Standard Linear Solid: Creep & Stress Relaxation

Test Type

Model Parameters

Equilibrium spring E_∞ 2.0 MPa

Maxwell-arm spring E₁ 4.0 MPa

Dashpot viscosity η 3.0 MPa·s

Step magnitude 50%

Simulation speed 1.0×

Response

Strain ε(t)

Strain ε

0.0%

Stress σ

0.00 MPa

Time t

0.00 s

How it works

This is the Standard Linear Solid (Zener model): a spring of modulus E_∞ in parallel with a Maxwell arm — a spring E₁ in series with a dashpot of viscosity η. It is the simplest lumped-element model that reproduces both creep and stress relaxation.

Governing ODE (total strain ε, total stress σ):
σ + (η/E1)·σ̇ = E∞ε + η(1+E∞/E1)·ε̇

Creep (σ = σ0, step at t=0):
ε(t) = σ0/E∞ - [σ0/E∞ - σ0/(E∞+E1)]·exp(-t/τc)
τc = η(E∞+E1) / (E∞·E1)

Relaxation (ε = ε0, step at t=0):
σ(t) = E∞ε0 + E1ε0·exp(-t/τr)
τr = η / E1

Creep — a constant stress is switched on; the spring E_∞ plus spring E₁ respond instantly, then the dashpot slowly lets the Maxwell arm creep further until strain saturates at σ0/E∞.
Relaxation — a constant strain is switched on and held; stress jumps to (E∞+E1)ε0 instantly, then decays toward the equilibrium value E∞ε0 as the dashpot bleeds off the load in the Maxwell arm.
The 3D model shows two real branches between the platens: a plain helical spring (E_∞) and a spring-in-series-with-dashpot arm (E₁, η) — the node between them moves according to the true internal stress split σ1 = σ − E∞ε.
This exact model describes polymers, biological tissue, asphalt and many viscoelastic solids under sustained load or held deformation.

Equilibrium spring E∞ Maxwell arm (E1 + dashpot η)

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