Aerothermoelastic Behavior of Flat and Curved Panels

Synonyms

Aerothermoelasticity; Edge movability; Hypersonic speed; Lyapunov first quantity; Nonlinear aerothermoelastic analysis; Piston aerodynamic theory; Stable/unstable LCO; Thermal loading and degradation

Definitions

Aeroelasticity (AE) is the science which studies the mutual interactions among inertial, elastic, and aerodynamic forces acting on structural members exposed to an airstream, and the influence of this study on design.

Aerothermoelasticity (ATE) is the science that studies the mutual interactions among inertial, elastic, and aerodynamic forces acting on structural members under the combined effect of aerodynamic heating and loading.

Overview

The panel flutter is a form of dynamic aeroelastic instability resulting from the interaction between the motion of a high-speed aerospace vehicle’s skin panel, typical of spacecrafts and missiles, and the aerodynamic loads exerted on that panel by air flowing past one side at supersonic or hypersonic speed and to still air on the other side. Often a skin panel encounters flutter and then a limit cycle oscillation (LCO), which is an oscillation bounded in amplitude. There have been many incidents reported in the literature dating back to the V-2 rocket of World War II, the X-15, the Saturn launch vehicle of the Apollo program, and continuing on to the present day [1–3].

One essential limitation of the linearized panel flutter analysis is that it gives information only up to the point of instability. Furthermore, the linearized analysis is restricted to cases where the aeroelastic response is small. Often this assumption is violated before the onset of instability. Thus, to study the behavior of aeroelastic systems in the proximity of the instability boundary including the postinstability region, the inherent nonlinearities of structural and aerodynamic nature must be accounted for. By using the Von Kármán large deflection plate theory combined with the linear piston theory aerodynamics (PTA), one of the most popular unsteady aerodynamic theories, it was recognized that geometrical nonlinearities due to moderate plate deflection, mainly creating mid-plane stretching forces, can play an important role in panel flutter [4]. The nonlinear panel dynamic response, due to large deformations and mutual interaction between the aerodynamic loading and high order panel modes, despite the deterministic nature of the panel equation, can be oscillatory, quasiperiodic, limit cycle, or random-like irregular chaotic [5]. Various nonlinearities can influence differently the character of the panel flutter boundary; these nonlinearities could be of several origins, including structural or geometric, thermodynamic, aerodynamic arising from flow characteristics, and material nonlinearities. Furthermore, in the presence of thermal effects, aerothermoelastic considerations have to be considered in the design of space reentry vehicles and high-speed aircraft, since theses effects may produce deformations, stresses, and changes in material properties that can dramatically affect their aeroelastic behavior. In this sense, the structural panels of supersonic/hypersonic flight vehicles can experience, among others, the thermal flutter instability generated by the combined influence of the thermal field, unsteady aerodynamic loads, elasticity of structures, and the dynamic effects.

The effect of panel heating is twofold. First, there is reduction in stiffness due to softening of the panel material; second, thermal stresses are generated due to mismatch in thermal expansion coefficients of the panel and support structure. These effects, in turn, affect the static and dynamic behaviors of the panel [6]. The bulk of literature dealing with flat and curved panels flutter is based on the stress-strain equations including shear wall and thermal effects [6–16]. In these works, quasi-steady first-order or nonlinear piston theories and Euler equations for unsteady aerodynamic have been considered. Other aerodynamic models, perhaps less computationally efficient, are also available and have been explored. In recent years, viscoelastic materials, such as some composite materials, have been widely used in the aerospace industry partly due to their inherent properties to reduce undesired vibrations.

Stability and vibration studies of plates and shells with initial geometric imperfections are of a significant importance in modern solid mechanics. These structures are rather sensitive to small deviations from their design shape. Experimental and numerical studies conducted so far have shown unambiguously that the basic cause of the discrepancy between theory and experiment is the initial deflections of the structure [7, 17, 18]. This means that generally, the influence of initial deflections has to be studied within the framework of large deflection theory, meaning primarily dynamic instability (including flutter) where imperfections play a vital part [18]. Small deviations of the shell’s surface from its idealized shape were also shown to drastically reduce its resistance to panel flutter, a dynamic instability of the shell, even though the deviations were only on the order of one shell thickness or less. Even the best manufacturing methods admit this magnitude of imperfection in the fabricated shell geometry [19]. Previous investigations have suggested that detailed studies are needed to better understand and explore the complex motions that can be encountered in the presence of various coupled nonlinearities. These studies are also needed when it comes to system identification and damage detection, since the vibration behavior of the system needs to be clearly understood.

Aerothermoelastic Analysis Methodology

Structural Modeling

To derive the aerothermoelastic governing equations of a curved panel, the geometrically nonlinear theory of infinitely long two-dimensional panels with some small initial curvature is usually considered. The classical von Kármán nonlinear strain-displacement relation for a general plate undergoing both extension and bending in conjunction with the Kirchhoff plate-hypothesis is adopted. The effects of thermal degradation and Kelvin’s model of structural damping can also be considered.

Let us consider an isotropic curved panel model (Fig. 1) with a width $a$, infinity long length $b$, thickness $h$, maximum rise height $H$, and constant radii curvature ${\Re_x}$ [20]. The thickness $h$ is small compared to the length $a$. In addition, $b$ is infinitely long as compared to $a$. The panel is supported on the sides $x = 0$ and $x = a$. These sides are fixed with respect to the longitudinal displacements.

Fig. 1

Two-dimensional panel with initial curvature

The displacements from the unstressed state of the panel’s mid-plane surface in the $x$ and $z$ directions are denoted by $u$ and $w$, and the total transverse displacement of a given mid-plane surface point after deformation is given by:

$$ {w_{{total}}}(x,t) = \hat{w}(x) + w(x,t) $$

(1)

Herein $\hat{w}(x)$ indicates the initial undeformed shape (initial geometric imperfection) of the mid-plane surface, while $w(x,t)$ corresponds to the transverse displacement of the mid-plane surface relative to its undeformed configuration. The strain ${\epsilon_x}$ of the mid-plane surface in the $x$-direction and based on the von Kármán assumption is given by [4]:

$$ {\epsilon_x} = {u_{,x }} + \frac{1}{2}{{({w_{\,,x }})}^2} + {w_{\,,x }}{{\hat{w}}_{\,,x }} - {w \left/ {\Re_x } \right.} $$

(2)

