An extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3) for free vibration analysis of cracked Reissner-Mindlin shells

Abstract

A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) was recently proposed and proven to be robust for free vibration analyses of Reissner-Mindlin shell. The method improves significantly the accuracy of the solution due to softening effect of the cell-based strain smoothing technique. In addition, due to using only three-node triangular elements generated automatically, the CS-FEM-DSG3 can be applied flexibly for arbitrary complicated geometric domains. However so far, the CS-FEM-DSG3 has been only developed for analyzing intact structures without possessing internal cracks. The paper hence tries to extend the CS-FEM-DSG3 for free vibration analysis of cracked Reissner-Mindlin shells by integrating the original CS-FEM-DSG3 with discontinuous and crack−tip singular enrichment functions of the extended finite element method (XFEM) to give a so-called extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3). The accuracy and reliability of the novel XCS-FEM-DSG3 for free vibration analysis of cracked Reissner-Mindlin shells are investigated through solving three numerical examples and comparing with commercial software ANSYS.

Keywords

cracked Reissner-Mindlin shell / free vibration analysis / cell-based smoothed discrete shear gap method (CS-FEM-DSG3) / extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3) / smoothed finite element methods (SFEM)

Introduction

Shell structures are widely used in many engineering disciplines such as civil, mechanical, aerospace, marine engineering, etc. They can appear in various structures like shell roofs, pressure vessels, pipelines, tubular joints, offshore oil tanks, aircraft, submarines, etc. Similar to other types of structures, shell structures suffer from many types of defects in their service life owing to external agents such as corrosion, chemical attack, degradation, and so on. Among them, the appearance of cracks in shell structures is one of the most popular causes and makes reduce significantly their loading capacities. When a crack appears in shell structures, local flexibility and anisotropy arise right at the crack location. As a result, the characteristics of shell structures would significantly differ from those of intact shells. The simulation of the behavior of flawed shell structures thus becomes very important to evaluate the effects of these imperfections on loading capacity and safety in designing shell structures.

From the realistic demand of deep understanding cracked shell behaviors, many studies were conducted in the past decades. For example, in 1988, Kwon [ 1] developed a set of singular finite elements for application to problems of fracture mechanics in general. This set contained elements with derivative singularity between 0 and 1 which helps describe constant strain field and satisfy the compatible condition along inter-element boundaries. Krawczuk et al. [ 2] analyzed the static deflections and natural frequencies of thin, rectangular shell structures with transverse, internal, open cracks. In this study, the crack was modeled by additional flexibility matrices calculated on the basic laws of fracture mechanics. Liu et al. [ 3] developed the formulation of crack shell based on the linearized shallow shell theory, by including the effect of the crack-face closure on the stress intensity factors. Vaziri et al. [ 4], in 2006, used linear eigenvalue analysis to formulate cracked cylindrical shells under combined internal pressure and axial compression. In this study, some insight into the effect of internal pressure on critical crack length was provided. Fu et al. [ 5], in 2012, proposed an approach based on the transverse or out-of-plane nonlinear dynamic response at the centers of crack edges in a cracked cylindrical shell structure. This approach helped avoid the problem of mesh selection at the crack tip in the finite element analysis for bulging factors and provided information about the influence of crack lengths as well as crack widths on responses. Among these proposed methods, eXtended Finite Element Method (X-FEM) proposed by Belytschko et al. [ 6− 8] is shown as one of the most robust and effective numerical methods for analyzing the behaviors of cracked structures. In this methodology, the enriched approximation is constructed from the interaction of the crack geometry with the mesh. In association with the level set method (LSM), the XFEM allows the entire crack to be represented independently of the mesh, and thus remeshing is not necessary in process of modeling crack growth. Banchene et al. [ 9] applied this method to study the free flexural vibrations of cracked plates using 4-noded field consistent enriched element. Natarajan et al. [ 10] also applied XFEM for free vibration analysis for FGM plates and obtained some good results. Besides, the enriched and extended formulations related to XFEM have been also employed by various methods for fracture analysis of plates and shells such as meshfree methods [ 11− 13], phantom node method [ 14], isogeometric analysis [ 15, 16], finite element methods [ 17− 20], etc. It can be seen from the literature that studies related to analysis of cracked shells using shear-locking-free triangular shell elements is somewhat limited. This paper hence will try to fill this gap by using a new three-node triangular shell element proposed recently for free vibration analysis of cracked Reissner-Mindlin shells.

In the other front of the development of numerical methods, Liu and Nguyen-Thoi et al. [ 21] have integrated the strain smoothing technique into the FEM to create a series of smoothed FEM (S-FEM) such as cell-based smoothed FEM (CS-FEM) [ 22− 24], node-based smoothed FEM (NS-FEM) [ 25− 27], edge-based smoothed FEM (ES-FEM) [ 28, 29], face-based smoothed FEM (FS-FEM) [ 30] and a group of alpha-FEM [ 31− 34]. Each of these smoothed FEM has different properties and has been used to produce desired solutions for a wide class of benchmark and practical mechanics problems. Several theoretical aspects of the S-FEM models have been provided in Refs. [ 35, 36]. The S-FEM models have also been further investigated and applied to various problems such as plates and shells [ 37− 50], piezoelectricity [ 51, 52], fracture mechanics [ 53], visco-elastoplasticity [ 54− 56], limit and shakedown [ 57, 58], and some other applications [ 59, 60], etc.

Among them, the cell-based smoothed discrete shear gap method CS-FEM-DSG3 [ 45], which is a combination of the CS-FEM and discrete shear gap method (DSG3) [ 61], is one of the most effective methods for static and free vibration analyses of Reissner-Mindlin shells. The CS-FEM-DSG3 is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature. Besides such advantages, the implementation of the CS-FEM-DSG3 is also simple since it only uses the three-node triangular elements which can be easily generated automatically, especially for arbitrary complicated geometric domains. Due to these outstanding properties, the CS-FEM-DSG3 then has been extended to analyze various problems such as flat shells [ 45], stiffened plates [ 43], FGM plates [ 62], composite plates [ 63], piezoelectricity plates [ 51], and dynamic responses of plates on the viscoelastic foundation [ 64, 65]. However so far, the CS-FEM-DSG3 has been only developed to analyze perfect structures without possessing internal cracks.

The paper hence extends the CS-FEM-DSG3 for free vibration analysis of cracked cylindrical shells by integrating itself with discontinuous and crack−tip singular enrichment functions of the XFEM to give a so-called extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3). In the formulation of XCS-FEM-DSG3 method, enrichment functions are used to model the discontinuities at crack locations. By using level-set method in the X-FEM, the finite element mesh can be discretized independently of the crack locations. The formulation of CS-FEM-DSG3 will be combined with the discontinuous enrichment functions and the singular functions of X-FEM to compute the stiffness matrices of elements cut by the crack. On the other hand, for un-cut elements, stiffness matrices are normally computed by the original CS-FEM-DSG3. The obtained results through three numerical examples are compared with that of commercial software ANSYS to illustrate the accuracy and robustness of the XCS-FEM-DSG3 in solving free vibration analysis of cracked cylindrical Reissner-Mindlin shells.

The paper is then organized as follows. Section 2 generally introduces the weakform of the Reissner-Mindlin shells. In Section 3, the formulation of XCS-FEM-DSG3 for cracked shells are presented. Section 4 performs numerical examples to verify the reliability and efficiency of the proposed method. Finally, some conclusions are withdrawn in Section 5.

