Patent Publication Number: US-8534570-B2

Title: Adaptive structures, systems incorporating same and related methods

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. Patent Application Ser. No. 11/556,988, filed Nov. 6, 2006, now U.S. Patent 7,913,928, issued Mar. 29, 2011, which claims the priority filing date of United States Provisional Patent Application Ser. No. 60/733,980 filed Nov. 4, 2005, the disclosure of each of which is hereby incorporated by reference herein in its entirety. 
    
    
     FIELD OF THE INVENTION 
     The present invention is related generally to adaptive structures, systems incorporating such adaptive structures and related methods. More particularly, the present invention is related to mechanisms and structures constructed and operative at the microscale that may be used individually or as part of a system for adapting and responding to environmental changes including time varying and location varying environmental parameters. 
     BACKGROUND OF THE INVENTION 
     There are numerous circumstances in which a structure experiences time varied and location varied state changes. Such state changes may be heavily influenced by the environment in which the structure is placed or is operating. For example, there are numerous structures that experience a change in temperature, with the temperature varying from one location of the structure to another (i.e., temperature gradients and localized heating or cooling of the structure), and wherein the structure experiences changes in temperature at different times. In other words, such a temperature or other condition is often transient and asymmetric in multiple dimensions. 
     Often, it is desirable to manage the temperature (or other parameter or condition) of such structures. However, efforts to control such parameters in a structure have conventionally included “blanket” approaches that are generally conservatively designed, often for worst-case scenarios. 
     For example, one structure that is desirably maintained within a certain temperature range includes the leading edge of a wing foil on supersonic or hypersonic aerospace vehicle. Similarly, the wall (or some other component) of an engine that is associated with high-speed aerospace vehicles experiences substantial temperature variations and may require some form of thermal management for effective operation of the vehicle. Depending, for example, on the current speed of the vehicle, the acceleration pattern of the vehicle and numerous other parameters, such surfaces and structures may experience very substantial temperature increases, with such increases sometimes occurring at a rather rapid pace and in a non-uniform manner. 
     Conventional cooling approaches for structures associated with, for example, high-speed aerospace vehicles, can be broadly classified as film cooling, where the wall is covered with a thin film of fresh coolant (often fuel), transpiration cooling, where the coolant is supplied uniformly through a porous wall, and wall cooling, where coolant flow convectively cools the back side of the wall. Conventionally, these approaches are implemented passively wherein one or more arrays of fixed orifices are supplied with a pressurized coolant without substantial control over the volume of coolant or the location to which such coolant is supplied. Such approaches can often present a number of challenges and problems. 
     During the trajectory of a typical hypersonic air-breathing vehicle (e.g., Mach 2-15), the heat transfer thermal loads can vary dramatically, from take-off to the mission altitude, as illustrated in  FIG. 1 .  FIG. 1 , which illustrates the large temporal variations of heat load on the external skin of a hypersonic vehicle for a typical mission, shows that the maximum convective heat transfer coefficient, “h” and the adiabatic wall temperature, T aw  occur at different times in the trajectory of such a vehicle (note that the speed of such a vehicle after 95 seconds is approximately Mach 3). Such large variations in thermal loads impose tremendous demands on a vehicle thermal management system, typically requiring the cooling system to be designed for a fixed worst-case trajectory point. Though this design approach may provide adequate cooling at the highest thermal loads point, it comes at the expense of potentially over-cooling at other “off-design” points. This approach is inefficient, requires an undesirably high volume of coolant and places stringent demands on the thermal management system for the entire mission. 
     Similar inefficiencies may result from the spatial distribution of thermal loads over the surface of the structure being cooled. Uncertainty in predicting the temperature at any particular location within a given structure (e.g., the inlet, combustor, or nozzle walls of an engine) will lead to conservative coolant flow rates for the entire structure. 
     Numerous other structures likewise function more effectively if maintained within a desired temperature range (or within other specified parameter limits) but suffer from similar inefficiencies in maintaining the desired parameters. For example, with respect to temperature or thermal management, other examples of structures, where it is desirable to maintain the structure within a specified temperature range includes gas turbine blades, nuclear reactors, combustors, heat exchangers, rocket engines and various components of aerospace vehicles including hypersonic vehicles. In actuality, the number of components and structures that require some kind of parameter (e.g., thermal) management, and wherein the parameter is transient and asymmetrical is virtually limitless. 
     It is an ongoing desire to improve management of other parameters that may be time or spatially varied (or both). It would be advantageous to improve the efficiency and effectiveness of such parameter management including the use of structures, systems and methods that are adaptive in nature. For example, it is an ongoing desire to improve thermal management of various structures that exhibit time varied or spatially varied temperature changes. 
     SUMMARY OF THE INVENTION 
     In accordance with various embodiments of the present invention, adaptive structures, systems incorporating such adaptive structures and related methods are provided. In accordance with one embodiment of the present invention, a thermal management system is provided. The system includes a skin panel having a first surface, a second surface and at least one micropore extending from the first surface and the second surface. A flow path extends between a source of coolant and the at least one micropore. At least one adaptive structure is associated with the at least one micropore and is configured to alter a flow rate of coolant through the at least one micropore responsive to temperature sensed by the at least one adaptive structure. 
     In accordance with another embodiment of the present invention, an adaptive structure is provided. The adaptive structure includes a first structure and at least one microstructure associated with the first structure. The at least one microstructure includes a microscale beam configured to be displaced relative to the first structure upon the adaptive structure being exposed to a specified temperature. 
     In accordance with yet another embodiment of the present invention, a method of cooling a structure is provided. The method includes providing a source of coolant and flowing coolant from the source to a skin panel associated with the structure and through at least one micropore formed in the skin panel. An adaptive structure is associated with the at least one micropore and a temperature is sensed by the adaptive structure. A flow of coolant through the at least one micropore is altered by the adaptive structure responsive to the sensed temperature. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
       The foregoing and other advantages of the invention will become apparent upon reading the following detailed description and upon reference to the drawings in which: 
         FIG. 1  is a graph related to heating issues and thermal management of an aerospace vehicle. 
         FIG. 2A  shows an aerospace vehicle in accordance with an embodiment of the present invention; 
         FIGS. 2B-2D  show progressively enlarged, partial cross-sections of a component or structure of the aerospace vehicle of  FIG. 2A ; 
         FIG. 2E  shows a cross-sectional view of a microstructure utilized in conjunction with the structure set forth in  FIGS. 2B-2D ; 
         FIG. 2F  shows a perspective view of another microstructure in accordance with another embodiment of the present invention; 
         FIG. 2G  shows a plan view of yet another microstructure in accordance with a further embodiment of the present invention; 
