MEASUREMENT OF PRESSURES 149 



with known velocity show that the maximum depression of the ball is a 

 function only of the energy of the piston before impact, a result sug- 

 gesting that the resistance of the ball to deformation is primarily the 

 result of work hardening. If the deformation S is measured, the aver- 

 age force acting during such a ballistic impact can be calculated from the 

 initial energy of the striking piston, and with the aid of such experiments 

 a calibration curve of force F versus plastic deformation S can be pre- 

 pared. The use of copper spheres has the practical advantage of giving 

 a very nearly linear calibration curve of the form F = kS for deforma- 

 tions up to .05 and .08 inches for the 5/32 inch and 3/8 inch spheres, 

 corresponding to static pressures of 3,000 and 10,000 lb. /in. 2. 



This is formally the same as a Hooke's law restoring force for a 

 spring, with the difference that there is a permanent set of the sphere 

 after the force is released rather than complete elastic recovery. The 

 response of the entire gauge up to the time of maximum deformation is 

 thus equivalent to that of a mass and spring subjected to a force applied 

 to the mass and the familiar differential eciuation of a linear oscillator 

 results for the motion of the piston : 



(5.1) m^ + kS = AP{t) 



w^here AP(t) is the force on the piston of area A for an applied pressure 

 P{t) and 771 is the effective mass of the gauge. This effective mass is 

 taken, in the theory developed by G. K. Hartmann (44), to be the sum 

 of the piston mass plus J^ the mass of the copper ball (by analogy with 

 the corresponding value for a spring of finite mass) plus a hydrodynamic 

 mass. This last term represents the inertia of water near the face of 

 the piston which moves with the piston, and is assumed by Hartmann 

 to be %pd'^ where d is the diameter of the piston face, this being the 

 value predicted by elementary hydrodynamics for the case of an in- 

 finite cylinder. 2 



The equation of motion (5.1) can be integrated for any assumed 

 pressure variation. We consider the approximation to a shock wave 

 of an initial peak followed by an exponential decay of the form 

 P{t) = Pme~^/^. With this assumption, the solution of Eq. (5.1) is 

 easily found by standard methods to be 



(5.2) S ^^"^ ^''^^' 



k 1 + {o^ey 



g-t/e J sin a)t — cos cct 



cot 



where 



0; = J- 



k 



m 



2 This value is derived on the assumption that the motion of the water near the 

 piston can be described as noncompressive flow. The formulation of the effective 

 pressure at a moving boundary, from which the correction is obtained, is given in 

 section 10.5. 



