Free vibration analysis of cracked thin plates by quasi-convex coupled isogeometric-meshfree method

Abstract

The free vibration analysis of cracked thin plates via a quasi-convex coupled isogeometric-meshfree method is presented. This formulation employs the consistently coupled isogeometric-meshfree strategy where a mixed basis vector of the convex B-splines is used to impose the consistency conditions throughout the whole problem domain. Meanwhile, the rigid body modes related to the mixed basis vector and reproducing conditions are also discussed. The mixed basis vector simultaneously offers the consistent isogeometric-meshfree coupling in the coupled region and the quasi-convex property for the meshfree shape functions in the meshfree region, which is particularly attractive for the vibration analysis. The quasi-convex meshfree shape functions mimic the isogeometric basis function as well as offer the meshfree nodal arrangement flexibility. Subsequently, this approach is exploited to study the free vibration analysis of cracked plates, in which the plate geometry is exactly represented by the isogeometric basis functions, while the cracks are discretized by meshfree nodes and highly smoothing approximation is invoked in the rest of the problem domain. The efficacy of the present method is illustrated through several numerical examples.

Keywords

meshfree method / isogeometric analysis / quasi-convex isogeometric-meshfree method / free vibration / cracked thin plate

Introduction

The study of free vibration characteristics of cracked thin plate structures is an important topic in structural engineering which has attracted noticeable research efforts [ 1− 8]. For example, Lynn and Kumbasar [ 1] used Green's functions to represent the transverse displacements of the simply supported rectangular plates with cracks. Stahl and Keer [ 2] formulated the problems of cracked thin plates as dual series equations and then reduced them to the homogeneous Fredholm integral equations of the second kind. Neku [ 3] established the Green’s functions for the simply-supported rectangular plate having a straight notch based on Lynn and Kumbasar’s approach [ 1] via Levy's form. Because of the discontinuities of the displacement and slope across the cracks, Solecki [ 4] constructed a solution for the vibration of a cracked plate by means of finite Fourier transformation of discontinuous functions. On the other hand, Hirano and Okazaki [ 5] used the Levy-Nadai’s form of solution to describe the mixed boundary conditions on the line of the crack with the weighted residual methods. Leissa et al. [ 6] provided the free vibration data for circular plates having V-notches. Liew et al. [ 7] presented a domain decomposition method for the analysis of cracked plates. In this approach, the continuities of displacement and slope are not satisfied at every point along the interconnecting boundaries. The vibration analysis of rectangular plates with side cracks was also investigated by Huang and Leissa [ 8] using the Ritz method.

Since the thin plate vibration is governed by the fourth order differential equation, which necessitates the employment of at least C¹ continuous approximation even within the Galerkin variational formulation. Construction of C¹ continuous approximation may introduce considerable complexity and difficulty in the finite element methods [ 9]. On the other hand, the meshfree methods [ 10− 17] and isogeometric analysis [ 18− 25] developed in recent years share many similarities [ 26] and both offer arbitrary order smoothing approximation that is very desirable for the thin plate analysis [ 27− 37]. Moreover, the node-based nature makes meshfree methods very suitable for the crack modeling. Belytschko and co-workers [ 38, 39] have proposed several methods for crack modeling, i.e., the widely used visibility criterion and the diffraction method. Rabczuk and Belyschko [ 40] also proposed a cracking particle approach which allows a unified treatment of arbitrarily oriented cracks. In the isogeometric analysis framework, Luycker et al. [ 41] showed that X-FEM in isogeometric analysis can produce solutions with higher order convergence rates for linear fracture mechanics problems. A combination of isogeometric analysis with T-splines and X-FEM was also presented for fracture analysis by Ghorashi et al. [ 42, 43]. Tran et al. [ 44] presented an extended isogeometric analysis for vibration of cracked functionally graded thick plates using higher-order shear deformation theory. An extended isogeometric analysis based on Kirchhoff−Love theory was carried out by Nguyan-Thanh et al. [ 45] for the thickness crack modeling in thin shells.

In this study, we perform the free vibration analysis of cracked thin plates by the quasi-convex coupled isogeometric-meshfree method. Here the quasi-convex coupled scheme implies the consistent blending of the convex isogeometric basis functions and the quasi-convex isogeometric enriched meshfree shape functions [ 46]. In this formulation, the geometry exactness of isogeometric analysis and the nodal placement flexibility of meshfree formulation are preserved. Meanwhile, the smoothing property of the isogeometric enriched meshfree shape approximation is fully employed for thin plate analysis. By importing the mixed reproducing conditions of the isogeometric B-spline basis functions into the reproducing kernel meshfree formulation, we obtain a set of quasi-convex shape functions which gives superior frequency accuracy [ 16] and thus are preferred in vibration analysis. It is noted that although the local maximum entropy based meshfree methods [ 12, 13] give a convex approximation, an iterative computation is required for this type approximation and furthermore its extension to higher order completeness may face severe difficulty. In Ref. [ 16], Wang and Chen have developed an arbitrary order quasi-convex reproducing kernel approximation. The present approach yields similar quasi-convex meshfree shape functions while using an alternative mixed basis vector. Furthermore, this formulation leads to a unified coupling of isogeometric and meshfree approximations based upon the mixed basis vector. In order to properly model the thin plate crack, the quasi-convex meshfree approximation in conjunction with the visibility criterion is invoked in the region surrounding the crack, and the rest part of the problem domain is represented by the isogeometric basis functions via an geometry exact way. The numerical results are compared with available reference results to verify the effectiveness of the proposed method.

