Static and free-vibration analyses of cracks in thin-shell structures based on an isogeometric-meshfree coupling approach

Abstract

This paper develops a coupling approach which integrates the meshfree method and isogeometric analysis (IGA) for static and free-vibration analyses of cracks in thin-shell structures. In this approach, the domain surrounding the cracks is represented by the meshfree method while the rest domain is meshed by IGA. The present approach is capable of preserving geometry exactness and high continuity of IGA. The local refinement is achieved by adding the nodes along the background cells in the meshfree domain. Moreover, the equivalent domain integral technique for three-dimensional problems is derived from the additional Kirchhoff–Love theory to compute the J-integral for the thin-shell model. The proposed approach is able to address the problems involving through-the-thickness cracks without using additional rotational degrees of freedom, which facilitates the enrichment strategy for crack tips. The crack tip enrichment effects and the stress distribution and displacements around the crack tips are investigated. Free vibrations of cracks in thin shells are also analyzed. Numerical examples are presented to demonstrate the accuracy and computational efficiency of the coupling approach.

Appendices

Appendix A: Derivatives of strain variables

The membrane strains and bending strains can be expressed when second order derivatives of displacements are emitted

$$\begin{aligned} \delta {\varepsilon _{\alpha \beta }}= & {} \frac{1}{2}\left( {\delta {\mathbf{{u}}_{,\alpha }} \cdot {\mathbf{{G}}_\beta } + \delta {\mathbf{{u}}_{,\beta }} \cdot {\mathbf{{G}}_\alpha }} \right) , \\ \delta {\kappa _{\alpha \beta }}= & {} -\,\delta {\mathbf{{u}}_{,\alpha \beta }} \cdot {\mathbf{{G}}_3} + \frac{1}{{{{\bar{j}}_0}}}\left( \delta {\mathbf{{u}}_{,1}} \cdot \left( {{\mathbf{{G}}_{\alpha ,\beta }} \times {\mathbf{{G}}_2}} \right) \right. \\&\quad \left. - \delta {\mathbf{{u}}_{,2}} \cdot \left( {{\mathbf{{G}}_{\alpha ,\beta }} \times {\mathbf{{G}}_1}} \right) \right) \\&\quad + \frac{{{\mathbf{{G}}_{\alpha ,\beta }} \cdot {\mathbf{{G}}_3}}}{{{{\bar{j}}_0}}}\left( {\delta {\mathbf{{u}}_{,1}} \cdot \left( {{\mathbf{{G}}_2} \times {\mathbf{{G}}_3}} \right) + \delta {\mathbf{{u}}_{,2}} \cdot \left( {{\mathbf{{G}}_3} \times {\mathbf{{G}}_1}} \right) } \right) \end{aligned}$$

The first derivative of membrane strains with respect to the nodal displacement ${u_r}$ is given as follows:

$$\begin{aligned} \frac{{\partial {\varepsilon _{\alpha \beta }}}}{{\partial {u_r}}} = \frac{1}{2}{\left( {{\mathbf{{g}}_\alpha } \cdot {\mathbf{{g}}_\beta } - {\mathbf{{G}}_\alpha } \cdot {\mathbf{{G}}_\beta }} \right) _{,r}} = \frac{1}{2}\left( {{\mathbf{{g}}_{\alpha ,r}} \cdot {\mathbf{{g}}_\beta } + {\mathbf{{g}}_\alpha } \cdot {\mathbf{{g}}_{\beta ,r}}} \right) . \end{aligned}$$

The second derivative of membrane strains is calculated as

$$\begin{aligned} \frac{{{\partial ^2}{\varepsilon _{\alpha \beta }}}}{{\partial {u_r}\partial {u_s}}} = \frac{1}{2}\left( {{\mathbf{{g}}_{\alpha ,r}} \cdot {\mathbf{{g}}_{\beta ,s}} + {\mathbf{{g}}_{\alpha ,s}} \cdot {\mathbf{{g}}_{\beta ,r}}} \right) . \end{aligned}$$

