FR2915806A1 - Polycrystalline model determining method for e.g. determining deformation on metal can, involves modifying number of representation grains or parameter of texture to adjust model based on comparison between theoretical and real behaviors - Google Patents

Abstract

The method involves determining a theoretical behavior of a solid material by simulation using a polycrystalline model of a stress on the material, where the material is represented by representation grains regrouping orientation crystalline grains. The theoretical behavior is compared with a real behavior obtained from a mechanical test base. Number of representation grains or parameter of a texture utilized by the polycrystalline model is modified to adjust the polycrystalline model based on the comparison.

Description

-1- Method for determining a polycrystalline model for

  represent the behavior of a solid material subject to mechanical stress

  The present invention relates to a method for determining a polycrystalline model intended to represent the behavior of a solid material subjected to mechanical stress. It also relates to a polycrystalline model determined by the method according to the invention and a use of this polycrystalline model to determine the effects of a mechanical stress on a piece made of a solid material.

  In many technical fields, it is necessary to determine the impact of mechanical stresses on parts, in particular metal parts, either in order to optimize manufacturing processes involving shaping operations, or in order to predict the behavior. parts service in complex conditions.

  The shaping processes concerned are the plastic deformation forming processes such as stamping sheets, drawing tubes, forging mechanical parts. These methods can have a significant impact on the properties of the constituent material of the parts concerned, and it is important to be able to predict this incidence so as to allow a priori assessment of the service behavior of the parts obtained. The service requirements of the parts are various and correspond to all the stresses that may be encountered, and for example to the effect of permanent efforts, the effect of forces applied gradually to higher or lower speeds, d applied forces applied periodically, applied forces under ambient temperature conditions or efforts applied under conditions of high or very low temperature. In order to evaluate the effects of these mechanical stresses on parts, numerical calculation models are used in known manner, in elastic or anelastic mode. These methods of calculation are, in particular, finite difference methods and Galerkin methods based on a decomposition on basic functions of which a particular type is the finite element method. By anelastic mode is meant any mode of behavior which is not reversible elastic, that is to say for example plastic, plastic depending on time, or viscoplastic, plastic depending on the temperature, plastic resulting from phase transformations . These calculations use a modeling of the behavior of the material under the effect of the stresses, represented by a law connecting the stresses to the deformations. Conventionally, macroscopic behavior laws are used that correspond to phenomenological representations of the phenomena of deformation, that is, representations that do not take into account the underlying fundamental physical phenomena. Such models have the advantage of using a relatively limited number of variables, which leads to calculations that are sufficiently small for them to be used in industrial design processes. On the other hand, this type of model has the disadvantage of not making it possible to adequately model the effect of the microstructure of the materials on their properties, for example on the anisotropic properties, and even more so on the evolution of these anisotropic properties under the effect of stresses and deformations. In order to overcome this drawback, material behavior models, called polycrystalline models, are used in known manner, in which the material is represented by a plurality of grains having the crystallographic structure and the properties of a grain of the material. Such models have the disadvantage of requiring the use of a very large number of variables. This large number of variables leads to very long computation times that do not allow the use of such models in an industrial context. To overcome this drawback, several researchers have proposed to greatly reduce the number of crystalline orientations in the polycrystalline models as taught in the document FR 03 09252. These models will be designated in the following reduced polycrystalline model.

  However, it has been found that it is not possible to obtain with the same determination of the model a satisfactory modeling of both the stresses and the strain rates, as well with full-text polycrystalline models as in known manner with reduced polycrystalline models. In particular, the Lankford ratio, which is the ratio of two rates of anelastic deformation in two perpendicular directions, is very different from that measured during simple tensile tests in the third direction perpendicular to the other two.