         FIG. 2H  shows an enlarged cross-sectional view of another portion of the structure shown in  FIG. 2B  including yet another microstructure in accordance with another embodiment of the present invention; 
         FIGS. 3A and 3B  show a perspective view and a side view, respectively, of a clamped-clamped beam with an eccentricity; 
         FIGS. 4A and 4B  show side views of a pinned-pinned simplification of a clamped-clamped buckling problem, showing pinned deflection v(x) in  FIG. 4A  and clamped deflection d(x) in  FIG. 4B ; 
         FIGS. 5A and 5B  illustrate equivalent loadings of pinned-pinned beams; 
         FIG. 6  shows nondimensional design curves for deflection, δ=d/h, as a function of temperature rise, θ=12 αΔT(L/πh) 2 , for various eccentricities, ε=e/h; 
         FIG. 7  shows nondimensional design curves for stress, Σ=(σ M /E)(L/h) 2 , as a function of temperature rise, θ=12αΔT(L/πh) 2 , for various eccentricities, ε=e/h; 
         FIG. 8  shows nondimensional stress components for a beam; 
         FIG. 9  shows inflection point location (θ*, δ*) as a function of fabricated beam eccentricity, ε; 
         FIGS. 10A and 10B  are schematics of an experimental setup; 
         FIGS. 11A and 11B  show central deflection versus temperature rise results for the various beam geometries plotted against the theoretical predictions for (a) e/h=0.05 and (b) e/h=0.0125; 
         FIGS. 12A  and 12B show nondimensional central deflection, 6, versus nondimensional temperature rise, θ, results for the various beam geometries plotted against the theoretical predictions for (a) ε=0.05 and (b) ε=0.0125; and 
         FIGS. 13A and 13B  show repeatability of test results. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Referring to  FIG. 2A , an aerospace vehicle  100  is shown that has one or more components, structures or surfaces that experience transient parameter changes, asymmetrical parameter changes, or both, during operation thereof. For example, an engine wall  102 , a nozzle wall or some other component associated with a propulsion system may experience varying temperatures during operation of the vehicle  100 . Other components, such as a leading edge of an air foil  104  (or various other surfaces) may likewise experience transient and asymmetrical temperature changes. As shown in  FIG. 2B , considering the wall  102  of an engine as an example, air flow  106  passes through a passage of the engine (such as a nozzle) at a rapid flow rate. In aerospace vehicles  100  that are designed, for example, to travel at a very high rate of speed, such as air breathing hypersonic vehicles (vehicles that travel at a rate of speed between approximately Mach 2 and Mach 15), temperatures within the engine may reach extreme levels and, therefore, require a thermal management system to transfer heat away from the engine wall  102  (or other structure) as generally indicated at  108 . 
     As shown in  FIG. 2C , a thermal management system in accordance with one embodiment of the present invention may include a source of coolant  110  that passes along a flow path  112  to be distributed along a portion of the engine wall  102 . The coolant  110  may be supplied as a pressurized coolant and, in one embodiment, may include fuel from the vehicle  100 . As shown in  FIGS. 2C and 2D , a structure, referred to herein as a skin or a skin panel  114 , is associated with the engine wall  102  and may include one or more micron sized pores  116  through which the coolant  110  may flow, as needed or desired, for cooling the engine wall  102 . In one embodiment, the skin panel  114  may be formed the exhibit a thickness of approximately  1  millimeter (mm) while the pores  116  may exhibit a diameter or cross-sectional width of approximately 100 to 200 micrometers (μm). 
     In accordance with an embodiment of the present invention, an adaptive structure may be associated with a pore  116  and utilized to control the flow of the coolant  110  therethrough. For example, as shown in  FIG. 2E , the adaptive structure  118  may be used to limit the flow of coolant  110  through an associated pore  116  in an adaptive manner responsive to, for example a temperature sensed by the adaptive structure  118 . Thus, adaptive structure  118  may be said to “sense” the surrounding temperature, even in the absence of a dedicated sensor element. In one embodiment, the adaptive structure  118  may include an actuator or microstructure (a structure formed at the microscale). The actuator may include a flap or a beam  120  formed of a desired material and exhibiting a desired shape such that, in reaction to an increase in temperature, and as indicated by a dashed line, the beam  120  deflects away from the opening  122  of its associated pore  116  (e.g., due to thermal expansion of the beam  120  or a portion thereof), enabling the flow rate of coolant  110  through the pore  116  to increase. Upon a decrease in temperature, the beam  120  may deflect back toward the opening  122  of the pore  116 , again changing the flow rate of coolant through the pore  116 . 
     The beam  120  may be configured to deflect, or be displaced, upon sensing of one or more threshold temperatures to provide a flow rate of coolant  110  that is adequate to transfer a desired amount of heat away from the structure (e.g., the engine wall  102 ) and maintain the structure within a specified temperature range. Moreover, not only may the beam  120  be configured to be actuated at a desired temperature (or within a desired temperature range), but the beam  120  may also be configured to be displaced in accordance with a desired relationship with the temperature variation of the structure (e.g., the engine wall  102 ). For example, the beam  120  may be configured to exhibit a linear or a nonlinear displacement relationship with the temperature variation of the associated structure depending on specific cooling needs. Such tailoring of the flap or the beam  120  may be accomplished, for example, by tailoring the geometric configuration of the flap or the beam  120 , by tailoring the material from which the flap or the beam  120  is formed, through the manner in which the flap or the beam  120  is coupled to an underlying material, or a combination of such techniques. 
     A plurality of such adaptive structures  118  formed in association with the skin panel  114  provides a self-regulating thermal management system and results in more efficient thermal management of a structure that exhibits, for example, a highly transient thermal profile. Such a thermal management system generally behaves similar to the human skin, and more particularly to the sweat gland, when it responds to a thermal stimulus by providing a tailored amount of cooling in a localized area based on need. In the case of the adaptive structure  118  or other embodiments of the present invention, the flow of coolant is automatically adjusted in response to an external flow-field heat flux. 
     Thus, a structure needing thermal management may have one or more surfaces covered with large arrays of adaptive structures  118  or other self-regulating coolant pores that control the flow rate of coolant for film, transpiration, or wall cooling. Each pore  116 , or sub-array of pores  116 , may independently control the flow rate of coolant  110  passing therethrough such that coolant flow rates vary with the thermal load to maintain the temperature of the structure within a desired operational range. Moreover, such adaptive structures  118 , because they are formed at the micron scale, introduce very minimal flow disturbance while saving considerable weight and providing fast response times as compared to conventional thermal management systems. 
     Referring briefly back to  FIG. 1 , if a conventional thermal management system were designed for the highest heat flux expected throughout a hypersonic vehicle mission (e.g., at approximately 6 seconds in  FIG. 1 ), it is believed that the resulting total heat load integrated over the 95 second mission would be overestimated by approximately 74%. A thermal management system configured with adaptive structures in accordance with one or more embodiments of the present invention would enable an aerospace vehicle to significantly reduce the amount of fuel required for cooling due to the use of coolant as needed, rather than based on conservative design parameters. 