Weakform of the Reissner−Mindlin shells

Shell structures are usually subjected to both membrane and bending forces. Hence, in this paper, the cylindrical shell is modeled by the flat shell element which is a combination of a membrane element and a bending plate element. In this flat shell element, the middle surface of shell is chosen as the reference plane that occupies a domain \mathrm{\Omega }\in {ℝ}^{\mathrm{3}} as shown in Fig. 1.

Let u, v, w be the displacements of the middle plane in the x, y, z directions, and {\beta }_x , {\beta }_y , {\beta }_z be the rotations of the middle plane around y-axis, x-axis, and z-axis respectively. Then, the unknown vector of six independent field variables at any point in the Reissner−Mindlin shell domain is written as, \mathbf{u}={\left[\begin{array}{llllll}u& v& w& {\beta }_x& {\beta }_y& {\beta }_z\end{array}\right]}^T , in local coordinate system oxyz.

According to the first-order shear deformation theory (FSDT), the membrane strain {\epsilon }_m , the bending strain \mathbf{\kappa } and the shear strain \gamma are respectively defined as

(1) {\mathbf{\epsilon }}_m=\left[\begin{array}{l}{u}_{,x}\\ v_{,y}\\ u_{,y}+v_{,x}\end{array}\right];\hspace{1em}\mathbf{\kappa }=\left[\begin{array}{l}{\beta }_{x,x}\\ {\beta }_{y,y}\\ {\beta }_{x,y}+{\beta }_{y,x}\end{array}\right];\hspace{1em}\mathbf{\gamma }=\left[\begin{array}{l}{\beta }_x+w_{,x}\\ {\beta }_y+w_{,y}\end{array}\right],

where the subscript “comma” represents the partial derivative with respect to the spatial coordinate succeeding it.

Using the Hamilton principle, the general weakform of the Reissner−Mindlin shells for free vibration problems can be expressed as

(2) {\displaystyle {\int }_{\mathrm{\Omega }}\left[\begin{array}{lll}\delta {({\mathit{\epsilon }}_m)}^T& \delta {(\mathit{\kappa })}^T& \delta {(\mathit{\gamma })}^T\end{array}\right]\left[\begin{array}{lll}{\mathbf{D}}_{\mathit{m}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& {\mathbf{D}}_b& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& {\mathbf{D}}_s\end{array}\right]\left[\begin{array}{l}{\mathit{\epsilon }}_m\\ \mathit{\kappa }\\ \mathit{\gamma }\end{array}\right]d\mathrm{\Omega }}+{\displaystyle {\int }_{\mathrm{\Omega }}\delta {(\mathbf{u})}^T\mathbf{m}\ddot{\mathbf{u}}d\mathrm{\Omega }}=\mathrm{0},

where {\mathbf{D}}_m, {\mathbf{D}}_b \mathrm{a}\mathrm{n}\mathrm{d} {\mathbf{D}}_s are the material matrices related to the membrane, the bending and the shear deformations, calculated respectively as

(3) {\mathbf{D}}_m={\displaystyle \frac{Eh}{(\mathrm{1}-v^{\mathrm{2}})}}\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];\hspace{1em}{\mathbf{D}}_b={\displaystyle \frac{E{h}^{\mathrm{3}}}{\mathrm{12}(\mathrm{1}-v^{\mathrm{2}})}}\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];\hspace{1em}{\mathbf{D}}_s=kh\left[\begin{array}{ll}\mu & \mathrm{0}\\ \mathrm{0}& \mu \end{array}\right],

in which E, h, v and μ are, respectively, Young’s modulus, shell thickness, Poisson ratio and shear modulus; k=\mathrm{5}/\mathrm{6} is the shear correction factor; and m is the mass matrix containing the mass density of the material \rho and is expressed as

(4) \mathbf{m}=\mathrm{diag}[\rho h, \rho h, \rho h, \rho h^{\mathrm{3}}/\mathrm{12}, \rho h^{\mathrm{3}}/\mathrm{12}, \mathrm{0}].

Formulation of the extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3) for cracked cylindrical shells

A brief on the formulation of the CS-FEM-DSG3 for shell structures

For the formulation of shell structures, the flat shell element is formulated first in local coordinate system oxyz, then, it is transformed to the global coordinate system OXYZ. In the formulation of the CS-FEM-DSG3, each triangular element {\mathrm{\Omega }}_e is further divided into three sub-triangles {\mathrm{\Delta }}_{\mathrm{1}}^{ } , {\mathrm{\Delta }}_{\mathrm{2}}^{ } and {\mathrm{\Delta }}_{\mathrm{3}}^{ } by connecting the central point O of the element to three field nodes as shown in Fig. 2. Then, in each sub-triangle, the stabilized DSG3 [ 61] is used to compute the strain field of shell. Finally the strain smoothing technique is applied on the whole triangular element to smooth the strains on these three sub-triangles. More details of the formulation of the CS-FEM-DSG3 can be found in [ 43, 45, 62, 64].

In the CS-FEM-DSG3, the displacement vector {\mathbf{d}}_O^e={\left[\begin{array}{lllll}{u}_O^e& v_O^e& w_O^e& {\beta }_{xO}^e& {\beta }_{yO}^e\end{array} {\beta }_{zO}^e\right]}^T at the central point O of an element is assumed to be the simple average of three displacement vectors {\mathbf{d}}_{\mathrm{1}}^e, {\mathbf{d}}_{\mathrm{2}}^e, {\mathbf{d}}_{\mathrm{3}}^e of three field nodes, and has the form of

(5) {\mathbf{d}}_O^e={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}({\mathbf{d}}_{\mathrm{1}}^e+{\mathbf{d}}_{\mathrm{2}}^e+{\mathbf{d}}_{\mathrm{3}}^e).

On the first sub-triangle {\mathrm{\Delta }}_{\mathrm{1}}^{ } (triangle O-1-2), the linear approximation {\mathbf{u}}^{e{\mathrm{\Delta }}_{\mathrm{1}}}={\left[\begin{array}{llllll}{u}^{e{\mathrm{\Delta }}_{\mathrm{1}}}\hfill & v^{e{\mathrm{\Delta }}_{\mathrm{1}}}\hfill & w^{e{\mathrm{\Delta }}_{\mathrm{1}}}\hfill & {\beta }_x^{e{\mathrm{\Delta }}_{\mathrm{1}}}\hfill & {\beta }_y^{e{\mathrm{\Delta }}_{\mathrm{1}}}\hfill & {\beta }_z^{e{\mathrm{\Delta }}_{\mathrm{1}}}\hfill \end{array}\right]}^T is constructed by

(6) {\mathbf{u}}^{e{\mathrm{\Delta }}_{\mathrm{1}}}=N_{\mathrm{1}}^{e{\mathrm{\Delta }}_{\mathrm{1}}}{\mathbf{d}}_O^e+N_{\mathrm{2}}^{e{\mathrm{\Delta }}_{\mathrm{1}}}{\mathbf{d}}_{\mathrm{1}}^e+N_{\mathrm{3}}^{e{\mathrm{\Delta }}_{\mathrm{1}}}{\mathbf{d}}_{\mathrm{2}}^e=N^{e{\mathrm{\Delta }}_{\mathrm{1}}}{\mathbf{d}}^{e{\mathrm{\Delta }}_{\mathrm{1}}},

where {\mathbf{d}}^{e{\mathrm{\Delta }}_{\mathrm{1}}}={\left[\begin{array}{lll}{\mathbf{d}}_O^e& {\mathbf{d}}_{\mathrm{1}}^e& {\mathbf{d}}_{\mathrm{2}}^e\end{array}\right]}^T is the vector of nodal degrees of freedom of the sub-triangle {\mathrm{\Delta }}_{\mathrm{1}} and {\mathbf{N}}^{e{\mathrm{\Delta }}_{\mathrm{1}}}=\left[\begin{array}{lll}{N}_{\mathrm{1}}^{e{\mathrm{\Delta }}_{\mathrm{1}}}& N_{\mathrm{2}}^{e{\mathrm{\Delta }}_{\mathrm{1}}}& N_{\mathrm{3}}^{e{\mathrm{\Delta }}_{\mathrm{1}}}\end{array}\right] contains the linear shape functions created by the sub-triangle {\mathrm{\Delta }}_{\mathrm{1}}^{ } in natural coordinate defined by

(7) N_{\mathrm{1}}=\mathrm{1}-\eta -\xi ; N_{\mathrm{2}}=\xi ; N_{\mathrm{3}}=\eta .

