. A treatise on the mathematical theory of elasticity . L. E. ? 3 34 SPECIFICATION OF STRAIN [CH. I Now, writing /3 for Jir - a, we have a;j=a;+2 tan a cos ^j3 (- ^sin il3+y cos ^^), i/2=y + 2 tan a sin J/3 (- « sin ^/3+y cos J|8);and we can observe that - ^2 sin 4 3+^2 cos ii3 = - ^ sin i^+y cos J/3,and that x^ cos J/3+y2 sin J0=.r cos J^ +y sin J^ + 2 tan a (- « sin Jj3+y cos J0). Hence, taking axes of X and F which are obtained from those of x and yby a rotation through ^tt —Ja in the sense from a; towards y, we see thatthe particle which was at (X, Y) is moved by the pure shear followed by
. A treatise on the mathematical theory of elasticity . L. E. ? 3 34 SPECIFICATION OF STRAIN [CH. I Now, writing /3 for Jir - a, we have a;j=a;+2 tan a cos ^j3 (- ^sin il3+y cos ^^), i/2=y + 2 tan a sin J/3 (- « sin ^/3+y cos J|8);and we can observe that - ^2 sin 4 3+^2 cos ii3 = - ^ sin i^+y cos J/3,and that x^ cos J/3+y2 sin J0=.r cos J^ +y sin J^ + 2 tan a (- « sin Jj3+y cos J0). Hence, taking axes of X and F which are obtained from those of x and yby a rotation through ^tt —Ja in the sense from a; towards y, we see thatthe particle which was at (X, Y) is moved by the pure shear followed bythe rotation to the point (Xa, Y^), where Za = Z + 2 tan «. F, Y^=Y. Thus every plane of the material which is parallel to the plane of (Z, z) slidesalong itself in the direction of the axis of Z through a distance proportionalto the distance of the plane from the plane of (Z, z). The kind of strain justdescribed is called a simple shear, the angle a is the angle of the shear,and 2 tan a is the amount of the Fig. 1. 3-5] BY MEANS OF DISPLACEMENT 35 Fig. 1 shows a square ABGD distorted by pure shear into a rhombusABCiy oitYie same area, which is then rotated into the position angle of the shear is AOA, and the angle AOX is half the complementof this angle. The lines AA, BE, CO, DD are parallel to OX and propor-tional to their distances from it. We shall find that all kinds of strain can be described in terms of simpleextension and simple shear, but for the discussion of complex states of strainand for the expression of them by means of simpler strains we require ageneral kinematical theory *. 4. Displacement. We have, in every case, to distinguish two states of a body—a first stateand a second state., The particles of the body pass from their positions inthe first state to their positions in the second state by a displacement. Thedisplacement may be such that the line joining any two particles of the bodyhas the same length in the second state as it has i
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