Vibrations of Skewed FGM Plates with Internal Cracks via MLS-Ritz

Method

*C. S. Huang1), Y. C. Fu2) and P. Y. Li2)

1), 2) Department of Civil Engineering, NCTU, Hsinchu, Taiwan

1) cshuang@mail.nctu.edu.tw

ABSTRACT

This study shows vibration analysis of an internally cracked functionally graded material (FGM) plate based on three-dimensional elasticity theory using the famous Ritz method with admissible functions that are constructed by the moving least square (MLS) method with enriched basis functions. The enriched basis functions consist of regular polynomial functions and crack functions that correctly account for the singular behaviors of stresses at the terminus edge fronts of the crack and the discontinuities of displacement across a crack. The accuracy and efficiency of the presented approach are demonstrated through convergence study and comparisons of the present results with those published data. The effects of the volume fraction of the constituents of FGM, crack length and skew angle on the frequencies of skewed cantilevered rhombic FGM plates with vertical internal cracks are investigated. 1. INTRODUCTION Functionally graded materials, which are heterogeneous composite materials and whose material properties vary continuously with a change in the volume fraction of its constituents, have been invented to overcome the abrupt change in material properties across the interface of conventional laminated composite materials and have found various applications in various engineering fields. The flat plate is a very common component in engineering structures. Sharp corners of openings, cutouts, welding and irregular loads very likely cause initiation and propagation of cracks. Redistribution of stresses in a cracked FGM plate causes dynamic characteristics markedly different from those for an intact plate, and it is interesting to investigate the vibrations of a cracked FGM plate.

Numerous studies of the vibrations of cracked plates have been conducted. Most of them consider homogeneous plates and utilize different plate theories. Based on the classical thin plate theory, Stahl and Keer (1972), Aggarwala and Ariel (1981) and Neku

1)

Professor 2) Graduate Student

(1982) applied integral equation techniques to investigate the vibrations of cracked rectangular plates; Qian et al. (1991), Krawczuk (1993) and Bachene et al. (2009) utilized finite element methods, while Yuan and Dickinson (1992), Liew et al. (1994) and Huang and Leissa (2009, 2011) employed the Ritz method. Based on the first-order shear deformable plate theory, finite element solutions (Ma and Huang 2001) and solutions obtained using the Ritz method (Lee and Lim 1991, Huang et al. 2011a) have also been proposed.

To analyze the vibrations of cracked FGM plates, Huang et al. (2011b, 2012, 2013) used Reddy’s plate theory and three-dimensional elasticity theory along with the conventional Ritz method with admissible functions covering the whole problem domain, while Yin et al. (2016) and Fantuzzi et al. (2016) proposed different finite element approaches based on the first-order shear deformable plate theory.

The main purpose of this work is to explore the possibility of applying the Ritz method with the admissible functions constructed by the moving least square method with enriched basis functions to determine the accurate vibration frequencies of skewed cantilevered FGM plates with vertical internal cracks. Three-dimensional elasticity is used, and the enriched basis functions include regular polynomial functions and crack functions that correctly account for the singular behaviors of stresses at the terminus edge fronts of the crack and the discontinuities of displacement across a crack. New sets of crack functions are proposed to help the Ritz method to recognize the existing of an internal crack in a plate. Numerous non-dimensional frequencies are reported herein for the first time for skewed cantilevered rhombic FGM plates with various skew angles and thickness-to-length ratios and having vertical central cracks with various lengths. 2. Geometric configuration and model of material properties Fig.1 shows a skewed cantilevered FGM plate (with side dimensions a-b, thickness h and skew angle ) with an internal crack and also displays the rectangular

coordinate (x, y, z) originating at the mid-plane of the plate. Two polar coordinates (r1,

1 ) and (r2, 2 ) originate at the two intersections of crack fronts with the mid-plane of the

plate, respectively. The internal crack has the angle of inclination and the length d.

The functionally graded material of interest is a mixture of metal and ceramic. The material properties (i.e., elastic modulus, E = E(z), Poisson’s ratio, = (z) and mass density, ρ = ρ(z)) vary as a simple power law in the parallelepiped thickness (i.e., the z direction in Fig. 1), as follows:

( ) ( ) ( )t b bP z P P V z P and 1

( ) ( )2

mzV z

h (1)

where P denotes a property of the material; tP and bP are the properties at the top (z

= h/2) and the bottom (z = -h/2) faces, respectively, and m is the parameter of volume fraction that governs the material variation profile in the thickness direction. The FGM plates considered herein are made of aluminum (Al) and ceramic (alumina (Al2O3)), whose material properties are given in Table 1.

