LETTERS IN APPLIED AND ENGINEERING SCIENCE, Vol. 3, pp. 435-439. Pergamon Press, Inc. Printed in the United States. ON THE COINCIDENCE OF THE PRINCIPAL AXES OF STRESS AND STRAIN IN ISOTROPIC ELASTIC BODIES R. C. Batra Engineering Mechanics Department University of Missouri Rolla, Missouri 65401 ABSTRACT It is shown that for isotropic elastic materials the principal axes of stress are also the principal axes of strain provided the empirical inequalities hold. 1. Introduction A general form of the constitutive equation for an unconstrained iso- tropic homogeneous elastic material is ([~J, Eqn. (47.9)) ~ = fO~ + fl~ + f-l~-l (1.1 where T is the Cauchy stress tensor, B is the left Cauchy-Green tensor with - - respect to an undistorted configuration, and the response coefficients f cx (cx = -1,0,1) are functions of the principal invariants of B. We note that B - - is a symmetric positive definite tensor and T is symmetric. It is an immedi- ate consequence of (1.1) that a proper vector of B is also a proper vector of T so that the principal axes of strain are also the principal axes of stress. That the converse need not be true unless some restrictions are imposed upon the response coefficients f is clear from the following special case of (1.1): cx ~ =fO (III) ~ . (1.2) In (1.2). III is the third principal invariant of B. The constitutive relat- ion (1.2) represents an elastic fluid and states that every vector is a prop- er vector of T but it gives no information about the proper vectors of B. - - 435