6 L04 — Thermodynamic Framework

Laws of Thermodynamics, Dissipation, and the FEM Stress-Update Context

Sign convention — heat flux. The heat flux vector $\mathbf{q}$ points in the direction of positive heat flow (outward from hot to cold). In the energy-balance equation, the term $-\nabla\cdot\mathbf{q}$ represents heat supplied to the body (so a net positive divergence of $\mathbf{q}$ means heat is leaving).

6.1 First Law of Thermodynamics

The rate of change of total energy equals the power input plus heat supply: \[ \dot{E} = \mathcal{P}_\text{ext} + \mathcal{Q}, \]

In local (strong) form with internal energy density $e$: \[ \rho\dot{e} = \boldsymbol{\sigma}:\mathbf{d} + \rho r - \nabla\cdot\mathbf{q} \]

where $r$ is the heat source density and $\mathbf{q}$ the heat flux.

6.2 Second Law — Entropy Inequality

The Clausius-Duhem inequality: \[ \rho\dot{\eta} \geq \frac{\rho r}{\theta} - \nabla\cdot\left(\frac{\mathbf{q}}{\theta}\right) \]

where $\eta$ is the specific entropy and $\theta > 0$ the absolute temperature.

Introducing the Helmholtz free energy $\Psi = e - \theta\eta$ and substituting: \[ \mathcal{D} = \boldsymbol{\sigma}:\mathbf{d} - \rho\dot{\Psi} - \rho\eta\dot{\theta} - \frac{1}{\theta}\mathbf{q}\cdot\nabla\theta \geq 0 \]

This is the dissipation inequality — the cornerstone for thermodynamically consistent constitutive models.

6.3 Isothermal Conditions and Free Energy Potentials

For isothermal processes ($\dot\theta = 0$, $\nabla\theta = \mathbf{0}$):

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

The free energy $\Psi$ depends on:

Elastic strain (recoverable energy storage)
Internal variables $\boldsymbol{\alpha}$ (irreversible processes: plastic strain, damage, etc.)

6.4 Constitutive Restrictions from Thermodynamics

For a hyperelastic material in material description, $\Psi = \hat{\Psi}(\mathbf{C})$: \[ \mathbf{S} = 2\rho_0\frac{\partial\hat{\Psi}}{\partial\mathbf{C}} \]

This guarantees $\mathcal{D} = 0$ (purely elastic, no dissipation).

For inelastic materials, the free energy depends on elastic strains and internal variables: \[ \Psi = \hat{\Psi}(\mathbf{C}^e, \boldsymbol{\alpha}) \] Dissipation $\mathcal{D} \geq 0$ constrains the evolution equations for $\boldsymbol{\alpha}$.

Notation — $\mathbf{C}^e$ . In this chapter, $\mathbf{C}^e = (\mathbf{F}^e)^T\mathbf{F}^e$ is the elastic part of the right Cauchy-Green tensor (a second-order kinematic quantity, from the multiplicative split $\mathbf{F} = \mathbf{F}^e\mathbf{F}^p$). This is distinct from the fourth-order elasticity tensor $\mathbb{C}^e$ (blackboard bold) used in L07–L09 plasticity chapters. Typography distinguishes them: $\mathbf{C}^e$ is bold upright (2nd-order), $\mathbb{C}^e$ is blackboard (4th-order).

6.5 The Stress-Update Problem in FEM

In a nonlinear FEM analysis, at each time/load step the constitutive law is called at each Gauss point to:

Input: strain increment $\Delta\boldsymbol{\varepsilon}$ (from displacement increment), previous stress $\boldsymbol{\sigma}_n$, previous internal variables $\boldsymbol{\alpha}_n$.
Output: updated stress $\boldsymbol{\sigma}_{n+1}$, updated internal variables $\boldsymbol{\alpha}_{n+1}$, consistent tangent modulus $\mathbb{C}_{n+1}$.

