| ∂w | − | ∂v | , | ∂u | − | ∂w | , | ∂v | − | ∂u |
| ∂y | ∂z | ∂z | ∂x | ∂x | ∂y |
vanish, rotational if any one of them is different from zero. The halves of these three quantities are the components of a vector quantity called the “rotation.”
15. Whether the strain is rotational or not, there is always one set of three linear elements issuing from any point which cut each other at right angles both before and after strain. If these directions are chosen as axes of x, y, z, the shearing strains eyz, ezx, exy vanish at this point. These directions are called the “principal axes of strain,” and the extensions in the directions of these axes the “principal extensions.”
16. It is very important to observe that the relations between components of strain and components of displacement imply relations between the components of strain themselves. If by any process of reasoning we arrive at the conclusion that the state of strain in a body is such and such a state, we have a test of the possibility or impossibility of our conclusion. The test is that, if the state of strain is a possible one, then there must be a displacement which can be associated with it in accordance with the equations (1) of § 11.
We may eliminate u, v, w from these equations. When this is done we find that the quantities exx, ... eyz are connected by the two sets of equations
| ∂²eyy | + | ∂²ezz | = | ∂²eyz |
| ∂z² | ∂y² | ∂y∂z |
(1)
| ∂²ezz | + | ∂²exx | = | ∂²ezx |
| ∂x² | ∂z² | ∂z∂x |
| ∂²exx | + | ∂²eyy | = | ∂²exy |
| ∂y² | ∂x² | ∂x∂y |
and