The subscript ${{( \cdot )}_{,x }}$ denotes the differentiation with respect to $x$. The bending equation of motion is given by [4, 21, 22]:

$$ {M_{\,,xx }} + {N_x}({w_{\,,xx }} + 1/{\Re_x}) + {P_z} = 0 $$

(3)

where ${N_x}$ represent the axial stress resultant, $M$ is the bending moment; furthermore, $M \equiv D\aleph$ where $D$ is the panel stiffness $\equiv {{E{h^3}}} / {{12(1 - {\upsilon^2})}}$, $E$ is the modulus of elasticity, $\upsilon$ is Poisson’s ratio, and $\aleph$ is the curvature change of the mid-plane surface $( \equiv - {w_{,xx }})$. The bending moment can be recast as:

$$ M = - D{w_{\,,xx }} $$

(4)

In (3), ${P_z}$ is the distributed load on the panel and can be expressed as follows:

$${P_z} = - {\rho_m}h{w_{\,,tt }} + P_z^A(x,t) + P_z^{stat }(x) + {\Delta}{B^T} $$

(5a)

The first term in the (5a) corresponds to the transverse inertial load, while the superscript ${{( \cdot )}^A}$ indicates an unsteady aerodynamic load and the superscript ${{( \cdot )}^{stat }}$ indicates an initial static load. ${B^T}$ is the thermal load defined as [8]:

$$ {B^T} = \frac{{E\alpha }}{{(1 - \upsilon )}}\int\limits_{- h/2}^{h/2 } {T(x,z)\;z\;dz} $$

(5b)

where $\alpha$ is the linear thermal expansion coefficient, $T(x,z)$ is the temperature increment from a free-stress temperature ${T_0}$, and ${\Delta}$ in (5a) is the Laplace operator. The material properties of the panel, $E$ and $\alpha$, are influenced by the thermal field as follows [7]:

$$\begin{array}{ll}E = {E_0} + {E_1}T = {E_0}(1 + {e_T}T),\cr \alpha = {\alpha_0} + {\alpha_1}T = {\alpha_0}(1 + {\alpha_T}T)\end{array}$$

(6a)

where

$$ {e_T} = {E_1}/{E_0} < 0,\quad {\alpha_T} = {\alpha_1}/{\alpha_0}> 0 $$

(6b)

In (6b), ${e_T}$ and ${\alpha_T}$ are the coefficients associated with the thermal degradation. As a result of the temperature dependence of the thermoelastic material properties and of the spatially distributed temperature field, the thermoelastic coefficients of the material become functions of the space variables, for example, $E \Rightarrow E(x)$ and $\alpha \Rightarrow \alpha (x)$. This implies that the structural panel presents a certain level of nonhomogeneity [4]. Typical aerospace panel, such as a fuselage section, wing and empennage panels, is usually solidly connected to structural members of the airframe. For this reason, it has been assumed that ${\sigma_x} \to \sigma$, that is, the tangential stresses act only in the $x$ -direction. Physically, this stress is generated by the constraint of the panel with the members of the airframe. This condition yields:

$$ {\sigma_x} = {N_x \left/ {h} \right.} = {(E(x) \left/ {(} \right.}1 - {\upsilon^2}))({\epsilon_x} + \upsilon {\epsilon_y}) = \sigma $$

(7)

Moreover, when the flight vehicle travels at high flight speeds regimes, due to aerodynamic heating, the skin panel temperature could potentially reach the high values of several hundred degrees. This effect can result in a lower value of the flutter instability boundary or in larger limit cycle amplitude at the same dynamic pressure. This implies that also the effect of the temperature should be carefully considered for more accurate results. This can be done including an in-plane tension $\sigma_x^T$, acting in the $x$ -direction, due to the temperature [23]:

$$ \sigma_x^T = - {(E(x) \left/ {(} \right.}1 - \upsilon ))\alpha (x)\;T $$

(8)

This implies that the total in-plane stress, in the $x$ -direction, can be expresses as:

$$ {\sigma_{{x\_{{total}}}}} = {\sigma_x} + \sigma_x^T $$

(9a)

By substituting (2) in (7), assuming ${\epsilon_y} = 0$, and making use of (8) in (9a), the total in-plane thermomechanical stress is obtained as:

$$\begin{array}{ll}{\sigma_{{x\_{{total}}}}} = {(E(x) \left/ {{\left( {1 - {\upsilon^2}} \right)}}\right.\Bigg[ {\bigg( {{u_{,x }} + \frac{1}{2}{{{({w_{,x }})}}^2} }}}\cr + {w_{,x }}{{\hat{w}}_{,x }} - {w \left/ {\Re_x } \right.} \bigg) - \alpha (x)(1 + \upsilon )T\Bigg]\end{array} $$

(9b)

Therefore, (3) becomes:

$$ (D/h){w_{,xxxx }} - {\sigma_{{x\_{{total}}}}}({w_{,xx }} + 1/{\Re_x}) - {P_z}/h = 0 $$

(10)

To evaluate the tangential stress component ${\sigma_{{x\_{{total}}}}}$, one expresses the average end-shortening ${\Delta_x}$ as [4, 7]:

$$ {\Delta_x} = \frac{1}{a}\int\limits_0^a {\mathop{u(x,t)}\nolimits_{{,x\,}} } dx $$

(11)

Solving (9b) for ${u_{,x }}$, one obtains:

$$\begin{array}{ll} {u_{,x }} = (1 - {\upsilon^2})({{{{\sigma_{{x\_total}}}}} \left/ {E} \right.}(x)) - \frac{1}{2}{{({w_{,x }})}^2}\\ - {w_{,x }}{{\hat{w}}_{,x }} + {w \left/ {\Re_x } \right.} + \alpha (x)(1 + \upsilon )T \end{array}$$

(12)

In particular, the operator $\frac{1}{a}\int\limits_0^a {( \cdot )dx}$ that appears in (11) is applied to (12) for the particular case of immovable edges $x = (0,a)$, that is, ${\Delta_x} = 0$, then the tangential stress ${\sigma_{{x\_{{total}}}}}$ yields:

$$\sigma {}_{{x\_{{total}}}} = \left[ {\frac{1}{{(1 - {\upsilon^2})\int\limits_0^a {E{(x)^{- 1 }}dx} }}} \right]\left[ \begin{array}{ll}{{{{\frac{1}{2}\int\limits_0^a {{({w_{,x }})}}^2}dx}}}\cr {+\int\limits_0^a {{w_{,x }}{{\hat{w}}_{,x }}dx} }\cr{- \int\limits_0^a {\frac{w}{\Re_x }dx} }\cr- \int\limits_0^a {\alpha (x)\left( {1 + \upsilon } \right)Tdx}\end{array} \right]$$

(13)

Structural Damping Independent of Time and Temperature

Structural damping for panels consists of both material damping and frictional damping acting at the panel supports. Support damping has not been considered here, and therefore, conservative results are likely to be obtained, that is, a lower value of the flutter speed and larger LCOs than the one would occur if this additional damping component would be accounted for [24]. The most widely used material-damping models are the linear viscous and hysteresis models. It has been proved that these damping models can significantly influence the flutter boundaries and it’s extremely dependent on the type of model employed. If only linear damping is considered, the work by Ellen [25] provides a useful classification of structural damping and showed which classes are always stabilizing using spatial derivatives arguments. The structural damping plays an important role in the flutter stability with low aerodynamic damping but would not affect significantly the flutter boundary with high aerodynamic damping. Fazelzadeh [22] showed that the structural damping reduced the panel domain of stability in linear analysis, whereas in nonlinear simulation, damping can have a stabilizing or destabilizing contribution.

