A Brief Introduction to Tensors and Their Properties
B.1. BASIC PROPERTIES OF TENSORS
B.1.1 Examples of Tensors
The gradient of a vector field is a good example of a tensor. Visualize a vector field: at every point in space, the field has a vector value . Let represent the gradient of u. By definition, G enables you to calculate the change in u when you move from a point x in space to a nearby point at :
G is a second order tensor. From this example, we see that when you multiply a vector by a tensor, the result is another vector.
This is a general property of all second order tensors. A tensor is a linear mapping of a vector onto another vector. Two examples, together with the vectors they operate on, are:
The stress tensor
where n is a unit vector normal to a surface, is the stress tensor and t is the traction vector acting on the surface.
The deformation gradient tensor
where dx is an infinitesimal line element in an undeformed solid, and dw is the vector representing the deformed line element.
B.1.2 Matrix representation of a tensor
To evaluate and manipulate tensors, we express them as components in a basis, just as for vectors. We can use the displacement gradient to illustrate how this is done. Let be a vector field, and let represent the gradient of u. Recall the definition of G
Now, let be a Cartesian basis, and express both du and dx as components. Then, calculate the components of du in terms of dx using the usual rules of calculus
We could represent this as a matrix product
From this we see that G can be represented as a matrix. The elements of the matrix are known as the components of G in the basis . All second order tensors can be represented in this form. For example, a general second order tensor S could be written as
You have probably already seen the matrix representation of stress and strain components in introductory courses.
Since S can be represented as a matrix, all operations that can be performed on a matrix can also be performed on S. Examples include sums and products, the transpose, inverse, and determinant. One can also compute eigenvalues and eigenvectors for tensors, and thus define the log of a tensor, the square root of a tensor, etc. These tensor operations are summarized below.
Note that the numbers , , … depend on the basis , just as the components of a vector depend on the basis used to represent the vector. However, just as the magnitude and direction of a vector are independent of the basis, so the properties of a tensor are independent of the basis. That is to say, if S is a tensor and u is a vector, then the vector
has the same magnitude and direction, irrespective of the basis used to represent u, v, and S.
B.1.3 The difference between a matrix and a tensor
If a tensor is a matrix, why is a matrix not the same thing as a tensor? Well, although you can multiply the three components of a vector u by any matrix,
the resulting three numbers may or may not represent the components of a vector. If they are the components of a vector, then the matrix represents the components of a tensor A, if not, then the matrix is just an ordinary old matrix.
To check whether are the components of a vector, you need to check how change due to a change of basis. That is to say, choose a new basis, calculate the new components of u in this basis, and calculate the new matrix in this basis (the new elements of the matrix will depend on how the matrix was defined. The elements may or may not change if they don’t, then the matrix cannot be the components of a tensor). Then, evaluate the matrix product to find a new left hand side, say . If are related to by the same transformation that was used to calculate the new components of u, then are the components of a vector, and, therefore, the matrix represents the components of a tensor.
B.1.4 Creating a tensor using a dyadic product of two vectors.
Let a and b be two vectors. The dyadic product of a and b is a second order tensor S denoted by
with the property
for all vectors u. (Clearly, this maps u onto a vector parallel to a with magnitude )
The components of in a basis are
Note that not all tensors can be constructed using a dyadic product of only two vectors (this is because always has to be parallel to a, and therefore the representation cannot map a vector onto an arbitrary vector). However, if a, b, and c are three independent vectors (i.e. no two of them are parallel) then all tensors can be constructed as a sum of scalar multiples of the nine possible dyadic products of these vectors.
B.2. OPERATIONS ON SECOND ORDER TENSORS
Let be a Cartesian basis, and let S be a second order tensor. The components of S in may be represented as a matrix
The representation of a tensor in terms of its components can also be expressed in dyadic form as
This representation is particularly convenient when using polar coordinates, as described in Appendix E.
(c) A.F. Bower, 2008