Tensor derivative (continuum mechanics)

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 value functions of vectors

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

∂ f ∂ v ⋅ u = D f ( v ) [ u ] = [ d d α   f ( v + α   u ) ] α = 0

for all vectors u.

Properties:

1) If f ( v ) = f 1 ( v ) + f 2 ( v ) then ∂ f ∂ v ⋅ u = ( ∂ f 1 ∂ v + ∂ f 2 ∂ v ) ⋅ u

2) If f ( v ) = f 1 ( v )   f 2 ( v ) then ∂ f ∂ v ⋅ u = ( ∂ f 1 ∂ v ⋅ u )   f 2 ( v ) + f 1 ( v )   ( ∂ f 2 ∂ v ⋅ u )

3) If f ( v ) = f 1 ( f 2 ( v ) ) then ∂ f ∂ v ⋅ u = ∂ f 1 ∂ f 2   ∂ f 2 ∂ v ⋅ u

Derivatives of vector valued functions of vectors

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

∂ f ∂ v ⋅ u = D f ( v ) [ u ] = [ d d α   f ( v + α   u ) ] α = 0

for all vectors u.

Properties: 1) If f ( v ) = f 1 ( v ) + f 2 ( v ) then ∂ f ∂ v ⋅ u = ( ∂ f 1 ∂ v + ∂ f 2 ∂ v ) ⋅ u 2) If f ( v ) = f 1 ( v ) × f 2 ( v ) then ∂ f ∂ v ⋅ u = ( ∂ f 1 ∂ v ⋅ u ) × f 2 ( v ) + f 1 ( v ) × ( ∂ f 2 ∂ v ⋅ u ) 3) If f ( v ) = f 1 ( f 2 ( v ) ) then ∂ f ∂ v ⋅ u = ∂ f 1 ∂ f 2 ⋅ ( ∂ f 2 ∂ v ⋅ u )

Derivatives of scalar valued functions of second-order tensors

Let f ( S ) be a real valued function of the second order tensor S . Then the derivative of f ( S ) with respect to S (or at S ) in the direction T is the second order tensor defined as

∂ f ∂ S : T = D f ( S ) [ T ] = [ d d α   f ( S + α   T ) ] α = 0

for all second order tensors T .

Properties: 1) If f ( S ) = f 1 ( S ) + f 2 ( S ) then ∂ f ∂ S : T = ( ∂ f 1 ∂ S + ∂ f 2 ∂ S ) : T 2) If f ( S ) = f 1 ( S )   f 2 ( S ) then ∂ f ∂ S : T = ( ∂ f 1 ∂ S : T )   f 2 ( S ) + f 1 ( S )   ( ∂ f 2 ∂ S : T ) 3) If f ( S ) = f 1 ( f 2 ( S ) ) then ∂ f ∂ S : T = ∂ f 1 ∂ f 2   ( ∂ f 2 ∂ S : T )

Derivatives of tensor valued functions of second-order tensors

Let F ( S ) be a second order tensor valued function of the second order tensor S . Then the derivative of F ( S ) with respect to S (or at S ) in the direction T is the fourth order tensor defined as

∂ F ∂ S : T = D F ( S ) [ T ] = [ d d α   F ( S + α   T ) ] α = 0

for all second order tensors T .

Properties: 1) If F ( S ) = F 1 ( S ) + F 2 ( S ) then ∂ F ∂ S : T = ( ∂ F 1 ∂ S + ∂ F 2 ∂ S ) : T 2) If F ( S ) = F 1 ( S ) ⋅ F 2 ( S ) then ∂ F ∂ S : T = ( ∂ F 1 ∂ S : T ) ⋅ F 2 ( S ) + F 1 ( S ) ⋅ ( ∂ F 2 ∂ S : T ) 3) If F ( S ) = F 1 ( F 2 ( S ) ) then ∂ F ∂ S : T = ∂ F 1 ∂ F 2 : ( ∂ F 2 ∂ S : T ) 4) If f ( S ) = f 1 ( F 2 ( S ) ) then ∂ f ∂ S : T = ∂ f 1 ∂ F 2 : ( ∂ F 2 ∂ S : T )

Gradient of a tensor field

The gradient, ∇ T , of a tensor field T ( x ) in the direction of an arbitrary constant vector c is defined as:

∇ T ⋅ c = lim α → 0 d d α   T ( x + α c )

The gradient of a tensor field of order n is a tensor field of order n+1.

