Asymptotic results for highly anisotropic spinning disks Ciprian D. - PDF document

Asymptotic results for highly anisotropic spinning disks Ciprian D. Coman University of Glasgow, Scotland ABSTRACT: The in-plane elastic instabilities experienced by a spinning anisotropic disk are captured through a classical boundary-layer strategy. It is assumed that the material is orthotropic with cylindrical symmetry, while the entire configuration is consistent with the plane-stress framework of linear elasticity. With the help of matched asymptotic expansions, simple analytic expressions are derived for the critical rotational speeds. 1 INTRODUCTION trifugal force is suitably modified to account for ra- dial displacements, certain rotational speeds lead to Boundary-layer methods represent a simple and unbounded values for the displacements and stresses efficient toolbox for finding approximate results in in isotropic disks. problems that exhibit singular dependence on small Recently, a similar situation was studied in (Port- parameters. Within the statics of elastic solids such nov et al. 2003): a polar orthotropic disk with an axis techniques have been mostly relevant to the mechan- of anisotropy coinciding with the main central axis of ics of thin-walled bodies (e.g., classical plates and inertia, but displaced relative to the axis of rotation by shells); by contrast, only relatively little attention a small amount. The investigation was carried out by has been payed to their relevance vis-´ a-vis structural assuming that the material of the disk is infinitely stiff members having strongly anisotropic material proper- in the azimuthal direction. A parallel study (Belov & ties. Notable exceptions include (Morland 1973, Pip- Portnov 2003) relaxed this assumption and presented kin 1973, Spencer 1974). a range of numerical results that were found to cor- The present contribution is motivated by recent relate well with the earlier findings of Portnov et al. technological advances regarding hoop-wound com- Our main objective is to clarify the asymptotic struc- posite flywheels having elastomeric resin and carbon ture that was left open in those works. fibres. Characterised by strengths comparable to their isotropic counterparts, these structural components 2 GOVERNING EQUATIONS are significantly lighter and allow much higher speeds A sketch of the setting that will be considered of rotation. For instance, in the case of composite in what follows appears in Figure 1 : a cylindrical disks based on carbon fibres in a flexible polyurethane rigid shaft whose cross-section has radius R 1 passes resin (Belov & Portnov 2003, Portnov et al. 2003), through the centre of an anisotropic disk of radius R 2 the ratio between the Young’s moduli in the azimuthal and thickness h . The shaft is perfectly bonded to the and radial directions is as large as 1 . 7 × 10 3 . Various disk and is rotated around its longitudinal axis with details regarding the design and production of such a constant angular velocity Ω ; possible out-of-plane composites are included in (Gabrys & Bakis 1997), bending of this configuration is ignored. where the readers will find pointers to the relevant lit- In polar coordinates the equilibrium is expressed erature as well. through the usual plane-stress equations, In this contribution we are concerned with a partic- ular stability situation involving the steady rotation of ∂σ rr ∂r + 1 ∂σ rθ ∂θ + 1 a flat disk, a problem that is described in many stan- r ( σ rr − σ θθ ) + P r = 0 , (1) r dard texts (Lekhnitskii 1968, Soedel 1981). In these classical treatments, the expression of the centrifugal ∂σ rθ ∂r + 1 ∂σ θθ ∂θ + 2 forces acting on the disk ignores the radial displace- rσ rθ + P θ = 0 , (2) r ment, and as a consequence instability is ruled out where P r and P θ represent the radial and, respec- right from the outset. Starting with (Brunelle 1971) it was found that when the expression of this cen- tively, the azimuthal components of the volumetric

