On Torsion of Functionally Graded Elastic Beams

The evaluation of tangential stress fields in linearly elastic orthotropic Saint-Venant beams under torsion is based on the solution of Neumann and Dirichlet boundary value problems for the cross-sectional warping and for Prandtl stress function, respectively. A skillful solution method has been recently proposed by Ecsedi for a class of inhomogeneous beams with shear moduli defined in terms of Prandtl stress function of corresponding homogeneous beams. An alternative reasoning is followed in the present paper for orthotropic functionally graded beams with shear moduli tensors defined in terms of the stress function and of the elasticity of reference inhomogeneous beams. An innovative result of invariance on twist centre is also contributed. Examples of functionally graded elliptic cross sections of orthotropic beams are developed, detecting thus new benchmarks for computational mechanics.


Introduction
Analyses of composite media are a well-investigated research field in structural mechanics.Theoretical noteworthy results also in the nonlinear range have recently contributed to several engineering applications, such as beam and plate theories [1][2][3], fracture mechanics [4][5][6][7][8], hyperelastic media [9][10][11], concrete systems [12][13][14][15][16], nonlocal models [17][18][19], homogenization [20][21][22], thermoelasticity [23][24][25], nanostructures [26][27][28][29][30], and limit analysis [31][32][33].In the context of the classical theory of elasticity, an innovative methodology for the analysis of beams was proposed by Saint-Venant [34,35], with the assumption that the normal interactions between longitudinal fibres vanish [36].Basic results about this model are collected in classical treatments [37][38][39][40][41][42][43][44] with a coordinate approach.Coordinate-free investigations can be found in [45][46][47][48].Nevertheless, analytical solutions of beams subjected to torsion can be obtained only for special cross-sectional geometries and shear moduli distributions.Exact solutions of functionally graded structures can be found in [49].However, finite element strategies are often adopted in order to get effective numerical results when exact solutions are not available; see, for example, [50][51][52][53].Alternatively, experimental methods are employed; see, for example, [54].Recently Ecsedi showed that, for functionally graded cross sections under torsion, with shear modulus defined by a positive function of the Prandtl stress function of a corresponding homogeneous cross-section, the warping is invariant and the stress function is expressed in terms of the one associated with the reference homogeneous cross section [55,56].Ecsedi's treatment is based on an integral transformation proposed by Kirchhoff in nonlinear heat conduction [57].An intrinsic reasoning is illustrated in the present paper, by performing a direct discussion of Neumann and Dirichlet boundary value problems for the cross-sectional warping and for the stress function of orthotropic composite beams under torsion.An invariance condition for the Cicala-Hodges centre is also assessed (see Section 3).Finally, new analytical solutions of functionally graded elliptic cross sections are constructed in Section 4. Basic results of Saint-Venant theory of linearly elastic orthotropic beams are collected in the next section.

