Symmetry of the Stress Tensor

Consider a rectangular cuboid of dimensions , , and .

It follows from Newton’s second law for rotational motion, that where is the resultant moment about the z-axis, is the moment of inertia (or rotational mass) about the -axis and is the angular acceleration.

From statics, we recall that the moment of inertia of a rectangular block about the centroidal -axis is where is the mass of the element. Expressing in terms of density, we can write where is the mass density of the material at the point.

The components of stress that contribute to are the shear stresses in the -plane. Body forces do not contribute to the moment. Thus, the total moment can be expressed as Equating this with the inertia expression, we obtain Neglecting higher-order small terms, we get which leads to

By applying the same logic for rotations about the x and y axes, we can prove that and . This means that in general and the stress tensor is always symmetric, and to specify it we need only 6 independent components (instead of 9). The result holds regardless of whether the body is at rest, in uniform motion, or accelerating. ¹ This result is known as Cauchy’s Second Law of Motion.