IMA Journal of Applied Mathematics (2007) 72, 556 562 doi:10.1093/imamat/hxm027 Advance Access publication on September 25, 2007 edicated with admiration to Robin Knops On the torsion of functionally graded anisotropic linearly elastic bars CORNELIUS O. HORGAN Structural and Solid Mechanics Program, epartment of Civil Engineering, University of Virginia, Charlottesville, VA 22904, USA [Received on 3 November 2006; accepted on 13 March 2007] The torsion of homogeneous, isotropic, linearly elastic cylindrical bars has been the subject of numerous investigations from theoretical, computational and applied viewpoints. It is well known that in this case, the circular shaft is the only one whose cross-section does not involve a warping displacement in the axial direction. For an inhomogeneous, isotropic, circular bar, there is also no warping of the cross-section provided that the shear modulus varies only in the radial direction. For general anisotropic materials, torsion induces bending and vice versa. For those anisotropic materials with at least one plane of elastic symmetry normal to the axial direction, pure torsion is possible. For homogeneous materials of this type and special inhomogeneous materials, it has been shown by various arguments that no warping occurs for bars of particular elliptical cross-section. Here, we provide a unified derivation of the foregoing results. Some discussion of torsional rigidities is also given. Keywords: torsion of inhomogeneous, linearly elastic bars; anisotropic functionally graded materials; warping of the cross-section; elliptical cross-sections; torsional rigidity. 1. Introduction The torsion of homogeneous, linearly elastic cylindrical shafts has been the subject of numerous studies from both a theoretical and an applied viewpoint. The treatment of this problem for isotropic materials in standard textbooks (see, e.g. Sokolnikoff, 1956) has served to introduce beginning elasticity students to elegant methods for solving the irichlet or Neumann boundary-value problems over the plane cross-section of the cylinder. Qualitative methods for the second-order elliptic partial differential equations that arise have been used to establish general properties of the solutions, e.g. in estimating torsional rigidities. See, e.g. the review article of Payne (1967) for a summary of classical results. The anisotropic problem for homogeneous and some inhomogeneous materials was treated by Lekhnitskii (1981) with emphasis on torsion plus bending for generally anisotropic bodies. In recent years, the rapid technological developments in the area of functionally graded materials has sparked considerable renewed interest in elasticity problems for inhomogeneous bodies, both isotropic and anisotropic. For the torsion of inhomogeneous, isotropic bars, Rooney & Ferrari (1995) and Horgan & Chan (1999) formulated the basic problem that arises in terms of a Prandtl stress function. The boundary-value problem is a irichlet problem for a Poisson equation with a single variable coefficient, namely, the shear modulus. Explicit results for axisymmetric torsion of a circular cylinder with shear modulus depending only on the radial coordinate were obtained in these papers. Furthermore, for general cross-sections, it was shown by Horgan & Chan (1999) and Horgan (2000) that the maximum shear stress no longer ownloaded from http://imamat.oxfordjournals.org/ at Pennsylvania State University on May 11, 2016 Email: coh8p@virginia.edu c The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
