Tensor derivative (continuum mechanics) explained

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]

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(v) be a real valued function of the vector v. Then the derivative of f(v) with respect to v (or at v) is the vector defined through its dot product with any vector u being

$$\backslash frac\backslash cdot\backslash mathbf\; =\; Df(\backslash mathbf)[\backslash mathbf\{u\}]\; =\; \backslash left[\backslash frac\{d\}\{d\backslash alpha\}~f(\backslash mathbf\{v\}\; +\; \backslash alpha~\backslash mathbf\{u\})\backslash right]\_$$

for all vectors u. The above dot product yields a scalar, and if u is a unit vector gives the directional derivative of f at v, in the u direction.

Properties:

1. If

f( v ) = f _{1(v)} + f _{2(v)} then (\partial f)/(\partial v) ⋅ u = \left( (\partial f _{1})/(\partial v) + (\partial f _{2})/(\partial v) \right) ⋅ u

1. If

f( v ) = f _{1(v)~} f _{2(v)} then (\partial f)/(\partial v) ⋅ u = \left( (\partial f _{1})/(\partial v) ⋅ u \right)~f _{2(v)} + f _{1 ( v )~\left( (\partial f _{2})/(\partial v)} ⋅ u \right)

1. If

f( v ) = f _{1(f} _{2(v))} then (\partial f)/(\partial v) ⋅ u = (\partial f _{1})/(~ \partial f _{2}) (\partial f _{2})/(\partial v) ⋅ u

Derivatives of vector valued functions of vectors

Let f(v) be a vector valued function of the vector v. Then the derivative of f(v) with respect to v (or at v) is the second order tensor defined through its dot product with any vector u being

$$\backslash frac\backslash cdot\backslash mathbf\; =\; D\backslash mathbf(\backslash mathbf)[\backslash mathbf\{u\}]\; =\; \backslash left[\backslash frac\{d\}\{d\backslash alpha\}~\backslash mathbf\{f\}(\backslash mathbf\{v\}\; +\; \backslash alpha~\backslash mathbf\{u\})\; \backslash right]\_$$

for all vectors u. The above dot product yields a vector, and if u is a unit vector gives the direction derivative of f at v, in the directional u.

Properties:

1. If

f ( v ) = f _{1(v)} + f _{2(v)} then (\partial f)/(\partial v) ⋅ u = \left( (\partial f _{1})/(\partial v) + (\partial f _{2})/(\partial v) \right) ⋅ u

1. If

f ( v ) = f _{1(v) x f} _{2(v)} then (\partial f)/(\partial v) ⋅ u = \left( (\partial f _{1})/(\partial v) ⋅ u \right) x f _{2(v)} + f _{1 ( v ) x \left( (\partial f _{2})/(\partial v)} ⋅ u \right)

1. If

f ( v ) = f _{1(f} _{2(v))} then (\partial f)/(\partial v) ⋅ u = (\partial f _{1})/(⋅ \left( \partial f _{2}) (\partial f _{2})/(\partial v) ⋅ 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$$\backslash frac:\backslash boldsymbol\; =\; Df(\backslash boldsymbol)[\backslash boldsymbol\{T\}]\; =\; \backslash left[\backslash frac\{d\}\{d\backslash alpha\}~f(\backslash boldsymbol\{S\}\; +\; \backslash alpha~\backslash boldsymbol\{T\})\backslash right]\_$$for all second order tensors \boldsymbol{T} .

Properties:

1. If

f(\boldsymbol{S}) = f _{1(\boldsymbol{S})} + f _{2(\boldsymbol{S})} then (\partial f)/(\partial \boldsymbol{S) }:\boldsymbol = \left(\frac + \frac\right):\boldsymbol

1. If

f(\boldsymbol{S}) = f _{1(\boldsymbol{S})~} f _{2(\boldsymbol{S})} then (\partial f)/(\partial \boldsymbol{S) }:\boldsymbol = \left(\frac:\boldsymbol\right)~f_2(\boldsymbol) + f_1(\boldsymbol)~\left(\frac:\boldsymbol \right)

1. If

f(\boldsymbol{S}) = f _{1(f} _{2(\boldsymbol{S}))} then (\partial f)/(\partial \boldsymbol{S) }:\boldsymbol = \frac~\left(\frac:\boldsymbol \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$$\backslash frac:\backslash boldsymbol\; =\; D\backslash boldsymbol(\backslash boldsymbol)[\backslash boldsymbol\{T\}]\; =\; \backslash left[\backslash frac\{d\}\{d\backslash alpha\}~\backslash boldsymbol\{F\}(\backslash boldsymbol\{S\}\; +\; \backslash alpha~\backslash boldsymbol\{T\})\backslash right]\_$$for all second order tensors \boldsymbol{T} .

