HVM Model Reference¶

This page provides the complete mathematical formulation and implementation details for the Hybrid Vitrimer Model (HVM).

Note

HVM builds on VLB transient network theory. For foundational distribution tensor derivations and the governing ODE, see VLB Transient Network Models. For shared protocol methodology (flow curve, SAOS, startup, etc.), see VLB Protocol Equations & Derivations. The D-network in HVM follows the same evolution equation as a single VLB transient network.

Quick Reference¶

Full name	Hybrid Vitrimer Model
Class	`HVMLocal`
Base class	`HVMBase` (extends VLBBase)
Registry names	`"hvm_local"`, `"hvm"`
Parameters	6 (base) + 2 (dissociative) + 2 (damage) = 6-10
TST coupling	`kinetics="stress"` (default) or `"stretch"`
ODE solver	diffrax Tsit5 with adaptive stepping
Bayesian	Full NLSQ -> NUTS pipeline

Notation Guide¶

Symbol	Meaning	Units	Typical Range
\(G_P\)	Permanent network modulus	Pa	\(10^2\) – \(10^6\)
\(G_E\)	Exchangeable network modulus	Pa	\(10^2\) – \(10^6\)
\(G_D\)	Dissociative network modulus	Pa	\(10^1\) – \(10^5\)
\(\nu_0\)	TST attempt frequency	1/s	\(10^8\) – \(10^{12}\)
\(E_a\)	Activation energy for BER	J/mol	40k – 150k
\(V_{act}\)	Activation volume	m³/mol	\(10^{-7}\) – \(10^{-4}\)
\(T\)	Temperature	K	250 – 450
\(k_d^D\)	Dissociative bond rate	1/s	\(10^{-3}\) – \(10^3\)
\(k_{BER}\)	Bond exchange rate	1/s	(computed from TST)
\(\boldsymbol{\mu}^E\)	E-network distribution tensor	–	(state variable)
\(\boldsymbol{\mu}^E_{nat}\)	E-network natural-state tensor	–	(state variable)
\(\boldsymbol{\mu}^D\)	D-network distribution tensor	–	(state variable)
\(D\)	Damage variable	–	[0, 1]
\(R\)	Universal gas constant	J/(mol K)	8.314

Three-Subnetwork Architecture¶

The HVM decomposes the total stress as:

\[\boldsymbol{\sigma} = \boldsymbol{\sigma}_P + \boldsymbol{\sigma}_E + \boldsymbol{\sigma}_D\]

Permanent (P) network:

Neo-Hookean elastic response from covalent crosslinks:

\[\boldsymbol{\sigma}_P = (1 - D) \, G_P (\mathbf{B} - \mathbf{I})\]

where \(\mathbf{B}\) is the left Cauchy-Green tensor and \(D\) is the damage variable. In simple shear at strain \(\gamma\):

\[\sigma_{P,xy} = (1 - D) \, G_P \, \gamma\]

Exchangeable (E) network:

Stress arises from the deviation of \(\boldsymbol{\mu}^E\) from its evolving natural state \(\boldsymbol{\mu}^E_{nat}\):

\[\boldsymbol{\sigma}_E = G_E (\boldsymbol{\mu}^E - \boldsymbol{\mu}^E_{nat})\]

This is the vitrimer hallmark: the natural state is not the identity tensor but evolves to track the current deformation via BER.

Dissociative (D) network:

Standard upper-convected Maxwell:

\[\boldsymbol{\sigma}_D = G_D (\boldsymbol{\mu}^D - \mathbf{I})\]

Evolution Equations (Simple Shear)¶

E-network distribution tensor (\(\boldsymbol{\mu}^E\)):

\[\begin{split}\dot{\mu}^E_{xx} &= 2 \dot{\gamma} \, \mu^E_{xy} + k_{BER}(\mu^E_{nat,xx} - \mu^E_{xx}) \\ \dot{\mu}^E_{yy} &= k_{BER}(\mu^E_{nat,yy} - \mu^E_{yy}) \\ \dot{\mu}^E_{xy} &= \dot{\gamma} \, \mu^E_{yy} + k_{BER}(\mu^E_{nat,xy} - \mu^E_{xy})\end{split}\]

E-network natural-state tensor (\(\boldsymbol{\mu}^E_{nat}\)):

\[\dot{\mu}^E_{nat,ij} = k_{BER}(\mu^E_{ij} - \mu^E_{nat,ij})\]

This coupled evolution is the vitrimer hallmark: the natural state continuously drifts toward the current state at rate \(k_{BER}\).

D-network distribution tensor (\(\boldsymbol{\mu}^D\)):

\[\begin{split}\dot{\mu}^D_{xx} &= 2 \dot{\gamma} \, \mu^D_{xy} - k_d^D(\mu^D_{xx} - 1) \\ \dot{\mu}^D_{yy} &= -k_d^D(\mu^D_{yy} - 1) \\ \dot{\mu}^D_{xy} &= \dot{\gamma} \, \mu^D_{yy} - k_d^D \, \mu^D_{xy}\end{split}\]

This is identical to the VLB single-network evolution.