From the mathematical point of view, structural damping independent of time and temperature can be introduced into the system by adding a term of the form $({g_{sb }}{{{{\partial^{j + 1 }}w}} \left/ {{\partial t\partial {x^j}}} \right.})$ to the bending terms of (10) and $({g_{sm }}{{{{\partial^{j + 1 }}w}} \left/ {{\partial t\partial {x^j}}} \right.})$ to the membrane terms of the (13). Herein, ${g_s}$ is structural damping coefficient and it is constant for viscous damping. ${g_{sb }}$ and ${g_{sm }}$ are bending and membrane damping coefficients, respectively. In the following, it will be assumed that these three damping coefficients are time and temperature independent. Based on the Kelvin’s model on elastic materials, $E(x)$ is replaced with the operator $E(x)(1 + {g_s}{\partial \left/ {\partial t } \right.})$ [26, 27]. By substituting (13) in (10), the aerothermoelastic bending governing equations becomes:

$$\begin{array}{ll} D\left( {1 + {g_{sb }}\frac{\partial }{\partial t }} \right){w_{,xxxx }} - \left( {1 + {g_{sm }}\frac{\partial }{\partial t }} \right)\cr \times\left[ {\frac{h}{{\left( {1 - {\upsilon^2}} \right)\int\limits_0^a {E{(x)^{- 1 }}dx} }}} \right] \cr \,\times \left[ \begin{array}{ll}{\frac{1}{2}\int\limits_0^a {{{{({w_{,x }})}}^2}dx} + \int\limits_0^a {{w_{,x }}{{\hat{w}}_{,x }}dx} \int\limits_0^a {\frac{w}{\Re_x }dx} }\cr {{{- (1 + \upsilon )\int\limits_0^a {\alpha (x)Tdx} }}}\end{array} \right]\\\times{({w_{,xx }} + 1/{\Re_x})+\, {\rho_m}h{w_{,tt }} - P_z^A\left( {x,t} \right) = P_z^{stat }(x) } \end{array}$$

(14a)

To improve accuracy and retain additional physics into the model, one could consider a thermoviscoelastic problem. Hilton [28] described the viscoelastic creep or relaxation functions and how this should be used in the case of thermal problems. Following Hilton one could recast the thermoviscoelastic problem using the following, general, constitutive equation:

$$\begin{array}{ll}{\sigma_{kl }}(x,t) = & \int\limits_{{- \infty}}^t {E_{klmn}^{* }[x,t,t',T(x,t' )]\;{\epsilon_{mn }}(x,t')\;{{\text d}}t'} \cr & - \int\limits_{{- \infty}}^t {E_{kl}^{{{T^{* }}}}[x,t,t',T(x,t' )]\;\alpha \,T(x,t')\;{{\text d}}t'}\end{array}$$

(14b)

where ${E^{* }}$ is the viscoelastic moduli, $T$ is the temperature function, and $\alpha$ is the thermal expansion coefficient. The viscoelastic moduli can be described by a Prony series [28]. The first integral is the contribution of the stress from ordinary strains, while the second integral is due to the thermal stresses. Both integrals are hereditary integrals meaning that a viscoelastic material has memory. Following the development, one could include this constitutive equation into the model by re-deriving the panel stiffness, $D$.

Aerodynamic Modeling

The fluid-structure interaction model used here is based on the nonlinear piston theory [29]. According to this theory, the radial aerodynamic pressure $p$ applied to the surface of the shell can be obtained by analogy with the instantaneous isentropic pressure on the face of a piston moving with velocity ${v_z}$ into a perfect gas which is confined in a one-dimensional channel; this pressure is given by:

$$\begin{array}{ll}{{{{p^{+ }}(x,t)}} \left/ {{{p_{\infty }}}} \right.} = {{\{ 1 + (\gamma - 1)[(\gamma - 1)/2]({v_z}/{c_{\infty }})\}}^{{{{{2\gamma }} \left/ {{(\gamma - 1)}} \right.}}}} \end{array}$$

(15)

In the analogy, the local transverse piston velocity (downwash velocity) ${v_z}$ normal to the panel and the undisturbed speed of sound $c{}_{\infty }$ may be expressed in terms of the panel transverse displacement $w(x,t)$ in order to obtain the radial aerodynamic pressure applied to the surface of the shell as a consequence of the external supersonic flow:

$${v_z} = {w_{{,\;t}}} + {U_{\infty }}{{[\hat{w} + w]}_{,x }};\quad c_{\infty}^2 = {{{\gamma \,{p_{\infty }}}} \left/ {{{\rho_{\infty }}}} \right.} $$

(16)

Herein ${p_{\infty }},{\rho_{\infty }},{U_{\infty }}$,and $\gamma$ are the pressure, air density, and air speed of the undisturbed flow and the isentropic gas coefficient ($\gamma = 1.4$ for dry air), respectively. To study the nonlinear panel flutter, in addition to the inclusion of geometrical nonlinearities, a nonlinear piston theory aerodynamics (PTA) model is used. PTA is a popular modeling technique for supersonic and hypersonic aeroelastic analyses [4]. Retaining, in the binomial expansions of (15), the terms up to and including ${{({v_z}/{c_{\infty }})}^3}$ yields the pressure formula for the PTA in the third-order approximation [7]:

$$\begin{array}{ll}{{{{p^{+ }}(x,t)}} \left/ {{{p_{\infty }}}} \right.} = \,1 + \gamma \left( {{v_z \left/ {{{c_{\infty }}}} \right.}} \right)\eta \cr + [\gamma (\gamma + 1)/4]{{[({v_z}/{c_{\infty }})\eta ]}^2} \cr + [\gamma (\gamma + 1)/12]{{[({v_z}/{c_{\infty }})\eta ]}^3}\end{array}$$