TORSION OF FUNCTIONALLY GRAE ANISOTROPIC LINEARLY ELASTIC BARS 557 necessarily occurs on the boundary of the cross-section as it always does in the homogeneous, isotropic case. This result has significant implications for the initiation of failure in the torsion of functionally graded bars. Bounds on the torsional rigidity and a discussion of an effective modulus theory were also given by Horgan & Chan (1999). More recent results on solutions for inhomogeneous, linear anisotropic elastostatics with application to the torsion of cylinders with trigonal symmetry are given by Fraldi & Cowin (2004). See also the earlier paper by Cowin (1987). We cite Ting et al. (2004) and the references cited therein for recent results on composite cylinders. The purpose of the present paper is to investigate some particular aspects of the torsion problem for inhomogeneous, anisotropic materials. For simplicity of the discussion, we confine attention to solid shafts so that the plane cross-section is simply connected. Extensions to the multiply connected case for hollow shafts are straightforward. One motivation for this work is provided by two recent papers on the issue of warping of the cross-section. It is well known that for the case of a homogeneous, isotropic material, the circular shaft is the only one whose cross-section does not involve a warping displacement in the axial direction. More generally, it was shown by Lekhnitskii (1981) that for an isotropic, inhomogeneous, circular shaft with a shear modulus that varies only in the radial direction, there is also no warping. Recently, it was shown by Chen (2004) that there exist homogeneous, anisotropic, elliptical shafts that also do not warp. Independently, it was demonstrated by Ecsedi (2004) that special classes of inhomogeneous, elliptical shafts have the same property. In this paper, we provide a unified derivation of the foregoing interesting results on warping of cross-sections in torsion. A brief discussion of some results for torsional rigidities is also given. 2. Formulation of the problem We will confine attention to the torsion of anisotropic solid cylindrical bars with simply connected cross-section. Since the torsion problem for the homogeneous, anisotropic case has been formulated by Lekhnitskii (1981) and given a comprehensive modern treatment in a recent paper by Chen & Wei (2005), here we merely summarize the equations for the inhomogeneous problem relevant to our purposes. On specialization to homogeneity, we recover the classical results. We use a Cartesian coordinate system with origin at the left end of the cylinder and x 3 -axis along the axis of the cylinder. The class of anisotropy considered has at least one plane of elastic symmetry normal to the x 3 -axis and so twisting will not induce bending (see, e.g. Lekhnitskii, 1981; Ting, 1996; Chen & Wei, 2005). Important anisotropies included are monoclinic and orthotropic symmetries. The displacement field is given by u 1 = τ x 2 x 3, u 2 = τ x 1 x 3, u 3 = τϕ(x 1, x 2 ), (2.1) where τ is the twist per unit length of the shaft and ϕ(x 1, x 2 ) is the warping function. We will use a convenient notation introduced by Chen & Wei (2005) for the homogeneous case. Thus, we write the symmetric matrix of elastic coefficients as ( ) μ55 (x) μ 45 (x) μ(x) =, (2.2) μ 45 (x) μ 44 (x) ownloaded from http://imamat.oxfordjournals.org/ at Pennsylvania State University on May 11, 2016 where x = (x 1, x 2 ). Thus, the non-zero stresses are given by τ 13 /τ = μ 55 (ϕ 1 x 2 ) + μ 45 (ϕ 2 + x 1 ), (2.3) τ 23 /τ = μ 45 (ϕ 1 x 2 ) + μ 44 (ϕ 2 + x 1 ), (2.4)
