Stochastic non-linear flutter of a panel subjected to random in-plane forces

—An analysis of non-linear flutter of a simply-supported panel exposed to supersonic gas flow and random in-plane forces is presented for two- and three-mode interactions. A first order quasi-steady state aerodynamic piston theory is used to model the aerodynamic loading. The Fokker-Planck equation is used to derive a general moment equation for two- and three-mode interactions. For stability analysis the moment equation is consistent and the mean square stability boundaries of the equilibrium are obtained in terms of the system parameters. The stability boundaries reveal common features to those predicted by the deterministic theory of panel nutter. For the non-linear response the moment equation is found inconsistent and a cumulant-neglect closure is used by setting cumulants of fifth and sixth orders to zero. This first order non-Gaussian closure is carried out to solve for the response statistics in terms of the air-to-plate mass ratio, aerodynamic pressure, modal damping, and in-plane random force spectral density. It is found that the non-Gaussian solution yields higher levels for the response statistics than those obtained by the Gaussian solution. The inclusion of more modes results in a reduction of the response levels and expands the stability region.     

Direct simulation of compressible turbulence in a shear flow

The purpose of this study is to investigate compressibility effects on the turbulence in homogeneous shear flow. We find that the growth of the turbulent kinetic energy decreases with increasing Mach number—a phenomenon which is similar to the reduction of turbulent velocity intensities observed in experiments on supersonic free shear layers. An examination of the turbulent energy budget shows that both the compressible dissipation and the pressure-dilatation contribute to the decrease in the growth of kinetic energy. The pressure-dilatation is predominantly negative in homogeneous shear flow, in contrast to its predominantly positive behavior in isotropic turbulence. The different signs of the pressure-dilatation are explained by theoretical consideration of the equations for the pressure variance and density variance. We previously obtained the following results for isotropic turbulence: first, the normalized compressible dissipation is of O(M _t ² ), and, second, there is approximate equipartition between the kinetic and potential energies associated with the fluctuating compressible mode. Both these results have now been substantiated in the case of homogeneous shear. The dilatation field is significantly more skewed and intermittent than the vorticity field. Strong compressions seem to be more likely than strong expansions.Dedicated to Professor J.L. Lumley on the occasion of his 60th birthday.This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665, U.S.A.     

A nonlinear composite plate theory

A general nonlinear theory for the dynamics of elastic anisotropic plates undergoing moderate-rotation vibrations is presented. The theory fully accounts for geometric nonlinearities (moderate rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. The theory accounts for transverse shear deformations by using a third-order theory and for extensionality and changes in the configuration due to in-plane and transverse deformations. Five third-order nonlinear partial-differential equations of motion describing the extension-extension-bending-shear-shear vibrations of plates are obtained by an asymptotic analysis, which reveals that laminated plates display linear elastic and nonlinear geometric couplings among all motions.     

Stokes Flow through an Array of Staggered Strips

The Stokes flow through a periodic array of thin staggered strips is studied. The method of eigenfunction expansion and collocation is used to obtain detailed flow and pressure fields. The permeabilities in the three principle directions are found to be different and their characters depend heavily on the geometry. Approximate formulas are also obtained.     

Damage accumulation and crack growth in bilinear materials with softening: Application of strain energy density theory

A pseudo-elastic damage-accumulation model is developed by application of the strain energy density theory. The three-point bending specimen is analyzed to illustrate the crack growth characteristics according to a linear elastic softening constitutive law that is typical of concrete materials. Damage accumulation is accounted for by the decrease of elastic modulus and fracture toughness. Both of these effects are assessed by means of the strain energy density functions in the elements around a slowly moving crack. The rate of change of the strain energy density factor S with crack growth as expressed by the relation dS/da = constant is shown to describe the failure behavior of concrete. Results are obtained for different loading steps that yield different slopes of lines in an S versus a (crack length) plot. The lines rotate about the common intersect in an anti-clockwise direction as the load steps are increased. The intersect shifts upward according to increase in the specimen size. In this way, the combined interaction of material properties, load steps and specimen geometry and size are easily analyzed in terms of the failure mode or behavior that can change from the very brittle to the ductile involving stable crack growth. An upper limit on specimen or structural size is established beyond which stable crack growth ceases to occur and failure corresponds to unstable crack propagation or catastrophic fracture. The parameters that control the failure mode are the threshold values of the strain energy density function (dW/dV)_c and the strain energy density factor S_c.     