Using the DSG3 formulation [ 61] for the sub-triangle {\mathrm{\Delta }}_{\mathrm{1}}^{ } , the membrane, bending and shear strains {\mathbf{\epsilon }}_m^{e{\mathrm{\Delta }}_{\mathrm{1}}} , {\mathbf{\kappa }}_{ }^{e{\mathrm{\Delta }}_{\mathrm{1}}} and {\mathbf{\gamma }}_{ }^{e{\mathrm{\Delta }}_{\mathrm{1}}} in the sub-triangle {\mathrm{\Delta }}_{\mathrm{1}}^{ } are then obtained, respectively, by

(8) {\mathit{\epsilon }}_m^{e{\mathrm{\Delta }}_{\mathrm{1}}}=\underset{{\mathbf{b}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}}{\underbrace{\left[\begin{array}{lll}{\mathbf{b}}_{\mathrm{1}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}& {\mathbf{b}}_{\mathrm{2}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}& {\mathbf{b}}_{\mathrm{3}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}\end{array}\right]}}\left[\begin{array}{l}{\mathbf{d}}_O^e\\ {\mathbf{d}}_{\mathrm{1}}^e\\ {\mathbf{d}}_{\mathrm{2}}^e\end{array}\right]\mathbf{=}{\mathbf{b}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}{\mathbf{d}}^{e{\mathrm{\Delta }}_{\mathrm{1}}},

(9) {\mathbf{\kappa }}^{e{\mathrm{\Delta }}_{\mathrm{1}}}=\underset{{\mathbf{b}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}}{\underbrace{\left[\begin{array}{lll}{\mathbf{b}}_{\mathrm{1}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}& {\mathbf{b}}_{\mathrm{2}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}& {\mathbf{b}}_{\mathrm{3}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}\end{array}\right]}}\left[\begin{array}{l}{\mathbf{d}}_O^e\\ {\mathbf{d}}_{\mathrm{1}}^e\\ {\mathbf{d}}_{\mathrm{2}}^e\end{array}\right]\mathbf{=}{\mathbf{b}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}{\mathbf{d}}^{e{\mathrm{\Delta }}_{\mathrm{1}}},

(10) {\mathbf{\gamma }}^{e{\mathrm{\Delta }}_{\mathrm{1}}}=\underset{{\mathbf{b}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}}{\underbrace{\left[\begin{array}{lll}{\mathbf{b}}_{\mathrm{1}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}& {\mathbf{b}}_{\mathrm{2}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}& {\mathbf{b}}_{\mathrm{3}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}\end{array}\right]}}\left[\begin{array}{l}{\mathbf{d}}_O^e\\ {\mathbf{d}}_{\mathrm{1}}^e\\ {\mathbf{d}}_{\mathrm{2}}^e\end{array}\right]\mathbf{=}{\mathbf{b}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}{\mathbf{d}}^{e{\mathrm{\Delta }}_{\mathrm{1}}},

where {\mathbf{b}}^{m{\mathrm{\Delta }}_{\mathrm{1}}} , {\mathbf{b}}^{b{\mathrm{\Delta }}_{\mathrm{1}}} and {\mathbf{b}}^{s{\mathrm{\Delta }}_{\mathrm{1}}} are the matrices of shape function derivatives related to membrane, bending and shear deformation of the triangular elements, calculated as

(11) {\mathbf{b}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}=\left[\begin{array}{llllll}b-c& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& d-a& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ d-a& b-c& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array} \begin{array}{llllll}c& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -d& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ -d& c& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array} \begin{array}{llllll}-b& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& a& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ a& -b& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right] ,

(12) {\mathbf{b}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}=\left[\begin{array}{llllll}\mathrm{0}& \mathrm{0}& \mathrm{0}& b-c& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& d-a& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& d-a& b-c& \mathrm{0}\end{array} \begin{array}{llllll}\mathrm{0}& \mathrm{0}& \mathrm{0}& c& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& -d& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& -d& c& \mathrm{0}\end{array} \begin{array}{llllll}\mathrm{0}& \mathrm{0}& \mathrm{0}& -b& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& a& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& a& -b& \mathrm{0}\end{array}\right] ,

(13) {\mathbf{b}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}{A}_e}}\left[\begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}b-c\\ d-a\end{array} \begin{array}{l}{A}_e\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ A_e\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}c\\ -d\end{array} \begin{array}{l}ac/\mathrm{2}\\ -ad/\mathrm{2}\end{array} \begin{array}{l}bc/\mathrm{2}\\ -bd/\mathrm{2}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}-b\\ a\end{array} \begin{array}{l}-bd/\mathrm{2}\\ ad/\mathrm{2}\end{array} \begin{array}{l}-bc/\mathrm{2}\\ ac/\mathrm{2}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array}\right],

with a=x_{\mathrm{2}}-x_{\mathrm{1}}; b=y_{\mathrm{2}}-y_{\mathrm{1}}; c=y_{\mathrm{3}}-y_{\mathrm{1}}; d=x_{\mathrm{3}}-x_{\mathrm{1}} where {\mathbf{x}}_i={\left[\begin{array}{ll}{x}_i\hfill & y_i\hfill \end{array}\right]}^T, i=\mathrm{1},\mathrm{2},\mathrm{3} are coordinates of three nodes in the local coordinate system and A_e is the area of the triangular element as shown in Fig. 3.

Substituting {\mathbf{d}}_O^e in Eq. (5) into Eqs. (8), (9) and (10), and then rearranging we obtain

(14) {\mathit{\epsilon }}_m^{e{\mathrm{\Delta }}_{\mathrm{1}}}=\underset{{\mathbf{B}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}}{\underbrace{\left[\begin{array}{lll}\frac{\mathrm{1}}{\mathrm{3}}{\mathbf{b}}_{\mathrm{1}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}+{\mathbf{b}}_{\mathrm{2}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}& {\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\mathbf{b}}_{\mathrm{1}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}+{\mathbf{b}}_{\mathrm{3}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}& {\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\mathbf{b}}_{\mathrm{1}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}\end{array}\right]}}\left[\begin{array}{l}{\mathbf{d}}_{\mathrm{1}}^e\\ {\mathbf{d}}_{\mathrm{2}}^e\\ {\mathbf{d}}_{\mathrm{3}}^e\end{array}\right]={\mathbf{B}}^{m{\mathrm{\Delta }}_{\mathrm{1}}}{\mathbf{d}}^e,