558 C. O. HORGAN where the subscript notation on ϕ denotes partial differentiation with respect to the corresponding x α variable. The equilibrium equations in the absence of body forces thus lead to the governing partial differential equation where and L[ϕ] = 0 on, (2.5) L = A/ + B/ (2.6) A = μ 55 (ϕ 1 x 2 ) + μ 45 (ϕ 2 + x 1 ), (2.7) B = μ 45 (ϕ 1 x 2 ) + μ 44 (ϕ 2 + x 1 ). (2.8) We will assume that the strain energy density is positive definite at all points in and so μ 55 (x) > 0, det μ > 0 on. (2.9) The partial differential equation (2.5) is thus a second-order elliptic partial differential equation with variable coefficients. The traction-free boundary condition on the lateral surface of the cylinder can then be written as An 1 + Bn 2 = 0 on, (2.10) where ( ) dx2 n = (n 1, n 2 ) = ds, dx 1 (2.11) ds is the unit outward normal on the boundary curve, with s denoting an arc length coordinate. On assuming sufficient regularity of the domain and sufficient smoothness of the elastic coefficients, standard results from the theory of linear partial differential equations show that classical solutions to (2.5) subject to (2.10) are unique to within a constant. Without loss of generality, we set this constant, which corresponds to a rigid translation along the axis, equal to zero. 3. Conditions for no warping to occur THEOREM 1 Suppose that the inhomogeneous, anisotropic, elastic cylinder with warping function governed by the boundary-value problem (2.5), (2.10) has an elasticity coefficient matrix (2.2) of the special form ( ) γ f (x) β f (x) μ(x) =, (3.1) β f (x) α f (x) where α, β and γ are constants such that γ > 0, α γ β 2 > 0. (3.2) Let f (x) be a sufficiently smooth positive-valued function such that the first-order differential equation is satisfied, where c(x 1, x 2 ) f + d(x 1, x 2 ) f = 0 on (3.3) c(x 1, x 2 ) = βx 1 γ x 2, d(x 1, x 2 ) = αx 1 βx 2. (3.4) ownloaded from http://imamat.oxfordjournals.org/ at Pennsylvania State University on May 11, 2016
TORSION OF FUNCTIONALLY GRAE ANISOTROPIC LINEARLY ELASTIC BARS 559 Then, no warping of the cross-section occurs if and only if is the ellipse αx1 2 2βx 1x 2 + γ x2 2 = k2 (k 0). (3.5) Proof. Suppose that no warping occurs so that ϕ 0 on. Then, from (2.7) and (2.8), we have A = μ 55 x 2 + μ 45 x 1, B = μ 45 x 2 + μ 44 x 1. (3.6) Thus, (2.5) and (2.6) imply that x 1 μ 45 x 2 μ 55 + x 1 μ 44 x 2 μ 45 = 0 on. (3.7) On using (3.1), it is easily verified that (3.7) is satisfied if and only if the inhomogeneity function f satisfies (3.3) and (3.4). The boundary condition (2.10), on using (2.11), (3.6) and (3.1), reduces to d(x)dx 1 c(x)dx 2 = 0 on, (3.8) where c and d are defined in (3.4). We note that the boundary condition (3.8) is independent of the inhomogeneity function f. On integration, it is easily seen that (3.8) holds if and only if is the ellipse (3.5). This completes the proof of the theorem. COROLLARY 1 (Homogeneous, anisotropic, elliptical cylinder) Suppose that the material is homogeneous so that μ is a constant matrix. Thus, we formally set f = 1 in the preceding and have ( ) γ β μ =, (3.9) β α where the constants α, β and γ are such that (3.2) holds. Since f = 1, (3.3) is identically satisfied and so we deduce that no warping of the cross-section occurs if and only if is the special ellipse (3.5). This result was obtained also by Chen & Wei (2005) on using an affine transformation of axes. Such transformations were proposed for the torsion problem by Sokolnikoff (1956) and used by Horgan & Miller (1994) for analogous problems in anti-plane shear (see Horgan (1995) for a review of anti-plane shear deformations for both linear and non-linear elasticity). For the special case of orthotropic symmetry where μ 45 = 0, i.e. β = 0 in (3.9), one finds that the ellipse (3.5) has semi-major and -minor axes related by b a = μ44. (3.10) μ 55 ownloaded from http://imamat.oxfordjournals.org/ at Pennsylvania State University on May 11, 2016 This result was also obtained by Chen (2004). We note that this result can also be established simply and directly from an explicit expression for the warping function for an elliptical shaft composed of an orthotropic material given on p. 283 of Lekhnitskii (1981) as ϕ = μ 55b 2 μ 44 a 2 μ 55 b 2 + μ 44 a 2 x 1x 2. (3.11) This function is zero when (3.10) holds. Of course, when μ 44 = μ 55 so that the material is isotropic, we recover from (3.10) the classical result for the circle.
