Derivatives, Gradients, and Tensor-Valued Functions
📽 Slides: Open presentation
Let \(f = f(\mathbf{F})\) be a scalar. Its derivative with respect to \(\mathbf{F}\) is defined via the directional derivative: \[ df = \frac{\partial f}{\partial F_{ij}}dF_{ij} = \frac{\partial f}{\partial \mathbf{F}}:\,d\mathbf{F} \]
so \(\frac{\partial f}{\partial \mathbf{F}}\) is a second-order tensor with components \(\frac{\partial f}{\partial F_{ij}}\).
The derivative of a second-order tensor \(\mathbf{F}\) with respect to a second-order tensor \(\mathbf{G}\) is a fourth-order tensor: \[ \mathbb{H} = \frac{\partial \mathbf{F}}{\partial \mathbf{G}}, \qquad H_{klij} = \frac{\partial F_{kl}}{\partial G_{ij}}. \]
Useful identity: derivative of a second-order tensor with respect to itself, \[ \frac{\partial \mathbf{A}}{\partial \mathbf{A}} = \mathbb{I}, \qquad \mathbb{I}_{ijkl} = \tfrac{1}{2}(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}). \]
For a symmetric tensor \(\mathbf{A}\), three principal invariants capture all geometric information (see A01 for notation):
\[ I_1(\mathbf{A}) = \operatorname{tr}(\mathbf{A}), \quad I_2(\mathbf{A}) = \tfrac{1}{2}[\operatorname{tr}(\mathbf{A})^2 - \operatorname{tr}(\mathbf{A}^2)], \quad I_3(\mathbf{A}) = \det(\mathbf{A}) \]
Many constitutive functions depend on \(\mathbf{A}\) only through its invariants, so we must compute derivatives like \(\partial f(I_1, I_2, I_3)/\partial \mathbf{A}\) by the chain rule:
\[ \frac{\partial f}{\partial \mathbf{A}} = \frac{\partial f}{\partial I_1}\frac{\partial I_1}{\partial \mathbf{A}} + \frac{\partial f}{\partial I_2}\frac{\partial I_2}{\partial \mathbf{A}} + \frac{\partial f}{\partial I_3}\frac{\partial I_3}{\partial \mathbf{A}} \]
Below we derive each invariant gradient from first principles.
Starting with the definition \(I_1(\mathbf{A}) = \operatorname{tr}(\mathbf{A}) = A_{ii}\), we use index notation:
\[ \frac{\partial I_1}{\partial A_{mn}} = \frac{\partial}{\partial A_{mn}}(A_{ii}) = \delta_{im}\delta_{in} = \delta_{mn} \]
In tensor form, with the identity tensor \(\mathbf{I}\): \[ \boxed{\frac{\partial I_1}{\partial \mathbf{A}} = \mathbf{I}} \]
Physical meaning: The trace is linear in \(\mathbf{A}\), so its gradient is constant.
Starting with \(I_2 = \tfrac{1}{2}(I_1^2 - \operatorname{tr}(\mathbf{A}^2))\), we differentiate each term:
\[ \frac{\partial}{\partial A_{mn}}\tfrac{1}{2}I_1^2 = I_1\frac{\partial I_1}{\partial A_{mn}} = I_1\delta_{mn} \]
For the second term, \(\operatorname{tr}(\mathbf{A}^2) = A_{ij}A_{ji}\): \[ \frac{\partial}{\partial A_{mn}}(A_{ij}A_{ji}) = \delta_{im}\delta_{jn}A_{ji} + A_{ij}\delta_{jm}\delta_{in} = A_{ni} + A_{im} = (A+A^T)_{mn} \]
Since \(\mathbf{A}\) is symmetric, \(\mathbf{A}^T = \mathbf{A}\): \[ \frac{\partial}{\partial A_{mn}}(A_{ij}A_{ji}) = 2A_{mn} \]
Combining: \[ \frac{\partial I_2}{\partial A_{mn}} = I_1\delta_{mn} - A_{mn} \]
In tensor form: \[ \boxed{\frac{\partial I_2}{\partial \mathbf{A}} = I_1\mathbf{I} - \mathbf{A}} \]
Physical meaning: This gradient involves both the trace and the tensor itself; it arises because \(I_2\) mixes first-order and second-order contractions.