(15) {\mathbf{\kappa }}^{e{\mathrm{\Delta }}_{\mathrm{1}}}=\underset{{\mathbf{B}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}}{\underbrace{\left[\begin{array}{lll}\frac{\mathrm{1}}{\mathrm{3}}{\mathbf{b}}_{\mathrm{1}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}+{\mathbf{b}}_{\mathrm{2}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}& {\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\mathbf{b}}_{\mathrm{1}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}+{\mathbf{b}}_{\mathrm{3}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}& {\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\mathbf{b}}_{\mathrm{1}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}\end{array}\right]}}\left[\begin{array}{l}{\mathbf{d}}_{\mathrm{1}}^e\\ {\mathbf{d}}_{\mathrm{2}}^e\\ {\mathbf{d}}_{\mathrm{3}}^e\end{array}\right]={\mathbf{B}}^{b{\mathrm{\Delta }}_{\mathrm{1}}}{\mathbf{d}}^e,

(16) {\mathbf{\gamma }}^{e{\mathrm{\Delta }}_{\mathrm{1}}}=\underset{{\mathbf{B}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}}{\underbrace{\left[\begin{array}{lll}\frac{\mathrm{1}}{\mathrm{3}}{\mathbf{b}}_{\mathrm{1}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}+{\mathbf{b}}_{\mathrm{2}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}& {\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\mathbf{b}}_{\mathrm{1}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}+{\mathbf{b}}_{\mathrm{3}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}& {\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\mathbf{b}}_{\mathrm{1}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}\end{array}\right]}}\left[\begin{array}{l}{\mathbf{d}}_{\mathrm{1}}^e\\ {\mathbf{d}}_{\mathrm{2}}^e\\ {\mathbf{d}}_{\mathrm{3}}^e\end{array}\right]={\mathbf{B}}^{s{\mathrm{\Delta }}_{\mathrm{1}}}{\mathbf{d}}^e.

Similarly, by using cyclic permutation, we easily obtain the membrane, bending and shear strains {\epsilon }_m^{e{\mathrm{\Delta }}_j} , {\mathbf{\kappa }}^{e{\mathrm{\Delta }}_j} , {\mathit{\gamma }}^{e{\mathrm{\Delta }}_j} and matrices {\mathbf{B}}^{m{\mathrm{\Delta }}_j} , {\mathbf{B}}^{b{\mathrm{\Delta }}_j} , {\mathbf{B}}^{s{\mathrm{\Delta }}_j} , j=\mathrm{2},\mathrm{3} , for the second sub-triangle {\mathrm{\Delta }}_{\mathrm{2}}^{ } (triangle O-2-3) and third sub-triangle {\mathrm{\Delta }}_{\mathrm{3}}^{ } (triangle O-3-1), respectively. Then, the strain smoothing technique is applied on the whole triangular element to smooth the strains. The elemental smoothed strains {\tilde{\mathit{\epsilon }}}_m^e , {\tilde{\kappa }}^e and {\tilde{\mathit{\gamma }}}^e are then expressed by

(17) {\tilde{\mathit{\epsilon }}}_m^e={\tilde{\mathbf{B}}}_m^e{\mathbf{d}}^e; {\tilde{\kappa }}^e={\tilde{\mathbf{B}}}_b^e{\mathbf{d}}^e; {\tilde{\mathit{\gamma }}}^e={\tilde{\mathbf{B}}}_s^e{\mathbf{d}}^e,

where {\tilde{\mathbf{B}}}_m^e, {\tilde{\mathbf{B}}}_b^e, {\tilde{\mathbf{B}}}_s^e are the smoothed strain gradient matrices, respectively, given by

(18) {\tilde{\mathbf{B}}}_i^e={\displaystyle \sum _{j=\mathrm{1}}^{\mathrm{3}}{\displaystyle \underset{{\mathrm{\Delta }}_j}{\int }{\mathbf{B}}_i^{{\mathrm{\Delta }}_j}{\mathrm{\Phi }}_e(\mathbf{x})d\mathrm{\Omega }};i=m,b,s};j=1,2,3,

in which {\mathrm{\Phi }}_e(\mathbf{x}) is a smoothing function presented as

(19) {\mathrm{\Phi }}_e(\mathbf{x})=\{\begin{array}{ll}\mathrm{1}/A_e\hfill & \mathbf{x}\in {\mathrm{\Omega }}_e\hfill \\ \mathrm{0}\hfill & \mathbf{x}\notin {\mathrm{\Omega }}_e\hfill \end{array},

with A^e is the area of the whole triangular element; {\mathbf{B}}_i^{{\mathrm{\Delta }}_j} is the strain gradient matrix with respect to the sub-triangular element {\mathrm{\Delta }}_j .

By substituting Eq. (17) to the general weakform of the Reissner−Mindlin shells in Eq. (2), the governing equation for free vibration analysis becomes

(20) (\mathbf{K}\mathbf{-}{\omega }^{\mathrm{2}}\mathbf{M})\mathbf{d}=\mathbf{0},

where \omega is the natural frequency; d is the nodal unknown vector of variables; \mathbf{M} \mathrm{and} \mathbf{K} are the global mass and stiffness matrices computed by assembling the elemental mass and stiffness matrices as

(21) {\mathbf{M}}^e={\mathbf{T}}^T{\displaystyle {\int }_{{\mathrm{\Omega }}_e}{\left(\mathbf{N}\right)}^T\mathbf{m}\mathbf{N}\mathrm{d}\mathrm{\Omega }}\mathbf{T},

(22) {\mathbf{K}}^e={\mathbf{T}}^T\left({\displaystyle {\int }_{{\mathrm{\Omega }}_e}{\tilde{\mathbf{B}}}_m^e{\mathbf{D}}_m_T{\tilde{\mathbf{B}}}_m^e\mathrm{d}\mathrm{\Omega }+} {\displaystyle {\int }_{{\mathrm{\Omega }}_e}{\tilde{\mathbf{B}}}_b^e{\mathbf{D}}_b_T{\tilde{\mathbf{B}}}_b^e\mathrm{d}\mathrm{\Omega }+}{\displaystyle {\int }_{{\mathrm{\Omega }}_e}{\tilde{\mathbf{B}}}_s^e{\mathbf{D}}_s_T{\tilde{\mathbf{B}}}_s^e\mathrm{d}\mathrm{\Omega }}\right)\mathbf{T},

where T is the transformation matrix of coordinates from the local coordinate system oxyz to the global coordinate system OXYZ as shown in Fig. 4.

It was suggested [ 66] that a stabilization term should be added to improve the accuracy of approximate solution and to stabilize shear force oscillations. In this modification, {\mathbf{D}}_s in Eq. (22) is replaced by {\widehat{\mathbf{D}}}_s as

(23) {\widehat{\mathbf{D}}}_s={\displaystyle \frac{k{h}^{\mathrm{3}}}{h^{\mathrm{2}}+\alpha h_e^{\mathrm{2}}}}\left[\begin{array}{ll}\mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}\end{array}\right],

in which h_e^{ } is the longest length of the element and \alpha is a positive constant [ 67].

From Eqs. (11) − (17) and (22), it can be seen that the values of stiffness matrix at the drilling degree of freedom {\beta }_z is equal to zero. This causes the singularity to the global stiffness matrix when all elements meeting at a node are coplanar. To overcome this problem, these null value in the stiffness corresponding to the drilling degree of freedom are replaced by approximate values which equal to 10⁻³ times the maximum diagonal value in the element stiffness matrix [ 60]. It also can be observed that the element stiffness matrix in the CS-FEM-DSG3 is no longer depend on the sequence of nodes in elements as in the conventional DSG3. This helps improve accuracy as well as the stability of the DSG3, especially for the coarse or distorted meshes.

