PULSE WAVES IN VISCO-ELASTIC TUBINGS 



III 



FIG. 2. Types of stress-strain relations for rubberlike materials. 



2. DYN.^MIC ELASTICITY AND VISCOSITY 



If we Stretch a strip of rubber, a piece of an artery, 

 or any one of the materials termed today as 

 "elastomers," either in a continuous manner or in 

 graduated steps by hanging on weights, we can 

 graphically present the results of such an experiment 

 in the form of a stress-strain curve. As a rule, such 

 curves will be more or less S-shaped (see fig. 2.4). 

 If we remove the strain in the same continuous or 

 stepwise fashion, we obtain a characteristic descending 

 curve of similar shape, which does not coincide with 

 the ascending curve and does not return to the zero 

 point of origin of the coordinate system. In other 

 words, when the force applied to the sample is re- 

 leased, a certain amount of stretch remains (at least 

 for some time). Recognizing its analogy to a similar 

 effect well known in magnetism, we term this be- 

 havior "hysteresis." 



This means that Hooke's law does not apply to 

 such materials because it is valid only when the 

 stress-strain relationship is reversible and can be 

 represented as a straight line passing through the 

 origin. 



In order to avoid these difficulties, we shall use a 

 rhythmic stretch of a sinusoidal type, so that the 

 length of the sample (i.e. the rubber strip) under 

 consideration will be given by 



'(0 = 'm -I- A/ sin uit 



(2.1) 



where /„ corresponds to the mean prestretch length. 

 If we record the corresponding stress simultaneously 

 with the length l{i), we obtain a closed curve which 

 approaches, more or less, a stretched-out ellipse when 

 the amplitude of stretch (J'2A/) is made sufficiently 

 small (fig. 2C). The long axis of that ellipse almost 

 coincides with the tangent drawn to the static stress- 

 strain curve at the point (/„ , 5„), S,n representing 

 the strain for the prestretched length /„ . 



The appearance of an ellipse means that a phase 

 shift occurs between stretch and strain, and that some 

 of the work done on the sample is transformed into 



heat. By this ellipsoidal characteristic which can be 

 recorded experimentally, the visco-elastic behavior 

 of the considered material in the state (/„i'„) will be 

 sufficiently well characterized, at least for our pur- 

 poses. 



For the purpose of interpretation, we shall make use 

 of the simplest mechanical model that will describe 

 the above-mentioned characteristic. It is made of a 

 spring, with the spring-constant /, and a dashpot 

 for damping connected in parallel with it (see fig. 3). 

 One end of the spring is considered to be fixed in 

 space, whereas the free end moves in a sinusoidal 

 manner, like the rubber strip mentioned above. If 

 we take as the origin of .v-coordinates the free end of 

 the spring at rest, we might describe the movement 

 of that end bv the formula 



X = xa sin mt 



(2.2) 



(For the actual sample, the coordinate x corresponds 

 to the displacement / — /,„.) The resulting strain .S" 

 for the model in the .v-direction will then be given as 

 the sum of the restoring force /■ .v and the force 

 R dxj dt needed to o\ercome friction 



S = /• X -I- R 



dx 



Jt 



(2.3) 



It follows from equation 2.3 that the strain .S' is also a 

 sinusoidal function of time. This can easily be seen 

 by inserting equation 2.2 into 2.3. We obtain in 

 this wa\- 



S = f- xtt sin (Jit -\- Ruixo cos u>t 



(2.4) 



The sum of a pure sin-function and a pure cos-fimc- 

 tion can always be put in the form of a pure sin- 

 function with an appropriate phase <p 



S = S„ sin (ut + ip) (2.5) 



For .S'u and (p we obtain in the usual manner 



S« = .v.,-/(i -I- ^.•=/?2//2)i -• tan,e = oiR/f (2.6) 



As with the experiment on rubber, the points (.?, x) 