Cartesian coordinates

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

If e 1 , e 2 , e 3 are the basis vectors in a Cartesian coordinate system, with coordinates of points denoted by ( x 1 , x 2 , x 3 ), then the gradient of the tensor field T is given by

∇ T = e i ⊗ ∂ T ∂ x i

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 v, and a second-order tensor field S .

∇ ϕ = ∂ ϕ ∂ x i   e i ∇ v = ∂ ( v j e j ) ∂ x i ⊗ e i = ∂ v j ∂ x i   e j ⊗ e i ∇ S = ∂ ( S j k e j ⊗ e k ) ∂ x i ⊗ e i = ∂ S j k ∂ x i   e j ⊗ e k ⊗ e i

Curvilinear coordinates

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

If g 1 , g 2 , g 3 are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by ( ξ 1 , ξ 2 , ξ 3 ), then the gradient of the tensor field T is given by (see for a proof.)

∇ T = ∂ T ∂ ξ i ⊗ g i

From this definition we have the following relations for the gradients of a scalar field ϕ , a vector field v, and a second-order tensor field S .

∇ ϕ = ∂ ϕ ∂ ξ i   g i ∇ v = ∂ ( v j g j ) ∂ ξ i ⊗ g i = ( ∂ v j ∂ ξ i + v k   Γ i k j )   g j ⊗ g i = ( ∂ v j ∂ ξ i − v k   Γ i j k )   g j ⊗ g i ∇ S = ∂ ( S j k   g j ⊗ g k ) ∂ ξ i ⊗ g i = ( ∂ S j k ∂ ξ i − S l k   Γ i j l − S j l   Γ i k l )   g j ⊗ g k ⊗ g i

where the Christoffel symbol Γ i j k is defined using

Γ i j k   g k = ∂ g i ∂ ξ j ⟹ Γ i j k = ∂ g i ∂ ξ j ⋅ g k = − g i ⋅ ∂ g k ∂ ξ j

Cylindrical polar coordinates

In cylindrical coordinates, the gradient is given by

∇ ϕ = ∂ ϕ ∂ r   e r + 1 r   ∂ ϕ ∂ θ   e θ + ∂ ϕ ∂ z   e z ∇ v = ∂ v r ∂ r   e r ⊗ e r + 1 r ( ∂ v r ∂ θ − v θ )   e r ⊗ e θ + ∂ v r ∂ z   e r ⊗ e z + ∂ v θ ∂ r   e θ ⊗ e r + 1 r ( ∂ v θ ∂ θ + v r )   e θ ⊗ e θ + ∂ v θ ∂ z   e θ ⊗ e z + ∂ v z ∂ r   e z ⊗ e r + 1 r ∂ v z ∂ θ   e z ⊗ e θ + ∂ v z ∂ z   e z ⊗ e z ∇ S = ∂ S r r ∂ r   e r ⊗ e r ⊗ e r + 1 r [ ∂ S r r ∂ θ − ( S θ r + S r θ ) ]   e r ⊗ e r ⊗ e θ + ∂ S r r ∂ z   e r ⊗ e r ⊗ e z + ∂ S r θ ∂ r   e r ⊗ e θ ⊗ e r + 1 r [ ∂ S r θ ∂ θ + ( S r r − S θ θ ) ]   e r ⊗ e θ ⊗ e θ + ∂ S r θ ∂ z   e r ⊗ e θ ⊗ e z + ∂ S r z ∂ r   e r ⊗ e z ⊗ e r + 1 r [ ∂ S r z ∂ θ − S θ z ]   e r ⊗ e z ⊗ e θ + ∂ S r z ∂ z   e r ⊗ e z ⊗ e z + ∂ S θ r ∂ r   e θ ⊗ e r ⊗ e r + 1 r [ ∂ S θ r ∂ θ + ( S r r − S θ θ ) ]   e θ ⊗ e r ⊗ e θ + ∂ S θ r ∂ z   e θ ⊗ e r ⊗ e z + ∂ S θ θ ∂ r   e θ ⊗ e θ ⊗ e r + 1 r [ ∂ S θ θ ∂ θ + ( S r θ + S θ r ) ]   e θ ⊗ e θ ⊗ e θ + ∂ S θ θ ∂ z   e θ ⊗ e θ ⊗ e z + ∂ S θ z ∂ r   e θ ⊗ e z ⊗ e r + 1 r [ ∂ S θ z ∂ θ + S r z ]   e θ ⊗ e z ⊗ e θ + ∂ S θ z ∂ z   e θ ⊗ e z ⊗ e z + ∂ S z r ∂ r   e z ⊗ e r ⊗ e r + 1 r [ ∂ S z r ∂ θ − S z θ ]   e z ⊗ e r ⊗ e θ + ∂ S z r ∂ z   e z ⊗ e r ⊗ e z + ∂ S z θ ∂ r   e z ⊗ e θ ⊗ e r + 1 r [ ∂ S z θ ∂ θ + S z r ]   e z ⊗ e θ ⊗ e θ + ∂ S z θ ∂ z   e z ⊗ e θ ⊗ e z + ∂ S z z ∂ r   e z ⊗ e z ⊗ e r + 1 r   ∂ S z z ∂ θ   e z ⊗ e z ⊗ e θ + ∂ S z z ∂ z   e z ⊗ e z ⊗ e z