The outline of this paper is organized as follows. Section 2 summarizes the isogeometric basis functions and meshfree shape functions. The construction of quasi-convex coupled isogeometric-meshfree approximation is illustrated in Section 3, which in Section 4 are employed to discretize the thin plate vibration equation. The numerical demonstration of the proposed method is given in Section 5. Finally conclusions are described in Section 6.

Isogeometric basis functions and meshfree shape functions

The quasi-convex coupled isogeometric-meshfree approximation is a consistently continuous mixture of the isogeometric basis functions and the meshfree shape functions. Thus we first briefly summarize these two approximation schemes. The most frequently used basis functions in isogeometric analysis are the non-uniform rational B-spline (NURBS) basis functions, say, R_a(\xi ) ’s , which are defined in a parametric space \xi \in [\mathrm{0},\mathrm{1}] . In general, a p-th order NURBS basis function R_a(\xi ) can be expressed as a linear combination of the B-spline basis functions N_a^p(\xi ) ’s multiplied with their corresponding weight {\varpi }_a ’s , i.e., in 1D case, we have [ 18]:

(1) R_a(\xi )={\displaystyle \frac{N_a^p(\xi ){\varpi }_a}{{\displaystyle \sum _b{N}_b^p(\xi ){\varpi }_b}}}.

A p-th order B-spline basis function N_a^p(\xi ) can be constructed recursively as follows [ 18]:

(2) N_a^p(\xi )={\displaystyle \frac{\xi -{\xi }_a}{{\xi }_{a+p}-{\xi }_a}}{N}_a^{p-\mathrm{1}}(\xi )+{\displaystyle \frac{{\xi }_{a+p+\mathrm{1}}-\xi }{{\xi }_{a+p+\mathrm{1}}-{\xi }_{a+\mathrm{1}}}}{N}_{a+\mathrm{1}}^{p-\mathrm{1}}(\xi ),\hspace{1em}p\ge \mathrm{1},

with

(3) N_a^{\mathrm{0}}(\xi )=\{\begin{array}{l}\mathrm{1},\hspace{1em}\xi \in [{\xi }_a,{\xi }_{a+\mathrm{1}})\hfill \\ \mathrm{0},\hspace{1em}\mathrm{otherwise}\hfill \end{array},

in which {\xi }_a ’s are the spline knots taken from a given knot vector k_{\mathit{\xi }} :

(4) k_{\xi }={\{{\xi }_{\mathrm{1}}=\mathrm{0},\cdots ,{\xi }_a,\cdots ,{\xi }_{n_k}=\mathrm{1}\}}^{\mathrm{T}},

where n_k represents the total number of knots.

By employing the tensor product of the basis functions in two parametric dimensions \xi and \eta , the 2D B-spline and NURBS basis functions are defined as:

(5) N_A(\mathit{\xi })=N_{ab}^p(\mathit{\xi })=N_a^p(\xi )N_b^p(\eta ),

(6) R_A(\mathit{\xi })={\displaystyle \frac{N_{ab}^p(\mathit{\xi }){\varpi }_{ab}}{{\displaystyle \sum _{c,d}{N}_{cd}^p(\mathit{\xi }){\varpi }_{cd}}}},

in which A=\{a,b\} , {\varpi }_{ab} is the 2D NURBS weight. It is noted that by definition, N_A(\mathit{\xi }) and R_A(\mathit{\xi }) meet the partition of unity condition:

(7) {\displaystyle \sum _A{N}_A(\mathit{\xi })}=\mathrm{1}, {\displaystyle \sum _A{R}_A(\mathit{\xi })}=\mathrm{1.}

Different from the B-spline basis functions, the starting point to construct the reproducing kernel meshfree shape functions is the p -th order reproducing conditions defined as follows [ 11]:

(8) {\displaystyle \sum _B{\mathrm{\Psi }}_B(\mathit{\xi }){\xi }_B^{\alpha }{\eta }_B^{\beta }}={\xi }^{\alpha }{\eta }^{\beta },\hspace{1em}\mathrm{0}\le \alpha +\beta \le p,

in which the meshfree shape function {\mathrm{\Psi }}_B(\mathit{\xi }) has the following form:

(9) {\mathrm{\Psi }}_B(\mathit{\xi })={\mathit{p}}^{\mathrm{T}}({\mathit{\xi }}_B)b(\mathit{\xi }){\phi }_s({\mathit{\xi }}_B-\mathit{\xi }),

with b(\mathit{\xi }) being the unknown coefficient vector. {\phi }_s({\mathit{\xi }}_B-\mathit{\xi }) is the kernel function of node B which has a compact support measured by \prime s\prime . Here the cubic B-spline kernel function is employed:

(10) {\phi }_s({\mathit{\xi }}_B-\mathit{\xi })=\phi ({\xi }_B-\xi )\phi ({\eta }_B-\eta ),

with

(11)

The p -th order monomial basis vector \mathit{p}(\mathit{\xi }) is defined as

(12) \mathit{p}(\mathit{\xi })={\{\mathrm{1},\xi ,\eta ,{\xi }^{\mathrm{2}},\xi \eta ,{\eta }^{\mathrm{2}},\cdots ,{\xi }^p,\cdots ,{\eta }^p\}}^{\mathrm{T}}.

