Frequency Response of Viscoelastic Damping Materials
Theory and Physics
What is a Viscoelastic Material?
Professor, how is the damping of viscoelastic materials different from structural damping?
Structural damping assumes $g$ is independent of frequency, but the damping of viscoelastic materials strongly depends on frequency and temperature. Rubber, polymers, adhesives, and damping materials are typical examples.
Complex Modulus
Stress-strain relationship for viscoelastic materials:
- $E'(\omega)$ — Storage modulus (stiffness. Stores energy)
- $E''(\omega)$ — Loss modulus (damping. Dissipates energy)
- $\eta(\omega) = E''/E'$ — Loss factor
So both stiffness and damping change with frequency, right?
Typical behavior:
- Low frequency — $E'$ is low (soft), $\eta$ is moderate
- Glass transition region — $E'$ increases sharply, $\eta$ peaks (maximum damping)
- High frequency — $E'$ is high (stiff), $\eta$ decreases
Temperature-Frequency Equivalence Principle (WLF Equation)
For viscoelastic materials, when temperature increases, the properties shift to the low-frequency side, and when temperature decreases, they shift to the high-frequency side. This equivalence is described by the Williams-Landel-Ferry (WLF) equation:
$a_T$ is the shift factor. This allows estimating properties at all temperatures from data at a single temperature.
So if we measure data at one temperature using DMA (Dynamic Mechanical Analysis), we can extrapolate to other temperatures using the WLF equation, right?
Create a master curve. Shift the DMA data at the reference temperature using the WLF equation to obtain $E'(\omega), \eta(\omega)$ over a wide frequency range. Input this into FEM.
Summary
Key points:
- $E^*(\omega) = E'(1+i\eta)$ — Frequency-dependent complex modulus
- Both $E'$ and $\eta$ depend on frequency and temperature — Peak at glass transition
- Temperature-frequency equivalence via WLF equation — Construction of master curve
- Input into FEM using direct method (frequency response) — Difficult to handle with modal methods
The memory effect of viscoelasticity was discovered by Maxwell in 1867
The memory effect of viscoelastic materials, where "strain depends on the history of stress," was discovered by James Clerk Maxwell in 1867 during his research on gas viscosity. Later, Kelvin-Voigt (1890) proposed a parallel model, and the "Zener (standard linear solid) model," which combines the Maxwell series and Kelvin parallel models, became the foundation for FEM implementation of structural damping. This model is still used today for characterizing the properties of 3M's damping sheets.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" 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, assuming "forces are applied slowly enough that acceleration can be ignored." 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's Hooke's law $F=kx$, the essence of the stiffness term. Now 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 surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall here: getting the load direction wrong. Intending "tension" but ending up with "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 vibrational 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 vibrational energy to improve ride comfort. What if damping were zero? Buildings would keep 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 behaviors 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 elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistencies when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations) |
| Elastic 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
Viscoelastic Materials in FEM
How are viscoelastic materials handled in FEM?
Model $E'(\omega), E''(\omega)$ using the Prony series:
$E_i, \tau_i$ are Prony series parameters. $N = 5 \sim 15$ terms cover a wide frequency range.
Abaqus
```
*VISCOELASTIC, FREQUENCY=PRONY
g_1, k_1, tau_1
g_2, k_2, tau_2
...
```
Nastran
```
MAT1, 1, ...
TABLEM1, 100, ...
$ Define frequency-dependent E' and η in a table
```
In Nastran, you can input directly via a table (frequency vs. E', η).
Can Abaqus's Prony series be fitted directly from DMA data?
Abaqus/CAE's *VISCOELASTIC, TEST DATA option allows input of measured $E'(\omega), E''(\omega)$ and automatically fits the Prony series. Very convenient.
Damping Material (CLD) Analysis
Analysis of CLD (Constrained Layer Damping):
1. Model the laminate: substrate (steel plate, etc.) + viscoelastic layer + constraining layer
2. Set frequency-dependent Prony series for the viscoelastic layer
3. Evaluate damping effect using direct method frequency response analysis
4. Confirm reduction of FRF peaks
Summary
Compress data to 1/100 using time-temperature superposition
The loss factor of viscoelastic materials changes with a combination of temperature and frequency. Using the time-temperature superposition principle via the Williams-Landel-Ferry (WLF) equation (1955), properties at any temperature and frequency can be predicted from measurement data at a reference temperature Tr alone. The frequency characteristics of interior materials (ethylene-propylene rubber based) in vehicles can be represented by a single master curve at 20°C reference for all conditions from -40°C to 100°C and 1Hz to 10kHz.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but low 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 important.
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 tangent stiffness matrix each iteration. Achieves quadratic convergence within the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using 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, apply it in small increments. The arc-length method (Riks method) can trace beyond extremum points in the load-displacement relationship.
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 estimate where to open it and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
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 "flexible curves"—can represent curved changes, dramatically improving accuracy even with the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Practical Aspects of Viscoelastic Damping
Damping material design is one of the most important applications in automotive NVH.
Damping Material Design Flow
1. Measure $E'(\omega, T), \eta(\omega, T)$ of the viscoelastic material via DMA testing
2. Construct master curve using WLF equation
3. Fit to Prony series
4. Optimize damping material placement and thickness in FEM
5. Confirm reduction of FRF resonance peaks
Practical Checklist
So damping material should be applied at the "location of maximum strain"?
Damping materials dissipate energy through shear deformation. Applying damping material at the location of maximum bending strain in the substrate (near fixed ends) maximizes effectiveness. Applying it at locations with low strain (near free ends) has little effect.
3M damping sheet reduces Shinkansen noise by 5dB
3M's damping material series "Dynamat" is a structure with a viscoelastic polymer (acrylic-based) laminated onto a steel sheet, converting bending waves at 100-1000Hz into thermal energy. An equivalent product to 3M Dynamat was applied to the underfloor steel plates of JR East's E5 series Shinkansen (2011), reducing interior structure-borne noise from track noise by approximately 5dB. It is only 2mm thick with an area density of 2.4kg/m². Due to cost advantages, it is widely adopted in mass-produced vehicles.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, you shop for ingredients (prepare CAD model), do the prep work (
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