Page 279 - Intro to Tensor Calculus

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I 8. Verify the equations defining the stress for plane strain in Cartesian coordinates are

E
σ xx = [(1 − ν)e xx + νe yy ]
(1 + ν)(1 − 2ν)
E
σ yy = [(1 − ν)e yy + νe xx ]
(1 + ν)(1 − 2ν)
Eν
σ zz = [e xx + e yy ]
(1 + ν)(1 − 2ν)
E
σ xy = e xy
1+ ν
σ yz = σ xz =0

I 9. Verify the equations defining the stress for plane strain in polar coordinates are

E
σ rr = [(1 − ν)e rr + νe θθ ]
(1 + ν)(1 − 2ν)
E
σ θθ = [(1 − ν)e θθ + νe rr ]
(1 + ν)(1 − 2ν)
νE
σ zz = [e rr + e θθ ]
(1 + ν)(1 − 2ν)
E
σ rθ = e rθ
1+ ν
σ rz = σ θz =0

I 10. Write out the independent components of Hooke’s generalized law for strain in terms of stress, and
stress in terms of strain, in Cartesian coordinates. Express your results using the parameters ν and E.
(Assume a linear elastic, homogeneous, isotropic material.)

I 11. Write out the independent components of Hooke’s generalized law for strain in terms of stress, and
stress in terms of strain, in cylindrical coordinates. Express your results using the parameters ν and E.
(Assume a linear elastic, homogeneous, isotropic material.)
I 12. Write out the independent components of Hooke’s generalized law for strain in terms of stress, and
stress in terms of strain in spherical coordinates. Express your results using the parameters ν and E. (Assume
a linear elastic, homogeneous, isotropic material.)

I 13. For a linear elastic, homogeneous, isotropic material assume there exists a state of plane strain in
Cartesian coordinates. Verify the equilibrium equations are

∂σ xx ∂σ xy
+ + %b x =0
∂x ∂y
∂σ yx ∂σ yy
+ + %b y =0
∂x ∂y
∂σ zz
+ %b z =0
∂z
Hint: See problem 14, Exercise 2.3.