The coefficient vector b(\mathit{\xi }) is solved by imposing the reproducing conditions. Substituting Eq.(9) into Eq.(8) yields:

(13) ℳ(\mathit{\xi })b(\mathit{\xi })=\mathit{p}(\mathit{\xi }),

where ℳ(\mathit{\xi }) is the moment matrix given by:

(14) ℳ(\mathit{\xi })={\displaystyle \sum _B{\mathit{p}}^{\mathrm{T}}({\mathit{\xi }}_B)p({\mathit{\xi }}_B){\phi }_s({\mathit{\xi }}_B-\mathit{\xi })}.

Thus we obtains b(\mathit{\xi })={ℳ}^{-\mathrm{1}}(\mathit{\xi })p(\mathit{\xi }) and finally the reproducing kernel meshfree shape function has the following expression:

(15) {\mathrm{\Psi }}_B(\mathit{\xi })={\mathit{p}}^T({\mathit{\xi }}_B){ℳ}^{-\mathrm{1}}(\mathit{\xi })p(\mathit{\xi }){\phi }_s({\mathit{\xi }}_B-\mathit{\xi }).

Quasi-convex coupled isogeometric-meshfree approximation

Here we focus on the development of isogeometric enriched quasi-convex meshfree approximation with particular reference to the cubic basis functions, which we shall use for the free vibration analysis of cracked thin plates. The quasi-convex coupled isogeometric-meshfree approximation is built upon the mixed reproducing or consistency conditions [ 47, 48]. It has been shown that the reproducing points for cubic B-spline basis functions are [ 47, 49]:

(16) \{\begin{array}{l}{\xi }_a^{[\mathrm{1}]}={\displaystyle \frac{{\xi }_{a+\mathrm{1}}+{\xi }_{a+\mathrm{2}}+{\xi }_{a+\mathrm{3}}}{\mathrm{3}}}\hfill \\ {\xi }_a^{[\mathrm{2}]}=\sqrt{{\displaystyle \frac{{\xi }_{a+\mathrm{1}}{\xi }_{a+\mathrm{2}}+{\xi }_{a+\mathrm{2}}{\xi }_{a+\mathrm{3}}+{\xi }_{a+\mathrm{3}}{\xi }_{a+\mathrm{1}}}{\mathrm{3}}}}\hfill \\ {\xi }_a^{[\mathrm{3}]}=\sqrt[\mathrm{3}]{{\xi }_{a+\mathrm{1}}{\xi }_{a+\mathrm{2}}{\xi }_{a+\mathrm{3}}}\hfill \end{array},

Thus the consistency conditions up to the {\mathrm{3}}^{rd} order take the following form:

(17) {\displaystyle \sum _a{N}_a^{\mathrm{3}}(\mathit{\xi }){({\xi }_a^{(\alpha )})}^{\alpha }}={\xi }^{\alpha },\hspace{1em}\mathrm{0}\le \alpha \le \mathrm{3.}

According to the reproducing conditions defined in Eqs. (16) and (17), we can derive the rigid body translation mode {\mathit{\phi }}_t and rotation mode {\mathit{\phi }}_r . Fig. 1(a) shows that there are four non-zero cubic B-spline basis functions, namely, N_{a-\mathrm{2}}^{\mathrm{3}}(\xi ) , N_{a-\mathrm{1}}^{\mathrm{3}}(\xi ) , N_a^{\mathrm{3}}(\xi ) and N_{a+\mathrm{1}}^{\mathrm{3}}(\xi ) associated with one generic element \xi \in [{\xi }_{a+\mathrm{1}},{\xi }_{a+\mathrm{2}}] , their corresponding displacement coefficients are denoted by d_{a-\mathrm{2}} , d_{a-\mathrm{1}} , d_a and d_{a+\mathrm{1}} . Without loss of generality, assume the element length be l_e and the coordinate origin be the element centroid, then the constant and linear displacement modes can be obtained from Eqs.(7) and (16) as:

(18) d_{a-\mathrm{2}}=d_{a-\mathrm{1}}=d_a=d_{a+\mathrm{1}}=\mathrm{1},

(19) \{\begin{array}{l}{d}_{a-\mathrm{2}}={\displaystyle \frac{{\xi }_{a-\mathrm{1}}+{\xi }_a+{\xi }_{a+\mathrm{1}}}{\mathrm{3}}}=-{\displaystyle \frac{\mathrm{3}{l}_e}{\mathrm{2}}}\hfill \\ d_{a-\mathrm{1}}={\displaystyle \frac{{\xi }_a+{\xi }_{a+\mathrm{1}}+{\xi }_{a+\mathrm{2}}}{\mathrm{3}}}=-{\displaystyle \frac{l_e}{\mathrm{2}}}\hfill \\ d_a={\displaystyle \frac{{\xi }_{a+\mathrm{1}}+{\xi }_{a+\mathrm{2}}+{\xi }_{a+\mathrm{3}}}{\mathrm{3}}}={\displaystyle \frac{l_e}{\mathrm{2}}}\hfill \\ d_{a+\mathrm{1}}={\displaystyle \frac{{\xi }_{a+\mathrm{2}}+{\xi }_{a+\mathrm{3}}+{\xi }_{a+\mathrm{4}}}{\mathrm{3}}}={\displaystyle \frac{\mathrm{3}{l}_e}{\mathrm{2}}}\hfill \end{array}.