TST Kinetics¶

Thermal BER rate (zero stress):

\[k_{BER,0} = \nu_0 \exp\!\left(-\frac{E_a}{R T}\right)\]

Stress-coupled BER rate (kinetics="stress"):

\[k_{BER} = k_{BER,0} \cosh\!\left(\frac{V_{act} \, \sigma_{VM}^E}{R T}\right)\]

where \(\sigma_{VM}^E\) is the von Mises stress of the E-network:

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

with \(\sigma^E_{ij} = \mu^E_{ij} - \mu^E_{nat,ij}\).

Stretch-coupled BER rate (kinetics="stretch"):

\[\lambda_E = \sqrt{\frac{\text{tr}(\boldsymbol{\mu}^E)}{3}} \qquad k_{BER} = k_{BER,0} \cosh\!\left(\frac{V_{act} \, G_E(\lambda_E - 1)}{R T}\right)\]

Cooperative Damage Shielding¶

When include_damage=True, the damage variable \(D \in [0, 1]\) evolves as:

\[\dot{D} = \Gamma_0 (1 - D) \max(0, \lambda_{chain} - \lambda_{crit})\]

where:

\(\Gamma_0\) is the damage rate coefficient
\(\lambda_{crit}\) is the critical stretch for damage onset
\(\lambda_{chain} = \sqrt{\text{tr}(\mathbf{B})/3}\) is the effective chain stretch

The \((1 - D)\) factor provides cooperative shielding: damage slows as the network degrades, preventing unphysical complete fracture.

Factor-of-2 in Relaxation¶

For the E-network under constant \(k_{BER}\), the stress difference \(\Delta\mu_{ij} = \mu^E_{ij} - \mu^E_{nat,ij}\) satisfies:

\[\dot{\Delta\mu}_{ij} = -2 k_{BER} \, \Delta\mu_{ij}\]

Therefore the E-network contribution to stress relaxes with effective time constant:

\[\tau_{E,eff} = \frac{1}{2 k_{BER,0}}\]

A naive Maxwell fit yields \(\tau_{fit} = \tau_{E,eff} = \tau_E / 2\). This is a fundamental vitrimer signature.

Protocol Summary¶

The HVM supports six rheological protocols. For complete derivations and closed-form solutions, see HVM Protocol Equations & Derivations.

Protocol	Method	Key Result
Flow Curve	Analytical	\(\sigma_E = 0\) at steady state; \(\sigma^{ss} = G_P\gamma + \eta_D\dot{\gamma}\)
SAOS	Analytical	Two Maxwell modes + \(G_P\) plateau; \(\hat{\tau}_E = 1/(2k_{BER,0})\)
Startup	ODE	TST creates stress overshoot; analytical constant-rate solution
Relaxation	ODE	Bi-exponential + \(G_P\) plateau; TST gives KWW-like decay
Creep	ODE	Two retardation times + vitrimer plastic creep \(\delta J(t)\)
LAOS	ODE	TST generates odd harmonics; \(N_1\) at \(2\omega\)

For the HVM vs VLB comparison and the thermodynamic framework, see HVM Advanced Theory & Numerical Methods.

Limiting Cases¶

Case	Parameters	Equivalent Model	Factory Method
Neo-Hookean	\(G_E=0, G_D=0\)	Pure elastic	`HVMLocal.neo_hookean(G_P)`
Maxwell / VLBLocal	\(G_P=0, G_E=0\)	Single Maxwell element (equivalent to `VLBLocal`)	`HVMLocal.maxwell(G_D, k_d_D)`
Zener (SLS)	\(G_E=0\)	Standard Linear Solid	`HVMLocal.zener(G_P, G_D, k_d_D)`
Pure vitrimer	\(G_P=0, G_D=0\)	E-network only	`HVMLocal.pure_vitrimer(G_E, ...)`
Partial vitrimer	\(G_D=0\)	Meng et al. (2019)	`HVMLocal.partial_vitrimer(G_P, G_E, ...)`
Full HVM	All \(> 0\)	Full 3-network	`HVMLocal()`

Parameter Table¶

Parameter	Default	Bounds	Units	Description
`G_P`	1e4	(0, 1e9)	Pa	Permanent network modulus (covalent crosslinks)
`G_E`	1e4	(0, 1e9)	Pa	Exchangeable network modulus (vitrimer bonds)
`nu_0`	1e10	(1e6, 1e14)	1/s	TST attempt frequency
`E_a`	80e3	(20e3, 200e3)	J/mol	Activation energy for bond exchange
`V_act`	1e-5	(1e-8, 1e-2)	m³/mol	Activation volume (mechanochemical coupling)
`T`	300	(200, 500)	K	Temperature (typically fixed)
`G_D`	1e3	(0, 1e8)	Pa	Dissociative network modulus (when `include_dissociative=True`)
`k_d_D`	1.0	(1e-6, 1e6)	1/s	Dissociative bond rate (when `include_dissociative=True`)
`Gamma_0`	1e-4	(0, 0.1)	1/s	Damage rate coefficient (when `include_damage=True`)
`lambda_crit`	2.0	(1.001, 10)	–	Critical stretch for damage onset (when `include_damage=True`)

Advanced Theory¶

For thermodynamic foundations (Helmholtz energy, Clausius-Duhem derivation), upper-convected kinematics, topological freezing, and numerical implementation details, see HVM Advanced Theory & Numerical Methods.

For troubleshooting, cross-protocol validation, and knowledge extraction workflows, see HVM Knowledge Extraction Guide.