Creep and stress relaxation: the Standard Linear Solid model

Time is a material axis

Polymer chains rearrange under load, so stiffness is a function of time: hold a polymer at constant stress and it creeps; hold it at constant strain and the stress relaxes. Both are signatures of the same underlying relaxation spectrum.

Spring–dashpot models make this concrete. A spring stores energy instantly; a dashpot dissipates it at a rate set by viscosity η. Their arrangements generate the canonical responses: in series (Maxwell) stress relaxes fully and creep never stops — a fluid with elasticity; in parallel (Kelvin–Voigt) creep is bounded but there is no instantaneous elasticity — a delayed solid.

The Standard Linear Solid

Real solid polymers show both an instantaneous modulus and a bounded creep. The Standard Linear Solid (Zener) model — a spring in parallel with a Maxwell arm — is the simplest model that captures this: relaxation decays from the glassy modulus E∞+E₁ to a rubbery plateau E∞ with time constant τ = η/E₁, and creep retards toward 1/E∞ with a longer retardation time.

One relaxation time is still a caricature: real polymers relax over many decades, described by a spectrum of times (Prony series) and shifted by temperature via time–temperature superposition. Those generalizations are on the roadmap; the single-τ models remain the right way to build intuition.

The equations, and the two time constants that are not the same

For the Standard Linear Solid — a spring E∞ in parallel with a Maxwell arm (spring E₁, dashpot η) — relaxation under a held strain ε₀ is σ(t) = ε₀·(E∞ + E₁·e^(−t/τ_σ)), with the relaxation time τ_σ = η/E₁. The modulus falls from a glassy E∞ + E₁ to a rubbery plateau E∞.

Creep under a held stress σ₀ uses the compliance J(t) = 1/E∞ − [E₁/(E∞(E∞ + E₁))]·e^(−t/τ_ε), and here is the part that catches people out: the retardation time τ_ε = η(E∞ + E₁)/(E∞·E₁) is not the same as τ_σ.

Divide one by the other and the algebra is clean: τ_ε/τ_σ = (E∞ + E₁)/E∞ — the ratio of the glassy modulus to the rubbery one. A material with a 3:1 modulus drop creeps three times more slowly than it relaxes. There is no single characteristic time for a viscoelastic material; there is one for each experiment, and quoting τ without saying which you measured is meaningless.

A worked example: a 3:1 modulus drop

Take E∞ = 1.0 GPa, E₁ = 2.0 GPa and η = 200 GPa·s. That gives a glassy modulus of 3.0 GPa falling to a rubbery 1.0 GPa, a relaxation time τ_σ = 200/2 = 100 s, and a retardation time τ_ε = 200 × 3/(1 × 2) = 300 s. Every number below comes from the simulation engine and matches the closed-form solution to five figures.

Relaxation, holding ε₀ = 0.5%: stress starts at 15.00 MPa (a modulus of 3.00 GPa), falls to 8.76 MPa after ~98 s, 5.51 MPa by ~299 s, and 5.03 MPa at 600 s — a modulus of 1.005 GPa, essentially the rubbery plateau after six time constants.

Creep, holding σ₀ = 15 MPa: strain starts at 0.50% — the same glassy 3.0 GPa response — then grows to 1.13% by 300 s, 1.45% by ~912 s and 1.50% at 1800 s, converging on σ₀/E∞ = 15/1000 = 1.5%.

Note the timescales. Relaxation is essentially finished by 600 s; creep needs 1800 s to reach the same fraction of its final value. Same material, same model, same η — three times slower, exactly as (E∞ + E₁)/E∞ predicts.

Design consequences

Creep governs long-term deflection of polymer parts under sustained load — snap fits relax, bolted joints lose preload, pressurized pipes strain toward failure. Design rules use creep modulus at the design lifetime rather than the short-term datasheet modulus, which can overpredict long-term stiffness several-fold near or above Tg.

The example above puts a number on that. The datasheet would quote 3.0 GPa, the glassy response you measure in a fast test. Divide the applied 15 MPa by the strain actually reached at 1800 s and the effective creep modulus is 1.00 GPa. Size that part on the datasheet number and it deflects three times further than you predicted — and 1800 s is half an hour, not a service life. This is the single most common way polymer parts disappoint in service.