Three wrong expressions in the paper (1) have been found. Equations (4) and (5) in the paper are written in the forms
$φIIz=−iφ′z,ψIIz=−izψ′z+2izφ¯′z,$
(1)
$uiII=yui,x−xui,y$
(2)
$σijII=yσij,x−xσij,y+12 ∫σij,xdy−12 ∫σij,ydxi,j=1,2.$
(3)

1 Complex potentials suggested by Muskhelishvili should be an analytic function (2). However, since the argument $z¯$ is involved in the second term of $ψIIz$ in Eq. (1), $ψIIz$ cannot be an analytic function. Therefore, $ψIIz$ in Eq. (1) is a wrong expression.

2 In the complex variable function method, the displacement components can be expressed as (2)
$2Gu+iv=κφz−zφ′z¯−ψz¯=κφz+z{−φ′z}¯−ψz¯$
(4)
where G is the shear modulus of elasticity, $κ=3−ν/1+ν$ is for the plane stress problem, $κ=3−4ν$ is for the plane strain problem, and ν is the Poisson’s ratio, and $φz$ and $ψz$ are two analytic functions.

Equation (4) reveals a rule that in a real displacement expression of plane elasticity, if the function after the elastic constant κ is $φz,$ the term after z in Eq. (4) should be $−φ′z¯.$

On the other hand, from Eq. (4) we have
$2G∂u∂x+i ∂v∂x=κφ′z−φ′z¯−zφ″z¯+ψ′z¯$
$2G∂u∂y+i ∂v∂y=i{κφ′z−φ′z¯+zφ″z¯+ψ′z¯}.$
(5)
Therefore, from Eqs. (2) and (5), the displacement components in Eq. (2) can be expressed as
$2GuII+ivII=2Gy∂u∂x+i ∂v∂x−x∂u∂y+i ∂v∂y=κ{−izφ′z}+z{iφ′z¯−z¯φ″z¯}−iz¯ψ′z¯.$
(6)
From the fact that
$−ddz {−izφ′z}¯=−iφ′z¯+z¯φ″z¯≠iφ′z¯−z¯φ″z¯$
(7)
and the rule mentioned above, the displacements $uII$ and $vII$ shown in Eq. (2) are not an elasticity solution. Therefore, the displacement shown in Eq. (2) is also a wrong expression.
3 In Eq. (3) an indefinite integral is used to express the stress components. In the continuum medium of elastic body, the integral should be path-independent. Also, it is well known that if a function $Fx,y$
$Fx,y=∫xo,yox,ypx,ydx+qx,ydy$
(8)
is a path independent integral, the following condition must be satisfied:
$∂px,y∂y=∂qx,y∂xor∂qx,y∂x−∂px,y∂y=0.$
(9)
If Eq. (3) were true, substituting $px,y=−σij,y/2$ and $qx,y=σij,x/2$ into Eq. (9) yields the following:
$∂2σij∂x2+∂2σij∂y2=0.$
(10)
However, the stress components $σij$ are not a harmonic function in general. Thus, the $σijII$ shown by Eq. (3) is also a wrong expression.
1.
Shi
,
J. P.
,
Liu
,
X. H.
, and
Li
,
J. M.
,
2000
, “
On the Relation Between the L-integral and the Bueckner Work-Conjugate Integral
,”
ASME J. Appl. Mech.
,
67
, pp.
828
829
.
2.
Muskhelishvili, N. I., 1953, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoof, Dordrecht, The Netherlands.