Viscoelasticity: Materials That Are Both Solid and Liquid

Neither a solid nor a liquid

An ideal elastic solid (a spring) stores energy and springs back instantly; an ideal viscous liquid (a dashpot) simply resists the rate of deformation and never springs back at all. Almost every real material — rubber, dough, biological tissue, molten polymer, silly putty — sits between the two, storing some energy elastically and dissipating some viscously depending on how fast you deform it. Viscoelasticity is the framework for describing that in-between behaviour, and the standard trick is to build it out of springs and dashpots combined in different arrangements.

Maxwell, Kelvin-Voigt, and the Standard Linear Solid

The Maxwell model puts a spring and a dashpot in series: under a constant strain the spring's stress relaxes away exponentially as the dashpot slowly creeps to relieve it, so it behaves like a solid on short timescales and a liquid on long ones — good for describing stress relaxation but it never fully recovers its original shape after flowing. The Kelvin–Voigt model puts them in parallel instead: applying a constant stress makes the material creep toward an equilibrium strain and then hold it, capturing delayed elastic recovery but not stress relaxation. The Standard Linear Solid (a Maxwell arm and a lone spring in parallel) combines both behaviours and is the simplest model that gets stress relaxation, creep, and full recovery all qualitatively right.

Maxwell (series):        stress relaxation, σ(t) = σ₀·e^(−t/τ),  τ = η/G₀
Kelvin-Voigt (parallel):  creep toward equilibrium strain, no relaxation
Standard Linear Solid:    Maxwell arm ∥ spring — relaxation AND recovery

τᴿ = relaxation time (η/G₀) sets the boundary between
"acts solid" (t ≪ τᴿ) and "acts liquid" (t ≫ τᴿ)

Two experiments that separate the models

Stress relaxation: hold the strain fixed and watch the stress. A pure elastic spring holds it forever; Maxwell and the SLS let it decay toward a plateau (zero for Maxwell, a finite value for the SLS) with a characteristic time constant τᴿ = η/G₀ — the ratio of viscosity to stiffness. Creep: hold the stress fixed and watch the strain grow. Kelvin-Voigt and the SLS approach a finite equilibrium strain asymptotically; pure Maxwell creeps forever at a constant rate because the dashpot never stops extending under constant load.

Oscillating instead: the complex modulus

Apply a small sinusoidal strain instead of a step, and the resulting stress oscillates at the same frequency but shifted in phase — in phase with strain for a pure elastic solid, 90° out of phase for a pure viscous liquid, somewhere in between for a viscoelastic material. Splitting the response into its storage modulus G′(ω) (the in-phase, elastic part) and loss modulus G″(ω) (the out-of-phase, viscous part) shows the same solid/liquid crossover as a function of frequency instead of time: at low frequency a Maxwell-type material behaves like a liquid (G″ dominates), and at high frequency it behaves like a solid (G′ dominates), crossing over near ω ≈ 1/τᴿ.

Why this matters beyond the lab

Rheology built on exactly this spring-dashpot bookkeeping governs how bread dough is kneaded and proofed, how asphalt resists rutting under a hot afternoon of traffic but stays rigid overnight, how biological tissue like cartilage cushions impact, and how 3D-printing polymers need to be fluid enough to extrude yet solid enough to hold their shape the instant they are deposited — all of it explained by where a material sits on the same solid-to-liquid spectrum this simulation lets you drag through directly.

Frequently asked questions

Try it live

Everything above runs in your browser — open Viscoelastic Fluid and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.