Therefore the translation mode {\mathit{\phi }}_t and rotational mode {\mathit{\phi }}_r as shown in Figs. 1(b) and 1(c) are.

(20) {\mathit{\phi }}_t={\left\{\begin{array}{llll}\mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}\end{array}\right\}}^{\mathrm{T}},

(21) {\mathit{\phi }}_r={\displaystyle \frac{l_e}{\mathrm{2}}}{\left\{\begin{array}{llll}-\mathrm{3}& -\mathrm{1}& \mathrm{1}& \mathrm{3}\end{array}\right\}}^{\mathrm{T}},

Similarly, by employing the tensor product rule, the rigid body translation and rotational modes {\mathit{\phi }}_t , {\mathit{\phi }}_{rx} and {\mathit{\phi }}_{ry} for an element as shown in Fig. 2 are:

(22) {\mathit{\phi }}_t={\left\{\begin{array}{llllllllllllllll}\mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}\end{array}\right\}}^{\mathrm{T}},

(23) {\mathit{\phi }}_{rx}={\displaystyle \frac{l_e}{\mathrm{2}}}{\left\{\begin{array}{llllllllllllllll}-\mathrm{3}& -\mathrm{3}& -\mathrm{3}& -\mathrm{3}& -\mathrm{1}& -\mathrm{1}& -\mathrm{1}& -\mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{3}& \mathrm{3}& \mathrm{3}& \mathrm{3}\end{array}\right\}}^{\mathrm{T}},

(24) {\mathit{\phi }}_{ry}={\displaystyle \frac{l_e}{\mathrm{2}}}{\left\{\begin{array}{llllllllllllllll}-\mathrm{3}& -\mathrm{1}& \mathrm{1}& \mathrm{3}& -\mathrm{3}& -\mathrm{1}& \mathrm{1}& \mathrm{3}& -\mathrm{3}& -\mathrm{1}& \mathrm{1}& \mathrm{3}& -\mathrm{3}& -\mathrm{1}& \mathrm{1}& \mathrm{3}\end{array}\right\}}^{\mathrm{T}}.

In the quasi-convex coupled isogeometric-meshfree approximation, the problem domain \mathrm{\Omega } , say, the plate mid-surface, is partitioned into three regions, namely, the isogeometric region {\mathrm{\Omega }}^i , the meshfree region {\mathrm{\Omega }}^m , and the coupled region {\mathrm{\Omega }}^c . The plate deflection w(x) is approximated as:

(25) w^h(x)={\displaystyle \sum _C{\phi }_C(\mathit{\xi })d_C},

in which {\phi }_C is the shape function that is defined in different regions as follows:

(26) {\phi }_C(\mathit{\xi })=\{\begin{array}{ll}{R}_{C(A)}(\mathit{\xi })\hfill & \mathit{\xi }\in {\mathrm{\Omega }}^i\hfill \\ R_{C(A)}(\mathit{\xi })+{\mathrm{\Psi }}_{C(B)}(\mathit{\xi })\hfill & \mathit{\xi }\in {\mathrm{\Omega }}^c\hfill \\ {\mathrm{\Psi }}_{C(B)}(\mathit{\xi })\hfill & \mathit{\xi }\in {\mathrm{\Omega }}^m\hfill \end{array},

where C is the global numbering of the basis or shape function, while A and B are the local numberings of isogeometric basis function and meshfree shape function. Eq. (25) is accompanied by the isogeometric exact geometry mapping:

(27) x={\displaystyle \sum _A{R}_A(\mathit{\xi })P_A},

where P_A denotes the control point associated with a plate geometry.

In order to establish the quasi-convex approximation framework, the following consistency or reproducing conditions are imposed on Eq. (25):

(28) {\displaystyle \sum _C{\phi }_C(\mathit{\xi })\mathit{p}({\mathit{\xi }}_C^{[•]})}=\mathit{p}(\mathit{\xi }),\hspace{1em}\mathit{\xi }\in \mathrm{\Omega },

in which \mathit{p}({\mathit{\xi }}^{[•]}) are the cubic mixed reproducing point vector [ 47]:

(29) \mathit{p}({\mathit{\xi }}^{[•]})=\{\mathrm{1},{\xi }^{[\mathrm{1}]},{\eta }^{[\mathrm{1}]},{({\xi }^{[\mathrm{2}]})}^{\mathrm{2}},{\xi }^{[\mathrm{1}]}{\eta }^{[\mathrm{1}]},{({\eta }^{[\mathrm{2}]})}^{\mathrm{2}},{({\xi }^{[\mathrm{3}]})}^{\mathrm{3}},{({\xi }^{[\mathrm{2}]})}^{\mathrm{2}}{\eta }^{[\mathrm{1}]},{\xi }^{[\mathrm{1}]}{({\eta }^{[\mathrm{2}]})}^{\mathrm{2}},{({\eta }^{[\mathrm{3}]})}^{\mathrm{3}}\}.