  The object of the present invention is to overcome these drawbacks by proposing a method for determining a polycrystalline model intended to represent the behavior of a solid material more precisely. Another object of the invention is to propose a method for determining a polycrystalline model, intended to represent the behavior of a solid material, requiring: less computing time and usable in the industry. The invention thus proposes a method for determining a polycrystalline model intended to represent the behavior of a solid material subjected to stresses, the solid material being represented by a plurality of grains, said to be of limited number, and each grouping together grains of crystalline orientations close to said solid material, the polycrystalline model using: a crystalline model representing the behavior of a grain of representation, a texture representative of the texture of the material, this texture comprising at least one orientation and a volume fraction for each grains of representation; The method according to the invention comprises, an optimization step comprising at least one iteration of the following operations: determination, by simulation through said polycrystalline model of a stress on the material, of the so-called theoretical behavior of the material; comparison of the theoretical behavior with a behavior, said real, representative of the material, obtained from a mechanical test base; readjustment of the polycrystalline model according to this comparison. By behavior is meant the mechanical deformations of the material, its texture, or any other modification occurring in the solid material. The method according to the invention further comprises an initialization step comprising - an initialization of the number of representation grains used by the polycrystalline model to represent the solid material, an initialization of the texture representative of the solid material. This initialization includes a choice of at least one texture or at least one combination of textures. The textures or combination of textures can be chosen according to the knowledge of the material and data relating to the solid material; and an initialization of at least one parameter of the crystalline model. Thus, the polycrystalline model determined by the method according to the invention makes it possible to represent the behavior of the solid material by using a limited number of grains of representation. Advantageously, the number of grain of representation is less than or equal to twenty. In the remainder of the description, the polycrystalline model and the texture may be designated by reduced polycrystalline model or reduced texture.

  The result of the comparison of the theoretical behavior with a real behavior obtained from mechanical tests may require a modification, during the readjustment step, of at least one of the parameters chosen from the following list: the number of grains of representation, - a parameter of the texture or combination of textures used during the simulation or a choice of at least one other texture or combination of textures, and a parameter of the crystalline model. To determine the polycrystalline model intended to represent the behavior of the solid material, the method according to the invention comprises a determination of the number of grains of representation used by the polycrystalline model. The method according to the invention may further comprise a determination of the texture most representative of the texture of said material. This determination can be either a choice among a plurality of textures, the most representative texture of the solid material, or a combination of several textures. The determination of the most representative texture of the solid material may comprise a determination of an orientation and / or a volume fraction of at least one representation grain. The method according to the invention may advantageously comprise a determination of a texture evolution model, said model comprising a calculation of a rotation of the crystal lattice of each grain of representation with respect to a reference mark. predetermined, for example a corotational marker. The method according to the invention may further comprise determining a progress of a phase transformation. For this, each grain of representation is assigned characteristics depending on the progress of the phase transformation. Moreover, since the polycrystalline model uses a crystalline model, the method according to the invention may further comprise a determination of the crystalline model representing, best, the behavior of a grain of representation. The behavior of a representation grain can be determined from a plurality of slip systems associated with the solid material. Moreover, the method according to the invention advantageously comprises a determination of a scale-change model comprising a homogenization model for the passage from the scale of the representation grain, called the microscopic scale, to the scale of the set of representation grains, called the macroscopic scale, and a location model for the passage from the macroscopic scale to the microscopic scale. In particular versions, the localization model may be an anisotropic model involving fourth-order tensors, and the crystalline model may be a model of microscopic behavior such as twinning. Furthermore, the mechanical test base can be a so-called extended test base obtained by enriching a real mechanical test base with at least one mechanical test, said theoretical, obtained from the tests. real mechanical Theoretical mechanical tests can be obtained by interpolation of real mechanical tests.

  Theoretical mechanical tests can also be obtained, by simulation of a mechanical stress on a polycrystalline aggregate of grains, modeled in finite elements, and whose texture is representative of the solid material, the behavior of each of the grains being represented by a crystalline model previously determined.

  The crystalline model can be determined by an optimization step comprising at least one iteration of the following operations: determination, by simulation on the polycrystalline aggregate, of the so-called simulated behavior of the material; comparison of simulated behavior with actual mechanical tests; readjustment, according to said comparison, of said crystalline model representing the behavior of each of said grains of the polycrystalline aggregate. The parameters of the reduced polycrystalline model are determined by identification, for example by means of a device using an optimization software. The parameters to be identified are those of the crystalline model, also called the crystalline microscopic behavior model, on the representation grain scale, those of the scale-change model, and the orientations and the volume fractions of the texture representation grains. reduced and if necessary the parameters of the pattern of evolution of the texture.