Properties:

1. If

\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F} _{1(\boldsymbol{S})} + \boldsymbol{F} _{2(\boldsymbol{S})} then (\partial \boldsymbol{F)/() }:\boldsymbol = \left(\frac + \frac\right):\boldsymbol

1. If

\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F} _{1(\boldsymbol{S}) ⋅ \boldsymbol{F}} _{2(\boldsymbol{S})} then (\partial \boldsymbol{F)/() }:\boldsymbol = \left(\frac:\boldsymbol\right)\cdot\boldsymbol_2(\boldsymbol) + \boldsymbol_1 (\boldsymbol) \cdot\left(\frac:\boldsymbol \right)

1. If

\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F} _{1(\boldsymbol{F}} _{2(\boldsymbol{S}))} then (\partial \boldsymbol{F)/() }:\boldsymbol = \frac:\left(\frac:\boldsymbol \right)

1. If

f(\boldsymbol{S}) = f _{1(\boldsymbol{F}} _{2(\boldsymbol{S}))} then (\partial f)/(\partial \boldsymbol{S) }:\boldsymbol = \frac:\left(\frac:\boldsymbol \right)

Gradient of a tensor field

The gradient,

\boldsymbol{\nabla}\boldsymbol{T} , of a tensor field \boldsymbol{T}( x ) in the direction of an arbitrary constant vector c is defined as:$$\backslash boldsymbol\backslash boldsymbol\backslash cdot\backslash mathbf\; =\; \backslash lim\_\; \backslash quad\; \backslash cfrac~\backslash boldsymbol(\backslash mathbf+\backslash alpha\backslash mathbf)$$The gradient of a tensor field of order n is a tensor field of order n+1.

Cartesian coordinates

If

e _{1,e} _{2,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$$\backslash boldsymbol\backslash boldsymbol\; =\; \backslash cfrac\; \backslash otimes\; \backslash mathbf\_i$$

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 v, and a second-order tensor field \boldsymbol{S} .

Curvilinear coordinates

See main article: Tensors in curvilinear coordinates.

If

g ^{1,g} ^{2,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.)$$\backslash boldsymbol\backslash boldsymbol\; =\; \backslash frac\backslash otimes\backslash mathbf^i$$

From this definition we have the following relations for the gradients of a scalar field

\phi , a vector field v, and a second-order tensor field \boldsymbol{S} .\Gamma ^{k} _{ij} is defined using$$\backslash Gamma\_^k~\backslash mathbf\_k\; =\; \backslash frac\; \backslash quad\; \backslash implies\; \backslash quad\; \backslash Gamma\_^k\; =\; \backslash frac\backslash cdot\backslash mathbf^k\; =\; -\backslash mathbf\_i\backslash cdot\backslash frac$$

Cylindrical polar coordinates

In cylindrical coordinates, the gradient is given by

\boldsymbol\mathbf =\quad &\frac~\mathbf_r \otimes \mathbf_r + \frac\left(\frac - v_\theta\right)~\mathbf_r \otimes \mathbf_\theta + \frac~\mathbf_r \otimes \mathbf_z \\ + &\frac~\mathbf_\theta \otimes \mathbf_r + \frac\left(\frac + v_r\right)~\mathbf_\theta \otimes \mathbf_\theta + \frac~\mathbf_\theta \otimes \mathbf_z \\ + &\frac~\mathbf_z\otimes\mathbf_r + \frac\frac~\mathbf_z \otimes\mathbf_\theta + \frac~\mathbf_z\otimes\mathbf_z \\

