HVM Advanced Theory & Numerical Methods

This page documents the thermodynamic foundations, kinematics, and numerical methods underlying the HVM. For the constitutive equations, see HVM Model Reference. For protocol derivations, see HVM Protocol Equations & Derivations.

Thermodynamic Framework

Helmholtz Free Energy

The total Helmholtz free energy density is the sum of contributions from each subnetwork, with damage coupling on the permanent network:

\[\Psi_{tot} = (1-D)\,\Psi_P(\mathbf{F}) + \Psi_E[\boldsymbol{\mu}^E, \boldsymbol{\mu}^E_{nat}] + \Psi_D[\boldsymbol{\mu}^D] + p(\det\mathbf{F} - 1)\]

Permanent network (Neo-Hookean, Gaussian chains):

\[\Psi_P(\mathbf{F}) = \frac{G_P}{2}\left(\text{tr}(\mathbf{B}) - 3\right)\]

where \(\mathbf{B} = \mathbf{F}\mathbf{F}^T\) and \(G_P = c_P k_B T\).

Exchangeable (vitrimer) network:

\[\Psi_E = \frac{G_E}{2}\,\text{tr}\!\left(\boldsymbol{\mu}^E - \boldsymbol{\mu}^E_{nat}\right)\]

The stress vanishes when \(\boldsymbol{\mu}^E = \boldsymbol{\mu}^E_{nat}\), not when \(\boldsymbol{\mu}^E = \mathbf{I}\). This distinction is the hallmark of associative exchange.

Dissociative (physical) network:

\[\Psi_D = \frac{G_D}{2}\,\text{tr}(\boldsymbol{\mu}^D - \mathbf{I})\]

Natural state is always \(\mathbf{I}\) (bonds reform stress-free).

Clausius-Duhem Derivation

The second law requires non-negative dissipation:

\[\mathcal{D} = \boldsymbol{\sigma}:\mathbf{D} - \dot{\Psi}_{tot} \geq 0\]

Expanding \(\dot{\Psi}_{tot}\) and collecting terms linear in \(\mathbf{L}\) identifies the Cauchy stress:

\[\boldsymbol{\sigma}_{tot} = (1-D) G_P (\mathbf{B} - \mathbf{I}) + G_E (\boldsymbol{\mu}^E - \boldsymbol{\mu}^E_{nat}) + G_D (\boldsymbol{\mu}^D - \mathbf{I}) - p\mathbf{I}\]

The remaining terms yield the dissipation from kinetic processes, each of which must be individually non-negative:

Exchangeable network dissipation:

\[\mathcal{D}_{exch} = \frac{G_E}{2} k_{BER} \text{tr}\!\left[(\boldsymbol{\mu}^E - \boldsymbol{\mu}^E_{nat})^2 \cdot (\boldsymbol{\mu}^E_{nat})^{-1}\right] \geq 0\]

Guaranteed non-negative because \((\boldsymbol{\mu}^E - \boldsymbol{\mu}^E_{nat})^2\) is positive semi-definite.

Dissociative network dissipation:

\[\mathcal{D}_{diss} = \frac{G_D}{2} k_d^D \text{tr}(\boldsymbol{\mu}^D - \mathbf{I})^2 \geq 0\]

Damage dissipation:

\[\mathcal{D}_{dam} = \Psi_P \dot{D} \geq 0\]

Satisfied because \(\Psi_P \geq 0\) and \(\dot{D} \geq 0\) (damage is irreversible).

Upper-Convected Kinematics

The evolution equations use the full velocity gradient \(\mathbf{L}\), not the symmetric part \(\mathbf{D}\) alone:

\[\dot{\boldsymbol{\mu}}^E = \mathbf{L}\boldsymbol{\mu}^E + \boldsymbol{\mu}^E\mathbf{L}^T + k_{BER}(\boldsymbol{\mu}^E_{nat} - \boldsymbol{\mu}^E)\]

The decomposition \(\mathbf{L} = \mathbf{D} + \mathbf{W}\) means:

\[\mathbf{L}\boldsymbol{\mu} + \boldsymbol{\mu}\mathbf{L}^T = \mathbf{D}\boldsymbol{\mu} + \boldsymbol{\mu}\mathbf{D} + \mathbf{W}\boldsymbol{\mu} - \boldsymbol{\mu}\mathbf{W}\]

The vorticity terms \(\mathbf{W}\boldsymbol{\mu} - \boldsymbol{\mu}\mathbf{W}\) provide the Jaumann co-rotational correction for rigid-body rotation.

