Deformation characteristics of functionally graded bio-composite plate using higher-order shear deformation kinematics

Abstract

Deformation behavior of functionally graded bio- composite plate structures subjected to uniform pressure are examined and presented. Here, biocompatible metals/alloys and ceramics are utilized as constituent materials throughout in the analysis. The material properties of functionally graded bio- composite plate are evaluated through power-law distribution based Voigt’s micromechanical scheme. The displacement field is defined in third-order shear deformation mid-plane kinematics. However, the motion equations are governed by minimizing total potential energy. The deflection responses are obtained in finite element framework using nine noded quadrilateral element. To confirm the correctness of the present finite element model, the present results are compared with the reported results. In addition, various numerical illustrations are demonstrated to exhibit the significance of different geometrical and material parameters on the deformation behaviour of functionally graded bio-composite plate structure, and discussed in detail.

Keywords

Bio-composites FGM Voigt’s scheme TSDT

1. Introduction

The capabilities of the advanced composite material always attracted the researchers because we can obtain the desired properties with change in the fiber orientation, volume fraction, and geometric parameters. Functionally graded materials (FGM) with smooth gradation of properties over the two surfaces which replicate skin and/or bone materials, can be play a vital role in bio-medical sectors.

In general, various displacement field theories are employed to develop such models for various analyses [1]. Thai et al. [2] analyzed the bending, vibration and buckling behavior of functionally graded (FG) plates by employing a new sinusoidal shear deformation theory with four unknowns only. Governing equations and material properties were obtained using Hamilton’s principle and power-law formulation, respectively. Reddy et al. [3] employed first order shear deformation theory (FSDT) to analyze the bending and stretching of FG solid and annular plates. Brischetto et al. [4] proposed a model based on Reissner’s mixed variation theorem (RMVT). FGM plates were transversely loaded and properties were graded according to the Legendre’s polynomial. Compelling results of RMVT shows the improvement when compared with the models based on principle of virtual displacement (PVD). Zidi et al. [5] employed a four variable refined plate theory without any shear correction factor for bending analyses of FGM plates experiencing hygro-thermo-mechanical load. Governing equations were obtained using principle of virtual displacement and material properties varied according to the simple rule of mixture. Croce and Venini [6] developed an extended model based on Reissner-Mindlin plate theory to manipulate the shear-locking phenomenon. Material properties were determined using power law formulation and variational principle derived the governing equations. Kar and Panda [7] obtained non-linear bending responses for FG spherical panels using higher-order finite element approach.

In the present investigation, the effect of various geometrical parameters on the bending responses of FG bio-composite plate is examined using Zirconia as ceramic and Titanium as metal materials. Deflection responses for FG bio-composite plate is obtained using third-order shear deformation mid- plane theory (TSDT). Voigt’s model with power law formulation is used to obtain the graded material properties. The governing equations are obtained using minimum total potential energy principle. A customized computer code is prepared based on FEM to obtain the bending responses. The proposed model is than, validated with the available literature to show its accuracy. Finally, the deflection of FGM plate is computed using the TSDT model by varying power law index ( $n$ ), aspect ratio ( $a r$ ) and thickness ( $h$ ) subjected to uniform loading.

Figure 1.

Geometrical description of FG bio-composite plate.

2. Mathematical formulation

In the present work, a FG bio-composite panel is considered with uniform thickness ‘ $h$ ’, length ‘ $a$ ’ and breadth ‘ $b$ ’ which is shown in Fig. 1.

2.1 Effective material properties

Effective material properties by employing Voigt’s model are as follows (Gibson et al., 1995):

$\displaystyle P=(P_{m}-P_{c})V_{fm}+P_{c}$ (1)

where, $P_{m}$ and $P_{c}$ are the metal and ceramic material properties respectively, $V_{fm}$ is the volume fraction of the evaluated by power law distribution (Shen et al. [8]) as follows:

$\displaystyle V_{fm}=\left|\left(\frac{z}{h}+\frac{1}{2}\right)\right|$ (2)

where, power law index ( $n$ ) must be within the limits (0 $\leqslant n<\infty$ ).

