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Principal Strain 105
3.9 Principal Strain
Since the strain tensor E is symmetric, therefore, (see Section 2B.18) there exists at least
three mutually perpendicular directions n 1,n 2,n3 with respect to which the matrix of E is
diagonal. That is
Geometrically, this means that infinitesimal line elements in the directions of 1^,112,113 remain
mutually perpendicular after deformation. These directions are known as the principal
directions of strain. The unit elongation along the principal direction (i.e., £^2^3) are the
eigenvalues of E, or principal strains, they include the maximum and the minimum normal
strains among ail directions emanating from the particle. For a given E, the principal strains
are to be found from the characteristic equation of E, i.e.,
where
The coefficients /i/2' an< 3 /3 are called the principal scalar invariants of the strain tensor.
3.10 Dilatation
The first scalar invariant of the infinitesimal strain tensor has a simple geometric meaning.
For a specific deformation, consider the three material lines that emanate from a single point
P and are in the principal directions. These lines define a rectangular parallelepiped whose
sides have been elongated from the initial dimension
to
an
where £±£2 d £3 are tne principal strains. Hence the change A(rfF) in this material volume
dVh
