Analysis of the Effect of the Homogenization Methods on the Shear Correction Factor of Functionally Graded Beams

Analysis of the Effect of the Homogenization Methods on the Shear Correction Factor of Functionally Graded Beams

Trung-Kien Nguyen

Faculty of Civil Engineering and Applied Mechanics University of Technical Education Ho Chi Minh City, Viet Nam

AbstractThis paper presents analysis of the effect of the homogenization methods on the shear correction factor of functionally graded beams. It is based on the first-order shear deformation beam theory in which the effective material properties of functionally graded beams are estimated by the homogenization schemes of Hill and Hashin-Shtrikman. The material property is assumed to be isotropic and varies through the beam depth according to a power-law form. Numerical results shows that the shear correction factor not only depends on material properties, but also on the homogenization schemes.

Keywords Functionally graded beam; Shear correction factor; Homogenization method.

1. INTRODUCTION

   Functionally Graded Material (FGM) is an advanced composite material whose properties vary continuously to avoid stress concentrations at interfaces found in laminated composites ([1]). The studies on behaviors of functionally graded (FG) beams rapidly become an active subject with many researches over the recent years. Many approaches have been used to study static, vibration and buckling behaviors of FG beams, only several references cited here ([2-7]). The FG beams based on the first-order shear deformation theory (FSDT) have been widely applied by its simplificity in calculation and programming. However it requires a convenient value of the shear correction factor. In practice, this coefficient has been taken by the five-sixth one as homogeneous beams. The contribution on this topic for FSDT beams can be found in [7-11]. Moreover, the estimation of effective elastic properties of functionally graded materials has been studied by many authors in which continuous models are most used, among Voigts approximation [12]. The works of [13-14] showed that the Mori-Tanakas scheme is convenient for estimating the effective properties of FGM in the matrix-inclusion region while the self-consistent is for the area having interconnected phases. Many more approximations of the effective elastic properties can be found within the literature ([15]). In continuous model, practically, FG beams are first homogenized with their effective moduli

   such as Youngs modulus, Poissons ratio, etc, and then the effective properties of the FG beams will be derived from

   The objective of this paper is to study the effect of the effective material properties on the shear correction factor of FG beams. Theory is based on the first-order shear deformation beam theory within which the effective moduli of the beam will be calculated by the Voigt and Reuss models, and Hashin-Shtrikmans estimates. The material elastic properties are supposed to be isotropic at each point and vary through the beam depth according to the power-law form.

2. THEORETICAL FORMULATION Consider a FG beam having the thickness h, section b h

   and length L as Fig. 1. The beam is made of a functionally

   graded material constituted by a mixture of two constituents whose material properties vary through the beam depth according to the volume fractions of the constituents. The material elastic properties are supposed to be isotropic at each point, the gravity is not taken into account.

   Fig. 1. Geometry of a functionally graded beam of ceramic-metal mixture.

   1. Shear correction factor of FSDT FG beams

      The axial strain and stress of the FSDT are related by the constitutive equation:

      xx x, z E z 0 x z x (1) where E(z) is Youngs modulus at location z, 0 , are the axial strain and curvature of the beam, respectively. These strains are related with the axial displacement u and rotation

      , x

      of the beam as follows: 0 u,x , where the comma indicates partial differentiation with respect to the coordinate subscript that follows. Moreover, the stress resultants (N, M)

      homogeneous beam theories. The analysis of the effect of homogenization methods on the shear correction factor of FG beams is not investigated.

      are associated to the axial stress

      constitutive relations:

      xx

      by the global

      N x A 0 B

      (2a)

      E z E E V z E

      (9a)

      11 11

      11 11

      M x B 0 D

      (2b)

      c m c m

      E z Ec Em

      (9b)

      where A11, B11, D11 are the stiffnesses of FG beams defined by:

      h/2

      EmVc z Ec 1Vc z

      where Ec, Em are Youngs modulus of the constituents, Vc is the volumic fraction of ceramic which is given by the power-

      A , B , D

      1, z, z2 E z dz

      (3)

      law distribution as follows:

      11 11 11

      p

      h/2

      Substituting (2a) and (2b) into (1) leads to:

      2z h

      Vc z 2h

      (10)

      x, z E za zb N b zd M

      (4)

      xx 11 11 11 11

      where p the material parameter which is positive. The

      where

      1. D

         D A

         1 ,

      2. B

      D A

      1 ,

      material distribution under consideration is plotted in Fig. 2.

