CONTINUUM MECHANICS for ENGINEERS

Upon setting the off-diagonal terms of Φqk to zero, that is, if
Φ , we obtain the solution proposed by Maxwell. By setting
12 = Φ23 = Φ31
= 0
the diagonal terms of Φqk to zero, namely, Φ we obtain the
11 = Φ22 = Φ33
= 0
solution proposed by Morera which is known by that name. It is interesting
to note that if all the components of Φqk except Φ33 are zero, that component
is the Airy stress function introduced in Section 6.7 as can be verified by
Eq 6.9-34. Although the potential Φqk provides us with a solution of the
equilibrium equations, that solution is not compatible with the Beltrami-
Michell equations except under certain conditions.
Problems
6.1 In general, the strain energy density W may be expressed in the form
*
W = εε (α,β = 1, …, 6)
C αβ α β
where is not necessarily symmetric. Show that this equation may
be rearranged to appear in the form
C *
αβ
W =
1
2 C εε αβ α β
where Cαβ is symmetric, so that now
∂ W
= C ε = σ
∂ε
β
αβ α β
in agreement with Eq 6.1-8.
6.2 Let the stress and strain tensors be decomposed into their respective
spherical and deviator components. Determine an expression for the
strain energy density W as the sum of a dilatation energy density
and a distortion energy density .
W( 1)
W( 2)
1 1
Answer: W = W ( +W S
1) ( 2)
= σε ii jj + ijηij 6 2
6.3 If the strain energy density W is generalized in the sense that it is
assumed to be a function of the deformation gradient components
instead of the small strain components, that is, if W = W(FiA), make