Size effect on cracked functional composite micro-plates by an XIGA-based effective approach

Abstract

Failure of structures and their components is one of major problems in engineering. Studies on mechanical behavior of functionally graded (FG) microplates with defects or cracks by effective numerical methods are rarely reported in literature. In this paper, an effective numerical model is derived based on extended isogeometric analysis (XIGA) for assessment of vibration and buckling of FG microplates with cracks. Based on the modified couple stress theory, the non-classical theory of Reissner–Mindlin plate is extended to capture microstructure, and thus, the size effect. In such theory, possessing C¹-continuity is straightforward with the high-order continuity of non-uniform rational B-spline. Due to the use of enrichments in XIGA, crack geometry is independent of the computational mesh. Numerical examples are performed to illustrate the effects of microplate aspect ratio, crack length, internal material length scale parameter, material distribution, and boundary condition on the mechanical responses of cracked FG microplates. The obtained results are compared with reference solutions and that shows that the frequency and buckling loads increases with decreasing the size of FG microplates and crack length. The convergence of the present method is also studied.

Appendices

Appendix A

To compute the stiffness matrices for elements enriched by the near tip asymptotic functions, the two-order derivatives of near tip asymptotic functions with respect to the global coordinate are determined. The crack-tip branch functions for the transverse normal rotations in the xz and yz planes of mid-plane are given as follows (Fig. 17):

$$\left[ {F_{l} \left( {r,\theta } \right),l = 1, \ldots ,4} \right] = \sqrt r \left[ {\begin{array}{*{20}c} {\sin \left( {\frac{\theta }{2}} \right)} & {\cos \left( {\frac{\theta }{2}} \right)} & {\sin \left( {\frac{\theta }{2}} \right)\sin \left( \theta \right)} & {\cos \left( {\frac{\theta }{2}} \right)\sin \left( \theta \right)} \\ \end{array} } \right]$$

(A.1)

Fig. 17

Global coordinate system and local coordinate system

The second-order derivatives of $F_{l} \left( {r,\theta } \right)$ with respect to the local crack tip coordinate system (x₁; x₂) shown in Fig. 17 are given by:

$$\left\{ \begin{aligned} & {\frac{{\partial^{2} F_{1} }}{{\partial x_{1}^{2} }} = \frac{1}{{4\sqrt {r^{3} } }}\sin \frac{3\theta }{2}} \\ & {\frac{{\partial^{2} F_{1} }}{{\partial x_{2}^{2} }} = - \frac{1}{{4\sqrt {r^{3} } }}\sin \frac{3\theta }{2}} \\ & \frac{{\partial^{2} F_{1} }}{{\partial x_{1} \partial x_{2} }} = - \frac{1}{{4\sqrt {r^{3} } }}\cos \frac{3\theta }{2} \end{aligned} \right.$$

(A.2)

$$\left\{ \begin{aligned} & {\frac{{\partial^{2} F_{2} }}{{\partial x_{1}^{2} }} = - \frac{1}{{4\sqrt {r^{3} } }}\cos \frac{3\theta }{2}} \\ & {\frac{{\partial^{2} F_{2} }}{{\partial x_{2}^{2} }} = \frac{1}{{4\sqrt {r^{3} } }}\cos \frac{3\theta }{2}} \\ & \frac{{\partial^{2} F_{2} }}{{\partial x_{1} \partial x_{2} }} = - \frac{1}{{4\sqrt {r^{3} } }}\sin \frac{3\theta }{2} \end{aligned} \right.$$

(A.3)

$$\left\{ \begin{aligned} \frac{{\partial^{2} F_{3} }}{{\partial x_{1}^{2} }} = \frac{1}{{4\sqrt {r^{3} } }}\sin \frac{3\theta }{2}\sin \theta \cos \theta + \frac{1}{{2\sqrt {r^{3} } }}\sin \theta \left( {\frac{3}{2}\cos \frac{3\theta }{2}\sin \theta + \sin \frac{3\theta }{2}\cos \theta } \right) \hfill \\ \frac{{\partial^{2} F_{3} }}{{\partial x_{2}^{2} }} = - \frac{1}{{4\sqrt {r^{3} } }}\sin \theta \left( {\sin \frac{3\theta }{2}\cos \theta + \sin \frac{\theta }{2}} \right) + \frac{1}{{2\sqrt {r^{3} } }}\cos \theta \left( {\frac{1}{2}\cos \frac{\theta }{2} + \frac{3}{2}\cos \frac{3\theta }{2}\cos \theta - \sin \frac{3\theta }{2}\sin \theta } \right) \hfill \\ \frac{{\partial^{2} F_{3} }}{{\partial x_{1} \partial x_{2} }} = \frac{1}{{4\sqrt {r^{3} } }}\sin \frac{3\theta }{2}\sin \theta \sin \theta - \frac{1}{{2\sqrt {r^{3} } }}\cos \theta \left( {\frac{3}{2}\cos \frac{3\theta }{2}\sin \theta + \sin \frac{3\theta }{2}\cos \theta } \right) \hfill \\ \end{aligned} \right.$$