This is the most subtle case. We use a perturbation argument. Let \(\mathbf{A} + \epsilon\mathbf{B}\) be a perturbed tensor, and expand the determinant:
\[ \det(\mathbf{A} + \epsilon\mathbf{B}) = \det\mathbf{A}\,\det(\mathbf{I} + \epsilon\mathbf{A}^{-1}\mathbf{B}) \]
For small \(\epsilon\), the characteristic polynomial of \(\epsilon\mathbf{A}^{-1}\mathbf{B}\) gives: \[ \det(\mathbf{I} + \epsilon\mathbf{A}^{-1}\mathbf{B}) = 1 + \epsilon\operatorname{tr}(\mathbf{A}^{-1}\mathbf{B}) + O(\epsilon^2) \]
Substituting back: \[ \det(\mathbf{A} + \epsilon\mathbf{B}) = \det\mathbf{A}\left(1 + \epsilon\operatorname{tr}(\mathbf{A}^{-1}\mathbf{B}) + O(\epsilon^2)\right) \]
Now, \(\operatorname{tr}(\mathbf{A}^{-1}\mathbf{B}) = (\mathbf{A}^{-T})_{ij}B_{ij}\), so the Fréchet derivative is: \[ \delta(\det\mathbf{A}) = \det\mathbf{A}\,(\mathbf{A}^{-T})_{ij}\delta A_{ij} \]
By the chain rule definition of the gradient: \[ \boxed{\frac{\partial I_3}{\partial \mathbf{A}} = \frac{\partial(\det\mathbf{A})}{\partial \mathbf{A}} = I_3\mathbf{A}^{-T} = (\det\mathbf{A})\mathbf{A}^{-T}} \]
Physical meaning: The derivative couples the determinant (volume change) with the inverse-transpose of the tensor. This makes sense dimensionally: to measure how volume changes with strain, we need the strain-inverse.
Invariant Derivatives (Quick Reference)
For a symmetric tensor \(\mathbf{A}\) with invariants \(I_1, I_2, I_3\):
\[ \begin{aligned} \frac{\partial I_1}{\partial \mathbf{A}} &= \mathbf{I} \\ \frac{\partial I_2}{\partial \mathbf{A}} &= I_1\mathbf{I} - \mathbf{A} \\ \frac{\partial I_3}{\partial \mathbf{A}} &= I_3\mathbf{A}^{-T} \end{aligned} \]
Use these with the chain rule to compute derivatives of any invariant-based function.
Suppose we want \(\partial(\operatorname{tr}(\mathbf{A}^2))/\partial\mathbf{A}\). Note that \(\operatorname{tr}(\mathbf{A}^2) = I_1^2 - 2I_2\).
By the chain rule: \[ \frac{\partial}{\partial\mathbf{A}}(I_1^2 - 2I_2) = 2I_1\frac{\partial I_1}{\partial\mathbf{A}} - 2\frac{\partial I_2}{\partial\mathbf{A}} \]
\[ = 2I_1\mathbf{I} - 2(I_1\mathbf{I} - \mathbf{A}) = 2I_1\mathbf{I} - 2I_1\mathbf{I} + 2\mathbf{A} = 2\mathbf{A} \]
This makes sense: \(\operatorname{tr}(\mathbf{A}^2)\) is bilinear in \(\mathbf{A}\), so the gradient scales linearly with \(\mathbf{A}\).
For a scalar field \(\phi(\mathbf{x})\): \[ \nabla\phi = \frac{\partial\phi}{\partial x_i}\mathbf{e}_i \]
For a vector field \(\mathbf{v}(\mathbf{x})\): \[ (\nabla\mathbf{v})_{ij} = \frac{\partial v_i}{\partial x_j} \quad \Rightarrow \quad \text{a second-order tensor} \]
Divergence of a vector: \(\nabla\cdot\mathbf{v} = \frac{\partial v_i}{\partial x_i}\)
Divergence of a second-order tensor: \((\nabla\cdot\mathbf{T})_i = \frac{\partial T_{ij}}{\partial x_j}\)
Gauss (Divergence) Theorem: \[ \int_\Omega \nabla\cdot\mathbf{v}\,dV = \int_{\partial\Omega} \mathbf{v}\cdot\mathbf{n}\,dA \]
For a tensor: \[ \int_\Omega \nabla\cdot\mathbf{T}\,dV = \int_{\partial\Omega} \mathbf{T}\mathbf{n}\,dA \]
These theorems convert body integrals to boundary integrals — essential for the weak form in FEM. We will expand on these theorems and their applications in Section “Gauss Divergence Theorem” later in this chapter.
