Prony Series Viscoelastic Model
Theory and Physics
Prony Series Viscoelasticity
Professor, the Prony series is the standard model for viscoelasticity, right?
The Prony series is a discretization of the generalized Maxwell model. It expresses the relaxation modulus as a sum of exponential functions:
$G_i$ is the shear modulus of each Maxwell element, and $\tau_i$ is the relaxation time. It describes the time-dependent behavior of rubber, polymers, and adhesives.
Summary
Historical Origin of the Prony Series
The name "Prony series" originates from the French mathematician Gaspard de Prony (1755–1839), but its application to viscoelastic relaxation functions was established in the 20th-century field of rheology. In the form E(t)=E∞+Σ Eᵢ exp(-t/τᵢ), it is mathematically equivalent to the parallel model of Maxwell elements (Generalized Maxwell) and can approximate any linear viscoelastic behavior.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, meaning "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being pulled" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, which assumes "acceleration can be ignored because forces are applied slowly". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That is Hooke's law $F=kx$, the essence of the stiffness term. So here's a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External force term (load term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (e.g., pressure, contact forces). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression"—it sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because the vibration energy is converted into heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity and creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load and modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation) |
| Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
FEM Settings for Prony Series
```
*VISCOELASTIC, TIME=PRONY
g1, k1, tau1
g2, k2, tau2
```
Or:
```
*VISCOELASTIC, FREQUENCY=PRONY
g1, k1, tau1
```
TIME = time domain, FREQUENCY = frequency domain.
Summary
Parameter Identification from Relaxation Curves
DMA (Dynamic Mechanical Analysis) or stress relaxation tests are used to identify Prony parameters. The E'(ω) and E''(ω) obtained from frequency sweeps are converted into a master curve using the time-temperature superposition principle (WLF equation: log aT=−C₁(T−Tref)/(C₂+T−Tref)), and discrete Prony coefficients are fitted via least squares using tools like Pronyseries.py. At least 8–12 τᵢ points are necessary to ensure accuracy.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with mid-side nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2–3 times. Recommended when stress evaluation is critical.
Full integration vs Reduced integration
Full integration: Risk of over-constraint (locking). Reduced integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates the tangent stiffness matrix each iteration. Achieves quadratic convergence within the convergence radius but has high computational cost.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Instead of applying the full load at once, applies it in small increments. The arc-length method (Riks method) can track beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative) than to search sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd-order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
Practical Checklist
Application to Automotive Vibration-Isolating Rubber
Prony series viscoelasticity is standard for vibration damping analysis of engine mount rubber (EPDM-based). The loss factor tanδ in the 10–1000Hz frequency range is adjusted by tuning τᵢ to match target values (typically 0.1–0.3). A workflow combining Abaqus frequency response analysis (*STEADY STATE DYNAMICS) with Prony input for virtual evaluation of NVH (Noise, Vibration, Harshness) performance is widely adopted by Japanese automakers.
Analogy of the Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Common Pitfalls for Beginners
Are you checking mesh convergence? Do you think "the calculation ran = the results are correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer will be far from reality. Confirm that results stabilize across at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is the same as "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling real-world constraint conditions is often the most critical step in the entire analysis.
Software Comparison
Tools
Prony Input Formats by Solver
Prony series input formats differ by solver. Abaqus inputs normalized relative modulus gᵢ=Eᵢ/E₀ and relaxation time τᵢ pairs, LS-DYNA's MAT_076 inputs E∞ and each Eᵢ as absolute values, MSC Marc uses modulus ratios and time constants, ANSYS uses Prony coefficient tables with shift functions. Conversion errors frequently occur when comparing analyses between different solvers, so always check whether normalization is used.
The Three Most Important Questions for Selection
- "What are you solving?": Does the necessary physical model/element type for the Prony series viscoelastic model have support? For example, the presence of LES support for fluids, or contact/large deformation capability for structures, makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUIs are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between an automatic transmission car (GUI) and a manual transmission car (script).
- "How far will you expand?": Selection considering future expansion of analysis scale (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technologies
Advanced
Combined Model with Hyperelasticity
For large deformation viscoelasticity like rubber, "visco-hyperelasticity" combining Prony+Mooney-Rivlin is used. In Abaqus, writing *HYPERELASTIC and *VISCOELASTIC in the same *MATERIAL block automatically applies cumulative formulation. The same method is applied in brain tissue surgery simulation (Schroeder 2011), reporting deformation prediction errors within 2mm.
Troubleshooting
Troubles
Time Increment and Prony Accuracy
For time integration of Prony viscoelasticity, a guideline is time increment Δt < τmin/10 relative to the minimum relaxation time τmin. If Δt is too large, short relaxation components cannot be integrated fully, leading to overestimation of storage modulus E'. In Abaqus, setting the maximum number of increments and DTMAX (maximum Δt) in the *VISCO step is crucial; forgetting these settings frequently causes divergence or inaccurate stress relaxation results.
If You Think "The Analysis Doesn't Match"
- First, take a deep breath—Panicking and randomly changing settings will only complicate the problem further.
- Create a minimal reproducible case—Reproduce the problem with the Prony series viscoelastic model in its simplest form. "Subtractive debugging" is most efficient.
- Change only one thing and re-run—Making multiple changes simultaneously makes it impossible to know what worked. The principle of "controlled experiment" same as in science.
- Return to the physics—If the calculation result is non-physical, like "an object floating against gravity", suspect a fundamental error in the input data.
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