Tensor derivative (continuum mechanics)

Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...

of scalars

Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

, vectors, and second-order tensor

Tensor

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

s with respect to second-order tensors are of considerable use in continuum mechanics

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

. These derivatives are used in the theories of nonlinear elasticity and plasticity

Plasticity

Plasticity may refer to:Science* Plasticity , in physics and engineering, plasticity is the propensity of a material to undergo permanent deformation under load...

, particularly in the design of algorithms for numerical simulations.

The directional derivative

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

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

.

Properties:

1) If

then

2) If

then

3) If

then

Derivatives of vector valued functions of vectors

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

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

(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

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

The gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

, of a tensor field

in the direction of an arbitrary constant vector

is defined as:

The gradient of a tensor field of order

is a tensor field of order

Cartesian coordinates

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 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

, a vector field

, and a second-order tensor field

Curvilinear coordinates

are the contravariant basis vectors in a curvilinear coordinate

Curvilinear coordinates

Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point. This means that one can convert a point given...

system, with coordinates of points denoted by (

), then the gradient of the tensor field

is given by (see for a proof.)

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

, a vector field

, 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

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

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

then the divergence of the field is a tensor of order

Cartesian coordinates

In a Cartesian coordinate system we have the following relations for the divergences of a vector field

and a second-order tensor field

Curvilinear coordinates

In curvilinear coordinates, the divergences of a vector field

and a second-order tensor field

are

Cylindrical polar coordinates

In cylindrical polar coordinates

Curl of a tensor field

The curl

Curl

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl is represented by a vector...

of an order-

tensor field

is also defined using the recursive relation

where

is an arbitrary constant vector and

is a vector field.

Curl of a first-order tensor (vector) field

Consider a vector field

and an arbitrary constant vector

. In index notation, the cross product is given by

where

is the permutation 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 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

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.

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

Recall that we can expand the determinant of a tensor in the form of
a characteristic equation in terms of the invariants

using
(note the sign of

)

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

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

, we get

Then, invoking the arbitrariness of

, we have

This implies that

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

Proof
Recall that Since , we can write Using the product rule for second order tensors we get or, Therefore,

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

Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...