  The optimization software compares model results with those of mechanical tests performed on the material. For this, the software calculates the response of the model for the same loadings as those applied to the material during the experimental tests. These calculations can be of the "material point" type, that is to say with a single calculation point, when it can be considered that the test structure remains homogeneous, for example a test on a single tensile test specimen. These calculations can also be of the structural calculation type, with a plurality of computation points, when the test structure does not give homogeneous stresses and deformations at the macroscopic scale, for example a test on a specimen having a cut. The experimental data for the studied material are generally limited in number, for example simple tensile tests in the directions L (rolling RD), T (transverse TD) and "LT" (at 45 in the plane LT) of a sheet laminated, or simple tensile tests in the L and T directions and a shear test in the L or T direction. This is not enough to properly identify an anisotropic model. In the example of a rolled sheet, tests every 15 in the LT plane are sometimes available (for example: Barlat, F., Aretz, Yoon, JW, Karabin, ME, Brem, JC, Dick, RE, 2005 Linear transformation-based anisotropic yield functions, International Journal of Plasticity 21, 1009-1039). This may still be insufficient to finely identify "reduced polycrystalline models", for which data every 5 is preferable, because of the reduced texture of these models. As for the non-LT test data, they are generally non-existent. In the method according to the invention, the finite element model of a polycrystalline aggregate is used to generate the experimental data necessary for the identification of the "reduced polycrystalline model" used. A polycrystalline aggregate is a set of grains. The construction of the aggregate model is known in itself. For example, it is possible to use a simple cubic aggregate consisting of cubic grains, for example at the number of 5 × 5 × 5 = 125 grains whose crystalline orientations are in agreement with the texture of the material. We can also build an aggregate model that better represents the grain morphology of the material. If the texture is not available for the studied material, one can use that of a nearby material, for example if it is a laminated aluminum alloy, the texture of another aluminum alloy of similar chemical composition, laminated and thermally treated under the same conditions. The experimental data generated will be all the better as the aggregate model will be refined and close to the actual material, at the cost of longer computing times. Aggregate models use fewer simplifying assumptions than polycrystalline models, and unlike them, they do not require more or less phenomenal scale-up models. In a polycrystalline aggregate model, the only parameters to be identified are those of the crystal behavior model. In a known manner, the parameters are identified by means of an optimization software applied to the finite element aggregate calculations of the available mechanical tests. Finite element aggregate calculations are then used to generate the experimental data needed to identify the "reduced polycrystalline model" used. For example, the aggregate model will be identified on simple tensile tests in directions L (rolling RD), T (transverse TD) and "LT" (at 45 in plane LT) of a rolled sheet, and will be then used to generate the simple tensile tests every 15 or better every 5 for the identification of the "reduced polycrystalline model". In this example, the generated data will be both the curves connecting the stress and strain in the direction of traction, and the curves connecting the two transverse deformations giving the evolution of the Lankford ratios. In exceptional cases where many tests are available, a simple interpolation of the curves can replace the aggregate calculations. In the example of a rolled sheet, if, for example, tests every 15 in the LT plane are available, it is possible, by interpolation, to generate data every 5. In the general case, the use of polycrystalline aggregate calculations is essential, especially if the extended test basis is generated by extrapolating actual experimental tests and not only by interpolation between these actual tests. If, for example, the part to be calculated is subjected to stresses close to pure shear, and the only available experimental tests are simple tensile tests, the shear data correspond to an extrapolation of the experimental data in simple tension. Aggregate calculations have some extrapolation capability because due to the plurality of "grain" orientations all slip systems and all parameters of the crystalline model used can be solicited by loadings in a restricted domain of types and amplitudes, for example by simple tensile tests. However, it is preferable if possible to limit the extrapolations and to have experimental tests representative of the mechanical stresses to which the parts will be computed, in terms of stresses, deformations, deformation rates and temperatures. All the parameters of the "reduced polycrystalline model" are identified on all the mechanical tests, including real tests and tests generated by simple interpolation and polycrystalline aggregate calculations. Given the large number of mechanical tests and the large number of parameters to be identified, the identification calculation times are reasonable only for "reduced polycrystalline models". It is the identification procedure that explains that the "reduced polycrystalline models" give a satisfactory modeling of both the stresses and the rates of deformation, unlike conventional polycrystalline models with complete textures, to which this approach can not be applied either in principle, because their textures are imposed, nor in practice, because the identification calculation times are much too long. The identified parameters of the crystalline model used in the "reduced polycrystalline model" are different from the identified parameters of the crystalline model used in the polycrystalline aggregate model. These two crystalline models may also be different, as well as the optimization software and finite element calculation software implemented for the "reduced polycrystalline model" on the one hand, the polycrystalline aggregate model of on the other hand, may be different. The aggregate model only serves as an intermediary in the identification process. The identification process and the process apply regardless of the models and software implemented, even if the quality of the results depends on the quality of the models and software. The polycrystalline model determined by the method according to the invention can be used for determining the effects of a thermomechanical stress on a part made of a solid material, and in particular a metal part. This use may comprise a determination of deformations and / or stresses generated by the thermomechanical stress at a plurality of points of the part by means of finite element modeling of the part. In general, the finite element method used is a mechanical or thermomechanical method of calculation involving time. The thermomechanical bias may include a bias associated with a plastic deformation forming operation such as stamping, drawing, extruding, cutting, drilling, rolling or forging. The method according to the invention can be used to evaluate the effect of these operations on the properties of the materials. The results obtained can be used, for example, in simulation models of motor vehicle crash tests.

