.. _hvm_advanced: ============================================== HVM Advanced Theory & Numerical Methods ============================================== This page documents the thermodynamic foundations, kinematics, and numerical methods underlying the HVM. For the constitutive equations, see :doc:`hvm`. For protocol derivations, see :doc:`hvm_protocols`. .. _hvm-thermodynamics: 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: .. math:: \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): .. math:: \Psi_P(\mathbf{F}) = \frac{G_P}{2}\left(\text{tr}(\mathbf{B}) - 3\right) where :math:`\mathbf{B} = \mathbf{F}\mathbf{F}^T` and :math:`G_P = c_P k_B T`. **Exchangeable (vitrimer) network:** .. math:: \Psi_E = \frac{G_E}{2}\,\text{tr}\!\left(\boldsymbol{\mu}^E - \boldsymbol{\mu}^E_{nat}\right) The stress vanishes when :math:`\boldsymbol{\mu}^E = \boldsymbol{\mu}^E_{nat}`, not when :math:`\boldsymbol{\mu}^E = \mathbf{I}`. This distinction is the hallmark of associative exchange. **Dissociative (physical) network:** .. math:: \Psi_D = \frac{G_D}{2}\,\text{tr}(\boldsymbol{\mu}^D - \mathbf{I}) Natural state is always :math:`\mathbf{I}` (bonds reform stress-free). Clausius-Duhem Derivation -------------------------- The second law requires non-negative dissipation: .. math:: \mathcal{D} = \boldsymbol{\sigma}:\mathbf{D} - \dot{\Psi}_{tot} \geq 0 Expanding :math:`\dot{\Psi}_{tot}` and collecting terms linear in :math:`\mathbf{L}` identifies the Cauchy stress: .. math:: \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:** .. math:: \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 :math:`(\boldsymbol{\mu}^E - \boldsymbol{\mu}^E_{nat})^2` is positive semi-definite. **Dissociative network dissipation:** .. math:: \mathcal{D}_{diss} = \frac{G_D}{2} k_d^D \text{tr}(\boldsymbol{\mu}^D - \mathbf{I})^2 \geq 0 **Damage dissipation:** .. math:: \mathcal{D}_{dam} = \Psi_P \dot{D} \geq 0 Satisfied because :math:`\Psi_P \geq 0` and :math:`\dot{D} \geq 0` (damage is irreversible). .. _hvm-upper-convected: Upper-Convected Kinematics =========================== The evolution equations use the full velocity gradient :math:`\mathbf{L}`, not the symmetric part :math:`\mathbf{D}` alone: .. math:: \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 :math:`\mathbf{L} = \mathbf{D} + \mathbf{W}` means: .. math:: \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 :math:`\mathbf{W}\boldsymbol{\mu} - \boldsymbol{\mu}\mathbf{W}` provide the Jaumann co-rotational correction for rigid-body rotation. The upper-convected derivative form is: .. math:: \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 :math:`\mathbf{L} = \dot{\gamma} \mathbf{e}_1 \otimes \mathbf{e}_2` with isotropic initial conditions, :math:`\mu_{22} = \mu_{33}`, which means :math:`\boldsymbol{\mu}` commutes with :math:`\mathbf{W}`, and the vorticity terms vanish. The :math:`\mathbf{L}` and :math:`\mathbf{D}` formulations then coincide -- this is why all protocol derivations in :doc:`hvm_protocols` use scalar ODEs. TST Kinetics Deep Dive ======================== Stress-Coupled vs Stretch-Coupled ----------------------------------- **Option A -- Stress-based (von Mises invariant, ``kinetics="stress"``):** .. math:: 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"``):** .. math:: 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 :math:`k_{BER,0}(T) = \nu_0 \exp(-E_a / k_B T)`. Von Mises Computation (Simple Shear) -------------------------------------- With :math:`\sigma^E_{ij} = G_E(\mu^E_{ij} - \mu^{E,nat}_{ij})`, the von Mises equivalent stress is: .. math:: \sigma_{VM}^E = G_E \sqrt{(\Delta_{xx})^2 + (\Delta_{yy})^2 - \Delta_{xx}\Delta_{yy} + 3(\Delta_{xy})^2} where :math:`\Delta_{ij} = \mu^E_{ij} - \mu^{E,nat}_{ij}`. **Square root singularity:** At :math:`\sigma_{VM} = 0`, the gradient of :math:`\cosh(\cdot)` can produce numerical issues. The implementation uses :math:`\sqrt{\max(x, 0) + 10^{-30}}` to guard against infinite gradients (see :ref:`hvm-numerical` below). .. _hvm-phenomenological-mode: Phenomenological Fast Mode =========================== For computational efficiency (avoiding ODE stiffness from the exponential stress-coupling), a linearized rate is available: .. math:: k_{BER}^{phen} = k_{BER,0}(T) \cdot \left[1 + \alpha\,(\text{tr}(\boldsymbol{\mu}^E) - 3)\right] where :math:`\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 :doc:`/models/vlb/vlb_advanced` for the analogous VLB Bell breakage mechanism. .. _hvm-temperature: Temperature & Topological Freezing ===================================== Arrhenius Structure ------------------- The TST rate directly produces the topological freezing temperature :math:`T_v`. Defining :math:`T_v` as the temperature where :math:`\tau_{BER} = 1/k_{BER,0}` exceeds :math:`10^3` s: .. math:: T_v = \frac{E_a}{k_B \ln(\nu_0 \cdot 10^3)} **Temperature regimes:** .. list-table:: :widths: 20 30 50 :header-rows: 1 * - Regime - Condition - Behavior * - Below :math:`T_v` - :math:`T < T_v` - Vitrimer behaves as thermoset (no exchange) * - Above :math:`T_v` - :math:`T > T_v` - Active BER, material flows * - Well above :math:`T_v` - :math:`T \gg T_v` - Fast exchange, approaches viscous liquid **Arrhenius shift factor** for time-temperature superposition: .. math:: \ln a_T = \frac{E_a}{k_B}\left(\frac{1}{T} - \frac{1}{T_{ref}}\right) If :math:`G_P` and :math:`G_E` have the entropic :math:`T`-scaling (:math:`G \propto T`), a vertical shift :math:`b_T = T_{ref}/T` is also needed. Dissociative Bond Temperature Dependence ----------------------------------------- The D-network rate can follow Arrhenius: .. math:: 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): .. math:: 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) .. _hvm-numerical: 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 :math:`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 :math:`\sqrt{\text{tr}(\boldsymbol{\mu}^E - \boldsymbol{\mu}^E_{nat})}`, which has infinite gradient at zero. The implementation uses: .. code-block:: python safe_stretch = jnp.sqrt(jnp.maximum(stretch_invariant, 0.0) + 1e-30) **Initial conditions:** All tensors at identity (:math:`\mu_{xx} = \mu_{yy} = 1`, :math:`\mu_{xy} = 0`), :math:`\gamma = 0`, :math:`D = 0`. References =========== 1. Vernerey, F.J., Long, R. & Brighenti, R. (2017). "A statistically-based continuum theory for polymers with transient networks." *J. Mech. Phys. Solids*, 107, 1--20. https://doi.org/10.1016/j.jmps.2017.05.016 2. 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