(17)

The linear term of this expression corresponds to Ackeret’s formula for the quasi-steady pressure on a thin profile in a supersonic flow field, whereas the quadratic term is from Busemann’s formula for ${M_{\infty }}>> 1$. In (17), the aerodynamic correction factor $\eta = {M_{\infty }}/\sqrt {{M_{\infty}^2 - 1}}$ enables one to extend the validity of the PTA to the entire low supersonic/hypersonic flight speed regime. Note that PTA provides results in excellent agreement with those based on the Euler solution and the CFL3D codes and with the exact unsteady supersonic aerodynamics theory [7]. Consider the flow only on the upper surface of the panel $U_{\infty}^{+ } \equiv {U_{\infty}}$ and ${M_{\infty}} = {U_{\infty}}/{c_{\infty }}$, that is, consider $U_{\infty}^{- } = 0$ and ${p^{-}} = {p_{\infty}}$; from (15) and (17), the aerodynamic pressure difference can be expressed as:

$$\begin{array}{lll}P_z^A(x.t) = & {p^{+ }} - {p_{\infty }} = {{\left. {\delta p} \right|}_{PTA}} = \cr& - \left( {{{{2{q_{\infty }}}} \left/ {{{M_{\infty }}}} \right.}} \right)\eta \left\{ {(1/{U_{\infty }}){w_{,t }}}\right.\cr & + {{{(\hat{w} + w)}}_{,x }}+ {[(1 + \gamma )/4]\eta {M_{\infty }} \hfill}\cr& \times {{{{[(1/{U_{\infty }}){w_{,t }}}} + {{{{{{{(\hat{w} + w)}}_{,x }}]}}^2}} \hfill}} \cr& + \left. {[(1 + \gamma)/12]{\eta^2}M_{\infty}^2\hfill} \right.\cr & \left. \times{{[(1/{U_{\infty }}){w_{,t }}}} + {{{{{{{(\hat{w} + w)}}_{,x }}]}}^3}}\right\}\end{array}$$

(18)

where the undisturbed dynamic pressure ${q_{\infty }} = {{{{\rho_{\infty }}U_{\infty}^2}} \left/ {2} \right.}$.

Thermal Loading

A linear temperature distribution $T$ throughout the panel thickness is considered:

$$ T(x,z) = \mathop{T}\limits^0 (x) + z\;\mathop{T}\limits^1 (x) $$

(19a)

Note that this temperature distribution was obtained via an exact analysis by Bolotin [21]. Using (5b) yields the thermal moment given by ${{{E\alpha {h^3}\mathop{{{T_{,xx }}}}\limits^1 }} \left/ {{12(1 - \upsilon )}} \right.}$. A membrane temperature distribution $\mathop{T}\limits^0 (x)$, implying $\mathop{T}\limits^1 (x) = 0$, will be considered. This temperature distribution can correspond to the steady-state flight regime of a high-speed aerospace vehicle. Such a representation of the temperature field is adopted here to reduce the problem to an eigenvalue one [8]. Specifically, $\mathop{T}\limits^0 (x)$ is expressed as:

$$ \mathop{T}\limits^0 (x) = \mathop{{T}}\limits^{* } \cos ({{{\pi \;x}} \left/ {a} \right.}) $$

(19b)

where $\mathop{{ T}}\limits^{* }$ is the temperature amplitude.

Aeroelastic Governing Equations

Substitution of (18) and (19) into (14a) and using the nondimensional parameters, which are presented in Appendix A, one can obtain the geometrically nonlinear aerothermoelastic governing equations of infinitely long curved panels in the form of $Q\{ \hat{\overline{W}},\overline{W}(\xi, \bar{t}\,)\} = 0$, where

$$\begin{array}{lll} Q\Big\{ \hat{\overline{W}},\overline{W}(\xi, \bar{t}\,)\Big\}& \equiv \,\underbrace{{\left( {1 + {g_{sb }}{\Omega_0}\frac{\partial }{{\partial \bar{t}}}} \right)\left( {1 + {\delta_e}e\mathop{{{\bar{{\text T}}}}}\limits^{* } {T_{cr }}} \right){{\overline{W}}_{{,\xi \xi \xi \xi \xi}}}}}_{{{{\text bending}}\;{{\text resistance} \ {\text with} \ {\text thermal} \ {\text degradation}}\;{{\text effect}}}} - \underbrace{{P_z^{stat}}}_{{{{\text pressure}}\;{{\text on}}\;{{\text plate}}}} \ - \ {\delta_{em }}\left( {1 + {g_{sm }}{\Omega_0}\frac{\partial }{{\partial \bar{t}}}} \right) \hfill \cr& \times\left[ {1/\int\limits_0^1 {\frac{1}{{(1 + {\delta_e}e\mathop{{{\bar{{\text T}}}}}\limits^{* } {T_{cr }})}}} d\xi } \right]\frac{12 }{{{{\bar{h}}^2}}}\underbrace{{\left[ {\frac{1}{2}\int\limits_0^1 {{{{({{\overline{W}}_{{,\xi }}})}}^2}d\xi } + \int\limits_0^1 {{{\overline{W}}_{{,\xi }}}\,{{{\hat{\overline{W}}}}_{{,\xi }}}d\xi - \frac{\hat{h}}{\bar{h}}\int\limits_0^1 {\overline{W}d\xi } } } \right]}}_{{{{\text in}} - {{\text plane}}\;{{\text force}}\;{{\text due}}\;{{\text to}}\;{{\text length}}\;{{\text change}}}}\, \left( {{{\overline{W}}_{{,\xi \xi }}} + \frac{\hat{h}}{\bar{h}}} \right) \cr&+ {\delta_{em }}\left( {1 + {g_{sm }}{\Omega_0}\frac{\partial }{{\partial \bar{t}}}} \right) \times \left[ {1/\int\limits_0^1 {\frac{1}{{(1 + {\delta_e}e\mathop{{{\bar{{\text T}}}}}\limits^{*}{T_{cr}})}}\, } d\xi} \right] \times \underbrace{{\frac{1}{{(1 - \upsilon )}}\left[ {\int\limits_0^1 {(1 + {\delta_{\alpha }}\alpha \mathop{{{\bar{{\text T}}}}}\limits^{* } {T_{cr }})\mathop{{{\bar{{\text T}}}}}\limits^{* } d\xi}} \right]}}_{{\text thermal} \ {\text degradation} \ {\text effect}}\,\cr& \times \left( {{{\overline{W}}_{{,\xi \xi }}} + \frac{\hat{h}}{\bar{h}}} \right) + \underbrace{{\pi^4 {{\overline{W}}_{{,\overline {tt}}}}}}_{{{{\text inertia}}\;{{\text effect}}}} + \underbrace{{\frac{{{M_{\infty }}{\pi^4}}}{{\bar{h}\bar{\rho }{{\bar{\Omega}}^2}}}\eta \left[ \begin{array}{lll} {\delta_{{a1\bar{t}}}}\frac{\bar{\Omega}}{{{M_{\infty }}}}{{\overline{W}}_{{,\bar{t}}}} + {\delta_{{a1\xi }}}({{{\hat{\overline{W}}}}_{{,\xi }}} + {{\overline{W}}_{{,\xi}}}) +\cr \frac{{1 + \gamma}}{4}\eta {M_{\infty}}{{\left( {{\delta_{{a2\bar{t}}}}\frac{\bar{\Omega }}{{{M_{\infty }}}}{{\overline{W}}_{{,\bar{t}}}} + {\delta_{{a2\xi}}}({{{\hat{\overline{W}}}}_{{,\xi }}} + {{\overline{W}}_{{,\xi}}})} \right)}^2} + \hfill \cr \frac{{1 + \gamma }}{12 }{\eta^2}M_{\infty}^2{{\left( {{\delta_{{a3\bar{t}}}}\frac{\bar{\Omega }}{{{M_{\infty }}}}{{\overline{W}}_{{,\bar{t}}}} + {\delta_{{a3\xi }}}({{{\hat{\overline{W}}}}_{{,\xi }}} + {{\overline{W}}_{{,\xi }}})} \right)}^3} \cr \end{array} \right]}}_{{{{\text aerodynamic}}\;{{\text loads} \ (3^{\text rd} \ {\text PTA})}}} = 0 \end{array}$$

