Tensores Parte 2
Tensores Parte 2
Tensores Parte 2
that is,
TRANSFORMATION MATRIX BETWEEN TWO
RECTANGULAR CARTESIAN COORDINATE SYSTEMS…
Where
Or
We note that
Example
TRANSFORMATION LAW FOR CARTESIAN COMPONENTS OF A VECTOR
or
TRANSFORMATION LAW FOR CARTESIAN COMPONENTS OF A TENSOR…
We can also express the unprimed components in terms of the primed components
If we premultiply the preceding equation with [Q] and post-multiply it with [Q]T
Since,
In indicial notation
TRANSFORMATION LAW FOR CARTESIAN COMPONENTS OF A TENSOR…
Example
TRANSFORMATION LAW FOR CARTESIAN COMPONENTS OF A TENSOR…
Example
Verificar con el ejemplo pág. 6.
TRANSFORMATION LAW FOR CARTESIAN COMPONENTS OF A TENSOR…
Example
TRANSFORMATION LAW FOR CARTESIAN COMPONENTS OF A TENSOR…
We see from the first example that we can calculate all nine components of a tensor T
with respect to from the matrix by using Eq. (2.18.5).
However, there are often times when we need only a few components. Then it is
more convenient to use
where denote the row matrix whose elements are the components of
with respect to the basis
Example
SYMMETRIC AND ANTISYMMETRIC TENSORS
that is,
that is,
SYMMETRIC AND ANTISYMMETRIC TENSORS
Any tensor T can always be decomposed into the sum of a symmetric tensor and an
antisymmetric tensor. In fact,
where
The diagonal elements of an antisymmetric tensor are always zero, and, of the
six nondiagonal elements, only three are independent, because T12 = -T21; T23 =
-T32 and T31 = -T13. Thus an antisymmetric tensor has really only three
components, just like a vector. More specifically, for every antisymmetric
tensor T:
This vector tA is called the dual or axial vector of the antisymmetric tensor.
THE DUAL VECTOR OF AN ANTISYMMETRIC TENSOR
or, in indicial
notation,
The calculations of dual vectors have several uses. For example, it allows us to
easily obtain the axis of rotation for a finite rotation tensor. In fact, the axis of
rotation is parallel to the dual vector of the antisymmetric part of the rotation
tensor. Also, it will be shown that the dual vector can be used to obtain the
infinitesimal angles of rotation of material elements under infinitesimal
deformation.
THE DUAL VECTOR OF AN ANTISYMMETRIC TENSOR
Example
THE DUAL VECTOR OF AN ANTISYMMETRIC TENSOR
Example
EIGENVALUES AND EIGENVECTORS OF A TENSOR
Consider a tensor T. If a is a vector that transforms under T into a vector parallel to
itself, that is,
A tensor may have infinitely many eigenvectors. In fact, since Ia = a, any vector is an
eigenvector for the identity tensor I, with eigenvalues all equal to unity. For the tensor
βI, the same is true except that the eigenvalues are all equal to β.
Some tensors only have eigenvectors in one direction. For example, for any rotation
tensor that effects a rigid body rotation about an axis through an angle not equal to an
integral multiple of π, only those vectors that are parallel to the axis of rotation will
remain parallel to themselves.
Let n be a unit eigenvector
thus,
trivial solution
This solution simply states the obvious fact that a = 0 satisfies the equation Ta = λa,
independent of the value of λ.
To find the nontrivial eigenvectors for T, we note that a system of homogeneous,
linear equations admits a nontrivial solution only if the determinant of its coefficients
vanishes. That is,
that is,
Example
Example
Example
Example
All the examples given here have three eigenvalues that are real. It can be
shown that if a tensor is real (i.e., with real components) and symmetric, then
all its eigenvalues are real.
If a tensor is real but not symmetric, then two of the eigenvalues may be
complex conjugates. The following is such an example.
Example
PRINCIPAL VALUES AND PRINCIPAL DIRECTIONS OF
REAL SYMMETRIC TENSORS
We shall encounter several real tensors (stress tensor, strain tensor, rate of
deformation tensor, etc.) that are symmetric.
The following significant theorem can be proven: The eigenvalues of any real
symmetric tensor are all real.
Thus, for a real symmetric tensor, there always exist at least three real eigenvectors,
which we shall also call the principal directions. The corresponding eigenvalues are
called the principal values.
PRINCIPAL VALUES AND PRINCIPAL DIRECTIONS OF
REAL SYMMETRIC TENSORS
Then, by definition,
In other words, if there are two distinct eigenvectors with the same eigenvalue, then there
are infinitely many eigenvectors (which form a plane) with the same eigenvalue. This
situation arises when the characteristic equation has a repeated root.
PRINCIPAL VALUES AND PRINCIPAL DIRECTIONS OF
REAL SYMMETRIC TENSORS
Suppose the characteristic equation has roots
Therefore there exist infinitely many sets of three mutually perpendicular principal
directions, each containing n3 and any two mutually perpendicular eigenvectors of
the repeated root λ.
PRINCIPAL VALUES AND PRINCIPAL DIRECTIONS OF
REAL SYMMETRIC TENSORS
Ejemplo:
We conclude that for every real symmetric tensor there exists at least one triad of
principal directions that are mutually perpendicular.
MATRIX OF A TENSOR WITH RESPECT TO PRINCIPAL DIRECTIONS
For a real symmetric tensor, there always exist three principal directions that are
mutually perpendicular. Let n1, n2 and n3 be unit vectors in these directions.
Then, using n1, n2 and n3 as base vectors, the components of the tensor are
Thus, the matrix is diagonal and the diagonal elements are the eigenvalues of T.
The principal values of a tensor T include the maximum and the minimum
values that the diagonal elements of any matrix of T can have.
we have
that is,
We also have
that is,
MATRIX OF A TENSOR WITH RESPECT TO PRINCIPAL DIRECTIONS
Thus, the maximum value of the principal values of T is the maximum value of
the diagonal elements of all matrices of T, and the minimum value of the
principal values of T is the minimum value of the diagonal elements of all
matrices of T.
It is important to remember that for a given T, there are infinitely many matrices
and therefore, infinitely many diagonal elements, of which the maximum
principal value is the maximum of all of them and the minimum principal value is
the minimum of all of them.
PRINCIPAL SCALAR INVARIANTS OF A TENSOR
The characteristic equation of a tensor T,
where
PRINCIPAL SCALAR INVARIANTS OF A TENSOR
Since by definition, the eigenvalues of T do not depend on the choices of the base vectors,
therefore the coefficients of
will not depend on any particular choices of basis. They are called the principal scalar
invariants of T.
We note that, in terms of the eigenvalues of T, which are the roots of