Page 265 - Intro to Tensor Calculus

259

~
THEOREM A general elastostatic solution of the equation (2.4.38) in terms of harmonic potentials φ,ψ
is
~
~
r
~u = grad (φ + ~ · ψ) − 4(1 − ν)ψ (2.4.39)
~
where φ and ψ are continuous solutions of the equations
~
~
−%~ · b 2 ~ %b
r
2
∇ φ = and ∇ ψ = (2.4.40)
4µ(1 − ν) 4µ(1 − ν)
r
with ~ = x ˆ e 1 + y ˆ e 2 + z ˆ e 3 a position vector to a general point (x, y, z) within the continuum.
Proof: First we write equation (2.4.38) in the tensor form
1 %
u i,kk + (u j,j ) ,i + b i =0 (2.4.41)
1 − 2ν µ
Now our problem is to show that equation (2.4.39), in tensor form,

u i = φ ,i +(x j ψ j ) ,i − 4(1 − ν)ψ i (2.4.42)

is a solution of equation (2.4.41). Toward this purpose, we differentiate equation (2.4.42)

(2.4.43)
u i,k = φ ,ik +(x j ψ j ) ,ik − 4(1 − ν)ψ i,k
and then contract on i and k giving

u i,i = φ ,ii +(x j ψ j ) ,ii − 4(1 − ν)ψ i,i . (2.4.44)

Employing the identity (x j ψ j ) ,ii =2ψ i,i + x i ψ i,kk the equation (2.4.44) becomes

u i,i = φ ,ii +2ψ i,i + x i ψ i,kk − 4(1 − ν)ψ i,i . (2.4.45)

By differentiating equation (2.4.43) we establish that

u i,kk = φ ,ikk +(x j ψ j ) ,ikk − 4(1 − ν)ψ i,kk

=(φ ,kk ) ,i +((x j ψ j ) ,kk ) − 4(1 − ν)ψ i,kk (2.4.46)
,i
=[φ ,kk +2ψ j,j + x j ψ j,kk ] − 4(1 − ν)ψ i,kk .
,i
We use the hypothesis
−%x j F j %F j
φ ,kk = and ψ j,kk = ,
4µ(1 − ν) 4µ(1 − ν)
and simplify the equation (2.4.46) to the form

u i,kk =2ψ j,ji − 4(1 − ν)ψ i,kk . (2.4.47)

Also by differentiating (2.4.45) one can establish that

u j,ji =(φ ,jj ) ,i +2ψ j,ji +(x j ψ j,kk ) ,i − 4(1 − ν)ψ j,ji

−%x j F j %x j F j
= +2ψ j,ji + − 4(1 − ν)ψ j,ji (2.4.48)
4µ(1 − ν) 4µ(1 − ν)
,i ,i
= −2(1 − 2ν)ψ j,ji .