2.2 Displacement field

The TSDT mid-plane kinetics is used in this study for the FG panel to derive the mathematical model [7]:

$\displaystyle\begin{rcases}u=u_{0}+zu_{1}+z^{2}u_{2}+z^{3}u_{3}\\ v=v_{0}+zv_{1}+z^{2}v_{2}+z^{3}v_{3}\\ w=w_{0}\end{rcases}$ (3)

where, $z$ is the thickness coordinate varies from $-h/2$ to $+h/2$ . $(u_{0},v_{0},w_{0})$ are the displacements in $x$ , $y$ and $z$ axis. $u_{1}$ and $v_{1}$ are the rotations about the $y$ and $x$ axis, respectively. Other higher order terms $u_{2}$ , $u_{3}$ , $v_{2}$ and $v_{3}$ are obtained by the Taylor series expansion which is defined in the mid-plane.

The equilibrium equation for static analysis is obtained by

using energy equations as mentioned in Kar and Panda [7] and presented as:

$\displaystyle[K]\{\delta\}=\{q\}$ (4)

where, is the global displacement and $[K]$ is the global stiffness matrix and $\{\delta\}$ is the global displacement vector.

3. Results and discussions

3.1 Convergence and validation

Deflection responses are evaluated for the present model to analyse the convergence behaviour and validate with the available literature. Bending behaviour of simply supported (SSSS) FG Bio-Composite square plate is shown in Fig. 2, which reveals that meshing with 36 elements is giving optimum responses. Deflection responses for a FG bio-composite panel under uniform load is obtained and compared with the results of Ganapati et al. [9]. Figure 3 shows that the present results are in good agreement with the literature. Slightly higher deflection responses reveal that our model is more flexible as more number of variables is taken into consideration.

3.2 Numerical experimentation

A customised code is prepared in MATLAB to analyse the bending behaviour of simply supported FG bio-composite panel ( $a=b=1,h=a/50,Q=100$ ) under uniform load. The material properties of titanium and zirconia are given in Table 1, which are used throughout the analyses.

Table 1
Elastic properties of bio-compatible materials [10]

	Properties
Materials	Young’s modulus E (GPa)	Poisson’s ratio ( $\nu$ )
Zirconia (ZrO ${}_{2}$ )	200	0.3
Titanium (Ti)	110	0.3

Figure 2.

Dimensionless deflection for simply supported FG bio-composite (Ti/ZrO ${}_{2}$ ) panel. ( $h=$ 1/50, $Q=$ 100).

Figure 3.

Dimensionless deflection for simply supported FG (Al/Al ${}_{2}$ O ${}_{3}$ ) panel.

Figure 4.

Variation of deflection with power law indices and aspect ratio for simply supported FG bio-composite panel. ( $h=$ a/50, $Q=$ 100).

Figure 5.

Variation of deflection with power-law indices and thickness ratio for simply supported FG bio-composite panel.

In Fig. 4 significance of aspect ratios on the bending responses of the FG bio-composite panel is revealed. Under uniform loading the deflection is reducing as the aspect ratio is increasing. Among all the cases, maximum deflection is obtained for the square panel. Figure 5 demonstrates that there is a significant increase in the deflection responses as thickness is reducing. It is also observed in Figs 4 and 5 that deflection is increasing as the power law index ( $n$ ) increases, which mean that material becomes less stiff as panel changes from ceramic rich to metal rich FG plate.

4. Conclusion

In the present numerical study, deformation characteristics of the FG bio-composite panel are examined under uniform loading. A customized MATLAB code is developed in finite element framework using TSDT kinematics. Two different bio-compatible materials such as Titanium and Zirconia are used, throughout in the analysis. However, overall material properties of FG bio-composite panel along the thickness direction are evaluated via power-law based Voigt’s material scheme. The present results are validated with the published results and the differences are found within the expected line. The numerical results reveal that the deflection parameters are higher in case of metal-rich, square and thin panels.

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