Discussion: “On the Relationship Between the L-Integral and the Bueckner Work-Conjugate Integral” (Shi, J. P., Liu, X. H., and Li, J., 2000 ASME J. Appl. Mech., 67, pp. 828–829)

Chen , Y. Z.; Lee , K. Y.

doi:10.1115/1.1491577

Three wrong expressions in the paper (1) have been found. Equations (⁴) and (⁵) in the paper are written in the forms

φ^{(II)} (z) = - i φ^{'} (z), ψ^{(II)} (z) = - iz ψ^{'} (z) + 2 iz {\bar{φ}}^{'} (z),

(1)

{u_{i}}^{(II)} = {yu}_{i, x} - {xu}_{i, y}

(2)

{σ_{ij}}^{(II)} = y σ_{ij, x} - x σ_{ij, y} + \frac{1}{2} \int σ_{ij, x} dy - \frac{1}{2} \int σ_{ij, y} dx (i, 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 $ψ^{(II)} (z)$ in Eq. (¹), $ψ^{(II)} (z)$ cannot be an analytic function. Therefore, $ψ^{(II)} (z)$ in Eq. (¹) is a wrong expression.

2 In the complex variable function method, the displacement components can be expressed as (2)

2 G (u + iv) = κ φ (z) - z \bar{φ^{'} (z)} - \bar{ψ (z)} = κ φ (z) + z {- \bar{φ^{'} (z)}} - \bar{ψ (z)}

(4)

where G is the shear modulus of elasticity,

κ = (3 - ν) / (1 + ν)

is for the plane stress problem,

κ = 3 - 4 ν

is for the plane strain problem, and ν is the Poisson’s ratio, and

φ (z)

and

ψ (z)

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 $φ (z),$ the term after z in Eq. (⁴) should be $- \bar{φ^{'} (z)} .$

On the other hand, from Eq. (⁴) we have

2 G (\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}) = (κ φ^{'} (z) - \bar{φ^{'} (z)}) - (z \bar{φ^{″} (z)} + \bar{ψ^{'} (z)})

2 G (\frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y}) = i {(κ φ^{'} (z) - \bar{φ^{'} (z)}) + (z \bar{φ^{″} (z)} + \bar{ψ^{'} (z)})} .

(5)

Therefore, from Eqs. (²) and (⁵), the displacement components in Eq. (²) can be expressed as

2 G (u^{(II)} + {iv}^{(II)}) = 2 G (y (\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}) - x (\frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y})) = κ {- iz φ^{'} (z)} + z {i (\bar{φ^{'} (z)} - z ¯ \bar{φ^{″} (z)})} - iz ¯ \bar{ψ^{'} (z)} .

(6)

From the fact that

- \bar{\frac{d}{dz} {- iz φ^{'} (z)}} = - i (\bar{φ^{'} (z)} + z ¯ \bar{φ^{″} (z)}) \neq i (\bar{φ^{'} (z)} - z ¯ \bar{φ^{″} (z)})

(7)

and the rule mentioned above, the displacements

u^{(II)}

and

v^{(II)}

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

F (x, y)

F (x, y) = \int_{(x_{o}, y_{o})}^{(x, y)} p (x, y) dx + q (x, y) dy

(8)

is a path independent integral, the following condition must be satisfied:

\frac{\partial p (x, y)}{\partial y} = \frac{\partial q (x, y)}{\partial x} or \frac{\partial q (x, y)}{\partial x} - \frac{\partial p (x, y)}{\partial y} = 0 .

(9)

If Eq. (³) were true, substituting

p (x, y) = - σ_{ij, y} / 2

and

q (x, y) = σ_{ij, x} / 2

into Eq. (⁹) yields the following:

\frac{\partial^{2} σ_{ij}}{\partial x^{2}} + \frac{\partial^{2} σ_{ij}}{\partial y^{2}} = 0 .

(10)

However, the stress components

σ_{ij}

are not a harmonic function in general. Thus, the

{σ_{ij}}^{(II)}

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.

by ASME

Discussion: “On the Relationship Between the L-Integral and the Bueckner Work-Conjugate Integral” (Shi, J. P., Liu, X. H., and Li, J., 2000 ASME J. Appl. Mech., 67, pp. 828–829)

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Discussion: “On the Relationship Between the L-Integral and the Bueckner Work-Conjugate Integral” (Shi, J. P., Liu, X. H., and Li, J., 2000 ASME J. Appl. Mech., 67, pp. 828–829)

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