Creep and Viscoelasticity

Why it matters

Time-dependent deformation matters in polymers and in metals operating at sufficiently high temperatures. Creep is increasing strain under sustained stress; stress relaxation is decreasing stress under sustained strain. Their constitutive models and initial conditions differ.

Creep and recovery

Classical metal creep may show primary (decreasing rate), secondary (approximately constant rate), and tertiary (accelerating damage) stages. Viscoelastic creep can include recoverable delayed strain, so “creep” does not always mean wholly permanent deformation.

Maxwell model: spring and dashpot in series

The elements share stress and their strains add. With spring modulus E and dashpot viscosity η:

ε̇ = σ̇/E + σ/η.

For an initially unloaded model subjected to a stress step σ₀ at t = 0, the spring jumps immediately to strain σ₀/E. For t > 0, ε(t) = σ₀/E + σ₀t/η. At fixed strain, stress relaxes exponentially with time constant η/E. This ideal model creeps indefinitely and does not reproduce every real material.

Kelvin–Voigt model: parallel elements

The elements share strain and their stresses add:

σ = Eε + ηε̇.

Under a constant stress applied at t = 0 with initial strain zero, ε(t) = (σ₀/E)[1 − exp(−Et/η)]. There is no instantaneous strain jump. This equation must not be labelled as the Maxwell model.

See Tampere University material-model lecture notes for spring–dashpot constitutive models.

Worked example

An illustrative Maxwell model has E = 200 GPa and η = 10¹⁴ Pa·s. A 50 MPa tensile stress step is applied at t = 0. Find the strain after 1000 s.

Elastic strain = 50 × 10⁶/(200 × 10⁹) = 0.00025.

Viscous strain = (50 × 10⁶)(1000)/10¹⁴ = 0.0005.

Total strain = 0.00075, or 0.075%. It is dimensionless. These are specified model parameters, not a validated creep law for steel at an unspecified temperature.

Common mistakes

• Swapping the series Maxwell and parallel Kelvin–Voigt equations.
• Setting total strain to zero immediately after a stress step in a Maxwell spring.
• Treating a dashpot viscosity as independent of temperature without justification.
• Accepting enormous computed strain while continuing to assume small-strain linear behavior.

Quick check

1. What remains fixed in a creep test? Stress or load, as specified.
2. What remains fixed in a relaxation test? Strain.
3. What are the units of η? Pa·s.