Abstract
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 , define a functional on such that , for each , where is the set of all displacement fields that correspond to the solutions of the torsion problem and depends only on the cross-section and the elastic properties of the considered cylinder. This 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 be a right cylinder of length with its cross-section , a multiply connected bounded regular region of with boundary , and let a system of rectangular Cartesian coordinates be introduced, with origin at the centroid of left end cross-section of as shown in Figure 1. Position vector of a generic point of with respect to is , where , , and are the unit base vectors. Let , be the bases of , and let be the mantle of according to Figure 1. The nonhomogeneous, monoclinic, piezoelectric cylinder is loaded only at its end cross-sections by tangential surface forces. The end tractions are specified aswhere the tangential surface forces and on () satisfy the following equations:
In (3) is the torque. Let denote the set of all displacement-electric potential fields 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 .
The aim of this paper is to derive a torque-generalized twist relationship which has the form where is the displacement field, 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 generalized twist is a functional defined on the solutions of the generalized (relaxed) torsion problem, that is, . 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 Ieşan [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 Ieşan [7, 8]. Day [4] defined the generalized twist for homogeneous isotropic elastic cylinder as the constant associated with Saint-Venant torsion field which approximates a mean square solution in the strain instead of exact solution. Podio-Guidugli [5] proved that more general result can be achieved if the strain norm is replaced by the energy norm.
In this paper, the concept of the generalized twist is introduced for the torsion of monoclinic, nonhomogeneous, piezoelectric cylinder. The material properties of the examined piezoelectric cylinder are smooth functions of the cross-sectional coordinates and ; they do not depend on the axial coordinate . For the generalized torsion of nonhomogeneous piezoelectric cylinder a global cross-sectional strain measure , which conforms to the concept of the rate of twist in the theory of Saint-Venant torsion, is introduced. The corresponding generalized twist , , is that constant which approximates most closely in mean square on . Truesdell’s problem for torsion of nonhomogeneous piezoelectric cylinders is solved by using the displacement-electric potential function formulation of Saint-Venant torsion [9, 10], the concept of the global cross-sectional strain measure and a reciprocal work theorem of the linear piezoelectricity [11], and methods given by Day [4] and Podio-Guidugli [5].
2. Governing Equations in Piezoelectricity
Consider a piezoelectric body occupying a three-dimensional domain with the boundary . The basic equations governing the elastic and electric fields in a linear piezoelectric material referring to a rectangular Cartesian coordinate system can be summarized in the following [12–14].
Strain-displacement and electric field-electric potential relationships:where is the displacement vector, is the strain tensor, is the electric potential, is the electric field vector, and is the three-dimensional gradient operator. In formula (5) of the strain-displacement relationship the dyadic product of two vectors is denoted by circle.
Constitutive equations:where is the stress tensor, is the electric displacement vector, is the elastic modulus tensor measured in a constant electric field, is the piezoelectric tensor, and is the dielectric tensor measured at constant strains. In (6) the scalar product is indicated by dot.
Equilibrium equations:In (7), is the body force vector per unit volume and is the intrinsic electric charge per unit volume.
Boundary conditions (BCs):where is the traction vector, is the surface charge, is the outward unit normal vector to surface , and the over barred quantities indicate given values and we have
Equations (5)–(7) under the boundary conditions (8)-(9) form a complete mathematical description of the coupled elastic and electric fields in a general anisotropic piezoelastic body. For an elastic material, there is no coupling of the elastic and electric fields, that is, the piezoelectric tensor . In this case, (5)–(7) will be decoupled between the two fields, yielding the usual elasticity equations and “Poisson equation” for the electric potential. Starting from (5)–(9) and by the use of symmetry properties of , , and [13–15]the next reciprocal work theorem can be derived [11]
in which and are two sets of the admissible solutions satisfying (5)–(9). Here, we note in (6) that tensors , , and are smooth functions of the coordinates , , and on .
3. Saint-Venant’s Torsion
The analytical solution of Saint-Venant’s torsion for nonhomogeneous, monoclinic, piezoelectric beams (Figure 1) originates from the next displacement and electric potential hypothesis [9, 10] where is the rate of twist with respect to the axial coordinate , is the torsion function, and the cross denotes the vectorial product of two vectors. Starting from the governing equations of piezoelectricity and following the derivation presented by Rovenski et al. [9, 10], we can derive the next formulas for stress tensor and electric displacement vector [16]
whereHere, is the two-dimensional gradient (del) operator.
is the matrix of elastic stiffness tensor [9, 10], where the matrix elements , , and are the shear rigidities measured under the conditions of constant electric field and they are written in compressed notations [17].
is the matrix of the tensor of piezoelectric constants [9, 10], where , , , and are the piezoelastic stress constants; they are written in compressed notations [17].
is the matrix of the dielectric tensor measured at constant strain field [9, 10].
Upper T in (15) indicates the operation of transpose.
From equilibrium equation (7) with and and the boundary conditions and on (Figure 1), it follows that Ecsedi and Baksa [16] and Rovenski et al. [9, 10] show that and are solutions of the following coupled nonhomogeneous Neumann’s boundary value problem defined on the cross-sectional domain (Figure 2):In (16)-(17), is the outer unit normal vector to the boundary curve (Figure 2).