(A.4)

$$\left\{ \begin{aligned} \frac{{\partial^{2} F_{4} }}{{\partial x_{1}^{2} }} = \frac{1}{{4\sqrt {r^{3} } }}\cos \frac{3\theta }{2}\sin \theta \cos \theta + \frac{1}{{2\sqrt {r^{3} } }}\sin \theta \left( {\cos \frac{3\theta }{2}\cos \theta - \frac{3}{2}\sin \frac{3\theta }{2}\sin \theta } \right) \hfill \\ \frac{{\partial^{2} F_{4} }}{{\partial x_{2}^{2} }} = - \frac{1}{{4\sqrt {r^{3} } }}\sin \theta \left( {\cos \frac{3\theta }{2}\cos \theta + \cos \frac{\theta }{2}} \right) - \frac{1}{{2\sqrt {r^{3} } }}\cos \theta \left( {\frac{1}{2}\sin \frac{\theta }{2} + \frac{3}{2}\sin \frac{3\theta }{2}\cos \theta - \cos \frac{3\theta }{2}\sin \theta } \right) \hfill \\ \frac{{\partial^{2} F_{4} }}{{\partial x_{1} \partial x_{2} }} = \frac{1}{{4\sqrt {r^{3} } }}\cos \frac{3\theta }{2}\sin \theta \sin \theta + \frac{1}{{2\sqrt {r^{3} } }}\cos \theta \left( {\frac{3}{2}\sin \frac{3\theta }{2}\sin \theta - \cos \frac{3\theta }{2}\cos \theta } \right) \hfill \\ \end{aligned} \right.$$

(A.5)

Using a vector transformation, the second-order derivatives of $F_{l} \left( {r,\theta } \right)$ with respect to the global coordinate system are given by:

$$\left\{ \begin{aligned} \begin{array}{*{20}c} {\frac{{\partial^{2} F_{\alpha } }}{{\partial x^{2} }} = \frac{{\partial^{2} F_{\alpha } }}{{\partial x_{1}^{2} }}\cos^{2} \beta - \frac{{\partial^{2} F_{\alpha } }}{{\partial x_{1} \partial x_{2} }}\sin 2\beta + \frac{{\partial^{2} F_{\alpha } }}{{\partial x_{2}^{2} }}\sin^{2} \beta } \\ {\frac{{\partial^{2} F_{\alpha } }}{{\partial y^{2} }} = \frac{{\partial^{2} F_{\alpha } }}{{\partial x_{1}^{2} }}\sin^{2} \beta + \frac{{\partial^{2} F_{\alpha } }}{{\partial x_{1} \partial x_{2} }}\sin 2\beta + \frac{{\partial^{2} F_{\alpha } }}{{\partial x_{2}^{2} }}\cos^{2} \beta } \\ \end{array} \hfill \\ \frac{{\partial^{2} F_{\alpha } }}{\partial x\partial y} = \frac{1}{2}\sin 2\beta \left( {\frac{{\partial^{2} F_{\alpha } }}{{\partial x_{1}^{2} }} - \frac{{\partial^{2} F_{\alpha } }}{{\partial x_{2}^{2} }}} \right) + \cos 2\beta \frac{{\partial^{2} F_{\alpha } }}{{\partial x_{1} \partial x_{2} }} \hfill \\ \end{aligned} \right.$$

(A.6)

Appendix B

The crack-tip branch functions for the mid-plane displacement components in the x, y, and z directions are expressed as follows:

$$\left[ {G_{l} \left( {r,\theta } \right),l = 1, \ldots ,4} \right] = r^{{\frac{3}{2}}} \left[ {\begin{array}{*{20}c} {{ \sin }\left( {\frac{\theta }{2}} \right)} & {\cos \left( {\frac{\theta }{2}} \right)} & {\sin \left( {\frac{3\theta }{2}} \right)} & {\cos \left( {\frac{3\theta }{2}} \right)} \\ \end{array} } \right]$$

(B.1)