      11 11 11 11

      1 B2

      11 11 11 11

      d D

      and D D 11 . Moreover, it is known

      0.5

      11 11

      11 11

      A11

      0.4

      that the transverse shear stress can be calculated from the equilibrium equation by taking in account (4), that leads to:

      0.3

      0.2

      xz x, z R z Q x

      where

      R z b11 A11z z d11B11z z

      z z

      A11z z h/ 2 E d , B11z z h/ 2 E d

      (5)

      (6)

      0.1

      z/h

      0

      -0.1

      -0.2

      -0.3

      p=0.5

      p=1 p=2 p=5

      Moreover, it is well-known that the FSDT beams require

      an convenient shear correction value to calculate the

      -0.4

      -0.5

      p=10

      transverse shear force. In the present study, a transverse shear stiffness of the FG beams is obtained by considering the balance of the shear deformation energy and then a shear correction coefficient is derived. Based on the previous works of the author in [7,10], the expression of the improved transverse shear stiffness is obtained:

      0 0.2 0.4 0.6 0.8 1

      V

      c

      Fig. 2. Variation of the volumic fraction of ceramic through the beam depth.

      Alternatively, the effective moduli of FG beams can be derived from the Hashin-Shtrikmans bounds ([16]) which are given by:

      2 1

      K K V

      h/2 b11 A11z z d11B11z z

      l m c ,

      As

      h/2

      G z

      dz

      (7)

      Kc Km

      1 1Vc

      K

      Kc Km

      4 / 3

      where G(z)=E(z)/2(1+(z)) is the transverse shear modulus at location z. A shear correction coefficient can be found as:

      l m

      m m

      Vc

      (11a)

      c m

      1 1V c m

      c

      k s

      As

      h /2

      (8)

      m fm

      G z dz

      K K V

      h /2

      u m c ,

      This coefficient is equal to 5/6 for homogeneous beam and

      Kc Km

      1 1V

      c

      Kc Km

      a priori varies with respect to the material variation, material contrast and different homogenization methods.

   2. Effective material properties of FG beams

   u m

   c m

   Km 4c / 3

   Vc

   1 1V c m

   (11b)

   It is well-known that the approximations of Voigt and Reuss based on the Hills principle ([12]) are the simplest ones for estimating the effective elastic properties of heterogeneous composite materials in which Voigts rule is most used for calculating the effective moduli of the FGM. The mixture of two ceramic-metal materials according to the Voigt and Reuss models through the beam depth are respectively given by:

   m fc

   where Km, Kc and m, c are the bulk and shear moduli of the metal and ceramic, respectively; Kl, Ku and l, u are the lower and upper bounds of bulk and shear moduli, respectively; f 9 K 8 / 6K with

   =m,c. It is noted that the relation (11a) is also known as the Mori-Tanakas estimate ([16]). The effective moduli are

   finally obtained as:

   3K 2 / 23K .

   E 9K / 3K ,

3. NUMERICAL RESULTS AND DISCUSSIONS

   A number of numerical examples will be carried out in this section to verify the accuracy of the present study and to investigate the effect of homogenization schemes on the shear correction factor of FG beams. It is assumed that the Poissons ratio is constant through the beam depth. The effect of the power-law index p, material contrast n= Ec/Em and homogenization methods on the shear correction factor is studied.

   value of p (p>3), highest curve is RM and lowest one is VM, and inversely for p<3. Figure 5 displays the variation of the shear coefficient with respect to the ratio of Youngs modulus for various homogenization schemes. It can be seen that the difference of the bounds of the shear correction factor increases with the material contrast, that confirms that the homogenization schemes based on the variational principles are suitable for small contrast of materials. This figure also shows that the Hashin-Shtrikmans bounds give the approximation more accurate than the Hills ones.

