METHOD FOR PRODUCING A PART FROM A WOVEN MATERIAL TAKING THE OFF-CENTERING INTO ACCOUNT

Abstract

Methods are provided for creating a component from a preform having a network of fibers having, after the shaping of the preform, an out-of-register angle. The methods include: defining an orthogonal local frame of reference, defining a natural local frame of reference, defining a linked frame of reference, expressing a tensor of the stiffnesses of the woven composite material in said natural local frame of reference, constructing a tensor of the deformations in the orthogonal local frame of reference, expressing, in the linked frame of reference, the tensor of the deformations, calculating a tensor of the stresses in the natural local frame of reference, expressing, in the orthogonal local frame of reference, the stresses tensor, expressing the stiffnesses tensor, constructing a tangent operator, establishing an optimized configuration for the network, and locally adapting the network before impregnating said network.

Claims

1. A method for producing a part made of a composite material from a woven preform intended to be shaped, the woven preform comprising woven warp fibers and weft fibers forming a network, said woven preform being intended to be impregnated with a polymer matrix so as to form a woven composite material, the network having, before the shaping of the preform, two preferred directions that are substantially perpendicular to each other, and having, after the shaping of the preform, at least one off-centering angle α, the woven composite material furthermore following, without any shaping of the preform, a known general behavior law (L), the method comprising: defining an orthogonal local frame of reference (R.sub.1) with respect to the network before the shaping of the preform; defining a natural local frame of reference (R.sub.2) with respect to the network after the shaping of the preform; defining a linked frame of reference (R.sup.2) with respect to the network; obtaining a tensor of a stiffnesses (C) of the woven composite material without shaping of the perform, expressed in the orthogonal local frame of reference (R.sub.1); constructing a deformation tensor of deformations (E) in the orthogonal local frame of reference (R.sub.1); expressing, in the linked frame of reference (R.sup.2), the deformation tensor of deformations (E) previously constructed in the orthogonal local frame of reference (R.sub.1); calculating a stress tensor of stresses (π) in the natural local frame of reference (R.sub.2) from the behavior law (L) that is dependent on a tensor of stiffnesses expressed in the natural local frame of reference (R.sub.2) and the deformation tensor expressed in the linked frame of reference R.sup.2, by using, as the tensor of the stiffnesses expressed in the natural local frame of reference (R.sub.2), the tensor of the stiffnesses (C) of the woven composite material without shaping of the preform, expressed in the orthogonal local frame of reference (R.sub.1), which is unchanged regardless of the off-centering angle; constructing a tangent operator, for a numerical solution using a finite-element method comprising components which are equal to those of the tensor of the stiffnesses previously expressed in the orthogonal local frame of reference (R.sub.1); establishing an optimized configuration for the network based upon at least the stress tensor expressed in the orthogonal local frame of reference (R.sub.1) before the shaping of the preform; locally adapting the fibers of the network during a weaving prior to impregnation of said network with the polymer matrix, so as to fix said fibers in the optimized configuration; placing the preform in a mold; impregnating the preform with the polymer matrix; and demolding the part.

2. The method according to claim 1, wherein the natural local frame of reference (R.sub.2) is attached to the preferred directions of the fibers of the off-centered network, the natural local frame of reference (R.sub.2) being non-orthogonal in the presence of the off-centering angle α, wherein the off-centering angle α is non-zero.

3. The method according to claim 1, wherein the natural local frame of reference (R.sub.2) is a covariant local frame of reference and wherein the linked frame of reference (R.sup.2) is the contravariant local frame of reference, dual of the natural local frame of reference (R.sub.2).

4. The method according to claim 1, wherein a passage from the orthogonal local frame of reference (R.sub.1) to the natural local frame of reference (R.sub.2) is made by means of a passage matrix (J.sup.T) defined as: $J^T=\left(\begin{array}{ccc}1& \sin \alpha & 0\\ 0& \cos \alpha & 0\\ 0& 0& 1\end{array}\right)$

5. The method according to claim 1, wherein the behavior law L is a linear elastic behavior law and wherein the tangent operator is a tensor of the elastic stiffnesses.

6. The method according to claim 1, wherein the part is a vane.

7. The method according to claim 1, wherein the part is a casing.

8. A fold of a dry three-dimensional woven preform comprising the network of fibers, said network having locally at least one area in which it is not orthogonal, this at least one area having been defined by the method according to claim 1.

