Kelvin–Voigt Model Calculator

Professional article

1) Why the Kelvin–Voigt model matters

Many solids show both instantaneous elasticity and time-dependent deformation. The Kelvin–Voigt model combines a spring and dashpot in parallel to capture creep under sustained load. In practice, it helps interpret short-duration loading tests, polymer fixtures, damping layers, and soft materials where strain rises gradually toward a steady value.

2) Parameters and typical magnitudes

The elastic modulus E controls the long-time stiffness, while viscosity η sets the rate of deformation. For many polymers, E can range from tens of MPa to several GPa, and η can span from 10² to 10⁸ Pa·s depending on temperature and formulation.

3) The characteristic time constant τ

The model has a single time constant τ = η/E. At t = τ, creep strain reaches about 63% of its final value. If τ is 0.2 s, deformation settles quickly; if τ is 200 s, the same load produces a much slower approach to steady strain.

4) Creep prediction under step stress

Under a step stress σ0, strain follows ε(t) = (σ0/E)(1 − e^{−t/τ}). This calculator can generate a time series to visualize the curve. For example, with E = 5 MPa and η = 2000 Pa·s, τ = 0.4 s, so steady strain is approached within a few seconds.

5) Stress from strain and strain-rate

When strain is prescribed, the constitutive law σ = Eε + η dε/dt separates elastic and viscous contributions. A rapid ramp increases dε/dt and raises stress beyond the purely elastic value. This is useful for estimating load spikes during fast actuation or impact-like deformation in viscoelastic components.

6) Dynamic response and phase lag

In harmonic tests, E* = E + iηω, so storage modulus is constant (E′=E) while loss modulus grows linearly with frequency (E″=ηω). The phase lag δ = arctan(E″/E′) quantifies damping. Small δ implies stiff response; larger δ implies stronger energy dissipation per cycle.

7) Energy loss per cycle

For sinusoidal strain amplitude ε0, dissipated energy density per cycle is W_d = πE″ε0². Because E″ scales with ω, increasing frequency raises damping losses even when E stays constant. This is important for vibration isolation, acoustic liners, and viscoelastic adhesives under cyclic loading.

8) Practical interpretation and limits

The Kelvin–Voigt model is best for small strains and materials with a single dominant relaxation timescale. It predicts creep well but does not reproduce stress relaxation under an ideal step strain. For broader behavior, engineers may use generalized models with multiple springs and dashpots fit to experimental data across temperature and frequency.

FAQs

1) What does this model represent physically?

It represents a spring and dashpot acting together, so the material resists deformation elastically and viscously at the same time. This captures gradual creep under constant stress.

2) Why is τ = η/E important?

τ sets how quickly creep approaches its steady value. At t = τ, strain reaches about 63% of the final strain for a step stress, so τ provides an immediate “speed” measure.

3) Can it model stress relaxation after a step strain?

Not well. The Kelvin–Voigt form mainly models creep. For stress relaxation, a Maxwell-type model or a generalized viscoelastic network is typically more appropriate.

4) How do I choose units safely?

Use consistent stress units for E and σ0, and keep η in Pa·s (or scaled versions). The calculator converts internally to SI, so mixed units are fine if selected correctly.

5) What does tan δ mean in dynamic mode?

tan δ = E″/E′ is the damping ratio indicator for harmonic loading. Larger tan δ means stronger viscous losses relative to stored elastic energy at that frequency.

6) Why does E″ increase with frequency?

Because E″ = ηω in this model. Higher frequency means faster deformation rate, so viscous stress rises proportionally, increasing the loss modulus and energy dissipation.

7) What input data works best for fitting E and η?

Combine a creep test to estimate τ and steady strain (σ0/E), then compute η = τE. Dynamic tests across frequencies can validate the fitted parameters using E″ trends.