  The method according to the invention can also be used to simulate the manufacture of metal cans such as cans, obtained by stamping, so as to determine their use properties taking into account phenomena such as stamping horns resulting from the anisotropy of the material.

  The mechanical stress may also include a load generated by the use in service of the metal part causing a phenomenon of creep and / or fatigue of the workpiece. These uses are only given by way of example and the person skilled in the art will be able to adapt the process to any particular situation involving phenomena of plasticity or creep. The method according to the invention differs from known numerical calculation methods by the way of identifying a "reduced polycrystalline model" of behavior of the materials then used in numerical calculations of industrial objects. Other features and advantages of the invention will become apparent upon consideration of the detailed description of non-limiting exemplary embodiments and the accompanying drawings in which: FIG. 1 is a representation of the schematic diagram of generating a base extensive testing; Figure 2 is a representation of a simple polycrystalline aggregate model used to generate experimental data; FIGS. 3a and 3b are representations of a refined model of aggregate used to generate experimental data; FIG. 4 is a representation of the schematic diagram for determining the reduced texture of a reduced polycrystalline model and the use of the model for industrial object calculations; FIG. 5 is a representation of the complete texture of aluminum 2090-T3, represented as a discretized pole figure; Figure 6 is a representation of the Lankford ratios all 15 of the reduced polycrystalline model identified every 15; Figure 7 is a representation of the Lankford ratios every 10 of the reduced polycrystalline model identified every 15; FIG. 8 is a representation of the Lankford ratios of the reduced polycrystalline model identified every 5; FIG. 9 is a representation of the constraints of the reduced polycrystalline model identified every 5; Figure 10 is a representation of the Lankford ratios of the reduced polycrystalline model re-identified every 15; FIG. 11 is a representation of the Lankford ratios of the reduced polycrystalline model re-identified every 45; FIG. 12 is a representation of the constraints of the reduced polycrystalline model re-identified every 45; FIG. 3 is a representation of the curves a11-E11 of the reduced polycrystalline model identified every 5; Fig. 4 is a representation of curves E22-c33 of the reduced polycrystalline pattern identified every 5; Fig. 15 is a representation of the reduced texture of aluminum 2090-T3 identified on the basis of extended testing; FIG. 16 is a representation of the poles of the aluminum 2090-T3 compared to the poles of the texture component S; Fig. 17 is a representation of the Lankford ratios of the reduced polycrystalline pattern identified every 5 with the imposed texture component S; FIG. 18 is a representation of the finite element mesh of the punching operation of a 2090-T3 aluminum thin sheet; - 12 -

