Aeroviscoelasticity Designer FGMs: Passive Control Through Tailored Functionally Graded Materials

Overview

This entry covers four distinct areas, namely, the interaction in a closed loop system of designer aerodynamics, of viscoelastic materials and structures, and of controls. The presence of varying temperatures not only induces thermal stresses but also strongly affects material properties. The effects of temperature on viscoelastic material properties as well as on flutter velocities and times to reach flutter conditions are discussed. It is shown that optimized FGM distribution can increase flutter velocities and lengthen the time to when flutter will occur.

Introduction

The confluence of designer aerodynamics, of viscoelastic materials and structures, and of controls in a closed loop dynamical system introduces several distinct problems in each of the four contributing areas as well as in their ensemble.

All functionally graded materials, or FGMs for short, are from a fundamental mechanics point of view nonhomogeneous materials where the property distributions are prescribed during the manufacturing phase. Such distributions may follow continuous and/or piecewise continuous functions. Other possible sources of inhomogeneities are dissimilar materials, composites, and temperature distributions. A striking example of artificially created dissimilar material FGMs is illustrated in Fig. 1 where thin layers of distinct materials are deposited in a prescribed fashion on a plate [1]. A comprehensive formulation (space limitations necessitate citing publications where expanded bibliographies can be found) of viscoelastic FGMs may be found in [2] and of aero-thermo-servo-elasticity with FGMs in [3]. A treatment of the differences and similarities between thermo-elasticity and thermo-viscoelasticity is given in [4].

Fig. 1

Five-layer functionally graded ZrO₂/NiCoCrAlYcoating [1]

Linear viscoelasticity has become a mature though not closed field [5]. However, much research remains to be undertaken in nonlinear viscoelasticity [6]. For a list of additional references, see [7].

Tailored aerodynamics have been introduced in publications on airfoil design such as [8–10], where airfoil surfaces are analytically generated to deliver prescribed performance characteristics of low drag, high L/D ratios, etc. Modern airfoil morphing reminiscent of the Wright brothers’ original plane also offers control possibilities as seen in [11–14].

Aeroelasticity is a mature field and is covered by a significant number of textbook starting with the everlasting classic [15] and including but not limited to [16–25]. Aero-viscoelasticity on the other hand is still an emerging field starting with [26] and [27] and at this time with no text books. A comprehensive bibliography of the subject as well as an analytical treatment of aero-servo-viscoelasticity may be found in [28].

Elastic designer materials are first described in [29] and viscoelastic ones in [30]. The formal analytical formulation based on calculus of variations is presented in [31].

Analysis

Consider a Cartesian coordinate system $x=\left\{ {{x_1},{x_2},{x_3}} \right\}=\left\{ {{x_i}} \right\}$, an FGM function ${\mathcal F}(x)$, and a temperature distribution $T(x,t)$. The linear anisotropic viscoelastic constitutive relations can be expressed as [4–6, 32]

$$\begin{array}{ll}{\sigma_{ij }}(x,t)\:=& \int\limits_{{-\infty}}^t {{E_{ijkl }}} \left[ {x,t,{t}^{\prime},\ \mathcal{F}(\it x),T(x,{t}^{\prime})} \right]\,\frac{{\partial {\varepsilon_{kl }}(x,{t}^{\prime})}}{{\partial {t}^{\prime}}}\:d{t}^{\prime}\cr& -\int\limits_{{-\infty}}^t {E_{ij}^T} \left[ {x,t,{t}^{\prime},\ \mathcal{F}(\it x),T(x,{t}^{\prime})} \right]\,\frac{{\partial \left[ {\alpha T(x,{t}^{\prime})} \right]}}{{\partial {t}^{\prime}}}\:d{t}^{\prime}\end{array}$$

(1)

or

$$\begin{array}{ll}{\varepsilon_{ij }}(x,t)=& \:\int\limits_{{-\infty}}^t {{C_{ijkl }}} \left[ {x,t,{t}^{\prime},\ {\mathcal F}(x),T(x,{t}^{\prime})} \right]\,\frac{{\partial {\sigma_{kl }}(x,{t}^{\prime})}}{{\partial {t}^{\prime}}}\:d{t}^{\prime}\:\cr& +\int\limits_{{-\infty}}^t {C_{ij}^T} \left[ {x,t,{t}^{\prime},\ {\mathcal F}(x),T(x,{t}^{\prime})} \right]\,\frac{{\partial \left[ {\alpha T(x,{t}^{\prime})} \right]}}{{\partial {t}^{\prime}}}\:d{t}^{\prime}\end{array}$$

