________________________________________________________________________
PROBLEM (12.31) Determine the strain-energy density and its components for the triaxial state of
stress shown in Fig. P12.31.
Given: 3
σ
= 30 MPa, 2
σ
= 40 MPa, 1
σ
= 100 MPa, E = 110 GPa,
ν
= 0.3
SOLUTION
110 42.3
2(1 0.3)
GG==
+Pa
Equation (12.17):
12 22 2
09
10 [30 40 100 2 0.3(30 40 40 100 30 100)]
2(110 10 )
U=++−××+×+
××
3
34.46 kJ m=
Equation (12.19) yields
12 23
09
(1 2 0.3)10 (30 40 100) 17.52
6(110 10 )
v
Uk
−×
=++=
×Jm
Equation (12.20):
12 222
09
10 (10 60 70 ) 16.94
12(42.3 10 )
d
Uk=++=
×
3
Jm
Check:
3
00 0
34.46
vd
UU U kJm=+=
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