DMTA Theory & Conversion¶

Physical Background¶

DMTA (Dynamic Mechanical Thermal Analysis) and DMA (Dynamic Mechanical Analysis) instruments apply tensile, bending, or compressive deformations to measure the complex Young’s modulus:

\[E^*(\omega) = E'(\omega) + i\,E''(\omega)\]

while rotational rheometers apply shear deformations to measure the complex shear modulus:

\[G^*(\omega) = G'(\omega) + i\,G''(\omega)\]

For isotropic, linear-viscoelastic materials these are connected by the complex Poisson’s ratio \(\nu^*\):

\[E^*(\omega) = 2\bigl(1 + \nu^*(\omega)\bigr)\,G^*(\omega)\]

A direct interchange \(E^* \leftrightarrow G^*\) is only valid under specific conditions. Understanding when the conversion is safe — and when it fails — is essential for correct DMTA analysis.

Important

The relaxation spectrum \(H(\tau)\) is a material property independent of deformation mode. Shear, tension, and bending all share the same spectrum — only the amplitude scale changes. Every OSCILLATION-capable model in RheoJAX is mathematically applicable to DMTA data after a modulus conversion, provided the conditions below are satisfied.

Validity Conditions for \(E^* \leftrightarrow G^*\)¶

The following four conditions must hold for the scalar conversion to be physically meaningful.

Condition 1 — Incompressibility (\(\nu \to 1/2\))

For a truly incompressible material the conversion simplifies to:

\[E^* = 3\,G^*\]

This is the most commonly invoked condition in soft matter and polymer physics. It applies when:

Rubbers and elastomers above \(T_g\) (bulk modulus \(K \gg G\), so \(\nu \approx 0.5\))
Hydrogels and biological soft tissues
Polymer melts and concentrated solutions in the terminal regime
Any system where volumetric deformation is energetically costly compared to shear (\(K/G \gg 1\))

The factor-of-3 conversion is exact in the incompressible limit and is widely used in rubber elasticity and XPCS/rheology cross-comparisons.

Condition 2 — Real, frequency-independent Poisson’s ratio

If \(\nu^* \approx \nu\) (real, not complex), the conversion remains clean:

\[E^* = 2(1 + \nu)\,G^*\]

This holds when volumetric relaxation is either absent or occurs on a very different timescale than the shear relaxation being probed. In practice:

Glassy polymers near and below \(T_g\): \(\nu \approx 0.33\text{--}0.40\), so \(E \approx 2.6\text{--}2.8\,G\)
Semi-crystalline polymers: \(\nu \approx 0.35\text{--}0.45\)

Here \(E^*\) and \(G^*\) are proportional but not equal — the conversion factor must be applied explicitly.

Condition 3 — Isotropy

The relation \(E = 2G(1+\nu)\) assumes linear elastic isotropy:

No fibre reinforcement, crystalline texture, or flow-induced anisotropy
Sample geometry must not induce multiaxial stress states that break the uniaxial / simple-shear assumption

For oriented polymer films, highly anisotropic nanocomposites, or liquid crystalline networks, this scalar conversion fails. The moduli must instead be treated as full compliance/stiffness tensors.

Condition 4 — Linear viscoelastic (LVE) regime

Both moduli must be measured strictly within the linear viscoelastic limit. If either the DMA or the shear rheometer applies a strain amplitude large enough to induce non-linear structural breakdown (Payne effect in filled networks, chain disentanglement), the linear conversion is no longer mathematically valid.

When the Conversion Fails¶

The conversion breaks down — and using \(E^* \approx 3G^*\) can be seriously wrong — in the following situations:

Situation	Why it fails
Near glass transition (\(T_g \pm 20\,°C\))	\(\nu^*\) becomes strongly frequency-dependent and complex; volumetric relaxation couples to shear
Semicrystalline polymers under tension	Crystalline lamellae introduce anisotropy; \(\nu\) varies with orientation
Foams, cellular solids	Compressibility is significant; \(K \sim G\)
Filled systems (high \(\varphi\) particles)	Compressive and shear reinforcement differ; effective \(\nu\) shifts
Highly crosslinked thermosets	\(\nu\) can drop to \(\sim 0.3\); \(E \approx 2.6\,G\), not \(3\,G\)

The most physically rich failure case is near \(T_g\), where the bulk modulus \(K\) relaxes on a different timescale than \(G\). In this regime:

\[\nu^*(\omega) = \frac{3K^*(\omega) - 2G^*(\omega)} {2\bigl(3K^*(\omega) + G^*(\omega)\bigr)}\]

is itself a complex, frequency-dependent quantity. The measured \(E^*\) from a tensile DMA mode encodes a mixture of \(K^*\) and \(G^*\) responses that cannot be cleanly separated without independent volumetric measurements.