Divergence of a tensor field

The divergence of a tensor field T ( x ) is defined using the recursive relation

( ∇ ⋅ T ) ⋅ c = ∇ ⋅ ( c ⋅ T T )   ; ∇ ⋅ v = tr ( ∇ v )

where c is an arbitrary constant vector and v is a vector field. If T is a tensor field of order n > 1 then the divergence of the field is a tensor of order n−1.

Cartesian coordinates

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 S .

∇ ⋅ v = ∂ v i ∂ x i ∇ ⋅ S = ∂ S k i ∂ x k   e i

Note that last relation can be found in reference under relation (1.14.13).

Note also that according to the same paper in the case of the second-order tensor field, we have:

∇ ⋅ S ≠ div ⁡ S = ∇ ⋅ S T .

Importantly, other written conventions for the divergence of a second-order tensor do exist. For example, the divergence of a second rank tensor can also be written as

∇ ⋅ S = ∂ S k i ∂ x i   e k

The difference stems from whether the differentiation is performed with respect to the rows or columns of S , and is conventional.

Curvilinear coordinates

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 S are

∇ ⋅ v = ( ∂ v i ∂ ξ i + v k   Γ i k i ) ∇ ⋅ S = ( ∂ S i k ∂ ξ i − S l k   Γ i i l − S i l   Γ i k l )   g k

Cylindrical polar coordinates

In cylindrical polar coordinates

∇ ⋅ v = ∂ v r ∂ r + 1 r ( ∂ v θ ∂ θ + v r ) + ∂ v z ∂ z ∇ ⋅ S = ∂ S r r ∂ r   e r + ∂ S r θ ∂ r   e θ + ∂ S r z ∂ r   e z + 1 r [ ∂ S θ r ∂ θ + ( S r r − S θ θ ) ]   e r + 1 r [ ∂ S θ θ ∂ θ + ( S r θ + S θ r ) ]   e θ + 1 r [ ∂ S θ z ∂ θ + S r z ]   e z + ∂ S z r ∂ z   e r + ∂ S z θ ∂ z   e θ + ∂ S z z ∂ z   e z

Curl of a tensor field

The curl of an order-n > 1 tensor field T ( x ) is also defined using the recursive relation

( ∇ × T ) ⋅ c = ∇ × ( c ⋅ T )   ; ( ∇ × v ) ⋅ c = ∇ ⋅ ( v × c )

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

v × c = e i j k   v j   c k   e i

where e i j k is the permutation symbol. Then,

∇ ⋅ ( v × c ) = e i j k   v j , i   c k = ( e i j k   v j , i   e k ) ⋅ c = ( ∇ × v ) ⋅ c

Therefore,

∇ × v = e i j k   v j , i   e k

Curl of a second-order tensor field

For a second-order tensor S

c ⋅ S = c m   S m j   e j

Hence, using the definition of the curl of a first-order tensor field,

∇ × ( c ⋅ S ) = e i j k   c m   S m j , i   e k = ( e i j k   S m j , i   e k ⊗ e m ) ⋅ c = ( ∇ × S ) ⋅ c

Therefore, we have

∇ × S = e i j k   S m j , i   e k ⊗ e m

Identities involving the curl of a tensor field

The most commonly used identity involving the curl of a tensor field, T , is

∇ × ( ∇ T ) = 0

This identity holds for tensor fields of all orders. For the important case of a second-order tensor, S , this identity implies that

∇ × S = 0 ⟹ S m i , j − S m j , i = 0

Derivative of the determinant of a second-order tensor

The derivative of the determinant of a second order tensor A is given by

∂ ∂ A det ( A ) = det ( A )   [ A − 1 ] T   .

