Abdel Wahab, Magd

Phu Nguyen, Vinh

Phung Van, Phuc

Tran Vinh, Loc

English

Ghent University Academic Bibliography

International Journal of Fracture Fatigue and Wear, Volume 2 127 Proceedings of the 3rd International Conference on  Fracture Fatigue and Wear, pp. 127-133, 2014VIBRATION ANALYSIS OF CRACKED PLATE USING HIGHER-ORDER SHEAR DEFORMATION THEORY  Loc V. Tran1, P. Phung-Van1,3, Vinh Phu Nguyen2, M. Abdel Wahab3 and H. Nguyen-Xuan 1 1 Division of Computational Mechanics, Ton Duc Thang University Ho Chi Minh City, Vietnam 2Cardi_University, Queen’s Buildings, The Parade, Cardi_CF24 3AA, Wales, UK 3Department of Mechanical Construction and Production, Faculty of Engineering and Architecture, Ghent University, Belgium Abstract: A novel and effective formulation that combines the eXtended IsoGeometric Approach (XIGA) and Higher-order Shear Deformation Theory (HSDT) is proposed to study the free vibration of cracked plates. XIGA utilizes the Non-Uniform Rational B-Spline (NURBS) functions with their inherent arbitrary high order smoothness, which permit the C1 requirement of the HSDT model. Two numerical examples are provided to show excellent performance of the proposed method compared with other published solutions in the literature.Keywords:  cracked plate; non-uniform rational b-spline; isogeometric analysis; higher-order shear deformation theory 1 INTRODUCTION  Plate structures play an increasing important role in engineering applications. They thus were researched since long time from static to dynamic and buckling analysis [1-5]. From the literature, these works are carried out for designing the plate structures without the presences of cracks or flaws. It is known that in service, the cracks are generated and grown when subjected to large cyclic loading. They lead to a reduction of the load carrying capacity of the structures. To clearly understand the dynamic behaviour of plate with initial cracks is hence too necessary. This problem has been interesting by many scientists with various methods: Stahl and Keer [6] and Liew et al. [7] with analytical solutions, Finite Element Method (FEM) [9], eXtended Finite Element Method (XFEM) [8,10], etc. Among them, XFEM is known as a robust numerical method, which uses enrichment functions to model discontinuities independent of the finite element mesh. Recently, Natarajan et al. [11] extended this method for dynamic analysis of FGM plate based on the FSDT model. This model is simple to implement and is applicable to both thick and thin FGM plates. However, the accuracy of solutions will be strongly dependent on the Shear Correction Factors (SCF) of which their values are quite dispersed through each problem.  In this paper, we develop the Higher-order Shear Deformation Theory (HSDT) model that includes higher-order terms in the approximation of the displacement field for modelling the plates. It is worth mentioning that this model requires C1-continuity of the generalized displacements leading to the second-order derivative of the stiffness formulation which causes some obstacles in standard C0 finite formulations. Fortunately, it is shown that such a C1-HSDT formulation can be easily achieved using a NURBS-based isogeometric approach [12]. In addition, to capture the discontinuous phenomenon in the cracked plates, the enrichment functions is incorporated with NURBS basic functions to create a novel method as so-called eXtended Isogeometric Analysis (XIGA) [13]. Herein, our study focuses on investigating the vibration of the cracked plates with an initial crack. Two numerical examples are given to show the performance of the proposed method and results obtained are compared to other published methods in the literature. International Journal of Fracture Fatigue and Wear, Volume 2 128 2 GOVERNING EQUATIONS FOR PLATES To consider the effect of shear deformation directly, the ﬁve-parameter displacement ﬁeld based on higher-order shear deformation theory is defined as ( )( )224 30 , 34 30 , 30( , , )( , , )( , )x xhy yhu x y z u zw z zv x y z v zw z z  w x y w       ββ= − + −= − + −=,   2 2h hz− ≤ ≤  (1) where 0 0, ,u v w  are the axial displacements and , x yβ β are the rotations in the x and y axes, respectively.  