Geometric and boundary effects. In a DMA tensile test of a short, thick sample the clamps restrict lateral contraction. This introduces shear stresses into the nominally pure-tension measurement, artificially inflating \(E^*\) and breaking the mathematical conversion unless aspect-ratio corrections are applied.

Practical Summary¶

Material Class	Safe?	\(\nu\)	\(E/G\) Factor
Rubbers, elastomers (\(T > T_g + 30\,\text{K}\))	Yes	0.50	3.0
Hydrogels, biopolymer networks	Yes	0.50	3.0
Polymer melts (terminal regime)	Yes	0.50	3.0
Glassy polymers (\(T < T_g - 30\,\text{K}\))	With caution	0.33–0.40	2.6–2.8
Semi-crystalline polymers	With caution	0.35–0.45	2.7–2.9
Near \(T_g\) (\(\pm 20\,°\text{C}\))	No	complex \(\nu^*(\omega)\)	—
Filled / composite systems	Verify independently	depends on \(\varphi\), morphology	—
Foams, cellular solids	No	\(K \sim G\)	—

Warning

Blindly applying \(E^* = 3G^*\) across a full temperature or frequency sweep through the glass transition will artificially skew the shape and width of the \(\tan\delta\) peak and the transition zone. For broad master curves spanning \(T_g\), see Viscoelastic Poisson’s Ratio below for mitigation strategies.

Deformation Modes¶

RheoJAX recognises four deformation geometries via the DeformationMode enum:

Mode	Measures	Typical Instrument
`SHEAR`	\(G^*\)	Rotational rheometer
`TENSION`	\(E^*\)	DMTA / DMA (film tension clamp)
`BENDING`	\(E^*\)	DMTA (3-point bending, cantilever)
`COMPRESSION`	\(E^*\)	DMA (compression clamp)

All tensile modes (TENSION, BENDING, COMPRESSION) measure \(E^*\) and share the same conversion factor. The instrument firmware handles geometry-specific corrections; the data reaching RheoJAX is always \(E^*\).

Conversion Utility¶

The rheojax.utils.modulus_conversion module provides:

from rheojax.utils.modulus_conversion import convert_modulus

# E* -> G* conversion (rubber, nu = 0.5)
G_star = convert_modulus(E_star, "tension", "shear", poisson_ratio=0.5)

# G* -> E* conversion
E_star = convert_modulus(G_star, "shear", "tension", poisson_ratio=0.35)

# Roundtrip is exact
E_recovered = convert_modulus(
    convert_modulus(E_star, "tension", "shear", poisson_ratio=0.4),
    "shear", "tension", poisson_ratio=0.4,
)

Automatic Conversion at fit()¶

The preferred workflow passes DMTA data directly to model.fit() with deformation_mode and poisson_ratio:

from rheojax.models import Maxwell

model = Maxwell()
model.fit(
    omega, E_star,
    test_mode='oscillation',
    deformation_mode='tension',
    poisson_ratio=0.5,
)

# Parameters are in G-space (model-native)
G0 = model.parameters.get_value("G0")  # NOT E0

# predict() returns E* automatically
E_pred = model.predict(omega, test_mode='oscillation')

Internally, BaseModel.fit() converts \(E^* \to G^*\) before calling the model’s _fit() method, and predict() converts the result back to \(E^*\). This means all 45 oscillation-capable models work without any model-level changes.

Viscoelastic Poisson’s Ratio¶

Note

RheoJAX currently implements only scalar (constant) Poisson’s ratio. The frequency-dependent \(\nu^*(\omega)\) described below is presented for theoretical context; see Future Extensions for the planned implementation.

The Poisson’s ratio of polymers is not constant through the glass transition. In the glassy state \(\nu \approx 0.35\); in the rubbery state \(\nu \approx 0.50\) (incompressible limit). The transition follows a relaxation function analogous to the modulus:

\[\nu(\omega) \approx \nu_\infty + \frac{\nu_0 - \nu_\infty}{1 + (i\omega\tau_\nu)^\beta}\]

where \(\nu_0 \approx 0.50\) (rubbery), \(\nu_\infty \approx 0.35\) (glassy), and \(\tau_\nu\) is the Poisson relaxation time.

The \(K^\)/\(G^\) Coupling Near \(T_g\)¶

In the glass transition region, the bulk modulus \(K^*\) and shear modulus \(G^*\) relax on different timescales. \(K^*\) typically has a weaker dispersion than \(G^*\) (volumetric relaxation is faster and narrower). This means the complex Poisson’s ratio:

\[\nu^*(\omega) = \frac{3K^*(\omega) - 2G^*(\omega)} {2\bigl(3K^*(\omega) + G^*(\omega)\bigr)}\]

has both a real and imaginary part that vary with frequency. Consequently, a tensile DMA measurement in this regime encodes a mixture of \(K^*\) and \(G^*\), and cannot be cleanly decomposed into shear-only information without an independent volumetric (e.g.dilatometric or ultrasonic) measurement.

