Simulate creep and stress relaxation of Maxwell, Kelvin-Voigt and SLS viscoelastic models. The spring-dashpot animation visualizes deformation, and storage modulus E' and tan δ vs frequency are computed in real time.
Model Selection
Material Parameters
Analysis Settings
Results
Relaxation time τ (s)
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E' @ ωτ=1
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E'' @ ωτ=1
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tanδ @ ωτ=1
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Creep (strain at constant stress) & Stress Relaxation (stress at constant strain)
Spring-Dashpot Model Animation — Creep Behavior
Animates model deformation under constant applied stress σ₀.
Dynamic Modulus E' and E'' vs Angular Frequency ω (log scale)
About the Viscoelasticity Simulator
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So, "viscoelasticity" means materials that have both elasticity and viscosity, right? Like rubber or plastic?
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Exactly! Simply put, it's a material that behaves like rubber — stretches quickly when pulled, but under a constant load over time, the deformation gradually increases. We represent it using a combination of a spring (instantaneous elasticity) and a dashpot (time-dependent viscosity). If you switch to the "Model Animation" tab, you can see the actual spring and dashpot moving under load.
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I don't understand the difference between Maxwell and Kelvin-Voigt. Both are a set of a spring and a dashpot, right?
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The difference is "series vs. parallel." Maxwell is in series: the spring deforms instantly, then the dashpot keeps flowing (creep continues indefinitely). Kelvin-Voigt is in parallel: the dashpot "delays" the spring's deformation, so creep eventually reaches an equilibrium value. Try both in the "Creep/Relaxation" tab. You'll clearly see Maxwell's strain increasing linearly, while KV saturates.
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The panel shows "relaxation time τ." What does it mean if this is large?
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Relaxation time τ = η/E is a measure of "how long it takes for stress to drop to about 37% (e⁻¹) of its initial value." A larger τ means slower relaxation over a longer time. For example, natural rubber has τ on the order of seconds to minutes, while metal creep can be hours or days. In the simulator, increasing the η slider increases τ, and you'll see the relaxation curve in the graph become gentler.
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In the "Dynamic Properties" tab, I see E' and E''. What does it mean that E' increases as frequency goes up?
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At high-speed vibrations (high frequency), the dashpot doesn't have time to flow, so only the spring works, making the material feel stiffer. At slow deformations (low frequency), the viscous component kicks in and it becomes softer. For tire rubber, during hard braking (high speed), it's stiffer for better braking force, while under a constant load over time (low speed), it deforms gradually. Switch to the SLS model and look at the "Dynamic Properties" tab — you'll see it converges to E∞ (equilibrium modulus) at low frequencies and to E at high frequencies.
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What are materials with a large tanδ used for?
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tanδ (loss tangent) is an indicator of how efficiently vibration energy is converted into heat. The larger it is, the better the damping and sound absorption performance. Specific applications include automotive mount parts (engine vibration isolation), anti-vibration rubber, and sound-absorbing materials. Conversely, for structural parts, you want a small tanδ to avoid heat generation and degradation under cyclic loading. Interestingly, tires have a trade-off: high tanδ is needed for wet grip, but low tanδ is needed to reduce rolling resistance (fuel efficiency).
tanδ reaches its maximum near $\omega\tau = 1$, where the excitation period is comparable to the relaxation time.
Frequently Asked Questions
It applies to all materials that exhibit time-dependent deformation behavior, such as polymer materials (plastics, rubber, elastomers), biological tissues (tendons, cartilage, blood vessels), asphalt, food gels, and clay. Even metals under high temperature and high stress can show viscoelastic (creep) behavior that cannot be ignored.
In the SLS model, if a load is applied for a long time, the deformation converges to a finite value. The elastic modulus at that convergence point is E∞ (equilibrium modulus). In the Maxwell model, flow continues indefinitely, so E∞ → 0, but in SLS, solid-like stiffness remains. By changing E∞ with the slider, you can see how the long-term creep behavior changes.
DMA is a test where a sinusoidal displacement is applied to a specimen, and the stress amplitude and phase difference are measured to determine E', E'', and tanδ. The horizontal axis ω in the "Dynamic Properties" tab corresponds to the DMA frequency (Hz × 2π), and the vertical axis corresponds to the DMA output values. By fitting the peak frequency and high-frequency asymptote of an actual measured DMA curve to the simulator's E and η, you can estimate model parameters.
Real polymer materials do not have a single relaxation time but a broad relaxation spectrum with multiple relaxation times superimposed. This is expressed in the form $G(t) = G_\infty + \sum_i G_i e^{-t/\tau_i}$, known as the Prony series. This tool uses the simplest model with one relaxation time, but it is useful as an introduction to conceptual understanding. In FEA (finite element method) software, Prony coefficients are often directly entered for material definitions.
For polymer materials, as temperature rises, the dashpot viscosity η decreases, and the relaxation time τ becomes shorter. This relationship is described by the "time-temperature superposition principle (WLF equation)," where long-term behavior at low temperatures is equivalent to short-term behavior at high temperatures. In other words, summer asphalt (high temperature, low τ) and winter asphalt (low temperature, high τ) are in different viscoelastic states.
In FEA software like Ansys or Abaqus, you input Prony series parameters (stiffness ratio gi and relaxation time τi) for material definitions. First, obtain E' and E'' data from DMA testing, then fit them to Prony coefficients to determine the input values. The E, η, and E∞ obtained from this tool can be used as a starting point for the SLS model and as initial values for fitting more complex multi-term Prony series.
What is Viscoelasticity Model Simulator?
Viscoelasticity Model Simulator is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Real-World Applications
Engineering Design: The concepts behind Viscoelasticity Model Simulator are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.
Select a viscoelastic model: Maxwell (spring-dashpot series), Kelvin-Voigt (parallel), or Standard Linear Solid (SLS) with relaxation time constant.
Enter elastic modulus E (GPa), infinite-frequency modulus Einf (GPa), viscosity eta (Pa·s), and initial stress sig0 (MPa).
Run creep or stress relaxation simulation; observe real-time storage modulus G', loss modulus G'', and tan delta (damping ratio) across frequency range 0.01–100 rad/s.
Worked Example
Polyurethane foam with E=0.8 GPa, Einf=0.2 GPa, eta=5×10⁷ Pa·s, sig0=0.5 MPa in SLS mode: At 1 rad/s, storage modulus G'=0.45 GPa, loss modulus G''=0.12 GPa, tan delta=0.27. Creep strain at 100 seconds reaches 2.3% under constant stress, while stress relaxation drops from 0.5 MPa to 0.35 MPa in 50 seconds, characteristic of polymer viscoelasticity.
Practical Notes
Maxwell model suits fluids with long relaxation times (tau=eta/E); Kelvin-Voigt better for solid-like materials with instantaneous deformation reversal.
SLS captures both instantaneous elasticity and delayed creep, essential for rubber compounds and epoxy adhesives in structural bonding.
Peak tan delta indicates transition region; high values (>0.5) suggest damping-critical applications like vibration isolation mounts.
For composites with fiber relaxation, use eta >10⁸ Pa·s to model secondary creep accurately over years.