Page 244 - Intro to Tensor Calculus

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EXERCISE 2.3

2
I 1. Assume an orthogonal coordinate system with metric tensor g ij =0 for i 6= j and g (i)(i) = h (no
i
summation on i). Use the definition of strain
1 1 t t
e rs = (u r,s + u s,r )= g rtu ,s + g st u ,r
2 2
and show that in terms of the physical components
e ij
e(ij)= no summation on i or j
h i h j
u(i)= h i u i no summation on i

there results the equations:
t
∂u t m
e ii = g it i + u no summation on i
∂x mi
∂u t ∂u t
2e ij = g it j + g jt i , i 6= j
∂x ∂x
3
∂ u(i) 1 X u(m) ∂ 2
e(ii)= + 2 h i no summation on i
∂x i h i 2h h m ∂x m
i m=1

h i ∂ u(i) h j ∂ u(j)
2e(ij)= + , no summation on i or j, i 6= j.
h j ∂x j h i h i ∂x i h j
I 2. Use the results from problem 1 to write out all components of the strain tensor in Cartesian coordinates.
Use the notation u(1) = u,u(2) = v,u(3) = w and

e(11) = e xx , e(22) = e yy , e(33) = e zz , e(12) = e xy , e(13) = e xz , e(23) = e yz

to verify the relations:

1 ∂v ∂u
∂u
e xx = e xy = +
∂x 2 ∂x ∂y

∂v 1 ∂u ∂w
e yy = e xz = +
∂y 2 ∂z ∂x
∂w 1 ∂w ∂v
e zz = e zy = +
∂z 2 ∂y ∂z
I 3. Use the results from problem 1 to write out all components of the strain tensor in cylindrical coordinates.
Use the notation u(1) = u r , u(2) = u θ , u(3) = u z and

e(11) = e rr , e(22) = e θθ , e(33) = e zz , e(12) = e rθ , e(13) = e rz , e(23) = e θz

to verify the relations:

1 1 ∂u r ∂u θ u θ
∂u r e rθ = + −
e rr = 2 r ∂θ ∂r r
∂r

1
1 ∂u θ u r ∂u z ∂u r
e θθ = + e rz = +
r ∂θ r 2 ∂r ∂z

∂u z 1
e zz = e θz = ∂u θ + 1 ∂u z
∂z 2 ∂z r ∂θ