Finite anti-plane shear of an infinite slab with a traction-free elliptical cavity: Bounds on the stress concentration factor

This paper provides an analytical approach for obtaining bounds on elastic stress concentration factors in the theory of finite anti-plane shear of homogeneous, isotropic, incompressible materials. The problem of an infinite slab with traction-free elliptical cavity subject to a state of finite simple shear deformation is considered. Explicit estimates are obtained for the maximum shear stress in terms of the cavity geometry, applied stress at infinity and constitutive parameters. The analysis is based on application of maximum principles for second-order quasilinear uniformly elliptic equations.     

Effective elasticity tensor of a periodic composite

The effective elasticity tensor of a composite is defined to be the four-tensor C which relates the average stress to the average strain. We determine it for an array of rigid spheres centered on the points of a periodic lattice in a homogeneous isotropic elastic medium. We first express C in terms of the traction exerted on a single sphere by the medium, and then derive an integral equation for this traction. We solve this equation numerically for simple, body-centered and face-centered cubic lattices with inclusion concentrations up to 90% of the close-packing concentration. For lattices with cubic symmetry the effective elasticity tensor involves just three parameters, which we compute from the solution for the traction. We obtain approximate asymptotic formulas for low concentrations which agree well with the numerical results. We also derive asymptotic results for C at high inclusion concentrations for arbitrary lattice geometries. We find them to be in good agreement with the numerical results for cubic lattices. For low and moderate concentrations the approximate results of Nemat-Nasseret al., also agree well with the numerical results for cubic lattices.     

An analysis of ductile rupture in notched bars

Ductile fracture in axisymmetric and plane strain notched tensile specimens is analyzed numerically, based on a set of elastic-plastic constitutive relations that account for the nucleation and growth of microvoids. Final material failure by void coalescence is incorporated into the constitutive model via the dependence of the yield function on the void volume fraction. In the analyses the material has no voids initially; but as the voids nucleate and grow, the resultant dilatancy and pressure sensitivity of the macroscopic plastic flow influence the solution significantly. Considering both a blunt notch geometry and a sharp notch geometry in the computations permits a study of the relative roles of high strain and high triaxiality on failure. Comparison is made with published experimental results for notched tensile specimens of high-strength steels. All axisymmetric specimens analyzed fail at the center of the notched section, whereas failure initiation at the surface is found in plane strain specimens with sharp notches, in agreement with the experiments. The results for different specimens are used to investigate the circumstances under which fracture initiation can be represented by a single failure locus in a plot of stress triaxiality vs effective plastic strain.     

Application of quasilinearization technique to the vertical channel flow and heat convection

The quasilinearization technique, also known as the generalized Newton-Raphson method, is employed to solve the problem of fluid injection through one side of a long vertical channel. With very approximate starting values, only a few iterations are needed to obtain a result with high accuracy. The results are compared with those obtained by the perturbation technique. Although the perturbation technique is perhaps the simplest and the most straight-forward method to achieve solutions to non-linear boundary value problems, the limitation on the magnitude of perturbed parameters hinders its applicability.     

Active feedback control of a beam subjected to a nonconservative force

This study examines the possibility of controlling through feedback a thin cantilevered beam subjected to a nonconservative follower force. A converging frequency flutter instability which occurs in this model is similar to classical bending-torsion flutter of an aircraft wing. Because of the similar nature of the instabilities, the beam under the follower force can be a useful vehicle for investigating the fundamental aspects of stabilization of wing flutter by feedback control. A modal approach is used for obtaining the mathematical model and control laws. A standard root locus technique for simple analytical models is also used to understand and explain the control of the beam. Experiments are carried out to verify the validity of this theoretical model. Good correlation is shown between theoretically and experimentally determined stability boundaries as well as for modal frequency and damping variation with follower force.