The torsional rigidity of the monoclinic, piezoelectric cylinder is denoted by and it can be computed as [9, 10]From (16)–(18) after some manipulations we get
In (19) is an arc-length defined on the boundary curve of the cross-sectional domain (Figure 2). Combination of (18) with (19) gives the next formula for On the other hand we have from (15) and the electric boundary condition formulated by (17)
Substitution of (21) into (20) yields a new formula for the torsional rigiditywhich shows that is always positive since and are second-order two-dimensional symmetric positive definite tensors [9, 10, 13, 14]. This statement is in accordance with the mechanical meaning of .
4. Generalized Torsion and Generalized Twist
A sufficiently smooth equilibrium displacement-electric potential field is a solution of the generalized torsion for the monoclinic piezoelectric cylindrical beams shown in Figure 1 if(i)(5)–(7) under the conditions , in are satisfied,(ii) and on ,(iii) and on and ,(iv) and , where is a given value.The set of all solutions of the generalized torsion for a monoclinic piezoelastic cylinder is denoted by . One of the solutions of the generalized torsion problem is Saint-Venant’s torsion (uniform torsion) whose torsion function-electric potential function formulation is summarized in Section 3 of this paper. It is evident that , .
The in-plane cross-sectional displacement for the cylindrical piezoelectric beam shown in Figure 1 is defined as . It is obvious in the case of Saint-Venant’s torsion that we have whereFor the generalized torsion we define a global cross-sectional strain measure as
Equation (24) formulates that the zero virtual work is done by on the displacement field for an arbitrary cross-section of the considered piezoelastic cylinder. From (18) and (24) it follows thatIf U is given by (23) then we have . The rate of twist for the generalized torsion, which depends on the axial coordinate, is defined by the following equation:
It is evident in the case of Saint-Venant torsion that . The generalized twist associated with is defined as the unique number that minimizes the functionthat is
Solution of minimization problem (28) is as follows:The kinematic interpretation of the generalized twist is given by (26) and (28). It is obvious for Saint-Venant’s torsion . Here, we note, for rigid body displacement field, that , where a and b are arbitrary constant vectors, and, for constant electric potential, This statement follows from the next equations
according to (15) and (16)-(17). A simple computation gives It is evident that than in this case .
In the next part of this section we prove the validity of formula (4). By the application of reciprocity theorem (12) for an arbitrary solution of generalized torsion and Saint-Venant’s torsion with unit value of the rate of twist we get By the use of definitions of , and of the generalized twist and of the next equations
from (32) we obtain the relationship between the applied torque and the generalized twist as
5. Example 1: A Generalized Torsion Problem of Piezoelectric Cylinder
Let us consider the next 3D boundary value problem of monoclinic piezoelectric cylindrical beam shown in Figure 1:For given values of , , , and the boundary value problem formulated by (35) has a unique solution. From the conditions of equilibrium it follows that the section forces
and the section momentsvanish and the torque
does not depend on the axial coordinate .
This mixed type 3D boundary value problem is a generalized torsional problem specified by the given surface displacements , and , . By the application of formulas (25), (29), and (34) we get the value of the torque transmitted by the beam in terms of , , , and without knowing the solution of the corresponding mixed type 3D piezoelectric boundary value problem.
6. Example 2: Generalized Twist for Hollow Circular Cylinder Made of Orthotropic Material
The cross-section of the considered hollow circular cylinder is shown in Figure 3. The hollow circular cylinder is made of orthotropic piezoelectric material. In this case . Let us suppose that and be [9]. In this case we have [16]In (41) the radius of inner boundary circle is denoted by and the radius of outer boundary circle is indicated by (Figure 3).
The next generalized torsion problem will be considered according to Example 1:A simple computation based on (25) gives the next result
Substitution results obtained above into (29) gives the expression of generalized twist for the considered torsional problemIn the case of Saint-Venant’s torsion we have ; that is .
7. Torque-Generalized Twist Relationship for End Loaded Piezoelectric Cylinders
The equilibrium displacement-electric potential fields for end loaded, nonhomogeneous, monoclinic, piezoelectric cylinder are defined as a solution of (5)–(7) under the next conditions The normal stresses and the axial components of the electric displacement at the end cross-sections satisfy the end load symmetry conditions which are formulated by (46) and (47). In this case we have the cross-sections of the considered monoclinic piezoelectric cylinder which are subjected to axial force , bending moment , and torque . The expressions of the axial force and the bending moment vector are as follows:
Following the deduction presented in Section 4 of this paper, it can be proven that if (45)–(48) are satisfied then the torque-twist relationship (34) is yet right in which is defined according to (25) and (29).
8. Conclusions
In this paper the generalized torsion of nonhomogeneous, monoclinic, piezoelectric cylinder is examined. By the use of displacement-electric potential functions formulation of Saint-Venant’s torsion and a reciprocal work theorem of linear piezoelectricity, a torque-generalized twist relationship is proven which gives the solution of a problem proposed by Truesdell.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was (partially) carried out in the framework of the 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.