In an orthonormal basis, the components of A 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

I 1 ( A ) = tr A I 2 ( A ) = 1 2 [ ( tr A ) 2 − tr A 2 ] I 3 ( A ) = det ( A )

The derivatives of these three invariants with respect to A are

∂ I 1 ∂ A = 1 ∂ I 2 ∂ A = I 1   1 − A T ∂ I 3 ∂ A = det ( A )   [ A − 1 ] T = I 2   1 − A T   ( I 1   1 − A T ) = ( A 2 − I 1   A + I 2   1 ) T

Derivative of the second-order identity tensor

Let 1 be the second order identity tensor. Then the derivative of this tensor with respect to a second order tensor A is given by

∂ 1 ∂ A : T = 0 : T = 0

This is because 1 is independent of A .

Derivative of a second-order tensor with respect to itself

Let A be a second order tensor. Then

∂ A ∂ A : T = [ ∂ ∂ α ( A + α   T ) ] α = 0 = T = I : T

Therefore,

∂ A ∂ A = I

Here I is the fourth order identity tensor. In index notation with respect to an orthonormal basis

I = δ i k   δ j l   e i ⊗ e j ⊗ e k ⊗ e l

This result implies that

∂ A T ∂ A : T = I T : T = T T

where

I T = δ j k   δ i l   e i ⊗ e j ⊗ e k ⊗ e l

Therefore, if the tensor A is symmetric, then the derivative is also symmetric and we get

∂ A ∂ A = I ( s ) = 1 2   ( I + I T )

where the symmetric fourth order identity tensor is

I ( s ) = 1 2   ( δ i k   δ j l + δ i l   δ j k )   e i ⊗ e j ⊗ e k ⊗ e l

Derivative of the inverse of a second-order tensor

Let A and T be two second order tensors, then

∂ ∂ A ( A − 1 ) : T = − A − 1 ⋅ T ⋅ A − 1

In index notation with respect to an orthonormal basis

∂ A i j − 1 ∂ A k l   T k l = − A i k − 1   T k l   A l j − 1 ⟹ ∂ A i j − 1 ∂ A k l = − A i k − 1   A l j − 1

We also have

∂ ∂ A ( A − T ) : T = − A − T ⋅ T T ⋅ A − T

In index notation

∂ A j i − 1 ∂ A k l   T k l = − A j k − 1   T l k   A l i − 1 ⟹ ∂ A j i − 1 ∂ A k l = − A l i − 1   A j k − 1

If the tensor A is symmetric then

∂ A i j − 1 ∂ A k l = − 1 2 ( A i k − 1   A j l − 1 + A i l − 1   A j k − 1 )

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

∫ Ω F ⊗ ∇ G d Ω = ∫ Γ n ⊗ ( F ⊗ G ) d Γ − ∫ Ω G ⊗ ∇ F d Ω

where F and G are differentiable tensor fields of arbitrary order, n 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 F is equal to the identity tensor, we get the divergence theorem

∫ Ω ∇ G d Ω = ∫ Γ n ⊗ G d Γ .

We can express the formula for integration by parts in Cartesian index notation as

∫ Ω F i j k . . . . G l m n . . . , p d Ω = ∫ Γ n p F i j k . . . G l m n . . . d Γ − ∫ Ω G l m n . . . F i j k . . . , p d Ω .

For the special case where the tensor product operation is a contraction of one index and the gradient operation is a divergence, and both F and G are second order tensors, we have

∫ Ω F ⋅ ( ∇ ⋅ G ) d Ω = ∫ Γ n ⋅ ( G ⋅ F T ) d Γ − ∫ Ω ( ∇ F ) : G T d Ω .

In index notation,

∫ Ω F i j G p j , p d Ω = ∫ Γ n p F i j G p j d Γ − ∫ Ω G p j F i j , p d Ω .