# Saint-Venant torsion of non-homogeneous orthotropic circular cylinder

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## Abstract

The object of this paper is the Saint-Venant torsion of a radially non-homogeneous, hollow and solid circular cylinder made of orthotropic piezoelectric material. The elastic stiffness coefficients, piezoelectric constants and dielectric constants have only radial dependence. This paper gives the solution of the Saint-Venant torsion problem for torsion function, electric potential function, Prandtl’s stress function and electric displacement potential function.

## Keywords

Piezoelectric Saint-Venant torsion Circular cylinder Non-homogeneous## 1 Introduction

Application of piezoelectric materials and structures has been increasing recently. Sensors and actuators are examples of active components made of piezoelectric materials which are used widely in smart structures. These structural components are often subjected to mechanical loading. The torsion of these structural members is an important task.

The Saint-Venant torsion of a homogeneous, isotropic elastic cylindrical body is a classical problem of elasticity [1, 2, 3], which is solved using a semi-inverse method by assuming a state of pure shear in the cylindrical body so that it gives rise to a resultant torque over the end cross sections. Extension of more complicated cases of anisotropic or non-homogeneous materials has been considered by Lekhnitskii [4, 5], Rooney and Ferrari [6], Davi [7], Bisegna [8, 9], Horgan and Chan [10], Rovenski et. al. [11, 12], Rovenski and Abramovich [13], Horgan [14], Ecsedi and Baksa [15, 16].

In this paper, the torsional deformation of radially non-homogeneous, piezoelectric, solid and hollow circular cylinders is studied. This study gives a non-trivial generalization of the results in paper [8], which deals with the Saint-Venant torsion of radially non-homogeneous anisotropic elastic circular cylinder.

The formulation of the Saint-Venant’s theory of uniform torsion for the homogeneous piezoelectric beams has been analyzed by Dave [7], Bisegna [8, 9] and Rovenski et al. [11, 12]. The papers of Bisegna [8, 9] use the Prandtl’s stress function and electric displacement potential function formulation for simply connected cross section. Davi [7] obtained a coupled boundary-value problem for the torsion function and for the electric potential function from a constrained three-dimensional static problem by the application of the usual assumptions of the Saint-Venant’s theory. Rovenski et al. [11, 12] give a torsion and electric potential function formulation of the Saint-Venant’s torsional problem for monoclinic homogeneous piezoelectric beams. In these papers [11, 12], a coupled Neumann problem is derived for the torsion and electric potential functions, where exact and numerical solutions for elliptical and rectangular cross sections are presented. Ecsedi and Baksa [15] give a formulation of the Saint-Venant torsional problem for homogeneous monoclinic piezoelectric beams in terms of Prandtl’s stress function and the electric displacement potential function. The Prandtl’s stress function and electric displacement potential function satisfy a coupled Dirichlet problem in the multiply connected cross section. A direct formulation and a variational formulation are developed in [15]. In another paper by Ecsedi and Baksa [16], a variational formulation is presented for the torsional deformation of homogeneous linear piezoelectric monoclinic beams. The variational formulation presented uses the torsion and electric potential functions as independent quantities of the considered variational functional. The mechanical meaning of the variational functional defined in [16] is also given. Examples illustrate the application of the presented variational functional. Rovenski and Abramovich apply a linear analysis to piezoelectric beams with non-homogeneous cross sections that consist of various monoclinic (piezoelectric and elastic) materials [13]. They give the solution procedure for extension, bending, torsion and shear. The developed method is illustrated by numerical examples [13]. Batra et al. [17] studied the electromechanical nonlinear deformations of homogeneous, transversely isotropic piezoelectric circular cylinder loaded on its end cross sections. In [17], the second-order constitutive equations are used and show that when the cylinder is deformed by applying pure torque and non-electric charges at the end cross sections the potential difference between the end cross sections is proportional to the square of twist.