A fundamental principle of mechanics states that material properties are independent of the observer. A quantity is called objective (or frame-indifferent) if its form does not change when observed from different (moving) reference frames related by a superposed rigid-body motion.
Consider two observers: one in a fixed frame and one in a frame undergoing a superposed rigid-body motion with orthogonal rotation tensor \(\mathbf{Q}(t)\) and translation \(\mathbf{c}(t)\):
\[ \mathbf{x}^* = \mathbf{Q}(t)\mathbf{x} + \mathbf{c}(t), \quad \mathbf{Q}^T(t)\mathbf{Q}(t) = \mathbf{I}, \quad \det\mathbf{Q} = 1 \]
Different tensor quantities transform differently:
Right Cauchy-Green tensor: Since \(\mathbf{F}^* = \mathbf{Q}\mathbf{F}\): \[ \mathbf{C}^* = (\mathbf{F}^*)^T\mathbf{F}^* = \mathbf{F}^T\mathbf{Q}^T\mathbf{Q}\mathbf{F} = \mathbf{F}^T\mathbf{F} = \mathbf{C} \] So \(\mathbf{C}\) is invariant (objective in the reference configuration).
Green-Lagrange strain: Similarly, \(\mathbf{E}^* = \frac{1}{2}(\mathbf{C}^* - \mathbf{I}) = \mathbf{E}\). Invariant.
Left Cauchy-Green (Finger) tensor: In the current configuration: \[ \mathbf{b}^* = \mathbf{F}^*(\mathbf{F}^*)^T = \mathbf{Q}\mathbf{F}\mathbf{F}^T\mathbf{Q}^T = \mathbf{Q}\mathbf{b}\mathbf{Q}^T \] So \(\mathbf{b}\) is objective (transforms as a proper 2nd-order tensor).
Almansi strain: Similarly, \(\mathbf{e}^* = \mathbf{Q}\mathbf{e}\mathbf{Q}^T\). Objective.
Cauchy stress (force per current area) transforms as: \[ \boldsymbol{\sigma}^* = \mathbf{Q}\boldsymbol{\sigma}\mathbf{Q}^T \] Objective.
1st Piola-Kirchhoff stress (force per reference area): \[ \mathbf{P}^* = \mathbf{Q}\mathbf{P} \] Two-point tensor (not fully objective; the right leg does not rotate because area is in the reference frame).
2nd Piola-Kirchhoff stress (purely material): \[ \mathbf{S}^* = \mathbf{S} \] Invariant (objective scalar-like in the reference frame).
An objective tensor \(\mathbf{T}\) remains objective under the superposed motion, but its material time derivative does not:
\[ \dot{\mathbf{T}}^* = \frac{d}{dt}[\mathbf{Q}(t)\mathbf{T}(t)\mathbf{Q}^T(t)] \]
Using the product rule: \[ \dot{\mathbf{T}}^* = \dot{\mathbf{Q}}\mathbf{T}\mathbf{Q}^T + \mathbf{Q}\dot{\mathbf{T}}\mathbf{Q}^T + \mathbf{Q}\mathbf{T}\dot{\mathbf{Q}}^T \]
If \(\dot{\mathbf{T}}^* = \mathbf{Q}\dot{\mathbf{T}}\mathbf{Q}^T\) were to hold, the first and third terms would vanish. But they don’t—they involve \(\dot{\mathbf{Q}}\).
Define the spin of the superposed frame: \[ \boldsymbol{\Omega} = \dot{\mathbf{Q}}^T\mathbf{Q} \] (skew-symmetric). Then: \[ \dot{\mathbf{T}}^* = \mathbf{Q}(\dot{\mathbf{T}} - \boldsymbol{\Omega}\mathbf{T} + \mathbf{T}\boldsymbol{\Omega}^T)\mathbf{Q}^T \]
The material time derivative \(\dot{\mathbf{T}}\) does not transform as an objective tensor.
