sm12 31

________________________________________________________________________
PROBLEM (12.31) Determine the strain-energy density and its components for the triaxial state of
stress shown in Fig. P12.31.
Given: σ 3 = 30 MPa, σ 2 = 40 MPa,
σ1
= 100 MPa,
E = 110 GPa,
ν
= 0.3
SOLUTION
G=
110
= 42.3 GPa
2(1 + 0.3)
Equation (12.17):
1012
U0 =
[302 + 402 + 1002 − 2 × 0.3(30 × 40 + 40 × 100 + 30 × 100)]
9
2(110 × 10 )
= 34.46 kJ m3
Equation (12.19) yields
(1 − 2 × 0.3)1012
U 0v =
(30 + 40 + 100) 2 = 17.52 kJ m3
9
6(110 ×10 )
Equation (12.20):
1012
U 0d =
(102 + 602 + 702 ) = 16.94 kJ m3
9
12(42.3 × 10 )
Check:
U 0 = U 0 v + U 0 d = 34.46 kJ m3
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.