# On Torsion of Functionally Graded Elastic Beams

Authors: Marina Diaco
Journal: Modelling and Simulation in Engineering
 (2016)

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## Abstract

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.


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## Body

## 1. 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 [1–3], fracture mechanics [4–8], hyperelastic media [9–11], concrete systems [12–16], nonlocal models [17–19], homogenization [20–22], thermoelasticity [23–25], nanostructures [26–30], and limit analysis [31–33]. 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 [37–44] with a coordinate approach. Coordinate-free investigations can be found in [45–48]. 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, [50–53]. 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.


## 2. Composite Saint-Venant Beams under Torsion

LetΩ 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 E:Ω↦R of beam’s longitudinal fibers, is denoted by r. The tensor R is the rotation by π/2 counterclockwise in the cross-sectional plane πΩ. Hence RT=R-1=-R and RR=-I. Tangential stresses can be expressed in terms of the warping function [34] ϕ:Ω↦R or of the stress function [58] Ψ:Ω↦R by the coordinate-free formulae [59](1)τα,r=Λrγr=αΛrRr+∇ϕr=-αR∇Ψr,where the scalar α is the twist, γ:Ω↦V is the elastic tangential strain, and Λ:Ω↦L(V;V) is the positive definite symmetric Lamé tensor field, V being the two-dimensional linear space of translations in πΩ. The warping field is the solution of the following Neumann-like problem [60]:(2)div⁡Λr∇ϕr=-div⁡ΛrRr,r∈Ω,Λr∇ϕr·nr=-ΛrRr·nr,r∈∂Ω,where n is the unit outward normal to the domain Ω. Prandtl stress function is the solution of the Dirichlet problem(3)div⁡RΛ-1R∇Ψr=2onΩ,Ψr=0on∂Ωo,Ψr=cion∂Ωi,ci∈R,i=1,2,…,n,where Ω is a multiply connected cross-section, with ∂Ωo exterior boundary and ∂Ωi boundary of the ith hole, being i=1,…,n and n≥0. The procedure for the evaluation of integration constants ci is illustrated in [59]. Note that the warping function ϕ:Ω↦R has been introduced above by assuming tacitly that the cross section undergoes a rotation about the pole G. Denoting by ϕC:Ω↦R the warping function corresponding to a cross-sectional rotation with respect to a point C∈πΩ, we get the formula(4)ϕC=ϕr+RrC·r-c,with rC position vector of C and c∈R. Tangential stress fields are independent of the rotation centre [40]. The twist centre Ctw and a particular value of the constant c were introduced in [61] by requiring that zeroth and first elastic moments of the scalar field ϕC:Ω↦R are zero(5)∫ΩErϕr-cdA=0,∫ΩErϕr+RrC·rrdA=o.The position of the twist centre is given by the formula(6)rCtw=RJGE-1∫ΩErϕrrdA,with JG(E)≔∫Ω‍E(r)r⊗rdA 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 Ctw coincides with the shear centre Ctimosh of Timoshenko beams [63], evaluated by the composite and orthotropic Saint-Venant beam theory. Hereafter, the point C≔Ctw≡Ctimoshwill be named the Cicala-Hodges centre. The next section provides a family of composite beams, generated by a Lamé tensor field Λ:Ω↦L(V;V), for which the warping field and the Cicala-Hodges centre are invariant.