\boldsymbol\boldsymbol =\quad &\frac~\mathbf_r\otimes\mathbf_r\otimes\mathbf_r + \frac~\mathbf_r \otimes \mathbf_r \otimes \mathbf_z + \frac\left[\frac{\partial S_{rr}}{\partial\theta} - (S_{\theta r} + S_{r\theta})\right]~\mathbf_r \otimes \mathbf_r\otimes\mathbf_\theta \\ + &\frac~\mathbf_r \otimes \mathbf_\theta \otimes \mathbf_r + \frac~\mathbf_r \otimes \mathbf_\theta \otimes \mathbf_z + \frac\left[\frac{\partial S_{r\theta}}{\partial\theta} + (S_{rr} - S_{\theta\theta})\right]~\mathbf_r \otimes \mathbf_\theta \otimes \mathbf_\theta \\ + &\frac~\mathbf_r \otimes \mathbf_z \otimes \mathbf_r + \frac~\mathbf_r \otimes \mathbf_z \otimes \mathbf_z + \frac\left[\frac{\partial S_{rz}}{\partial \theta} - S_{\theta z}\right]~\mathbf_r \otimes \mathbf_z \otimes \mathbf_\theta \\ + &\frac~\mathbf_\theta \otimes \mathbf_r \otimes \mathbf_r + \frac~\mathbf_\theta \otimes \mathbf_r \otimes \mathbf_z + \frac\left[\frac{\partial S_{\theta r}}{\partial\theta} + (S_{rr} - S_{\theta\theta})\right]~\mathbf_\theta \otimes \mathbf_r \otimes \mathbf_\theta \\ + &\frac~\mathbf_\theta \otimes \mathbf_\theta \otimes \mathbf_r + \frac~\mathbf_\theta \otimes \mathbf_\theta \otimes \mathbf_z + \frac\left[\frac{\partial S_{\theta\theta}}{\partial\theta} + (S_{r\theta} + S_{\theta r})\right]~\mathbf_\theta \otimes \mathbf_\theta \otimes \mathbf_\theta \\ + &\frac~\mathbf_\theta \otimes \mathbf_z \otimes \mathbf_r + \frac~\mathbf_\theta \otimes \mathbf_z \otimes \mathbf_z + \frac\left[\frac{\partial S_{\theta z}}{\partial\theta} + S_{rz}\right]~\mathbf_\theta \otimes \mathbf_z \otimes \mathbf_\theta \\ + &\frac~\mathbf_z \otimes \mathbf_r \otimes \mathbf_r + \frac~\mathbf_z \otimes \mathbf_r \otimes \mathbf_z + \frac\left[\frac{\partial S_{zr}}{\partial \theta} - S_{z\theta}\right]~\mathbf_z \otimes \mathbf_r \otimes \mathbf_\theta \\ + &\frac~\mathbf_z \otimes \mathbf_\theta \otimes \mathbf_r + \frac~\mathbf_z \otimes \mathbf_\theta \otimes \mathbf_z + \frac\left[\frac{\partial S_{z\theta}}{\partial\theta} + S_{zr}\right]~\mathbf_z \otimes \mathbf_\theta \otimes \mathbf_\theta \\ + &\frac~\mathbf_z \otimes \mathbf_z \otimes \mathbf_r + \frac~\mathbf_z \otimes \mathbf_z \otimes \mathbf_z + \frac~\frac~ \mathbf_z \otimes \mathbf_z \otimes \mathbf_\theta\end

Divergence of a tensor field

The divergence of a tensor field

\boldsymbol{T}( x ) is defined using the recursive relation$$(\backslash boldsymbol\backslash cdot\backslash boldsymbol)\backslash cdot\backslash mathbf\; =\; \backslash boldsymbol\backslash cdot\backslash left(\backslash mathbf\backslash cdot\backslash boldsymbol^\backslash textsf\backslash right)\; ~;\backslash qquad\; \backslash boldsymbol\backslash cdot\backslash mathbf\; =\; \backslash text(\backslash boldsymbol\backslash mathbf)$$

where c is an arbitrary constant vector and 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.

Cartesian coordinates

In a Cartesian coordinate system we have the following relations for a vector field v and a second-order tensor field

\boldsymbol{S} .

where tensor index notation for partial derivatives is used in the rightmost expressions. Note that$$\backslash boldsymbol\backslash cdot\backslash boldsymbol\; \backslash neq\; \backslash boldsymbol\backslash cdot\backslash boldsymbol^\backslash textsf.$$

For a symmetric second-order tensor, the divergence is also often written as^[4]

The above expression is sometimes used as the definition of

\boldsymbol{\nabla} ⋅ \boldsymbol{S} in Cartesian component form (often also written as\operatorname{div}\boldsymbol{S} ). Note that such a definition is not consistent with the rest of this article (see the section on curvilinear co-ordinates).

The difference stems from whether the differentiation is performed with respect to the rows or columns of

\boldsymbol{S} , and is conventional. This is demonstrated by an example. In a Cartesian coordinate system the second order tensor (matrix) S is the gradient of a vector function v .

