Search

An investigation of heuristic failure theories based on the Shanley-Ryder (S-R) stress ratio formulations and their extensions to ratios of stress invariants is undertaken. The deterministic structural margin (MS) and factor of safety (FoS) are re-examined and their multiple valued relevances to probabilistic analyses based on statistical loads as well as statistical material failure properties are discussed. MS's have no unique meaning in stochastic analyses while factors of safety effectively increase allowable loads thereby decreasing failure probabilities. [ABSTRACT FROM AUTHOR]

The application of chaos theory to forces inducing rolling maneuvers by acting on linear or nonlinear temperature dependent viscoelastic wings with actual widely scattered random material properties is investigated in detail. Published experimental data shows widely scattered 1-D viscoelastic moduli and small excursions from average or beginning values - the basis for chaos theory - is realistically inapplicable and would simply constitute a purely academic exercise excursion. While conventional approaches have been associated with applied forces, such as probability theory, to characterize these deterministic and stochastic properties, chaos theory is offered as an additional alternate generalized protocol to characterize viscoelastic responses to chaotic flight loads. At this time unfortunately no deterministic, stochastic or chaotic experimental data seems to be available, nor experimental data on aero-viscoelastic structural responses to flight vehicle maneuvers at any temperature. [ABSTRACT FROM AUTHOR]

The present paper is a continuation and expansion of a previous Part I [1] emphasizing material failure modeling and visualization protocols in conjunction with aero - elastic/viscoelastic phenomena. Failure is defined in terms of the Shanley-Ryder stress ratio relations and through the generalization based on the first three stress invariant ratios. Both deterministic and stochastic stress and strain as well as failure analyses are carried out. Visualization examples of multi-dimensional failure surfaces depending on possibly more than 35 elastic and 47 viscoelastic parallel coordinates is formulated, presented and discussed. Separate analyses are formulated for elastic and viscoelastic combined unsymmetrical bendingtorsion during level flight and for vehicle rolling motion. The overall bending degrees of freedom considered are plunging, in plane and chord-wise motions. Bending-torsion effects on and changes in angles of attack due the rolling velocity as well as the infiuence of moving shear centers and neutral axes and of material failures are considered during simultaneous occurrences. The final goal is to establish conditions for bending and torsional flutter, torsional divergence, control effectiveness and ultimate survival time of the wing due to material failures and structural instabilities (buckling) with future extensions to the entire vehicle under the rubric of system of systems approach, leading to a single pair of critical velocities and frequencies including material failure effects. The original Shanley-Ryder stress ratio failure criterion as well as its successor three stress invariant formulations are utilized. The latter has the advantage of having an unlimited number of arbitrary coefficients to be used to in fitting analytical expressions to stochastic experimental data. The multi-D numerical example results are displayed as a single figure of multiple 2-D parallel coordinates (1 1-cor(is) as opposed to numerous simultaneous, but separate, 2-D trace plots of a multi-D aeroelastic/aero-viscoelastic combined stability, buckling and material failure surface. In the present analyses, the use of Inselberg's parallel coordinates protocols clearly demonstrates in a single graph the individual and collective influences of many parameters on critical velocities without recourse to multiple 3-D critical surface representations. The critical velocities are also displayed as separate 2-D traces for each of the divers parameters, as well as in H-coords renderings. A small sample of randomly chosen subsonic wing parametric variation calculations show that compared to the free standing bending-torsion configuration, combinations involving plunging and in-plane bending, control effectiveness and reversal, with or without positive or negative roll velocities, and absent or present Timoshenko effects, produce substantially altered fiutter velocities and their paired frequencies. Finally, it noteworthy to mention that failure conditions can be improved by introducing morphing to the structure. Morphing [2], 13] airfoil shapes, although not treated here in detail, can produces significant improvements in aerodynamic, aeroelastic, aero-viscoelastic and structural performances, and delay as well as reduce their failure probabilities - all with relatively little energy expenditures. Additionally, the optimizing process that leads to the design of high performance Liebeck airfoils [4] contributes in equal measures. [ABSTRACT FROM AUTHOR]

