in the sim is linear in this sense. Linearity is an approximation that holds at small strain. Polymer melts leave it where the network deforms enough that its own relaxation times change; filled [[Elastomer|elastomers]] leave it earlier, through the softening of the filler network under repeated strain. Beyond that limit the relaxation modulus depends on strain as well as time, and a single `E(t)` no longer suffices. The nonlinear models below keep as much linear structure as possible while admitting that dependence. ## Dynamic modulus The most informative experiment on a viscoelastic material is a small sinusoidal strain at [[Angular_frequency|angular frequency]] *ω*. The stress is sinusoidal too, at the same frequency, but leads the strain by a phase angle *δ* between 0° (perfectly elastic) and 90° (perfectly viscous). Splitting it into a part in phase with the strain and a part in phase with the strain rate gives the storage modulus `E'` and the loss modulus `E''`; their ratio is the loss tangent, `tan(delta) = E''/E'`, the sim's headline readout. `E'` measures the energy stored and returned per cycle; `E''` the energy dissipated. The arrangement is the mechanical twin of a driven damped [[Harmonic_oscillator|harmonic oscillator]], whose displacement also lags the drive by an angle that decides the power delivered; *University Physics* develops that case, phase running from 0 through 90° at [[Resonance|resonance]] to 180°, in its chapter on [[Oscillation|oscillations]].[^openstax-up1-15-6][^openstax-up1-15-5] For a single relaxation time the loss tangent has the Debye form the sim plots, `tan(delta) = Delta·omega·tau/(1 + omega²·tau²)`, with `Delta` a dimensionless strength. It is symmetric in `log(omega·tau)` and peaks where `omega·tau = 1`. At the sim's 1 Hz, ω = 6.283 rad/s, so the peak falls at τ = 0.159 s; with Δ = 1 the readout is 0.500 there, 0.155 at τ = 1 s and 0.0626 at τ = 0.01 s (derived). Putting that peak at the frequency a part must damp is the design principle behind vibration mounts and [[Noise_control|noise-control]] layers. Real polymers have peaks several decades wide, because they have a spectrum of relaxation times rather than one, so the sim's single-time Debye curve is ILLUSTRATIVE. ## Constitutive models of linear viscoelasticity Linear models are built from two elements: a Hookean spring of modulus *E*, obeying `sigma = E·eps`,[^openstax-up1-12-3] and a Newtonian dashpot of viscosity *η*, obeying `sigma = eta·(d eps/dt)`. Every combination yields a linear differential equation between stress and strain whose solutions are sums of exponentials, and defines a relaxation time `tau = eta/E` from the pair it contains.[^ferry-models] ### Maxwell model Spring and dashpot in series. Both carry the same stress and their strains add, giving `d eps/dt = (1/E)·(d sigma/dt) + sigma/eta`. Under a step of strain the stress decays exactly as `sigma(t) = sigma_0·exp(-t/tau)`, falling to 36.8 % of its initial value in one relaxation time and 5.0 % in three (derived). Under a step of stress it gives an elastic jump and then steady flow at `sigma_0/eta` that never stops, so the Maxwell element is a liquid: right about relaxation, wrong about creep. ### Kelvin–Voigt model Spring and dashpot in parallel. Now the strains are equal and the stresses add, giving `sigma = E·eps + eta·(d eps/dt)`. Under a step of stress the strain rises as `eps(t) = (sigma_0/E)·(1 - exp(-t/tau))`, reaching 63.2 % of its final value in one relaxation time and 95.0 % in three (derived). Because the dashpot blocks any instantaneous strain, the model misses the elastic jump a real solid has and predicts an infinite stress spike under a step of strain. It is a solid: right about creep, wrong about relaxation. ### Standard linear solid model Adding a second spring in parallel with a Maxwell element — or in series with a Kelvin–Voigt element — repairs both defects at once. The [[Standard_linear_solid_model|standard linear solid]] has an unrelaxed modulus `E_u`, a relaxed modulus `E_r`, and one relaxation time