\boldsymbol \cdot \left[\left(\boldsymbol{\nabla} \mathbf{v} \right)^\textsf{T} \right] &= \boldsymbol \cdot \left(v_ ~\mathbf_i \otimes \mathbf_j \right) = v_ ~\mathbf_i \cdot \mathbf_i \otimes \mathbf_j = \boldsymbol^ v_ ~\mathbf_j = \boldsymbol^ \mathbf\end

The last equation is equivalent to the alternative definition / interpretation

$$\backslash begin\; \backslash left(\backslash boldsymbol\; \backslash cdot\; \backslash right)\_\backslash text\; \backslash left(\backslash boldsymbol\; \backslash mathbf\; \backslash right)\; =\; \backslash left(\backslash boldsymbol\; \backslash cdot\; \backslash right)\_\backslash text\; \backslash left(v\_\; ~\backslash mathbf\_i\; \backslash otimes\; \backslash mathbf\_j\; \backslash right)\; =\; v\_\; ~\backslash mathbf\_i\; \backslash otimes\; \backslash mathbf\_j\; \backslash cdot\; \backslash mathbf\_j\; =\; \backslash boldsymbol^2\; v\_i\; ~\backslash mathbf\_i\; =\; \backslash boldsymbol^2\; \backslash mathbf\backslash end$$

Curvilinear coordinates

See main article: Tensors in curvilinear coordinates. In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field

\boldsymbol{S} are

More generally,

Cylindrical polar coordinates

In cylindrical polar coordinates

\boldsymbol\cdot\boldsymbol =\quad &\frac~\mathbf_r + \frac~\mathbf_\theta + \frac~\mathbf_z \\

+ &\frac\left[\frac{\partial S_{\theta r}}{\partial \theta} + (S_{rr} - S_{\theta\theta})\right]~\mathbf_r + \frac\left[\frac{\partial S_{\theta\theta}}{\partial\theta} + (S_{r\theta} + S_{\theta r})\right]~\mathbf_\theta + \frac\left[\frac{\partial S_{\theta z}}{\partial\theta} + S_{rz}\right]~\mathbf_z \\

+ &\frac~\mathbf_r + \frac~\mathbf_\theta + \frac~\mathbf_z\end

Curl of a tensor field

The curl of an order-n > 1 tensor field

\boldsymbol{T}( x ) is also defined using the recursive relation$$(\backslash boldsymbol\backslash times\backslash boldsymbol)\backslash cdot\backslash mathbf\; =\; \backslash boldsymbol\backslash times(\backslash mathbf\backslash cdot\backslash boldsymbol)\; ~;\backslash qquad\; (\backslash boldsymbol\backslash times\backslash mathbf)\backslash cdot\backslash mathbf\; =\; \backslash boldsymbol\backslash cdot(\backslash mathbf\backslash times\backslash mathbf)$$where c is an arbitrary constant vector and v is a vector field.

Curl of a first-order tensor (vector) field

Consider a vector field v and an arbitrary constant vector c. In index notation, the cross product is given by$$\backslash mathbf\; \backslash times\; \backslash mathbf\; =\; \backslash varepsilon\_~v\_j~c\_k~\backslash mathbf\_i$$where

\varepsilon _{ijk} is the permutation symbol, otherwise known as the Levi-Civita symbol. Then,$$\backslash boldsymbol\backslash cdot(\backslash mathbf\; \backslash times\; \backslash mathbf)\; =\; \backslash varepsilon\_~v\_~c\_k\; =\; (\backslash varepsilon\_~v\_~\backslash mathbf\_k)\backslash cdot\backslash mathbf\; =\; (\backslash boldsymbol\backslash times\backslash mathbf)\backslash cdot\backslash mathbf$$Therefore,$$\backslash boldsymbol\backslash times\backslash mathbf\; =\; \backslash varepsilon\_~v\_~\backslash mathbf\_k$$

Curl of a second-order tensor field

For a second-order tensor

\boldsymbol{S} $$\backslash mathbf\backslash cdot\backslash boldsymbol\; =\; c\_m~S\_~\backslash mathbf\_j$$Hence, using the definition of the curl of a first-order tensor field,$$\backslash boldsymbol\backslash times(\backslash mathbf\backslash cdot\backslash boldsymbol)\; =\; \backslash varepsilon\_~c\_m~S\_~\backslash mathbf\_k\; =\; (\backslash varepsilon\_~S\_~\backslash mathbf\_k\backslash otimes\backslash mathbf\_m)\backslash cdot\backslash mathbf\; =\; (\backslash boldsymbol\backslash times\backslash boldsymbol)\; \backslash cdot\; \backslash mathbf$$Therefore, we have$$\backslash boldsymbol\backslash times\backslash boldsymbol\; =\; \backslash varepsilon\_~S\_~\backslash mathbf\_k\backslash otimes\backslash mathbf\_m$$