Thus, by using Eq. (26), the consistency conditions of Eq. (28) implies:

(30) \{\begin{array}{ll}{\displaystyle \sum _A{R}_A(\mathit{\xi })\mathit{p}({\mathit{\xi }}_A^{[•]})}=\mathit{p}(\mathit{\xi })\hfill & \mathit{\xi }\in {\mathrm{\Omega }}^i\hfill \\ {\displaystyle \sum _A{R}_A(\mathit{\xi })\mathit{p}({\mathit{\xi }}_A^{[•]})}+{\displaystyle \sum _B{\mathrm{\Psi }}_B(\mathit{\xi })\mathit{p}({\mathit{\xi }}_B^{[•]})}=\mathit{p}(\mathit{\xi })\hfill & \mathit{\xi }\in {\mathrm{\Omega }}^c\hfill \\ {\displaystyle \sum _B{\mathrm{\Psi }}_B(\mathit{\xi })\mathit{p}({\mathit{\xi }}_B^{[•]})}=\mathit{p}(\mathit{\xi })\hfill & \mathit{\xi }\in {\mathrm{\Omega }}^m\hfill \end{array}.

The first condition of Eq. (30) is identical to Eq. (17) when uniform weights are used. The second condition of Eq. (30) provides us a consistently coupled isogeometric-meshfree approximation in {\mathrm{\Omega }}^c , where the meshfree shape function {\mathrm{\Psi }}_B(\mathit{\xi }) takes the reproducing kernel form:

(31) {\mathrm{\Psi }}_B(\mathit{\xi })={\mathit{p}}^{\mathrm{T}}({\mathit{\xi }}_B^{[•]})\mathit{b}(\mathit{\xi }){\phi }_s({\mathit{\xi }}_B^{[\mathrm{1}]}-\mathit{\xi }).

Substituting Eq.(31) into the second condition of Eq. (30) gives:

(32) \mathit{b}(\mathit{\xi })={\overline{ℳ}}^{-\mathrm{1}}(\mathit{\xi })\left[\mathit{p}(\mathit{\xi })-{\displaystyle \sum _A{R}_A(\mathit{\xi })\mathit{p}({\mathit{\xi }}_A^{[•]})}\right],

with \overline{ℳ}(\mathit{\xi }) being:

(33) \overline{ℳ}(\mathit{\xi })={\displaystyle \sum _B\mathit{p}({\mathit{\xi }}_B^{[•]})p^T({\mathit{\xi }}_B^{[•]}){\phi }_s({\mathit{\xi }}_B^{[\mathrm{1}]}-\mathit{\xi })}.

Finally the meshfree shape functions in the coupled region becomes:

(34) {\mathrm{\Psi }}_B(\mathit{\xi })={\mathit{p}}^{\mathrm{T}}({\mathit{\xi }}_B^{[•]}){\overline{ℳ}}^{-\mathrm{1}}(\mathit{\xi })\left[\mathit{p}(\mathit{\xi })-{\displaystyle \sum _A{R}_A(\mathit{\xi })\mathit{p}({\mathit{\xi }}_A^{[•]})}\right]{\phi }_s({\mathit{\xi }}_B^{[\mathrm{1}]}-\mathit{\xi }).

While in the pure meshfree region {\mathrm{\Omega }}^m , Eq. (34) reduces to:

(35) {\mathrm{\Psi }}_B(\mathit{\xi })={\mathit{p}}^{\mathrm{T}}({\mathit{\xi }}_B^{[•]}){\overline{ℳ}}^{-\mathrm{1}}(\mathit{\xi })\mathit{p}(\mathit{\xi }){\phi }_s({\mathit{\xi }}_B^{[\mathrm{1}]}-\mathit{\xi }).

A close comparison of Eqs. (35) and (15) shows that the mixed basis vector is employed for the reproducing kernel meshfree formulation. We call {\mathrm{\Psi }}_B(\mathit{\xi }) in Eq. (35) that meets the third condition in Eq. (30) a quasi-convex meshfree shape function since the mixed basis vector employed herein yields a nice quasi-convex property, which can be clearly seen from Fig. 3. Meanwhile, because of the convex property of the isogeometric basis functions, the proposed coupled formulation constitutes a quasi-convex approximation framework which is preferred for structural vibration analysis.

Discrete formulation for vibration analysis of cracked thin plates

The free vibration of a cracked thin plate is governed by the following fourth order eigenvalue problem:

(36) D{\nabla }^{\mathrm{4}}w(x)-\rho h{\omega }^{\mathrm{2}}w(x)=\mathrm{0},\hspace{1em}x\in \mathrm{\Omega }

where w(x) is the plate deflectional vibration mode, \rho is the material density, h is the plate thickness, \omega is the circular frequency, D=E{h}^{\mathrm{3}}/\mathrm{12}(\mathrm{1}-{\nu }^{\mathrm{2}}) is the flexural rigidity with Young’s modulus E and Poisson ratio v, and \mathrm{\Omega } denotes the mid-surface of the thin plate. Eq. (36) is subjected to proper boundary conditions.

The weak form corresponding to Eq. (36) can be stated as:

(37) {\displaystyle \underset{\mathrm{\Omega }}{\int }\delta {\mathit{\kappa }}^{\mathrm{T}}D\mathit{\kappa }\mathrm{d}\mathrm{\Omega }}-{\displaystyle \underset{\mathrm{\Omega }}{\int }\delta w\rho hw\mathrm{d}\mathrm{\Omega }}=\mathrm{0},

where \mathit{\kappa }(x) is the plate curvature that can be expressed by the deflection w(x) as:

(38) \mathit{\kappa }(x)=\left\{\begin{array}{l}{w}_{,xx}(x)\\ w_{,yy}(x)\\ \mathrm{2}{w}_{,xy}(x)\end{array}\right\}.