A brief on extended finite element method (X-FEM) and its integration to CS-FEM-DSG3

Based on the idea of partition unity of Babuška et al. [ 68], Belytschko and Black [ 6] proposed the extended finite element method (X-FEM) for solving linear elastic fracture mechanic problems. The key point of the X-FEM is adding discontinuous and singular enrichment functions besides conventional expansion of the displacement field in order to capture the effects of cracks or other local discontinuities within the mesh. In the XFEM, the mesh is first generated normally without considering the presence of cracks or discontinuities. This helps overcome the difficulties of remeshing as in conventional FEM when solving problems related to crack, especially crack propagation. Then, the cracks or other discontinuities are modeled by the generalized Heaviside function and singularly asymptotic functions. This section will present a brief overview of the X-FEM for shells. Further formulation can be found at [ 6, 69− 72] for interested readers.

In general, the field variables are approximated by

(24) {\mathbf{u}}^h(\mathbf{x})={\displaystyle \sum _{i\in N^{\mathrm{FEM}}}{\mathbf{N}}_i}(\mathbf{x}){\mathbf{d}}_i+\mathrm{e}\mathrm{n}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{s},

where {\mathbf{N}}_i(\mathbf{x}) are standard finite element shape functions; {\mathbf{d}}_i are nodal variables associated with i^th node; N^FEM is the number of nodes in the whole mesh.

For cracked shells, the finite element approximation of shell displacements and section rotations is expressed in the form [ 73] of

(25) (u^h,v^h,w^h)(\mathbf{x})=\underset{u^h,v^h,w_{\mathrm{.cont}}^h}{\underbrace{\sum _{i\in N^{\mathrm{FEM}}}{N}_i(\mathbf{x})(u_i,v_i,w_i)}}+\underset{u^h,v^h,w_{\mathrm{.split}}^h}{\underbrace{\sum _{j\in N^H}{N}_j(\mathbf{x})H(\mathbf{x})(b_j^u,b_j^v,b_j^w)}}+\underset{u^h,v^h,w_{\mathrm{.tip}}^h}{\underbrace{\sum _{k\in N^B}{N}_k(\mathbf{x})\left({\displaystyle \sum _{l=1}^4{G}_l(r,\theta )(c_{\mathrm{kl}}^u,c_{\mathrm{kl}}^v,c_{\mathrm{kl}}^w)}\right)}},

(26) ({\beta }_x^h,{\beta }_y^h,{\beta }_z^h)(\mathbf{x})=\underset{{\beta }_x^h,{\beta }_y^h,{\beta }_{z.\mathrm{cont}}^h}{\underbrace{\sum _{i\in N^{\mathrm{FEM}}}{N}_i(\mathbf{x})({\beta }_{xi},{\beta }_{yi},{\beta }_{zi})}}+\underset{{\beta }_{x,}^h,{\beta }_y^h{\beta }_{z\mathrm{.split}}^h}{\underbrace{\sum _{j\in N^H}{N}_j(\mathbf{x})H(\mathbf{x})(b_j^{{\beta }_x},b_j^{{\beta }_y},b_j^{{\beta }_z})}}+\underset{{\beta }_x^h,{\beta }_y^h{\beta }_{z\mathrm{.tip}}^h}{\underbrace{\sum _{k\in N^B}{N}_k(\mathbf{x})\left({\displaystyle \sum _{l=1}^4{F}_l(r,\theta )(c_{kl}^{{\beta }_x},c_{kl}^{{\beta }_y},c_{kl}^{{\beta }_z})}\right)}}.

In Eqs. (25) and (26), {\mathbf{u}}_{\mathrm{cont}}={\left[\begin{array}{llllll}{u}_{\mathrm{.cont}}^h\hfill & v_{\mathrm{.cont}}^h\hfill & w_{\mathrm{.cont}}^h\hfill & {\beta }_{x\mathrm{.cont}}^h\hfill & {\beta }_{y\mathrm{.cont}}^h\hfill & {\beta }_{z\mathrm{.cont}}^h\hfill \end{array}\right]}^{\mathrm{T}} is the continuous displacement and rotation fields; {\mathbf{u}}_{\mathrm{split}}={\left[\begin{array}{llllll}{u}_{\mathrm{.split}}^h\hfill & v_{\mathrm{.split}}^h\hfill & w_{\mathrm{.split}}^h\hfill & {\beta }_{x\mathrm{.split}}^h\hfill & {\beta }_{y\mathrm{.split}}^h\hfill & {\beta }_{z\mathrm{.split}}^h\hfill \end{array}\right]}^{\mathrm{T}} is the displacement and rotation field of elements split by cracks; {\mathbf{u}}_{\mathrm{tip}}={\left[\begin{array}{llllll}{u}_{\mathrm{.tip}}^h\hfill & v_{\mathrm{.tip}}^h\hfill & w_{\mathrm{.tip}}^h\hfill & {\beta }_{x\mathrm{.tip}}^h\hfill & {\beta }_{y\mathrm{.tip}}^h\hfill & {\beta }_{z\mathrm{.tip}}^h\hfill \end{array}\right]}^{\mathrm{T}} is displacement and rotation field of elements containing crack tips. N^FEM, N^H, N^B are, respectively, the total number of nodes of the mesh, number of enriched nodes whose support is split by cracks and number of enriched nodes whose support contains crack tips as shown in Fig. 5; {\mathbf{d}}_i={\left[\begin{array}{llllll}{u}_i\hfill & v_i\hfill & w_i\hfill & {\beta }_{xi}\hfill & {\beta }_{yi}\hfill & {\beta }_{zi}\hfill \end{array}\right]}^{\mathrm{T}} is the nodal unknown vector associated with the continuous part of the finite solution, {\mathbf{b}}_j={\left[\begin{array}{llllll}{b}_j^u\hfill & b_j^v\hfill & b_j^w\hfill & b_j^{{\beta }_x}\hfill & b_j^{{\beta }_y}\hfill & b_j^{{\beta }_z}\hfill \end{array}\right]}^{\mathrm{T}} is the nodal enriched degree of freedom vector related to Heaviside function H(\mathbf{x}) , and {\mathbf{c}}_{kl}={\left[\begin{array}{llllll}{c}_{kl}^u\hfill & c_{kl}^v\hfill & c_{kl}^w\hfill & c_{kl}^{{\beta }_x}\hfill & c_{kl}^{{\beta }_y}\hfill & c_{kl}^{{\beta }_z}\hfill \end{array}\right]}^{\mathrm{T}} is the nodal enriched degree of freedom vector associated with the elastic asymptotic crack-tip functions F_l(r,\theta ) or G_l(r,\theta ) .

The first term of enrichment involves a Heaviside jump function H(x) whose value is+ 1 above the crack and −1 below the crack and can be expressed as

(27) H(\mathbf{x})=\{\begin{array}{ll}+\mathrm{1},\hfill & \mathrm{if} (\mathbf{x}-\mathbf{x}*)\cdot \mathbf{n}\ge \mathrm{0};\hfill \\ -\mathrm{1},\hfill & \mathrm{otherwise},\hfill \end{array}

where x is a sample (or Gauss) point; x* (lies on the crack) is the closest point to x; and n is the unit outward normal to the crack at x* as shown in Fig. 6.