9. A turbomachine part made of woven composite material produced by shaping a woven preform, said woven composite material comprising the network of fibers impregnated with a polymer matrix, said network having, prior to shaping of the woven preform, at least one area in which it is not orthonormal, this at least one area having been defined by the method according to claim 1.

10. The method according to claim 1, further comprising: expressing, in the orthogonal local frame of reference (R.sub.1), the tensor of the stresses (π) calculated beforehand using the behavior law (L); expressing the tensor of the stiffness C, already expressed in the natural local frame of reference (R.sub.2), in the orthogonal local frame of reference (R.sub.1); and constructing a tangent operator for a numerical solution using the finite-element method comprising components determined from those of the tensor of the stiffness previously expressed in the orthogonal local frame of reference (R.sub.1).

11. The method according to claim 10, wherein the components of the tangent operator are equal to those of the tensor of the stiffness previously expressed in the orthogonal local frame of reference (R.sub.1).

Description

BRIEF DESCRIPTION OF FIGURES

[0037] Further characteristics and advantages of the invention will become apparent from the following detailed description, for the understanding of which reference is made to the attached drawings in which:

[0038] FIG. 1 is a front view of a conventional loom allowing to weave a woven preform,

[0039] FIG. 2 is a schematic cross-sectional view of a woven preform prior to shaping said preform,

[0040] FIG. 3 is a turbomachine vane produced by means of a shaping and an impregnation of a matrix of a preform as shown in the preceding figures,

[0041] FIG. 4a is a schematic view of the deformations of the network of the preform once it has been shaped,

[0042] FIG. 4b is a similar view to that of FIG. 4a, but the changes in the off-centering angle α are shown numerically,

[0043] FIG. 5 is a schematic view of the change in the off-centering angle α before and after shaping the preform,

[0044] FIG. 6 is an illustration of a set of frames of reference based on the local orthonormal frame of reference according to the invention,

[0045] FIG. 7 is a schematic summary of the first four steps of the method according to the invention,

[0046] FIG. 8 is a series of schematically drawn turbomachine vanes produced by means of the method according to the present invention.

DETAILED DESCRIPTION OF THE INVENTION

[0047] The method proposed in the present invention consists in, firstly, modelling the part 10 to be manufactured, for example a fan vane of a turbomachine. The modelling is carried out using, for example, a calculation software by the method of the computer-aided finite-elements and equipping said computer.

[0048] This part 10 is manufactured from the shaping of a woven preform 12. This woven preform 12 comprises woven fibres and is conventionally, as illustrated in FIG. 1, woven on a loom allowing to obtain either a preform from a single lap or fold, or from a plurality of laps or folds that are arranged together to constitute a conformed preform. The fold or the folds that form the preform are said to be dry. Indeed, the fold or the folds are not yet impregnated with a matrix intended to densify the woven preform.

[0049] At the loom outlet, the preform 12 thus has a set of fibres woven along two preferred directions that are substantially perpendicular to each other, as visible in FIG. 1. A distinction is made between two types of fibres or yarns: the warp fibres 16a which extend in the direction of the weaving 17 and the weft fibres 16b which extend substantially perpendicular to the weaving direction 17, and thus to the warps 16a. This set of warps 16a and wefts 16b thus forms a substantially orthogonal network 18, as seen in FIG. 2.

[0050] The woven preform 12 is then shaped, (as seen in FIG. 3) to give, after matrix impregnation and curing, the part 10. The shaping is typically carried out in an injection mold in which the matrix is injected. However, as illustrated in FIGS. 4a and 4b, it can be seen that this shaping of the preform 12 induces a series of deformations of the network 18. The network 18 is no longer orthogonal: it is off-centered. That is, a off-centering angle α has occurred between the initial direction (pre-shaping) and final direction (post-shaping) of the weft fibres 16b (see FIG. 5). After the shaping, the weft fibres 16b of the network 18 are no longer perpendicular to the weaving direction 17. It can be seen in FIG. 4b, that the off-centering angle α varies locally along the surface of the part 10. FIG. 5 shows the evolution of the network 18 before shaping of the preform 12 (zone Z.sub.1—orthonormal) and after shaping of the preform 12 (zone Z.sub.2—off-centered).