Consider a hypoelastic law \(\dot{\boldsymbol{\sigma}} = \mathbb{H}:\mathbf{d}\). If the frame rotates (\(\boldsymbol{\Omega} \neq 0\)) but the stress does not change in the rotating frame (\(\dot{\boldsymbol{\sigma}}^* = 0\)), then:
\[ \dot{\boldsymbol{\sigma}} = \mathbf{Q}(-\boldsymbol{\Omega}\boldsymbol{\sigma} + \boldsymbol{\sigma}\boldsymbol{\Omega}^T)\mathbf{Q}^T \neq 0 \]
This would induce spurious strains \(\mathbf{d} \neq 0\) even though the material state is unchanged in the rotating frame—a physically inadmissible result.
Solution: Use objective rate tensors that automatically subtract off the spin contributions.
The Jaumann rate removes the spin of the continuum itself. Let \(\mathbf{W} = \operatorname{skew}(\mathbf{l})\) be the continuum spin tensor (with axial vector \(\mathbf{w}\) such that \(\mathbf{W}\mathbf{v} = \mathbf{w}\times\mathbf{v}\)):
\[ \boxed{\overset{\triangledown}{\mathbf{T}} = \dot{\mathbf{T}} - \mathbf{W}\mathbf{T} + \mathbf{T}\mathbf{W}} \]
Proof of objectivity: \[ \overset{\triangledown}{\mathbf{T}}^* = \dot{\mathbf{T}}^* - \mathbf{W}^*\mathbf{T}^* + \mathbf{T}^*(\mathbf{W}^*) \]
where \(\mathbf{W}^* = \mathbf{Q}\mathbf{W}\mathbf{Q}^T\) (since \(\mathbf{l}^* = \mathbf{Q}(\mathbf{l}-\boldsymbol{\Omega})\mathbf{Q}^T\), the skew part gives \(\mathbf{W}^* = \mathbf{Q}\mathbf{W}\mathbf{Q}^T\)). Substituting:
\[ \overset{\triangledown}{\mathbf{T}}^* = \mathbf{Q}(\dot{\mathbf{T}} - \boldsymbol{\Omega}\mathbf{T} + \mathbf{T}\boldsymbol{\Omega}^T)\mathbf{Q}^T - \mathbf{Q}\mathbf{W}\mathbf{Q}^T\mathbf{Q}\mathbf{T}\mathbf{Q}^T + \mathbf{Q}\mathbf{T}\mathbf{Q}^T\mathbf{Q}\mathbf{W}\mathbf{Q}^T \]
\[ = \mathbf{Q}[\dot{\mathbf{T}} - \boldsymbol{\Omega}\mathbf{T} + \mathbf{T}\boldsymbol{\Omega}^T - \mathbf{W}\mathbf{T} + \mathbf{T}\mathbf{W}]\mathbf{Q}^T \]
The \(\boldsymbol{\Omega}\) terms cancel if \(\boldsymbol{\Omega}\) is skew (which it is), leaving:
\[ \overset{\triangledown}{\mathbf{T}}^* = \mathbf{Q}[\dot{\mathbf{T}} - \mathbf{W}\mathbf{T} + \mathbf{T}\mathbf{W}]\mathbf{Q}^T = \mathbf{Q}\overset{\triangledown}{\mathbf{T}}\mathbf{Q}^T \]
Objective!
Physical interpretation: The Jaumann rate removes the rigid rotation of the continuum. In a frame that co-rotates with the material, the rate is computed in that rotating frame.
The Oldroyd rate uses the full velocity gradient (not just spin):
\[ \boxed{\overset{\triangle}{\mathbf{T}} = \dot{\mathbf{T}} - \mathbf{L}\mathbf{T} - \mathbf{T}\mathbf{L}^T} \]
where \(\mathbf{L} = \mathbf{l} = \dot{\mathbf{F}}\mathbf{F}^{-1}\) is the velocity gradient.
Proof (sketch): Under the superposed motion, \(\mathbf{L}^* = \mathbf{Q}(\mathbf{L}-\boldsymbol{\Omega})\mathbf{Q}^T\), so:
\[ \overset{\triangle}{\mathbf{T}}^* = \mathbf{Q}(\dot{\mathbf{T}} - \boldsymbol{\Omega}\mathbf{T} + \mathbf{T}\boldsymbol{\Omega}^T - \mathbf{L}\mathbf{T} - \mathbf{T}\mathbf{L}^T)\mathbf{Q}^T \]
Again the \(\boldsymbol{\Omega}\) spin terms cancel (skew-symmetric), and:
\[ \overset{\triangle}{\mathbf{T}}^* = \mathbf{Q}\overset{\triangle}{\mathbf{T}}\mathbf{Q}^T \]
Objective!