   TABLE I. COMPARISON OF THE SHEAR CORRECTION FACROR

   0.9

   Shear correction factor

   0.8

   0.7

   0.6

   RM

   LHS

   p	Method	n
   		2	6	10	20
   0.5	Present (VM)	0.8401	0.8458	0.8471	0.8478
   	Present (RM)	0.8320	0.7848	0.7465	0.6846
   	Present (UHS)	0.8371	0.8370	0.8360	0.8353
   	Present (LHS)	0.8359	0.8119	0.7846	0.7328
   	[7]	0.8402	0.8458	0.8471	0.8479
   2	Present (VM)	0.8095	0.7662	0.7563	0.7580
   	Present (RM)	0.8023	0.7219	0.6799	0.6239
   	Present (UHS)	0.8058	0.7401	0.7137	0.6960
   	Present (LHS)	0.8045	0.7211	0.6727	0.6058
   	[7]	0.8095	0.7662	0.7563	0.7580
   5	Present (VM)	0.7891	0.6641	0.5919	0.5043
   	Present (RM)	0.7980	0.7335	0.7026	0.6617
   	Present (UHS)	0.7923	0.6779	0.6053	0.5043
   	Present (LHS)	0.7938	0.7067	0.6619	0.6026
   	[7]	0.7891	0.6643	0.5923	0.5046
   10	Present (VM)	0.7990	0.6746	0.5861	0.4521
   	Present (RM)	0.8097	0.7678	0.7474	0.7194
   	Present (UHS)	0.8034	0.7025	0.6255	0.4983
   	Present (LHS)	0.8053	0.7406	0.7063	0.6596
   	[7]	0.7990	0.6746	0.5861	0.4521

   0.5

   0.4

   0

   n 10

   20 0

   UHS

   VM

   20

   15

   10

   5 p

   Fig. 3. Variation of the shear correction factor with respect to the power-law index and material contrast.

   0.9

   Shear correction factor

   0.85

   0.8

   0.75

   0.7

   VM RM UHS LHS

   0.65

   0 5 10 15 20

   p

   Fig. 4. Variation of the shear correction factor with respect to the power-law index with n=6.

   Table I presents the shear correction coefficients derived from Voigt model (VM), Reuss model (RM), lower bound of Hashin-Shtrikman (LHS) and upper bound of Hashin- Shtrikman (UHS) . They are compared to results obtained from [7]. It can be seen that the present results using VM are similar with ones of [7] while there are differences with other models. It proves that the shear coefficients not only depend on the power-law index and material contrast, but also on homogenization methods for estimating the effective material properties of FG beams.

   Furthermore, the effects of the power-law index p,

   0.9

   0.85

   Shear correction factor

   0.8

   0.75

   0.7

   0.65

   0.6

   0.55

   0.5

   0.45

   1

   VM RM UHS LHS

   2 4 6 8 10 12 14 16 18 20

   n

   material contrast n and homogenization schemes on the shear correction factor are plotted in Figs. 3, 4 and 5. Four interaction-curves observed in Fig. 3 show that for a specific

   Fig. 5. Variation of the shear correction factor with respect to the material contrast with p=10.

4. CONCLUSIONS

This paper analyzed the effects of the effective material properties on the shear correction factor of FG beams. It is based on the first-order shear deformation beam theory within which the effective moduli of the beam are calculated by the Voigt and Reuss models, and Hashin-Shtrikmans estimates. The material elastic properties are supposed to be isotropic and varying through the beam depth according to the power- law form. Numerical results showed that the shear correction factor not only depends on the material distribution, material contrast but also on homogenization methods. The approximation schemes based on bounds are not accurate anymore for high material contrasts and the Hashin- Shtrikmans bounds give the approximation more accurate than Hills ones.

ACKNOWLEDGMENT

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 107.02-2012.07. The authors gratefully acknowledge this support.

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