Research Article
The Generalized Twist for the Torsion of Piezoelectric Cylinders
István Ecsedi and Attila Baksa
Institute of Applied Mechanics, University of Miskolc, Egyetemv´ aros, Miskolc 3515, Hungary
Correspondence should be addressed to Attila Baksa; attila.baksa@uni-miskolc.hu
Received 20 March 2017; Accepted 11 May 2017; Published 15 June 2017
Academic Editor: Dimitrios E. Manolakos
Copyright © 2017 Istv´ an Ecsedi and Attila Baksa. Tis is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
In the classical theory of elasticity, Truesdell proposed the following problem: for an isotropic linearly elastic cylinder subject to
end tractions equipollent to a torque , defne a functional (u) on such that =(u), for each u ∈, where is the set of all
displacement felds that correspond to the solutions of the torsion problem and depends only on the cross-section and the elastic
properties of the considered cylinder. Tis problem has been solved by Day. In the present paper Truesdell’s problem is extended
to the case of piezoelastic, monoclinic, and nonhomogeneous right cylinders.
1. Introduction
Let =×]0,[ be a right cylinder of length with its cross-
section , a multiply connected bounded regular region
of R
2
with boundary , and let a system of rectangular
Cartesian coordinates be introduced, with origin at the
centroid of lef end cross-section of as shown in Figure 1.
Position vector of a generic point of with respect to is
r =
1
e
1
+
2
e
2
+
3
e
3
= R +
3
e
3
, where e
1
, e
2
, and e
3
are the unit base vectors. Let
1
= {(
1
,
2
) ∈ ,
3
= 0},
2
= {(
1
,
2
) ∈ ,
3
= } be the bases of , and let
3
= × [0,] be the mantle of according to Figure 1.
Te nonhomogeneous, monoclinic, piezoelectric cylinder is
loaded only at its end cross-sections by tangential surface
forces. Te end tractions are specifed as
p
=
(
1
,
2
) e
1
+
(
1
,
2
) e
2
on
(=1,2), (1)
where the tangential surface forces
=
(,) and
=
(,) on
(=1,2) satisfy the following equations:
∫
(
1
,
2
) d=∫
(
1
,
2
) d=0,
(=1,2),
(2)
=∫
2
(
1
2
−
2
2
) d
=−∫
1
(
1
1
−
2
1
) d.
(3)
In (3) is the torque. Let
denote the set of all dis-
placement-electric potential felds that correspond to the
solution of the torsion problem for a prescribed value of . In
this relaxed torsion problem the pointwise assignment of the
terminal tangential tractions is replaced by the corresponding
value of the resultant torque .
Te aim of this paper is to derive a torque-generalized
twist relationship which has the form
=(u,), (4)
where u is the displacement feld, is the electric potential,
and is a positive constant which depends only on the
geometry of the cross-section and the material properties of
the considered piezoelastic cylinder. Moreover, the general-
ized twist (u,) is a functional defned on the solutions of
the generalized (relaxed) torsion problem, that is, (u,) ∈
. For isotropic, homogeneous, linearly elastic cylinder
the torque-generalized twist relationship as a problem was
formulated by Truesdell [1–3]. Day [4], Podio-Guidugli
[5], and Ies ¸an [6–8] presented the solution of Truesdell’s
problem for extension, bending, and torsion of isotropic,
linearly elastic, homogeneous cylinder. A detailed analysis
and the solution of Truesdell’s problem for anisotropic,
homogeneous/nonhomogeneous elastic and Cosserat elastic
cylinders were presented by Ies ¸an [7, 8]. Day [4] defned
the generalized twist for homogeneous isotropic elastic
cylinder as the constant associated with Saint-Venant torsion
Hindawi
Modelling and Simulation in Engineering
Volume 2017, Article ID 3634520, 7 pages
https://doi.org/10.1155/2017/3634520