Physical interpretation: The Oldroyd rate accounts for the full deformation (strain + rotation) in the current configuration. It is the Lie derivative along the velocity field, and is commonly used in viscoelasticity and finite-strain plasticity.
The Truesdell rate additionally corrects for volume change:
\[ \boxed{\overset{\circ}{\mathbf{T}} = \dot{\mathbf{T}} - \mathbf{L}\mathbf{T} - \mathbf{T}\mathbf{L}^T + (\operatorname{tr}\mathbf{L})\mathbf{T}} \]
The term \((\operatorname{tr}\mathbf{L})\mathbf{T}\) (proportional to the volumetric strain rate \(\operatorname{tr}\mathbf{d}\)) restores objective status when volume changes couple to stress evolution.
Objectivity proof: Similar to Oldroyd, with \(\operatorname{tr}\mathbf{L}\) being objective.
Physical interpretation: Used in hyperelastic models where the strain energy couples to both distortion and volume. The Truesdell rate ensures that hydrostatic stress changes are handled correctly.
Objective Rate Tensors
| Rate | Formula | Use Case |
|---|---|---|
| Jaumann | \(\overset{\triangledown}{\mathbf{T}} = \dot{\mathbf{T}} - \mathbf{W}\mathbf{T} + \mathbf{T}\mathbf{W}\) | Small-strain plasticity, materials with negligible convection |
| Oldroyd | \(\overset{\triangle}{\mathbf{T}} = \dot{\mathbf{T}} - \mathbf{L}\mathbf{T} - \mathbf{T}\mathbf{L}^T\) | Viscoelasticity, finite-strain models with convection |
| Truesdell | \(\overset{\circ}{\mathbf{T}} = \dot{\mathbf{T}} - \mathbf{L}\mathbf{T} - \mathbf{T}\mathbf{L}^T + (\operatorname{tr}\mathbf{L})\mathbf{T}\) | Hyperelasticity with volumetric coupling |
All three transform as objective 2nd-order tensors under superposed rigid motions, making them suitable for frame-indifferent constitutive laws.
Consider a material element undergoing simple shear at constant shear rate \(\dot{\gamma}\):
\[ \mathbf{v} = \dot{\gamma}y\mathbf{e}_1, \quad \mathbf{L} = \begin{pmatrix} 0 & \dot{\gamma} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]
The spin and strain rate are: \[ \mathbf{W} = \frac{1}{2}\begin{pmatrix} 0 & \dot{\gamma} & 0 \\ -\dot{\gamma} & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad \mathbf{D} = \frac{1}{2}\begin{pmatrix} 0 & \dot{\gamma} & 0 \\ \dot{\gamma} & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]
For a stress state \(\boldsymbol{\sigma} = \sigma_{12}\mathbf{e}_1\otimes\mathbf{e}_2 + \sigma_{12}\mathbf{e}_2\otimes\mathbf{e}_1\) (shear only), the material time derivative is \(\dot{\boldsymbol{\sigma}} = \dot{\sigma}_{12}(\mathbf{e}_1\otimes\mathbf{e}_2 + \mathbf{e}_2\otimes\mathbf{e}_1)\).
The Jaumann rate modifies this by: \[ \overset{\triangledown}{\boldsymbol{\sigma}}_{12} = \dot{\sigma}_{12} - \mathbf{W}\boldsymbol{\sigma}\cdot\mathbf{e}_2\otimes\mathbf{e}_1 + \cdots \]
This removes the spurious stress oscillations that would arise from the material element spinning without actual shear strain changing—a well-known issue in nonlinear dynamics simulations.