The second additional part of the enriched displacement field involves set of branch functions G_l,F_l to model the asymptotic features of the displacement field at the crack tip and defined by

(28) \begin{array}{l}{G_l(r,\theta )|}_{l=\mathrm{1}}^{\mathrm{4}}\equiv \left\{r^{\mathrm{3}/\mathrm{2}}\sin {\displaystyle \frac{\theta }{\mathrm{2}}},r^{\mathrm{3}/\mathrm{2}}\cos {\displaystyle \frac{\theta }{\mathrm{2}}},r^{\mathrm{3}/\mathrm{2}}\sin {\displaystyle \frac{\mathrm{3}\theta }{\mathrm{2}}},r^{\mathrm{3}/\mathrm{2}}\cos {\displaystyle \frac{\mathrm{3}\theta }{\mathrm{2}}}\right\}\hfill \\ {F_l(r,\theta )|}_{l=\mathrm{1}}^{\mathrm{4}}\equiv \left\{r\sin {\displaystyle \frac{\theta }{\mathrm{2}}},r\cos {\displaystyle \frac{\theta }{\mathrm{2}}},r\sin {\displaystyle \frac{\theta }{\mathrm{2}}}\sin \theta ,r\cos {\displaystyle \frac{\theta }{\mathrm{2}}}\sin \theta \right\}\hfill \end{array},

where (r,\theta ) are polar coordinates in the local coordinate system with the origin at the crack tip.

When these enrichment functions are implemented to the CS-FEM-DSG3, the governing equation for free vibration of cracked shell now becomes

(29) (\mathbf{K}\mathbf{-}{\omega }^{\mathrm{2}}\mathbf{M})e=\mathbf{0},

where \omega is the natural frequency; \mathbf{e}=\left[\begin{array}{lll}\mathbf{d}\hfill & {\mathbf{b}}_j\hfill & {\mathbf{c}}_{kl}\hfill \end{array}\right] is the nodal unknown vector of variables (for both classical and enriched ones). \mathbf{M}\mathrm{and}\mathbf{K} are the global mass, stiffness matrices computed by assembling the element mass and stiffness matrices. For standard elements (unenriched elements), these {\mathbf{M}}^e\mathrm{and}{\mathbf{K}}^e are normally calculated using the CS-FEM-DSG3 as stated in Section 3.1. For elements associated with cracks (enriched elements), these matrices are defined by follows. Note that in the following equations, superscript d, b, c are respectively present for matrices related to un-enriched nodal degree of freedom, nodal degree of freedom enriched by Heaviside function b_j and nodal degree of freedom enriched by elastic asymptotic crack-tip functions c_kl.

(30) M^e=\left[\begin{array}{lll}{M}^{dd}& M^{db}& M^{dc}\\ M^{bd}& M^{bb}& M^{bc}\\ M^{cd}& M^{cb}& M^{cc}\end{array}\right];\hspace{1em}{\mathbf{K}}^e=\left[\begin{array}{lll}{\mathbf{K}}^{dd}& {\mathbf{K}}^{db}& {\mathbf{K}}^{dc}\\ {\mathbf{K}}^{dc}& {\mathbf{K}}^{bb}& {\mathbf{K}}^{bc}\\ {\mathbf{K}}^{cd}& {\mathbf{K}}^{cb}& {\mathbf{K}}^{cc}\end{array}\right],

in which

(31) {\mathbf{M}}^{rs}={\displaystyle {\int }_{{\mathrm{\Omega }}_e}{({\mathbf{N}}^r)}^T\mathbf{m} {\mathbf{N}}^{s}h\mathrm{d}\mathrm{\Omega }} (r,s=d,b,c),

(32) {\mathbf{K}}^{rs}={\displaystyle \underset{{\mathrm{\Omega }}_e}{\int }{\left({\tilde{\mathbf{B}}}_m^r\right)}^T{\mathbf{D}}_m{\tilde{\mathbf{B}}}_m^{r}d\mathrm{\Omega }}+{\displaystyle \underset{{\mathrm{\Omega }}_e}{\int }{\left({\tilde{\mathbf{B}}}_b^s\right)}^T{\mathbf{D}}_b{\tilde{\mathbf{B}}}_b^{s}d\mathrm{\Omega }} +{\displaystyle \underset{{\mathrm{\Omega }}_e}{\int }{\left({\tilde{\mathbf{S}}}^r\right)}^T{\mathbf{D}}_s{\tilde{\mathbf{S}}}^{s}d\mathrm{\Omega }}(r,s=d,b,c),

where {\mathrm{\Omega }}_e indicates the domain of the element; {\mathbf{N}}^{r,s} are the matrices of shape functions given by

(33) \begin{array}{l}{\mathbf{N}}^{\mathit{d}}=\left[\begin{array}{lll}{\mathbf{N}}_{\mathrm{1}}& {\mathbf{N}}_{\mathrm{2}}& {\mathbf{N}}_{\mathrm{3}}\end{array}\right];{\mathbf{N}}^b=\left[\begin{array}{lll}{\mathbf{N}}_{\mathrm{1}}H& {\mathbf{N}}_{\mathrm{2}}H& {\mathbf{N}}_{\mathrm{3}}H\end{array}\right];\hfill \\ {\mathbf{N}}^{\mathit{c}}=\left[\begin{array}{llll}{\mathbf{N}}^{c\mathrm{1}}\hfill & {\mathbf{N}}^{c\mathrm{2}}\hfill & {\mathbf{N}}^{c\mathrm{3}}\hfill & {\mathbf{N}}^{c\mathrm{4}}\hfill \end{array}\right] \mathrm{where} {\mathbf{N}}^{c\alpha }=\left[\begin{array}{lll}{\mathbf{N}}_{\mathrm{1}}{F}_{\alpha }& {\mathbf{N}}_{\mathrm{2}}{F}_{\alpha }& {\mathbf{N}}_{\mathrm{3}}{F}_{\alpha }\end{array}\right];\alpha =\mathrm{1}÷\mathrm{4}\hfill \end{array}.

{\tilde{B}}_m^r,{\tilde{B}}_m^r,{\tilde{\mathbf{S}}}^r,r=d,b,c , are the matrices of shape function derivatives related to membrane, bending and shear deformation calculated as followings. Note that in these equations, the notation A\circ B represents Hadamard product of the two matrices A and B.

(34) {\mathbf{B}}_m^d={\tilde{\mathbf{B}}}_m^e;{\mathbf{B}}_m^b=\mathbf{H}\circ {\tilde{\mathbf{B}}}_m^e;{\mathbf{B}}_m^c=\left[\begin{array}{llll}{\mathbf{F}}_{\mathrm{1}}\circ {\tilde{\mathbf{B}}}_m^e& {\mathbf{F}}_{\mathrm{2}}\circ {\tilde{\mathbf{B}}}_m^e& {\mathbf{F}}_{\mathrm{3}}\circ {\tilde{\mathbf{B}}}_m^e& {\mathbf{F}}_{\mathrm{4}}\circ {\tilde{\mathbf{B}}}_m^e\end{array}\right],

(35) {\mathbf{B}}_b^d={\tilde{\mathbf{B}}}_b^e;{\mathbf{B}}_b^b=\mathbf{H}\circ {\tilde{\mathbf{B}}}_b^e;{\mathbf{B}}_b^c=\left[\begin{array}{llll}{\mathbf{F}}_{\mathrm{1}}\circ {\tilde{\mathbf{B}}}_b^e\hfill & {\mathbf{F}}_{\mathrm{2}}\circ {\tilde{\mathbf{B}}}_b^e\hfill & {\mathbf{F}}_{\mathrm{3}}\circ {\tilde{\mathbf{B}}}_b^e\hfill & {\mathbf{F}}_{\mathrm{4}}\circ {\tilde{\mathbf{B}}}_b^e\hfill \end{array}\right],