The consistent (algorithmic) tangent $\mathbb{C} = \frac{\partial\boldsymbol{\sigma}}{\partial\boldsymbol{\varepsilon}}$ is required for the global Newton-Raphson loop to converge quadratically.

6.6 Generalized Standard Materials

A material is a Generalized Standard Material (GSM, Halphen & Son 1975) if:

A free energy $\Psi(\boldsymbol{\varepsilon}^e, \boldsymbol{\alpha})$ governs reversible response.
A convex dissipation potential $\phi(\dot{\boldsymbol{\alpha}})$ governs irreversible evolution:

\[ \boldsymbol{A} = -\frac{\partial\Psi}{\partial\boldsymbol{\alpha}}, \qquad \dot{\boldsymbol{\alpha}} \in \partial_{\boldsymbol{A}}\phi^*(\boldsymbol{A}) \]

Most standard models (J2 plasticity, linear isotropic damage) fit this framework and inherit thermodynamic consistency automatically.

6.7 Material Frame-Indifference (Objectivity)

The principle of material frame-indifference states that the constitutive law of a material must be independent of the observer.

The material’s response should not depend on whether the observer is stationary, translating at constant velocity, or rotating relative to the material.

Mathematically, if we apply a rigid body motion (superimposed translation and rotation) to the entire system, the form of the constitutive equation must remain the same when expressed in terms of the transformed variables in the new frame.

For a quantity like the Cauchy stress tensor $\boldsymbol{\sigma}$ and deformation gradient $\mathbf{F}$, transformation under a change of frame with time-dependent rotation tensor $\mathbf{Q}(t)$ gives: \[ \boldsymbol{\sigma}^* = \mathbf{Q}(t) \boldsymbol{\sigma} \mathbf{Q}(t)^T, \quad \mathbf{F}^* = \mathbf{Q}(t) \mathbf{F} \]

Frame-indifference requires: if $\boldsymbol{\sigma} = \mathbb{F}(\mathbf{F}, \ldots)$, then $\boldsymbol{\sigma}^* = \mathbb{F}(\mathbf{F}^*, \ldots)$, leading to: \[ \mathbf{Q}(t) \mathbb{F}(\mathbf{F}, \ldots) \mathbf{Q}(t)^T = \mathbb{F}(\mathbf{Q}(t)\mathbf{F}, \ldots) \]

This must hold for all orthogonal tensors $\mathbf{Q}(t)$.

Consequence: The free energy cannot depend on $\mathbf{F}$ directly, but only through objective combinations like: - Right Cauchy-Green tensor: $\mathbf{C} = \mathbf{F}^T\mathbf{F}$ - Green-Lagrange strain: $\mathbf{E} = \frac{1}{2}(\mathbf{C} - \mathbf{I})$ - Left stretch tensor: $\mathbf{U}$ from polar decomposition $\mathbf{F} = \mathbf{R}\mathbf{U}$

So we write: $\Psi = \tilde{\Psi}(\mathbf{C}, \theta)$ or $\Psi = \hat{\Psi}(\mathbf{E}, \theta)$ instead of $\Psi = \Psi(\mathbf{F}, \theta)$.

6.8 Coleman-Noll Procedure

The Coleman-Noll procedure is a systematic method to derive restrictions on constitutive equations by combining:

Material Frame-Indifference (objectivity)
Second Law of Thermodynamics (Clausius-Duhem inequality)

The Key Idea:

The procedure exploits the fact that superimposed rigid motions (especially rotations) can be chosen arbitrarily at any instant. By: - Assuming a general functional form for constitutive relations - Applying the mathematical statement of frame-indifference - Choosing specific, convenient rotations to simplify - Invoking arbitrariness of thermodynamic rates

We can deduce necessary conditions on the form of material functions.

