Tensor derivative (continuum mechanics)

The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.^[1]

The directional derivative provides a systematic way of finding these derivatives.^[2]

1 Derivatives with respect to vectors and second-order tensors
2 Gradient of a tensor field
- 2.1 Cartesian coordinates
- 2.2 Curvilinear coordinates
  - 2.2.1 Cylindrical polar coordinates
3 Divergence of a tensor field
- 3.1 Cartesian coordinates
- 3.2 Curvilinear coordinates
  - 3.2.1 Cylindrical polar coordinates
4 Curl of a tensor field
5 Derivative of the determinant of a second-order tensor
6 Derivatives of the invariants of a second-order tensor
7 Derivative of the second-order identity tensor
8 Derivative of a second-order tensor with respect to itself
9 Derivative of the inverse of a second-order tensor
10 Integration by parts
11 References
12 See also

Derivatives with respect to vectors and second-order tensors

The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.

Derivatives of scalar valued functions of vectors

Let $f(\mathbf{v})$ be a real valued function of the vector $\mathbf{v}$ . Then the derivative of $f(\mathbf{v})$ with respect to $\mathbf{v}$ (or at $\mathbf{v}$ ) in the direction $\mathbf{u}$ is the vector defined as

$\frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = Df(\mathbf{v})[\mathbf{u}] = \left[\frac{d }{d \alpha}~f(\mathbf{v} %2B \alpha~\mathbf{u})\right]_{\alpha = 0}$

for all vectors $\mathbf{u}$ .

Properties:

1) If $f(\mathbf{v}) = f_1(\mathbf{v}) %2B f_2(\mathbf{v})$ then $\frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = \left(\frac{\partial f_1}{\partial \mathbf{v}} %2B \frac{\partial f_2}{\partial \mathbf{v}}\right)\cdot\mathbf{u}$

2) If $f(\mathbf{v}) = f_1(\mathbf{v})~ f_2(\mathbf{v})$ then $\frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = \left(\frac{\partial f_1}{\partial \mathbf{v}}\cdot\mathbf{u}\right)~f_2(\mathbf{v}) %2B f_1(\mathbf{v})~\left(\frac{\partial f_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)$

3) If $f(\mathbf{v}) = f_1(f_2(\mathbf{v}))$ then $\frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = \frac{\partial f_1}{\partial f_2}~\frac{\partial f_2}{\partial \mathbf{v}}\cdot\mathbf{u}$

Derivatives of vector valued functions of vectors

Let $\mathbf{f}(\mathbf{v})$ be a vector valued function of the vector $\mathbf{v}$ . Then the derivative of $\mathbf{f}(\mathbf{v})$ with respect to $\mathbf{v}$ (or at $\mathbf{v}$ ) in the direction $\mathbf{u}$ is the second order tensor defined as

$\frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = D\mathbf{f}(\mathbf{v})[\mathbf{u}] = \left[\frac{d }{d \alpha}~\mathbf{f}(\mathbf{v} %2B \alpha~\mathbf{u})\right]_{\alpha = 0}$

for all vectors $\mathbf{u}$ .

Properties:

1) If $\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{v}) %2B \mathbf{f}_2(\mathbf{v})$ then $\frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = \left(\frac{\partial \mathbf{f}_1}{\partial \mathbf{v}} %2B \frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\right)\cdot\mathbf{u}$

2) If $\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{v})\times\mathbf{f}_2(\mathbf{v})$ then $\frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = \left(\frac{\partial \mathbf{f}_1}{\partial \mathbf{v}}\cdot\mathbf{u}\right)\times\mathbf{f}_2(\mathbf{v}) %2B \mathbf{f}_1(\mathbf{v})\times\left(\frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)$

3) If $\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{f}_2(\mathbf{v}))$ then $\frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = \frac{\partial \mathbf{f}_1}{\partial \mathbf{f}_2}\cdot\left(\frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)$

Derivatives of scalar valued functions of second-order tensors

Let $f(\boldsymbol{S})$ be a real valued function of the second order tensor $\boldsymbol{S}$ . Then the derivative of $f(\boldsymbol{S})$ with respect to $\boldsymbol{S}$ (or at $\boldsymbol{S}$ ) in the direction $\boldsymbol{T}$ is the second order tensor defined as

$\frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = Df(\boldsymbol{S})[\boldsymbol{T}] = \left[\frac{d }{d \alpha}~f(\boldsymbol{S} %2B \alpha~\boldsymbol{T})\right]_{\alpha = 0}$

for all second order tensors $\boldsymbol{T}$ .