The strains of the mid-surface deformation are given by ( )( )224 30 1 234 21hhz z zz + + −        −   ε κε= (2) Using the Hamilton principle, the weak form for free vibration analysis of a plate can be expressed as: d d dT b T s Tδ δ δΩ Ω ΩΩ + Ω = Ω  D D u mu ε ε γ γ (3) where bD and sD are the material matrices Herein, the mass matrix m is calculated according to consistent form as follow 1 2 42 3 54 5 6I I II I II I I 	 = 	 	  m  with  ( ) ( ) ( )( )2 2 2 2/2 4 4 42 3 3 33 3 3/2 1, , , , , dhi e h h hhI z z z z z z z z z zρ−= − − − (4)and  1 0 ,2 1 0 2 , 33, ; ;0 0x xy yu wv wwββ              = = = − =                    uu u u u uu (5) 3 AN EXTENDED ISOGEOMETRIC CRACKED PLATE FORMULATION 3.1 A brief of B-spline/NURBS functions A knot vector { }1 2 1, ,..., n pξ ξ ξ + += is defined as a sequence of knot value i Rξ ∈ , 1,...i n p= + . If the first and the last knots are repeated p+1 times, the knot vector is called an open knot. Through open knot, the B-spline basis functions ( ),i pN ξ are defined by the following recursion formula To present exactly some curved geometries (e.g. circles, cylinders, spheres, etc.) the Non-Uniform Rational B-Splines (NURBS) functions are used. Be different from B-spline, each control point of NURBS has additional value called an individual weightiw  [12]. Then the NURBS functions can be expressed as( )( ),,i iii iN wRN wξ ηξ η=(7) ( ) ( ) ( )1, , 1 1, 11 1i pii p i p i pi p i i p iN N Nξ ξξ ξξ ξ ξξ ξ ξ ξ+ +− + −+ + + +−−= +− −as p = 0, ( ) 1,0 10 otherwisei iiifNξ ξ ξξ +< < =   (6) International Journal of Fracture Fatigue and Wear, Volume 2 129 3.2 Cracked plate formulation eXtended IsoGeometric Analysis (XIGA) To capture the local discontinuous and singular fields, the new enriched functions are added according to idea of XFEM as follow: ( ) ( ) ( )enrh std enr enrI I J JI S J SR R∈ ∈= + u x  q  q (8) Here, the NURBS basis functions are utilized instead of the Lagrange polynomials to create a new numerical procedure – so-called eXtended IsoGeometric Analysis (XIGA) [13,14]. enrJR are the enrichment functions associated with node J located in enriched domain enrS  which is splitted up two parts including: a set Sc for Heaviside enriched control points and a set Sf for crack tip enriched control points as shown in Fig. 1.                           Fig. 1 Illustration of the nodal sets Sc, Sf for a quadratic NURBS mesh To describe the discontinuous displacement field, the enrichment function is given by ( )( ) ( ) ( ) ( ) ,enr cJ J JR R H H J S= − ∈  x x (9) where ( )H x is the Heaviside function and the singularity field near crack tip is modified by the branching functions (see more details in  Ref [10]) as follow( )41( ) ( ) ( , ) ( , ) ,enr fJ J L L J JLR R G r G r J Sθ θ= = − ∈    (10) where 3/20 01/23 3sin cos sin cos , ,2 2 2 2( , )sin cos sin sin cos cos ,2 2 2 2Lx xr u v wG rrθ θ θ θθθ θ θ θθ θ β β   	   =  	   for variablesfor variables(11) in which r and θ are polar coordinates in the local crack tip coordinate system. 3.3 Cracked plate formulation based on HSDT Now, using the displacement field approximated in Eq. (8), the strain matrices including in-plane and shear strains can be rewritten as: ( ) ( ) ( ) ( )1 20 1 21m n TT T T TTT T T T m b b ss I I I I II×=   =  	   B B B B qε  κ ε(12) in which the unknown vector q contains both displacements and enriched DOFs, andstd enr  B = B B(13) Heaviside enriched control pointsCrack tip enriched control pointsCrack pathPhysical mesh Control pointsControl point of set Sc Control point of set SfInternational Journal of Fracture Fatigue and Wear, Volume 2 130 where stdB  and enrB  are the standard and enriched strain matrices of B .Substituting Eq. (2) with relation in Eq. (12) into Eq. (3), the formulations of free vibration problem can be rewritten as follow:  ( )2ω− =K M d 0 (14) where ω +∈ R are the natural frequency and the global stiffness matrix K is given by: { } { }1 2 1 2 dTm b b b m b b sT s sΩ= + ΩK B B B D B B B B D B (15) 4 NUMERICAL RESULTS  First, let study the natural frequencies of a thin isotropic cracked plate (ν=0.3) with dimension L W h× ×  has an initial crack at center as shown in Fig. 2a.  