Table 1 Material properties of the FGM components

Material

Properties

E (GPa) Poisson’s ratio ( ) (kg/m3)

Aluminum (Al) 70 0.3 2702

Alumina (Al2O3) 380 0.3 3800

1

y

x

a

bd

2r2θ

2α

1r1θ

x

β

(a) top view

Ceramic surface

Metal surface

h2

h2 x

z

(b) side view

Fig. 1 A skewed cantilevered FGM plate with an internal crack

3. Theoretical Formulation

In using the Ritz method, the dynamic characteristics of plates are predicted by minimizing the energy functional

Vmax - Tmax, (2)

where Vmax is the maximum strain energy and Tmax is the maximum kinetic energy in simple harmonic motion.

Based on three-dimensional elasticity theory, a plate vibrating harmonically with circular frequency and amplitudes Ui (x, y, z) (i = 1, 2, 3) along the x, y and z

coordinate directions, respectively,

2 2 2 2 2

max 1, 2, 3, 1, 2, 3, 1, 2,

1{ ( )( ) ( )[2( ) 2( ) 2( ) ( )

2x y z x y z y x

V

V z U U U G z U U U U U

2 2

2, 3, 3, 1,( ) ( ) ]}z y x zU U U U dV , (3a)

22 2 2

max 1 2 3( )( )2

V

T z U U U dV , (3b)

where

( ) ( )( )

(1 ( ))(1 2 ( ))

z E zz

z z,

( )( )

[2(1 ( )]

E zG z

z and the subscript comma denotes

partial differentiation with respect to the coordinate defined by the variable after the comma.

Displacement amplitude functions, Ui (x,y,z), are expressed as

1

1 1

( , ) Np Nz

k

ik i

i k

U u x y z

( 1, 2, 3) (4)

where ( , )i x y are the shape functions associated with node i in x-y plane and are

constructed by the moving least square (MLS) method (Liu 2003) with enriched basis

functions. The proposed enriched basis functions for ( , )i x y are

T T

p( , )( , )T

cg x y p p p (5)

where T 2 2

p (1, , , , , )x y x xy yp , cp are sets of crack functions and

( , )g x y ensure

that the shape functions satisfy the essential boundary conditions of the problem of interest along the faces except for the top and bottom faces. When a cantilevered plate

as shown in Fig. 1 is under consideration, ( , ) xg x y .

To properly describe the stress singularity behavior at the terminus edge fronts of

a crack and show the displacement discontinuity crossing the crack, the proposed cp

are

1/2 1/2 2

1 2 2 1 1 1 1

1/2 1/2 2

2 1 1 2 2 2 2

( sin ( / 2) cos(1/ 2) , cos 3 / 2 , sin(1/ 2) , sin 3 / 2 ,

sin ( / 2) cos(1/ 2) , cos 3 / 2 , sin(1/ 2) , sin 3 / 2 ) for =1, 2,

T

c r r

r r

p

(6a)

1/2 2 1/2 3/2 2 3/2

3 2 2 1 1 2 2 1 1

1/2 2 1/2 3/2 2 3/2

1 1 2 2 1 1 2 2

( sin ( / 2) sin( / 2), sin ( / 2) cos( / 2),

sin ( / 2) sin( / 2), sin ( / 2) cos( / 2) ),

T

c r r r r

r r r r

p (6b)

where j . The sets of crack functions given in Eqs. (6) and (7) show the

stress singularity order (1/ ) r at the terminus edge fronts of a crack, which agree

with the findings from the asymptotic solutions in Hartranft and Sih (1969). In constructing the admissible functions using MLS, the following weight function

is employed because its first derivatives with respect to x and y are continuous,

2 3

2 3

2 14 4 ,

3 2

4 4 1, 4 4 1,

3 3 2

0 1 ,

I I

d d if d

W x x y y d d d if d

if d

(8)

where ( , )i ix y is the coordinates of node i, 2 2

m

( ) ( )i ix x y yd

d

and dm is the

size of the support. Substituting Eq. (4) into Eqs. (3) and minimizing with respect to coefficients

iku yield 3 p zN N linear algebraic equations, which form a generalized eigenvalue

problem with eigenvalue 2 .

4. Convergence study

The accuracy and efficiency of the presented approach are demonstrated through convergence study and comparisons of the present results with those published data.

In the following analyses, the nodes for constructing ( , )i x y are uniformly distributed

inside the domain, [0.005 , 0.995 ]x a a and [0.005 , 0.995 ]y b b with x =

0.99a/Nd,a and y 0.99b/Nd,b, where Nd,a and Nd,b are parameters to be prescribed

to define the distance between two adjacent nodes (see Fig. 2). When a=b, set Nd,a = Nd,b =Nd.