(36) {\mathbf{S}}^d={\tilde{\mathbf{B}}}_s^e;{\mathbf{S}}^b=\mathbf{S}\circ {\tilde{\mathbf{B}}}_s^e;{\mathbf{S}}^c=\left[\begin{array}{llll}{\mathbf{G}}_{\mathrm{1}}\circ {\tilde{\mathbf{B}}}_s^e\hfill & {\mathbf{G}}_{\mathrm{2}}\circ {\tilde{\mathbf{B}}}_s^e\hfill & {\mathbf{G}}_{\mathrm{3}}\circ {\tilde{\mathbf{B}}}_s^e\hfill & {\mathbf{G}}_{\mathrm{4}}\circ {\tilde{\mathbf{B}}}_s^e\hfill \end{array}\right],

where H,F_{\alpha },S,G_{\alpha };\alpha =1÷4 are matrices containing partial derivative of enrichment functions which is defined as follow:

(37) \mathbf{H}=\left[\begin{array}{l}{H}_{,x}\\ \mathrm{0}\\ H_{,y}\end{array} \begin{array}{l}\mathrm{0}\\ H_{,y}\\ H_{,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{H}_{,x}\\ \mathrm{0}\\ H_{,y}\end{array} \begin{array}{l}\mathrm{0}\\ H_{,y}\\ H_{,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{H}_{,x}\\ \mathrm{0}\\ H_{,y}\end{array} \begin{array}{l}\mathrm{0}\\ H_{,y}\\ H_{,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{H}_{,x}\\ \mathrm{0}\\ H_{,y}\end{array} \begin{array}{l}\mathrm{0}\\ H_{,y}\\ H_{,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{H}_{,x}\\ \mathrm{0}\\ H_{,y}\end{array} \begin{array}{l}\mathrm{0}\\ H_{,y}\\ H_{,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{H}_{,x}\\ \mathrm{0}\\ H_{,y}\end{array} \begin{array}{l}\mathrm{0}\\ H_{,y}\\ H_{,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array}\right],

(38) {\mathbf{F}}_{\alpha }=\left[\begin{array}{l}{F}_{\alpha ,x}\\ \mathrm{0}\\ F_{\alpha ,y}\end{array} \begin{array}{l}\mathrm{0}\\ F_{\alpha ,y}\\ F_{\alpha ,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{F}_{\alpha ,x}\\ \mathrm{0}\\ F_{\alpha ,y}\end{array} \begin{array}{l}\mathrm{0}\\ F_{\alpha ,y}\\ F_{\alpha ,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{F}_{\alpha ,x}\\ \mathrm{0}\\ F_{\alpha ,y}\end{array} \begin{array}{l}\mathrm{0}\\ F_{\alpha ,y}\\ F_{\alpha ,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{F}_{\alpha ,x}\\ \mathrm{0}\\ F_{\alpha ,y}\end{array} \begin{array}{l}\mathrm{0}\\ F_{\alpha ,y}\\ F_{\alpha ,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{F}_{\alpha ,x}\\ \mathrm{0}\\ F_{\alpha ,y}\end{array} \begin{array}{l}\mathrm{0}\\ F_{\alpha ,y}\\ F_{\alpha ,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{F}_{\alpha ,x}\\ \mathrm{0}\\ F_{\alpha ,y}\end{array} \begin{array}{l}\mathrm{0}\\ F_{\alpha ,y}\\ F_{\alpha ,x}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array}\right],

(39) \mathbf{S}=\left[\begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{H}_{,x}\\ H_{,y}\end{array} \begin{array}{l}H\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ H\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{H}_{,x}\\ H_{,y}\end{array} \begin{array}{l}H\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ H\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{H}_{,x}\\ H_{,y}\end{array} \begin{array}{l}H\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ H\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array}\right],

(40) {\mathbf{G}}_{\alpha }=\left[\begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{G}_{\alpha ,x}\\ G_{\alpha ,y}\end{array} \begin{array}{l}{F}_{\alpha }\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ F_{\alpha }\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{G}_{\alpha ,x}\\ G_{\alpha ,y}\end{array} \begin{array}{l}{F}_{\alpha }\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ F_{\alpha }\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array} \begin{array}{l}{G}_{\alpha ,x}\\ G_{\alpha ,y}\end{array} \begin{array}{l}{F}_{\alpha }\\ \mathrm{0}\end{array} \begin{array}{l}\mathrm{0}\\ F_{\alpha }\end{array} \begin{array}{l}\mathrm{0}\\ \mathrm{0}\end{array}\right],

in which {\tilde{\mathbf{B}}}_m^e,{\tilde{\mathbf{B}}}_b^e\mathrm{and}{\tilde{\mathbf{B}}}_s^e are strain gradient matrices related to membrane, bending and shear deformation of triangular element which are calculated as Eq. (18).

Methodology of the coupling XCS-FEM-DSG3

As presented above, the CS-FEM-DSG3 is extended for free vibration analysis of cracked FGM shells by integrating the CS-FEM-DSG3 itself with discontinuous and crack−tip singular enrichment functions of the X-FEM. This novel element is hence called the extended cell-based smoothed discrete shear gap shell element (XCS-FEM-DSG3).

In summary of the formulation of XCS-FEM-DSG3 for cracked shell, first, the mesh is generated independently of cracks location. After that, the formulation of CS-FEM-DSG3 is used to calculate strain gradient matrices {\tilde{\mathbf{B}}}_m,{\tilde{\mathbf{B}}}_b,{\tilde{\mathbf{B}}}_s of all elements without any notion about cracks as in Eq. (18). Then, these matrices are enriched with discontinuous Heaviside function or elastic asymptotic crack-tip functions appropriately as in Eqs. (34)-(36) to calculate {\tilde{\mathbf{B}}}_m^r,{\tilde{\mathbf{B}}}_m^r,{\tilde{\mathbf{S}}}^r;r=d,b,c . The global mass, force and stiffness matrices \mathbf{M},\mathbf{K} are then computed using Eqs. (30)-(32).

In numerical implementation of XCS-FEM-DSG3, these strain gradient and global matrices are computed at the Gauss points using numerical Gauss quadrature. As a result, it is necessary to define the number of necessary Gauss points for each type of elements as shown in Fig. 5. In general, the elements can be classified into five types such as

1) Tip element is the element containing a crack tip. All nodes belonging to a tip element are enriched with the near-tip fields of Eq. (28).

2) Split elements are elements that completely cut by the crack. Their nodes are enriched by the discontinuous Heaviside function displayed in Eq. (27).

3) Tip blending elements are elements which is around tip elements. Some of these elements’ nodes are enriched with the near-tip fields and others are not enriched at all.

4) Split blending elements are elements neighboring split elements. Some of these elements’ nodes are enriched with the discontinuous function, and others are not enriched at all.

5) Standard elements are elements that are in none of the above categories. None of their associated nodes are enriched.

In the numerical implementation [ 74], the tip elements and split elements are first subdivided into 3 or 5 sub-triangles as shown in Fig. 7. Then the number of necessary Gauss points for five types of elements, as shown in Fig. 8, can be defined as below:

Tip elements: 65 Gauss points for each triangular sub-element.

Split elements: 3 Gauss points for each triangular sub-element.

Tip blending elements: 13 Gauss points.

Split blending elements: 1 Gauss point.

Standard elements: 1 Gauss point.

Numerical results

In this section, a cracked shell with geometry as shown in Fig. 9 with different crack locations is investigated to illustrate the robustness and effectiveness of the XCS-FEM-DSG3 for free vibration of cracked shell analysis. Material parameters of the shell is given by the elastic modulus E=\mathrm{2.1}\times {\mathrm{10}}^{\mathrm{11}}(N/m^{\mathrm{2}}) , Poisson’s ratio \nu =\mathrm{0.3} and material mass density \rho =\mathrm{7860}(\mathrm{kg}/m^{\mathrm{3}}) . For each crack location, free vibration behaviors of the shell are analyzed and compared to that of the commerical software ANSYS in order to confirm the accuracy of XCS-FEM-DSG3. In the analysis performed by the ANSYS, the “shell 63” element is used to model and analyze the problem.

