CONTROLLED BISTABILITY

The present invention relates to the field of morphing structures and, more specifically, to a method for creating a shell having multiple stable equilibrium shapes or configurations at a predetermined range of applied load values. Such shell is specially suitable to produce aerodynamic surfaces: indeed, a structure with these characteristics is advantageously able to autonomously modify its shape according to the external conditions, in order to maximize aerodynamic performances or the aerodynamic resistance to advancement in all operating conditions.

A comprehensive overview of the technological possibilities offered by the currently known morphing structures can be found in recently published papers ^[1,2 ’ ^3] , even if with relevant emphasis to aircraft moving appendices.

Several of the advanced proposals, some of which are being tested, envisage the use of multi -stable components .

Indeed, these latter allow for a significant reduction in the actuation costsy since they do not require additional energy contribution besides the one needed to switch between the stable equilibrium configurations. To this end, various actuation strategies have been studied and tested, e.g. via SMA ( Shape Memory Alloys ) and MFC (Macro- Fiber Composites ) ^[4,5] ; the serious difficulties encountered in both cases, in terms of integration (SMA) and of actuation power (MFC), have suggested the use of more complex techniques or expedi ents ^[6,7,8] .

External actuation is necessary in all cases, while reversibility remains difficult to achieve.

With regard to the mathematical modelling of the physical phenomena involved, the relevant scientific literature is almost entirely limited to the case of free boundary conditions on all the shell edges ^[9101112] , even if mul ti -stabi 1 i ty is particularly sensitive to boundary conditions and they are common in the technological applications.

Only a recent paper published by Inventors ^[13] studied explicitly the case of clamped shells.

It is the primary purpose of the present invention to suggest a method for making a structure (such as a shell, a profile or a sheet) having “tunable bistability”. The term “tunable bistability” means a structure which has one or more stable equilibrium configurations according to the load to which it is subjected.

A second purpose of the invention is to provide a simple additional mechanism to tune the multi-stable behavior of the structure by modifying the curvature of the clamp in operating conditions.

These and other purposes will be better understood from the following detailed description. This description and the associated figures are intended to give a simple, non-exhausti ve , example of a shell satysfying the claimed stability properties.

In particular:

figure la illustrates the configuration of the structure without external loads;

figure lb illustrates the possible configurations of the structure in the bistability region;

figure lc illustrates the equilibrium paths as the load F varies;

figure 2 illustrates the geometric parameters which describe the initial stress- free shape of the shel 1 ;

figure 3 illustrates the stability diagrams with respect to the geometric design parameters;

figure 4 illustrates a shell prototype built and tested at Dipartimento di Ingegneri a Struttura 7e e Ceotecnica (University of Rome “La Sapienza”);

figure 5 illustrates the stability limits as functions of the clamp curvature-^

figure 6 illustrates the natural configuration of the free shell corresponding to point N in Fig. 3.

An example of the suggested bistable structure is described with reference to the figures listed above.

In its natural stress-free configuration the shell has the shape shown in Figure 2. Once clamped along the edge x _t = 0 highlighted in blue, in absence of load the shell assumes the shape shown in Figure la. If the initial shape is appropriately chosen (as specified below) the shell is prestressed (meaning that each part thereof has a given level of stored elastic energy) but monostabl e.

Then, the behavior under load of the shell represented is shown in Figure lc in the plane (F, q) with F a load parameter and q a configuration parameter, as outlined below: 1. subject to a load, e.g. an aerodynamic pressure, the shell deforms following the equilibrium branch /i;

2. When F = F the branch becomes unstable. By increasing the load even further, the shell leaves branch/^ for branch G ₂ ; the transition takes place with an abrupt change of shape. The branch G ₂ is stable for F>F ₂ . Therefore, for F ₂ £F£F ₁ , the shell has two stable equilibrium configurations, Figure lb;

3. by increasing the load, the shell deforms following the branch G ₂ ;

4. by decreasing the load, the shell returns to the branch when F < F ₂ and takes the configuration O as the load is completely removed.

The non-linear equilibrium path of Figure lc is completely defined by the shapes that the shell assumes during the steps of loading and unloading (i.e. at low- and high-load regimes) and by the critical transition loads F and F ₂ which are the design requi remen ts.