     Referring back to  FIGS. 2A-2E , the self-regulating skin panel  114 , including the adaptive structures  118 , may be fabricated using microfabrication processes utilized in the semiconductor and microeletromechanical systems (MEMS) industries as will be appreciated by those of ordinary skill in the art. Such processes may include, for example, film deposition, lithographic patterning, and etching to create planar arrays of devices or adaptive structures having micron (or micrometer) scale features on or through, for example, a Si or SiC substrate. The skin panels  114  may be mounted to coolant distribution webs, which may be integrally formed with the structure that is being cooled (e.g., the engine wall  102 ) such as, for example, through fusion bonding, brazing, or glass bead fusing. 
     As will be appreciated by those of ordinary skill in the art, MEMS technology allows the integration of a sensor, an actuator, and electronics on an area smaller than the size of a pupil. The use of MEMS or semiconductor fabrication processes to provide adaptive structures in accordance with various embodiments of the present invention provides various advantages. For example, batch fabrication at large volumes conventionally results in cost reduction. Additionally, as previously noted, the small structure or device size that is produced provides weight savings, localized response, minimal flow disturbance and rapid response. Moreover, the use of such structures or devices provides significant redundancy since thousands of MEMS or MEMS-like structures can be fabricated on a single chip having a surface area of only 1 mm x 1 mm. Additionally, such structures and devices consume very little power. Indeed, in certain embodiments, wherein the material properties of the beam  120  of the adaptive structure  118 , no power is consumed by operation of such structures or devices. Various other advantages are offered by use of microscaled structures and devices as will be apparent to those of ordinary skill in the art. 
     Referring briefly to  FIG. 2F , a perspective view of another adaptive structure  118 ′ is shown. The adaptive structure  118 ′ includes an actuator in the form of an elongated slender beam  120 ′ that, rather than being free at one end (or cantilevered at a single end thereof), is constrained at both ends, causing the beam to buckle in the middle upon actuation thereof. Thus, when in a nonactuated state, such as when the temperature of an associated structure is below a threshold temperature, the actuator or beam  120 ′ remains in an unactuated state (as shown in dashed lines) to prevent, or at least limit, the flow of coolant  110  through an associated pore (not shown in  FIG. 2F ). When actuated, the actuator or beam  120 ′ buckles such that the central portion thereof deflects away from the skin panel  114  to increase the flow rate of coolant through the associated pore. It is additionally noted that, in the embodiment shown in  FIG. 2F , the actuator or beam  120 ′ is not located on the surface of the skin panel  114  that is exposed to the heat source, but is located on an opposing side thereof. 
     Referring now to  FIG. 2G , a plan view is shown of another adaptive structure  118 ″ that is formed at the microscale. The adaptive structure  118 ″ includes a beam having a main body portion  130  sized and configured to cover, or at least substantially cover, the opening  122  of an associated pore  116 . One or more lateral displacement members  132  are coupled with the main body portion  130  and attached to, or otherwise integrated with, the skin panel  114 . The displacement members  132  may be appendages from the main body portion  130  and substantially integrally formed therewith, or they may be coupled to the main body portion  130  by some other means. 
     When the structure (e.g., engine wall  102  ( FIG. 2A )) to which the adaptive structure  118 ″ is attached exhibits a temperature below a threshold level, the displacement members  132  remain in a first, nonactuated position (shown in dashed lines) such that the main body portion  130  covers, or at least partially covers, the opening  122  of the pore  116  to prevent or limit flow of coolant  110  therefrom. Upon reaching or exceeding a threshold temperature, the displacement members  132  deflect laterally relative to the pore  116  to expose a larger area of the opening  122  and increase the flow rate of the coolant  110 . It is noted that the adaptive structure  118 ″ deflects laterally (i.e., in a direction that is generally parallel with respect to a surface of skin panel  114  with which the adaptive structure  118 ″ is associated). This is in contrast to the embodiment previously described wherein the beam  120  is displaced away from or toward the surface of the skin panel  114 . 
     Referring now to  FIG. 2H , an enlarged portion of air foil  104  is shown which incorporates an adaptive structure  140  in accordance with yet another embodiment of the present invention. While the previously described embodiments included adaptive structures that could, if desired, be exposed to a flow path (such as an air flow path within a portion of an engine), it may be desirable at times to provide cooling in a manner such that the coolant does not enter into such a flow path and such that the adaptive structure is likewise not exposed to such a flow path. One example of such a cooling technique includes impingement cooling or wall cooling wherein coolant is sprayed against (or otherwise contacts) a backside or a surface of the structure that is relatively remote from the source of heat to which the structure is exposed. 
     For example, the leading edge of structure such as an air foil  104  tends to experience significant temperature increases in high-speed aerospace vehicles. In some instances, it may be desirable to cool the air foil  104 , or portions thereof, without exposing the adaptive structure  140  or other components to the air stream  142 , which is passing over the air foil  104 . Thus, a thermocouple  144  or other device may be embedded in or otherwise be associated with a portion of the air foil  104  to sense a change in temperature experienced by the air foil  104  at a particular location. Due to the thermoelectric effect employed by the thermocouple  144  (generally the effect of converting a temperature differential to an electrical voltage and vice versa, as will be appreciated by those of ordinary skill in the art), an electrical voltage may be established via an electrical conductor  146  to actuate a beam  148  or other displaceable component of the adaptive structure  140 . 
     The beam  148  may be disposed within a coolant flow path  150  such that, prior to actuation (such as shown by dashed lines) the beam  148  or a portion thereof prevents, or at least limits, the flow of coolant  110  that is to impinge upon the back surface  152  of the air foil  104  or other structure. Upon application of the voltage from the thermocouple  144 , the beam  148  is displaced (such as indicated by solid lines) such that the flow rate of coolant  110  is increased. 
     In one embodiment, the displaceable component or beam  148  may be formed from a shape memory alloy (SMA). In one particular embodiment, the displaceable component or beam  148  may be formed of an SMA material including nickel and titanium. One more specific example includes a material commercially known as Nitinol, which includes approximately 55% nickel by weight and approximately 45% titanium by weight. As discussed in further detail below, the SMA is a material that is deformed and then, when heated, returns to its original form or shape. 
     It is noted that, because various components of the adaptive structure  140  are not exposed to, for example, the flow path of the air stream  142  or some other harsh environment, that a wider range of materials may be considered for use in providing and operating the adaptive structure  140 . 
     Functionally, the adaptive structure  140  can be viewed as a combination of a sensor, an actuator, and a control subsystem. Using MEMS technology, these functions may be integrated on a chip to provide autonomous, local thermal management. As already mentioned, numerous approaches are possible to implement such a device or adaptive structure for sensing, actuation, and control. 
     For film and transpiration cooling, the adaptive structure may often be exposed to harsh environments and, therefore, traditional electronic circuitry may not be desirable or feasible if the adaptive structure were to be directly collocated with the sensor and actuator device. Instead, a thermomechanical control approach may be used (i.e., without electronics) such as has been described with particular reference to  FIGS. 2E and 2F . For example, a local increase in wall temperature will lead to a mechanical deformation (via thermal expansion), which in turn affects the coolant flow rate. Not using analog or digital electronics may limit the control algorithms to the simplest strategies, such as proportional control. However, the thermomechanical implementation can be extremely simple, reliable, completely autonomous, require no external wiring, and be suitable for use in high temperature environments. 