D is the elastic plate constitutive matrix:

(39) D=D\left[\begin{array}{lll}\mathrm{1}& \nu & \mathrm{0}\\ \nu & \mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& (\mathrm{1}-\nu )/\mathrm{2}\end{array}\right].

By invoking the quasi-convex coupled isogeometric-meshfree approximation of the plate deflection according to Eq. (25), the approximate curvature vector of Eq. (38) takes the following form:

(40) {\mathit{\kappa }}^h(\mathit{x})={\displaystyle \sum _C{B}_C(\mathit{\xi })d_C},

with

(41) B_C(\mathit{\xi })=\left\{\begin{array}{l}{\phi }_{C,xx}(\mathit{\xi })\\ {\phi }_{C,yy}(\mathit{\xi })\\ \mathrm{2}{\phi }_{C,xy}(\mathit{\xi })\end{array}\right\}.

Introducing Eqs.(25) and (40) into the weak form of Eq.(37) leads to the classical discrete matrix equation for plate vibration:

(42) [K-{({\omega }^h)}^{\mathrm{2}}\mathit{{\rm M}}]d=\mathbf{0}\mathbf{,}

where {\omega }^h is the approximate frequency, \mathit{{\rm M}} and K are the mass and stiffness matrices and their corresponding components are given by:

(43) M_{CD}={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{\Phi }}_C\rho h{\mathrm{\Phi }}_D\mathrm{d}\mathrm{\Omega }},\hspace{1em}{K}_{CD}={\displaystyle \underset{\mathrm{\Omega }}{\int }{B}_C^{\mathrm{T}}D{B}_D\mathrm{d}\mathrm{\Omega }}.

Since isogeometric basis functions and meshfree shape functions are both smooth and continuous, it is difficult to deal with cracks where deflectional discontinuity occurs. Here in order to properly model the plate cracks, the plate geometry is exactly represented by the isogeometric approximation, while the meshfree approximation based upon the visibility criterion [ 38, 39] is employed for the crack discretization. Fig. 4 describes the quasi-convex meshfree shape functions crossing a crack, where the meshfree shape functions are almost positive and the desired jumps of shape functions are obtained at the crack locations. Of course, special crack tip enrichment functions can be added to further improve the solution accuracy [ 39], nonetheless, in this study to make the stiffness and mass matrices better conditioning, we do not use extra enrichment functions and numerical examples do show very favorable solution accuracy using this standard formulation.

Numerical examples

Rectangular cracked plates

Consider a simply supported rectangular plate as shown in Fig. 5(a). The plate geometry and material nomenclatures are: length L=\mathrm{20} , width W , length/width ratio L/W=\mathrm{2} , thickness h=\mathrm{0.1} , density \rho =\mathrm{6000} , Young’s modulus E=\mathrm{2}\times {\mathrm{10}}^{\mathrm{11}} , and Poisson's ratio \nu =\mathrm{0.3} . This plate has a horizontal side crack at b/W=\mathrm{0.5} from the right edge with different lengths α, where L, W, α, and b are defined in Fig.5 (a). Numerical results are presented for the first three symmetric and anti-symmetric non-dimensional frequency parameters \lambda 's with \lambda =\omega L^{\mathrm{2}}\sqrt{\rho h/D} . A normalized support size of 3.0 for the kernel function is employed for all the numerical results presented herein.

Different crack lengths with a/L=\mathrm{0.2}, \mathrm{0.5}, \mathrm{0.8} are considered. The corresponding quasi-convex coupled isogeometric-meshfree discretizations for these different crack lengths are given in Figs. 6, 8 and 10, where a local meshfree node refinement surrounding the crack tip is used to better model the plate cracks. In the refined region 9-point Gauss quadrature is used in each quadrilateral integration element formed by four neighboring nodes, while 5-point Gauss quadrature is employed in the rest region. Tables 1-3 list the convergence of the non-dimensional frequency parameters for the plates with different lengths of cracks, in which A and S denote symmetric and anti-symmetric modes, respectively. For comparison, the results of Stahl and Keer [ 2] are included in these Tables as the reference. It can be seen that the results by the proposed approach are in very good agreement with the reference solutions. Moreover, the first six vibration modes for the three cracked plates are plotted in Figs. 7, 9 and 11, which demonstrate that the cracks with deflectional discontinuities are well modeled by the present approach.

Square cracked plates

We also consider a simply supported square plate with a central crack as shown in Fig. 5(b). The plate geometry is L=\mathrm{10} and other geometry and material parameters are the same as those of the previous rectangular plate. Two kinds of central crack lengths, i.e., a/L=\mathrm{0.4}, \mathrm{0.8} are studied and the corresponding quasi-convex coupled isogeometric-meshfree discretizations are depicted in Figs. 12 and 14. The results for the non-dimensional frequencies are tabulated in Tables 4-5. Once again, very favorable agreement is observed between the numerical results and the reference solutions given by Stahl and Keer [ 2]. Meanwhile, the first six vibration modes of this square cracked plate with different crack lengths are plotted in Figs. 13 and 15, where both the discontinuous deflection across the crack and the smoothing deflection in the rest plate domain are properly simulated.