Identities involving the curl of a tensor field

The most commonly used identity involving the curl of a tensor field,

\boldsymbol{T} , is$$\backslash boldsymbol\backslash times(\backslash boldsymbol\backslash boldsymbol)\; =\; \backslash boldsymbol$$This identity holds for tensor fields of all orders. For the important case of a second-order tensor, \boldsymbol{S} , this identity implies that$$\backslash boldsymbol\backslash times(\backslash boldsymbol\backslash boldsymbol)\; =\; \backslash boldsymbol\; \backslash quad\; \backslash implies\; \backslash quad\; S\_\; -\; S\_\; =\; 0$$

Derivative of the determinant of a second-order tensor

The derivative of the determinant of a second order tensor

\boldsymbol{A} is given by$$\backslash frac\backslash det(\backslash boldsymbol)\; =\; \backslash det(\backslash boldsymbol)~\backslash left[\backslash boldsymbol\{A\}^\{-1\}\backslash right]^\backslash textsf\; ~.$$

In an orthonormal basis, the components of

\boldsymbol{A} can be written as a matrix A. In that case, the right hand side corresponds the cofactors of the matrix.

Derivatives of the invariants of a second-order tensor

The principal invariants of a second order tensor are

The derivatives of these three invariants with respect to

\boldsymbol{A} are

Derivative of the second-order identity tensor

Let

\boldsymbol{ 1 } be the second order identity tensor. Then the derivative of this tensor with respect to a second order tensor \boldsymbol{A} is given by$$\backslash frac:\backslash boldsymbol\; =\; \backslash boldsymbol:\backslash boldsymbol\; =\; \backslash boldsymbol$$This is because \boldsymbol{ 1 } is independent of \boldsymbol{A} .

Derivative of a second-order tensor with respect to itself

Let

\boldsymbol{A} be a second order tensor. Then$$\backslash frac:\backslash boldsymbol\; =\; \backslash left[\backslash frac\{\backslash partial\; \}\{\backslash partial\; \backslash alpha\}\; (\backslash boldsymbol\{A\}\; +\; \backslash alpha~\backslash boldsymbol\{T\})\backslash right]\_\; =\; \backslash boldsymbol\; =\; \backslash boldsymbol:\backslash boldsymbol$$

Therefore,$$\backslash frac\; =\; \backslash boldsymbol$$

Here

\boldsymbol{ I } is the fourth order identity tensor. In index notation with respect to an orthonormal basis$$\backslash boldsymbol\; =\; \backslash delta\_~\backslash delta\_~\backslash mathbf\_i\backslash otimes\backslash mathbf\_j\backslash otimes\backslash mathbf\_k\backslash otimes\backslash mathbf\_l$$

This result implies that$$\backslash frac:\backslash boldsymbol\; =\; \backslash boldsymbol^\backslash textsf:\backslash boldsymbol\; =\; \backslash boldsymbol^\backslash textsf$$where$$\backslash boldsymbol^\backslash textsf\; =\; \backslash delta\_~\backslash delta\_~\backslash mathbf\_i\backslash otimes\backslash mathbf\_j\backslash otimes\backslash mathbf\_k\backslash otimes\backslash mathbf\_l$$

Therefore, if the tensor

\boldsymbol{A} is symmetric, then the derivative is also symmetric and we get$$\backslash frac\; =\; \backslash boldsymbol^\; =\; \backslash frac~\backslash left(\backslash boldsymbol\; +\; \backslash boldsymbol^\backslash textsf\backslash right)$$where the symmetric fourth order identity tensor is$$\backslash boldsymbol^\; =\; \backslash frac~(\backslash delta\_~\backslash delta\_\; +\; \backslash delta\_~\backslash delta\_)\; ~\backslash mathbf\_i\backslash otimes\backslash mathbf\_j\backslash otimes\backslash mathbf\_k\backslash otimes\backslash mathbf\_l$$