     Multiple sensing and actuation approaches are contemplated in conjunction with embodiments of the present invention. For example, as already described, differential thermal expansion may be used as a mechanical actuation and control mechanism. In such a case, thermal expansion, due to temperature gradients, results in non-uniform expansion of the adaptive structure (or actuating component thereof). Such a mechanism may be implemented using SiC and used in high temperature environments. Additionally, such an approach enables the adaptive structure to exhibit a substantially linear response. It is noted, however, the differential expansion conventionally results in rather small deflections and such a mechanism is rather sensitive to temperature distributions. 
     Bimorph thermal expansion is similar to differential thermal expansion, but utilizes multiple materials having different thermal expansion coefficients so as to induce non-uniform deformation. Deflections of a larger magnitude, as compared to differential expansion, are possible using bimorph expansion and such a mechanism is less sensitive to temperature distribution. However, response of bimorph structures is moderately nonlinear and construction of a structure based on bimorph expansion is more involved because it inherently requires the use of two high temperature materials (e.g., two different metallic materials). 
     Shape memory alloys, briefly discussed hereinabove, are unique materials that undergo a material property phase transformation in their crystal structure that is temperature dependent. This phase transformation is responsible for the shape memory and superelastic properties of these alloys. These metallic materials possess the ability to return to some previously defined shape or size when subjected to certain temperature characteristics. SMAs can be plastically deformed at some relatively low temperature and can then be returned to their original shape once exposed to some higher temperature. When the SMA is heated above its transformation temperature, it can recover a preset shape and size; upon cooling, the SMA returns to an alternate shape. 
     Considering the above example of NiTi, the transformation temperature for NiTi (or Nitinol) ranges from 30° F. to 250° F. and can occur either by direct heat or by applying an electric current that generates heat (ohmic heating). Thin film NiTi can also be superelastic and, in a certain state, behaves like rubber in that it is capable of attaining extreme angles and may be deformed into small shapes. In its opposite state, the NiTi material resorts to a different shape and is very rigid. NiTi is very flexible and possesses the largest energy density of any active material, generating a large force during the shape changing process. 
     Relatively large deflections are possible using SMAs (as compared to the differential expansion and the bimorph expansion mechanisms) and temperature distribution is less critical. However, operating temperatures may be limited when using SMAs and, furthermore, structures formed of SMAs tend to exhibit a highly nonlinear response. 
     With regard to sensing approaches, examples include direct thermal sensing, conductive thermal sensing and thermoelectric approaches. Direct thermal sensing includes exposing the adaptive structure, or at least the actuator component thereof, directly to the heat source (e.g., hot gases). Such an approach provides relative certainty in the results of the sensing. However, direct sensing, depending on the environment in which the sensor is exposed, may require the use of high temperature materials in the adaptive structure. The terms “sense,” “sensing” and variants thereof are used herein in a non-limiting manner, to indicate the passive or active recognition by an element or feature or a combination of elements or features of an adaptive structure, of one or more stimuli in the form of variations in selected parameters such as, for example, temperature, such recognition being usable to initiate or vary a response by the adaptive structure. 
     Conductive thermal sensing includes sensing heat that is conducted from a surface of the structure being cooled to the actuator or sensor of the adaptive structure. While there is relatively less certainty in sensing the actual temperature, such an approach enables the adaptive structure to operate at a relatively lower temperature. 
     Thermoelectric sensing includes sensing the temperature of a structure, for example, at or near a surface that is exposed to the heat source using a thermocouple or similar device to establish an electric potential. Such an approach enables the sensing to take place relatively remotely from the actuating component (e.g., the displaceable member) of the adaptive structure, but it may also limit operating temperatures to some degree. 
     While the various embodiments described herein have generally been discussed in the context of providing a thermal management system for a high-speed aerospace vehicle, it is noted that the present invention, including the various embodiments of adaptive structures, may be utilized in a number of different contexts and applications. 
     Embodiments of the present invention may be used in numerous other thermal management applications including, for example, gas turbine cooling, nuclear reactors, combustors, heat exchangers, rocket engines, or even cooling of electronic components such as microchips. 
     Embodiments of the present invention may also be used for in applications other than thermal management systems. For example, various types of flow controls may be managed by embodiments of the present invention. One example includes control of impulse thrust of, for example, a missile or rocket, enabling directional thrust to be effected through a desired surface of the missile or rocket and providing the rocket or missile with a high degree of maneuverability. 
     Other embodiments of the present invention may include positioning adaptive structures having, for example, flaps as an actuating device on an air foil. At certain times during flight, it may become desirable to perturb the boundary layer about the air foil. Such adaptive structures may be used to effect such a perturbation. 
     In yet another embodiment, such adaptive microstructures may be utilized in sensing flow separation about an air foil and responding to such flow separation in a desired manner. 
     EXAMPLE 
     Considering a long slender beam as an actuation device in an adaptive structure, a long slender beam in compression will exhibit a lateral deflection as the loading approaches a critical value and the beam becomes unstable. A beam with a perfectly symmetric cross section will buckle in a discontinuous manner at the critical load. A perfectly symmetric cross section, however, is a theoretical approximation. In reality, a compressive member will have some imperfection or asymmetry that leads to a continuous nonlinear deflection. Accordingly, the buckling of compressed beams with a designed eccentricity has been investigated, focusing on the regime of small eccentricity ratios, e/h→0 (e being the eccentricity or offset and h being the thickness of the beam), for the specific geometry shown in  FIGS. 3A and 3B . 
     The Elastic Curve and the Secant Formulation 
     An elastic analysis of clamped-clamped beams under thermal loading was carried out with the assumption of small curvatures. Due to symmetry, a clamped-clamped beam of length 2L buckling under a compressive force can be analyzed as a pinned-pinned beam of length L under the same loading. 
     The pinned ends correspond to inflection points in the clamped beam, where the internal moment is null. The resulting deflections can be extended accordingly to the clamped-clamped case as shown in  FIGS. 4A and 4B . 
     The clamped eccentric beam in  FIGS. 3A and 3B  can also be simplified as a pinned beam; in this case, the inflection points of the beam coincide with the eccentricity locations. More specifically, the point of zero moment in the beams is located at half the eccentric height, e/2. The resultant loading and deflection of the beam is, therefore, symmetric about this point. Using this simplification, the elastic curve and the state of stress have been analyzed. 
     The pinned beam-column with a compressive load, P, applied at an eccentric distance, e/2, is statically equivalent to an axially loaded beam with an additional moment, M 0 =Pe/2, applied at the end points as shown in  FIGS. 5A and 5B . 
     Assuming shallow beam curvatures but considering the moment induced by lateral deflection of the beam, the elastic curve for the beam is given as: 

