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10). volume of a small block of natural dimensions is 6x(i -\-a)xdy(i -\- P)x = dxdydz(i dz(i -\-y) 4-« + + /^ J') neglecting products of «, fi and y. Then it follows that the ratio of the increase of volume to the original volume is (a-\- fi -{- y)- Call this A. 20) . ^) and n are constants, n being the modulus of E and In terms of rigidity. 22) j^ du dv dw 25. Relations We between stresses and external with a rectangular block of dimensions . '"3) ^Si + 8y + T. forces. x x dy dx dx having its centre at x, y, 2, and take the means stresses on the faces to be the stresses at the middle of the face concerned.

Hence Again, [x,y -{- since \ dy, z), S'o it and S'^g are supposed to act at (x,y follows in the same way — \dy,z) and that 5S. ,=^<5;.. force on the block in the direction (P"i-P,) Similarly the forces in the direction and dy. dy S'3, S"3 are OX due to P\ and P''^ is dx dy dz. dx OX due to the stresses S'o, S"2 RELATIONS BETWEEN STRESS AND STRAIN (S"2 — S'2) dx dy = — ^ 25 dx dydz, 0% and (S'^g Then the total — S'^)dxdz=—^dxdydz. re zero. _Bhv ^^'^7) ;t-^^2' /2-^^2; /a-^^2 t denoting the time measured from any particular instant.

Elastic body may be set vibrating in different sorts of ways^ Any and these vibrations always die out after a short time. Since the damping effect on the body that is, it is vibrations usually take place in air the whole of the might be attributed to the action of the air ; possible that the whole of the energy in the vibrating body into the air by means of air waves, such as those that is dissipated produce sound. it appears to be well established by experiment that the air alone not sufficient to account for the whole of the damping effect; that, in fact, there is a sort of internal frictional resistance to the relative motion of the particles of the body, this resistance depending on the But is relative velocities, The property and increasing as these relative velocities increase.

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