The described behavior depends on the prestress field acting in the shell in its clamped configuration O.

A peculiar feature of the present invention is precisely the choice of the prestress field that ensures the desired behaviour when varying the external applied forces.

It is suggested to induce such prestress state by clamping a shell having an appropriate (stress-free) initial shape. Indeed, if the clamping action forces the shell to vary its Gaussian curvature, even locally, the level of prestress induced thereof is proportional to the thickness of the shell ; since the bending stiffness is proportional to the cube of the thickness, for thin shells this prestress results in sizeable variations in curvature and shape.

Specifically, in order to induce the appropriate prestress field in the shell , the following method is appl i ed :

1. A predetermined curvature is imposed on a portion of the shell edge. This part of the shell edge is then clamped.

The prestress obtained by applying the clamp depends on a finite number of design parameters Pi = {h _i=i . m _i=i ...;c} , with h _t , mi being representative of the shape of the initial stress-free configuration of the shell and of the material , respectively, and c the curvature of the clamp.

For example, shells whose natural stress-free configuration is of the type shown in Figure 2 have average surface given by the function

in this case the design parameters representative of the shape of the free shell are: h _Q , the curvature of the edge x _t = 0 which will then be clamped to induce the prestress; h _f , the curvature of the opposite edge, which will remain free; L ₂ /L _x , the planform aspect ratio.

For composite material shells, the representative design parameters of the material are the character st cs of the elementary ply and the lamination sequence. Again as an example, for an eight-layer anti -symmetri cal laminate of the type \a/-a/-a/a/-a/a/a/-a\, the design parameters are: Ei _, E _{2 ΐ} 2 _, ⁿ _ΐ 2. the properties of the elementary ply; t _t , the thickness of the elementary ply; a, the lamination angle.

Thus, in this case, the prestress in the shell will depend on the choice of ten design parameters Of these, the last, i.e. the curvature of the clamp cto be assigned to the edge x _t = 0 can be modified in operation.

A discrete shell model is defined by projecting the non-linear shell model of Marguerre-von Karman onto an appropriate finite dimensional space. The projection infers a discrete model with a few number of configuration parameters q _t , (i < 5) and does not not significantly alter the elastic energy of the shell; models of this type are described, for example, in scientific papers [12, 13]. For example, a suitable polynomial approximation of the elastic energy of the shell including membrane and bending contributions is found by choosing a polynomial form to describe the shell shape in the generic configuration and following the procedure described in detail in [13]. In this case, the configuration parameters are the coefficients q = . q _n }of the polynomials describing the curvature of the shell. Together with the design parameters, they define the elastic energy of the discrete shell model. 3. With the discrete representation of the elastic energy E(q;p)obtai ned it is possible to compute the stability maps in the space of the design parameters; for every choice of the design parameters; this allows to determine the number and the shape of the stable equilibrium configurations that the clamped shell has at each value of the applied load. For example, having fixed the material parameters (the properties of the elementary ply and the lamination sequence), the aspect ratio and the curvature of the clamp, it is possible to minimize the polynomial shape that approximates the elastic energy with respect to the configuration parameters and to determine, for each possible choice of the shell initial curvatures h ₀ h _f of the edges x _t = 0,L _x (Fi gure 2), and for each value of the applied load, the number, the shape, the elastic energy, etc. , of the stable equilibrium configurations that the shell exhibits after clamping, i .e. the design objectives. These informations can be summarized in plots of the type shown in Figure 3, which relate objectives and project parameters (in this case, curvatures for different load values.

4. Finally, the design parameters are chosen so as to provide the prestress needed to obtain the desired response in terms of project requirements. For example, the response described in Figure 1 is the one associated with point N of Figure 3, i .e. the choice of the project parameters h _Q = H,h _f = 0 which correspond to the initial stress-free configuration of Figure 6. The response in Figure 1 is obtained by considering the particular choice of the clamping curvature c = 0 (that is, a flat clamp). By varying the latter, it is possible to tune either the value and ratio between the critical loads F ₁ F ₂ , as well as the shape of the stable equilibrium configurations, even during operation. For example, for point N the critical loads vary with the curvature of the clamp as shown in Figure 5.

