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 definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.
Let be a real valued function of the vector . Then the derivative of with respect to (or at ) in the direction is the vector defined as
for all vectors .
1) If then
2) If then
3) If then
Let be a vector valued function of the vector . Then the derivative of with respect to (or at ) in the direction is the second order tensor defined as
for all vectors .
Let be a real valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the second order tensor defined as
for all second order tensors .
Let be a second order tensor valued function of the second order tensor . Then the derivative of with respect to (or at ) in the direction is the fourth order tensor defined as
for all second order tensors .
The gradient, , of a tensor field in the direction of an arbitrary constant vector is defined as:
The gradient of a tensor field of order n is a tensor field of order n + 1.
If are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by (x1,x2,x 3), then the gradient of the tensor field is given by
The vectors and can be written as and . Let . In that case the gradient 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 φ, a vector field , and a second-order tensor field .
From this definition we have the following relations for the gradients of a scalar field φ, a vector field , and a second-order tensor field .
where the Christoffel symbol is defined using
In cylindrical coordinates, the gradient is given by
The divergence of a tensor field is defined using the recursive relation
where is an arbitrary constant vector and 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.
In a Cartesian coordinate system we have the following relations for the divergences of a vector field and a second-order tensor field .
In curvilinear coordinates, the divergences of a vector field and a second-order tensor field are
The curl of an order-n > 1 tensor field is also defined using the recursive relation
where is an arbitrary constant vector and is a vector field.
Consider a vector field and an arbitrary constant vector . In index notation, the cross product is given by
where eijk is the permutation symbol. Then,
For a second-order tensor
Hence, using the definition of the curl of a first-order tensor field,
Therefore, we have
The most commonly used identity involving the curl of a tensor field, , is
This identity hold for tensor fields of all orders. For the important case of a second-order tensor, , this identity implies that
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 . In that case, the right hand side corresponds the cofactors of the matrix.
Let be a second order tensor and let . Then, from the definition of the derivative of a scalar valued function of a tensor, we have
Recall that we can expand the determinant of a tensor in the form of a characteristic equation in terms of the invariants I1,I2,I 3 using (note the sign of λ)
Using this expansion we can write
Recall that the invariant I1 is given by
Invoking the arbitrariness of we then have
The principal invariants of a second order tensor are
The derivatives of these three invariants with respect to are
|From the derivative of the determinant we know that
For the derivatives of the other two invariants, let us go back to the characteristic equation
Using the same approach as for the determinant of a tensor, we can show that
Now the left hand side can be expanded as
Expanding the right hand side and separating terms on the left hand side gives
If we define I0: = 1 and I4: = 0, we can write the above as
Collecting terms containing various powers of λ, we get
Then, invoking the arbitrariness of λ, we have
This implies that
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 .
Let be a second order tensor. Then
Here is the fourth order identity tensor. In index notation with respect to an orthonormal basis
This result implies that
Therefore, if the tensor is symmetric, then the derivative is also symmetric and we get
where the symmetric fourth order identity tensor is
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
Since , we can write
Using the product rule for second order tensors