Mapping between material (Lagrangian) and spatial (Eulerian) descriptions:
| Operation | Formula |
|---|---|
| Push-forward of covariant 2-tensor | \(\varphi_*\mathbf{S} = \mathbf{F}^{-T}\mathbf{S}\mathbf{F}^{-1}\) |
| Pull-back of contravariant 2-tensor | \(\varphi^*\boldsymbol{\tau} = \mathbf{F}^T\boldsymbol{\tau}\mathbf{F}\) |
| Lie derivative | \(\mathfrak{L}_v\mathbf{b} = \mathbf{F}\,\frac{d}{dt}(\mathbf{F}^{-1}\mathbf{b}\mathbf{F}^{-T})\,\mathbf{F}^T\) |
Consider a body deforming between: - Reference configuration \(\mathcal{\Omega}_0\) with coordinates \(\mathbf{X}\) - Current configuration \(\mathcal{\Omega}_t\) with coordinates \(\mathbf{x}\)
The mapping: \(\mathbf{x} = \boldsymbol{\varphi}(\mathbf{X}, t)\)
The deformation gradient: \[ \mathbf{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \nabla_X \mathbf{x} \]
is a two-point tensor relating infinitesimal material line elements: \[ d\mathbf{x} = \mathbf{F} \cdot d\mathbf{X} \]
Key property: \(J = \det(\mathbf{F}) > 0\) (volume ratio between configurations).
The right Cauchy-Green deformation tensor: \[ \mathbf{C} = \mathbf{F}^T \cdot \mathbf{F} \] Symmetric, positive-definite, describes deformation in the reference configuration (material description).
The left Cauchy-Green (Finger) deformation tensor: \[ \mathbf{b} = \mathbf{F} \cdot \mathbf{F}^T \] Symmetric, positive-definite, describes deformation in the current configuration (spatial description). Lowercase per the canonical convention (A.3.1, A.4): \(\mathbf{b}\) lives on the current configuration, while the uppercase \(\mathbf{C}\) lives on the reference configuration.
Relationship: \(\mathbf{b} = \mathbf{F} \cdot \mathbf{C} \cdot \mathbf{F}^{-T}\) (spatial push-forward of \(\mathbf{C}\)).
Green-Lagrange strain tensor (material description): \[ \mathbf{E} = \frac{1}{2}(\mathbf{C} - \mathbf{I}) = \frac{1}{2}(\mathbf{F}^T \cdot \mathbf{F} - \mathbf{I}) \]
Almansi strain tensor (spatial description): \[ \mathbf{e} = \frac{1}{2}(\mathbf{I} - \mathbf{b}^{-1}) = \frac{1}{2}(\mathbf{I} - \mathbf{F}^{-1} \cdot \mathbf{F}^{-T}) \]
Relationship: Pull-back of Almansi to reference config gives Green-Lagrange: \[ \mathbf{E} = \mathbf{F}^T \cdot \mathbf{e} \cdot \mathbf{F} \]
Small-strain limit: For small displacements \(\mathbf{u}\) with \(|\nabla \mathbf{u}| \ll 1\): \[ \boldsymbol{\varepsilon} = \frac{1}{2}(\nabla \mathbf{u} + (\nabla \mathbf{u})^T) = \text{sym}(\nabla \mathbf{u}) \]
The deformation gradient admits a unique decomposition: \[ \mathbf{F} = \mathbf{R} \cdot \mathbf{U} = \mathbf{V} \cdot \mathbf{R} \]
where: - \(\mathbf{R}\): orthogonal rotation tensor (\(\mathbf{R}^T \mathbf{R} = \mathbf{I}\), \(\det(\mathbf{R}) = 1\)) - \(\mathbf{U}\): right stretch tensor (symmetric, positive-definite, in reference config) - \(\mathbf{V}\): left stretch tensor (symmetric, positive-definite, in current config)
Relations: \[ \mathbf{U} = \sqrt{\mathbf{C}}, \quad \mathbf{V} = \sqrt{\mathbf{b}}, \quad \mathbf{V} = \mathbf{R} \mathbf{U} \mathbf{R}^T \]
Spectral decomposition: \(\mathbf{C}\) and \(\mathbf{b}\) share eigenvalues \(\lambda_i^2\) (principal stretches squared), but eigenvectors differ — \(\mathbf{N}_i\) (reference, uppercase) for \(\mathbf{C}\), \(\mathbf{n}_i\) (current, lowercase) for \(\mathbf{b}\): \[ \mathbf{C} = \sum_{i=1}^{3} \lambda_i^2 \mathbf{N}_i \otimes \mathbf{N}_i, \quad \mathbf{b} = \sum_{i=1}^{3} \lambda_i^2 \mathbf{n}_i \otimes \mathbf{n}_i \] with \(\mathbf{n}_i = \mathbf{R} \mathbf{N}_i\).