Fig. 2 The plate with a center crack: (a) model; (b) mesh of 21x21 cubic NURBS elements The relation between non-dimension frequencies 2ˆ /L h Dω ω ρ= and crack length ratio according to mesh 21x21 as shown in Fig2b is reported in Table1. The obtained results from XIGA are in good agreement compared to both analytical solution [6,7] and XFEM [8] using 20x20 nine-node Lagrange elements. For clearer vision, the comparison of first five frequencies between the present result and that of Stahl [6] and Liew [7] is depicted in Fig.3. It is revealed that the frequencies decrease via increase in crack length ratio. For example, the values of frequency according to change of mode from 1 to 5 drop up to 18.3%, 67.3%, 5.3%, 40.8% and 23.7% of its initial values corresponding to an intact plate, respectively. It is concluded that the magnitude of the frequency according to anti-symmetric modes through the y-axis, which is perpendicular with cracked path (e.g. mode 2, mode 4, shown in Fig. 4), is much more affected by the crack length. Here, the discontinuous displacement is shown clearly along crack path. (a) (b) International Journal of Fracture Fatigue and Wear, Volume 2 131 Table 1 Non dimensional natural frequency of an isotropic square plate with central crack L/h=1000 Mode Source Crack length ratio a/L  Number   0 0.2 0.4 0.5 0.6 0.8 11 Stahl [6] 19.7390 19.3050 18.2790 17.7060 17.1930 16.4030 16.1270Liew [7] 19.7400 19.3800 18.4400 17.8500 17.3300 16.4700 16.1300XFEM [8] 19.7390 19.3050 18.2780 17.7070 17.1800 16.4060 16.1330XIGA 19.7392 19.3846 18.4617 17.8989 17.3576 16.5157 16.13452 Stahl [6] 49.3480 49.1700 46.6240 43.0310 37.9780 27.7730 16.1270Liew [7] 49.3500 49.1600 46.4400 42.8200 37.7500 27.4300 16.1300XFEM [8] 49.3480 49.1810 46.6350 43.0420 37.9870 27.7530 17.8260XIGA 49.3501 49.1906 47.1197 44.7124 39.3469 29.1186 16.13453 Stahl [6] 49.3480 49.3280 49.0320 48.6970 48.2230 47.2560 46.7420Liew [7] 49.3500 49.3100 49.0400 48.7200 48.2600 47.2700 46.7400XFEM [8] 49.3480 49.3240 49.0320 48.6850 48.2140 47.2010 46.7340XIGA 49.3501 49.3292 49.0903 48.6328 48.3547 47.3448 46.73764 Stahl [6] 78.9570 78.9570 78.6020 77.7330 75.5810 65.7320 46.7420Liew [7] 78.9600 78.8100 78.3900 77.4400 75.2300 65.1900 46.7400XFEM [8] 78.9550 78.9450 78.6000 77.7100 75.5790 65.7150 49.0990XIGA 78.9589 78.9452 78.6507 77.6642 76.0779 67.2308 46.73765 Stahl [6] 98.6960 93.9590 85.5100 82.1550 79.5880 76.3710 75.2850Liew [7] 98.7000 94.6900 86.7100 83.0100 80.3200 76.6000 75.2800XFEM [8] 98.6980 93.8930 85.4500 82.1080 79.5560 76.3510 75.2750XIGA 98.7292 94.5834 86.6987 82.7347 80.3835 76.7866 75.2823Fig. 3 Variation of first five mode frequencies via length crack ratio           Mode 1                         Mode 2                           Mode 3                          Mode 4                          Mode 5 Fig. 4 The first five mode shapes of simply supported plate having center crack with a/L =0.8  !  !"  !#  !$  !% &! &'" "'( ('# #''  !"#$% !"#$& !"#$' !"#$!%&'$()#(*+,()*+,$)*-.!$*/0$1-*23$0.#4$5)#6#7-$891:9; !  !"  !#  !$  !% &! ' $ ( % ) &   !"#$%$&'()*$+,#-$./#0#1'$23&435 !"#$!%&'$()#(*+,6/(78$/(',!$(9+ !"#$:International Journal of Fracture Fatigue and Wear, Volume 2 132 Next, an annular plate with uniform thickness h, outer radius R and inner one r as shown in Fig. is studied. The plate is made of Al/Al2O3 FGM with properties : Ec=70 GPa , Em=380 GPa , νc=νm=0.3, ρc=2707 kg/m3, ρm=3800 kg/m3. The FGM is homogenized according to the Mori-Tanaka scheme with the effective values of Young’s modulus E and Poisson’s ratio ν  are calculated from the effective bulk and shear modulus [4] as below : 9 3 2,3 2(3 )e e e ee ee e e eK KEK Kµ µνµ µ−= =+ + (16) Table 2 tabulates the frequency parameter of the annular Al/Al2O3 plates via outer radius to inner radius ratio R/r and radius to thickness ratio R/h according to n=1. It is concluded that the frequency parameters decrease sequentially following to increase in inner radius to outer radius ratio r/R. To enclose this section the first four mode shapes of annular FGM plate are depicted in Fig. 6. Table 2 The frequency parameter 2( ) / /c cR r h Eω ω ρ= −  of an annular