Mae 640 Lec6

Continuum mechanics – MAE 640 Summer II – 2009

Dr. Konstantinos Sierros 263 ESB new add [email protected]

Eigenvalues and eigenvectors of tensors • It is useful to regard a tensor as an operator that changes a vector into another vector (by means of the dot product). • In this regard, it is interesting to find out whether certain vectors have only their lengths, and not their orientation, changed when operated upon by a given tensor (i.e., seek vectors that are transformed into multiples of themselves). • If such vectors exist, they must satisfy the following equation;

x is called characteristic vector, principal planes, or eigenvector associated with A

The parameter λ is called an characteristic value, principal value, or eigenvalue, and it characterizes the change in length of the eigenvector x after it has been operated upon by A.

Eigenvalues and eigenvectors of tensors

Since x can be expressed as x = I · x

Because this is a homogeneous set of equations for x, a nontrivial solution (i.e., vector with at least one component of x is nonzero) will not exist unless the determinant of the matrix [A − λI] is zero

The vanishing of this determinant yields an algebraic equation of degree n, called the characteristic equation, for λ when A is a n × n matrix.

Eigenvalues and eigenvectors of tensors • For a second-order tensor Φ the characteristic equation yields three eigenvalues λ1, λ2, and λ3 • At least one of the eigenvalues must be real. The other two may be real and distinct, real and repeated, or complex conjugates. • In a Cartesian system, the characteristic equation associated with a second order tensor can be expressed in the following form;

• I1, I2, and I3 are the invariants Φ as defined in the previous class • The invariants can also be expressed in terms of the eigenvalues as following;

Example 1 Determine the eigenvalues and eigenvectors of the following matrix;

Problem 2.1 Find the equation of a line (or a set of lines) passing through the terminal point of a vector A and in the direction of vector B.

Problem 2.3 Prove the following vector identity without the use of a coordinate system

Problem 2.6 Given the following components;

Determine the following;