Derivative of the inverse of a second-order tensor

Let

\boldsymbol{A} and \boldsymbol{T} be two second order tensors, then$$\backslash frac\; \backslash left(\backslash boldsymbol^\backslash right)\; :\; \backslash boldsymbol\; =\; -\; \backslash boldsymbol^\backslash cdot\backslash boldsymbol\backslash cdot\backslash boldsymbol^$$In index notation with respect to an orthonormal basis$$\backslash frac~T\_\; =\; -\; A^\_~T\_~A^\_\; \backslash implies\; \backslash frac\; =\; -\; A^\_~A^\_$$We also have$$\backslash frac\; \backslash left(\backslash boldsymbol^\backslash right)\; :\; \backslash boldsymbol\; =\; -\; \backslash boldsymbol^\backslash cdot\backslash boldsymbol^\backslash textsf\backslash cdot\backslash boldsymbol^$$In index notation$$\backslash frac~T\_\; =\; -\; A^\_~T\_~A^\_\; \backslash implies\; \backslash frac\; =\; -\; A^\_~A^\_$$If the tensor \boldsymbol{A} is symmetric then$$\backslash frac\; =\; -\backslash cfrac\backslash left(A^\_~A^\_\; +\; A^\_~A^\_\backslash right)$$

Integration by parts

Another important operation related to tensor derivatives in continuum mechanics is integration by parts. The formula for integration by parts can be written as$$\backslash int\_\; \backslash boldsymbol\backslash otimes\backslash boldsymbol\backslash boldsymbol\backslash ,d\backslash Omega\; =\; \backslash int\_\; \backslash mathbf\; \backslash otimes\; (\backslash boldsymbol\backslash otimes\backslash boldsymbol)\backslash ,d\backslash Gamma\; -\; \backslash int\_\; \backslash boldsymbol\backslash otimes\backslash boldsymbol\backslash boldsymbol\backslash ,d\backslash Omega$$

where

\boldsymbol{F} and \boldsymbol{G} are differentiable tensor fields of arbitrary order, n is the unit outward normal to the domain over which the tensor fields are defined, ⊗ represents a generalized tensor product operator, and \boldsymbol{\nabla} is a generalized gradient operator. When \boldsymbol{F} is equal to the identity tensor, we get the divergence theorem$$\backslash int\_\backslash boldsymbol\backslash boldsymbol\backslash ,d\backslash Omega\; =\; \backslash int\_\; \backslash mathbf\backslash otimes\backslash boldsymbol\backslash ,d\backslash Gamma\; \backslash ,.$$

We can express the formula for integration by parts in Cartesian index notation as$$\backslash int\_\; F\_\backslash ,G\_\backslash ,d\backslash Omega\; =\; \backslash int\_\; n\_p\backslash ,F\_\backslash ,G\_\backslash ,d\backslash Gamma\; -\; \backslash int\_\; G\_\backslash ,F\_\backslash ,d\backslash Omega\; \backslash ,.$$

For the special case where the tensor product operation is a contraction of one index and the gradient operation is a divergence, and both

\boldsymbol{F} and \boldsymbol{G} are second order tensors, we have$$\backslash int\_\; \backslash boldsymbol\backslash cdot(\backslash boldsymbol\backslash cdot\backslash boldsymbol)\backslash ,d\backslash Omega\; =\; \backslash int\_\; \backslash mathbf\backslash cdot\backslash left(\backslash boldsymbol\backslash cdot\backslash boldsymbol^\backslash textsf\backslash right)\backslash ,d\backslash Gamma\; -\; \backslash int\_\; (\backslash boldsymbol\backslash boldsymbol):\backslash boldsymbol^\backslash textsf\backslash ,d\backslash Omega\; \backslash ,.$$

In index notation,$$\backslash int\_\; F\_\backslash ,G\_\backslash ,d\backslash Omega\; =\; \backslash int\_\; n\_p\backslash ,F\_\backslash ,G\_\backslash ,d\backslash Gamma\; -\; \backslash int\_\; G\_\backslash ,F\_\backslash ,d\backslash Omega\; \backslash ,.$$

See also

• Covariant derivative
• Ricci calculus

Notes and References

1. J. C. Simo and T. J. R. Hughes, 1998, Computational Inelasticity, Springer
2. J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity, Dover.
3. R. W. Ogden, 2000, Nonlinear Elastic Deformations, Dover.
4. Book: Hjelmstad. Keith. Fundamentals of Structural Mechanics. 2004. Springer Science & Business Media. 9780387233307. 45.