[0051] Once shaped, the preform 12 is typically impregnated with a polymer matrix and then cured in an autoclave to form the woven composite material part 10. A composite material is defined as a woven preform 12 impregnated with a polymer matrix. This woven composite material 14 has known mechanical properties. These mechanical properties are expressed by a known general behavior law L. For example, it can be a linear elastic behavior law. It is important to note that the behavior law mentioned here characterizes the behavior of the woven composite material 14 (preform and matrix), not the preform itself. The mechanical behavior of a woven composite material 14 is influenced by the off-centering angles α. Similarly, the mechanical behavior of a woven composite material is different from that of a preform (dry fibrous reinforcement).

[0052] These off-centering angles α induce a variation in the mechanical properties of the woven composite material 14 as a function of the different areas Z.sub.1, Z.sub.2 of the part 10. These variations in mechanical properties induce weaknesses and require the design of parts 10 with some oversized parts so as to compensate for the mechanical weaknesses of the off-centered woven composite material 14. An “off-centered woven composite material 14” is referred as a woven composite material whose woven preform 12 have a fibres network 18 with a non-zero off-centering angle α. In other words, a woven composite material 14 with a woven preform 12 having an off-centered network 18 is referred to as a “off-centered woven composite material 14”.

[0053] As with any network, the orientations of the fibres 16a, 16b of the network 18 can be expressed by decomposition on the vectors of a base. In mathematics, a base of a vector space V is a free family of vectors of V which generates V. We can therefore express the directions of the fibres 16a, 16b of the initial (pre-shaping) network 18 in a base B.sub.1 of an orthogonal local frame of reference R.sub.1. This allows to define, among other things, a general behavior law L. This general behavior law L can be classically composed of tensors having numerical values of the coordinates in the considered base B.sub.1.

[0054] In order to remedy these problems of oversizing, the present invention involves modeling the behavior of the woven composite material 14 off-centered in a base B.sub.2 of a local frame of reference R.sub.2 called natural (or off-centered). This natural local frame of reference R.sub.2 is related to the directions of the fibres 16, 16b of the network 18 after off-centering. This modelling is done in the same way as the modelling of the behavior of the orthotropic woven composite material 14 (with zero off-centering angle α) in the base B.sub.1 of the orthogonal local frame of reference R.sub.1. Indeed, we consider that the network 18 keeps its material symmetries in the natural local frame of reference R.sub.2: we consider that the off-centered network 18 behaves like an orthotropic material in the natural local frame of reference R.sub.2. This natural local frame of reference R.sub.2 thus allows the behavior of an off-centered composite material 14 to be modeled by a set of reliable, so-called definitive, behavior laws, regardless of the different local off-centering angles α. This modeling implies that: [0055] the components of the various tensors (projected in the natural frame of reference R.sub.2 and) involved in said general behavior law L are identical for the orthotropic and off-centered woven composite materials 14 (i.e. before and after shaping of the preform 12), [0056] only the bases B.sub.1, B.sub.2 of the frames of reference R.sub.1, R.sub.2 on which these tensors are projected are different, [0057] as with the fibres 16a, 16b of the original network 18, the off-centered base B.sub.2 is no longer orthogonal in the presence of off-centering.

[0058] The shaping of the woven preform 12 of the part 10 to be produced is modelled in such a way as to predict locally the deformations and the off-centering angles α of the network 18 of fibres 16a, 16b as a function of the shaping of the preform 12. In the context of the present invention, this modelling is geometric and is obtained by a numerical simulation of the shaping of the preform following an algorithm of the improved net. Then, the off-centering angles allow the modeling of the part 10 by the finite-element method.

[0059] In a first step, the orthogonal local frame of reference R.sub.1 is defined with respect to the network 18 before shaping the preform 12.

[0060] In a second step of the method, the natural local frame of reference R.sub.2 is defined. This definition of the frame of reference R.sub.2 allows to express a tensor of the stiffnesses C of the off-centered woven composite material 14. In the case of a non-off-centered woven composite material 14, this tensor of the stiffnesses C is conventionally defined in the orthogonal local frame of reference R.sub.1. The components of the tensor of the stiffnesses C are known in the orthogonal local frame of reference R.sub.1. Any tensor of the stiffnesses is obtained experimentally by experimental testing on a woven composite material (in the form of a specimen) and in the orthogonal local frame of reference R.sub.1 (without off-centering). Each tensor of the stiffnesses is related to a defined material. In the case of an off-centered woven composite material, the components of the tensor C are assumed to be known and unchanged (or invariant) in the natural frame of reference R.sub.2 regardless of the value of the off-centering angle α. This point (expression of the tensor of the stiffnesses C of the woven composite material 14 in said natural local frame of reference R.sub.2) is the core and the novelty of the proposed technical solution.