Properties:

1) If $f(\boldsymbol{S}) = f_1(\boldsymbol{S}) %2B f_2(\boldsymbol{S})$ then $\frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \left(\frac{\partial f_1}{\partial \boldsymbol{S}} %2B \frac{\partial f_2}{\partial \boldsymbol{S}}\right):\boldsymbol{T}$

2) If $f(\boldsymbol{S}) = f_1(\boldsymbol{S})~ f_2(\boldsymbol{S})$ then $\frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \left(\frac{\partial f_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)~f_2(\boldsymbol{S}) %2B f_1(\boldsymbol{S})~\left(\frac{\partial f_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right)$

3) If $f(\boldsymbol{S}) = f_1(f_2(\boldsymbol{S}))$ then $\frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \frac{\partial f_1}{\partial f_2}~\left(\frac{\partial f_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right)$

Derivatives of tensor valued functions of second-order tensors

Let $\boldsymbol{F}(\boldsymbol{S})$ be a second order tensor valued function of the second order tensor $\boldsymbol{S}$ . Then the derivative of $\boldsymbol{F}(\boldsymbol{S})$ with respect to $\boldsymbol{S}$ (or at $\boldsymbol{S}$ ) in the direction $\boldsymbol{T}$ is the fourth order tensor defined as

$\frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = D\boldsymbol{F}(\boldsymbol{S})[\boldsymbol{T}] = \left[\frac{d }{d \alpha}~\boldsymbol{F}(\boldsymbol{S} %2B \alpha~\boldsymbol{T})\right]_{\alpha = 0}$

for all second order tensors $\boldsymbol{T}$ .

Properties:

1) If $\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{S}) %2B \boldsymbol{F}_2(\boldsymbol{S})$ then $\frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = \left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}} %2B \frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}\right):\boldsymbol{T}$

2) If $\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{S})\cdot\boldsymbol{F}_2(\boldsymbol{S})$ then $\frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = \left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)\cdot\boldsymbol{F}_2(\boldsymbol{S}) %2B \boldsymbol{F}_1(\boldsymbol{S})\cdot\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right)$

3) If $\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{F}_2(\boldsymbol{S}))$ then $\frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = \frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{F}_2}:\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right)$

4) If $f(\boldsymbol{S}) = f_1(\boldsymbol{F}_2(\boldsymbol{S}))$ then $\frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \frac{\partial f_1}{\partial \boldsymbol{F}_2}:\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right)$

Gradient of a tensor field

The gradient, $\boldsymbol{\nabla}\boldsymbol{T}$ , of a tensor field $\boldsymbol{T}(\mathbf{x})$ in the direction of an arbitrary constant vector $\mathbf{c}$ is defined as:

$\boldsymbol{\nabla}\boldsymbol{T}\cdot\mathbf{c} = \left.\cfrac{d}{d\alpha}~\boldsymbol{T}(\mathbf{x}%2B\alpha\mathbf{c})\right|_{\alpha=0}$

The gradient of a tensor field of order $n$ is a tensor field of order $n%2B1$ .

Cartesian coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

If $\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3$ are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by ( $x_1, x_2, x_3$ ), then the gradient of the tensor field $\boldsymbol{T}$ is given by

$\boldsymbol{\nabla}\boldsymbol{T} = \cfrac{\partial{\boldsymbol{T}}}{\partial x_i}\otimes\mathbf{e}_i$

Proof

The vectors $\mathbf{x}$ and $\mathbf{c}$ can be written as $\mathbf{x} = x_i~\mathbf{e}_i$ and $\mathbf{c} = c_i~\mathbf{e}_i$ . Let $\mathbf{y}�:= \mathbf{x} %2B \alpha~\mathbf{c}$ . In that case the gradient is given by