(2)

The fundamental difference between elastic and viscoelastic constitutive relations is the fact that the elastic ones are algebraic, while the viscoelastic relations belong to the integral-differential species. Additionally, there remains the most significant matter of the temperature dependence of relaxation moduli E _ijkl and creep compliances C _ijkl . When one eliminates all relaxation/creep influences at elevated temperatures from Young’s modulus experimental measurements, the remainder shows little variations of elastic moduli with temperature [33–35]. Viscoelastic metal and polymer matrix relaxation moduli, on the other hand, show extreme sensitivity to temperature due to real material variations in viscosity coefficients of approximately one order of magnitude per 20 °C – see Fig. 2. The additional most significant effect of this temperature dependence is to change the kernel functions in the hereditary integrals from $E(x,t-{t}^{\prime})$ to $E(x,t,{t}^{\prime})=E\left[ {x,t,{t}^{\prime},{\mathcal F}(x),T(x,{t}^{\prime})} \right]$ thus destroying the convenient properties of the convolution integrals.

Fig. 2

Viscoelastic viscosity temperature dependence

A large class of viscoelastic materials, known as thermo-rheologically simple materials (TSMs), has behavioral responses that admit the presence of the WLF (also known as the Williams-Landel-Ferry shift factor/function) material property shift function a _T [36] empirically defined as

$$ {\rm log}{_e}\left[ {{a_T}(t)} \right]\:=\:\frac{{{C_1}(T-{T_0})}}{{{C_2}+T-{T_0}}} $$

(3)

with T ₀ a conveniently chosen constant reference temperature. It may, but need not, be equated to the rest temperature at which the thermal expansions T vanish. In this entry, the same T ₀ is used. An empirical well-working model for TSMs defines an associated reduced time $\xi (x,t)$ as [37]$\mathcal{F}$

$$\begin{array}{ll} \xi (x,t)&=\int\limits_0^t {{a_T}} \left[ {T(x,s)} \right]\:ds \\& =\:\int\limits_0^t {\exp } \left( {\frac{{{C_1}\left[ {T(x,s)-{T_0}} \right]}}{{{C_2}+T(x,s)-{T_0}}}} \right)ds\\ \quad \quad \xi \:\in \:[0,\infty ] \end{array}$$

(4)

and reduces all relaxation moduli curves at many diverse temperatures to a single master relaxation curve for each TSM with $\hat{E}(x,\xi )=E\left[ {x,t,{t}^{\prime},\ {\mathcal F}(x),T(\it x,{t}^{\prime})} \right]$ versus ξ. By the above definition, it follows that at $T={T_0}$, ${a_T}=1$ and $\xi ({T_0})\equiv t$.

Further, examinations reveal that

$$\begin{array}{ll}\int\limits_{{-\infty}}^t {{E_{ijkl }}} \left[ {x,t,{t}^{\prime},\ {\mathcal F}(x),T(x,{t}^{\prime})} \right]\:\frac{{\partial {\sigma_{kl }}(x,{t}^{\prime})}}{{\partial {t}^{\prime}}}d{t}^{\prime}\cr\equiv \:\int\limits_{{-\infty}}^{{\xi (x,t)}} {\mathop{\widehat{E}}\nolimits_{ijkl}} \left[ {x,\xi (x,t)-{\xi}^{\prime}} \right]\:\frac{{\partial \mathop{\hat{\sigma}}\nolimits_{kl }(x,{\xi}^{\prime})}}{{\partial {\xi}^{\prime}}}\:d{\xi}^{\prime}\end{array}$$

(5)

and thus the convolution integrals are restored in the ξ-space. However, any success at recapturing an elastic-viscoelastic correspondence principle in the ξ-space is thwarted by the fact that the x _i derivatives acquire variable coefficients due to the ξ transformations, to whit

$$ \begin{array}{ll}\frac{\partial }{{\partial {x_i}}}\:=& \underbrace{{\frac{{\partial \xi (x,t)}}{{\partial {x_i}}}}}\frac{\partial }{{\partial \xi }}\cr& =\,{Z_i}(x,t) \cr&={Z}_i(x,\xi ) \end{array}$$

(6)

The transformation into the ξ-space mandates that

$$\begin{array}{ll}{E_{ijkl }}(x,t)& \:\equiv \:{{\hat{E}}_{ijkl }}(x,\xi )\\&=\:E_{ijkl}^{\infty }(x)\:+\:\sum\limits_{n=1}^{{{N_{\underline{ijkl}}}}}\,{E_{{\underline{ijkl}n}}} (x) \\ & \quad \times \exp \left( {-\frac{{\xi (x,t)}}{{\tau_{{\underline{ijkl}n}}^0(x)}}} \right) \end{array}$$

(7)

Equation (5) should be preferentially used in the governing relations as they simplify the “bookkeeping” and numerical solutions when used.

