CONTINUUM MECHANICS for ENGINEERS

FIGURE P4.24
Strain gauge rosette.
(a) the components of the infinitesimal strain tensor ε, and the infinitesimal
rotation tensor ω.
(b) the principal values of εε, at the point (1, 1, 0).
Answer: (a) ε11 = ε22 = ε33 = 0, ε12 = AX3, ε13 = A(X2 + 2X1) and
1
ε 23 = AX 3
ω 11 = ω 22 = ω 33 = 0, ω 12 = –ω 21 = AX 3
1
ω13 = –ω31 = AX2 – AX1 and ω23 = –ω32 = AX 2
3
(b) ε (I) = A, ε (II) = 0, ε (III) = – A
4.24 A 45° strain rosette measures longitudinal strains along the X1, X2, and axes shown in the sketch. At point P the strains recorded are
3
2 3
2
′
X 1
ε11 = 6 × 10 –4 , ε22 = 4 × 10 –4 and = 8 × 10 –4
′
Determine the shear strain γ12 at O, together with , and verify that
′
ε11 ′ ε22 ε11 + ε22 = + . See Eq 4.7-25.
4.25 By a direct expansion of Eq 4.7-29, 2ωi = εijk ωkj, show that ω1 = ω32 =
–ω23, etc. Also, show that only if A is a very small constant does the
mapping
x1 = X1 – AX2 + AX3 ′
x 2 = X 2 – AX 3 + AX 1
x 3 = X 3 – AX 1 + AX 2
2 1
2
represent a rigid body rotation (E ≡ 0). Additionally, determine the
infinitesimal rotation tensor ωij in this case; from it, using the result
proven above, deduce the rotation vector ωi. Answer: ω = A( eˆ + eˆ +
eˆ
)
1 2 3
ε 11
1
2
ε 22