Viscoelasticity

Why viscoelasticity matters

An ideal elastic solid obeys Hooke's law: stress is proportional to strain, the response is instantaneous, and every joule you put in comes back out. An ideal viscous fluid obeys Newton's law: stress is proportional to strain rate, and every joule is dissipated as heat. Real polymers, rubbers, gels, asphalt, and living tissue sit between these two idealizations. Their molecules are long chains that need time to slide, uncoil, and rearrange, so the material's stiffness is not a single number — it depends on the timescale of loading. This single fact reshapes how you design with these materials.

• Plastic parts that sag. A polypropylene shelf loaded at 20 % of its short-term yield stress can deflect two to three times its initial deflection over years — pure creep. Load-bearing plastic parts are sized against a creep modulus at the service life, not the tensile modulus from a fast test.
• Bolted joints and gaskets. A PTFE or elastomer gasket loses clamp force by stress relaxation; the bolt preload bleeds away and the joint leaks. Relaxation, not creep, is the failure mode here.
• Tires and rolling resistance. Every revolution, tread rubber is loaded and unloaded, and the hysteresis loop's area becomes heat. This rolling hysteresis is the dominant contributor to rolling resistance and a major fuel-economy lever; "silica" tread compounds are tuned to lower tan δ at rolling frequencies while keeping it high at braking frequencies.
• Damping and isolation. Engine mounts, seismic bearings, and constrained-layer damping tapes exploit high loss modulus to convert vibration into heat. The whole point is to dissipate energy in the hysteresis loop.
• Biomechanics. Tendon, ligament, cartilage, skin, and arterial wall are viscoelastic. They stiffen under fast impact (protecting against sudden loads) and relax under sustained posture. Cartilage relies on poroviscoelastic fluid flow for lubrication and load sharing.
• Asphalt and coatings. Pavement rutting is high-temperature, slow-load creep; low-temperature cracking is fast-load brittleness — the same binder, two ends of the time–temperature map.

How it works, step by step

The engineer's toolkit is a small alphabet of springs and dashpots. A spring of modulus E gives stress σ = Eε (Hooke). A dashpot of viscosity η gives stress σ = η dε/dt (Newton). Connecting them in series or parallel builds every classic model.

1. Maxwell model — spring in series with dashpot. Both elements carry the same stress; strains add. Hold the strain fixed and the stress decays exponentially,
   σ(t) = σ₀ e^−t/τ, with relaxation time τ = η / E.
   This captures stress relaxation and viscoelastic fluids: under constant load the dashpot flows without bound, so the material never stops creeping.
2. Kelvin–Voigt model — spring in parallel with dashpot. Both elements share the same strain; stresses add. Apply a constant stress σ₀ and the strain climbs toward a finite limit,
   ε(t) = (σ₀ / E)(1 − e^−t/τ), with retardation time τ = η / E.
   This captures creep and viscoelastic solids, but it cannot show instantaneous elastic jump or true relaxation.
3. Standard Linear Solid (Zener model). A spring in parallel with a Maxwell arm (three elements). It shows an instant elastic response, bounded creep to an equilibrium strain, and stress relaxation to a finite residual stress — a realistic minimum for a viscoelastic solid.
4. Generalized model / Prony series. Real materials relax over many decades of time, so a single τ is never enough. A Prony series sums many Maxwell arms:
   E(t) = E_∞ + Σ_i E_i e^−t/τ_i.
   Finite-element codes fit measured relaxation data to a Prony series to simulate creep, relaxation, and damping.
5. Frequency domain. Drive the material with a sinusoidal strain ε = ε₀ sin(ωt) and the stress comes out phase-shifted by δ. Splitting it gives the complex modulus
   E* = E′ + iE″, where E′ = (σ₀/ε₀) cos δ is the storage modulus (elastic, in-phase) and E″ = (σ₀/ε₀) sin δ is the loss modulus (viscous, out-of-phase). Their ratio is the damping factor tan δ = E″/E′.
6. Energy loss per cycle. The hysteresis loop's enclosed area is the dissipated energy density,
   ΔW = π E″ ε₀² (per unit volume, per cycle), and the fractional loss is ΔW / W_stored = 2π tan δ.
7. Temperature and rate. Because relaxation is molecular motion, raising temperature is equivalent to lengthening the timescale. Time–temperature superposition shifts curves along log-time by a factor a_T given, above the glass transition, by the WLF equation:
   log a_T = −C₁(T − T₀) / (C₂ + T − T₀), with "universal" constants C₁ ≈ 17.4 and C₂ ≈ 51.6 K when T₀ = T_g.

Symbol and unit reference

Symbol	Meaning	SI units
σ	Stress	Pa (N/m²)
ε	Strain	dimensionless (m/m)
E	Spring (elastic) modulus	Pa
η	Dashpot viscosity	Pa·s
τ = η/E	Relaxation / retardation time	s
J(t)	Creep compliance (strain/stress)	Pa⁻¹
E(t)	Relaxation modulus (stress/strain)	Pa
E′, E″	Storage, loss modulus	Pa
tan δ	Damping / loss factor	dimensionless
aT	WLF time–temperature shift factor	dimensionless

Maxwell vs. Kelvin–Voigt vs. Standard Linear Solid

The three basic linear models each capture part of reality. Choose the one whose behavior matches your loading history.