Table 2 shows the convergence of non-dimensional frequencies for a cantilevered FGM square plate with h/b=0.1 made of aluminum (Al) and ceramic (alumina (Al2O3)) with a vertical central crack having crack length equal to 0.5b. The parameter of volume

fraction ( m ) is set 0.5. The non-dimensional frequency parameter 2( / ) / c cb h E ,

where subscript “c” denotes ceramic, is considered. The results were obtained using Np=Nd*Nd with Nd=15, 20, 25, 30 and 40, Nz=3, 4 and 5, and the supports of the shape functions for in-plane and out-of-plane displacements equal to 0.3b and 0.8b, respectively. Table 2 illustrates that the numerical results converge from the upper bounds as Nd and Nz increase. The results obtained using Nd=30 and Nz=5 excellently agree with the results of Huang et al. (2012), who utilized the Ritz method with the admissible functions consisting of orthogonal polynomials and crack functions. Notably, since the admissible functions in Huang et al. (2012) include very high order of polynomials and trigonometric functions and cover the whole domain of plate, variables with 128-bit precision (with approximately 34 decimal digit accuracy) were used in their computer programs to avoid the numerical difficulties before the solutions converge.

5. Numerical Results

After the present solutions are validated, the proposed approach was utilized to investigate the vibrations of skewed cantilevered rhombic FGM plates having central vertical cracks with various lengths and different the volume fraction of the constituents. The following results were obtained using Nd= 30 and Nz=5.

b

d,bΔy = 0.99b / N

d,aΔx = 0.99a / N

a

Fig. 2 Distribution of nodal points

Table 2 Convergence of frequency parameters 2( / ) / c cb h E for a cantilevered

square FGM plate (h/b = 0.1) having a vertical central crack

Mode dN @ ( ) 3zN ;[ ] 4zN ; { } 5zN Huang et al. (2012) 15 20 25 30 40

1

(0.6463) (0.6456) (0.6451) (0.6450) (0.6448)

0.6423 [0.6437] [0.6430] [0.6425] [0.6425] [0.6423]

{0.6436} {0.6429} {0.6424} {0.6423} {0.6421}

2

(1.582) (1.579) (1.579) (1.578) (1.578)

1.561 [1.565] [1.562] [1.561] [1.561] [1.561]

{1.564} {1.562} {1.561} {1.561} {1.560}

3

(3.427) (3.418) (3.410) (3.409) (3.406)

3.365 [3.386] [3.377] [3.371] [3.370] [3.367]

{3.385} {3.376} {3.369} {3.368} {3.365}

4

(3.965) (3.962) (3.961) (3.961) (3.960)

3.960 [3.964] [3.961] [3.960] [3.959] [3.959]

{3.964} {3.961} {3.959} {3.959} {3.958}

5

(4.856) (4.853) (4.851) (4.851) (4.851)

4.808 [4.815] [4.813] [4.811] [4.811] [4.810]

{4.814} {4.812} {4.810} {4.809} {4.809}

Table 3 Frequency parameters 2( / ) /c cb h E for skewed cantilevered rhombic FGM

plates with vertical central cracks of various lengths (h/b = 0.1)

Table 3 lists the first five non-dimensional frequency parameters of skewed

cantilevered rhombic FGM plates with h/b=0.1 and skew angle 0 and 15o o and

having vertical central cracks with crack lengths d/b=0, 0.1, 0.3 and 0.5, while Table 4

shows the results for plates with 30 o and h/b=0.1 and 0.2. In both tables, the

volume fraction m =0, 0.2 and 5 are under consideration. Notably, “*” denotes in-plane

m d b Mode

1 2 3 4 5

0o

0

0 1.041 2.444 6.102 6.598* 7.730

0.1 1.038 2.439 6.053 6.579* 7.704

0.3 1.019 2.419 5.726 6.422* 7.567

0.5 0.9794 2.397 5.189 6.093* 7.398

(0.9800) (2.398) (5.185) (6.096*) (7.398)

0.2

0 0.9666 2.273 5.676 6.285* 7.190

0.1 0.9641 2.268 5.631 6.268* 7.167

0.3 0.9464 2.250 5.328 6.119* 7.039

0.5 0.9098 2.231 4.829 5.806* 6.882

(0.9100) (2.231) (4.826) (5.828*) (6.822)

5

0 0.6838 1.594 3.966 4.285* 5.026

0.1 0.6819 1.591 3.934 4.273* 5.009

0.3 0.6690 1.576 3.718 4.172* 4.919

0.5 0.6423 1.561 3.368 3.959* 4.809

(0.6423) (1.561) (3.365) (3.960*) (4.809)