The second-order derivatives of $G_{l} \left( {r,\theta } \right)$ with respect to the local crack tip coordinate system (x₁; x₂) shown in Fig. 17 are given by:

$$\left\{ \begin{aligned} \frac{{\partial^{2} G_{1} }}{{\partial x_{1}^{2} }} = \frac{1}{2\sqrt r }\sin \frac{\theta }{2}\cos \theta \left( {\cos \theta - \frac{1}{2}} \right) - \frac{1}{\sqrt r }\sin \theta \left( {\frac{1}{2}\cos \frac{\theta }{2}\cos \theta - \sin \frac{\theta }{2}\sin \theta - \frac{1}{4}\cos \frac{\theta }{2}} \right) \hfill \\ \frac{{\partial^{2} G_{1} }}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} = \frac{1}{2\sqrt r }\sin \frac{\theta }{2}\sin \theta \left( {\cos \theta - \frac{1}{2}} \right) + \frac{1}{\sqrt r }\cos \theta \left( {\frac{1}{2}\cos \frac{\theta }{2}\cos \theta - \sin \frac{\theta }{2}\sin \theta - \frac{1}{4}\cos \frac{\theta }{2}} \right) \hfill \\ \frac{{\partial^{2} G_{1} }}{{\partial x_{2}^{2} }} = \frac{1}{2\sqrt r }\sin \theta \left( {\sin \frac{\theta }{2}\sin \theta + \frac{1}{2}\cos \frac{\theta }{2}} \right) + \frac{1}{\sqrt r }\cos \theta \left( {\cos \theta \sin \frac{\theta }{2} + \frac{1}{2}\sin \theta \cos \frac{\theta }{2} - \frac{1}{4}\sin \frac{\theta }{2}} \right) \hfill \\ \end{aligned} \right.$$

(B.2)

$$\left\{ \begin{aligned} \frac{{\partial^{2} G_{2} }}{{\partial x_{1}^{2} }} = \frac{1}{2\sqrt r }\cos \frac{\theta }{2}\cos \theta \left( {\cos \theta + \frac{1}{2}} \right) + \frac{1}{\sqrt r }\sin \theta \left( {\frac{1}{2}\sin \frac{\theta }{2}\cos \theta + \cos \frac{\theta }{2}\sin \theta + \frac{1}{4}\sin \frac{\theta }{2}} \right) \hfill \\ \frac{{\partial^{2} G_{2} }}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} = \frac{1}{2\sqrt r }\cos \frac{\theta }{2}\sin \theta \left( {\cos \theta + \frac{1}{2}} \right) - \frac{1}{\sqrt r }\cos \theta \left( {\frac{1}{2}\sin \frac{\theta }{2}\cos \theta + \cos \frac{\theta }{2}\sin \theta + \frac{1}{4}\sin \frac{\theta }{2}} \right) \hfill \\ \frac{{\partial^{2} G_{2} }}{{\partial x_{2}^{2} }} = \frac{1}{2\sqrt r }\sin \theta \left( {\sin \frac{\theta }{2}\sin \theta + \frac{1}{2}\cos \frac{\theta }{2}} \right) + \frac{1}{\sqrt r }\cos \theta \left( {\cos \theta \sin \frac{\theta }{2} + \frac{1}{2}\sin \theta \cos \frac{\theta }{2} - \frac{1}{4}\sin \frac{\theta }{2}} \right) \hfill \\ \end{aligned} \right.$$

(B.3)

$$\left\{ \begin{aligned} \frac{{\partial^{2} G_{3} }}{{\partial x_{1}^{2} }} = - \frac{3}{4\sqrt r }\sin \frac{\theta }{2} \hfill \\ \frac{{\partial^{2} G_{3} }}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} = \frac{3}{4\sqrt r }\cos \frac{\theta }{2} \hfill \\ \frac{{\partial^{2} G_{3} }}{{\partial x_{2}^{2} }} = \frac{3}{4\sqrt r }\sin \frac{\theta }{2} \hfill \\ \end{aligned} \right.$$

(B.4)

$$\left\{ \begin{aligned} \frac{{\partial^{2} G_{4} }}{{\partial x_{1}^{2} }} = \frac{3}{4\sqrt r }\cos \frac{\theta }{2} \hfill \\ \frac{{\partial^{2} G_{4} }}{{\partial x_{1}^{{}} \partial x_{2}^{{}} }} = \frac{3}{4\sqrt r }\sin \frac{\theta }{2} \hfill \\ \frac{{\partial^{2} G_{4} }}{{\partial x_{2}^{2} }} = - \frac{3}{4\sqrt r }\cos \frac{\theta }{2} \hfill \\ \end{aligned} \right.$$

(B.5)

Using a vector transformation, the second-order derivatives of $G_{l} \left( {r,\theta } \right)$ with respect to the global coordinate system are obtained.