In this paper, the deformation of circular cylinders made of orthotropic, radially non-homogeneous piezoelectric material is studied by means of Saint-Venant’s theory of uniform torsion. The elastic stiffness coefficients, piezoelectric constants and dielectric constants depend only on the radial coordinate. The dependence of material parameters is either described by smooth functions of radial coordinate as in the case of functionally graded materials [18, 19], or the material parameters are piecewise smooth functions of the radial coordinate as in the case of radially layered circular cylinders.

## 2 Formulation of Saint-Venant torsional problem

*L*. Let \(A_{1}\) and \(A_{2}\) be the bases and \(A_{3}=\partial A \times (0,L)\) the mantle of

*B*. The cross section

*A*is given in the Cartesian coordinate frame

*Oxyz*

*Oxyz*is supposed to be chosen in such a way that the

*Oz*-axis is parallel to the generators of the cylindrical boundary surface segments \(A_{3}=A_{3}^{'}\cup A_{3}^{''}\) (Fig. 1). The plane

*Oxy*contains the terminal cross section \(A_{1}\). The position of the end cross section \(A_{2}\) is given by \(z=L\). A point

*P*in \(B=B\cup A_{1}\cup A_{2}\cup A_{3}\) is indicated by the vector \(\varvec{r} = x\varvec{e}_{x} + y \varvec{e}_{y}+z\varvec{e}_{z}=\varvec{R}+z\varvec{e}_{z}\), where \(\varvec{e}_{x}\), \(\varvec{e}_{y}\) and \(\varvec{e}_{z}\) are the unit vectors of the coordinate system

*Oxyz*(Figs. 1, 2).

*V*of the twisted cylindrical bar can be represented as [11, 13]

## 3 Solution of the torsion problem

### Theorem 1

### Proof

## 4 Shearing stresses and electric displacement field

*T*and the rate of twist \(\vartheta \) is characterized by mechanical torsional rigidity \(S_{M}\) which is defined as

*T*, the circumferential component of the electric displacement vector can be computed from the following equation:

- (a)
The torsion function \(\omega =\omega (x,y)\) and the electric potential function \(\phi =\phi (x,y)\) are independent of the inhomogeneity of the cross section.

- (b)
For a given torque

*T*, the stress field is independent of the material parameters (\(A_{44}\), \(A_{55}\), \(e_{15}\), \(e_{24}\), \(\kappa _{11}\) and \(\kappa _{22}\)) and it depends only on the non-homogeneity of the considered circular cross section.

## 5 Prandtl’s stress function, electric displacement potential function

*U*does not depend on the polar angle \(\varphi \). In the next part of this paper, we consider only a simply connected (solid) cross section, that is \(R_{1}=0\), \(R_{2}=R\). It is known that the Prandtl’s stress function satisfies the homogeneous boundary condition:

## 6 Layered non-homogeneous circular cross section

## 7 Example

The dependence of Prandtl’s stress function *U* from the graded index \(\alpha \) is shown in Fig. 8. For \(\alpha =-1\), \(\alpha =-0.5\), \(\alpha =0\), \(\alpha =0.5\) and \(\alpha =1\), the graphs of electric displacement \(D_{\varphi }\) are shown in Fig. 9. The electric torsional rigidity as a function of \(\alpha \) is shown in Fig. 10. The dependence of electric displacement potential function *H* from \(\alpha \) is illustrated in Fig. 11.

## 8 Conclusions

The torsion function and the electric potential function are independent of the cross-sectional inhomogeneity.

For the given torque, the stress field is independent of the material parameters; it depends only on the non-homogeneity of the cross section.

## Notes

### Acknowledgements

Open access funding provided by University of Miskolc (ME). The described article/presentation/study was carried out as part of the EFOP-3.6.1-16-00011 “Younger and Renewing University—Innovative Knowledge City—institutional development of the University of Miskolc aiming at intelligent specialization” project implemented in the framework of the Szechenyi 2020 program. The realization of this project is supported by the European Union, co-financed by the European Social Fund. This research was (partial) carried out in the framework of Center of Excellence of Innovative Engineering Design and Technologies at the University of Miskolc and supported by the National Research Development and Innovation Office—NKFIH K115701.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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