$\begin{align} \boldsymbol{\nabla}\boldsymbol{T}\cdot\mathbf{c} & = \left.\cfrac{d}{d\alpha}~\boldsymbol{T}(x_1%2B\alpha c_1, x_2 %2B \alpha c_2, x_3 %2B \alpha c_3)\right|_{\alpha=0} \equiv \left.\cfrac{d}{d\alpha}~\boldsymbol{T}(y_1, y_2, y_3)\right|_{\alpha=0} \\ & = \left [\cfrac{\partial{\boldsymbol{T}}}{\partial y_1}~\cfrac{\partial y_1}{\partial \alpha} %2B \cfrac{\partial{\boldsymbol{T}}}{\partial y_2}~\cfrac{\partial y_2}{\partial \alpha} %2B \cfrac{\partial{\boldsymbol{T}}}{\partial y_3}~\cfrac{\partial y_3}{\partial \alpha} \right]_{\alpha=0} = \left [\cfrac{\partial{\boldsymbol{T}}}{\partial y_1}~c_1 %2B \cfrac{\partial{\boldsymbol{T}}}{\partial y_2}~c_2 %2B \cfrac{\partial{\boldsymbol{T}}}{\partial y_3}~c_3 \right]_{\alpha=0} \\ & = \cfrac{\partial{\boldsymbol{T}}}{\partial x_1}~c_1 %2B \cfrac{\partial{\boldsymbol{T}}}{\partial x_2}~c_2 %2B \cfrac{\partial{\boldsymbol{T}}}{\partial x_3}~c_3 \equiv \cfrac{\partial{\boldsymbol{T}}}{\partial x_i}~c_i = \cfrac{\partial{\boldsymbol{T}}}{\partial x_i}~(\mathbf{e}_i\cdot\mathbf{c}) = \left[\cfrac{\partial{\boldsymbol{T}}}{\partial x_i}\otimes\mathbf{e}_i\right]\cdot\mathbf{c} \qquad \square \end{align}$

Since the basis vectors do not vary in a Cartesian coordinate system we have the following relations for the gradients of a scalar field $\phi$ , a vector field $\mathbf{v}$ , and a second-order tensor field $\boldsymbol{S}$ .

$\begin{align} \boldsymbol{\nabla}\phi & = \cfrac{\partial\phi}{\partial x_i}~\mathbf{e}_i \\ \boldsymbol{\nabla}\mathbf{v} & = \cfrac{\partial (v_j \mathbf{e}_j)}{\partial x_i}\otimes\mathbf{e}_i = \cfrac{\partial v_j}{\partial x_i}~\mathbf{e}_j\otimes\mathbf{e}_i\\ \boldsymbol{\nabla}\boldsymbol{S} & = \cfrac{\partial (S_{jk} \mathbf{e}_j\otimes\mathbf{e}_k)}{\partial x_i}\otimes\mathbf{e}_i = \cfrac{\partial S_{jk}}{\partial x_i}~\mathbf{e}_j\otimes\mathbf{e}_k\otimes\mathbf{e}_i \end{align}$

Curvilinear coordinates

Main article: Curvilinear coordinates

Note: the Einstein summation convention of summing on repeated indices is used below.

If $\mathbf{g}^1,\mathbf{g}^2,\mathbf{g}^3$ are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by ( $\xi^1, \xi^2, \xi^3$ ), then the gradient of the tensor field $\boldsymbol{T}$ is given by (see ^[3] for a proof.)

$\boldsymbol{\nabla}\boldsymbol{T} = \cfrac{\partial{\boldsymbol{T}}}{\partial \xi^i}\otimes\mathbf{g}^i$

From this definition we have the following relations for the gradients of a scalar field $\phi$ , a vector field $\mathbf{v}$ , and a second-order tensor field $\boldsymbol{S}$ .

$\begin{align} \boldsymbol{\nabla}\phi & = \cfrac{\partial\phi}{\partial \xi^i}~\mathbf{g}^i \\ \boldsymbol{\nabla}\mathbf{v} & = \cfrac{\partial (v^j \mathbf{g}_j)}{\partial \xi^i}\otimes\mathbf{g}^i = \left(\cfrac{\partial v^j}{\partial \xi^i} %2B v^k~\Gamma_{ik}^j\right)~\mathbf{g}_j\otimes\mathbf{g}^i = \left(\cfrac{\partial v_j}{\partial \xi^i} - v_k~\Gamma_{ij}^k\right)~\mathbf{g}^j\otimes\mathbf{g}^i\\ \boldsymbol{\nabla}\boldsymbol{S} & = \cfrac{\partial (S_{jk}~\mathbf{g}^j\otimes\mathbf{g}^k)}{\partial \xi^i}\otimes\mathbf{g}^i = \left(\cfrac{\partial S_{jk}}{\partial \xi_i}- S_{lk}~\Gamma_{ij}^l - S_{jl}~\Gamma_{ik}^l\right)~\mathbf{g}^j\otimes\mathbf{g}^k\otimes\mathbf{g}^i \end{align}$

where the Christoffel symbol $\Gamma_{ij}^k$ is defined using

$\Gamma_{ij}^k~\mathbf{g}_k = \cfrac{\partial \mathbf{g}_i}{\partial \xi^j} \quad \implies \quad \Gamma_{ij}^k = \cfrac{\partial \mathbf{g}_i}{\partial \xi^j}\cdot\mathbf{g}_k = -\mathbf{g}_i\cdot\cfrac{\partial \mathbf{g}^k}{\partial \xi^j}$

