00033___ebfeef01262096bb578dc0f0490b0a1c - 12 Chap. 1...

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Unformatted text preview: 12 Chap. 1 INTRODUCTION a l e l.2 E L 0 . c 0) Q ‘0 Time I Time Fig. 1 2 3 Relaxation function, of ( a) Maxwell, (b) Voigt, ( c) Kelvin solid. .:. For t he Kelvin solid, a similar interpretation is applicable. The constant rEis the time of relaxation of load under the condition of constant deflection [see Eq. ( lo)], whereas the constant ro is the time of relaxation of deflection under the condition of constant load [see Eq. ( 4)]. As t -+ 00, t he dashpot is completely relaxed, and the load-deflection relation becomes that of the springs, as is characterized by the constant E R in Eqs. ( 4) a nd ( 10). Therefore, E R is called the relaxed elastic modulus. Load-deflection relations such as ( 1)-(3) were proposed t o extend the classical theory of elasticity to include anelastic phenomena. Lord Kelvin (Sir William Thomson, 1824-1907), on measuring the variation of the rate of dissipation of energy with frequency of oscillation in various materials, showed the inadequacy of the Maxwell and Voigt equations. A more successful generalization using mechanical models was first made by John H. Poynting (1852-1914) a nd Joseph John Thomson ( “J.J.,”1856-1940) i n their book Properties of M atter (London: C . Griffin and Co., 1902). 1.3. SINUSOIDAL OSCILLATIONS IN A VISCOELASTIC MATERIAL I t is interesting to examine the relationship between the load and the deflection in a body when it is forced t o perform simple harmonic oscillations. A simple harmonic oscillation can be described by the real or imaginary part of a complex variable. Thus, if Fo is a complex variable FO= A eib = A(cos 4 + i sin d ), t hen a simple harmonic oscillatory force can be written as ( 1) F ( t ) = FoeiWt A (cos4 + i sin4)(coswt + i sinwt) = = A cos(wt + 4) + i A sin(wt + 4) . ...
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This note was uploaded on 01/04/2012 for the course ENG 501 taught by Professor Thomson during the Fall '05 term at MIT.

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