INTERACTION OF POLYMERS AND MECHANICAL WAVES 309 



From those aspects al)()\-e the cmreiit results of (lyuamics studies will 

 he re\'ie\ve(l. 



POLYMER solids: over-ai,l ,m kcua ntcs 



Solid polymers will denote ruhhers and plastics in the state iu which 

 they are technically used. This is usually their most complex form, with 

 inter- and intra-molecular factors undistin<>;uished. Thus, se])aration and 

 identification of the main relaxation processes are difficult or impossible. 

 IIo\ve\'er, it is interesting to consider typical values of modulus and 

 \-isc()sity as related to chemical structure, in the I'ange of fr(v|nencies 

 corresponding to extrusion rates, and stresses in actual use. 



These values of dynamic modulus and viscosity are distinct from the 

 usual cjuantities in the literature. The usual expressions are for longi- 

 tudinal (sound) waves, and give dynamic Young's modulus 



E* = Er - iE. 



E-2 measures the out of phase part of the force-displacement relation, 

 and Eo = w- ("effective viscosity coefficient"). Now, the general elastic 

 constants are X + 2/i, with X = Lame's constant and (jl = shear modu- 

 lus. Here. 



X -f- 2m = /C + l/x, 



with K = bulk modulus. Alternately, 



SK 3X + 2m 



X + M X +M 



However, in general the present results lead to the simpler shear modu- 

 lus M- Further the energy losses studied are expressible directly as the 

 usual shear viscosity 



Previous comprehensive studies of the dynamics of rubbers over sig- 

 nificant frequency ranges have yielded loss factors either written as 

 E-2 El (see above),'* or as a function of the shear viscosity based on 

 Stoke's assumption that the compressional (dilatational) viscosity is 

 zero.^" But as Nolle'^ and Ivey, Mrowca and Guth^° clearly re(;ognize, 

 recent work has strongly manifested the presence of compressional vis- 

 cosity in simple liquids"' as well as polymeric ones."" " Hence, the pres- 

 ent understanding relating molecular structure to viscosity, plasticity 

 and visco-elasticity is unsuitable for interpreting mechanical wave mo- 

 tion more complex than in shear, unless shear constants are also known. 