  FIG. 19 is a representation of the stresses 611 on the lower surface of the sheet - FIG. 20 is a representation of the stresses 622 on the lower surface of the sheet - FIG. 21 is a representation of the stresses 612 on the lower surface of the sheet - FIG. Figure 22 is a representation of the force-displacement curves of the punch; and Fig. 23 is a representation of the space mechanical tests {611, 622, 612} shown on the plastic flow surface of Tresca. In order to make the invention better understood, we will consider the particular case of the finite element calculation method. We will first of all recall some notions known about the calculations by this method and on

  The models of behavior of polycrystalline materials, then we will explain how, according to the invention, we identify the parameters of a reduced performance polycrystalline model. In a conventional way, to simulate by a finite element calculation the behavior of an object subjected to mechanical stresses, one

  20 builds a representation of the material object considered using a mesh, and identifies the external mechanical stresses to which it will be subjected. Then, in a manner known per se, iteratively calculates the stresses and deformations at successive instants separated by a time step Δt. These calculations involve a model of

  Behavior of the material. In a finite element calculation, at a given instant t, the macroscopic stress tensor E and the macroscopic strain tensor E which can be decomposed into a tensor of the elastic deformations are known for each of the mesh computation points. Ee and a tensor of anelastic deformations EP. By way of example, the following are considered the simplest polycrystalline models known as "medium fields", in which the tensor of the microscopic plastic deformations eP and the tensor of the microscopic stresses a are constant in each grain of = 8

  representation. Using a localization model allowing the passage from the macroscopic scale to the microscopic scale, one determines the tensor of the stresses o- in each of the grains of representation of the polycrystalline model.

  Using an anelastic behavior model such as a plastic or viscoplastic model (such models are known per se), and representation grain gliding systems, the tensor of the microscopic strain rates is determined. s "of each grain of = 8 representation.

  By integrating the deformation velocities over a time interval At, the microscopic plastic deformations sg are calculated at time t + At.

  By using a homogenization model allowing the passage from the microscopic scale to the macroscopic scale, we then determine the tensors of the E deformations and the macroscopic E stresses at the instant t + At. We can then start again the calculation and thus to determine the evolution over time of the stresses and deformations in the studied object.

  To implement a numerical calculation method of the type just described, three reference points are used:

  an orthonormal global reference (L1, L2, L3), or laboratory reference, which is fixed and which serves as a reference for the description of the object to be studied. Depending on the nature of the object studied, this marker allows you to locate each point

  of the object by its Cartesian coordinates or by its cylindrical coordinates, or by its spherical coordinates.

  - Local macroscopic orthonormal landmarks, associated with each of the points of the studied object. Thus, at each point, and in particular at each point for which a calculation is made, a local coordinate system is associated.

