Contents

Tensors in Solid MechanicsEdit

A sound understanding of tensors and tensor operation is essential if you want to read and understand modern papers on solid mechanics and finite element modeling of complex material behavior. This brief introduction gives you an overview of tensors and tensor notation. For more details you can read A Brief on Tensor Analysis by J. G. Simmonds, the appendix on vector and tensor notation from Dynamics of Polymeric Liquids - Volume 1 by R. B. Bird, R. C. Armstrong, and O. Hassager, and the monograph by R. M. Brannon. An introduction to tensors in continuum mechanics can be found in An Introduction to Continuum Mechanics by M. E. Gurtin. Most of the material in this page is based on these sources.

NotationEdit

The following notation is usually used in the literature:

 

MotivationEdit

A force   has a magnitude and a direction, can be added to another force, be multiplied by a scalar and so on. These properties make the force   a vector.

Similarly, the displacement   is a vector because it can be added to other displacements and satisfies the other properties of a vector.

However, a force cannot be added to a displacement to yield a physically meaningful quantity. So the physical spaces that these two quantities lie on must be different.

Recall that a constant force   moving through a displacement   does   units of work. How do we compute this product when the spaces of   and   are different? If you try to compute the product on a graph, you will have to convert both quantities to a single basis and then compute the scalar product.

An alternative way of thinking about the operation   is to think of   as a linear operator that acts on   to produce a scalar quantity (work). In the notation of sets we can write

 

A first order tensor is a linear operator that sends vectors to scalars.

Next, assume that the force   acts at a point  . The moment of the force about the origin is given by   which is a vector. The vector product can be thought of as an linear operation too. In this case the effect of the operator is to convert a vector into another vector.

A second order tensor is a linear operator that sends vectors to vectors.

According to Simmonds, "the name tensor comes from elasticity theory where in a loaded elastic body the stress tensor acting on a unit vector normal to a plane through a point delivers the tension (i.e., the force per unit area) acting across the plane at that point."

Examples of second order tensors are the stress tensor, the deformation gradient tensor, the velocity gradient tensor, and so on.

Another type of tensor that we encounter frequently in mechanics is the fourth order tensor that takes strains to stresses. In elasticity, this is the stiffness tensor.

A fourth order tensor is a linear operator that sends second order tensors to second order tensors.

Tensor algebraEdit

A tensor   is a linear transformation from a vector space   to  . Thus, we can write

 

More often, we use the following notation:

 

I have used the "dot" notation in this handout. None of the above notations is obviously superior to the others and each is used widely.

Addition of tensorsEdit

Let   and   be two tensors. Then the sum   is another tensor   defined by

 

Multiplication of a tensor by a scalarEdit

Let   be a tensor and let   be a scalar. Then the product   is a tensor defined by

 

Zero tensorEdit

The zero tensor   is the tensor which maps every vector   into the zero vector.

 

Identity tensorEdit

The identity tensor   takes every vector   into itself.

 

The identity tensor is also often written as  .

Product of two tensorsEdit

Let   and   be two tensors. Then the product   is the tensor that is defined by

 

In general  .

Transpose of a tensorEdit

The transpose of a tensor   is the unique tensor   defined by

 

The following identities follow from the above definition:

 

Symmetric and skew tensorsEdit

A tensor   is symmetric if

 

A tensor   is skew if

 

Every tensor   can be expressed uniquely as the sum of a symmetric tensor   (the symmetric part of  ) and a skew tensor   (the skew part of  ).

 

Tensor product of two vectorsEdit

The tensor (or dyadic) product   (also written  ) of two vectors   and   is a tensor that assigns to each vector   the vector  .

 

Notice that all the above operations on tensors are remarkably similar to matrix operations.

Spectral theoremEdit

The spectral theorem for tensors is widely used in mechanics. We will start off by definining eigenvalues and eigenvectors.

Eigenvalues and eigenvectorsEdit

Let   be a second order tensor. Let   be a scalar and   be a vector such that

 

Then   is called an eigenvalue of   and   is an eigenvector .

A second order tensor has three eigenvalues and three eigenvectors, since the space is three-dimensional. Some of the eigenvalues might be repeated. The number of times an eigenvalue is repeated is called multiplicity.

In mechanics, many second order tensors are symmetric and positive definite. Note the following important properties of such tensors:

  1. If   is positive definite, then  .
  2. If   is symmetric, the eigenvectors   are mutually orthogonal.

For more on eigenvalues and eigenvectors see Applied linear operators and spectral methods.

Spectral theoremEdit

Let   be a symmetric second-order tensor. Then

  1. the normalized eigenvectors   form an orthonormal basis.
  2. if   are the corresponding eigenvalues then  .

This relation is called the spectral decomposition of  .

Polar decomposition theoremEdit

Let   be second order tensor with  . Then

  1. there exist positive definite, symmetric tensors  ,  and a rotation (orthogonal) tensor   such that  .
  2. also each of these decompositions is unique.

Principal invariants of a tensorEdit

Let   be a second order tensor. Then the determinant of   can be expressed as

 

The quantities   are called the principal invariants of  . Expressions of the principal invariants are given below.