 
     where v is the pinned-pinned deflection, I is the beam moment of inertia and E is the modulus of elasticity. The resultant problem becomes: 
                             ⅆ   2     ⁢   v       ⅆ     x   2         +       (     P   EI     )     ⁢   v       =     -     Pe     2   ⁢           ⁢   EI           ⁢     
     ⁢         v   ⁡     (   0   )       =       v   ⁡     (   L   )       =   0       ,             (   2   )               
which has the following solution:
 
     
       
         
           
             
               
                 
                   
                     v 
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         e 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               tan 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     L 
                                     2 
                                   
                                   ⁢ 
                                   
                                     
                                       P 
                                       EI 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       P 
                                       EI 
                                     
                                   
                                   ⁢ 
                                   x 
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     P 
                                     EI 
                                   
                                 
                                 ⁢ 
                                 x 
                               
                               ) 
                             
                           
                           - 
                           1 
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     As seen in  FIGS. 4A and 4B , the central deflection of the associated clamped-clamped problem, d, is twice that of the central deflection of the pinned-pinned problem, v(x=L/2), hence: 
                   d   =       2   ⁢           ⁢     v   ⁡     (     x   =     L   /   2       )         =       e   [       sec   (       L   2     ⁢       P   EI         )     -   1     ]     .               (   4   )               
Maximum Stress
 
     A buckled beam under compressive loading is subjected to both axial and bending stress, the maximum of which is compressive and located at the midpoint on the lower surface of the beam, as drawn in  FIGS. 4A and 4B . The maximum stress can be written as the sum of these two components: 
     
       
         
           
             
               
                 
                   
                     σ 
                     M 
                   
                   = 
                   
                     
                       
                         σ 
                         A 
                       
                       + 
                       
                         σ 
                         B 
                       
                     
                     = 
                     
                       
                         P 
                         bh 
                       
                       + 
                       
                         
                           h 
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             I 
                           
                         
                         ⁢ 
                         
                           
                              
                             
                               M 
                               ⁡ 
                               
                                 ( 
                                 
                                   x 
                                   = 
                                   
                                     L 
                                     / 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                              
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Using the magnitude of the internal moment at the midpoint, as given by Eq. (1): 
                            M   ⁡     (     x   =     L   /   2       )            =       P   ⁡     (       e   2     +     v   ⁡     (     x   =     L   /   2       )         )       =       (     Pe   2     )     ⁢     sec   ⁡     (       L   2     ⁢       P   /   EI         )             ,           (   6   )               
yields the maximum stress in the buckled beam:
 
