Page:Love, A.E.H. - A treatise on the mathematical theory of elasticity (1920).djvu/58

Now, writing $\beta$ for ${\frac {1}{2}}\pi -a$, we have

$x_{2}=x+2\tan a\cos {\frac {1}{2}}\beta (-x\sin {\frac {1}{2}}\beta +y\cos {\frac {1}{2}}\beta )$,
$y_{2}=y+2\tan a\sin {\frac {1}{2}}\beta (-x\sin {\frac {1}{2}}\beta +y\cos {\frac {1}{2}}\beta )$;

and we can observe that

$-x_{2}\sin {\frac {1}{2}}\beta +y_{2}\cos {\frac {1}{2}}\beta =-x\sin {\frac {1}{2}}\beta +y\cos {\frac {1}{2}}\beta$,

and that

$x_{2}\cos {\frac {1}{2}}\beta +y_{2}\sin {\frac {1}{2}}\beta =x\cos {\frac {1}{2}}\beta +y\sin {\frac {1}{2}}\beta +2\tan a(-x\sin {\frac {1}{2}}\beta +y\cos {\frac {1}{2}}\beta )$.

Hence, taking axes of X and F which are obtained from those of x and y by a rotation through ${\frac {1}{4}}\pi -{\frac {1}{2}}\alpha$ in the sense from x towards y, we see that the particle which was at (X, Y) is moved by the pure shear followed by the rotation to the point (X₂, Y₂), where

$X_{2}=X+2\tan \alpha .Y$, $Y_{2}=Y$.

Thus every plane of the material which is parallel to the plane of (X, z) slides along itself in the direction of the axis of X through a distance proportional to the distance of the plane from the plane of (X, z). The kind of strain just described is called a "simple shear," the angle α is the "angle of the shear," and 2 tan α is the "amount of the shear."

Fig. 1.