(20a)

A quick look at (20a) may suggest that only the explicit terms in $\hat{w}$ have to be included in order to obtain the equation of motion which takes into account the effects of imperfection [30]. Panels with sinusoidal curvature, in nondimensional form, may also be approximated by a sinusoidal function, in this case:

$$ \hat{\overline{W}} = {{\hat{w}} \left/ {h} \right.} = \hat{\delta}\left\{ {\sum\limits_{p = 1}^n {q_p \sin (p\pi \xi )} } \right\} $$

(20b)

Herein, ${q_p}$ is the amplitude of geometric imperfection. To identify the effects of geometrical imperfection, edge movability, aerodynamic and thermal terms, various tracers have been adopted in the (20a) and (20b). The tracers ${\delta_e}$ and ${\delta_{\alpha }}$ identify the terms associated with the thermal degradation of the elastic modulus and the coefficient of thermal expansion, respectively. ${\delta_{em }} \in [0,1]$ identifies the degree of edge movability, where ${\delta_{em }} = 1$ indicates immovable edges. Movable edges can be simulated by assuming the panel is supported at the edges $\xi = 0$ and $\xi = 1$ by springs. In a later section of this entry, discussion of dynamic degree of movability simulation, a progressive damage, will be considered. The tracer $\hat{\delta } \in [0,1]$ indicates the implication of geometrical imperfection. The tracer ${\delta_{ajk }}$ has three indices: The first index $(a)$ identifies aerodynamic contribution, the second index $(j)$ identifies the degrees of linearity, (${1} \equiv {{\text linear}}$, ${2} \equiv {{\text quadratic}}$, and ${3} \equiv {{\text cubic}}$), while the third index $(k)$ represents the derivatives of the $\overline{W}$ with respect to $\bar{t}$ or $\xi$.

Solution Methodology

In the present work, Galerkin’s method and direct numerical integration DNIT will be considered to solve the integro-differential equation (18) to evaluate the structural response and the character of the curved panel flutter boundary with thermoelastic-elastic properties. For the simply supported panels on $\xi = 0, \;1$, it is required that $\overline{W} = {{\overline{W}}_{{,\xi \xi }}} = 0$. For these conditions, a solution can be found in the form:

$$ \overline{W}(\xi, \bar{t}\,) = \sum\limits_{j = 1}^n {\psi_j (\bar{t})} {{\,\bar{\,\phi}}_j}(\xi ) $$

(21)

where $n$ number of harmonic modes, $n \leq \infty$; ${{\,\bar{\,\phi}}_j}(\xi )$ are assumed orthogonal shape functions and ${\psi_j}(\bar{t})$ are unknown generalized coordinates that depend on time. The assumed functions ${{\,\bar{\,\phi}}_j}(\xi )$ are chosen to satisfy the boundary conditions. To fulfill such conditions, the mode shape functions ${{\,\bar{\,\phi}}_j}(\xi ) = \sin ({\lambda_j}\xi )$ and ${\lambda_j} = j\pi, \;j = 1,\,2, \cdot\cdot\cdot$ are considered. Cleary, the assumed approximate solution is not exactly the same as the unknown exact solution. Consequently, (21) will not satisfy the partial differential equations (PDE) (20a), that is, $Q(\xi, \bar{t}) = Q\left\{ {\hat{\overline{W}},\sum\limits_{j = 1}^n {\psi_j (\bar{t}){{\,\bar{\,\phi}}_j}(\xi )} } \right\} = {R_e}(\xi, \bar{t}) \ne 0$, where ${R_e}(\xi, \bar{t})$ is the residual function that results from the use of the approximate solution. Multiplying the residual by the basic function ${\,{\bar{\,\phi}}_r}(\xi ) = \sin (r\pi \xi )$ with $r = 1,2, \ldots, n \leq \infty$ and integrating over the panel length, $\xi$ from 0 to 1, and imposing the result to be 0, one obtains:

$$ \int\limits_0^1 {R_e (\xi, \bar{t})} {{\,\bar{\,\phi}}_r}(\xi ){{\text d}}\xi = 0 $$

(22)

As a result of (22), a set of nonlinear, simultaneous ordinary differential equations with respect to the series in (21), and function of geometrical imperfection (20b) can be obtained:

$$\begin{array}{ll}\frac{{{{{\text d}}^2}{\psi_r}}}{{{{\text d}}{{\bar{t}}^2}}} + g\frac{{{{\text d}}{\psi_r}}}{{{{\text d}}\bar{t}}} + {F_r}({\psi_j},{M_{\infty }},\mathop{{{\bar{{\text T}}}}}\limits^{* } ) = 0,\quad j,r = 1,2,3, \ldots \end{array}$$

