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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

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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

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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

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

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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

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

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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

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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

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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

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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

Proof

The vectors x and c can be written as and . Let y := x + αc. 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 , a vector field v, and a second-order tensor field .

Curvilinear coordinates

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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

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In cylindrical coordinates, the gradient is given by

Divergence of a tensor field

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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

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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[4]

The above expression is sometimes used as the definition of in Cartesian component form (often also written as ). 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 , and is conventional. This is demonstrated by an example. In a Cartesian coordinate system the second order tensor (matrix) is the gradient of a vector function .

The last equation is equivalent to the alternative definition / interpretation[4]

Curvilinear coordinates

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

More generally,


Cylindrical polar coordinates

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In cylindrical polar coordinates

Curl of a tensor field

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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

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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

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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

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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

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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.

Proof

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

The determinant of a tensor can be expressed in the form of a characteristic equation in terms of the invariants using

Using this expansion we can write

Recall that the invariant is given by

Hence,

Invoking the arbitrariness of we then have

Derivatives of the invariants of a second-order tensor

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The principal invariants of a second order tensor are

The derivatives of these three invariants with respect to are

Proof

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

Hence or,

Expanding the right hand side and separating terms on the left hand side gives

or,

If we define and , 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

Derivative of the second-order identity tensor

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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

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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

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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

Proof

Recall that

Since , we can write

Using the product rule for second order tensors

we get or,

Therefore,

Integration by parts

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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,

See also

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References

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  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. ^ a b Hjelmstad, Keith (2004). Fundamentals of Structural Mechanics. Springer Science & Business Media. p. 45. ISBN 9780387233307.