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.
The directional derivative provides a systematic way of finding these derivatives.

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 be a real valued function of the vector v. Then the derivative of f with respect to v is the vector defined through its dot product with any vector u being
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 then
  2. If then
  3. If then

Derivatives of vector valued functions of vectors

Let f be a vector valued function of the vector v. Then the derivative of f with respect to v is the second order tensor defined through its dot product with any vector u being
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 then
  2. If then
  3. If then

Derivatives of scalar valued functions of second-order tensors

Let be a real valued function of the second order tensor. Then the derivative of with respect to in the direction is the second order tensor defined as
for all second order tensors.
Properties:
  1. If then
  2. If then
  3. If then

Derivatives of tensor valued functions of second-order tensors

Let be a second order tensor valued function of the second order tensor. Then the derivative of with respect to in the direction is the fourth order tensor defined as
for all second order tensors.
Properties:
  1. If then
  2. If then
  3. If then
  4. If then

Gradient of a tensor field

The gradient,, of a tensor field in the direction of an arbitrary constant vector c is defined as:
The gradient of a tensor field of order n is a tensor field of order n+1.

Cartesian coordinates

If are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by, then the gradient of the tensor field is given by
Since the basis vectors do not vary in a Cartesian coordinate system we have the following relations for the gradients of a scalar field, a vector field v, and a second-order tensor field.

Curvilinear coordinates

If are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by, then the gradient of the tensor field is given by
From this definition we have the following relations for the gradients of a scalar field, a vector field v, and a second-order tensor field.
where the Christoffel symbol is defined using

Cylindrical polar coordinates

In cylindrical coordinates, the gradient is given by

Divergence of a tensor field

The divergence of a tensor field is defined using the recursive relation
where c is an arbitrary constant vector and v is a vector field. If 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.
where tensor index notation for partial derivatives is used in the rightmost expressions. Note that
For a symmetric second-order tensor, the divergence is also often written as
The above expression is sometimes used as the definition of
in Cartesian component form. Note that such a definition is not consistent with the rest of this article.
The difference stems from whether the differentiation is performed with respect to the rows or columns of, and is conventional. This is demonstrated by an example. In a Cartesian coordinate system the second order tensor is the gradient of a vector function.
The last equation is equivalent to the alternative definition / interpretation

Curvilinear coordinates

In curvilinear coordinates, the divergences of a vector field v and a second-order tensor field are
More generally,

Cylindrical polar coordinates

In cylindrical polar coordinates

Curl of a tensor field

The curl of an order-n > 1 tensor field is also defined using the recursive relation
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
where is the permutation symbol, otherwise known as the Levi-Civita symbol. Then,
Therefore,

Curl of a second-order tensor field

For a second-order tensor
Hence, using the definition of the curl of a first-order tensor field,
Therefore, we have

Identities involving the curl of a tensor field

The most commonly used identity involving the curl of a tensor field,, is
This identity holds for tensor fields of all orders. For the important case of a second-order tensor,, this identity implies that

Derivative of the determinant of a second-order tensor

The derivative of the determinant of a second order tensor is given by
In an orthonormal basis, the components of 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 are

Derivative of the second-order identity tensor

Let be the second order identity tensor. Then the derivative of this tensor with respect to a second order tensor is given by
This is because is independent of.

Derivative of a second-order tensor with respect to itself

Let be a second order tensor. Then
Therefore,
Here is the fourth order identity tensor. In index notation with respect to an orthonormal basis
This result implies that
where
Therefore, if the tensor is symmetric, then the derivative is also symmetric and we get
where the symmetric fourth order identity tensor is

Derivative of the inverse of a second-order tensor

Let and be two second order tensors, then
In index notation with respect to an orthonormal basis
We also have
In index notation
If the tensor is symmetric then

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
where and are differentiable tensor fields of arbitrary order, is the unit outward normal to the domain over which the tensor fields are defined, represents a generalized tensor product operator, and is a generalized gradient operator. When is equal to the identity tensor, we get the divergence theorem
We can express the formula for integration by parts in Cartesian index notation as
For the special case where the tensor product operation is a contraction of one index and the gradient operation is a divergence, and both and are second order tensors, we have
In index notation,