Principal Values and Directions of Symmetric Second Order Tensors

59 Since the two fundamental tensors in continuum mechanics are of the second order and symmetric (stress and strain), we examine some important properties of these tensors.

60 For every symmetric tensor Tij defined at some point in space, there is associated with each direction (specified by unit normal nj) at that point, a vector given by the inner product vi = Tij nj (1.79)

1.2 Tensor

If the direction is one for which vi is parallel to ni, the inner product may be expressed as

and the direction ni is called principal direction of Tij. Since ni = Sijnj, this can be rewritten as

which represents a system of three equations for the four unknowns ni and A.

(T11 - A)ni + Ti2n2 + T13n3 = 0 T21.n1 + (T22 - A)n2 + T23n3 = 0 T3ini + T32n2 + (T33 - A)n = 0

To have a non-trivial slution (ni = 0) the determinant of the coefficients must be zero,

61 Expansion of this determinant leads to the following characteristic equation the roots are called the principal values of Tj and

It IIT IIIT

ij ij

are called the first, second and third invariants respectively of T.j.

62 It is customary to order those roots as Ai > A2 > A3

63 For a symmetric tensor with real components, the principal values are also real. If those values are distinct, the three principal directions are mutually orthogonal.

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