For Hyperelastic Materials:

Assume the thermodynamic state is determined by $\mathbf{F}$ and $\theta$. We postulate: \[ \Psi = \hat{\Psi}(\mathbf{F}, \theta), \quad \boldsymbol{\sigma} = \hat{\boldsymbol{\sigma}}(\mathbf{F}, \theta), \quad \eta = \hat{\eta}(\mathbf{F}, \theta) \]

Using the dissipation inequality (neglecting heat flux): \[ \mathbf{P}:\dot{\mathbf{F}} - \rho_0(\dot{\Psi} + \eta \dot{\theta}) \geq 0 \]

where $\mathbf{P} = J\boldsymbol{\sigma}\mathbf{F}^{-T}$ is the first Piola-Kirchhoff stress.

From objectivity: $\Psi = \tilde{\Psi}(\mathbf{C}, \theta)$ with $\dot{\mathbf{C}} = \dot{\mathbf{F}}^T\mathbf{F} + \mathbf{F}^T\dot{\mathbf{F}}$.

The rate of free energy is: \[ \dot{\Psi} = \frac{\partial\tilde{\Psi}}{\partial\mathbf{C}} : \dot{\mathbf{C}} + \frac{\partial\tilde{\Psi}}{\partial\theta}\dot{\theta} \]

Substituting into the dissipation inequality: \[ \mathbf{P}:\dot{\mathbf{F}} - \rho_0 \left( \frac{\partial\tilde{\Psi}}{\partial\mathbf{C}} : (\dot{\mathbf{F}}^T\mathbf{F} + \mathbf{F}^T\dot{\mathbf{F}}) + \frac{\partial\tilde{\Psi}}{\partial\theta}\dot{\theta} + \eta\dot{\theta} \right) \geq 0 \]

Using tensor identities: \[ \frac{\partial\tilde{\Psi}}{\partial\mathbf{C}} : (\dot{\mathbf{F}}^T\mathbf{F} + \mathbf{F}^T\dot{\mathbf{F}}) = 2\left(\mathbf{F}\frac{\partial\tilde{\Psi}}{\partial\mathbf{C}}\right) : \dot{\mathbf{F}} \]

The inequality becomes: \[ \left( \mathbf{P} - 2\rho_0\mathbf{F}\frac{\partial\tilde{\Psi}}{\partial\mathbf{C}} \right) : \dot{\mathbf{F}} - \rho_0\left( \frac{\partial\tilde{\Psi}}{\partial\theta} + \eta \right) \dot{\theta} \geq 0 \]

6.9 Coleman-Noll Argument: Exploiting Arbitrariness

This inequality must hold for any kinematically and thermodynamically admissible process—meaning arbitrary $\dot{\mathbf{F}}$ and $\dot{\theta}$.

For the temperature rate $\dot{\theta}$:

If the coefficient $\rho_0(\frac{\partial\tilde{\Psi}}{\partial\theta} + \eta)$ were non-zero, we could choose $\dot{\mathbf{F}} = \mathbf{0}$ and select $\dot{\theta}$ with opposite sign to violate the inequality.

Therefore, the coefficient must be zero: \[ \eta = -\frac{\partial\tilde{\Psi}}{\partial\theta} \]

This is the standard thermodynamic relation for entropy.

For the deformation gradient rate $\dot{\mathbf{F}}$:

With the temperature term now zero, we need: \[ \left( \mathbf{P} - 2\rho_0\mathbf{F}\frac{\partial\tilde{\Psi}}{\partial\mathbf{C}} \right) : \dot{\mathbf{F}} \geq 0 \]

For a non-dissipative elastic material, this coefficient must also be zero: \[ \mathbf{P} = 2\rho_0\mathbf{F}\frac{\partial\tilde{\Psi}}{\partial\mathbf{C}} \]

Converting to Cauchy Stress:

Using the relation $\boldsymbol{\sigma} = J^{-1}\mathbf{P}\mathbf{F}^T = (\rho/\rho_0)\mathbf{P}\mathbf{F}^T$:

\[ \boldsymbol{\sigma} = 2\rho\mathbf{F}\frac{\partial\tilde{\Psi}(\mathbf{C}, \theta)}{\partial\mathbf{C}}\mathbf{F}^T \]

Alternatively, using Green-Lagrange strain $\mathbf{E} = \frac{1}{2}(\mathbf{C} - \mathbf{I})$:

\[ \boldsymbol{\sigma} = \rho\mathbf{F}\frac{\partial\hat{\Psi}(\mathbf{E}, \theta)}{\partial\mathbf{E}}\mathbf{F}^T \]

Stress is derived from a potential (the free energy) by differentiation with respect to strain.