The upper-convected derivative form is:

\[\overset{\nabla}{\boldsymbol{\mu}}^E \equiv \dot{\boldsymbol{\mu}}^E - \mathbf{L}\boldsymbol{\mu}^E - \boldsymbol{\mu}^E\mathbf{L}^T = k_{BER}(\boldsymbol{\mu}^E_{nat} - \boldsymbol{\mu}^E)\]

Simple shear as a special case: For \(\mathbf{L} = \dot{\gamma} \mathbf{e}_1 \otimes \mathbf{e}_2\) with isotropic initial conditions, \(\mu_{22} = \mu_{33}\), which means \(\boldsymbol{\mu}\) commutes with \(\mathbf{W}\), and the vorticity terms vanish. The \(\mathbf{L}\) and \(\mathbf{D}\) formulations then coincide – this is why all protocol derivations in HVM Protocol Equations & Derivations use scalar ODEs.

TST Kinetics Deep Dive

Stress-Coupled vs Stretch-Coupled

Option A – Stress-based (von Mises invariant, ``kinetics=”stress”``):

\[f(\boldsymbol{\sigma}^E) = \sqrt{\tfrac{3}{2} \boldsymbol{\sigma}^E:\boldsymbol{\sigma}^E}\]

Appropriate when the exchange barrier is reduced by total stress magnitude. Isotropic and simple to evaluate.

Option B – Stretch-based (chain stretch invariant, ``kinetics=”stretch”``):

\[f(\boldsymbol{\mu}^E) = G_E \sqrt{\text{tr}(\boldsymbol{\mu}^E - \boldsymbol{\mu}^E_{nat})}\]

Measures elastic stretch of exchangeable chains relative to their current natural state. More appropriate when force along the chain backbone directly lowers the barrier (Bell model picture).

In the zero-stress limit, both reduce to the thermal rate \(k_{BER,0}(T) = \nu_0 \exp(-E_a / k_B T)\).

Von Mises Computation (Simple Shear)

With \(\sigma^E_{ij} = G_E(\mu^E_{ij} - \mu^{E,nat}_{ij})\), the von Mises equivalent stress is:

\[\sigma_{VM}^E = G_E \sqrt{(\Delta_{xx})^2 + (\Delta_{yy})^2 - \Delta_{xx}\Delta_{yy} + 3(\Delta_{xy})^2}\]

where \(\Delta_{ij} = \mu^E_{ij} - \mu^{E,nat}_{ij}\).

Square root singularity: At \(\sigma_{VM} = 0\), the gradient of \(\cosh(\cdot)\) can produce numerical issues. The implementation uses \(\sqrt{\max(x, 0) + 10^{-30}}\) to guard against infinite gradients (see Numerical Implementation below).

Phenomenological Fast Mode

For computational efficiency (avoiding ODE stiffness from the exponential stress-coupling), a linearized rate is available:

\[k_{BER}^{phen} = k_{BER,0}(T) \cdot \left[1 + \alpha\,(\text{tr}(\boldsymbol{\mu}^E) - 3)\right]\]

where \(\alpha = V_{act} G_E / (2 k_B T)\) maps TST parameters to the phenomenological enhancement coefficient. This is a first-order Taylor expansion valid for small deformations.

See VLB Advanced Theory & Numerical Methods for the analogous VLB Bell breakage mechanism.

Temperature & Topological Freezing

Arrhenius Structure

The TST rate directly produces the topological freezing temperature \(T_v\). Defining \(T_v\) as the temperature where \(\tau_{BER} = 1/k_{BER,0}\) exceeds \(10^3\) s:

\[T_v = \frac{E_a}{k_B \ln(\nu_0 \cdot 10^3)}\]

Temperature regimes:

Regime

Condition

Behavior

Below \(T_v\)

\(T < T_v\)

Vitrimer behaves as thermoset (no exchange)

Above \(T_v\)

\(T > T_v\)

Active BER, material flows

Well above \(T_v\)

\(T \gg T_v\)

Fast exchange, approaches viscous liquid

Arrhenius shift factor for time-temperature superposition:

\[\ln a_T = \frac{E_a}{k_B}\left(\frac{1}{T} - \frac{1}{T_{ref}}\right)\]

If \(G_P\) and \(G_E\) have the entropic \(T\)-scaling (\(G \propto T\)), a vertical shift \(b_T = T_{ref}/T\) is also needed.