between them; it relaxes from `E_u` to `E_r` instead of to zero, and creeps from a finite initial strain to a finite final one. It is the smallest model qualitatively right for a crosslinked solid, and its loss tangent is the Debye peak above, with `Delta` set by the ratio `E_u/E_r`. ### Jeffreys model A dashpot in parallel with a Maxwell element gives the Jeffreys model, the liquid-like counterpart of the standard linear solid. Its extra viscous channel keeps the material flowing at long times while preserving a finite instantaneous response. It is the usual minimum model for dilute polymer solutions, where the solvent supplies that dashpot. ### Burgers model A Maxwell element in series with a Kelvin–Voigt element gives the Burgers model, four parameters and two characteristic times. It reproduces the three regimes a creep test on asphalt or a polymer actually shows: an instantaneous elastic strain, a decaying transient, and a steady flow that lasts as long as the load. Its recovery is equally realistic — instant spring-back, slow anelastic recovery, permanent set. ### Generalized Maxwell model No real material has one relaxation time. The generalized Maxwell model puts many Maxwell elements in parallel, with a lone spring for the equilibrium modulus, so the relaxation modulus becomes `E(t) = E_inf + sum_i E_i·exp(-t/tau_i)`. With enough terms it fits any measured curve, and the fitted set `{E_i, tau_i}` is the discrete relaxation spectrum — the form finite-element codes use, and the reason real loss peaks are broad rather than Debye-sharp.[^ferry-models] ## Constitutive models for nonlinear viscoelasticity Nonlinear models keep Boltzmann's memory integral but make the response depend on how much the material has been deformed, not only on how long ago. They must also be written so the predicted stress does not depend on the observer's motion, which for a flowing material means replacing ordinary time derivatives with frame-indifferent ones. ### Second-order fluid The second-order fluid is the first correction to a Newtonian liquid in a slow-flow expansion. Two normal-stress coefficients join the viscosity, so it predicts the rod-climbing and die-swell effects a Newtonian model cannot produce at all, while staying valid only for slow flows. ### Upper-convected Maxwell model Replacing the time derivative in the Maxwell equation with the upper-convected derivative gives the simplest frame-indifferent viscoelastic liquid: constant viscosity, but a first normal-stress difference growing as the square of the shear rate, the elastic signature of a melt. ### Oldroyd-B model Adding a solvent viscosity to the upper-convected Maxwell model gives the Oldroyd-B model, a case of the general framework J. G. Oldroyd set out in 1950 for writing rheological equations of state in an invariant form.[^oldroyd1950] It is also the continuum limit of a dilute suspension of Hookean dumbbells, which keeps it the reference model in computational rheology despite its unbounded extensional viscosity. ### Wagner model The Wagner model is the linear memory integral multiplied by a damping function of the strain, fitted to data. It is the workhorse for melts in processing, where strains are large and a separable strain-and-time form is accurate enough.[^wagner1976] ### Prony series A Prony series is the discrete relaxation spectrum written as a fitting form, `E(t) = E_inf + sum_i E_i·exp(-t/tau_i)` — the generalized Maxwell model as numerics rather than diagram. Finite-element packages take viscoelastic data as Prony coefficients, usually one term per decade, shifted in temperature by the rule below.[^ferry-models] ## Effect of temperature Raising the temperature of a polymer does not change the shape of its viscoelastic response so much as slide it along the time axis. This is [[Time–temperature_superposition|time–temperature superposition]]: curves measured at different temperatures shift horizontally by a factor `a_T` onto a single master curve spanning far more decades than any one measurement.