Page 287 - Intro to Tensor Calculus

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I 48. Find the stress σ rr ,σ rθ and σ θθ in an infinite plate with a small circular hole, which is traction free,
when the plate is subjected to a pure shearing force F 12 . Determine the maximum stress.

I 49. Show that in terms of E and ν

E(1 − ν) Eν E
C 1111 = C 1122 = C 1212 =
(1 + ν)(1 − 2ν) (1 + ν)(1 − 2ν) 2(1 + ν)
I 50. Show that in Cartesian coordinates the quantity

2
2
S = σ xx σ yy + σ yy σ zz + σ zz σ xx − (σ xy ) − (σ yz ) − (σ xz ) 2
1
is a stress invariant. Hint: First verify that in tensor form S = (σ ii σ jj − σ ij σ ij ).
2
I 51. Show that in Cartesian coordinates for a state of plane strain where the displacements are given by
u = u(x, y),v = v(x, y)and w = 0, the stress components must satisfy the equations

∂σ xx ∂σ xy
+ + %b x =0
∂x ∂y
∂σ yx ∂σ yy
+ + %b y =0
∂x ∂y

−% ∂b x ∂b y
2
∇ (σ xx + σ yy )= +
1 − ν ∂x ∂y
I 52. Show that in Cartesian coordinates for a state of plane stress where σ xx = σ xx (x, y), σ yy = σ yy (x, y),
σ xy = σ xy (x, y)and σ xz = σ yz = σ zz = 0 the stress components must satisfy

∂σ xx ∂σ xy
+ + %b x =0
∂x ∂y
∂σ yx ∂σ yy
+ + %b y =0
∂x ∂y

2
∇ (σ xx + σ yy )= − %(ν +1) ∂b x + ∂b y
∂x ∂y