6.10 Significance of Coleman-Noll Results

The Coleman-Noll procedure provides:

Rigorous Foundation: Constitutive models are guaranteed to be consistent with fundamental physical laws (objectivity and thermodynamics).
Restricted Possibilities: The set of admissible material model forms is greatly reduced, guiding rational development.
Identification of Variables: Shows that material response depends on objective strain measures (like $\mathbf{C}$ or $\mathbf{E}$), not the full deformation gradient.
Hyperelasticity Condition: A material is hyperelastic if and only if stress can be derived from a free energy potential—no independent dissipation beyond the potential itself.

Entropy Relations: Entropy is not an independent variable but determined by free energy derivatives.
Inelastic Dissipation: When inelastic processes occur (plasticity, damage), they appear as additional dissipation terms beyond zero, constrained to be non-negative by the dissipation inequality. Evolution equations for internal variables must be chosen to maintain $\mathcal{D} \geq 0$.

6.11 Mathematical Identity: Chain Rule for Free Energy

Goal: Show that the free energy rate depends on strain rate via a specific relationship with the deformation gradient.

To prove: \[ \frac{\partial\tilde{\Psi}}{\partial\mathbf{C}} : \dot{\mathbf{C}} = \left( 2\mathbf{F}\frac{\partial\tilde{\Psi}}{\partial\mathbf{C}} \right) : \dot{\mathbf{F}} \]

Key Properties: - $\mathbf{C} = \mathbf{F}^T\mathbf{F}$ (symmetric) - $\dot{\mathbf{C}} = \dot{\mathbf{F}}^T\mathbf{F} + \mathbf{F}^T\dot{\mathbf{F}}$ (symmetric) - $\mathbf{S}_\psi = \frac{\partial\tilde{\Psi}}{\partial\mathbf{C}}$ is symmetric - Tensor trace cyclic property: $\text{tr}(\mathbf{ABC}) = \text{tr}(\mathbf{CAB})$

Proof: Start with LHS using $\mathbf{A} : \mathbf{B} = \text{tr}(\mathbf{A}^T\mathbf{B})$: \[ \text{LHS} = \mathbf{S}_\psi : (\dot{\mathbf{F}}^T\mathbf{F} + \mathbf{F}^T\dot{\mathbf{F}}) \]

For the first term, using trace cyclic property and symmetry: \[ \mathbf{S}_\psi : (\dot{\mathbf{F}}^T\mathbf{F}) = \text{tr}(\mathbf{S}_\psi\dot{\mathbf{F}}^T\mathbf{F}) = \text{tr}(\mathbf{F}\mathbf{S}_\psi\dot{\mathbf{F}}^T) = (\mathbf{F}\mathbf{S}_\psi) : \dot{\mathbf{F}} \]

For the second term: \[ \mathbf{S}_\psi : (\mathbf{F}^T\dot{\mathbf{F}}) = \text{tr}(\mathbf{F}\mathbf{S}_\psi\dot{\mathbf{F}}^T) = (\mathbf{F}\mathbf{S}_\psi) : \dot{\mathbf{F}} \]

Adding both terms: \[ \text{LHS} = 2(\mathbf{F}\mathbf{S}_\psi) : \dot{\mathbf{F}} = \left( 2\mathbf{F}\frac{\partial\tilde{\Psi}}{\partial\mathbf{C}} \right) : \dot{\mathbf{F}} \quad \checkmark \]