[^ferry-tts] A short hot test therefore stands in for a long test at service temperature, which is how decade-scale creep predictions are made in an afternoon. Above the glass transition the shift factor follows the Williams–Landel–Ferry equation, `log10(a_T) = -C1·(T - T0)/(C2 + T - T0)`, with reference temperature `T0`; referred to the glass transition temperature the constants take the often-quoted approximate values C1 = 17.44 and C2 = 51.6 K.[^wlf1955][^ferry-tts] The consequence is steep. Twenty kelvin above `T_g` the equation gives `log10(a_T) = -4.87`, a shift factor of 1.3 × 10⁻⁵, so a relaxation that took a day at `T_g` takes about a second (derived); thirty kelvin above, the factor is 3.9 × 10⁻⁷ (derived). That is the sim's temperature slider, marked ILLUSTRATIVE because it applies the universal constants rather than constants fitted to a named polymer. Far above `T_g` the WLF form gives way to an [[Arrhenius_equation|Arrhenius]] dependence with a constant [[Activation_energy|activation energy]], and below `T_g` the material falls out of equilibrium and ages, so no single shift rule applies.[^ferry-tts] ## Viscoelastic creep Creep is the experiment run at constant stress. The strain climbs along `eps(t) = sigma_0·J(t)`, with `J(t)` the creep compliance; for a crosslinked solid it approaches a limit, for a melt it grows without one. Engineers use the compliance directly: a [[Thermoplastic|thermoplastic]] bracket designed against the short-term modulus will sag over years, so design charts give an apparent modulus at the service life and temperature, read off a shifted master curve.[^ferry-tts][^callister-visco] Unloading separates the recoverable and unrecoverable parts. The instantaneous elastic strain returns at once, the delayed elastic strain over roughly the retardation time, and whatever flowed through the free dashpot stays as permanent set — the Burgers model's four elements, one per observable. The distinction matters for gaskets, where the requirement is not that the part keep its shape but that it keep its sealing force: that is relaxation, governed by `E(t)` rather than `J(t)`, and bolted [[Nylon|polymer]] joints lose preload the same way. The metals analogue is the same phenomenon at high homologous temperature, where a turbine blade under constant load extends over thousands of hours, controlled by diffusion rather than chain motion. ## Measurement Characterization means measuring a relaxation or creep curve in time, or the dynamic moduli over frequency, then shifting temperature to extend the range. ### Shear rheometry A rotational rheometer holds the sample between a cone and a plate or between parallel plates and either rotates one surface (steady or step shear, for relaxation and creep) or oscillates it through a small angle (dynamic testing, for `E'`, `E''` and `tan(delta)`). A temperature-controlled oven supplies the second axis, and a frequency sweep repeated at several temperatures is the raw material for a master curve. The instrument measures shear quantities `G'` and `G''`; the tensile moduli quoted above exceed them by a factor near 3 for an incompressible rubber, set by the [[Poisson's_ratio|Poisson's ratio]]. ### Extensional rheometry Processing flows are often extensional rather than shear — fibre spinning, film blowing, blow moulding — and for a nonlinear material extensional viscosity cannot be deduced from shear data. Extensional rheometers stretch a molten filament between counter-rotating drums or let one thin under its own surface tension; they are hard to run because constant strain rate means stretching exponentially fast. ### Other methods Dynamic mechanical analysis applies the oscillatory experiment to a solid bar in bending or tension and reports `tan(delta)` against temperature; the peak locates the glass transition and is the standard way to measure it mechanically.