Page 262 - Intro to Tensor Calculus

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A contraction of the strain produces the dilatation

1
e rr = (u r,r + u r,r )= u r,r (2.4.31)
2
From the dilatation we calculate the covariant derivative

e kk,j = u k,kj . (2.4.32)

Employing the strain relation from equation (2.4.28), we calculate the covariant derivative

1
e ij,j = (u i,jj + u j,ij ). (2.4.33)
2
These results allow us to express the covariant derivative of the stress in terms of the displacement field. We
find
σ ij,j = µ [u i,jj + u j,ij ]+ λδ ij u k,kj
(2.4.34)
or σ ij,j =(λ + µ)u k,ki + µu i,jj .
Substituting equation (2.4.34) into the linear momentum equation produces the Navier equations:

(λ + µ)u k,ki + µu i,jj + %b i = %f i , i =1, 2, 3. (2.4.35)

In vector form these equations can be expressed

~
~
2
(λ + µ)∇ (∇· ~u)+ µ∇ ~u + %b = %f, (2.4.36)
~
~
where ~u is the displacement vector, b is the body force per unit mass and f is the acceleration. In Cartesian
coordinates these equations have the form:
2 2 2 2

∂ u 1 ∂ u 2 ∂ u 3 2 ∂ u i
(λ + µ) + + + µ∇ u i + %b i = % 2 ,
∂x 1 ∂x i ∂x 2 ∂x i ∂x 3 ∂x i ∂t
for i =1, 2, 3, where
2 2 2
2
∇ u i = ∂ u i 2 + ∂ u i 2 + ∂ u i 2 .
∂x 1 ∂x 2 ∂x 3
The Navier equations must be satisfied by a set of functions u i = u i (x 1 ,x 2 ,x 3 ) which represent the
displacement at each point inside some prescribed region R. Knowing the displacement field we can calculate
the strain field directly using the equation (2.4.28). Knowledge of the strain field enables us to construct the
corresponding stress field from the constitutive equations.
In the absence of body forces, such as gravity, the solution to equation (2.4.36) can be represented
in the form ~u = ~u (1) + ~u (2) , where ~u (1) satisfies div ~u (1) = ∇· ~u (1) = 0 and the vector ~u (2) satisfies
curl~u (2) = ∇× ~u (2) =0. The vector field ~u (1) is called a solenoidal field, while the vector field ~u (2) is
~
called an irrotational field. Substituting ~u into the equation (2.4.36) and setting b =0, we find in Cartesian
coordinates that
2 (1) 2 (2)
∂ ~u ∂ ~u (2) 2 (1) 2 (2)
% + =(λ + µ)∇ ∇· ~u + µ∇ ~u + µ∇ ~u . (2.4.37)
∂t 2 ∂t 2