  macroscopic (X1, X2, X3). This landmark is an orthonormal landmark whose position and orientation with respect to the global landmark can change as the studied object deforms. - 14 - - local microscopic orthonormal references associated with each of the representation grains of the polycrystalline model and which serve to describe the microscopic deformations of each grain of representation. The orientation of the microscopic local coordinate system (x1, x2r x3) of a representation grain associated with a point of the studied object represents the orientation of this representation grain with respect to the macroscopic local coordinate system of the considered point. Each representation grain is associated with slip systems, models of crystalline behavior of the material for each of the sliding systems, an orientation of the grain of representation with respect to a reference linked to the object to be studied, and finally a volume fraction f. Whatever the crystallographic nature of the solid material under consideration, the grain of representation is attributed to sliding systems. In a known manner, each system is defined by the normal to a sliding plane and by a sliding direction, which makes it possible to associate with each sliding system an orientation matrix. In a general and known manner, other deformation mechanisms, such as twinning, can be modeled on the scale of the representation grain, but they are not described here. If n, is the vector normal to the plane of the slip system of index s, and ls the vector of the sliding direction of this sliding system, the orientation matrix is by definition: ms = (nl, + 1, n) / 2 (Eq.1) If, for example, the crystallographic nature of the solid material in question is cubic with centered faces, we consider in a known manner M = 12 slip systems defined as follows in the microscopic local coordinate system (x1, x2) , x3) of a grain of representation: _ (1,1,1) 1, - (- 1,0,1) n, = (1,1,1) 12 = (0, -1,1) n , = (1,1,1) 13 = (-1,1,0) n, = (1,1,1,1) 14 = (-1,0,1) n5 = (1,1,1,1) 15 = (0,1,1) n6 = (1,1,1,1) 1, = (1,1,0) n, _ (-1,1,1) 17 - (0, -1,1) n8 = (- 1,1,1) 18 = 0,1,0) n9 = (-1,1,1) 19 = (1,0,1) n, 9 = (1,1, -1) 1 0 - (- 1,1,0) n = (1,1,1) -1 (1,0,1) n, 2 = (1,1, -1) 112 - (0,1,1 (Eq.2) For other crystalline structures, for example cubic centered or hexagonal, is defined in a similar and known manner sliding systems. If s represents the sliding speed in the direction 1 of the sliding system s, the tensor of the plastic strain rates of the grain of representation is written: 12 s; To determine the sliding speeds Ys on each of the sliding systems, a crystal behavior model 10 is used which, in known manner, involves the resolved splitting on the system. For example, the model of crystalline behavior is described in a known manner by the equations (5) to (8) below, but any other model 15 crystalline can be used. y '= 1) * Sign (rs) (Eq.5) n ù 0, -1; / r (Eq.6) vs = Max K 12 r, = Ro + QEH ,, [lxexp (bbv,)] (Eq.7) Equation (6) above involves 2 parameters of viscosity n and 20 K. Equation (7) uses 3 "work hardening parameters" Ro, Q and b.

  In equation (7), the "work hardening die" H ,, employs in known manner 6 other work hardening parameters h, at h6.