Volume change: A volume element transforms as \[ dV = J \, dV_0, \quad J = \det(\mathbf{F}) \]
Isochoric/isovolumetric decomposition: \[ \mathbf{F} = J^{1/3} \bar{\mathbf{F}}, \quad \det(\bar{\mathbf{F}}) = 1 \]
where \(\bar{\mathbf{F}}\) represents the shape-changing (distortional) part and \(J^{1/3}\) represents the volumetric part.
Area change (Nanson’s formula): Surface normal transforms as \[ \mathbf{n} \, dA = J \mathbf{F}^{-T} \cdot \mathbf{N} \, dA_0 \]
where \(\mathbf{N}\) and \(\mathbf{n}\) are normals in reference and current configurations.
Velocity gradient: \[ \mathbf{l} = \dot{\mathbf{F}} \mathbf{F}^{-1} = \frac{\partial \mathbf{v}}{\partial \mathbf{x}} \]
where \(\mathbf{v} = \dot{\mathbf{x}}\) is the material velocity.
Decomposition into symmetric and skew-symmetric parts: \[ \mathbf{l} = \mathbf{d} + \mathbf{w} \]
Volumetric strain rate: \[ \dot{J} = J \, \text{tr}(\mathbf{d}) = J \, \text{div}(\mathbf{v}) \]
Cauchy (true) stress \(\boldsymbol{\sigma}\): Force per current area. \[ \text{Transformation: } \boldsymbol{\sigma}^* = \mathbf{Q} \boldsymbol{\sigma} \mathbf{Q}^T \quad \text{(objective)} \]
Nominal stress (1st Piola-Kirchhoff) \(\mathbf{P}\): Force per reference area (one-point tensor). \[ \mathbf{P} = J \boldsymbol{\sigma} \mathbf{F}^{-T}, \quad \text{Transformation: } \mathbf{P}^* = \mathbf{Q} \mathbf{P} \quad \text{(two-point, not objective)} \]
2nd Piola-Kirchhoff stress \(\mathbf{S}\): Conjugate to Green strain (material description). \[ \mathbf{S} = J \mathbf{F}^{-1} \boldsymbol{\sigma} \mathbf{F}^{-T}, \quad \text{Transformation: } \mathbf{S}^* = \mathbf{S} \quad \text{(invariant/objective)} \]
Kirchhoff stress: \[ \boldsymbol{\tau} = J \boldsymbol{\sigma} = \mathbf{F} \mathbf{S} \mathbf{F}^T \]
As defined in Section “Objective Rates,” a spatial tensor \(\mathbf{T}\) is objective (frame-indifferent) if, under a superimposed rigid motion \(\mathbf{Q}(t)\): \[ \mathbf{T}^* = \mathbf{Q}(t) \mathbf{T} \mathbf{Q}^T(t) \]
Problem with material time derivative: The rate \(\dot{\mathbf{T}}\) of an objective tensor is not objective: \[ \dot{\mathbf{T}}^* \neq \mathbf{Q} \dot{\mathbf{T}} \mathbf{Q}^T \]
Instead: \[ \dot{\mathbf{T}}^* = \mathbf{Q}(\dot{\mathbf{T}} - \mathbf{\Omega}\mathbf{T} + \mathbf{T}\mathbf{\Omega}^T)\mathbf{Q}^T \]
where \(\mathbf{\Omega} = \dot{\mathbf{Q}}^T \mathbf{Q}\) is the spin of the rotating frame.
Building on the integral theorems introduced earlier (Section “Integral Theorems”), we now explore their applications in depth. For a region \(\Omega\) with boundary \(\partial\Omega\) and outward normal \(\mathbf{n}\):
Vector form: \[ \int_\Omega \nabla \cdot \mathbf{v} \, dV = \int_{\partial\Omega} \mathbf{v} \cdot \mathbf{n} \, dA \]
Tensor form: \[ \int_\Omega \nabla \cdot \mathbf{T} \, dV = \int_{\partial\Omega} \mathbf{T} \mathbf{n} \, dA \]
Essential for: - Deriving equilibrium equations from stress divergence - Formulating weak forms in finite element methods - Relating volume and surface integrals in conservation laws
Example: From \(\int_\Omega (\nabla \cdot \boldsymbol{\sigma}) \, dV = -\rho\int_\Omega \mathbf{a} \, dV\) (equilibrium without body force), we get: \[ \int_{\partial\Omega} \boldsymbol{\sigma} \mathbf{n} \, dA = -\rho \int_\Omega \mathbf{a} \, dV \]
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