Cracked annular plate

The last problem considered here is the vibration of the cracked annular plate problem as described in Fig. 16. This annular has an outer radius r_o=\mathrm{5} and an inner radius r_i=\mathrm{2} . The material properties are the same as those in the previous two examples. The annular plate is simply supported and has a centralized crack with a length of a=(r_o-r_i)/\mathrm{2} . The quasi-convex coupled isogeometric-meshfree discretizations of this cracked annular plate are plotted in Fig. 17. It is noted that the annular shape is exactly modeled by the isogeometric mapping. Table 6 lists the first six non-dimensional frequency parameters \lambda 's, where \lambda is defined as \omega {(r_o-r_i)}^{\mathrm{2}}\sqrt{\rho h/D} . Since no analytical solutions are available for this problem, a refined finite element model with 158112 discrete Kirchhoff triangular thin plate elements is used to compute the reference solutions. It is observed that the results using quasi-convex coupled isogeometric-meshfree analysis compares very well with the reference solutions. It again proved that the present numerical solutions yields excellent converged results for the curved boundary problems. Furthermore, the first six vibration modes for the cracked annular plate are well demonstrated in Fig. 18.

Conclusions

A quasi-convex coupled isogeometric-meshfree analysis of vibrations of cracked thin plates was presented. This method employs a mixed basis vector for the consistency or reproducing conditions throughout the problem domain. In the isogeometric region, these consistency conditions recover the monomial reproducing conditions of isogeometric basis functions, while in the coupled and meshfree regions, they yields quasi-convex coupled and meshfree shape functions. Consequently a quasi-convex arbitrary order smoothing approximation was achieved all over the problem domain, which is very desirable for the free vibration analysis. Subsequently this quasi-convex approximation was used to model the cracked thin plate problems, which guarantees a C¹ formulation. In order to properly model a cracked thin plate, the isogeometric approximation is adopted for the exact geometry description, while the discontinuous deflection field across a crack is represented by meshfree nodes based on the visibility criterion. The smoothing transition between the isogeometric and meshfree regions is ensured by the proposed method. Numerical results demonstrated that the present approach is very effective for cracked plate vibration analysis and yields excellent frequency accuracy.

References

[1] Lynn P P, Kumbasar N. Free vibration of thin rectangular plates having narrow cracks with simply supported edges. Developments in Mechanics, 1967, 4: 911–928

[2] Stahl B, Keer L M. Vibration and stability of cracked rectangular plates. International Journal of Solids and Structures, 1972, 8(1): 69–91

[3] Nezu K. Free vibration of a simply-supported rectangular plate with a straight through-notch. Bulletin of the Japan Society of Mechanical Engineers, 1982, 25(199): 16–23

[4] Solecki R. Bending vibration of a simply supported rectangular plate with a crack parallel to one edge. Engineering Fracture Mechanics, 1983, 18(6): 1111–1118

[5] Hirano Y, Okazaki K. Vibration of cracked rectangular plates. Bulletin of the Japan Society of Mechanical Engineers, 1980, 23(179): 732–740

[6] Leissa A W, McGee O G, Huang C S. Vibration of circular plates having V-notches or sharp radial cracks. Journal of Sound and Vibration, 1993, 161(2): 227–239

[7] Liew K M, Hung K C, Lim M K. A solution method for analysis of cracked plates under vibration. Engineering Fracture Mechanics, 1994, 48(3): 393–404

[8] Huang C S, Leissa A W. Vibration analysis of rectangular plates with side cracks via the Ritz method. Journal of Sound and Vibration, 2009, 323(3−5): 974–988

[9] Zienkiewicz O C, Taylor R L. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, 2005

[10] Belytschko T, Lu Y Y, Gu L. Element-free Gakerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256

[11] Liu W K, Jun S, Zhang Y F. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 1995, 20(8−9): 1081–1106

[12] Sukumar N. Construction of polygonal interpolants: a maximum entropy approach. International Journal for Numerical Methods in Engineering, 2004, 61(12): 2159–2181

[13] Arroyo M, Ortiz M. Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. International Journal for Numerical Methods in Engineering, 2006, 65(13): 2167–2202

[14] Rabczuk T, Samaniego E. Discontinuous modelling of shear bands using adaptive meshfree methods. Computer Methods in Applied Mechanics and Engineering, 2008, 197(6): 641–658

[15] Chen J S, Chi S W, Hu H Y. Recent developments in stabilized Galerkin and collocation meshfree methods. Computer Assisted Mechanics and Engineering Sciences, 2011, 18: 3–21

[16] Wang D, Chen P. Quasi-convex reproducing kernel meshfree method. Computational Mechanics, 2014, 54(3): 689–709

[17] Kwok O L A, Guan P C, Cheng W P, Sun C T. Semi-Lagrangian reproducing kernel particle method for slope stability analysis and post-failure simulation. KSCE Journal of Civil Engineering, 2015, 19(1): 107–115

[18] Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39−41): 4135–4195

[19] Cottrell J A, Hughes T J R, Reali A. Studies of refinement and continuity in isogeometric structural analysis. Computer Methods in Applied Mechanics and Engineering, 2007, 196(41): 4160–4183

[20] De Luycker E, Benson D J, Belytschko T, Bazilevs Y, Hsu M C. X‐FEM in isogeometric analysis for linear fracture mechanics. International Journal for Numerical Methods in Engineering, 2011, 87(6): 541–565