The hereditary and/or convolution integrals associated with linear viscoelastic material constitutive relations based on Prony series [1] characterization are recast into ordinary nonconvolution time integrals, which can be more efficiently evaluated analytically and numerically. Application of this protocol greatly reduces computational time, CPU usage and memory requirements used to solve linear temperature dependent and/or independent viscoelastic problems involving integral-differential equations with variable coefficients. The formulation includes temperature dependent materials with time and space dependent temperatures as well as stresses due to thermal expansions. Approximate approaches for dealing with time dependent temperatures are derived and solutions to non-convolution integral equations as well as to differential equations with variable coefficients are formulated. Relaxation time consistent relations are derived for isotropic viscoelastic materials. Applications of Galerkin and Runge-Kutta methods to viscoelastic solutions are discussed and evaluated. These protocols include solutions to IODEs and IPDEs with variable coefficients. An illustrative algorithm to be used in conjunction with differential equation solvers such as MATLAB's™ ODE45 has been developed which allows for numerical solutions simultaneously in both real and reduced time spaces without approximations of linear and nonlinear integral differential equations with variable or without coefficients. Similar protocols could be readily extended to other software such as MATHEMATICA™, MAPLE™, etc. [ABSTRACT FROM AUTHOR]

We derive approximate, closed-form expressions for the cycle-averaged forces produced by flapping wings operating in a regime similar to birds and small unmanned aerial vehicles. The model is 2-D and intended mainly as an aid to performance and stability analysis, and control design. The model accounts for the nonlinear behavior of lift at high angles of attack, corrections for unsteadiness, as well as an elementary expression for drag. As an elementary application of the model, we determine the conditions under which the power consumption is minimized and those under which the range is maximized. We demonstrate how the model can be employed gainfully for stability analysis and control design. [ABSTRACT FROM AUTHOR]

Large deflection analyses of nonhomogeneous nonlinear elastic or viscoelastic columns (elastica and viscoelastica) with initial curvature and with deflection dependent cross sectional areas subjected to follower loads are formulated. Column end shortening due to both curvature and compressive loads is taken into account and the governing coupled nonlinear differential equations are solved by successive approximation methods (SAM) and numerically by a double Runge-Kutta iterative shooting approach. Various solution protocols are analyzed and evaluated. The effects of end shortening and follower load angle on elastic and viscoelastic columns are discussed. The hierarchal setting between column instability and material failures is examined. [ABSTRACT FROM AUTHOR]

The permissible forms and nature of linear viscoelastic functionally graded inhomogeneous material (VFGM) functions are formulated and examined in terms of consistency principles. It is shown that no separable temporal and spatial VFGM isotropic or anisotropic inhomogeneous modulus or compliance function is admissible nor can the VFGM solutions be cast into similarly separable constructions. However, finite or infinite sums of individual separable functions are admissible combinations provided each series term in its own form satisfies boundary and initial conditions, and provided one considers such an expressions as a mathematical model without necessarily any affinity to generalized Kelvin or Maxwell models. [ABSTRACT FROM AUTHOR]

Comparison analyses are conducted for straight and curved viscoelastic beams based on da Vinci-Euler-Bernoulli-Winkler strength of materials prescriptions and on rigorous formulations in terms of 3-D field equations. The concepts of elastic straight and curved beam analyses are well established. In the present paper, these geometric principles are extended to linear viscoelastic materials. The beams are considered subjected to un-symmetric bending as well as to thermal expansions. The distinct conditions leading to shear center and neutral axis motions in time, previously observed in straight beams are investigated. The applicability of the elastic-viscoelastic correspondence principle (analogy) in terms of integral transforms is demonstrated by application to several simulations. Comparison studies are based on pure bending loading configurations as these offer the least distinct geometric deformation patterns and focus primarily on the influence of constitutive relations. [ABSTRACT FROM AUTHOR]