The first derivative of the normal vector ${\mathbf{{g}}_3}$ with respect to the nodal displacement ${u_r}$ is given as follows:

$$\begin{aligned} {\mathbf{{g}}_{3,r}} = \frac{{{{\vec {\mathbf{g}}}_{3,r}}{{\bar{g}}_3} - {{\vec {\mathbf{g}}}_3}{{\bar{g}}_{3,r}}}}{{{{\bar{g}}_3}^2}},\quad {{{\bar{g}}_3} = \left| {{g_1} \times {g_2}} \right| } ,\quad {{\vec {\mathbf{g}}}_3} = {\mathbf{{g}}_1} \times {\mathbf{{g}}_2}, \end{aligned}$$

where ${\bar{g}_3}$ represents the length of ${\vec {\mathbf{g}}_3}$, and the derivatives of ${\bar{g}_{3,r}}$ and ${\vec {\mathbf{g}}_{3,r}}$ are calculated:

$$\begin{aligned} {\bar{g}_{3,r}} = \frac{{{{\vec {\mathbf{g}}}_3}{{\vec {\mathbf{g}}}_{3,r}}}}{{{{\bar{g}}_3}}}\begin{array}{*{20}{c}},&{{{\vec {\mathbf{g}}}_{3,r}} = {\mathbf{{g}}_{1,r}} \times {\mathbf{{g}}_2}} \end{array} + {\mathbf{{g}}_1} \times {\mathbf{{g}}_{2,r}}. \end{aligned}$$

The derivative of ${\mathbf{{g}}_{\alpha ,\beta }}$ with respect to the nodal displacement ${u_r}$ is given as follows:

$$\begin{aligned} {\mathbf{{g}}_{\alpha ,\beta ,r}} = \mathop \sum \limits _{I = 1}^{NS} {\varPhi _{I,\alpha \beta }}\left( \xi \right) {u_{I,r}}. \end{aligned}$$

The first derivatives of bending strains ${\kappa _{\alpha \beta }}$ with respect to the nodal displacement ${u_r}$ can be given as follows:

$$\begin{aligned} {\kappa _{\alpha \beta ,r}} = {\left( {{\mathbf{{G}}_{\alpha ,\beta }} \cdot {\mathbf{{G}}_3} - {\mathbf{{g}}_{\alpha ,\beta }} \cdot {\mathbf{{g}}_3}} \right) _{,r}} = - {\mathbf{{g}}_{\alpha ,\beta ,r}} \cdot {\mathbf{{g}}_3} - {\mathbf{{g}}_{\alpha ,\beta }} \cdot {\mathbf{{g}}_{3,r}}. \end{aligned}$$

The second derivatives of bending strains can be further obtained as follows:

$$\begin{aligned} {\kappa _{\alpha \beta ,rs}}= & {} {\left( {{\mathbf{{G}}_{\alpha ,\beta }} \cdot {\mathbf{{G}}_3} - {\mathbf{{g}}_{\alpha ,\beta }} \cdot {\mathbf{{g}}_3}} \right) _{,rs}} \nonumber \\= & {} - \left( {{\mathbf{{g}}_{\alpha ,\beta ,r}} \cdot {\mathbf{{g}}_{3,s}} + \mathbf{{g}}_{\alpha ,\beta ,s}} \cdot {\mathbf{{g}}_{3,r}} + {{\mathbf{{g}}_{\alpha ,\beta }} \cdot {\mathbf{{g}}_{3,rs}}} \right) . \end{aligned}$$

Appendix B: Asymptotic fields near a crack tip

1.1 B.1. In-plane enrichment

The displacement and membrane stress components at a crack tip can be expressed in polar coordinates $(r,\theta )$ as follows [68]:

$$\begin{aligned} \left\{ {\begin{array}{c} {{u_1}} \\ {u{}_2} \end{array}} \right\}= & {} \frac{{{K_{I}}}}{{2\mu }}\sqrt{\frac{r}{{2\pi }}} \left\{ {\begin{array}{c} {\cos \frac{\theta }{2}\left( {2\frac{{1 - \nu }}{{1 + \nu }} + 2{{\sin }^2}\frac{\theta }{2}} \right) } \\ {\sin \frac{\theta }{2}\left( {\frac{4}{{1 + \nu }} - 2{{\cos }^2}\frac{\theta }{2}} \right) } \end{array}} \right\} \\&\quad + \frac{{{K_{II}}}}{{2\mu }}\sqrt{\frac{r}{{2\pi }}} \left\{ {\begin{array}{*{20}{c}} {\sin \frac{\theta }{2}\left( {\frac{4}{{1 + \nu }} + 2{{\cos }^2}\frac{\theta }{2}} \right) } \\ { - \cos \frac{\theta }{2}\left( {2\frac{{1 - \nu }}{{1 + \nu }} - 2{{\sin }^2}\frac{\theta }{2}} \right) } \end{array}} \right\} \\ \left\{ {\begin{array}{*{20}{c}} {{\sigma _{11}}}\\ {{\sigma _{12}}}\\ {{\sigma _{22}}} \end{array}} \right\}= & {} \frac{{{K_I}}}{{\sqrt{2\pi r} }}\cos \frac{\theta }{2}\left\{ {\begin{array}{*{20}{c}} {1 - \sin \frac{\theta }{2}\sin \frac{{3\theta }}{2}}\\ {\sin \frac{\theta }{2}\cos \frac{{3\theta }}{2}}\\ {1 + \sin \frac{\theta }{2}\sin \frac{{3\theta }}{2}} \end{array}} \right\} \\&\quad + \frac{{{K_{II}}}}{{\sqrt{2\pi r} }}\cos \frac{\theta }{2}\left\{ {\begin{array}{*{20}{c}} {\sin \frac{\theta }{2}\left( {2 + \cos \frac{\theta }{2}\cos \frac{{3\theta }}{2}} \right) }\\ {cos\frac{\theta }{2}\left( {1 - \sin \frac{\theta }{2}\sin \frac{{3\theta }}{2}} \right) }\\ {\sin \frac{\theta }{2}\cos \frac{\theta }{2}\cos \frac{{3\theta }}{2}} \end{array}} \right\} \end{aligned}$$