Composite Saint-Venant Beams under Torsion
Let Ω be the simply or multiply connected cross section of an orthotropic and linearly elastic Saint-Venant composite beam under torsion.Position of a point in Ω, with respect to the centre G of the Young moduli  : Ω  → R of beam's longitudinal fibers, is denoted by r.The tensor R is the rotation by /2 counterclockwise in the cross-sectional plane  Ω .Hence R  = R −1 = −R and RR = −I.Tangential stresses can be expressed in terms of the warping function [34]  : Ω  → R or of the stress function [58] Ψ : Ω  → R by the coordinate-free formulae [59]  (, r) = Λ (r)  (r) = Λ (r) (Rr + ∇ (r)) where the scalar  is the twist,  : Ω  →  is the elastic tangential strain, and Λ : Ω  → (; ) is the positive definite symmetric Lamé tensor field,  being the two-dimensional linear space of translations in  Ω .The warping field is the solution of the following Neumann-like problem [60]: where n is the unit outward normal to the domain Ω.Prandtl stress function is the solution of the Dirichlet problem where Ω is a multiply connected cross-section, with Ω  exterior boundary and Ω  boundary of the th hole, being  = 1, . . .,  and  ≥ 0. The procedure for the evaluation of integration constants   is illustrated in [59].Note that the warping function  : Ω  → R has been introduced above by assuming tacitly that the cross section undergoes a rotation about the pole G. Denoting by  C : Ω  → R the warping function corresponding to a cross-sectional rotation with respect to a point C ∈  Ω , we get the formula with r C position vector of C and  ∈ R. Tangential stress fields are independent of the rotation centre [40].The twist centre C tw and a particular value of the constant  were introduced in [61] by requiring that zeroth and first elastic moments of the scalar field  C : Ω  → R are zero The position of the twist centre is given by the formula with J G () fl ∫ Ω (r)r ⊗ r  bending stiffness and ⊗ tensor product.An equivalent definition of twist centre was proposed by Trefftz [62] in energetic terms.In [59] it was shown that the twist centre C tw coincides with the shear centre C sh timo of Timoshenko beams [63], evaluated by the composite and orthotropic Saint-Venant beam theory.Hereafter, the point C fl C tw ≡ C sh timo will be named the Cicala-Hodges centre.The next section provides a family of composite beams, generated by a Lamé tensor field Λ : Ω  → (; ), for which the warping field and the Cicala-Hodges centre are invariant.

Proposition 2. The relationship between Prandtl stress functions corresponding to Lamé tensor fields
Proof.Resorting to Proposition 1 we get  −1 =   .Then the equivalences hold

Examples
Let us provide some analytical solutions of functionally graded orthotropic beams under torsion with elliptic cross sections.Inertia principal axes {, } with origin in the centre G of the Euler moduli field  : Ω  → R will be adopted in the sequel.Position vector r and rotation R are written as whence Rr = [−, ]  .Torsional warping of elliptic composite beams with Lamé tensor field, is provided by the formula [40] ∈ ]0, +∞[ and ,  lengths of the ellipse semidiameters.Plots of the shear modulus   and of the warping  are provided in Figures 1  and 2.
Cartesian components of the tangential strain field  and stress function Ψ 1 are given by the formulae as depicted in Figures 3 and 4. As shown in Section 3, Lamé tensor field Λ 1 generates a sequence of composite beams under torsion for which warping function and Cicala-Hodges centre are invariant, and the relevant stress functions are given by Proposition 2. Analytical solutions of the following composite elliptic beams under torsion are discussed.The former is characterized by Lamé shear moduli described by the tensor field Λ 2 = (Ψ 1 + )Λ 1 , with  ∈ ]0, +∞[.The corresponding stress function is given by the formula Ψ 2 = (1/2)Ψ 2 1 + Ψ 1 , as assessed in Proposition 2. The latter is characterized by Lamé shear moduli described by the tensor field Λ 3 = (exp ∘ Ψ 2 )Λ 2 , where exp denotes the exponential function.The corresponding stress function is given by the formula Ψ 3 = (exp∘Ψ 2 )−1, as assessed in Proposition 2. Stress functions and tangential stresses are depicted in Figures 5,6,7,and 8.It is worth noting that if the Euler moduli scalar field  is assumed to be the same in the examples discussed above, then  warping functions and Cicala-Hodges centres are invariant, as prescribed by Propositions 1 and 3.

Conclusions
The outcomes of the present paper may be summarized as follows.
(i) Neumann and Dirichlet boundary value problems for the cross-sectional warping and for Prandtl stress function of linearly elastic, orthotropic composite beams under torsion have been examined.
(ii) Invariance conditions for the warping and for the Cicala-Hodges shear centre of simply and multiply connected cross sections have been established.
(iii) The relationship between Prandtl stress functions of orthotropic composite beams with invariant warping has been assessed.
(iv) Examples have been developed for orthotropic composite beams with elliptic cross section, providing thus also new benchmarks for numerical analyses.