     
       
         
           
             
               
                 
                   
                     σ 
                     M 
                   
                   = 
                   
                     
                       
                         P 
                         bh 
                       
                       ⁡ 
                       
                         [ 
                         
                           1 
                           + 
                           
                             3 
                             ⁢ 
                             
                               ( 
                               
                                 e 
                                 / 
                                 h 
                               
                               ) 
                             
                             ⁢ 
                             
                               sec 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     L 
                                     2 
                                   
                                   ⁢ 
                                   
                                     
                                       P 
                                       / 
                                       EI 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Equations (4) and (7) define the beam central deflection and maximum stress as a function of axial load. An additional relation is needed to relate the axial force, P, to the average beam temperature rise, ΔT. 
     Stress-Strain-Temperature Relationship 
     Consider the stress-strain relationship of a heated beam restrained from expansion in the axial direction: 
     
       
         
           
             
               
                 
                   
                     σ 
                     A 
                   
                   = 
                   
                     
                       P 
                       bh 
                     
                     = 
                     
                       
                         E 
                         ⁡ 
                         
                           [ 
                           
                             
                               α 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               T 
                             
                             - 
                             
                               ɛ 
                               ′ 
                             
                           
                           ] 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Here α is the difference in the coefficient of thermal expansion between the beam and the substrate, ΔT is the average temperature rise of the beam, σ A  is the axial stress and ε′ is the strain due to beam elongation: 
                       ɛ   ′     =       l   -   L     L       ,           (   9   )               
where l is defined as the deformed beam length, which is given by:
 
     
       
         
           
             
               
                 
                   l 
                   = 
                   
                     
                       ∫ 
                       0 
                       L 
                     
                     ⁢ 
                     
                       
                         
                           1 
                           + 
                           
                             
                               ( 
                               
                                 
                                   ⅆ 
                                   v 
                                 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ⅆ 
                           x 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     The assumption of shallow beam curvatures, which can be written as dv/dx&lt;&lt;1, has already been asserted previously in this analysis. Accordingly, the integrand in Eq. (10) can be simplified to: 
                         1   +       (       ⅆ   v       ⅆ   x       )     2         ≅     1   +       1   2     ⁢       (       ⅆ   v       ⅆ   x       )     2           ,           (   11   )               
and the strain term in Eq. (8) can, therefore, be rewritten as:
 
     
       
         
           
             
               
                 
                   
                     ɛ 
                     ′ 
                   
                   ≅ 
                   
                     
                       1 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         L 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         L 
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 ⅆ 
                                 v 
                               
                               
                                 ⅆ 
                                 x 
                               
                             
                             ) 
                           
                           2 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ⅆ 
                             x 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Knowing v(x) from Eq. (3), both the derivative and integral in Eq. (12) can be evaluated. Dropping the approximate equality, combining Eq. (8) and Eq. (12) and rearranging terms gives: 
                     Δ   ⁢           ⁢   T     =       P     α   ⁢           ⁢   Ebh       ⁢             [     1   +       3   4     ⁢       (     e   /   h     )     2     ⁢     {                       tan   ⁡     (       L   2     ⁢       P   /   EI         )       ⁢     cos   ⁡     (     2   ⁢           ⁢   L   ⁢       P   /   EI         )           L   ⁢       P   /   EI           +                   tan   2     ⁡     (       L   2     ⁢       P   /   EI         )       ⁢             [     1   +       sin   ⁡     (     2   ⁢           ⁢   L   ⁢       P   /   EI         )         2   ⁢           ⁢   L   ⁢       P   /   EI             ]     +                         [     1   -       sin   ⁡     (     2   ⁢           ⁢   L   ⁢       P   /   EI         )         2   ⁢           ⁢   L   ⁢       P   /   EI             ]           }         ]     ,                 (   13   )               
which defines the relationship between applied axial load and average temperature rise of the beam.
 
Nondimensional Design Curves
 
     Collectively, Eq. (4), (7) and (13) fully describe the thermomechanical behavior of doubly clamped eccentric beams. For convenience, several nondimensional parameters can be defined to simplify these equations. Recalling the definition of the critical load, P cr , as the force at which a theoretically perfect beam (e=0) will buckle: 
                       P   er     =           π   2     ⁢   EI       L   2       =         π   2     ⁢     Ebh   3         12   ⁢           ⁢     L   2             ,           (   14   )               
a Critical Temperature Rise, ΔT cr , can be defined by evaluating Eq. (8) at the critical load, noting that for a perfect beam prior to buckling, there is no deflection and, therefore, no associated strain term, ε′:
 
     
       
         
           
             
               
                 
                   
                     σ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       T 
                       er 
                     
                   
                   = 
                   
                     
                       
                         P 
                         er 
                       
                       
                         α 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         Ebh 
                       
                     
                     = 
                     
                       
                         1 
                         
                           12 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           α 
                         
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 h 
                               
                               L 
                             
                             ) 
                           
                           2 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Utilizing Eq. (14) and (15) and by simple examination of Eq. (4), (7) and (13), nondimensional forms of deflection (δ), eccentricity (ε), axial load (η), maximum compressive stress (Σ) and temperature rise (θ) have been defined respectively as: 
     
       
         
           
             
               
                 
                   δ 
                   = 
                   
                     d 
                     / 
                     h 
                   
                 
               
               
                 
                   
                     ( 
                     16 
                     ) 
                   
                   ⁢ 
                   
                     - 
                   
                   ⁢ 
                   
                     ( 
                     20 
                     ) 
                   
                 
               
             
             
               
                 
                   ɛ 
                   = 
                   
                     e 
                     / 
                     h 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   η 
                   = 
                   
                     
                       
                         π 
                         2 
                       
                       ⁢ 
                       
                         
                           P 
                           / 
                           
                             P 
                             cr 
                           
                         
                       
                     
                     = 
                     
                       
                         L 
                         2 
                       
                       ⁢ 
                       
                         
                           P 
                           / 
                           EI 
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   ∑ 
                   