15o

0

0 1.073 2.496 6.335 6.631* 7.477

0.1 1.070 2.490 6.285 6.612* 7.464

0.3 1.050 2.470 5.951 6.445* 7.390

0.5 1.008 2.446 5.392 6.091* 7.262

0.2

0 0.9964 2.321 5.895 6.318* 6.957

0.1 0.9937 2.316 5.849 6.300* 6.945

0.3 0.9749 2.298 5.539 6.141* 6.875

0.5 0.9366 2.277 5.021 5.805* 6.757

5

0 0.7047 1.628 4.108 4.314* 4.860

0.1 0.7027 1.624 4.076 4.301* 4.852

0.3 0.6888 1.610 3.858 4.190* 4.804

0.5 0.6609 1.593 3.495 3.961* 4.721

deformation dominated modes. In Table 3, the parenthesized results were obtained by Huang et al. (2012) using

three-dimensional elasticity theory and the conventional Ritz method, while the parenthesized results in Table 4 were obtained by McGee and Butalia (1994) utilizing higher-order shear deformable plate theory and the finite element approach. Comparisons of the present results with those published results reveal the correctness and accuracy of the present results.

Tables 3 and 4 show that, as expected, the non-dimensional frequency parameters decrease with the increase of crack length. A crack with d/b=0.1 only reduce the first five frequencies by less than 1%, while a d/b=0.5 crack can reduce frequencies by 15%.

The frequencies decrease with the increase of the volume fraction m because increasing m reduces the stiffness more than it does the mass of plate. The first four frequencies increase as the skew angle increases, except for some results for d/b=0.5. Table 4 displays the dimensionless frequency parameters decrease with the increase of thickness h because h is involved in the definition of the non-dimensional frequency parameter.

6. CONCLUSIONS

Proposed herein is a three-dimensional elasticity-based Ritz procedure to obtain accurate vibration frequencies of skewed cantilevered rhombic FGM plates with vertical internal cracks. The admissible functions are constructed by MLS with proposed sets of enriched basis functions, which consists of regular polynomials and crack functions that properly describe the 3-D stress singularities at the terminus edge fronts of the crack and show displacement discontinuities across the crack.

The efficiency of the proposed solutions has been substantiated through convergence study and comparisons with the published results. The present 3-D approach has been employed to investigate the effects of volume fraction ( m ), crack length ratios (d/b), thickness-to-length ratios (h/b) and skew angles ( ) on the vibration

frequencies of skewed cantilevered rhombic FGM plates with vertical central cracks. Most of the results are first shown in the literature and can be used as standard to judge the accuracy of other numerical methods and various plate theories.

ACKNOWLEDGEMENT

The authors would like to thank the Ministry of Science and Technology of the Republic of China, Taiwan, for financially supporting this research under Contract No. MOST 103-2221-E-009-040-MY3.

Table 4 Frequency parameters 2( / ) / c cb h E for skewed cantilevered rhombic FGM

plates with =30o and having vertical central cracks

h b m d b Mode

1 2 3 4 5

0.1

0

0 1.171 2.692 6.684* 7.063 7.364

(1.168) (2.686) (6.692*) (7.047) (7.354)

0.1 1.167 2.686 6.662* 7.010 7.361

0.3 1.143 2.664 6.465* 6.652 7.336

0.5 1.096 2.627 6.026 6.043* 7.252

0.2

0 1.087 2.504 6.367* 6.581 6.852

0.1 1.084 2.499 6.345* 6.532 6.849

0.3 1.062 2.478 6.150* 6.207 6.825

0.5 1.018 2.445 5.611 5.760* 6.748

5

0 0.7681 1.754 4.315* 4.601 4.786

0.1 0.7658 1.751 4.300* 4.569 4.786

0.3 0.7493 1.735 4.159* 4.348 4.769

0.5 0.7176 1.710 3.838 3.988* 4.714

0.2

0

0 1.132 2.451 3.349* 5.966 6.499

0.1 1.128 2.444 3.338* 5.919 6.496

0.3 1.098 2.410 3.240* 5.605 6.473

0.5 1.041 2.355 3.029* 5.067 6.347

0.2

0 1.053 2.286 3.192* 5.586 6.070

0.1 1.049 2.280 3.181* 5.543 6.067

0.3 1.021 2.249 3.087* 5.249 6.045

0.5 0.9695 2.199 2.886* 4.745 5.969

5

0 0.7378 1.570 2.182* 3.757 4.135

0.1 0.7348 1.566 2.175* 3.728 4.133

0.3 0.7145 1.542 2.112* 3.529 4.118

0.5 0.6766 1.503 1.977* 3.184 4.063

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