[0061] During a third step, a tensor of the deformations E in the orthogonal local frame of reference R.sub.1 is first defined or constructed. The tensor of the deformations is provided by a person skilled in the art and/or preferably by a software used to perform the modeling by the method of the finite-elements. The tensor of the deformations is known in the mathematical sense. In a second time, the tensor of the deformations E is expressed in a linked frame of reference R.sup.2. The linked frame of reference R.sup.2 is defined with respect to the fibre network. The expression or the calculation is performed by means of a J.sup.T passage matrix as shown in FIG. 7. Here, this linked frame of reference R.sup.2 is the contravariant local frame of reference R.sup.2, dual of the natural local frame of reference R.sup.2, called covariant frame of reference. Finally, the behavior law of the woven composite material is used to, by means of the tensors C and E, calculate a tensor of the stresses π in the natural local frame of reference R.sub.2.

[0062] During a fourth step, the tensor of the stresses π obtained above is expressed in the orthogonal local frame of reference R.sub.1 by means of a passage matrix J.sup.T.

[0063] During a fifth step, a tangent operator (necessary element) is constructed for a numerical solution by the method of the finite-elements comprising components which are equal to those of the tensor of the stiffnesses previously expressed in the orthogonal local frame of reference R.sub.1 In particular, and in other words, the components of the tensor of the stiffnesses C in the orthogonal local frame of reference R.sub.1 are calculated numerically. In the case of a linear finite-element calculation, the tangent operator is equal to C expressed in R.sub.1. In the case of a nonlinear finite-element calculation, the expression of the tangent operator is more complex and depends on the nature of the nonlinearity.

[0064] In order to allow a simplified understanding of the first five steps of the method of the present invention, the case of the dimension 2 is developed in the following. However, the technical solution proposed by the present invention remains entirely applicable in dimension 3.

[0065] More precisely, we consider the orthogonal local frame of reference R.sub.1 represented on FIG. 6 by the two vectors dX.sub.1 and dX.sub.2. The orthogonal local frame of reference R.sub.1 is written mathematically as R.sub.1=dX.sub.1−dX.sub.2. It corresponds to the frame of reference in which the deformations (tensor E) and the stresses (tensor π) must be expressed when using the method of the finite-elements. These deformations E and these stresses π are provided respectively as input and output of the general behavior law L.

[0066] We then consider the natural local frame of reference R.sub.2. The natural local frame of reference R.sub.2 is written mathematically R.sub.2=dM.sub.1−dM.sub.2. It is represented in FIG. 6 by the two vectors dM.sub.1 and dM.sub.2. The natural local frame of reference R.sub.2 is the frame of reference attached to the preferred directions of the fibres 16a, 16b of the off-centered network 18. The natural local frame of reference R.sub.2 is non-orthogonal in the presence of a non-zero off-centering angle α, i.e. when a is different from 0.

[0067] In mathematics, a collection of reference elements, one of which is designated as the origin, is called a “frame of reference”, these elements allowing any object in a given set to be designated in a simple manner. In geometry, a frame of reference is used to define the coordinates of each point. The frames of reference are used, for example, to represent data graphically.

[0068] The off-centering angle α is thus defined as the angle formed between dM.sub.2 and dX.sub.2 (see FIG. 6).

[0069] With the definition of the off-centering angle α shown in FIG. 6, the passage matrix J.sup.T from the orthogonal local frame of reference R.sub.1 to the natural local frame of reference R.sub.2 has the expression:

[00002]$J^T=\left(\begin{array}{ccc}1& \sin \alpha & 0\\ 0& \cos \alpha & 0\\ 0& 0& 1\end{array}\right)$

[0070] Recall that the result of the product of two tensors should not depend on the frames of reference in which they are expressed. This is the principle of objectivity of the physical laws. For this, it is therefore necessary that the two tensors E and C are expressed in dual bases.

[0071] More generally, in mathematics, the space of the linear forms on V is called the “dual space of a vector space V”. A particular type of linear application is called “linear forms”. A linear application (also called linear operator or linear transformation but many authors reserve the word “transformation” for those that are bijective) is an application between two vector spaces over a body K or two modules over a ring that respects the addition of the vectors and the scalar multiplication defined in these vector spaces or modules, or, in other words, that “preserves the linear combinations”.