Following the unified approach to viscoelastic Timoshenko beams (spars) formulated in [1], similar inclusive analyses are formulated for elastic and viscoelastic combined unsymmetrical bending-torsion during level flight and for vehicle rolling motion. The overall bending degrees of freedom considered are plunging, in plane and chord-wise motions. Bending-torsion effects on and changes in angles of attack due the rolling velocity as well as the influence of moving shear centers and neutral axes and of material failures are considered during simultaneous occurrences. The final goal is to establish conditions for bending and torsional flutter, torsional divergence, control effectiveness and ultimate survival time of the wing due to material failures and structural instabilities (buckling) with future extensions to the entire vehicle under the rubric of system of systems (SoS) approach, leading to a single pair of critical velocities and frequencies including material failure effects. A new stress invariant stochastic generalization to the original Shanley-Ryder stress ratio failure criterion is derived and utilized. The latter has the advantage of having an unlimited number of arbitrary coefficients to be used to in fitting analytical expressions to stochastic experimental data. The multi-D numerical example results are displayed as a single figure of multiple 2-D parallel coordinates (ǁ-coords), as opposed to numerous simultaneous, but separate, 2-D traces of a multi-D aeroelastic/aero-viscoelastic combined stability, buckling and material failure surface. In the present analyses, the use of ǁ-coords clearly graphically demonstrates the individual and collective influences of many parameters on critical velocities without recourse to a multidimensional critical surface representation. The critical velocities are also displayed as separate 2-D traces for each of the divers parameters, as well as in ǁ-coords renderings. A small sample of randomly chosen subsonic wing parametric variation calculations show that compared to the free standing bending-torsion configuration, combinations involving plunging and inplane bending, control effectiveness and reversal, with or without positive or negative roll velocities, and absent or present Timoshenko effects, produce substantially altered flutter velocities and their paired frequencies. [ABSTRACT FROM AUTHOR]

A modified truncated Maclaurin series (MMS) formulation is presented in order to gain additional probability density function (PDF) parameters over the usual two or three associated with classical PDFs, such as Gauss, log-normal, Weibull, beta, etc. Such a protocol results in a higher fidelity match of experimental or other observational data due to the practically almost unlimited increase in number of available PDF parameters. Additionally, such a power series expression is readily analytically integrable in order to produce its associated cumulative distribution function (CDF), expectation and probability as well as all statistical moments. An illustrative example is included to exhibit a pattern of series convergence and errors of least square (LSQ) statistical data fits. Additionally, the relation between the Fischer sufficiency index, LSQ fit errors and number of series parameters is also investigated. Results have direct aerospace applications in failure probability analysis, statistical characterization of material properties, optimal design of systems of systems, random loads, sizing, thermal effects, etc. [ABSTRACT FROM AUTHOR]

A unified bending/shear beam (spar) theory has been formulated by merging a number of previously completed theoretical segments into a comprehensive analytical treatment of linear non-homogenous viscoelastic Timoshenko beams (spars) with stochastic properties and random dynamic loads including shear center and neutral axis spatial and temporal motions due to bending, and including realistic physical starting load transients. These inverse problem analyses are framed entirely in terms of relaxation moduli or creep compliances, excluding any dependence on Poisson's ratios. The deterministic and stochastic effects of unequal tension and compression relaxation moduli/compliances on normal and shear bending stresses are derived and evaluated. The influences of all these phenomena on combined bending and shear stress distributions, structural instabilities, material failures and structural survival times also are formulated and discussed. [ABSTRACT FROM AUTHOR]

Analytical and experimental deterministic and statistical protocols are formulated for the constitutive linear relations that characterize viscoelastic media. These proceedings are achieved in the real time space in terms of moduli and/or compliances without any inclusion of viscoelastic Poisson's ratios. Independent experimental determinations of linear viscoelastic material properties of three distinct polymers include single conditions of creep, relaxation, and constant strain and extensometer time rates - except for starting transients. Statistical dynamic data for the instantaneous modulus and quasi-static data for relaxation moduli are analytically and numerically merged to produce relaxation modulus and creep compliance expressions containing properly evaluated parameters. The combined actual starting load and displacement transient and subsequent time histories are tracked, recorded and incorporated into the analyses to produce moduli and compliances based on actual continuous loading-time sequences. Prony series and generalized continuous relaxation time spectral modulus and compliance functions are derived and discussed. Four distinct probability models of material properties and temperature are postulated and evaluated for two sets of real viscoelastic materials (polymers). Statistical sufficiency considerations are included in the analyses. [ABSTRACT FROM AUTHOR]

This paper presents two characterization methods that are used to describe viscoelastic behavior. In the Prony series method, a discrete representation of the generalized Kelvin model is used, whereas the spectral method uses a continuous distribution. The Prony series coefficients are determined through the linear least squares technique. The Elastic-Viscoelastic Correspondence Principle and the Laplace transform are used in the spectral function approach, which utilizes a selected distribution function that has the potential to describe a wide range of viscoelastic material behavior. Characterization exercises from both methods used strain histories from short-term unidirectional tensile creep experiments that were obtained at three temperatures below the glass transition temperature of a vinyl ester polymer. Experimental strains in both the longitudinal and transverse directions were measured using the digital image correlation technique and linear viscoelastic material properties were obtained without using Poisson's ratios. The measured data was subsequently used to determine the tensile creep compliance function for each test configuration. The potential and limitations of each modeling approach are discussed. [ABSTRACT FROM AUTHOR]