With the method described above it is possible to:

A) design the shapes that the shell must take during the loading process according to the performance requi rements ;

B) set the load values at which the transition occurs, by choosing a point that has only one stable configuration for F = 0, two for F ₂ £ F £ F and again one for F > F

C) tune the critical transition loads also during structural operation;

D) identify the optimized shape of the shell when free from constraints.

In order to achieve the latter goal, according to a further peculiar characteristic of the invention, a clamping with tunable curvature is provided: by changing the curvature of the clamp the prestress induced in the shell is modified, and its structural response therewith, so as to meet the performance requirements of the moment. By way of non-limiting example, the curves in Figure 5 represent the law according to which the critical transition loads F and F ₂ vary as the curvature assigned to the clamp varies (this latter normalised with respect to the initial curvature of the edge to be clamped). It is worth noting that also with modest variations in the curvature of the clamping it is possible to radically change the structural response, moving the critical load values closer or further apart.

In other words: once the performance requirements that the shell must garantee have been decided (for example, the optimal geometrical configurations to be assumed in the various speed regimes and the critical velocity values), the method in the present invention makes it possible to determine the prestress field, that is, the initial geometry, the constitutive properties (e. g. the type of elementary ply and the lamination sequence) and the curvature of the clamp, to be chosen so that the performance requirements can be fulfilled; if necessary these latter can also be modified, even during the operation of the structure, simply by appropriately modifying the curvature, i. e. the shape, of the clamp.

The prototype in Figure 4 was made for experimental purposes: in Figure 4a, the free shell, before the application of the clamp (the shape is of the type shown in Figure 2, with h _Q - 15 nr ¹ , h _f = 0 nr ¹ ); in Figure 4b, the configuration assumed by the shell after the imposition of the clamp, in this case flat (c = 0) ; in Figure 4c and 4d, the two stable equilibrium configurations that the shell can assume for F ₂ £ F £ .

The shell of this invention can adapt its shape autonomously to the load applied in order to meet predeterm ned performance requ rements and maximize structural efficiency (e.g. it can be used as a component to improve the aerodynamic performance of a vehicle). The shell has a shape at low-load regimes (e.g. the most suitable for low speeds) and a different shape, even significantly different, at high-load regimes (e.g. the most suitable for high speeds), wherein the change of shape occurs spontaneously as the load varies.

By virtue of the above, the present invention provides at least the following advantages:

1. no actuating energy other than that supplied by the load directly acting on the shell is required; therefore, it is not necessary to provide actuators;

2. unlike traditional bistable structures, the shell returns to its original shape when the load is removed with no need for external intervention: in all cases, the equilibrium path is closed, i.e. the reversibility of the transformation is ensured;

3. the design concerns both the shapes required during loading and unloading and the critical transition load values, i.e. all the parameters that define the equilibrium path;

4. the possibility of varying the curvature of the clamp makes it possible to modify stable shapes and transition loads, even while the structure is in operati on .

Because of its ability to be efficient under radically different operating conditions, the present invention has many possibilities for use in various areas of industrial engineering, e.g. in the manufacture of morphing aerodynamic appendages, and in civil engineering, e.g. in the manufacture of building envelopes and adaptive ventilation systems. In the field of industrial engineering, in particular, the invention is a reliable and cost-effective solution. Indeed, it is not necessary to supply power to the system nor is it necessary to use gear systems or other means, because:

A. the transition between the different structural forms is induced by external load itself;

B. the maintenance of each of them is ensured by its stabi 1 i ty.

In conclusion, it is useful to point out that, according to the present invention, the shell is not initially prestressed: only with the application of the constraint the appropriate prestress field is induced within it.

Inter alia , this prestress is a function of the natural stress-free shape of the shell, i.e. the shape of the shell when it is free of external constraints, i.e. before it is clamped at one end.

Furthermore, according to the invention, the shape of the shell in the stress-free configuration is a fundamental parameter to be optimized (along with the constitutive properties of the material), which is not given a priori but has to be chosen on the basis of the design requirements: the critical transition loads and the optimal shell shapes for low- and high- load regimes.

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