1 := 1 A 11 αρ , Ω � n � ν α + 1 R 2 A 12 1 = − A 21 1 := ρ , β 1 := 1 R 1 A 22 βρ , � 1 � ν 2 α + n 2 h A 11 0 := − β + µ ρ 2 + λ, Ω � n Figure 1. Two views of the spinning disk. � ν 2 α + 1 A 12 0 = A 21 0 := − β + µ ρ 2 , inertial forces, whose expressions are discussed in � 1 � ν 2 � � + 1 (Belov & Portnov 2003). The in-plane displacements A 22 n 2 0 := − α + µ ρ 2 . in the radial and azimuthal directions will be denoted β by u ≡ u ( r,θ ) and v ≡ v ( r,θ ) , respectively. Using Here, ν ≡ ν θ is the Poisson’s ratio in the hoop direc- Hooke’s orthotropic constitutive law and the geomet- tion, α := 1 − ν 2 /µ , and the dash indicates differen- rical relations linking the strains to these displace- tiation with respect to ρ . The Fourier mode number ments, it is possible to express the problem in the n ∈ N present in some of the above expressions is ar- form of four linear PDE’s for the dependent variables bitrary. [ σ rr , σ rθ , u, v ] ; see (Coman 2010b) for full details. For further reference, the requisite boundary condi- A further judicious choice of non-dimensionalisation tions are recorded below: at ρ = η , singles out the key combination of parameters in those equations, namely, u = v = 0 , (5) µ := E θ β := E r while at ρ = 1 , , , (3) E r G rθ u ′ + ν ( u + nv ) = 0 , v ′ − ( nu + v ) = 0 . (6) where E r , E θ denote the Young’s moduli in the radial and hoop directions, respectively; G rθ represents the The asymptotic regime of interest is characterised by usual shear modulus characterising changes of angle between the r − and θ − directions. Also, the rescaled µ ≫ 1 , β = O (1) , n = O (1) . radial variable becomes ρ ≡ r/R 2 and the new annu- lar geometry is described by ( ρ,θ ) ∈ [ η, 1] × [0 , 2 π ) . 3 OUTER PROBLEM Given the linear nature of the problem it seems con- Direct numerical simulations of Equation 4, subjected venient to seek solutions in the form of simple Fourier to the constraints recorded in Equations 5 & 6, show series in θ , with variable amplitudes in the radial di- that in the limit µ ≫ 1 both u ( ρ ) and v ( ρ ) exhibit rection. This is helped by expressing P r in a similar sharp changes within a small region adjacent to ρ = 1 . fashion and neglecting P θ altogether. The eigenvalue We expect the behaviour of the spinning disk away in the present problem, which we shall agree to call from that region to be somewhat simpler than the pre- λ , is proportional to Ω 2 . dictions of the fourth-order differential equations sat- After lengthy (but routine) algebraic manipula- isfied by either displacement field. This expectation is tions, it transpires that the stresses can be eliminated confirmed by adopting an ansatz of the form out of the governing equation, leaving us with two coupled ODE’s with variable coefficients for the two � u � u 0 � u 1 � u 2 � � � � µ − 1 / 2 + µ − 1 + ... , = + displacement fields mentioned earlier. We found it v v 0 v 1 v 2 convenient to cast this equation as out λ = λ 0 + λ 1 µ − 1 / 2 + λ 2 µ − 1 + ... , A 2 u ′′ + A 1 ( ρ ) u ′ + A 0 ( ρ ; λ ) u = 0 , η < ρ < 1 , (4) where the unknowns on the right-hand sides are de- where u ≡ [ u ( ρ ) ,v ( ρ )] , A k = ( A ij k ) ( k = 0 , 1 , 2 and termined sequentially as outlined below. Simple cal- i,j = 1 , 2 ) are 2 × 2 matrices whose components have culations reveal that u 0 satisfies a Bessel-type equa- the following definitions, tion, and thus can be expressed in terms of the Bessel functions of the first and second kind, 2 := 1 2 = 1 A 11 A 12 2 = A 21 A 22 α , 2 = 0 , β , u 0 ( ρ ) = C 1 J ζ (Λ ρ ) + C 2 Y ζ (Λ ρ ) , (7)