that there does not exist an elastic potential in VorLrerra’s hysteresis, 
so that dissipation of energy takes place. 
Starting from the equations of motion: 
vi) Ou Watn dbnl OLNE 
e(2—5) | a Tou pena OD 
TE, bed TELT Ee IRE 
(3) 
the relation has been derived (see e.g. RrEMANN-Wreger. Die part. 
diff. Gleich. der math. Physik. IL § 65): 
Work done by the forces of mass X, Y, and Z acting on the 
volume elements + work done by the surface pressures — change 
in kinetic energy — 
‘ OY,, dy. pn 
— SAA GSE eo \ 1 — ( 1 
fle Ot Tha Ot _| te ve 
Then follows from the second law of thermodynamics that: 
lt ; t O¥11 t Òy.s 1 
C ij za Ot 12 Ot 2+. | ae 
represents the potential energy gained in the time dt inside the space 
a in consequence of elastic tensions, so that we get 
Change in the potential energy = dT" = Zn dyn 
th 
Substituting in this the expressions for the tensions corrected for 
linear hysteresis according to VOLTERRA, we get: 
; 
dT = T(z days Yrs | Cyan + SL] DE wijs (et) yrs(©) dr] dyn 
th rs th 2 rs 
0 
Formerly the w’s were zero, and the reasoning ran: 
If there is to be an elastic potential 7”, the righthand member must 
be a total differential; this requires a set of fifteen equations, con- 
ditions of integrability in the 36 coefficients, viz. bij, = 6s ; when 
they were fulfilled, a function 7’ could be solved as elastic 
potential. 
We shall now prove that the conditions of integrability cannot 
be fulfilled. To this end we consider: 
th rs 
t 
aT! => i > bih|rs Vrs (t) In 5 Won [rs (tr) Yrs (7) dr } dyn . (J) 
"i rs 
Following Vouterra’s fundamental idea | divide the interval from 
1) The tih's are as always the elastic tensions, the yii’s the quantities of 
deformation. 