Principal invariants of  

 

Note that   is an eigenvalue of   if and only if

 

The resulting equations is called the characteristic equation and is usually written in expanded form as

 

Cayley-Hamilton theoremEdit

The Cayley-Hamilton theorem is a very useful result in continuum mechanics. It states that

Cayley-Hamilton theorem

If   is a second order tensor then it satisfies its own characteristic equation

 

Index notationEdit

All the equations so far have made no mention of the coordinate system. When we use vectors and tensor in computations we have to express them in some coordinate system (basis) and use the components of the object in that basis for our computations.

Commonly used bases are the Cartesian coordinate frame, the cylindrical coordinate frame, and the spherical coordinate frame.

A Cartesian coordinate frame consists of an orthonormal basis   together with a point   called the origin. Since these vectors are mutually perpendicular, we have the following relations:

 

Kronecker deltaEdit

To make the above relations more compact, we introduce the Kronecker delta symbol

 

Then, instead of the nine equations in (1) we can write (in index notation)

 

Einstein summation conventionEdit

Recall that the vector   can be written as

 

In index notation, equation (2) can be written as

 

This convention is called the Einstein summation convention. If indices are repeated, we understand that to mean that there is a sum over the indices.

Components of a vectorEdit

We can write the Cartesian components of a vector   in the basis   as

 

Components of a tensorEdit

Similarly, the components   of a tensor   are defined by

 

Using the definition of the tensor product, we can also write

 

Using the summation convention,

 

In this case, the bases of the tensor are   and the components are  .

Operation of a tensor on a vectorEdit

From the definition of the components of tensor  , we can also see that (using the summation convention)

 

Dyadic productEdit

Similarly, the dyadic product can be expressed as

 

Matrix notationEdit

We can also write a tensor   in matrix notation as

 

Note that the Kronecker delta represents the components of the identity tensor in a Cartesian basis. Therefore, we can write

 

Tensor inner productEdit

The inner product   of two tensors   and   is an operation that generates a scalar. We define (summation implied)

 

The inner product can also be expressed using the trace :

 

Proof using the definition of the trace below :

 
 

Trace of a tensorEdit

The trace of a tensor is the scalar given by

 

The trace of an N x N-matrix is the sum of the components on the downward-sloping diagonal.

Magnitude of a tensorEdit

The magnitude of a tensor   is defined by

 

Tensor product of a tensor with a vectorEdit

Another tensor operation that is often seen is the tensor product of a tensor with a vector. Let   be a tensor and let   be a vector. Then the tensor cross product gives a tensor   defined by

 

Permutation symbolEdit

The permutation symbol   is defined as

 

Identities in tensor algebraEdit

Let  ,   and   be three second order tensors. Then

 

Proof:

It is easiest to show these relations by using index notation with respect to an orthonormal basis. Then we can write

 

Similarly,

 

Tensor calculusEdit

Recall that the vector differential operator (with respect to a Cartesian basis) is defined as

 

In this section we summarize some operations of   on vectors and tensors.

The gradient of a vector fieldEdit

The dyadic product   (or  ) is called the gradient of the vector field  . Therefore, the quantity   is a tensor given by

 

In the alternative dyadic notation,

 

'Warning: Some authors define the   component of   as  .

The divergence of a tensor fieldEdit

Let   be a tensor field. Then the divergence of the tensor field is a vector   given by

 

To fix the definition of divergence of a general tensor field (possibly of higher order than 2), we use the relation

 

where   is an arbitrary constant vector.

The Laplacian of a vector fieldEdit

The Laplacian of a vector field is given by

 

Tensor IdentitiesEdit

Some important identities involving tensors are:

  1.  .
  2.   .
  3.   .
  4.   .
  5.   .
  6.   .

Integral theoremsEdit

The following integral theorems are useful in continuum mechanics and finite elements.

The Gauss divergence theoremEdit

If   is a region in space enclosed by a surface   and   is a tensor field, then

 

where   is the unit outward normal to the surface.

The Stokes curl theoremEdit

If   is a surface bounded by a closed curve  , then

 

where   is a tensor field,   is the unit normal vector to   in the direction of a right-handed screw motion along  , and   is a unit tangential vector in the direction of integration along  .

The Leibniz formulaEdit

Let   be a closed moving region of space enclosed by a surface  . Let the velocity of any surface element be  . Then if   is a tensor function of position and time,

 

where   is the outward unit normal to the surface  .

Directional derivativesEdit

We often have to find the derivatives of vectors with respect to vectors and of tensors with respect to vectors and tensors. The directional directive provides a systematic way of finding these derivatives.

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 vectorsEdit

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 vectorsEdit

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 tensorsEdit

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 tensorsEdit

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  

3) If   then  

Derivative of the determinant of a tensorEdit

Derivative of the determinant of a 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 tensorEdit

Derivatives of the principal invariants of a 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 identity tensorEdit

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 tensor with respect to itselfEdit

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 tensorEdit

Derivative of the inverse of a 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,

 

RemarksEdit

The boldface notation that I've used is called the Gibbs notation. The index notation that I have used is also called Cartesian tensor notation.