Numerical analysis of moving orthotropic thin plates
Introduction
In the paper, textile, plastic film, and other industries involving moving thin materials, stress analysis is essential for the control of wrinkle, flutter, and sheet break. Similar vibration problems for moving plate and shell structures can also be found in aeronautic and aerospace structures, computer hard disk drives, and rotating blades in turbomachinery. Although the mechanical behavior of axially moving materials, in general, has been studied for many years, little information is available on the stress analysis including the transverse shear of moving orthotropic thin plates, such as paper sheets. Following the early work of Mote[18], significant research efforts have been directed toward studying the linear and nonlinear dynamical behavior of an axially moving string model31, 32, 33. In order to consider the dynamical behavior of moving materials coupled with surrounding fluid–air, in practice, one often ignores boundary layer shear forces and introduces different added mass expressions for the Coriolis and centrifugal intertia terms[21]. For the purpose of attenuation and guidance, coupling problems between moving material and fluid–air bearings have also been studied20, 25, 29, 34. More recently, Heinrich and Connolly[11] carried out a three-dimensional analysis of an axially moving tape coupled with bearings, by using the classical Kirchhoff thin plate assumptions. The importance and practical applications of stress analysis of moving materials were also discussed by Lee and Ng[16], where they applied a mode method with the Kirchhoff thin plate theory for isotropic materials and Lagrangian kinematic descriptions to avoid convective terms. It has been indicated in Ref.[30] that when the axial tension is not predominant, or the bending stiffness of the plate is significant, to accurately predict the transverse shear, as well as in-plane stress distributions of the moving thin plate and shell structures, reliable numerical formulations are needed. So far, no work has been documented form predicting the stress distributions of moving orthotropic thin plates.
Since the inception of finite element methods, many plate bending formulations and elements have been investigated; a proper review of such topics is available in Refs.1, 13, 23, 24. It has been widely recognized that mixed plate formulation based on the Mindlin–Reissner theory can effectively eliminate “shear locking” and predict accurately and reliably stress levels in thin plates4, 5, 9. Recently, the MITC elements have been proven, numerically, to satisfy the inf-sup condition[14]. The basic difficulty in choosing proper orders of interpolation for out-of-plane displacement, section rotations, and transverse shear strains which result in nonlocking behavior and optimal convergence of the element is summarized in Refs.3, 4, 5.
The purpose of this paper is to extend the mixed plate formulation, with MITC elements, to the analyses of axially moving orthotropic thin plates. In the following sections, we first briefly summarize the mathematical models, the corresponding assumptions, and the governing equations. Next, we present the mixed finite element formulation and some generic numerical examples. Although no mathematical theory is available to prove rigorously that with the incorporation of the axial motion, and consequently the loss of coerciveness due to gyroscopic inertia terms, the proposed formulation with MITC elements is as reliable as it is for the case of stationary plates and shells, we employ test examples for both frequency and transient dynamics analyses to confirm, numerically, the reliability and accuracy of the proposed formulation for moving orthotropic thin plates.
Section snippets
Plate displacement, strain, and stress
In a three-dimensional elasticity theory, there are six stress components that are expressed in terms of six strain components through material constitutive laws. For the general plate section illustrated in Fig. 1, the top and bottom faces of the plate (at z=±d/2) are considered to be free from tangential traction, but are under normal pressures p+ and p−, i.e.
For convenience, we set p=p−−p+. We recognize that in some cases we may have
Governing equations
From the force equilibrium equations, i.e., τij,j+fi= ρ(d2ui/dt2), where fi is the body force; ρ is the mass density; and the indices i (i=1, 2, 3) represent the x, y, and z directions, we obtain
Of course, as indicated in Eq. (15), the only body force considered in this paper is the gravitational force in the negative z direction. Furthermore, in terms of out-of-plane displacement w, and the two rotations βx
Mixed plate formulation
In the mixed formulation based on the Mindlin–Reissner plate theory, we account for the effects of inertia by applying the Hamilton's principle with the following definition of the kinetic energy, where Ω represents the midsurface area. In actuality, we include in the body forces the following inertia terms
Defining the spaces Θ=(H10(2 and W=H10(), and assuming that a force function fw is given in L2(), and a moment function
Numerical examples
To demonstrate the capabilities and reliability of the proposed formulation based on the Mindlin–Reissner plate theory, we analyze a typical orthotropic plate (a paper sheet as depicted in Fig. 3) with the follow ing physical parameters: ρ=700 kg m−3; d=0.7 mm; L=2 m; B=8 m; Ex=7.44 GPa; Ey=3.47 GPa; vxy=0.149; Gxy=2.04 GPa; Gxy=0.099 GPa; Gyz= 0.137 GPa.
To evaluate the critical axial velocity of the moving plate, we have computed the first natural frequency ω with the proposed finite element
Conclusion
The proposed mixed formulation based on the Mindlin–Reissner theory for a moving orthotropic plate is a natural extension of the reliable mixed formulation for stationary plates and shells. We consider the numerical tests used in this work to be comprehensive, and the natural frequencies, mode shapes, along with the five stress band plots, appear to be sufficient to demonstrate the accuracy and reliability of the proposed formulation. The employed MITC4 plate elements are demonstrated
Acknowledgements
The author would like to thank the Institute of Paper Science and Technology and its Member Companies for their support, and Professor F. Bloom for constructive comments and suggestions.
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