By Academician Leonid Brekhovskikh, Dr. Valery Goncharov (auth.)
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Then a torsional wave propagates along the rod. The strain in each section x is specified by the twist angle lp = lp(x, t) about the axis. The stress in the same section is specified by the torque M(x, t). 14) we can easily relate these two quantities. In fact, consider an element of the rod bounded by two sections a very small distance Ax apart. Its twist angle is Alp = lp(x + Ax) - lp(x) ~ Ax alp/ax. 14) we obtain M(x) = f Alp = /1(na 4 /2)Alp/Ax, or in the limit Ax --+ 0, na4 alp M(x) = /1 2 ax.
The strain in each section x is specified by the twist angle lp = lp(x, t) about the axis. The stress in the same section is specified by the torque M(x, t). 14) we can easily relate these two quantities. In fact, consider an element of the rod bounded by two sections a very small distance Ax apart. Its twist angle is Alp = lp(x + Ax) - lp(x) ~ Ax alp/ax. 14) we obtain M(x) = f Alp = /1(na 4 /2)Alp/Ax, or in the limit Ax --+ 0, na4 alp M(x) = /1 2 ax. 19) The net torque acting on the element Ax is equal to the difference of the torques at the two sections: 24 2.
I) Absolutely rigid boundary. Particle displacement is forbidden: u(xo,t) = o. 6) In practice, this case can be realized if an end face of the rod is stuck to a massive wall of a material with very large Young's modulus. ii) Absolutely soft (free) boundary, say, a rod in a rather rarefied medium or vacuum. The forces (stresses) vanish at such a boundary: F/sIXQ = Eou/oxlxQ = 0, or ou/oxlxQ = o. 7) An interface with air is a good approximation to an absolutely soft boundary. iii) Contact between two rods with equal cross sections glued together with different material constants Pl' El and P2' E 2, respectively.
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