[0072] In order to express two tensors E and C in dual bases (one with respect to the other) (in particular C is in a base and E is expressed in a base dual to that of C), the mathematical formalism followed in the proposed technical solution is that of the linear algebra. The linear algebra also allows to access to the notions of covariant base and contravariant base. These notions of covariant base and contravariant base are applied to the case of off-centered woven composite material 14 discussed in the present invention: thus, the natural local frame of reference R.sub.2 associated with the base B.sub.2 (represented by the vectors dM.sub.1 and dM.sub.2 in FIG. 6) is defined as covariant. This frame of reference changes with α, and is merged with the orthogonal frame of reference R.sub.1 (represented by the vectors dX.sub.1 and dX.sub.2 in FIG. 6) when the off-centering angle α is zero. We also define a dual frame of reference R.sup.2. This frame of reference is defined as the dual frame of reference of the natural local frame of reference R.sub.2. This dual frame of reference R.sup.2 is associated with a so-called contravariant base B.sup.2 (represented by the vectors dM.sub.1 and dM.sub.2 in FIG. 6). This dual frame of reference R.sup.2 also changes with the off-centering angle α, and is also merged with the orthogonal local frame of reference R.sub.1 when the off-centering angle α is zero. So mathematically we define the dual frame of reference R.sup.2 as R.sup.2=dM.sup.1−dM.sup.2. Each of its vectors dM.sup.i of the dual frame of reference R.sup.2 is orthogonal to the vector dM.sup.1 of the corresponding natural local frame of reference R.sub.2, with i≠j=1,2. Note that a frame of reference is identical to its dual if it is orthogonal. Note also that when a basis is orthogonal, it is confused with its dual base and the covariant B.sub.2 and contravariant B.sup.2 bases are identical.

[0073] In order to better explain the present invention, a concrete example of the first five steps of the method of the present invention, based on the notions defined above, is proposed in the following. The example is based on a particular general behavior law L: a linear elastic behavior law. In the absence of off-centering, recall that this law allows us to calculate the tensor of the stresses π by double contracted product of the tensor of the stiffnesses C and the tensor of the deformations E. We thus express π=C:E. The five steps detailed below are illustrated in FIG. 7. [0074] Step 1: the shaping of the woven reinforcement of the part 10 to be produced is modelled so as to predict locally the deformations and the off-centering angles α of the network 18 of fibres 16a, 16b according to the shaping of the preform 12. [0075] (in summary, this is how we obtain the off-centering angles α) [0076] Step 2: The components of the tensor of the stiffnesses of the material C are assumed to be known and unchanged in the natural local frame of reference R.sub.2 regardless of the value of the off-centering angle α. The tensor C is thus expressed in the natural local frame of reference R.sub.2. [0077] (in summary, the frames of reference R.sub.2 and R.sup.2 are constructed as a function of the previously obtained off-centering angles α and C is expressed) [0078] Step 3: the tensor of the deformations E is provided (by a person skilled in the art) as an input to the behavior law L in the orthogonal local frame of reference R.sub.1. The tensor E is then expressed in the dual frame of reference R.sup.2 with E=JEJ.sup.T (with J.sup.T, the matrix of passage from the orthogonal local frame of reference R.sub.1 to the natural local frame of reference R.sub.2). The tensor E being expressed in the dual frame of reference R.sup.2 and the tensor C being expressed in the natural local frame of reference R.sub.2, the tensors C and E are thus expressed in dual frames of reference (R.sub.2 and R.sup.2): their tensor product is thus objective, in the sense of the principle of objectivity of the physical laws. Thus, we can calculate the components of the tensor of the stresses π (π=C:E) in the covariant base B.sub.2 of the natural local frame of reference R.sub.2. [0079] (in summary, we express the tensor E in the frame of reference R.sup.2 (referred to as contravariant), then the behavior law is used to obtain π in R.sub.2 (referred to as covariant)) [0080] Step 4: the tensor of the stresses π, which has been calculated beforehand, is expressed in the orthogonal local frame of reference R.sub.1 (expression expected by the person skilled in the art) with π=JπJ.sup.T (with J.sup.T, the passage matrix from the orthogonal local frame of reference R.sub.1 to the natural local frame of reference R.sub.2). [0081] Step 5: Numerical solution by means of a tangent operator. The components of the tangent operator are calculated by means of the general behavior law L. In the present case, these components are calculated in the orthogonal local frame of reference R.sub.1. In the case of a general linear elastic behavior law L, the tangent operator is equal to the tensor of the elastic stiffnesses and its components in the frame of reference R.sub.1 are calculated by a change of base operation applied to a fourth order tensor whose expression is the following:

[v(p,q),v(r,s)]=J.sup.T[p,i].Math.J.sup.T[q,j].Math.J.sup.T[r,k].Math.J.sup.T[s,l].Math.[v(i,j),v(k,l)],

with: p, q, r, s, i, j, k, l being integer indices each in [1,2,3], J.sup.T being the matrix of passage from R.sub.1 to R.sub.2 (defined above) and v being the function [1,2,3].sup.2.fwdarw.[1,2,3,4,5,6] allowing to make the link between the components of a 3×3×3×3 tensor and the components of the same tensor written in a 6×6 matrix form thanks to the exploitation of the two minor and major symmetries, properties which the tensor of the elastic stiffnesses possesses. In other words, the tensor of the stiffnesses C (which had already been expressed in the frame of reference R.sub.2), is expressed in the orthogonal local frame of reference R.sub.1. This step is illustrated in FIG. 7 by the arrow number 4 which symbolizes the passage from the natural local frame of reference R.sub.2 to the orthogonal local frame of reference R.sub.1.

[0082] This allows to access to the tensor of the stresses π in the orthogonal local frame of reference R.sub.1 and thus predict the local stress state of the off-centered woven composite material 14, regardless of the value of the off-centering angle α. The impact of the off-centering angle α can then be anticipated and the fibres 16a, 16b of the network 18 of the preform 12 can be oriented prior to impregnation with the polymer matrix. The last five steps of the method of the present application are thus: [0083] establishing an optimized configuration of the network 18 of fibres 16a, 16b as a function at least of the stress tensor expressed in the orthogonal local frame of reference R.sub.1, before shaping of the preform 12; the orientation of the fibres is optimized everywhere in the digital profile to improve the mechanical response of the final part that will be obtained, [0084] locally adapting the fibres 16a, 16b of the network 18 before the impregnation of said network 18 of fibres 16a, 16b by the matrix, so as to fix the fibres 16a, 16b in the optimized configuration before the shaping of the preform 12; this adaptation takes place here during the weaving of the preform by taking into account the optimized configuration of the network upstream, [0085] shaping the preform 12 in a mold, after its placing in the mold; alternatively, the placing of the preform is done in the mold, after it has been shaped, [0086] impregnating the preform with a polymer matrix (e.g. resin), [0087] demolding the part 10 after curing the preform 12 impregnated with polymer matrix.

[0088] In this way, the mechanical properties of the part 10 can be predicted at each location and the weaving of the preform 12 can be adapted. This adaptation of the weaving can be done by a local rearrangement of the directions of the and/or a localized modification of the thickness of the fibres 16a, 16b and/or their spacing, for example. This adaptation is then fixed, before the impregnation of the preform by the polymer matrix. This allows the expected properties of the woven composite material 14 to be maintained despite the shaping, and allows the dimensional margins associated with the uncertainty of the mechanical properties of the woven composite material 14 after shaping the preform 12 to be avoided. In other words, the loom is reparametrized to produce a fibrous preform whose orientation of the weft and warp fibres allow to anticipate the behavior of the woven composite material with off-centering angles.

[0089] It can be seen in FIG. 8 that when the off-centering angle α is changed along the surface of the part 10, a difference is observed on the modal response of said part 10 (here a fan vane). For example, the modal response is determined by subjecting the part to waves (e.g. sound): the vibrations of the part, in response to these waves, are measured via sensors and/or cameras. Depending on the percentage of off-centering, the vibratory propagations are different in the part as shown by the lines in the vanes drawn in FIG. 8. The vibratory response is differently compliant to different modes and the part can thus be characterized in predictions of its behavior in particular under aeroelastic loads.

[0090] The technical solution presented here has very practical advantages. It can be applied to any type of general behavior law L and its implementation is simple and fast. The use of the method is immediate and requires no identification, with the off-centering angle α being the only additional input data required. Moreover, this approach has no impact on the calculation time, the transformations on the tensors of the deformations and the stresses 3×3 being almost instantaneous.