Practical impact: The \(\tan\delta\) peak from a tensile DMA test is shifted in both frequency and height relative to the \(\tan\delta\) from a shear rheometer. This is a real physical difference, not a calibration artefact.

Application to Vitrimers and Exchangeable Networks¶

For vitrimer networks modelled with HVM (Hybrid Vitrimer Model) or HVNM (Hybrid Vitrimer Nanocomposite Model), the incompressible approximation \(E^* = 3G^*\) is typically safe in the rubbery plateau and terminal regimes. However, caution is required near the topology freezing temperature \(T_v\), where bond-exchange kinetics can contribute to volumetric relaxation and cause \(\nu\) to drift from the incompressible limit.

Important

RheoJAX currently assumes constant \(\nu\). For a master curve spanning the full glass transition, this introduces a systematic error of up to ~11% in modulus (factor 3.0 / 2.7). Recommended practice:

Rubbery plateau analysis (\(T \gg T_g\)): use \(\nu = 0.50\)
Glassy plateau analysis (\(T \ll T_g\)): use \(\nu = 0.35\)
Broad master curve through \(T_g\): use \(\nu = 0.40\) (compromise) or fit both plateaus separately

See Future Extensions for the planned frequency-dependent \(\nu\) feature.

Kramers–Kronig Relations¶

The storage and loss moduli are not independent. For any linear viscoelastic material, they are connected by the Kramers–Kronig (KK) integral transforms:

\[E'(\omega) - E_\infty = \frac{2}{\pi} \int_0^\infty \frac{u\,E''(u)}{u^2 - \omega^2}\,du\]

\[E''(\omega) = -\frac{2\omega}{\pi} \int_0^\infty \frac{E'(u) - E_\infty}{u^2 - \omega^2}\,du\]

Practical consequence: A model that fits \(E'(\omega)\) perfectly should automatically fit \(E''(\omega)\) well, and vice versa. A large discrepancy between \(R^2(E')\) and \(R^2(E'')\) signals either:

The model is too simple (e.g., single-mode Zener for a broad transition)
The data has systematic errors (different instruments for E’ and E’’)
The material exhibits thermorheological complexity (no clean TTS)

Relaxation Spectrum Representations¶

The relaxation spectrum \(H(\tau)\) is the fundamental material function connecting time-domain and frequency-domain behaviour:

\[E(t) = E_\infty + \int_0^\infty H(\tau)\,e^{-t/\tau}\,d\ln\tau\]

\[E'(\omega) = E_\infty + \int_0^\infty H(\tau)\, \frac{\omega^2\tau^2}{1 + \omega^2\tau^2}\,d\ln\tau\]

\[E''(\omega) = \int_0^\infty H(\tau)\, \frac{\omega\tau}{1 + \omega^2\tau^2}\,d\ln\tau\]

Two representations are common in RheoJAX:

Discrete (Prony series) — GeneralizedMaxwell:

\[H(\tau) = \sum_{i=1}^{N} E_i\,\delta(\tau - \tau_i) \quad\Rightarrow\quad E^*(\omega) = E_\infty + \sum_{i=1}^{N} \frac{E_i\,(i\omega\tau_i)}{1 + i\omega\tau_i}\]

Continuous power-law — Fractional models (FZSS, FMM, …):

\[H(\tau) \sim \tau^{-\alpha} \quad\Rightarrow\quad E^*(\omega) \sim (i\omega)^\alpha\]

Per-Model \(E^*(\omega)\) Expressions¶

Since \(E^* = 2(1+\nu)\,G^*\), every model’s \(G^*(\omega)\) formula directly gives \(E^*(\omega)\). Key expressions:

Maxwell (single relaxation time):

\[E^*(\omega) = 2(1+\nu)\,G_0 \frac{i\omega\tau}{1 + i\omega\tau}\]

Zener (Standard Linear Solid):

\[E^*(\omega) = 2(1+\nu)\left[G_e + G_m \frac{i\omega\tau}{1 + i\omega\tau}\right]\]

Fractional Zener (Solid–Solid):

\[E^*(\omega) = 2(1+\nu)\,G_e\, \frac{1 + (i\omega\tau)^\alpha}{1 + (G_e/G_g)(i\omega\tau)^\alpha}\]

where \(\alpha \in (0, 1)\) is the fractional order controlling the breadth of the relaxation distribution.

Generalized Maxwell (\(N\)-mode Prony series):

\[E^*(\omega) = E_\infty + \sum_{i=1}^{N} \frac{E_i\,(i\omega\tau_i)}{1 + i\omega\tau_i}\]

When modulus_type='tensile', parameters \(E_i\) are fitted directly. When modulus_type='shear' with deformation_mode='tension', \(G_i\) are fitted and \(E^* = 2(1+\nu)G^*\) is applied at the prediction boundary.