Page 285 - Intro to Tensor Calculus

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I 36. Combine the results from problems 30,31,32 and 33 and write the Navier equations of equilibrium
in Cartesian coordinates. Alternatively, write the stress-strain relations (2.4.29(b)) in terms of physical
components and then use these results, together with the results from Exercise 2.3, problems 2 and 14, to
derive the Navier equations.

I 37. Combine the results from problems 30,31,32 and 34 and write the Navier equations of equilibrium
in cylindrical coordinates. Alternatively, write the stress-strain relations (2.4.29(b)) in terms of physical
components and then use these results, together with the results from Exercise 2.3, problems 3 and 15, to
derive the Navier equations.
I 38. Combine the results from problems 30,31,32 and 35 and write the Navier equations of equilibrium
in spherical coordinates. Alternatively, write the stress-strain relations (2.4.29(b)) in terms of physical
components and then use these results, together with the results from Exercise 2.3, problems 4 and 16, to
derive the Navier equations.
~
I 39. Assume %b = −gradV and let φ denote the Airy stress function defined by
2
∂ φ
σ xx =V +
∂y 2
2
∂ φ
σ yy =V +
∂x 2
2
∂ φ
σ xy = −
∂x∂y
(a) Show that for conditions of plane strain the equilibrium equations in two dimensions are satisfied by the
above definitions. (b) Express the compatibility equation
2 2 2
∂ e xx ∂ e yy ∂ e xy
+ =2
∂y 2 ∂x 2 ∂x∂y
in terms of φ and V and show that
1 − 2ν 2
4
∇ φ + ∇ V =0.
1 − ν
I 40. Consider the case where the body forces are conservative and derivable from a scalar potential function
such that %b i = −V ,i . Show that under conditions of plane strain in rectangular Cartesian coordinates the
1 2
2
compatibility equation e 11,22 + e 22,11 =2e 12,12 can be reduced to the form ∇ σ ii = ∇ V , i =1, 2
1 − ν
involving the stresses and the potential. Hint: Differentiate the equilibrium equations.
i
m i
i
I 41. Use the relation σ =2µe + λe δ and solve for the strain in terms of the stress.
j
m j
j
I 42. Derive the equation (2.4.26) from the equation (2.4.23).
I 43. In two dimensions assume that the body forces are derivable from a potential function V and
i
ij
%b = −g V ,j . Also assume that the stress is derivable from the Airy stress function and the potential
ij
function by employing the relations σ ij = im jn u m,n + g V i,j,m,n =1, 2where u m = φ ,m and

pq is the two dimensional epsilon permutation symbol and all indices have the range 1,2.

(a) Show that im jn (φ m ) ,nj =0.
i
(b) Show that σ ij = −%b .
,j
(c) Verify the stress laws for cylindrical and Cartesian coordinates given in problem 20 by using the above
ij
expression for σ . Hint: Expand the contravariant derivative and convert all terms to physical compo-
1
ij
ij
nents. Also recall that = √ e .
g