                     = 
                     
                       
                         
                           σ 
                           M 
                         
                         E 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             L 
                             h 
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   θ 
                   = 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         T 
                       
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           T 
                           er 
                         
                       
                     
                     = 
                     
                       12 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             T 
                             ⁡ 
                             
                               ( 
                               
                                 L 
                                 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   h 
                                 
                               
                               ) 
                             
                           
                           2 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Nondimensional forms of the main relations Eq. (4), (7) and (13) are obtained by rearranging and substituting in Eq. (16)-(20): 
     
       
         
           
             
               
                 
                   δ 
                   = 
                   
                     ɛ 
                     ⁡ 
                     
                       [ 
                       
                         
                           sec 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           η 
                         
                         - 
                         1 
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   ∑ 
                   
                     = 
                     
                       
                         η 
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             ( 
                             
                               1 
                               / 
                               3 
                             
                             ) 
                           
                           + 
                           
                             ɛ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sec 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             η 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   θ 
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               η 
                             
                             π 
                           
                           ) 
                         
                         2 
                       
                       ⁡ 
                       
                         [ 
                         
                           1 
                           + 
                           
                             
                               3 
                               4 
                             
                             ⁢ 
                             
                               ɛ 
                               2 
                             
                             ⁢ 
                             
                               { 
                               
                                 
                                   
                                     tan 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     η 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     cos 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     4 
                                     ⁢ 
                                     η 
                                   
                                   
                                     2 
                                     ⁢ 
                                     η 
                                   
                                 
                                 + 
                                 
                                   
                                     tan 
                                     2 
                                   
                                   ⁢ 
                                   
                                     η 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           
                                             sin 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             4 
                                             ⁢ 
                                             η 
                                           
                                           
                                             4 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             η 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     
                                       
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         4 
                                         ⁢ 
                                         η 
                                       
                                       
                                         4 
                                         ⁢ 
                                         η 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               } 
                             
                           
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     This set of nondimensional equations was solved numerically using MATLAB® to eliminate the nondimensional axial load, η. Curves for central beam deflection, δ, maximum compressive stress, Σ, and its corresponding stress components are shown in  FIGS. 6-8 , respectively, as a functional of temperature rise, θ. 
     At low temperature rises, θ&lt;&lt;1, the beam behavior is dominated by axial compression; the beam deflection and stress increase linearly with θ. At high temperatures, θ&gt;1, bending begins to be appreciable, leading to large deflections and, therefore, large strain. In this range, the strain term dominates, limiting the beam to finite deflections. At intermediate temperatures between these two regions, 0.5&lt;θ&lt;1, the shape of the deflection and stress curves are more sensitive to c and can exhibit very nonlinear behavior as seen in  FIG. 6  and  FIG. 7 . 
     The curves of deflection as a function of temperature rise shown in  FIG. 6  pass through an inflection point, denoted here by a circle. This is the point of maximum slope and the boundary between positive and negative concavity of the temperature-induced deflection. This makes the inflection point a design parameter of interest for implementing buckled beams into adaptive structures such as thermally actuated MEMS devices. Accordingly, the location of this point at various eccentricities has been solved numerically utilizing MATLAB®. 
     First, let δ* and θ* define, respectively, the nondimensional deflection and temperature rise of the beam at the inflection point. Using this notation, the location of the inflection point has been solved and plotted as a function of eccentricity in  FIG. 9 . 
       FIGS. 6 and 9  show that for a perfectly symmetric beam, ε=0, there is zero deflection (δ=0) up until buckling occurs at the critical temperature, θ≠1. The inflection point is therefore at (δ*, θ*)=(0, 1). For imperfect beams, ε≠0, continuous nonlinear deflections are predicted and the point of maximum slope varies as shown in  FIG. 9 . 
       FIGS. 6 ,  7  and  9  provide succinct nondimensional design curves for the implementation of thermally actuated buckled beams in MEMS systems. These curves, along with the preceding analysis, capture the complex and highly nonlinear behavior exhibited in thermally buckled beams. The beam shape, central deflection and state of stress have all been modeled as they vary with temperature rise and eccentricity. 
     A beam such as shown in  FIGS. 3A and 3B  may be fabricated using semiconductor fabrication and MEMS fabrication processes as will be appreciated by those of ordinary skill in the art. More specifically, a process for fabricating such a beam was described in the provisional application from which the present application claims priority, the disclosure of which provisional application has been incorporated by reference herein. 
     Several  300  μm wide beams were fabricated corresponding to two different eccentricity ratios, e/h, and five different slenderness ratios, L/h. These beams were then measured and compared to the design curves presented earlier. Table 1 lists all the geometries fabricated for the presently described example while table 2 lists process conditions for nickel plating the fabricated beams. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Beam Geometries 
               
            
           
           
               
               
               
               
            
               
                 Beam 
                 Length (L) 
                 Thickness (h) 
                 Eccentricity (e) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 A 
                 1000  
                 μm 
                 30 μm 
                 1.5  
                 μm 
               
               
                 B 
                 2000  
                 μm 
                 30 μm 
                 1.5  
                 μm 
               
               
                 C 
                 3000  
                 μm 
                 30 μm 
                 1.5  
                 μm 
               
               
                 D 
                 2000  
                 μm 
                 60 μm 
                 0.75  
                 μm 
               
               
                 E 
                 3000  
                 μm 
                 60 μm 
                 0.75  
                 μm 
               
               
                 F 
                 4000  
                 μm+ 
                 60 μm 
                 0.75  
                 μm 
               
               
                   
               
            
           
         
       
     
     Thickness, eccentricity and surface roughness measurements were taken using a profilometer. The surface roughness was ≦1 μm, while the variation in thickness across a single beam was measured to be ≦2 μm for the beams tested in this work. 
     The nickel electroplating process was optimized to create a deposition with both low tensile residual stress as well as high yield strength. The plating conditions used in this work, as listed in Table 2, resulted in a deposition rate of approximately 7 μm/hr. 
                     TABLE 2                  Process conditions for nickel electroplating       with a sulfur activated anode &amp; mechanical agitation                             Composition   Operating Conditions                                                     500    g/L   Ni (SO 3 NH 2 ) 2     pH value   4-4.5           30    g/L   H 3 BO 3     Temperature   35° C.           3    g/L   Laurel Sulfate   Cur. Dens.   10 mA/cm 2                          
Experimental Setup
 