560 C. O. HORGAN COROLLARY 2 (Inhomogeneous, anisotropic, elliptical cylinder) It is easily shown by using the method of characteristics that the general solution of c(x 1, x 2 ) f + d(x 1, x 2 ) f = 0 on, (3.12) where c(x 1, x 2 ) = βx 1 γ x 2, d(x 1, x 2 ) = αx 1 βx 2, (3.13) is given by an inhomogeneity function of the form f (αx1 2 2βx 1x 2 + γ x2 2 ). (3.14) Thus, we find that no warping of the cross-section occurs for the ellipse (3.5) if the elasticity matrix has the form (3.1) with f given by (3.14). This result was established directly by Ecsedi (2004). For the special case of orthotropic symmetry, i.e. β = 0 in (3.1), we obtain the analogue of (3.10) for the inhomogeneous material with f = f (αx1 2 + γ x2 2 ) so that the ellipse has semi-major and -minor axes related by b α a = γ. (3.15) Of course, when γ = α so that the material is inhomogeneous and isotropic, the preceding result reduces to a result due to Lekhnitskii (1981) for a circle, namely that no warping occurs for an inhomogeneous, isotropic circular cylinder with variable shear modulus of the form μ = μ(r), where r is the radial polar coordinate. See, e.g. Horgan & Chan (1999) and Rooney & Ferrari (1995) for further discussion of such axisymmetric problems. REMARK 1 The class of inhomogeneous materials defined by (3.1) is such that μ(x) = f (x)μ h, where μ h is the matrix of elastic constants for the corresponding homogeneous material. See the recent paper by Fraldi & Cowin (2004) for an interesting general treatment of inhomogeneous elastostatic problems of this type. 4. Torsional rigidity The torsional rigidity (see, e.g. Lekhnitskii, 1981) is defined in terms of the applied twisting moment M as K = M/τ = 1 (x 1 τ 23 x 2 τ 13 )da. (4.1) τ On multiplying the differential equation (2.5) by ϕ, integrating over and using the divergence theorem and the boundary condition (2.10), it can be readily shown that K = (μ 44 x1 2 2μ 45x 1 x 2 + μ 55 x2 2 )da (μ 44 ϕ1 2 + 2μ 45ϕ 1 ϕ 2 + μ 55 ϕ2 2 )da. (4.2) By virtue of (2.9), the second integral in (4.2) is non-negative and so it follows from (4.2) that K (μ 44 x1 2 2μ 45x 1 x 2 + μ 55 x2 2 )da (4.3) ownloaded from http://imamat.oxfordjournals.org/ at Pennsylvania State University on May 11, 2016
TORSION OF FUNCTIONALLY GRAE ANISOTROPIC LINEARLY ELASTIC BARS 561 with equality if and only if ϕ 0 on. (4.4) The isoperimetric inequality (4.3) is the counterpart for generally inhomogeneous, anisotropic bars of the celebrated result for the homogeneous, isotropic case, namely K μi 0, I 0 = (x1 2 + x2 2 )da, (4.5) with equality if and only if ϕ 0 on. See, e.g. Payne (1967) for a detailed discussion of this result. In (4.5), μ denotes the constant shear modulus for the isotropic solid. For the case of homogeneous or piecewise constant anisotropic (composite) bars, the result (4.3), (4.4) was obtained by Chen & Wei (2005). On using the results of the theorem, we find that for the class of inhomogeneities of the form (3.1), it follows that equality holds in (4.3) if and only if is the ellipse (3.5). In this case, we find from (4.3) and (3.1) that K = (αx1 2 2βx 1x 2 + γ x2 2 ) f (x)da, (4.6) where the inhomogeneity function f has the form (3.15). For the special subcase of orthotropic symmetry where β = 0 in (4.6) and (3.15), we have K = (αx1 2 + γ x2 2 ) f (αx2 1 + γ x2 2 )da, (4.7) where the ellipse has semi-axes related as in (3.15). When α = γ so that the material is isotropic, this reduces to the result of Lekhnitskii (1981) for a circle, namely K = r 2 μ(r)da, (4.8) where μ(r) is the shear modulus that depends only on the radial coordinate. Specific examples