Behavior	Maxwell (series)	Kelvin–Voigt (parallel)	Standard Linear Solid
Instantaneous elastic jump	Yes (spring)	No	Yes
Creep under constant σ	Unbounded (flows forever)	Bounded → σ/E	Bounded to equilibrium
Stress relaxation under constant ε	Full decay to 0	None (no relaxation)	Decay to residual stress
Represents a	Viscoelastic fluid	Viscoelastic solid	Realistic solid
Elements	2 (in series)	2 (in parallel)	3
Best for	Relaxation, flow, polymer melts	Creep, filled elastomers	General solids, tendon, gasket

Representative material properties

Order-of-magnitude values at room temperature and ~1 Hz. Storage modulus and tan δ move strongly with temperature and frequency, so treat these as anchors, not spec-sheet numbers.

Material	Storage modulus E′ (≈1 Hz, 25 °C)	tan δ	Notes
Natural rubber (vulcanized)	1–3 MPa	0.03–0.1	Low loss — bounces; tires tune this
Butyl rubber (IIR)	1–5 MPa	0.3–1.0	High loss — damping mounts, vibration
Polypropylene (semicrystalline)	1.3–1.6 GPa	0.03–0.06	Creeps notably above 20 % yield
Polycarbonate (glassy)	2.2–2.4 GPa	~0.01–0.02	Stiff, tough, low damping at RT
PMMA near Tg (~105 °C)	drops 10²–10³×	peaks ~1–2	Glass transition tan δ peak
Human tendon (collagen)	~1 GPa (fast)	0.05–0.15	Stiffens with strain rate
Articular cartilage	0.5–1 MPa (equilib.)	rate-dependent	Poroviscoelastic, fluid-driven
Bitumen / asphalt binder	10⁴–10⁹ Pa	0.5 → >5	Spans fluid-to-glass over T range

Worked example: relaxation of a bolted elastomer gasket

A rubber O-ring is compressed to a fixed strain to seal a flange (constant strain — a relaxation problem). Model it as a Maxwell element with instantaneous modulus E = 3 MPa and viscosity η = 3 × 10⁸ Pa·s. The relaxation time is

τ = η / E = 3 × 10⁸ Pa·s ÷ 3 × 10⁶ Pa = 100 s.

The sealing stress decays as σ(t) = σ₀ e^−t/τ. After one relaxation time (t = τ = 100 s) the stress is σ₀ e⁻¹ = 0.368 σ₀ — the seal has already lost 63 % of its contact pressure. After t = 3τ = 300 s it is σ₀ e⁻³ = 0.05 σ₀, only 5 % left. A pure Maxwell element relaxes to zero, which is exactly why real gaskets are chosen and modeled as a Standard Linear Solid: the parallel equilibrium spring holds a residual sealing stress σ_∞ so the joint does not leak. The design lesson: specify the elastomer and preload against the residual stress at end-of-life, and re-torque critical joints.

Now the dynamic side. Suppose the same rubber, sealing a pump housing, sees a cyclic strain of amplitude ε₀ = 0.02 at a frequency where E″ = 0.3 MPa. The energy dissipated per unit volume per cycle is

ΔW = π E″ ε₀² = π × (3 × 10⁵ Pa) × (0.02)² ≈ 377 J/m³ per cycle.

At 50 Hz that is ~1.9 × 10⁴ W/m³ of heat generated inside the rubber — enough to raise its temperature, which lowers its modulus, which changes E″, a self-heating feedback that governs the fatigue life of dynamically loaded elastomers.

Common misconceptions and failure modes

• "The modulus is one number." For a viscoelastic material stiffness depends on rate, time, and temperature. Quote a creep modulus at the service time, or E′ at the service frequency and temperature.
• "Creep and relaxation are different phenomena." They are two faces of the same time dependence — one holds stress and watches strain, the other holds strain and watches stress. A material that creeps also relaxes.
• "Kelvin–Voigt can model a gasket." Kelvin–Voigt shows no stress relaxation and no instantaneous elastic response, so it cannot describe a losing-preload seal. Use at least a Standard Linear Solid.
• "Faster loading is safer." Faster loading makes a polymer stiffer and more brittle. A part that flexes safely when bent slowly can shatter under impact — the same reason cold plastics crack.
• "Hysteresis is always bad." Loop area is wasted heat in a tire but the entire mechanism of a damper. Whether you want high or low tan δ depends on the job.
• "Elastic recovery is instantaneous." Set a viscoelastic part under load for a long time and it shows delayed recovery and permanent set; the strain does not spring back at once when released.
• "Room-temperature data is enough." Near the glass transition, E′ can fall by two to three orders of magnitude and tan δ spikes. A part rated at 25 °C can lose most of its stiffness at 60 °C if T_g is nearby.

Measuring viscoelasticity

Three complementary tests map the behavior. A creep test applies constant stress and records strain versus log-time to give creep compliance J(t). A relaxation test applies constant strain and records the decaying stress to give relaxation modulus E(t). A dynamic mechanical analysis (DMA) test applies a small oscillating strain across frequency and temperature sweeps and records amplitude ratio and phase lag to give E′, E″, and tan δ directly. DMA temperature sweeps are the standard way to locate the glass transition — seen as a steep drop in E′ and a sharp peak in tan δ — and are governed by standards such as ASTM D4065 and ASTM E1640. Time–temperature superposition then stitches short high-temperature runs into a master curve spanning many decades of time you could never wait out directly.