Two separate sets of nonlinear 3-D constitutive relations are modeled in terms of (1) elastic strain invariants and (2) viscoelastic strain and strain rate invariants each assembled in their own multidimensional Maclaurin series. Characterization of material properties is formulated and evaluated. Fundamental difficulties associated with the nature of the nonlinear functions defining the stressstrain relations are examined. Several protocols for joint analytical and experimental viscoelastic material characterization and wave motion analyses are presented. As an illustrative problem, the 1-D wave propagation phenomenon is studied in detail. It is shown that the wave front velocities in linear elasticity and viscoelasticity are invariant with respect to the magnitude and character of the forcing function. In nonlinear elastic and viscoelastic media, the specific characteristics of the constitutive relation nonlinearities also do not influence the wave front velocity, but impact stresses and strains. Both must be evaluated on a case by case basis for each specific material properties and impact load-time functions. The nonlinear analyses are carried out using Poincare's successive approximation method (SAM). The use of SAM successively leads to linear elastic PDEs and to linear viscoelastic IPDEs respectively for each and every unknown approximation term of the stress and displacement wave series as well as the series representing the propagation velocity. The influences of the impact force loading pattern and of the nonlinear elastic and viscoelastic contributions on responding wave motions and wave front velocities and amplitudes are analyzed and evaluated. In particular, both the classical and distributed Dirac delta function impact loads are analyzed in detail and found to be incompatible models compared to the real world in certain areas in bothlinear and nonlinear elastic and viscoelastic media. An actual impact force pattern is then modeled and analyzed in the same medium and it was found to also work well with the SAM. [ABSTRACT FROM AUTHOR]

The advent of new structural concepts employing composites in primary load carrying aerospace structures in UAVs, MAVs, Boeing 787s, Airbus A380s, etc., necessitates the inclusion of flexibility as well as viscoelasticity in static structural and aero-viscoelastic analyses. Differences and similarities between aeroelasticity and aero-viscoelasticity have been investigated in [6], An investigation is undertaken regarding the dependence and sensitivity of aerodynamic and stability derivatives to elastic and viscoelastic structural flexibility and as to time dependent flight and maneuver velocities. Longitudinal, lateral and directional stabilities are investigated. It has been a well established fact that elastic lifting surfaces are subject to loss of control effectiveness and control reversal at certain flight speeds, which depend on aerodynamic, structural and material properties [6]. Such elastic analyses are extended to linear viscoelastic materials under quasi-static, dynamic, and sudden and gradual loading conditions. [ABSTRACT FROM AUTHOR]

Aeroelastic and aero-viscoelastic phenomena arising from the high flexibility of modern flight vehicles are examined, and governing relations are formulated and solved. In particular, the :ime dependent flight velocities associated with maneuvers and with in-plane bending are considered, which necessitate new derivations of the Theodorsen function, unsteady aerodynamic relations and ;quations of motion. Under these conditions, simple harmonic motion (SHM) is no longer achievable md different flutter criteria based directly on motion stability are presented. The viscoelastic problem, af importance to composites and other media, is formulated in terms of integral partial differential îquations with variable nonlinear coefficients. Their solutions and evaluations are discussed in detail. 3ne interesting departure from linear elastic and/or viscoelastic constant flight velocity responses emerged which, for some configurations, indicates the presence of flutter in the in-plane bending node while the other bending (plunging) and torsional modes both remain stable. [ABSTRACT FROM AUTHOR]

An analytical study is undertaken to ascertain the effects on elastic and viscoelastic flutter of (1) starting transients due to changes in flight velocity, (2) boundary conditions and (3) unsteady aerodynamic force definitions (spatial deflection derivatives). Lifting surface and panel flutter are considered in the steady-state velocity range after transient effects and varied boundary condition influences are also studied. In addition to the various unsteady aerodynamic theories, the effects due to the presence or absence of spatial deflection derivatives on pressure distributions are also examined. Necessary and sufficient conditions for the onset of flutter in the form of simple harmonic motion are also formulated and evaluated. [ABSTRACT FROM AUTHOR]