     The beam central deflection, d, was measured experimentally with an optical probe for the six microfabricated nickel beams listed in Table 1. The beam temperature was controlled using a thin film heater and a thermocouple. A schematic of the test setup  200  used is shown in  FIGS. 10A and 10B  which, generally, included an optical probe  202 , a nickel beam on silicon  204 , an aluminum plate  206 , a thin film heater  208  and a thermocouple  210 . 
     Results and Discussion 
     The beam central deflection versus temperature rise is plotted against the theoretical predictions for all six beams.  FIGS. 11A and 11B  show graphs corresponding to the two eccentricity ratios considered in this work. Beams A, B and C were electroplated on the same wafer and have an eccentricity ratio of e/h=0.05. Similarly, beams D, E and F were fabricated together and have an eccentricity ratio of e/h=0.0125. 
     The theoretical results shown here are obtained by evaluating the nondimensional predictions for each beam&#39;s specific geometry and adjusting to account for the residual stress. The designed tensile residual stress in the beams will lead to an actuation temperature offset. A small rise in temperature will be required to overcome the tensile stress imparted during microfabrication. The resultant temperature offsets were used to determine the residual tensile stress in the nickel deposition for the two sets of beams fabricated. Beams A, B and C were offset by a temperature difference of approximately 27° C., while beams D, E and F were offset by a temperature difference of approximately 31° C. The corresponding residual tensile stresses are 5 MPa (megapascals) and 62 MPa, respectively, for the two sets. These stresses are in good agreement with the predicted values. A high purity bath of the composition and operating conditions used in this work has been reported to produce depositions with residual tensile stresses of 55 MPa or less. 
     Accounting for the temperature offset caused by the residual tensile stress, the ΔT used in the calculation of θ in Eq. (20) can been defined as the temperature rise above the zero stress state, rather than ambient conditions. Accordingly the data for the three beams at a common eccentricity ratio, ε=e/h, will collapse to a single nondimensional curve and can be compared against the predictions. 
       FIGS. 12A and 12B  show relatively good agreement between the theoretical predictions and the measured deflections. When presented nondimensionally, the 30 μm beams (A, B and C) show a larger amount of scatter in the data than that of the 60 μm beams (D, E and F). This is explained by the sensitivity of the optical probe measurements; the thinner beams have smaller deflections leading to higher percent errors. 
     It can also be seen that the predictions are less accurate in the temperature range of 0.5&lt;θ&lt;1. As noted previously, the shape of the deflection curve in this region is very sensitive to the asymmetry of the beam, modeled in this work by the eccentricity ratio ε=e/h. This would suggest that modeling of the beams with a designed imperfection, in the form of an eccentricity, has not completely captured the true imperfections of the beams tested. Two actual imperfections in the beams, the surface roughness and the thickness variation, were both measured to be on the order of the designed eccentricities. This seems to be a logical source of differences between the predictions and test results in this particular temperature range. 
     Hysteresis and Yield 
     The beams tested in the current work were scanned with a profilometer before and after thermal actuation to examine the onset and effect of plastic deformation. The central deflection of the beams after returning back to ambient conditions is listed in Table 3 along with the calculated maximum stress experienced by each beam. 
     A nickel electroplating bath of the composition and operating conditions used in this work has been reported to produce depositions with yield strengths of 400 MPa to 600 MPa. Table  3  shows the onset of an appreciable hysteresis effect occurring around 450 MPa for the beams tested in this work. 
     Table 3—Permanent deflection due to plastic deformation and the corresponding Calculated maximum stress for each beam geometry 
                     TABLE 3                  Permanent deflection due to plastic deformation and the corresponding       calculated maximum stress for each beam geometry                             Permanent Deflection   Calculated Maximum Stress       Beam   (μm)   (MPa)                                 A   1.25   400       B   1.00   150       C   0.50   75       D   10.25   450       E   4.00   250       F   2.00   150                    
Repeatability
 
     Two of the low stress beams exhibiting negligible hysteresis effects were additionally tested to examine the repeatability of the deflection measurements.  FIGS. 13A and 13B  show good repeatability of the test results for beams actuated with negligible plastic deformation. 
     Nomenclature 
     E modulus of elasticity 
     l moment of inertia 
     L pinned-pinned length 
     M moment 
     P axial force 
     ΔT temperature rise 
     b beam width 
     d clamped-clamped central deflection 
     e eccentricity or offset 
     h beam thickness 
     deformed beam length 
     v pinned-pinned lateral deflection 
     Greek 
     α difference in CTE of beam and substrate 
     δ nondimensional deflection, d/h 
     δ* nondimensional deflection at the inflection point 
     ε nondimensional eccentricity, e/h 
     ε′ strain 
     η nondimensional axial force, ½√{square root over (P/EI)} 
     θ nondimensional temperature rise, 12αΔT( L /πh) 2    
     θ* nondimensional temperature rise at the inflection point 
     σ compressive stress 
     Σnondimensional maximum compressive stress, 
     
       
         
           
             
               
                 σ 
                 M 
               
               E 
             
             ⁢ 
             
               
                 ( 
                 
                   L 
                   / 
                   h 
                 
                 ) 
               
               2 
             
           
         
       
     
     Subscripts 
     A axial 
     B bending 
     M maximum 
     cr critical 
     While the invention may be susceptible to various modifications and alternative forms, specific embodiments have been shown by way of example in the drawings and have been described in detail herein. However, it should be understood that the invention is not intended to be limited to the particular forms disclosed. For example, while the adaptive structure was described in various embodiments as including a beam, the beam may be configured to exhibit various shapes. Additionally, other configurations besides beams are contemplated. For example, a bellows microstructure may be utilized. Thus, the invention includes all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the following appended claims.