  The crystalline behavior of the grains of representation which has just been described makes it possible to model the microscopic behavior of the material. But, as we saw earlier, to model the behavior of an object, it is necessary to be able to go from the microscopic level to the macroscopic level and vice versa. For this, we use models called "scaling models" whose general principle is known to those skilled in the art, but an example will be described now. The relations allowing the passage from the macroscopic scale to the microscopic scale and vice versa are respectively: a) the model of localization for the passage of the macroscopic scale on the microscopic scale: NB = lfgfl (Egs8et9) x = l = g (Eq.10) the passage of the scale N (Eq 11) (Eq 12) (Eq 13) In these equations: E is the tensor of the macroscopic constraints. E "is the tensor of the macroscopic plastic deformations Ee is the tensor of the macroscopic elastic deformations 6 is the tensor of the microscopic stresses in the grain of = g representation of index G. 20 eP is the tensor of the microscopic plastic deformations in the = g of representation of index g, ZP) is the second invariant of the tensor speed of the microscopic plastic deformations e "in the representation grain of index g. = g is a variable in the representation grain of index g. = g 25 fg is the volume fraction of the representation grain of index g. 6 = E + C (Bu / 3) where g = g; (b) the homogenization relations for microscopic at the macroscopic scale: N E = Ifg6 with -g g = 1 g = 1 1fg = 1 N E`'f; g = g g = 1 Ee = 1 1 v v 2p - 1 + v = - 17 - N is the number of representation grains. 1u is the shear modulus of the material. v is the Poisson's ratio of the material. C and D are 2 parameters specific to the material, identifiable from tests. In known manner, it is possible and recommended in the case of a cyclic loading to add to the equations (6) and (7) kinematic hardening variables. In known manner, it is possible and recommended in the case of a long-term loading of relaxation or creep type to add equations (6) and (7) as well as the equation of location (10) of the terms restoration by time. The corresponding equations are not detailed here. When the deformations exceed 20% approximately, it is recommended, even essential for a precise calculation, to model the evolution of the texture of the material, as well in the methodology of determination of the parameters of the "reduced polycrystalline model" as in the use of this model for the calculations of industrial structures. To model the evolution of the texture of a polycrystalline material, that is to say the evolution of the orientations of the "grains", we consider in known manner the rotation tensor Q of the crystal lattice of a grain of representation , that is to say the rotation of its microscopic local coordinate system (x1, x2, x3), with respect to a "corotational" reference mark. Several models of texture evolution are described in the scientific literature. In the most widely used model, the rotational speed of the crystal lattice is given by: M QQ '= w = û ~ q Yr (Eq.14) where M is the number of sliding systems and where for each system: q = (l, n, ûn, 4l,) / 2 (Eq.15) The evolutions of the vectors ns and l are calculated. that is, the evolution of the texture: (Eq.16) - The corotational marker follows the "average" rotation of the material R. In known manner, its rotational speed with respect to the laboratory mark (L1, L2, L3) is calculated from the velocity gradients of the material points of the deformable solid body: S2 = RR T = (grade - grad `u) 12 (Eq.17) As for the mechanism of deformation by twinning, it implies particular models for the evolution of the texture, known elsewhere and which are not described here. To model the behavior of a specific material, it is then necessary to choose the representation grains of the polycrystalline model such that these representation grains represent the behavior of a grain aggregate of the material, taking into account the actual distribution of the orientations of the materials. grains in the material, that is, the texture of the material. To make this choice, one chooses the number of grains of representation, their orientations and their voluminal fractions so that the model obtained has a behavior well representative of the real material. To represent the orientation of the grains of representation, we consider a macroscopic local coordinate system (X1, X2, X3) and a microscopic local coordinate system (x1, x2, x3) bound to each grain of representation. The orientation of a representation grain is defined as the orientation of its microscopic local coordinate system (x1r x2, x3) relative to the macroscopic local coordinate system (X1, X2, X3). In a manner known per se, it is considered that the microscopic local coordinate system (x1, x2, x3) is deduced from the macroscopic local coordinate system (X1r X2, X3) by three rotations: a rotation of angle cp1 of the reference (X1 , X2, X3) around the axis X3 which leads to a reference (X'1, X'2, X3), - a rotation of angle around the axis X'1 which leads to a reference mark 30 (X '1, X "2, X'3) - an angle rotation (P2 of the reference (X'1, X" 2, X'3) around the axis X'3 which leads to a reference (X " 1, X "12, X'3), this last reference is the microscopic local coordinate system (xi, x2, x3) .- 19 - The microscopic local coordinate system (xi, x2, x3) is thus defined by the three angles of rotation ((pi, (h, (p2) which made it possible to obtain these angles are usually called "angles of Euler." This methodology of choice of the grains of representation will be illustrated by an example that the person skilled in the art will be able to transpose to any particular case.In known situations, the texture of an anisotropic material may be approximately represented by a combination of "texture components", i.e., particular orientations known per se. In the example, which concerns cold-rolling aluminum sheets, the texture components, defined by 3 Euler angles in the local macroscopic coordinate system (X1r X2, X3) = (L, T, S) = (RD, TD, ND), are mainly the components "Tin" (35, 45, 0), "S" (60, 32, 65) and "Cube" (0, 0, 0). Since the plates have an orthotropic symmetry in the macroscopic coordinate system, each of these orientations ((pi, 4), 92) is associated with 3 other symmetrical orientations (- (pi, (1), -92), (-91, - 4), -92) and ((pi, -4), (p2) .The texture components Etain and S thus have 4 grains of representation each, the texture component Cube has only one grain of representation. Annealed and recrystallized aluminum, other textural components appear: "Goss" (0, 45, 0) and "Copper" (90, 35, 45) The crystallographic structure of aluminum is cubic with centered faces. the steel sheets with a cubic centric crystallographic structure appear the texture components (0, 0, 45), (0, 35, 45), (0, 54.7, 45), (30, 54.7, 45) and (0, 45 , 0).