Free vibration analysis of cylindrical shell with middle crack along y-axis

This section investigates the free vibration behavior of a cracked shell possesing middle crack along y-axis as shown in Fig. 10 and compares to that of ANSYS sofware to illustrate the robustness of XCS-FEM-DSG3 in free vibration analysis of cracked shell. The shell behavior with different crack lengths is analyzed to further investigate the effect of crack length on shell structures’ vibration response.

The results of first five natural frequency values with different values of crack lengths are shown in Table 1. From the table, it can be seen that results by the XCS-FEM-DSG3 agree very well with those by commercial software ANSYS. In fact, all relative errors for cases of crack lengths are less than 5%. It can also be observed from the table that crack length significantly affects free vibration behavior of cracked shell. The natural frequency values decrease largely when crack lengths increase for all vibration modes. This phenomenon is understandable because the presence and propagation of crack cause the decrease of shell’s stiffness which results in decrease of natural frequency values. This hence illustrates the robustness and reliability of XCS-FEM-DSG3 for free vibration analysis of cracked cylindrical shell.

The results of free vibration analysis with crack length l = 0.3 m at different mesh sizes and crack location are presented in Table 2 to investigate the convergence of XCS-FEM-DSG3. From the table, it can be seen that when the mesh size increases, the natural frequency values decrease in a stable rate. This again illustrates the stability and reliability of XCS-FEM-DSG3. At the same mesh size of 40 × 40, the results by XCS-FEM-DSG3 are pretty worse than that of ANSYS, but the results brought by XCS-FEM-DSG3 are very close to those by ANSYS at the mesh 60 × 60. It’s worthy to note that the XCS-FEM-DSG3 uses only low order triangular elements, while ANSYS uses the nine-node quarter-point quadratic element which is a high order quadratic element. As a result, it can be seen that XCS-FEM-DSG3 is a simple and robust method for free vibration analysis of cracked cylindrical shell.

The first three vibration mode shapes of both intact shell and cracked shell with crack length l = 0.3 m are presented respectively in Fig. 11 and Fig. 12 in comparison with those by commercial software ANSYS. From the figures, it can be seen that vibration mode shapes are changed considerably with the presence of crack. This shows that the crack has significant impact on cylindrical shell’s vibration behavior. As a result, the presence of crack should be taken into account seriously in process of analysis for shell structures. It also can be seen that the free vibration mode shapes by XCS-FEM-DSG3 are closely similar to those by ANSYS in all cases. This further illustrates the suitability and accuracy of XCS-FEM-DSG3 for free vibration analysis of cracked cylindrical shell.

Free vibration analysis of cylindrical shell with side crack along y-axis

Free vibration behavior of the cracked cylindrical shell with side crack along x-axis as shown in Fig. 13 is examined. The numerical results are compared with those by ANSYS to illustrate the effectiveness of XCS-FEM-DSG3 in solving free vibration of shell strucutres.

The first five natural frequencies of side cracked shell with different crack lengths are presented in Table 3 in comparison with those by ANSYS. As shown in the table, all relative error values between results by XCS-FEM-DSG3 and those by ANSYS is smaller than 5%. This further illustrates the reliability and accuracy of XCS-FEM-DSG3 in analyzing free vibration behavior of crack shell. In good agreement with the case of middle crack along y-axis, the presence and propagation of crack cause the frequency values to decrease significantly. This illustrates the important influence of crack on shell structures’ free vibration behavior.

The frequency values of first five vibration modes are presented in Table 4 with crack length 0.3 and different mesh sizes. As seen in the table, the results converge in a stable rate. With the same mesh size, the difference between results by the XCS-FEM-DSG3 and those by ANSYS are relatively small. In addition, the first three vibration mode shapes are shown in Fig. 14 in comparison to those by ANSYS. It can be seen that the mode shapes are significantly different to mode shapes of the intact structure. This is because the presence of crack at shell’s side significantly cause shell stiffness to decrease considerably right at crack location. It is this localized decrease that cause free vibration modes to be asymetric which far different from that of the intact structure. This further confirms the considerable impact of crack on cylindrical shell. As presented in the figure, the mode shapes by XCS-FEM-DSG3 agree well with that of ANSYS. The accuracy and reliability of XCS-FEM-DSG3 in free vibration analysis for cracked cylindrical shell is hence further verified.

Free vibration analysis of cylindrical shell with middle crack along x-axis

The final numerical example is the cylindrical shell with middle crack along x-axis as shown in Fig. 15. As in two examples presented above, free vibration behavior of the shell is analyzed with various crack length l=\mathrm{0.1}÷\mathrm{0.3} m to further investigate the effect of crack on shell structures.

The first five frequency values are shown in Table 5 compared to that from ANSYS software for different cases of crack length. For this case, almost all the relative errors between XCS-FEM-DSG3 and ANSYS are less than 3%. The difference is understandable because the XCS-FEM-DSG3 used only low order triangular element while ANSYS used nine-node quarter-point quadratic elements. As predicted, the high order elements used by ANSYS would give better results compared to lower order elements. This one more time strengthens the conclusion that XCS-FEM-DSG3 is a simple, robust and appropriate method for free vibration analysis of cracked cylindrical shell.

With different mesh size, the first five frequency values of the XCS-FEM-DSG3 are presented as in Table 6 and compared to that of ANSYS. The stability of XCS-FEM-DSG3 is further proven by the fact that the change rate of natural frequency values through different mesh sizes is stable for all vibration modes. Figure 16 demonstrates the first three mode shapes of cylindrical shell with crack length l= 0.3 m compared with that of ANSYS. It can be seen that the mode shapes with presence of crack along x-axis are slightly different to that of intact structures. However, the difference is still considerable, especially for mode 1. This shows that middle crack along x-axis might has less impact on free vibration behavior of cylindrical shell than middle and side crack along y-axis. The good agreement in mode shapes between XCS-FEM-DSG3 and ANSYS in Fig. 16 illustrates the accuracy and reliability of XCS-FEM-DSG3 for free vibration analysis of cracked cylindrical shell.

Conclusion

In the paper, the XFEM is integrated into the CS-FEM-DSG3 to give a so-called XCS-FEM-DSG3 for free vibration analysis of cracked cylindrical shell structures. In the process of analysis, the stiffness matrix of uncut elements is formulated by the conventional CS-FEM-DSG3, while stiffness matrix of elements cut by crack is calculated by the XCS-FEM-DSG. In the formulation of XCS-FEM-DSG3, the discontinuity and localization of crack in shell structures are formulated by enrichment functions. Three numerical example of free vibration analysis of cracked cylindrical shell with different crack locations and crack lengths are solved and compared with the commercial software ANSYS to illustrate the accuracy and reliability of XCS-FEM-DSG3. The numerical results show that the XCS-FEM-DSG3 is a simple, robust and appropriate method for free vibration analysis of cracked Reissner-Mindlin cylindrical shells.

In general, the novel method possesses three main advantages including: (i) the mesh is generated independent of cracks which helps avoid remeshing as in conventional FEM; (ii) the presence of CS-FEM-DSG3 in analyzing the cylindrical shell’s behaviors helps overcome the shearlocking phenomenon and improve the accuracy of solutions considerably. This is due to softening effect of the cell-based strain smoothing technique; and (iii) the method is very flexible to apply for arbitrary complicated geometric domains due to using only three-node triangular elements which can be easily generated automatically.

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