CONTINUUM MECHANICS for ENGINEERS

Answer: (a)
(b)
2 2
(c) (1) a1a 2a 3 = 1, (2) 9a1+ a2= 2a
4.30 A homogeneous deformation is defined by
x 1 = αX 1 + βX 2, x 2 = –αX 1+ βX 2, x 3 = µX 3
where α, β and µ are constants.
Determine
(a) the magnitudes and directions of the principal stretches
(b) the matrix representation of the rotation tensor R
(c) the direction of the axis of the rotation vector, and the magnitude
of the angle of rotation.
2 2 2 2 2 2
Answer: (a) Λ Λ , ,
( 1)
= = 2α
Λ Λ
eˆ
( 2)
= = 2β
eˆ
(b)
l2
= 9a
+ a + a
cosθ =
(c) ; Φ = 45°
4.31 Consider the deformation field
x1 = X1 – AX2 + AX3 ˆ n= Iˆ
3
2
1
x 2 = X 2 – AX 3 + AX 1
x 3 = X 3 – AX 1 + AX 2
where A is a constant. Show that the principal values of the right
stretch tensor have a multiplicity of two, and that the axis of the
rotation tensor is along Nˆ = ˆ ˆ ˆ ( I1+ I2 + I3)
/ 3 . Determine the matrix
of the rotation vector together with the angle of rotation Φ.
2
Answer: Λ ( 1) = 1, Λ( 2) = Λ(
3)
= 1+ 3A=
β
Φ = cos –1 (1/β)
2
2
2a
1
2
3
2
4a
+ a
( 1)
1
2
2
⎡ 1/ 2 1/ 2 0⎤
⎢
⎥
Rij = ⎢−1/
2 1/ 2 0⎥
⎢
⎥
⎣
0 0 1
⎦
( 2 )
A A
Rij
A A
A A
=
⎡ β + 2 β −1−3 β − 1+ 3 ⎤
1 ⎢
⎥
⎢
β − 1+ 3 β + 2 β −1−3 3β
⎥
⎣
⎢β
−1−3 β − 1+ 3 β
+ 2 ⎦
⎥
2
3
2 2 2
Λ( 3)
= Λ = µ
eˆ
( 3 )