where $\mu = E/2(1 + \nu )$ is the shear modulus.

1.2 B.2. Out-of-plane enrichment

The internal stress due to bending can be computed by [56]

$$\begin{aligned} \begin{aligned} \left\{ {\begin{array}{*{20}{c}} {{\sigma _{rr}}} \\ {{\sigma _{r\theta }}} \\ {{\sigma _{\theta \theta }}} \end{array}} \right\}&= \frac{{{k_{1}}}}{{\sqrt{2r} }}\frac{{{x_3}}}{{2h}}\frac{1}{{3 + \nu }}\left\{ {\begin{array}{*{20}{c}} {\left( {3 + 5\nu } \right) \cos \frac{\theta }{2} - \left( {7 + \nu } \right) \cos \frac{{3\theta }}{2}} \\ { - \left( {1 - \nu } \right) \sin \frac{\theta }{2} + \left( {7 + \nu } \right) \sin \frac{{3\theta }}{2}} \\ {\left( {5 + 3\nu } \right) \cos \frac{\theta }{2} + \left( {7 + \nu } \right) \cos \frac{{3\theta }}{2}} \end{array}} \right\} \\&\quad +\, \frac{{{k_{2}}}}{{\sqrt{2r} }}\frac{{{x_3}}}{{2h}}\frac{1}{{3 + \nu }}\left\{ {\begin{array}{*{20}{c}} { - \left( {3 + 5\nu } \right) \sin \frac{\theta }{2} + \left( {5 + 3\nu } \right) \sin \frac{{3\theta }}{2}} \\ {\left( { - 1 + \nu } \right) \cos \frac{\theta }{2} + \left( {5 + 3\nu } \right) \cos \frac{{3\theta }}{2}} \\ { - \left( {5 + 3\nu } \right) \left( {\sin \frac{\theta }{2} + \sin \frac{{3\theta }}{2}} \right) } \end{array}} \right\} \\ \left\{ {\begin{array}{*{20}{c}} {{\sigma _{r3}}}\\ {{\sigma _{\theta 3}}} \end{array}} \right\}&= \frac{1}{{{{\left( {2r} \right) }^{3/2}}}}\frac{h}{2}\frac{1}{{\left( {3 + \nu } \right) }}\left[ {1 - {{\left( {\frac{{2{x_3}}}{h}} \right) }^2}} \right] \\&\quad \left\{ {\begin{array}{*{20}{c}} { - {k_1}\cos \left( {\theta /2} \right) + {k_2}\sin \left( {\theta /2} \right) }\\ { - {k_1}\sin \left( {\theta /2} \right) + {k_2}\cos \left( {\theta /2} \right) } \end{array}} \right\} \\ {\sigma _{33}}&= 0 \end{aligned} \end{aligned}$$

where $x_3$ is the out-of-plane component of the current coordinate vector. The out-of-plane displacement is given

$$\begin{aligned} {u_3}= & {} \frac{{{{\left( {2r} \right) }^{\frac{3}{2}}}\left( {1 - {\nu ^2}} \right) }}{{2Eh\left( {3 + \nu } \right) }}\left\{ {{k_{1}}\left[ {\frac{1}{3}\left( {\frac{{7 + \nu }}{{1 - \nu }}} \right) \cos \frac{{3\theta }}{2} - \cos \frac{\theta }{2}} \right] }\right. \\&\quad \left. + {{k_{2}}\left[ { - \frac{1}{3}\left( {\frac{{5 + 3\nu }}{{1 - \nu }}} \right) \sin \frac{{3\theta }}{2} + \sin \frac{\theta }{2}} \right] } \right\} \end{aligned}$$