## 3. Invariances

Let us consider a sequence of tensor fieldsΛ1,Λ2,…,Λn generated by a Lamé tensor field Λ1:Ω↦L(V;V) and by a sequence of positive scalar functions h1,h2,…,hn; that is, hi:Xi⊆R↦]0,+∞[, according to the rule:(7)Λn=hn-1∘Ψn-1Λn-1,n≥2,with Ψn-1:Ω↦R Prandtl stress function associated with the torsion tangential stress field involving Lamé tensor field Λn-1:Ω↦L(V;V). The sequence of Lamé fields Λ1,Λ2,…,Λn induces a sequence of Neumann-like PDE problems P1,P2,…,Pn for the warping field, defined by(8)Pn-1)div⁡Λn-1γn-1r=0,inΩ,Λn-1rγn-1r·nr=0,on∂Ω,with γn-1(r)=Λn-1-1(r)τn-1(r)=αRr+∇ϕn-1(r). The following results hold true.Proposition 1.
Neumann-like PDE problemsP1,P2,…,Pn provide, to within an additive constant, the same solution ϕ:Ω↦R.Proof.
Letϕn-1 be the solution, to within a constant, of the problem Pn-1. Since Λn=(hn-1∘Ψn-1)Λn-1, problem Pn takes the form:(9)Pn)div⁡hn-1∘Ψn-1Λn-1γnr=0,inΩ,Λn-1rγnr·nr=0,on∂Ω.Resorting to the formula(10)div⁡hn-1∘Ψn-1Λn-1γnr=hn-1∘Ψn-1rdiv⁡Λn-1γnr+dhΨn-1r∇Ψn-1r·Λn-1γnrand setting ϕn(r)=ϕn-1(r)+c with c∈R, we get γn=γn-1, so that the problem Pn may be rewritten as(11)Pn)hn-1∘Ψn-1rdiv⁡Λn-1γn-1r+dhΨn-1r∇Ψn-1r·Λn-1γn-1r=0,inΩ,Λn-1rγn-1r·nr=0,on∂Ω.Recalling the relation τn-1(α,r)=Λn-1(r)γn-1(r)=-αR∇Ψn-1(r), we infer that ∇Ψn-1(r)·(Λn-1γn-1(r))=0 and the problem Pn collapses into the one Pn-1. The result follows.Proposition 2.
The relationship between Prandtl stress functions corresponding to Lamé tensor fieldsΛn-1 and Λn=(hn-1∘Ψn-1)Λn-1 is expressed by the formula Ψn=Hn-1∘Ψn-1, with Hn-1 antiderivative of hn-1 such that Ψn is identically zero on the cross-sectional exterior boundary ∂Ωo.Proof.
Resorting to Proposition1 we get γn-1=γn. Then the equivalences hold(12)Λn-1-1τn-1=Λn-1τn⟺Λn-1-1-αR∇Ψn-1=hn-1∘Ψn-1Λn-1-1-αR∇Ψn,whence (hn-1∘Ψn-1)∇Ψn-1=∇(Hn-1∘Ψn-1)=∇Ψn which gives Ψn=Hn-1∘Ψn-1, with Hn-1 antiderivative of hn-1 such that Ψn is identically zero on the cross-sectional exterior boundary ∂Ωo.Proposition 3.
LetE:Ω↦]0,+∞[ be the Euler moduli scalar field of orthotropic and composite beams whose Lamé fields are described by the sequence Λ1,Λ2,…,Λn=Λ1,h1∘Ψ1Λ1,…,hn-1∘Ψn-1Λn-1. For these beams, the location of the Cicala-Hodges centre is invariant.Proof.
The result follows by the formula providing the twist centre positionrCtw=RJGE-1∫ΩE(r)ϕ(r)rdA and by Proposition 1.


## 4. Examples

Let us provide some analytical solutions of functionally graded orthotropic beams under torsion with elliptic cross sections. Inertia principal axesx,y with origin in the centre G of the Euler moduli field E:Ω↦R will be adopted in the sequel. Position vector r and rotation R are written as(13)r=xy,R=0-110,whence Rr=-y,xT. Torsional warping of elliptic composite beams with Lamé tensor field,(14)Λ1x,y=μxμxyμxyμy=-k1k2a2y2+b2x2-a2b2a2k2+b2k1+k3k100k2,is provided by the formula [40] ϕx,y=-a2k2-b2k1/a2k2+b2k1xy+c, with c∈R,k1,k2,k3∈]0,+∞[ and a,b lengths of the ellipse semidiameters. Plots of the shear modulus μx and of the warping ϕ are provided in Figures 1 and 2.Figure 1
Shear modulusμx; a=2, b=1; k1=k3=1, k2=2. Color spectrum: μx.Figure 2
Cross-sectional torsional warpingϕ; a=2, b=1; k1=1, k2=c=2. Color spectrum: ϕ.Cartesian components of the tangential strain fieldγ and stress function Ψ1 are given by the formulae(15)γα,x,y=γxzγyz=αRr+∇ϕr=2αa2k2+b2k1-a2k2yb2k1x,Ψ1x,y=k1k2a2y2+b2x2-a2b2a2k2+b2k112k1k2a2y2+b2x2-a2b2a2k2+b2k1-k3,as depicted in Figures 3 and 4. As shown in Section 3, Lamé tensor field Λ1 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 Λ2=(Ψ1+k)Λ1, with k∈]0,+∞[. The corresponding stress function is given by the formula Ψ2=1/2Ψ12+kΨ1, as assessed in Proposition 2. The latter is characterized by Lamé shear moduli described by the tensor field Λ3=(exp∘Ψ2)Λ2, where exp denotes the exponential function. The corresponding stress function is given by the formula Ψ3=(exp∘Ψ2)-1, as assessed in Proposition 2. Stress functions and tangential stresses are depicted in Figures 5, 6, 7, and 8.Figure 3
Tangential strain field per unit twistγ/α; a=2, b=1; k1=k3=1, k2=2. Color spectrum: γ(x,y)≔[γxz2(x,y)+γyz2(x,y)]1/2 for α=1.Figure 4
Stress functionΨ1; a=2, b=1; k1=k3=1, k2=c=2. Color spectrum: Ψ1.Figure 5
Stress functionΨ2; a=2, b=1; k1=k3=1, k2=c=2. Color spectrum: Ψ2.Figure 6
Stress functionΨ3; a=2, b=1; k1=k3=1, k2=c=2. Color spectrum: Ψ3.Figure 7
Tangential stress field per unit twistτ2/α; a=2, b=1; k1=k3=1, k2=2. Color spectrum: τ2(x,y) for α=1.Figure 8
Tangential stress field per unit twistτ3/α; a=2, b=1; k1=k3=1, k2=2. Color spectrum: τ3(x,y) for α=1.It is worth noting that if the Euler moduli scalar fieldE 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.


## 5. 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.



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