(23)

The ${F_r}({\psi_j},{M_{\infty }},\mathop{{{\bar{{\text T}}}}}\limits^{* } )$ functions can be represented as:

$$\begin{array}{ll}{F_r}({\psi_j},{M_{\infty }},\mathop{{{\bar{{\text T}}}}}\limits^{* } ) = & F_r^{(l) }({\psi_j},{M_{\infty }},\mathop{{{\bar{{\text T}}}}}\limits^{*} ) + F_r^{(a) }({\psi_j},{M_{\infty }}) \cr& + F_r^{(th) }({\psi_j},{M_{\infty }},\mathop{{{\bar{{\text T}}}}}\limits^{*}) + F_r^{(s)}({\psi_j},{M_{\infty }})\end{array}$$

(24)

where $F_r^{(l) }({\psi_j},{M_{\infty }},\mathop{{{\bar{{\text T}}}}}\limits^{* } )$ are linear functions and $F_r^{(a) }({\psi_j},{M_{\infty }})$, $F_r^{(th) }({\psi_j},{M_{\infty }})$ and $F_r^{(s) }({\psi_j},{M_{\infty }})$ are functions including the aerodynamic, thermal, and structural nonlinearities, respectively.

Panel Stability in the Vicinity of the Flutter Boundary via Lyapunov First Quantity

From the mathematical point of view, the benign or catastrophic character of the flutter boundary can be revealed via determination of the nature of the supercritical or subcritical Hopf-Bifurcation, as featured by the nonlinear aeroelastic system [31, 32]. The system of governing equations obtained from (22) is converted to a system of four differential equations in state-space form expressed generically as:

$$ \frac{{d{x_j}}}{dt } = \sum\nolimits_{m = 1}^n {a_m^{(\,j) }{x_m} + {P_j}\left( {x_1, {x_2},{x_3},{x_4}} \right); \;j = \overline {1,4} } $$

(25)

For the present case, the functions ${P_j}\left( {x_1, {x_2},{x_3},{x_4}} \right)$ include both the structural and aerodynamic nonlinear terms as well as the thermal damage terms. Equation (25) can be presented in a form that can then be used toward the evaluation of the Lyapunov first quantity (LFQ), that is, $L\left( {M_F } \right)$. Considering the solution of the linearized counterpart of (25) under the form ${x_j} = {A_j}{e^{\omega t }}$, one obtains the characteristic equation:

$$ {\omega^4} + p{\omega^3} + q{\omega^2} + r\omega + s = 0 $$

(26)

As a reminder, for steady motion, the equilibrium is stable in Lyapunov’s sense if the real parts of all the roots of the characteristic equation are negative. Such an analysis can be done by applying Routh-Hurwitz’s criterion. For sufficiently small values of the speed, all the roots of the characteristic equation are in the left half-plane of the complex variable, and the zero solution of the system is asymptotically stable. On the same boundary, the two roots of the characteristic equation are purely imaginary and the remaining two are complex conjugate and remain also in the left half-plane of the complex variable (Hopf-Bifurcation conditions). The nature of the LCO that provides important information on the behavior of the aeroelastic system in the vicinity of the flutter boundary can be examined by the nature of the Hopf-Bifurcation of the associated nonlinear aeroelastic system. Figure 2 presents several pertinent scenarios; V = V_F defines the flutter boundary that can be determined via a linearized analysis. The nonlinear approach to the problem enables one to determine the aeroelastic behavior in the vicinity of the flutter boundary. As a result of the nonlinear analysis, one can determine the aeroelastic behavior of the structure for a flight speed in the vicinity of the flutter speed V_F. In this sense, curve 2 corresponds to a stable LCO (supercritical Hopf-Bifurcation (H-B)) and curve 3 to an unstable LCO (subcritical Hopf-Bifurcation). In order to identify the benign and catastrophic portions of the stability boundary, it is necessary to solve the problem of stability for the system of equations in the state-space form in the critical case of a pair of pure imaginary roots and to determine the sign of the LFQ [32].

Fig. 2

Character of the flutter boundary in the terms of LCOs amplitudes; H-B Hopf-Bifurcation

The flutter critical boundary is benign (i.e., yields stable LCO), or is catastrophic, yielding unstable LCO, if the following inequalities,

$$ L\left( {M_F } \right) < 0,{ \ {\text and} \ }L\left( {M_F } \right)> 0 $$

(27)

are fulfilled, respectively. The combination of effects from the structural and aerodynamic nonlinearities, the thermal load and thermal damage, significantly affects the character of the flutter boundary. In the region of the benign flutter boundary, one can slightly exceed the flutter critical speed ${M_F}$ without catastrophic failure of the panel, and as a result, the amplitude of the transverse deflection remains limited. Conversely, in the region of catastrophic flutter boundary, an explosive type of flutter can occur.

Aerothermoelastic Behaviors of Panel

A number of numerical simulations are presented in this section. A linear analysis is performed first. The numerical simulation considers as a test case study #1, an aluminum cylindrical panel whose mechanical properties and geometric parameters are: $E = 7 \times {10^{10 }}\,{{\text Pa}}$, $\upsilon = 0.3$, ${\rho_m} = 3,\,000\,{{\text kg}}/{{{\text m}}^3}$, $a =1\,{{\text m}}$, $R = 10\,{{\text m}}$, ${R \left/ {h} \right.} = 1,\,000$, ${\rho_{\infty }} = 1.225\,{{\text kg}}/{{{\text m}}^3}$, ${c_{\infty }} = 340.4\,{{\text m}}/{{\text s}}$, $\gamma = 1.4$, $\eta = 1$, $P_z^{stat } = 0$, ${\delta_{em }} = 1$ and ${q_1} = 0$. As a result, considering four modes, and without thermal degradation, the Mach flutter is ${M_F} = 4.2$, and the flutter frequency is ${\omega_F} = 3.7\,{{\text rad}}/{{\text s}}$. Figure 3 reveals the implications of the curvature ratio on the normalized flutter dynamic pressure of the infinitely long cylindrical panel and compares it with that of its finite length counterpart, ${\lambda_{{\text F}}} \equiv 2{q_{\infty }}{a^3}/D$. The results obtained from the present analysis using four and six modes are compared with the four mode solution of Dowell [33, 34] and very good agreements are reached.

Fig. 3

Comparison of flutter dynamic pressure versus the curvature ratio (Case #1)

In Fig. 4, the effects of the geometric imperfection on the flutter boundary are highlighted along with the variation of curvature ratio. The results reveal that the effect of the imperfections, represented in terms of ${q_1}$, depends on the curvature ratio and the symmetric or asymmetric shape of the imperfection.