pate via inner radius to outer radius ratio r/R and radius to thickness ratio R/h according to n=1. R/h r/R Mode number 1 2 3 4 5 2 0 1.2786 1.7682 2.3336 2.7352 2.8058 0.2 0.8438 1.0109 1.7316 1.8588 1.9021 0.5 0.5516 0.5896 0.7308 0.8458 0.9817 0.8 0.2760 0.2771 0.2800 0.2896 0.2905 10 0 1.8480 3.5185 4.0473 5.9916 6.3512 0.2 1.1563 1.5352 3.1932 4.1404 4.8849 0.5 0.8621 0.9560 1.3533 1.6388 2.3101 0.8 0.6470 0.6540 0.6730 0.6975 0.7545 20 0 1.8379 3.1941 4.1533 6.1526 6.4956 0.2 1.1793 1.598 3.3309 4.4139 5.2045 0.5 0.8877 0.9954 1.4507 1.7937 2.5902 0.8 0.7279 0.7371 0.7655 0.7999 0.8775 100 0 1.8649 3.3264 4.2398 6.3270 6.7435 0.2 1.1922 1.6442 3.4202 4.6187 5.3415 0.5 0.8973 1.0154 1.5007 1.8656 2.6187 0.8 0.7646 0.7753 0.8124 0.8567 0.9362    Mode 1 Mode 2 Mode 3 Mode 4 Fig. 6 The first four mode shapes of the annular plate with r/R=0.8, R/h=10. Fig. 5 The model of annular plate. International Journal of Fracture Fatigue and Wear, Volume 2 133 5 CONCLUSIONS In this paper, a novel and effective formulation based on combining XIGA and HSDT has been applied to dynamic analysis of the cracked plates. The present method, utilizing NURBS basis functions, allows us to achieve easily the smoothness with arbitrary continuity order compared with the traditional FEM. Consequently, it naturally fulfills the C1-continuity of HSDT model. Furthermore, the special enrichment functions are applied for describing the singularity behaviors along the crack. The obtained results are in excellent agreement with others in the literature. 6 REFERENCES [1]  J.N. Reddy, Analysis of functionally graded plates. International Journal for Numerical Methods in Engineering 47, 663–684, 2000. [2]  S.V. Senthil, R.C. Batra, Exact solution for thermoelastic deformations of functionally graded thick rectangular plates. AIAA Journal 40,1021–1033, 2002. [3]  H. Nguyen-Xuan, Loc V. Tran, T. Nguyen-Thoi, H.C. Vu-Do, Analysis of functionally graded plates using an edge-based smoothed finite element method. Composite Structures 93, 3019-3039, 2011. [4] Loc V. Tran, A.J.M. Ferreira, H. Nguyen-Xuan, Isogeometric approach for analysis of functionally graded plates using higher-order shear deformation theory. Composite Part B: Engineering 51, 368-383, 2013. [5]  Loc V. Tran, Chien H. Thai, H. Nguyen-Xuan, An isogeometric finite element formulation for thermal buckling analysis of functionally graded plates, Finite Element in Analysis and Design 73, 65-76, 2013. [6]  B. Stahl, L. Keer, Vibration and stability of cracked rectangular plates. Int J Solids Struct 8, 69–91,1972. [7]  K.M. Liew, K.C. Hung, M.K. Lim, A solution method for analysis of cracked plates under vibration, Engineering Fracture Mechanics 48, 393–404, 1994. [8] M. Bachene, R. Tiberkak, S. Rechak, Vibration analysis of cracked plates using the extended finite element method, Archive of Applied Mechanics 79, 249-262, 2009. [9]  G. Qian, S.  Gu, J. Jiang, A ﬁnite element model of cracked plates and application to vibration problems. Comput Struct 39, 483–487, 1991. [10] J. Dolbow, N. Moës, T. Belytschko, Modeling fracture in Mindlin–Reissner plates with the extended ﬁnite element method. Int J Solids Struct 37, 7161–7183, 2000. [11] S.Natarajan, P.M. Baiz, S.Bordas, T.Rabczuk, P. Kerfriden, Natural frequencies of cracked FGM plates by the extended ﬁnite element method, Composite Structures 93, 3082–3092, 2011. [12] T.J.R. Hughes, J.A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194, 4135–4195, 2009. [13] S.S. Ghorashi, N. Valizadeh, and S. Mohammadi, Extended isogeometric analysis for simulation of stationary and propagating cracks. International Journal for Numerical Methods in Engineering, 89, 1069–1101, 2012. [14] E. De Luycker, D.J. Benson, T. Belytschko, Y. Bazilevs, and M.C. Hsu, X-FEM in isogeometric analysis for linear fracture mechanics. International Journal for Numerical Methods in Engineering 87,541–565,2011. 

Vibration analysis of cracked plate using higher-order shear deformation theory

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Vibration analysis of cracked plate using higher-order shear deformation theory

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