[21] Bazilevs Y, Hsu M C, Scott M A. Isogeometric fluid-structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines. Computer Methods in Applied Mechanics and Engineering, 2012, 249: 28–41

[22] Wang D, Liu W, Zhang H. Novel higher order mass matrices for isogeometric structural vibration analysis. Computer Methods in Applied Mechanics and Engineering, 2013, 260: 92–108

[23] Thai C H, Nguyen-Xuan H, Bordas S P A, Nguyen-Thanh N, Rabczuk T. Isogeometric analysis of laminated composite plates using the higher-order shear deformation theory. Mechanics of Advanced Materials and Structures, 2014, 22(6): 451–469

[24] Elguedj T, Hughes T J R. Isogeometric analysis of nearly incompressible large strain plasticity. Computer Methods in Applied Mechanics and Engineering, 2014, 288: 388–416

[25] Zuo B Q, Huang Z D, Wang Y W, Wu Z J. Isogeometric analysis for CSG models. Computer Methods in Applied Mechanics and Engineering, 2015, 285: 102–124

[26] Wang D, Xuan J. An improved NURBS-based isogeometric analysis with enhanced treatment of essential boundary conditions. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37−40): 2425–2436

[27] Krysl P, Belytschko T. Analysis of thin plates by the element-free Galerkin method. Computational Mechanics, 1995, 16: 1–10

[28] Krysl P, Belytschko T. Analysis of thin plates by the element-free Galerkin method. Computational Mechanics, 1995, 16: 1–10

[29] Li S, Lu H, Han W, Liu W K, Simkins D C. Reproducing kernel element method, Part II. Global conforming Im/Cn. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12−14): 953–987

[30] Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for nonlinear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548

[31] Wang D, Peng H. A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis of thin plates. Computational Mechanics, 2013, 51(6): 1013–1029

[32] Kiendl J, Bletzinger K U, Linhard J, Wüchner R. Isogeometric shell analysis with Kirchhoff−Love elements. Computer Methods in Applied Mechanics and Engineering, 2009, 198(49): 3902–3914

[33] Zhang H, Wang D, Xuan J. Non-uniform rational B spline-based isogeometric finite element analysis of thin beams and plates. Chinese Quarterly of Mechanics, 2010, 31: 469–477

[34] Benson D J, Bazilevs Y, Hsu M C, Hughes T J R. A large deformation, rotation-free, isogeometric shell. Computer Methods in Applied Mechanics and Engineering, 2011, 200(13): 1367–1378

[35] Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, Wüchner R, Bletzinger K U, Bazilevs Y, Rabczuk T. Rotation free isogeometric thin shell analysis using PHT-splines. Computer Methods in Applied Mechanics and Engineering, 2011, 200(47): 3410–3424

[36] Shojaee S, Izadpanah E, Valizadeh N, Kiendl J. Free vibration analysis of thin plates by using a NURBS-based isogeometric approach. Finite Elements in Analysis and Design, 2012, 61: 23–34

[37] Echter R, Oesterle B, Bischoff M. A hierarchic family of isogeometric shell finite elements. Computer Methods in Applied Mechanics and Engineering, 2013, 254: 170–180

[38] Wang D, Liu W, Zhang H. Superconvergent isogeometric free vibration analysis of Euler-Bernoulli beams and Kirchhoff plates with new higher order mass matrices. Computer Methods in Applied Mechanics and Engineering, 2015, 286: 230–267

[39] Organ D, Fleming M, Terry T, Belytschko T. Continuous meshless approximations for nonconvex bodies by diffraction and transparency. Computational Mechanics, 1996, 18(3): 225–235

[40] Belytschko T, Fleming M. Smoothing, enrichment and contact in the element-free Galerkin method. Computers & Structures, 1999, 71(2): 173–195

[41] Rabczuk T, Belytschko T. Cracking particles: a simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[42] De Luycker E, Benson D J, Belytschko T, Bazilevs Y, Hsu M C. X-FEM in isogeometric analysis for linear fracture mechanics. International Journal for Numerical Methods in Engineering, 2011, 87(6): 541–565

[43] Ghorashi S S, Valizadeh N, Mohammadi S. Extended isogeometric analysis for simulation of stationary and propagating cracks. International Journal for Numerical Methods in Engineering, 2012, 89(9): 1069–1101

[44] Ghorashi S S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146

[45] Tran L V, Nguyan V P, Wahab M A, Nguyan-Xuan H. An extended isogeometric analysis for vibration of cracked FGM plates using higher-order shear deformation theory, arXiv preprint arXiv:1403.0306, 2014

[46] Nguyen-Thanh N, Valizadeh N, Nguyen M N, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on Kirchhoff-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291

[47] Zhang H, Wang D. An isogeometric enriched quasi-convex meshfree formulation with application to material interface modeling. Engineering Analysis with Boundary Elements, 2015, 60: 37–50

[48] Wang D, Zhang H. A consistently coupled isogeometric-meshfree method. Computer Methods in Applied Mechanics and Engineering, 2014, 268: 843–870

[49] Zhang H, Wang D, Liu W. Isogeometric-meshfree coupled analysis of Kirchhoff plates. Advances in Structural Engineering, 2014, 17(8): 1159–1176

[50] Marsden M J. An identity for spline functions with applications to variation-diminishing spline approximation. Journal of Approximation Theory, 1970, 3(1): 7–49

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg