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Two-dimensional solution for electro-mechanical behavior of functionally graded piezoelectric hollow cylinder

دوشنبه, ۱۲ شهریور ۱۳۹۷، ۰۷:۳۰ ق.ظ

J.Jafari FesharakiaV.Jafari Fesharakib

 

 

https://www.sciencedirect.com/science/article/pii/S0307904X12000340

 

Two-dimensional solution for electro-mechanical behavior of functionally graded piezoelectric hollow cylinder

 

https://doi.org/10.1016/j.apm.2012.01.019Get rights and content

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Abstract

In this paper, the general theoretical analysis for a hollow cylinder made of functionally graded piezoelectric material subjected to two-dimensional electromechanical load, is developed. The material properties, except the Poisson’s ratio, are assumed to vary with the power law function through the thickness of the cylinder. The mechanical and electrical displacements are assumed to be a function of radial and circumferential directions. By using the separation of variables method and complex Fourier series, the Navier equations in terms of displacements are derived and solved.

Keywords

Two-dimensional

Electric field

Piezoelectric

Functionally graded materials

Cylinder

Analytical solution

1. Introduction

Functionally graded materials (FGMs) have attracted widespread attention in recent years. A FGM is usually a combination of two material phases that the material properties continuously vary along certain directions. This continuous transition allows the certain multiple properties without any mechanically weak interface. This makes FGM suitable for many specific applications in advanced structures and modern technologies. Several investigators represent analytical solutions about these kinds of materials [1], [2], [3], [4]. These works often carried out theoretical analyses of the symmetry condition problems but in many cases such as actuators, the two-dimensional conditions must be considered. Jabbari et al. [5] presented the general solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to non-axisymmetric steady-state loads. They used the separation of variables and complex Fourier series to solve the heat conduction and Navier equations. Eslami et al. [6] obtained an exact solution for thermal and mechanical stresses in a functionally graded thick sphere. Dai and Fu [7]developed the magneto thermo elastic interactions in hollow structures of functionally graded material subjected to mechanical loads. Alibeigloo [8] presented the exact solution for thermo-elastic response of functionally graded rectangular plates. Yas and Sobhani Aragh [9] studied the three-dimensional analysis for thermo elastic response of functionally graded fiber reinforced cylindrical panel. Tianhu et al. [10]investigated a two-dimensional generalized thermal shock problem for a half-space in electro magneto-thermo elasticity. They used the Laplace transform and numerical Laplace inversion to formulate and solve the generalized electro magneto-thermo elastic. By using the Legendre polynomials and the system of Euler differential equations, Poultangari et al. [11] solved the problem of functionally graded hollow spheres under non-axisymmetric thermo-mechanical loads. Peng and Li [12] showed the thermal stress in rotating functionally graded hollow circular disks. Ootao and Tanigawa [13]presented the transient thermo elastic problem of functionally graded thick strip due to non uniform heat supply. The Laplace transform and finite difference methods are used to analyze the magneto thermo elastic problem of multilayered conical shells subjected to magnetic and vapor fields by Lee [14]. Jabbari et al. [15]used the generalized Bessel function to investigate the axisymmetric mechanical and thermal stresses in thick short length FGM cylinders. Alibeigloo [16] studied the thermo elasticity analysis of functionally graded beam with integrated surface piezoelectric layers. Asghari and Ghafoori [17] investigated the three-dimensional elasticity solution for functionally graded rotating disks. Khoshgoftar et al. [18] presented the thermo elastic analysis of a thick walled cylinder made of functionally graded piezoelectric material. By using the separation of variables Dube et al. depicted the exact solution for electro thermo mechanical radially polarized circular cylindrical shell panel in cylindrical bending under electrostatic excitation and thermal fields [19]. Dumir et al. presented an analytical solution for piezoelectric orthotropic cylindrical panel in cylindrical bending with simply supported boundary conditions [20]. They used the Fourier series to satisfying the simply supported boundary conditions along longitudinal edges.

In this paper, a hollow cylinder made of functionally graded piezoelectric material (FGPM) subjected to non- axisymmetric electrical and mechanical loads is considered. The functionally graded piezoelectric material properties of the cylinder are assumed to be varied along radial direction by power functions in r. By using the complex Fourier series, The Navier equations in terms of displacements are derived and solved, and by this kind of analysis, many stresses and displacements or combination of them in boundary conditions is accepted.

2. Basic formulations of the problem

Consider a thick hollow cylinder made of functionally graded piezoelectric material as shown in Fig. 1 with inner and outer radius of a and b respectively. u and vare the displacement components in the radial (r) and circumferential (θ) directions respectively, thus in the cylindrical coordinate the relations between the strain and displacements are(1)εrr=∂u∂r,εθθ=1r∂v∂θ+ur,εrθ=121r∂u∂θ+∂v∂r-vr.The stress–strain relations for the FGPM hollow cylinder can be obtained from(2)Trr=k11εrr+k12εθθ+e11∂φ∂r,Tθθ=k21εrr+k22εθθ+e21∂φ∂r,Trθ=2k31εrθ+e311r∂φ∂θ,where Tij and ɛij (i, j = r, θ) are the stress and strain tensors respectively and φ is the electric potential. kij and eij (i, j = 1, 2, 3) are elastic and piezoelectric coefficients respectively. The electrical displacements in radial and circumferential directions, which include the strain and electrical field, can be written as(3)Dr=e11εrr+e21εθθ-q41∂φ∂r,Dθ=2e31εrθ-q511r∂φ∂θ,where Di (i = r, θ) and qi1 (i = 4, 5) are the electric displacement and dielectric constant respectively. All material coefficients are graded through the radial direction thus the material properties are functions of ras(4)kii=kii0rβ,eii=eii0rβ,qii=qii0rβ(i,j=1,2,3),where the superscript zero denotes the corresponding value at the outer surface of the FGPM hollow cylinder, and β is the power-law indices of the material.

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Fig. 1. Geometric model for FGPM hollow cylinder under two dimensional electro mechanical load.

The equilibrium equations of the FGPM hollow cylinder in the radial and circumferential directions, irrespective of the body force and the inertia terms and the charge equation of electrostatic, are expressed as(5)∂Trr∂r+1r∂Trθ∂θ+Trr-Tθθr=0,∂Trθ∂r+1r∂Tθθ∂θ+2Trθr=0,∂Dr∂r+1r∂Dθ∂θ+Drr=0.Using Eqs. (1), (2), (3), (4), (5), three coupled governing differential equations for the problem are obtained(6)∂2u∂r2+d1r∂u∂r+d2r2u+d3r2∂2u∂θ2+d4r∂2v∂r∂θ+d5r2∂v∂θ+d6∂2φ∂r2+d7r2∂2φ∂θ2+d8r∂φ∂r=0,∂2v∂r2+d9r∂v∂r-d9r2v+d10r2∂2v∂θ2+d11r∂2u∂r∂θ+d12r2∂u∂θ+d13r2∂φ∂θ+d14r∂2φ∂r∂θ=0,∂2φ∂r2+d15r2∂2φ∂θ2+d9r∂φ∂r+d16∂2u∂r2+d17r∂u∂r+d18r2u+d19r2∂2u∂θ2+d20r∂2v∂r∂θ+d21r2∂v∂θ=0,where the constants d1 to d21 are given in the appendix.

3. Solution of the problem

To solve Eq. (6) consider the complex Fourier series for displacements u(r, θ) and v(r, θ) and electric potential φ(r, θ) as(7)u(r,θ)=∑n=-∞∞un(r)einθ,v(r,θ)=∑n=-∞∞vn(r)einθ,φ(r,θ)=∑n=-∞∞φn(r)einθ,where un(r), vn(r) and φn(r) are the coefficients of the complex Fourier series of un(r, θ), vn(r, θ) and φn(r, θ), respectively, and are(8)un(r)=12π∫-ππu(r,θ)e-nθdθ,vn(r)=12π∫-ππv(r,θ)e-inθdθ,φn(r)=12π∫-ππφ(r,θ)e-inθdθ.Substituting Eq. (7) into Eq. (6) yields(9)un″+d1run′+d2r2un-d3r2n2un+d4rvn′in+d5r2vnin+d6φn″-d7r2n2φn+d8rφn′=0,(10)vn″+d9rvn′-d9r2vn-d10r2n2vn+d11run′in+d12r2unin+d13r2φnin+d14rφn′in=0,(11)φn″-d15r2n2φn+d9rφn′+d16un″+d17run′+d18r2un-d19r2n2un+d20rvn′in+d21r2vnin=0.Eqs. (9), (10), (11) are a system of ordinary differential equations with non-constant coefficients. The solution of the this system are assumed as(12)un(r)=Arξ,vn(r)=Brξ,φn(r)=Crξ,where A, B and C are the unknown constants and by using the specified boundary conditions is determined. Substituting Eq. (12) into Eqs. (9), (10), (11), yields(13)[ξ(ξ-1)+d1ξ+d2-n2d3]A+[d4ξ+d5]inB+[d6ξ(ξ-1)-d7n2+d8ξ]C=0,[d11ξ+d12]inA+[ξ(ξ-1)+d9ξ-d9+n2d10]B+[d13+d14ξ]inC=0,[d16ξ(ξ-1)+d17ξ+d18-n2d19]A+[d20ξ+d21]inB+[ξ(ξ-1)-d15n2+d9ξ]C=0,Eq. (13) are a system of algebraic equations. For obtaining the nontrivial solution of the equations, the determinant of system should be equal to zero. So the six roots, ξn1 to ξn6 for the equations, are achieved and the general solutions are(14)un(r)=∑j=16Anjrξnj,vn(r)=∑j=16NnjAnjrξnj,φn(r)=∑j=16MnjAnjrξnj,where Nnj is the relation between constant Anj and Bnj, and Mnj is the relation between constants Anj and Cn and is obtained from Eq. (13) as(15)Nnj=[ξ(ξ-1)+d1ξ+d2-n2d3]·[d13+d14ξ]in-[d6ξ(ξ-1)-d7n2+d8ξ]·[d11ξ+d12]in[d6ξ(ξ-1)-d7n2+d8ξ]·[ξ(ξ-1)+d9ξ-d9+n2d10]-[d4ξ+d5]in[d13+d14ξ]in,(16)Mnj=[d4ξ+d5]in[d16ξ(ξ-1)+d17ξ+d18-n2d19]-[ξ(ξ-1)+d1ξ+d2-n2d3]·[d20ξ+d21]in[d6ξ(ξ-1)-d7n2+d8ξ]·[d20ξ+d21]in-[d4ξ+d5]in[ξ(ξ-1)-d15n2+d9ξ],j=1,…,6.If n = 0, because of Eq. (10) is independent of the Eqs. (9), (11), the coefficients Nnj and Mnj are undefined, and for this case, Eqs. (9), (10), (11) are reduced to(17)u0″+d1ru0′+d2r2u0+d6φ0″+d8rφ0′=0,(18)v0″+d9rv0′-d9r2v0=0,(19)φ0″+d9rφ0′+d16u0″+d17ru0′+d18r2u0=0,where the subscript zero indicates the solution for n = 0. Two equations (17), (19) are a system of ordinary differential equations and the solution of this system is considered as(20)u0(r)=A0rξ0,φ0(r)=C0rξ0.Substituting Eq. (20) into Eqs. (17), (19), yields(21)(ξ02-ξ0+d1ξ0+d2)A0+(d6ξ02-d6ξ0+d8ξ0)C0=0,(d16ξ02-d16ξ0+d17ξ0+d18)A0+(ξ02-ξ0+d9ξ0)C0=0.To obtain the nontrivial solution of Eq. (21), the determinant of system should be equal to zero. So the four roots, ξ01 to ξ04, are achieved and the general solutions are(22)u0(r)=∑j=14A0jrξ0j,φ0(r)=∑j=14R0jA0jrξ0j,where(23)R0j=-(ξ02-ξ0+d1ξ0+d2)(d6ξ02-d6ξ0+d8ξ0).For n = 0 Eq. (18) is a decoupled ordinary differential equation and the solution of this equation is considered as(24)v0=∑j=56A0jrξ0j,where(25)ξ05,06=1-d9±(d9-1)2+4d92.Thus by substituting Eqs. (14), (22), (24) into Eq. (7), the general solutions for u(r, θ), v(r, θ) and φ(r, θ) are expressed as(26)u(r,θ)=∑j=14A0jrξ0j+∑n=-∞,n≠0∞∑j=16Anjrξnjeinθ,(27)v(r,θ)=∑j=56A0jrξ0j+∑n=-∞,n≠0∞∑j=16NnjAnjrnjξeinθ,(28)φ(r,θ)=∑j=14R0jA0jrξ0j+∑n=-∞,n≠0∞∑j=16MnjAnjrξnjeinθ.Substituting Eqs. (26), (27), (28) into Eq. (1), the strains are obtained as(29)εrr=∑j=14A0jξ0jrξ0j-1+∑n=-∞,n≠0∞∑j=16Anjξnjrξnj-1einθ,(30)εθθ=∑j=14A0jrξ0j-1+∑n=-∞,n≠0∞∑j=16Anjrξnj-1einθ+∑n=-∞,n≠0∞∑j=16NnjAnjrξnj-1ineinθ,(31)εrθ=12∑n=-∞,n≠0∞∑j=16Anjrξnjineinθ+∑j=56A0jξ0jrξ0j-1+∑n=-∞,n≠0∞∑j=16NnjAnjξnjrξnj-1einθ-∑j=56A0jrξ0j-1+∑n=-∞,n≠0∞∑j=16NnjAnjrξnj-1einθ.Substituting Eqs. (29), (30), (31) into Eq. (2) and by using Eq. (4), (28), the stress components are obtained as(32)Trr=k110rβ∑j=14A0jξ0jrξ0j-1+∑n=-∞,n≠0∞∑j=16Anjξnjrξnj-1einθ+k120rβ-1∑n=-∞,n≠0∞∑j=16NnjAnjrξnjineinθ+k120rβ-1∑j=14A0jrξ0j+∑n=-∞,n≠0∞∑j=16Anjrξnjeinθ+e110rβ∑j=14R0jA0jξ0jrξ0j-1+∑n=-∞,n≠0∞∑j=16MnjAnjξnjrξnj-1einθ,(33)Tθθ=k210rβ∑j=14A0jξ0jrξ0j-1+∑n=-∞,n≠0∞∑j=16Anjξnjrξnj-1einθ+k220rβ-1∑n=-∞,n≠0∞∑j=16NnjAnjrξnjineinθ+k220rβ-1∑j=14A0jrξ0j+∑n=-∞,n≠0∞∑j=16Anjrξnjeinθ+e210rβ∑j=14R0jA0jξ0jrξ0j-1+∑n=-∞,n≠0∞∑j=16MnjAnjξnjrξnj-1einθ,(34)Trθ=k310rβ-1∑n=-∞,n≠0∞∑j=16Anjinrξnjeinθ+k310rβ∑j=56β0jξ0jrξ0j-1+∑n=-∞,n≠0∞∑j=16NnjAnjξnjrξnj-1einθ-k310rβ-1∑j=56β0jrξ0j+∑n=-∞,n≠0∞∑j=16NnjAnjrξnjeinθ+e310rβ-1∑n=-∞,n≠0∞∑j=16MnjAnjinrξnjeinθ.Substituting Eqs. (29), (30), (31) into Eq. (3) and utilizing Eqs. (4), (28), the electrical displacements in radial and circumferential directions are obtained as follows:(35)Dr=e110rβ∑j=14A0jξ0jrξ0j-1+∑n=-∞,n≠0∞∑j=16Anjξnjrξnj-1einθ+e210rβ-1∑n=-∞,n≠0∞∑j=16NnjAnjrξnjineinθ+e210rβ-1∑j=14A0jrξ0j+∑n=-∞,n≠0∞∑j=16Anjrξnjeinθ-q410rβ∑j=14R0jA0jξ0jrξ0j-1+∑n=-∞,n≠0∞∑j=16MnjAnjξnjrξnj-1einθ,(36)Dθ=e310rβ-1∑n=-∞,n≠0∞∑j=16Anjinrnjξeinθ+e310rβ∑j=56β0jξ0jrξ0j-1+∑n=-∞,n≠0∞∑j=16NnjAnjξnjrξnj-1einθ-e310rβ-1∑j=56β0jrξ0j+∑n=-∞,n≠0∞∑j=16NnjAnjrξnjeinθ-q510rβ-1∑n=-∞,n≠0∞∑j=16MnjAnjinrξnjeinθ.It is recalled that Eqs. (26), (27), (28), (29), (30), (31), (32), (33), (34), (35), (36) contain six unknown constants Anj(j = 1, … , 6) and therefore to evaluate these constants, six boundary conditions that may be either the given displacements or stresses, or combinations are required. Expanding the given boundary conditions in complex Fourier series gives(37)xj(θ)=∑n=-∞∞Xj(n)einθ,j=1,…,6,where(38)Xj(n)=12π∫-ππxj(θ)e-inθdθ,j=1,…,6.With the help of Eqs. (37) and (38), and substituting the six boundary conditions into Eqs. (26)–(36), the constants Anj are calculated.

4. Results and discussion

To examine the proposed solution, three different examples are considered. In the first one, to validate the results, the boundary conditions, geometry and material properties are adopted from Ref. [21]. Fig. 2 shows the electric potentials for various values of material inhomogeneity β. The graphs are identical to those reported in Figs. 10–12 of Ref. [21] for the functionally graded piezoelectric hollow cylinder without any rotation (Ω = 0).

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Fig. 2. Electric potential for different values of material inhomogeneity (Example 1).

Let us consider a thick hollow cylinder made of functionally graded piezoelectric material with inner radius a = 1 m and outer radius b = 1.2 m for second and third examples. In the second example problem we consider the effect of two-dimensional electric potential, while the third example is chosen to show the effect of two-dimensional electro-mechanical behavior of functionally graded piezoelectric material. The material chosen for this purpose is PZT-4 and the material constants are taken as [22]k110=k220=139GPa,k120=k210=78GPa,k310=7.63GPa,e110=-5.2C/m2,e210=-5.2C/m2,e310=12.7C/m2,q410=6.5×10-9C2/Nm2,q510=6.5×10-9C2/Nm2.In this example, consider the inside and outside boundary conditions are expressed as:φ(a,θ)=100cos2θ,φ(b,θ)=0,Trr(a,θ)=0,Trr(b,θ)=0,u(b,θ)=0,v(b,θ)=0.For a given power-law indices of the material, β = 1, Fig. 3 shows electric potential distribution in the wall thickness along the radius and circumferential directions. It is noted that, due to the given boundary conditions the electric potential follow the pattern of the electric potential distribution at the internal surface of the cylinder. The radial and circumferential displacements in the cylinder are shown in Fig. 4, Fig. 5. Fig. 6, Fig. 7, Fig. 8, show the distribution of the radial, shear and circumferential stresses in the cross section of the cylinder. The radial stress is zero at the internal surface due to the assumed boundary conditions. The electric radial and circumferential displacements in the FGPM hollow cylinder are shown in Fig. 9, Fig. 10. It is note that all components of stresses, displacements, electric potential and electric displacement follow a harmonic pattern in the cross section of the cylinder.In the third example, the previous thick-walled cylinder subjected to two-dimensional electric potential and mechanical pressure at inner surface. The electric, stress and displacement boundary conditions are assumed to byφ(a,θ)=50cos2θ,φ(b,θ)=0,Trr(a,θ)=2×105cos2θ,Trr(b,θ)=0,u(b,θ)=0,v(b,θ)=0.Fig. 11 shows the electric potential distribution in the cylinder that satisfies the given boundary conditions. It can be seen that the electric potential arise at the middle of the wall of cylinder. Fig. 12, Fig. 13 show the variation of the radial and circumferential displacement distributions along the FGPM hollow cylinder. It can be seen easily from this figures that the boundary conditions which are considered were satisfied along the cylinder. By comparing Fig. 4, Fig. 12 for radial displacements one notes that the pressure has more apparent effect on radial displacement along the thickness of the cylinder. The radial, shear and circumferential stresses are depicted in Fig. 14, Fig. 15, Fig. 16.

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Fig. 3. Electric potential distribution in the FGPM hollow cylinder (Example 2).

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Fig. 4. Radial displacement in the FGPM hollow cylinder (Example 2).

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Fig. 5. Circumferential displacement in the FGPM hollow cylinder (Example 2).

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Fig. 6. Radial stress in the FGPM hollow cylinder (Example 2).

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Fig. 7. Shear stress in the FGPM hollow cylinder (Example 2).

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Fig. 8. Circumferential stress in the FGPM hollow cylinder (Example 2).

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Fig. 9. Radial electric displacement in the FGPM hollow cylinder (Example 2).

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Fig. 10. Circumferential electric displacement in the FGPM hollow cylinder (Example 2).

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Fig. 11. Electric potential distribution in the FGPM hollow cylinder (Example 3).

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Fig. 12. Radial displacement in the FGPM hollow cylinder (Example 3).

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Fig. 13. Circumferential displacement in the FGPM hollow cylinder (Example 3).

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Fig. 14. Radial stress in the FGPM hollow cylinder (Example 3).

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Fig. 15. Shear stress in the FGPM hollow cylinder (Example 3).

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Fig. 16. Circumferential stress in the FGPM hollow cylinder (Example 3).

It can be seen that the mechanical and electrical loads together, change the radial, shear and circumferential stresses along the thickness of the cylinder effectively. Fig. 17, Fig. 18 demonstrate the electric radial and circumferential displacements due to the given boundary conditions. So by considering the special boundary conditions and material properties for a FGPM hollow cylinder, the mechanical and electrical displacements and stresses can be controlled and optimized to design and use this kind of structures.

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Fig. 17. Radial electric displacement in the FGPM hollow cylinder (Example 3).

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Fig. 18. Circumferential electric displacement in the FGPM hollow cylinder (Example 3).

5. Conclusions

This paper presents the analytical solution for the two-dimensional electro mechanical behavior of a hollow cylinder made of FGPM. The material properties are assumed to vary with a power law function along the thickness of cylinder. The method of solution is based on the direct method and by using the complex Fourier series, the Navier equations were solved. The advantage of this method is its generality and from mathematical point of view, any type of the mechanical and electrical boundary conditions can be considered without any restrictions. By using this method and considering the special boundary conditions and material properties for a FGPM hollow cylinder, the mechanical and electrical displacements and stresses can be controlled and optimized to design and use this kind of structures.

Appendix A

d1=(1+β)k110+k120-k210k110,d2=βk120-k220k110,d3=k310k110,d4=k120+k310k110,d5=βk120-k310-k220k110,d6=e110k110,d7=e310k110,d8=(1+β)e110-e210k110,d9=1+β,d10=k220k310,d11=k310+k210k310,d12=(1+β)k310+k220k310,d13=(1+β)e310k310,d14=e310+e210k310,d15=q510q410,d16=e110-q410,d17=(1+β)e110+e210-q410,d18=βe210-q410,d19=e310-q410,d20=e210+e310-q410,d21=βe210-e310-q410.

References

 

 

https://www.sciencedirect.com/science/article/pii/S0307904X12000340

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