Dissociative Bond Temperature Dependence

The D-network rate can follow Arrhenius:

\[k_d^D(T) = k_{d,0}^D \exp\!\left(-\frac{E_a^D}{k_B T}\right)\]

For force-dependent dissociation (Bell-Evans model):

\[k_d^D(\boldsymbol{\mu}^D) = k_{d,0}^D \exp\!\left(-\frac{E_a^D - V_{act}^D \|\boldsymbol{\sigma}^D\|}{k_B T}\right)\]

Numerical Implementation

ODE solver: diffrax Tsit5 (explicit 5th-order Runge-Kutta) with PIDController adaptive stepping (rtol=1e-8, atol=1e-10).

Stiffness handling: TST stress-coupling can make the ODEs stiff at high shear rates (large \(k_{BER}\) variations within a timestep). The explicit Tsit5 solver handles moderate stiffness; for extreme cases, reduce shear rate or switch to kinetics="stretch" (smoother coupling).

Note

Implicit solvers (e.g., Kvaerno5) were tested but produce TracerBoolConversionError due to lineax LU transpose checks during JAX tracing. Tsit5 is the recommended solver.

Square-root guard: The BER rate computation involves \(\sqrt{\text{tr}(\boldsymbol{\mu}^E - \boldsymbol{\mu}^E_{nat})}\), which has infinite gradient at zero. The implementation uses:

safe_stretch = jnp.sqrt(jnp.maximum(stretch_invariant, 0.0) + 1e-30)

Initial conditions: All tensors at identity (\(\mu_{xx} = \mu_{yy} = 1\), \(\mu_{xy} = 0\)), \(\gamma = 0\), \(D = 0\).

References

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  2. Vernerey, F.J. (2018). “Transient response of nonlinear polymer networks: A kinetic theory.” J. Mech. Phys. Solids, 115, 230–247. https://doi.org/10.1016/j.jmps.2018.02.018 PDF

  3. Vernerey, F.J., Brighenti, R., Long, R. & Shen, T. (2018). “Statistical Damage Mechanics of Polymer Networks.” Macromolecules, 51(17), 6609–6622. https://doi.org/10.1021/acs.macromol.8b01052

  4. Meng, F., Saed, M.O. & Terentjev, E.M. (2019). “Elasticity and Relaxation in Full and Partial Vitrimer Networks.” Macromolecules, 52(19), 7423–7429. https://doi.org/10.1021/acs.macromol.9b01123

  5. Shen, T., Song, Z., Cai, S. & Vernerey, F.J. (2021). “Nonsteady fracture of transient networks: The case of vitrimer.” PNAS, 118(29), e2105974118. https://doi.org/10.1073/pnas.2105974118

  6. Wagner, R.J., Hobbs, E. & Vernerey, F.J. (2021). “A network model of transient polymers: exploring the micromechanics of nonlinear viscoelasticity.” Soft Matter, 17, 8742. https://doi.org/10.1039/D1SM00753J PDF

  7. Lamont, S.C., Mulderrig, J., Bouklas, N. & Vernerey, F.J. (2021). “Rate-Dependent Damage Mechanics of Polymer Networks with Reversible Bonds.” Macromolecules, 54(23), 10801–10813. https://doi.org/10.1021/acs.macromol.1c01943

  8. Meng, F., Saed, M.O. & Terentjev, E.M. (2022). “Rheology of vitrimers.” Nature Communications, 13, 5753. https://doi.org/10.1038/s41467-022-33321-w

  9. Vernerey, F.J., Rezaei, B. & Lamont, S.C. (2024). “A kinetic theory for the mechanics and remodeling of transient anisotropic networks.” J. Mech. Phys. Solids, 190, 105713. https://doi.org/10.1016/j.jmps.2024.105713

  10. Wagner, R.J. & Silberstein, M.N. (2025). “A foundational framework for the mesoscale modeling of dynamic elastomers and gels.” J. Mech. Phys. Solids, 194, 105914. https://doi.org/10.1016/j.jmps.2024.105914

  11. Karim, M.R., Vernerey, F. & Sain, T. (2025). “Constitutive Modeling of Vitrimers and Their Nanocomposites Based on Transient Network Theory.” Macromolecules, 58(10), 4899–4912. DOI: 10.1021/acs.macromol.4c02872 PDF

  12. Alkhoury, K. & Chester, S.A. (2025). “A chemo-thermo-mechanically coupled theory of photo-reacting polymers: Application to modeling photo-degradation with irradiation-driven heat transfer.” J. Mech. Phys. Solids, 197, 106050. https://doi.org/10.1016/j.jmps.2025.106050

  13. White, Z.T., Smith, A.M. & Vernerey, F.J. (2025). “Mechanical cooperation between time-dependent and covalent bonds in molecular damage of polymer networks.” Communications Physics, 8, 265. DOI: 10.1038/s42005-025-02192-0 PDF