929 
For maximum deformation: 
° Fa 
x= = sin ot, - x, sin ot, =0 
and hence 
ot, = (I-49) 
Putting Eqe (I-49) into Eq- (I-48), 
In = 32 - x, (I-50) 
Equation (I-50) is important in the interpretation of ball crusher gauge 
behavior, and will be considered in detail. 
Reviewing preceding derivations, the deformation produced by a step 
wave F H(t) is given by Eq. (I-11): 
(I-11) 
The deformation produced by a force F applied infinitely slowly is given by 
Eqe (I-37 ) 3 
(I-37) 
ae | 
(This assumes that rate of strain is not significant in the production of the 
deformatione The assumption is manifestly incorrect, but a consideration of 
these idealized results is nevertheless illuminating.) 
The deformation obtained under the dynamic conditions implicit in 
Eqe (I-11) is twice that obtained due to the same “static” force in Eqe (I-37)« 
This is due to the fact that in the former case the piston acquires kinetic 
energy in the initial stages of deformation, sausing it to “overshoot” and 
give a final deformation 2F/ke 
Now returning to the consideration of Eqe (I-50): When an initial 
deformation exists, the piston cannot acquire kinetic energy in the initial 
stages; in fact, Eqe (I-50) shows that the piston cannot begin to move unless 
the applied force exceeds twice the step force which would originally have been 
necessary to produce the deformation x,» In other words, if x, had been pro- 
duced initially by a step force Fi» so that 
ca Darr a (I-51) 