While the convolution integrals are restored in the ξ-space, the variable coefficients generated by the x derivatives negate any possibility of applying the elastic-viscoelastic correspondence principle (EVCP) in either real time t or in reduced time ξ. Table 1 summarizes these phenomena.

Table 1 Elastic and viscoelastic thermal coupling

The FGM function can be expressed as a series in the finite $x$ domain, such as for instance

$$ {\mathcal F}(x)=\sum\limits_{m=0}^{\mathcal{M}} {\sum\limits_{n=0}^{\mathcal{N}} {\sum\limits_{k=0}^{\mathcal{K}} {{A_{mnk }}\,x_1^m\,x_2^n\,x_3^k} } } $$

(8)

Then let

$$\begin{array}{ll}{ \mathcal S}\:=\:\left\{ {{{\mathcal{S}}_1},{{\mathcal{S}}_2},\cdots, {{\mathcal{S}}_{{{{\mathcal{S}}_s}}}}} \right\}\:=\:\left\{ {{{\mathcal{S}}_{\ell }}} \right\}\\ \quad \ell =1,2,\cdots, {{\mathcal{S}}_s} \end{array}$$

(9)

be the set of parameters (constants) representing ${\mathrm{{E}}_{ijkln }},\tau_{ijkln}^0,{A_{mnk }}$, geometry, sizing, weight, cost, etc., and in the case of composites fiber orientation, number of plies, volume ratios, stacking sequences, etc., to be optimized.

Further, let $u(x,t)$ be generalized displacements representing rigid body motion, spanwise and chordwise bending, torsion, etc., which leads to sets of governing relations of motion in the generating form

$$\begin{array}{ll}{{\mathcal{L}}_u}(x,t,S)\:=\:\underbrace{{m\,\frac{{{\partial^2}u(x,t)}}{{\partial {t^2}}}}}_{{=\,{\rm inertia}\;({{{\rm T}}_1})}}\cr +\:\underbrace{{c\,\frac{{\partial u(x,t)}}{{\partial t}}}}\\= \rm external \rm mechanical \rm damping\,(T_2)\; \cr+\:\underbrace{{\int\limits_{{-\infty}}^t {{{\hat{D}}_{kl }}} \left[ {x,\xi (x,t)-{\xi}^{\prime},\ {\mathcal F}(x)} \right]\,\frac{{\partial {{\hat{\epsilon}}_{kl }}(x,{\xi}^{\prime})}}{{\partial {\xi}^{\prime}}}\:d{\xi}^{\prime}}}_{{=\,{\rm internal}\;{\rm viscoelastic}\;{\rm restoring}\;{\rm force}\;({{{\rm T}}_3})}} \cr\:-\:\underbrace{{\int\limits_{{-\infty}}^t {{{\hat{D}}^T}} \left[ {x,\xi (x,t)-{\xi}^{\prime},\,{\mathcal F}(x)} \right]\,\frac{{\partial \left[ {\widehat{{\alpha T}}(x,{\xi}^{\prime})} \right]}}{{\partial {\xi}^{\prime}}}\:d{\xi}^{\prime}}}_{{=\,{\rm internal}\;{\rm thermal}\;{\rm expansion}\;{\rm force}\;({{{\rm T}}_4})}}\cr=\:\underbrace{{{F_V}(x,t)}}\\ \begin{array}{ll} =\,{\rm vibratory} \; {\rm force}\;({{{\rm T}}_5}) \end{array} \cr \begin{array}{ll} {F_{SC }}\bigg( {x,t,u\left( {x,t} \right),\frac{{\partial \left( {u\left( {x,t} \right)} \right]}}{{\partial t}},} \\ {\frac{{{\partial^2}\left( {u\left( {x,t} \right)} \right]}}{{\partial {t^2}}},\int\limits^t {u\left( {x,{t}^{\prime}} \right)d{t}^{\prime}} \bigg)}\end{array} \\{{=\, {\rm differential}\ {\rm and}\ {\rm integral}\ {\rm servo}\ {\rm control}\ {\rm force}\;({{{\rm T}}_6})}} \cr \:+\:\underbrace{{{F_A}\left( {x,t,u(x,t),\frac{{\partial \left( {u(x,t)} \right]}}{{\partial t}},\frac{{{\partial^2}\left[ {u(x,t)} \right]}}{{\partial {t^2}}}} \right)}}_{{=\,{\rm aerodynamic}\;{\rm forces}\;({{{\rm T}}_7})}}\end{array}$$

