CONTINUUM MECHANICS for ENGINEERS

an expression for u i in terms of the proposed potentials into the Navier
equations we obtain the governing equations for the appropriate potentials.
Often such potentials are harmonic, or bi-harmonic functions. We shall
present three separate methods for arriving at solutions of the Navier equations.
The method used in our first approach rests upon the well-known theorem
of Helmholtz which states that any vector function that is continuous and
finite, and which vanishes at infinity, may be resolved into a pair of components:
one a rotation vector, the other an irrotational vector. Thus, if the curl
of an arbitrary vector a is zero, then a is the gradient of a scalar φ, and a is
irrotational, or as it is sometimes called, solenoidal. At the same time, if the
divergence of the vector a is zero, then a is the curl of another vector ψ, and
is a rotational vector. Accordingly, in keeping with the Helmholtz theorem,
we assume that the displacement field is given by
(6.9-7)
where φ ,i is representative of the irrotational portion, and curl ψ the rotational
portion. Substituting this displacement vector into Eq 6.9-4 with b i in that
equation taken as zero, namely
we obtain
which reduces to
ui = φ, i + εipqψq, p
( ) = 0
µ u + λ + µ u
ijj , jji ,
( ) + ( + ) = 0
µφ + µε ψ + λ + µ φ λ µ ε ψ
, ijj ipq q, pjj , ijj jpq q, pji
( λ + 2µ ) φ + µε ψ = 0
, ijj ipq q, pjj
since ε ψ = 0.
In coordinate-free notation Eq 6.9-10 becomes
jpq q, pji
λ µ φ µ ψ
+
2 2
( 2 ) + × = 0
(6.9-8)
(6.9-9)
(6.9-10)
(6.9-11)
Any set of φ and ψ which satisfies Eq 6.9-10 provides (when substituted into
Eq 6.9-7) a displacement field satisfying the Navier equation, Eq 6.9-8.
Clearly, one such set is obtained by requiring φ and ψ to be harmonic
2
2
φ = 0
ψ =
0
(6.9-12a)
(6.9-12b)