is a fundamental principle in the analysis of strain that in the immediate neighbourhood of any point, whatever may be the nature of the distortion, the strain is sensibly homogeneous; in other words, whilst we can imagine types of distortion in which the faces and edges of a cube become curved, and in which opposite faces are unequally strained, it is permissible, when the dimensions of the cube are indefinitely diminished, to neglect these effects and to assume that the strained cube is a parallelopiped. This implies, according to the preceding investigation, that the most general type of strain at any point in a material may be described by specifying values of three extensions and three angles of shear. These six quantities are termed the "components of strain." Strains Expressed in Terms of Displacement.—i8. On the assumption that the displacements u, v, w are everywhere small, it may be shown (see article ELASTICITY) that the extension in the direction Ox will be given by Similar expressions hold for the other four components of strain. Thus the six components of strain are not really inde pendent : they can all be expressed in terms of three component displacements u, v, w, when these are given as functions of x, y, z, the co-ordinates of a particle in the unstrained material. Transformation of the Components of Strain.—i9. Given the values of u, v, w, we can evidently calculate the component displacements, u', v', in any other three perpendicular
tions Ox', Oy', Oz', and hence we can de duce expressions, e.g., for the extension in the direction Ox' and for the shear in the (z', x') plane. This is the problem of "transformation of strain-components": the general formulae will be found in the article ELASTICITY, and it will suffice here to give a relatively simple example.
Consider a square block ABCD which undergoes a simple shearing strain of mag nitude 7, as shown in fig. 7. The diagonals AC, BD will evidently remain perpendicular to one another, and the angle A'BD' will be
to the first order of small 2 quantities, the sides AB, BC, CD, DA will retain their original length 1. Hence, the strained length BD' will be given by 2/ cos (45 — - = A/2 • 1 (1+
2 ; very nearly, sincey is small.
That is to say, the fractional extension of BD is The fractional extension of AC may be shown in the same way to be —17. So we see that a state of strain represented by equal and opposite extensions of amounts e and —e in two directions at right angles implies a simple shearing strain of amount 2e in an element whose sides are equally inclined to these directions.
The fundamental theorem in the analysis of strain may now be stated :—Through any point in the material, however complicated may be the state of strain at that point, three lines can be found, each perpendicular to the other two, which were also perpendicular to one another initially, when the material was unstrained. In other words, a very small rectangular block of material, whose edges were originally paral lel to these lines, will remain rectangular after strain. The most general type of distortion may be specified by fixing these three directions, and the extension which corresponds to each' : the ex tensions are called "principal extensions," and the directions "principal axes of strain." In the two-dimensional example just considered, the diagonals AC, BD are the principal axes of strain.