Surface tension-induced stress concentration around a nanosized hole of arbitrary shape in an elastic half-plane

Abstract

This paper studies surface tension-induced stress concentration around a nanosized hole of arbitrary shape inside an elastic half-plane. Of particular interest is the maximum hoop stress on the hole’s boundary with relation to the point of maximum curvature and the distance between the hole and the free surface of the half-plane. The shape of the hole is characterized by a conformal mapping which maps the exterior of the hole onto the exterior of the unit circle in the image plane. On using the technique of conformal mapping and analytic continuation, the complex potentials of the half-plane are expressed in a series form with unknown coefficients to be determined by Fourier expansion method. Detailed numerical results are shown for elliptical, triangular, square and rectangular holes. Two basic conclusions are that the hoop stress increases with decreasing hole size and the maximum hoop stress generally appears nearby but not exactly at the point of maximum curvature. In addition, it is shown that the hoop stress nearby the point of maximum curvature on the hole’s boundary increases rapidly with decreasing distance between the hole and the free surface of the half-plane. On the other hand, if the distance between the hole and the free surface is more than three times the hole size, the effect of the free surface on the stress concentration around the hole is ignorable and the elastic half-plane can be treated approximately as an elastic whole plane.

Appendix

1.1 Proof of the mapping Eq. (9)

Firstly, since the conformal mapping function (8) is bijective, there exist one-to-one corresponding z ₁ and ξ ₁ satisfying

$$ z_{1} = \omega (\xi_{1} ) = R_{0} + R\left( {\xi_{1} + \sum\limits_{j = 1}^{M} {m_{j} \xi_{1}^{ - j} } } \right), \, z_{1} \in \varOmega_{1} , \, \left| {\xi_{1} } \right| \ge 1 $$

(35)

Let z ₂ and ξ ₂ be the conjugates of z ₁ and ξ ₁, respectively. So according to the conjugate of Eq. (35), the one-to-one corresponding z ₂ and ξ ₂ satisfy

$$ z_{2} = \overline{\omega }(\xi_{2} ) = \overline{R}_{0} + \overline{R}\left( {\xi_{2} + \sum\limits_{j = 1}^{M} {\overline{m}_{j} \xi_{2}^{ - j} } } \right), \, z_{2} \in \varOmega_{2} , \, \left| {\xi_{2} } \right| \ge 1 $$

(36)

which shows that the mapping function (9) is bijective from the exterior of the unit circle in the ξ ₂-plane to Ω₂ in the z-plane.

Then, the derivative of the conformal mapping function (8) should satisfy

$$ \forall \left| {\xi_{1} } \right| \ge 1, \, \omega^{\prime}(\xi_{1} ) = R\left( {1 - \sum\limits_{j = 1}^{M} {jm_{j} \xi_{1}^{ - j - 1} } } \right) \ne 0 $$

(37)

which means

$$ \forall \left| {\xi_{2} } \right| = \left| {\overline{\xi }_{2} } \right| \ge 1, \, \omega^{\prime}(\overline{\xi }_{2} ) = R\left( {1 - \sum\limits_{j = 1}^{M} {jm_{j} \overline{\xi }_{2}^{ - j - 1} } } \right) \ne 0 $$

(38)

Conjugating Eq. (38) leads to

$$ \forall \left| {\xi_{2} } \right| \ge 1, \, \overline{\omega^{\prime}}(\xi_{2} ) = \overline{R}\left( {1 - \sum\limits_{j = 1}^{M} {j\overline{m}_{j} \xi_{2}^{ - j - 1} } } \right) \ne 0 $$

(39)

Finally, one can conclude from Eqs. (36) and (39) that Eq. (9) is a mapping from the exterior of the unit circle in the ξ ₂-plane to Ω₂ in the z-plane.

1.2 Derivation of the curvature K

In the z-plane, the point-wise coordinates (x, y) on Γ₁ can be written in terms of ω(σ) as

$$ x = \text{Re} \left[ {\omega (\sigma )} \right], \, y = \text{Im} \left[ {\omega (\sigma )} \right], \, \sigma = e^{i\theta } $$

(40)

The first and second order derivatives of x and y with respect to θ can be expressed as

$$ \left\{ {\begin{array}{l} {\frac{dx}{d\theta } = \text{Re} \left[ {i\sigma \omega^{\prime}(\sigma )} \right] = - \text{Im} \left[ {\sigma \omega^{\prime}(\sigma )} \right]} \\ {\frac{dy}{d\theta } = \text{Im} \left[ {i\sigma \omega^{\prime}(\sigma )} \right] = \text{Re} \left[ {\sigma \omega^{\prime}(\sigma )} \right]} \\ \end{array} } \right. $$

(41)

$$ \left\{ {\begin{array}{l} {\frac{{d^{2} x}}{{d\theta^{2} }} = - \text{Im} \left[ {i\sigma \left( {\omega^{\prime}(\sigma ) + \sigma \omega^{\prime\prime}(\sigma )} \right)} \right] = - \text{Re} \left[ {\sigma \omega^{\prime}(\sigma ) + \sigma^{2} \omega^{\prime\prime}(\sigma )} \right]} \\ {\frac{{d^{2} y}}{{d\theta^{2} }} = \text{Re} \left[ {i\sigma \left( {\omega^{\prime}(\sigma ) + \sigma \omega^{\prime\prime}(\sigma )} \right)} \right] = - \text{Im} \left[ {\sigma \omega^{\prime}(\sigma ) + \sigma^{2} \omega^{\prime\prime}(\sigma )} \right]} \\ \end{array} } \right. $$

(42)

Finally, the signed curvature K can be given by the curvature formula as

$$ K = \frac{{\frac{dx}{d\theta }\frac{{d^{2} y}}{{d\theta^{2} }} - \frac{dy}{d\theta }\frac{{d^{2} x}}{{d\theta^{2} }}}}{{\left[ {\left( {\frac{dx}{d\theta }} \right)^{2} + \left( {\frac{dy}{d\theta }} \right)^{2} } \right]^{3/2} }} = \frac{{\left| {\omega^{\prime}(\sigma )} \right|^{2} + \text{Re} \left[ {\sigma \overline{{\omega^{\prime}(\sigma )}} \omega^{\prime\prime}(\sigma )} \right]}}{{\left| {\omega^{\prime}(\sigma )} \right|^{3} }} $$

(43)