Cylindrical polar coordinates

In cylindrical coordinates, the gradient is given by

$\begin{align} \boldsymbol{\nabla}\phi & = \cfrac{\partial \phi}{\partial r}~\mathbf{e}_r %2B \cfrac{1}{r}~\cfrac{\partial \phi}{\partial \theta}~\mathbf{e}_\theta %2B \cfrac{\partial \phi}{\partial z}~\mathbf{e}_z\ \\ \boldsymbol{\nabla}\mathbf{v} & = \cfrac{\partial v_r}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_r %2B \cfrac{1}{r}\left(\cfrac{\partial v_r}{\partial \theta} - v_\theta\right)~\mathbf{e}_r\otimes\mathbf{e}_\theta %2B \cfrac{\partial v_r}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_z \\ & %2B \cfrac{\partial v_\theta}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_r %2B \cfrac{1}{r}\left(\cfrac{\partial v_\theta}{\partial \theta} %2B v_r \right)~\mathbf{e}_\theta\otimes\mathbf{e}_\theta %2B \cfrac{\partial v_\theta}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_z \\ & %2B \cfrac{\partial v_z}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_r %2B \cfrac{1}{r}\cfrac{\partial v_z}{\partial \theta}~\mathbf{e}_z\otimes\mathbf{e}_\theta %2B \cfrac{\partial v_z}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_z\\ \boldsymbol{\nabla}\boldsymbol{S} & = \frac{\partial S_{rr}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_r %2B \cfrac{1}{r}\left[\frac{\partial S_{rr}}{\partial \theta} - (S_{\theta r}%2BS_{r\theta})\right]~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta %2B \frac{\partial S_{rr}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\ & %2B \frac{\partial S_{r\theta}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r %2B \cfrac{1}{r}\left[\frac{\partial S_{r\theta}}{\partial \theta} %2B (S_{rr}-S_{\theta\theta})\right]~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta %2B \frac{\partial S_{r\theta}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\ & %2B \frac{\partial S_{rz}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_r %2B \cfrac{1}{r}\left[\frac{\partial S_{rz}}{\partial \theta} -S_{\theta z}\right]~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta %2B \frac{\partial S_{rz}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_z \\ & %2B \frac{\partial S_{\theta r}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_r %2B \cfrac{1}{r}\left[\frac{\partial S_{\theta r}}{\partial \theta} %2B (S_{rr}-S_{\theta\theta})\right]~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta %2B \frac{\partial S_{\theta r}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\ & %2B \frac{\partial S_{\theta\theta}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r %2B \cfrac{1}{r}\left[\frac{\partial S_{\theta\theta}}{\partial \theta} %2B (S_{r\theta}%2BS_{\theta r})\right]~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta %2B \frac{\partial S_{\theta\theta}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\ & %2B \frac{\partial S_{\theta z}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_r %2B \cfrac{1}{r}\left[\frac{\partial S_{\theta z}}{\partial \theta} %2B S_{rz}\right]~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta %2B \frac{\partial S_{\theta z}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_z \\ & %2B \frac{\partial S_{zr}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_r %2B \cfrac{1}{r}\left[\frac{\partial S_{zr}}{\partial \theta} - S_{z\theta}\right]~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta %2B \frac{\partial S_{zr}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\ & %2B \frac{\partial S_{z\theta}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r %2B \cfrac{1}{r}\left[\frac{\partial S_{z\theta}}{\partial \theta} %2B S_{zr}\right]~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta %2B \frac{\partial S_{z\theta}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\ & %2B \frac{\partial S_{zz}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_r %2B \cfrac{1}{r}~\frac{\partial S_{zz}}{\partial \theta}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta %2B \frac{\partial S_{zz}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_z \end{align}$

Divergence of a tensor field

The divergence of a tensor field $\boldsymbol{T}(\mathbf{x})$ is defined using the recursive relation

$(\boldsymbol{\nabla}\cdot\boldsymbol{T})\cdot\mathbf{c} = \boldsymbol{\nabla}\cdot(\mathbf{c}\cdot\boldsymbol{T}) ~;\qquad\boldsymbol{\nabla}\cdot\mathbf{v} = \text{tr}(\boldsymbol{\nabla}\mathbf{v})$

where $\mathbf{c}$ is an arbitrary constant vector and $\mathbf{v}$ is a vector field. If $\boldsymbol{T}$ is a tensor field of order $n > 1$ then the divergence of the field is a tensor of order $n-1$ .