Fig. 4

Effect of the imperfections on the flutter speed versus the curvature ratio (Case #1)

To have a clear and accurate view of the complex behavior of the aerothermoelastic system, the nonlinear dynamic behavior has been numerically simulated as a case study #2 for a monolithic titanium (Ti-6Al-4V) panel. A cylindrical panel whose mechanical properties [35] ($T = 294.15{ \ {\text K}}$) and geometric parameters are ${E_0} = 110.352 \times {10^9}\,{{\text Pa}}$, $\upsilon = 0.31$, ${\alpha_0} = 4.85\,^*{10^{- 5 }}/{{{\text c}}^{{\text o}}}$, ${\rho_m} = 4,430\,{{\text kg}}/{{{\text m}}^3}$, $a \,= \,1\,{{\text m}}$, ${\Re_x} = 10\,{{\text m}}$, $h = 0.01\,{{\text m}}$, ${\rho_{\infty }} = 1.225\,{{\text kg}}/{{{\text m}}^3}$, ${c_{\infty }} = 340.4\,{{\text m}}/{{\text s}}$, $\gamma = 1.4$, $\eta = 1$, $P_z^{stat } = 0$, ${\delta_{em }} = 1$, ${e_T} = - 6.5764 \times {10^{- 4 }}/{{\text K}}$, ${\alpha_T} = {3}{.07085} \times {1}{0^{- 4 }}/{{\text K}}$, ${q_1} = 0$, ${g_{sb }} = {g_{sm }} = 0$ and the initial conditions are ${{\overline{W}}_1} = 0.1,\;{{\overline{W}}_r} = {{\dot{\overline{W}}}_r} = 0$ has been considered. From these data, the following parameters are obtained: $\bar{h} = 0.01,\,\hat{\,h} = 0.001,H = {a^2 \left/ {{(8{\Re_x}}} \right.}) = 0.0125\,{{\text m}}$, $H/h = 1.25,\;{\Omega_0} \approx 150{ \ (1}/\operatorname{s} )$, $t = 5 \operatorname {s},\bar{\rho } \approx 3616,{ \ {\text and} \ }{T_{cr }} = 1.90\;{{\text K}}$. To investigate the effect of degree of edge movability on the linear flutter Mach number for a system without thermal degradation, damping, and geometrically perfect, Figs. 5 and 6 show the frequency coalescence, and the flutter speed for selected ${\delta_{em }}$, respectively. It appears that the flutter speed is obtained from the coalescence of the two consecutive eigen-frequencies and this speed increases when the degree of edges movability increases, implying lower values of ${\delta_{em }}$. The edge constraint effect can induce earlier flutter. This is due to the reduction in the in-plane forces, and to the panel curvature effect.

Fig. 5

Frequency coalescence for different values of ${\delta_{em }}$

Fig. 6

Flutter speed versus ${\delta_{em }}$

For the dynamic analysis, the nondimensional time integration was carried out from $\bar{t} = 0$ to $\bar{t} \approx 750$ time units and only the last 50 units have been retained for the bifurcation representation. The analysis was performed with no damping on the system. The linear Mach flutter (without thermal degradation) is ${M_F} = 6.6$, as shown in Fig. 7. It is also shown that for static partial degree of edge movability, for example, ${\delta_{em }} = 1;0.9;0.8;{ \ {\text and} \ }0.75$, the flutter speed increases, meaning that the system will exhibit LCO at higher Mach numbers.

Fig. 7

Bifurcation diagram of the aerothermoelastic curved panel (Case #2) with respect to the variation of flight Mach number and static edge degree movability (without thermal degradation and damping)

In the absence of structural damping and thermal degradation, the nonlinear dynamic simulation of the system exposed to geometrically imperfect ${q_1} = 0.005$ has been determined as shown in Fig. 8 for different values of ${\delta_{em }}$ (1; 0.9 and 0.8).

Fig. 8

Bifurcation diagram of the aerothermoelastic curved panel (Case #2) with respect to the variation of flight Mach number and static edge degree movability (without thermal degradation and damping) under the effect of imperfections

Figure 9 shows the time histories (a, b, d, e, f, h, and i) and phase portraits (c, g, and j) of the considered system without damping for different flight Mach number. Imperfection can increase the LCO amplitude of the nonlinear oscillatory skin panel motion as shown in Fig. 9c, j, g or damp out as shown in Fig. 9d depending on the fluid-structure interaction behavior and on ${\delta_{em }}$.

Fig. 9

Time histories and phase portraits of the aerothermoelastic curved skin panel (Case #2) for different flight Mach number $({g_{sb }} = {g_{sm }} = 0,{\delta_{em }} = 1,{q_1} = 0.005)$

To consider the effect of heated panel, a wall temperature has been computed as follows: $\mathop{T}\limits^{* } = {T_w} = {T_{\infty }} + {R_f}[(1 - \gamma )/2]M_{\infty}^2$ where ${R_f} = \sqrt {\Pr } \approx 0.3$ [16]. The maximum material temperature was limited to $\mathop{T}\limits^{* } = {T_w}{} \approx {810 \ {\text K}}$ [35] to prevent thermal buckling. Within this constrain, in the case of heated panels, the time simulation was interrupted at $M_{\infty } = 5.4$. Figure 10 shows the bifurcation diagram when the thermal degradation has been considered. It clearly appears that the thermal degradation reduces the flutter speed. Furthermore, limit cycles appear at speeds as low as $M_{\infty } \approx 3.5$ due to the temperature-dependent material degradation effect, while unheated panel will exhibit LCOs at $M_{\infty }> 6.6$ (linear flutter Mach number). In addition, in the case of heated panels, LCOs with large amplitude are present, as compared to the case of unheated panel, and are growing at faster rate with jumps in amplitudes above $M_{\infty }> 4.5$. Decreasing the static partial edge movable from immovable ${\delta_{em }} = 1$ toward ${\delta_{em }} = 0$ has a significant effect on the shifting of the nonlinear flutter boundaries and the LCO behavior.