(10)

where the ${{\hat{D}}_{kl }}$ and ${{\hat{D}}^T}$ are differential spatial operators specific to the appropriate u component for beam and plate bending, torsion, etc. For instance, for Euler-Bernoulli beam bending, it is

$${D_{1111 }}(x)\:=\frac{{{\partial^2}}}{{\partial x_1^2}}\left\{ {E\left[ {x,t,{\mathcal F}(x),T(x,t)} \right]\frac{{{\partial^2}}}{{\partial x_1^2}}} \right\} $$

(11)

In order to examine the stability behavior of (10), it is advantageous to proceed in the following manner:

• Express the solution functions in terms of series

  $$ u\left( {x,t} \right)=\sum\limits_{m=1}^{{{M^u}}} {{A_m}(t)\:f_m^u(x)} $$

  (12)

  where each term $f_m^u(x)$ satisfies the BCs.

• Apply Galerkin’s method and eliminate the x dependence resulting in integral ordinary differential equations of the type

  $$\begin{array}{ll}{C_3}\frac{{{d^3}U(t)}}{{d{t^3}}}\:+\:\underbrace{{\left( {{S_2}+{A_2}+{C_2}} \right)}}_{{=\;{B_2}}}\frac{{{d^2}U(t)}}{{d{t^2}}}\cr +\:\underbrace{{\left( {{S_1}+{A_1}+{C_1}} \right)}}_{{=\;{B_1}}}\frac{dU(t) }{dt } \cr +\underbrace{{\left( {\underbrace{{{S_0}}}_{\mathrm{ elastic}}+{A_0}+{C_0}} \right)}}_{{={\,B_0}}}U(t)\cr +\underbrace{{\int\limits_{{-\infty}}^t {{{\hat{S}}_I}\left[ {\xi (t)-{\xi}^{\prime}} \right]\frac{{d\hat{U}\left( {\xi^{\prime}} \right)}}{{d{\xi}^{\prime}}}d{\xi}^{\prime}}}}_{{\rm viscoelastic}\ {\rm including}\ {\rm temperature}\ {\rm dependence}}\cr +\underbrace{{{C_I}\int\limits^t {U\left( {t^{\prime}} \right)d{t}^{\prime}=0}}}_{{\rm integral}\ \rm{\rm controller}}\end{array}$$

  (13)

where

$$\begin{array}{ll}{S_n},{S_I}\: =\:\mathrm{structural}\ \:\mathrm{coefficient}\mathrm{s},\cr{S_2}\: =\:\mathrm{ mass}\ \:\mathrm{ coefficient} \cr {A_n}(V)\: =\:\mathrm{ aerodynamic}\:\ {\rm coefficients} \cr \begin{array}{ll}{C_n}\: =\:\mathrm{ servo}-\mathrm{ control}\;\mathrm{ coefficients}\;\mathrm{ for}\cr \qquad\mathrm{ differential}\;\mathrm{controllers}\end{array}\cr{C_I}\: =\:\mathrm{ integral}\:\mathrm\ {\rm controller}\ \:\mathrm{ coefficient}\end{array}$$

The presence of temperature functions $T(x,t)$ or T(t), but not T(x), in the ${{\hat{S}}_I}\left[ {\xi (t)} \right]$ functions precludes the possibility of solutions $U(t)\sim {\rm exp}\left[ {(d+\imath\,\omega )t} \right]$. Consequently, the customary flutter criterion of simple harmonic motion (SHM) when $d({V_F},{\omega_F})=0$ no longer represents an attainable flutter criterion. Instead an alternate viscoelastic one must be enforced, such that

$$\begin{array}{ll}{\rm viscoelastic}\:\Rightarrow \cr \quad\left\{ \begin{array}{ll}{\mathop{\lim}\limits\begin{array}{ll} t\to {t_F} \cr V\to {V_F} \end{array}\left\{ {u\left( {x,t,V} \right)} \right\}\to \:\infty } \cr{{\rm or}} \cr {\mathop{\lim}\limits\begin{array}{ll} t\to {t_F} \cr V\to {V_F} \end{array}\left\{ {\frac{{\partial u\left[ {x,t,V} \right]}}{{\partial t}}} \right\}\:\to \infty } \cr\end{array} \right. \end{array}$$

