CONTINUUM MECHANICS for ENGINEERS

where α, β and µ are constants, calculate the Lagrangian finite strain
tensor E. Show that if α = cos θ, β = sin θ and µ = 1 the strain is zero
and the mapping corresponds to a rigid body rotation of magnitude
θ about the X 3 axis.
Answer:
2 2 ⎡α
+ β −1
0 0 ⎤
1 ⎢
2 2
⎥
EAB = ⎢ 0 α + β −1
0
2
⎥
⎢
2
⎣ 0 0 µ − 1⎥
⎦
4.17 Given the deformation defined by
2
x1 = X1, x = X + X , x3 = X3 2 2
(a) Sketch the deformed shape of the unit square OABC in the plane
X1 = 0.
(b) Determine the differential vectors dx (2) and dx (3) which are the
deformed vectors resulting from dX (2) = dX (2) and dX (3) =
dX (3) , respectively, that were originally at corner C.
(c) Calculate the dot product dx (2) ⋅ dx (3) , and from it determine the
change in the original right angle between dX (2) and dX (3) Î2 Î3 at C due
to the deformation.
(d) Compute the stretch Λ at B in the direction of the unit normal
( )
Answer: (b) dx (2) = dX (2) , dx (3) = dX (3)
ê ˆ ˆ
2 e + e
(c) ∆θ = –45°
(d) Λ
( ˆ = 25 .
N )
4.18 Given the deformation expressed by
2
x1 = X1 + , x 2 = X 2, x 3 = X 3 –
AX 2
where A is a constant (not necessarily small), determine the finite
strain tensors E and e, and show that if the displacements are small
so that x ≈ X, and if squares of A may be neglected, both tensors
reduce to the infinitesimal strain tensor ε.
⎡ 0 Ax2
0 ⎤
Answer:
⎢
⎥
εij =
⎢
Ax2 0 −Ax2⎥
⎣
⎢ 0 − Ax
⎦
⎥
2 0
4.19 For the infinitesimal homogeneous deformation xi = Xi + Aij Xj where
the constants Aij are very small, determine the small strain tensor ε,
1
2
Nˆ = Iˆ + Iˆ
/
3
2 3 2
.
( 2 3)
2
AX2