Fig. 10

Bifurcation diagram of the aerothermoelastic curved panel (Case #2) with respect to the variation of flight Mach number and static edge degree movability (with thermal degradation and no damping)

Effect of dynamic partial edge degree movability on the behavior of the nonlinear aerothermoelastic system (Case #2) is highlighted in Fig. 11. Herein ${\delta_{em }}$ has been considered for linear, quadratic, and cubic variations with the time simulation, that its ${\delta_{em }} = a + bt,\;a + b{t^2},$ and $a + b{t^3}$, or in dimensionless form ${\delta_{em }} = a + b\bar{t}/{\Omega_0},a + b{{(\bar{t}/{\Omega_0})}^2},$ and $a + b{{(\bar{t}/{\Omega_0})}^3}$, where $a$ and $b$ are constants. The analysis also considers that the edge might start moving at a predefined ${t_0}$ during the time simulation of the nonlinear dynamic system. This simulates the dynamic change in edge movability occurring while the system has already exhibited an LCO. The bifurcation diagrams with respect to the variation of flight Mach number, assuming ${q_1} = 0$, ${g_{sb }} = {g_{sm }} = 0$, ${t_0} = 0 \operatorname {s}\,$, are presented in the subsequent figures. The values $a = 1,b = - 0.04{({\text linear})}$, $b = - 0.008{({\text quadratic})}$, $b = - 0.0016{({\text cubic})}$ have been selected to represent the edge condition from immovable, ${\delta_{em }} = 1\;\,\,$, to partially movable, ${\delta_{em }} = 0.8$, in finite time. In addition, the edge condition from partial movable, ${\delta_{em }} = 0.8$, to immovable ${\delta_{em }}{({\text linear})} = 0.8 + 0.04t$ to reach ${\delta_{em }} = 1$ in finite time has been considered as well, along with the conditions ${t_0} = 2.5$ and $4.5 \operatorname {s}\,$. All these selected conditions have been presented in Fig. 11a–f), respectively. Comparing these results with the one in Fig. 9 (case when ${\delta_{em }} = 0.8$), no significant change has been revealed from Fig. 11a–c in the behavior of the system or the amplitude of LCO when different models of ${\delta_{em }}$ are implemented in the simulation. In Fig. 11d, when the condition of linear variation is from partially movable to immovable, the nonlinear flutter boundary decreases with time, up to ${M_{\infty }} \approx 3.6$, significantly smaller in comparison with ${M_{\infty }} \approx 4.3$ obtained for the case of static edge movability ${\delta_{em }} = 0.8$. In addition, different LCO behavior is obtained and it is evident from comparing Fig. 11a with Fig. 11d. For the case when ${t_0} = 2.5 \operatorname {s}$ (half of the simulation time), the system exhibits an LCO behavior, Fig. 11e, similar to the one shown in Fig. 11a. However, as ${t_0}$ increases, there are changes in the LCO behavior. When ${t_0} = 4.5 \operatorname {s}$ (near the end of the time simulation), the nonlinear flutter boundary is significantly affected, evident from comparing results displayed in Fig. 11a, e, f. These simulations show that the degree of edge mobility is an important effect to consider in the flutter and post-flutter behavior of high-speed panels, and the amplitude of oscillations of the panel at the time this structural change is triggered will promote a different post-flutter behavior.

Fig. 11

Bifurcation diagram of the aerothermoelastic curved panel (Case #2) with respect to the variation of flight Mach number, degrees of movability, and its starting time

Additional numerical simulations are presented for a Ti-8 Mn [6] infinitely long thin-flat panel (see Fig. 12).

Fig. 12

Spacecraft on reentry mission. The panels are exposed to high temperature field

Figures 13 and 14 depict the LFQ and in this context, the benign and catastrophic post-flutter scenarios are highlighted. In these plots, the effects of the structural and aerodynamic nonlinearities considered in conjunction with that of the temperature and the thermal damage on stable/unstable LCO are emphasized. With the increase of the thermal field, the transition from benign boundary ($L\left( {M_F } \right) < 0$) toward catastrophic flutter boundary ($L\left( {M_F } \right)> 0$) occurs at lower values of the flight speed (Fig. 13). This reveals that the temperature exerts a detrimental effect not only on the flutter boundary but on the character of the flutter boundary as well. It also clearly appears that the aerodynamic nonlinearities are, in general, destabilizing. In addition, the effect of the damage on the elastic modulus is prevalent (Fig. 14), and, as a result, the occurrence of the catastrophic flutter boundary is shifted toward lower values of the flight speed.

Fig. 13

Benign and catastrophic flutter boundary. No thermal damage

Fig. 14

Benign and catastrophic flutter boundary. Thermal damage included

Consideration About the Aerothermoelastic Behavior of Panel

A number of results related to the dynamic simulation of infinitely long thin-walled circular cylindrical panels featuring initial geometric imperfections and taking into consideration the thermal field and degradation due to its operation at supersonic/hypersonic speed have been presented. In this context, the implications of structural and aerodynamic nonlinearities, on the LCOs and on the character, benign or catastrophic of the panel flutter critical boundary, have been examined. The static and dynamic edge movability conditions simulating the propagation of supports degradation have been considered to explore the effect produced on the aerothermoelastic system responses. The dynamic response is either suppressed or evolves into an LCO, depending on the thermal degradation, imperfection, static or dynamic condition of edge movability, as well as the time when the edge constrain change is triggered. With the increase of the supersonic/hypersonic flight speed, when the aerodynamic nonlinearities become prevalent, the flutter boundary becomes catastrophic, irrespective of the presence of structural nonlinearities. It was also shown that the effect of temperature and thermal degradation are invariably detrimental in the sense of reducing the flutter speed and of rendering the flutter boundary a catastrophic one. In addition, as a by-product of this analysis, conclusions on the effects of the temperature field coupled with those of the thermal degradation on the eigen-frequency and flutter boundary have been outlined.

Appendix A: Dimensionless Parameters

$\overline{W} = w/a{ \ }$

$\hat{\overline{W}} = \hat{w}/a{ \ }$

$\xi = x/a{ \ }$

$\bar{t} = t{\Omega_0}{ \ }$

${\Omega_0} = {{(\pi /a)}^2}\sqrt {{D/{\rho_m}h}} { \ }$

${\bar{\Omega}} = {\Omega_0}a/{c_{\infty }}$

$\bar{h} = h/a{ \ }$

$\hat{h} = h/{\Re_x}$

$P_z^{stat } = {\Delta}P_z^{stat }(x){a^4}/{D_0}h$

${T_{cr }} = {D_0}/Eh{a^2}{\alpha_0}{ \ }$

$\bar{\rho } = ({\rho_m}/{\rho_{\infty }})$

$H \approx {a^2}/(8{\Re_x})$

$\mathop{\tau}\limits^{* } = \mathop{T}\limits^{* } /{T_{cr}}$

$\mathop T\limits^* = \mathop{\tau}\limits^{* } \cos (\pi \xi)$