[^callister-visco] Indentation and impulse methods reach small or awkward samples, and ultrasonic measurement reaches megahertz frequencies where the same material behaves as glass — one `E(t)` seen through different windows, which is the practical content of linearity. ## See also - [[Maxwell_model]] - [[Kelvin–Voigt_material]] - [[Standard_linear_solid_model]] - [[Time–temperature_superposition]] - [[Creep_(deformation)]] - [[Rubber_elasticity]] - [[Glass_transition]] - [[Polymer]] ## References [^maxwell1867]: Maxwell, J. C. (1867). "On the dynamical theory of gases." *Philosophical Transactions of the Royal Society of London* 157: 49–88. (No DOI given here; page to pin against the journal record.) [^boltzmann1874]: Boltzmann, L. (1874). "Zur Theorie der elastischen Nachwirkung." *Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien*. (Volume, pages and DOI to pin against the academy record.) [^ferry-models]: Ferry, J. D. (1980). *Viscoelastic Properties of Polymers*, 3rd ed. New York: Wiley. Chapters on the phenomenological treatment of linear viscoelasticity: spring-and-dashpot models, the discrete relaxation spectrum and the generalized Maxwell form. Not a Portal Book; pages to pin. [^ferry-linear]: Ferry, J. D. (1980). *Viscoelastic Properties of Polymers*, 3rd ed. Chapters on the Boltzmann superposition principle and the equivalence of the relaxation modulus and the creep compliance in the linear range. Not a Portal Book; pages to pin. [^ferry-tts]: Ferry, J. D. (1980). *Viscoelastic Properties of Polymers*, 3rd ed. Chapters on the dependence of viscoelastic behaviour on temperature: reduced variables, the shift factor, master curves, and the WLF and Arrhenius regimes. Not a Portal Book; pages to pin. [^wlf1955]: Williams, M. L.; Landel, R. F.; Ferry, J. D. (1955). "The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids." *Journal of the American Chemical Society* 77 (14): 3701–3707. (No DOI given here; page to pin against the journal record.) The approximate universal constants C1 = 17.44 and C2 = 51.6 K are quoted here from Ferry's tabulation. [^oldroyd1950]: Oldroyd, J. G. (1950). "On the formulation of rheological equations of state." *Proceedings of the Royal Society A* 200. (Pages and DOI to pin against the journal record.) [^wagner1976]: Wagner, M. H. (1976). "Analysis of time-dependent non-linear stress-growth data for shear and elongational flow of a low-density branched polyethylene melt." *Rheologica Acta* 15. (Pages and DOI to pin against the journal record.) [^callister-visco]: Callister, W. D.; Rethwisch, D. G. (2010). *Materials Science and Engineering: An Introduction*, 8th ed. Hoboken: Wiley. Chapter 15, Characteristics, Applications and Processing of Polymers — the viscoelastic relaxation modulus, its temperature dependence, and dynamic mechanical measurement of the glass transition. Not a Portal Book; page to pin. [^openstax-up1-12-3]: Ling, S. J.; Sanny, J.; Moebs, W. (2016). *University Physics Volume 1*. OpenStax (Portal Book 077). Chapter 12, §12.3 Stress, Strain, and Elastic Modulus, chapter pp. 565–610 (section page to pin). https://openstax.org/books/university-physics-volume-1/pages/12-3-stress-strain-and-elastic-modulus [^openstax-up1-15-5]: Ling, S. J.; Sanny, J.; Moebs, W. (2016). *University Physics Volume 1*. OpenStax (Portal Book 077). Chapter 15, Oscillations, §15.5 Damped Oscillations, chapter pp. 721–762 (section page to pin). https://openstax.org/books/university-physics-volume-1/pages/15-5-damped-oscillations [^openstax-up1-15-6]: Ling, S. J.; Sanny, J.; Moebs, W. (2016). *University Physics Volume 1*. OpenStax (Portal Book 077). Chapter 15, Oscillations, §15.6 Forced Oscillations, chapter pp. 721–762 (section page to pin). https://openstax.org/books/university-physics-volume-1/pages/15-6-forced-oscillations ### Bibliography - Ling, S. J.; Sanny, J.; Moebs, W. (2016). *University Physics Volume 1*. OpenStax. Portal Book 077. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-1 - Ferry, J. D. (1980). *Viscoelastic Properties of Polymers*, 3rd ed. Wiley. Not a Portal Book. - Callister, W. D.; Rethwisch, D. G. (2010). *Materials Science and Engineering: An Introduction*, 8th ed. Wiley. 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