## Abstract

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 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 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 (jj–cords) 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 jj–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 flutter velocities and their paired frequencies. Finally, it noteworthy to mention that failure conditions can be improved by introducing morphing to the structure. Morphing [2], [3] 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.

Original language | English (US) |
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Pages (from-to) | 513-547 |

Number of pages | 35 |

Journal | Mathematics in Engineering, Science and Aerospace |

Volume | 12 |

Issue number | 2 |

State | Published - 2021 |

## Keywords

- failures
- linear aero–elasticity/viscoelasticity
- linear da Vinci–Euler–Bernoulli and Timoshenko elastic and viscoelastic beams (spars)
- realistic starting transient loads
- shear center and neutral axis motions
- stochastic linear elastic and viscoelasticity
- survival times

## ASJC Scopus subject areas

- Modeling and Simulation
- Aerospace Engineering
- Applied Mathematics