Tensor derivative (continuum mechanics)

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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[edit]

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[edit]

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) in the direction u is the vector defined as

for all vectors u.

Properties:

1) If then

2) If then

3) If then

Derivatives of vector valued functions of vectors[edit]

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) in the direction u is the second order tensor defined as

for all vectors u.

Properties:
1) If then
2) If then
3) If then

Derivatives of scalar valued functions of second-order tensors[edit]

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 .

Properties:
1) If then
2) If then
3) If then

Derivatives of tensor valued functions of second-order tensors[edit]

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 .

Properties:
1) If then
2) If then
3) If then
4) If then

Gradient of a tensor field[edit]

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[edit]

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


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[edit]

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


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 (see [3] for a proof.)

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[edit]

In cylindrical coordinates, the gradient is given by

Divergence of a tensor field[edit]

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[edit]

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

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

Note that last relation can be found in reference [4] under relation (1.14.13). Note also that according to the same paper in the case of the second-order tensor field, we have:

Curvilinear coordinates[edit]

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

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

Cylindrical polar coordinates[edit]

In cylindrical polar coordinates

Curl of a tensor field[edit]

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[edit]

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. Then,

Therefore,

Curl of a second-order tensor field[edit]

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[edit]

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

Derivative of the determinant of a second-order tensor[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

Domain , its boundary and the outward unit normal

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,

References[edit]

  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. ^ Ogden, R. W., 2000, Nonlinear Elastic Deformations, Dover.
  4. ^ http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/Chapter_1_Vectors_Tensors/Vectors_Tensors_14_Tensor_Calculus.pdf

See also[edit]