________________________________________________________________________
PROBLEM (12.29) An element in three-dimensional state of stress is subjected to stresses as shown
in Fig. P12.29. Determine the strain-energy density and its components at the element.
Given: E = 70 GPa ,
= 1/3
SOLUTION
2(1 ) 70 2.5 28GE GPa
=+= =
Fig. P12.29:
140 .
Pa
=
22
2,3 50 30 50 30 (20)
22
σ
−− −−
⎛⎞
=± +
⎜⎟
⎝⎠
or
23
17.6 62.4
Pa MPa
σ
=− =−
Equation (12.17) yields
12 222
09
10 {40 17.6 62.4 2(0.25)
27010
U=++−
××
[40( 17.6) ( 17.6)( 62.4) 40( 62.4)]}×− +− − +−
3
48.96 kJ m=
Equation (12.19) gives
12 23
09
(1 2 4)10 [40 17.6 62.4] 1.905
6(70 10 )
v
Uk
−
=−−=
×Jm
Equation (12.20):
12 22
2
+
09
10 [(40 17.6) ( 17.6 62.4) 102.4 ]
12(28 10 )
d
U=++−+
×
3
47.05 kJ m=
Check:
3
00 0
48.96
vd
UU U kJm=+=
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