Bending of a nanoplate with strain-dependent surface stress containing two collinear through cracks

Abstract

The bending fracture problem of two thickness-through collinear cracks of equal length in a flexible nanoplate with surface stress is analyzed. Using the Kirchhoff thin plate theory together with surface elasticity theory, a mixed boundary value problem is given for applied bending moment, twisting moment and shear force, and solved by use of the Fourier integral transform. Singular integral equations are obtained for each case and analytic solutions are determined in closed form for the case of constant loading. Exact singular elastic fields including the moments, effective shear force, and bulk stress components along the crack line for each case are presented in terms of the complete elliptical integrals. The stress intensity factors for in-plane stresses exhibit a usual inverse square-root singularity and depend on both surface and bulk material properties, while the intensity factors of the anti-plane shear stress and of the effective shear force admit an $r^{-3/2}$ singularity, r being the distance from the closest crack tip. The influences of the material properties and the space between two cracks on fracture parameters are illustrated graphically. Surface phase with positive material properties has a shielding effect and that with negative material properties has an anti-shielding effect on crack growth.

Appendix A

It is helpful to express the surface strain components in terms of surface stress components. For the present study, in the case of $\sigma _{0}=0$, we have

$$\begin{aligned} \sigma _{xx}^{s}&=\left( \lambda ^{s}+2\mu ^{s}\right) \varepsilon _{xx} ^{s}+\lambda ^{s}\varepsilon _{yy}^{s}, \end{aligned}$$

(A.1)

$$\begin{aligned} \sigma _{yy}^{s}&=\left( \lambda ^{s}+2\mu ^{s}\right) \varepsilon _{yy} ^{s}+\lambda ^{s}\varepsilon _{xx}^{s}. \end{aligned}$$

(A.2)

After some calculations, the expressions for the surface strains $\varepsilon _{xx}^{s}$ and $\varepsilon _{yy}^{s}$ are expressed as

$$\begin{aligned} \varepsilon _{xx}^{s}&=\frac{\lambda ^{s}+2\mu ^{s}}{4\mu ^{s}\left( \lambda ^{s}+\mu ^{s}\right) }\left( \sigma _{xx}^{s}-\frac{\lambda ^{s} }{\lambda ^{s}+2\mu ^{s}}\sigma _{yy}^{s}\right) , \end{aligned}$$

(A.3)

$$\begin{aligned} \varepsilon _{yy}^{s}&=\frac{\lambda ^{s}+2\mu ^{s}}{4\mu ^{s}\left( \lambda ^{s}+\mu ^{s}\right) }\left( \sigma _{yy}^{s}-\frac{\lambda ^{s} }{\lambda ^{s}+2\mu ^{s}}\sigma _{xx}^{s}\right) . \end{aligned}$$

(A.4)

Therefore, the relationships between a pair of the surface Lame constants $\lambda ^{s}$ and $\mu ^{s}$ and a pair of surface Young’s modulus $E_{s}$ and surface Poisson’s ratio $\nu _{s}$ are written as follows

$$\begin{aligned} E_{s}&=\frac{4\mu ^{s}\left( \lambda ^{s}+\mu ^{s}\right) }{\lambda ^{s} +2\mu ^{s}},\text { }\nu _{s}=\frac{\lambda ^{s}}{\lambda ^{s}+2\mu ^{s}}, \end{aligned}$$

(A.5)

$$\begin{aligned} \lambda ^{s}&=\frac{E_{s}\nu _{s}}{1-\nu _{s}^{2}},\ \ \mu ^{s}=\frac{E_{s} }{2\left( 1+\nu _{s}\right) }. \end{aligned}$$

(A.6)

It is easily found that the above relationships are different from their classical counterparts. Furthermore, the surface strain can be given by

$$\begin{aligned} \varepsilon _{\alpha \alpha }^{s}&=\frac{1}{2\left( \lambda ^{s}+\mu ^{s}\right) }\sigma _{\alpha \alpha }^{s}=\frac{1-\nu _{s}}{E_{s}}\sigma _{\alpha \alpha }^{s}\ \ \end{aligned}$$

(A.7)

$$\begin{aligned} \sigma _{\alpha \alpha }^{s}&=2\left( \lambda ^{s}+\mu ^{s}\right) \varepsilon _{\alpha \alpha }^{s}=\frac{E_{s}}{1-\nu _{s}}\varepsilon _{\alpha \alpha }^{s}. \end{aligned}$$

(A.8)

Obviously, if requiring the surface Lame constants to be positive, we find that surface Poisson’s ratio ranges between $-1$ and 1. If permitting negative surface Lame constants (see e.g., [37, 38]), surface Poisson’s ratio exceeds the above-stated range. In fact, such values of surface Poisson’s ratio have been reported for nanoplates in [39].