Issue Section:
Discussion
Keywords:
elasticity, shear modulus, stress analysis, integral equations
Topics:
Displacement, Elasticity, Integral equations, Shear modulus, Stress analysis (Engineering), Stress
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.
Copyright © 2002
 by ASME