(14)

These instability conditions can be determined from the solution’s non-converging series or from a single unbounded amplitude in the solution series or through limit cycle analyses when applicable. In any case, (14) points to the fact that under variable temperatures viscoelastic flutter conditions are dictated by a combination of velocity (V _F ) and critical time or time to flutter (t _F ).

Figure 3 describes typical conditions according to the stability prescription (14). The ultimate viscoelastic velocity that can be reached is in this case the viscoelastic flutter velocity, and the graph depicts its reduced value compared to the equivalent elastic one. For a given lifting surface, both of these values will, of course, vary with altitude, trim angle, T(t), etc. The time t _F is the time at which the viscoelastic flutter velocity occurs, and it is paired with a flutter velocity V _F , which in these cases are not eigenvalues. For a constant T, Fig. 3 would have roughly the same shape but different values. In general an increase in temperature decreases both V _F and t _F shifting the curve toward the origin. Conversely, a decrease in temperature has an opposite delaying effect.

Fig. 3

Viscoelastic flutter velocity

The designer material formulation is basically that of an inverse problem solved through the calculus of variations [31]. The optimization is subject to prescribed constraints based on cost, weight, t _F , V _F , and some of the parameters S enumerated after (9). Formally the constraints can be stated as

$$ {\mathcal C}\left( {\mathcal S} \right)=0 $$

(15)

where S is a subset of the entire ensemble S. After (13) are solved for the U(t), the temporal influence is eliminated by specifying t _F and hence U(t _F ), or any other convenient time, or an average U _ave value, such as

$$ {U_{ave }}(S)\:=\:\int\limits_0^{{{t_F}}} {\frac{U(t) }{{{t_F}}}} dt $$

(16)

The next step is to solve the now remaining algebraic relations for each S _m from

$$ \frac{\partial }{{\partial {S_m}}}\left\{ {{{\mathcal{L}}_u}(S)+\lambda \,{\mathcal C}\left( {\mathcal S} \right)} \right\}=0 $$

(17)

where λ is a Lagrangian multiplier [38]. The protocol is summarized in the flow chart of Fig. 4.

Fig. 4

Designer flow chart

Discussion and Conclusions

Figure 5 depicts in a normalized fashion the effects of constant temperatures on V _F and t _F . As can be expected, an increase in temperature brings with it higher relaxation and creep rates and, therefore, both decrease in value as the temperature is elevated. The converse is seen as cooling effects take place.

Fig. 5

Temperature effects on viscoelastic flutter velocity

The control that can be exercised on flutter and times to flutter at one constant temperature can be seen in Fig. 6. From left to right, the first curve representing a lifting surface with optimized homogeneous viscoelastic properties yields the shortest t _F and the lowest flutter velocities. When designer FGMs are applied to the same geometric surface, V _F s and t _F s are increased and flutter conditions are improved.

Fig. 6

FGM influence on viscoelastic flutter velocity

The use of FGM passive control principles is extremely attractive for UAVs and MAVs, where the lifting surfaces are light weight and more importantly highly flexible. Their limited mission scope compared to a more complex fighter or transport flight vehicle makes them ideally suited for a priori built in FGM distributions.

Of course these designer material studies create materials with hypothetical elastic or viscoelastic optimize properties. The next step, not part of these studies, is to develop manufacturing techniques to produce such materials to designer/tailored properties specifications. In [39], analyses are presented which relate material chemical structure to polymer properties. It offers a partial path to the inverse manufacturing quandary.

Finally, in [7], case analyses are developed to extend the designer material concepts to the entire vehicle. The possibility of carrying out the solution of possibly some 800,000,000 simultaneous algebraic equations for an estimated set of necessary parameters will materialize when the University of Illinois at Urbana-Champaign NSF/NCSA Blue Waters^TM sustained petascale supercomputer comes online in late 2012 [40, 41].