of such radial inhomogeneities are discussed in Horgan & Chan (1999). On returning to the general inhomogeneous case, we see from (4.2) that upper and lower bounds for the second integral will yield upper and lower bounds for K. Such issues for the homogeneous isotropic case have been the subject of numerous investigations and have application well beyond the theory of torsion. Some results of this type for the functionally graded case are given in Rooney & Ferrari (1995) and Horgan & Chan (1999) for isotropic materials and Chen & Wei (2005) for composite anisotropic materials. 5. Concluding remarks In this note, our primary interest has been on warping of the cross-section and so we have focussed attention on the formulation of the torsion problem in terms of a Neumann-type boundary-value problem for the warping function. The divergence structure of the governing equation (2.5) ensures that one can also formulate the basic boundary-value problem as a irichlet-type problem for an analogue of the Prandtl stress function. Such an approach was used by Horgan & Chan (1999) for the functionally graded isotropic case. It is well known that such an alternative formulation is generally more convenient than that in terms of the warping function and can lead to further results but we shall not pursue this here. ownloaded from http://imamat.oxfordjournals.org/ at Pennsylvania State University on May 11, 2016
562 C. O. HORGAN REFERENCES CHEN, T. (2004) A homogeneous elliptical shaft may not warp under torsion. Acta Mech., 169, 221 224. CHEN, T. & WEI, C.-J. (2005) Saint-Venant torsion of anisotropic shafts: theoretical frameworks, extremal bounds and affine transformations. Q. J. Mech. Appl. Math., 58, 269 287. COWIN, S. C. (1987) Torsion of cylinders with shape intrinsic orthotropy. J. Appl. Mech., 109, 778 782. ECSEI, I. (2004) Elliptic cross section without warping under torsion. Mech. Res. Commun., 31, 147 150. FRALI, M. & COWIN, S. C. (2004) Inhomogeneous elastostatic problem solutions constructed from stressassociated homogeneous solutions. J. Mech. Phys. Solids, 52, 2207 2233. HORGAN, C. O. (1995) Anti-plane shear deformations in linear and nonlinear solid mechanics. SIAM Rev., 37, 53 81. HORGAN, C. O. (2000) Extremal stresses in boundary-value problems for functionally graded linearly elastic materials. Proceedings of the ISIMM Symposium on Trends in the Application of Mathematics to Mechanics (STAMM 2000) (P. E. O onoghue & J. N. Flavin eds). Paris: Elsevier, pp. 100 107. HORGAN, C. O. & CHAN, A. M. (1999) Torsion of functionally graded isotropic linearly elastic bars. J. Elast., 52, 181 199. HORGAN, C. O. & MILLER, K. L. (1994) Antiplane shear deformations for homogeneous and inhomogeneous anisotropic linearly elastic solids. J. Appl. Mech., 61, 23 29. LEKHNITSKII, S. G. (1981) Theory of Elasticity of an Anisotropic Elastic Body. Moscow: Mir Publishers. PAYNE, L. E. (1967) Isoperimetric inequalities and their applications. SIAM Rev., 9, 453 488. ROONEY, F. T. & FERRARI, M. (1995) Torsion and flexure of inhomogeneous elements. Compos. Eng., 5, 901 911. SOKOLNIKOFF, I. S. (1956) Mathematical Theory of Elasticity. New York: McGraw Hill. TING, T. C. T. (1996) Anisotropic Elasticity: Theory and Applications. Oxford: Oxford University Press. TING, T. C. T., CHEN, T. & LI, K. S. (2004) Torsion of an isotropic shaft of arbitrary cross-section embedded with multicoated or graded circular cylinders of cylindrically orthotropic materials. Q. J. Mech. Appl. Math., 57, 347 362. ownloaded from http://imamat.oxfordjournals.org/ at Pennsylvania State University on May 11, 2016