  These texture components are known to those skilled in the art, which may in particular situations determine other components from the experimental texture of the material. All of these texture components, known or determined in specific cases, constitute the initial texture of the material in the step of identifying the parameters of the reduced polycrystalline model. In the laminated aluminum example, there will be 3 angular parameters ((pi, 4), 92) to be identified for the component S and 3 angular parameters 4'1, 4) ', (p'2) to be identified for the tin component. A "Cube45" texture component (0, 45, 0 ") suggested by the experimental texture can be added to the Cube component, and it can be considered that these two components with a single grain of representation have no angle Euler can also be separated into 2 or 4 grains of representation each, in which case it will be necessary to identify the corresponding Euler angles If, for example, an identification is carried out with the "texture 1 "consisting of 6 = 4 + 1 + 1 grains of representation of the 3 components S, Cube and Cube45, we will have 5 texture parameters to identify: the volume fractions f2 and f3 of the Cube and Cube components 45, and the 3 angles of Euler (91, (h, (p2) of the component S whose volume fraction is f1 = 1- f2 - f3 A second identification will be carried out with the "texture 2" consisting of 6 grains of representation of the 3 components Etain, Cube and Cube45, with 5 texture settings at i to identify: the volume fractions f2 and f3 of the Cube and Cube 45 components, and the 3 Euler angles (9'1, (h ', (p'2) of the tin component whose volume fraction is f1 = 1- f2 - f3. A third identification will be carried out with the "texture 3" consisting of 8 = 4 + 4 grains of representation of the 2 components S and Etain, with 7 texture parameters to be identified: the volume fraction f1 and the 3 angles of Euler ((pl). , (h, (p2) of the S component, and the 3 Euler angles 4'1, (h ', (p'2) of the tin component whose volume fraction is f2 = 1- f1. the results of the identifications, other combinations of texture 4, texture 5, ..., may be used The principle of obtaining an extended test base from a reduced test base is shown schematically in the figure 1. In this figure, reduced mechanical test basis represents the actual tests on the material, where such tests are sufficiently numerous, for example simple tensile tests every 15 in the plane of a sheet, a simple interpolation between these tests. tests, on the right of the diagram, makes it possible to generate intermediate tests, for example all 5, which are added to the reduced base to form a first version of an extensive mechanical test base. In the opposite case, and especially when it is necessary to extrapolate outside the field covered by the reduced base, for example to generate biaxial tensile tests or shear tests, a finite element model (EF) of an aggregate is used. The first step consists of generating the geometry of the aggregate, for example by constructing an aggregate consisting of cubic finite elements as in FIG. 2. In a second step, a crystalline orientation is affected. at each point of calculation of the finite elements, by distributing the orientations of the complete experimental texture of the material. The behavior of the material in all these calculation points is that of a crystalline model, whose parameters are initialized in a third step. The finite element calculation software is then able to calculate the response of the polycrystalline aggregate to all the mechanical stresses corresponding to the real tests of the reduced base. In a fourth step, the results of these calculations are compared to the results of the tests by an optimization software, in an iterative process which leads in a fifth step to the determination of the parameters of the crystalline model. In a sixth step, the aggregate model thus determined is used in the finite element software with new mechanical stresses, corresponding, for example, to biaxial traction or shearing if these are the target conditions. In a seventh step, the results of these calculations are added to the test database to form the extended base of mechanical tests.

  FIG. 2 shows an example of a finite element model of a polycrystalline aggregate mentioned in FIG. 1. This is a simple model consisting of 6 × 6 × 6 = 216 cubic finite elements oriented along the directions (X 1, X 2, X3) = (L, T, S) = (RD, TD, ND) of a rolled sheet. In each finite element, the crystalline behavior of the material is identical.

  On the other hand, a different crystalline orientation is assigned to each finite element. For example, from the experimental texture discretized in N = 216 orientations ((pl, (1), 92) of identical volume fractions f = 1/216, orientations are drawn randomly and assigned in a predefined order to the elements If N is different from 216 and if the volume fractions f are not uniformly equal to 1/216, we can use known methods to better represent the experimental texture, for example if the aim is to generate simple traction in a direction in the plane LT and making an angle 0 with the direction L = RD, as shown in Figure 2, is imposed in known manner on the 6 faces of the cube "boundary conditions" corresponding For example, one can impose homogeneous boundary conditions corresponding to the macroscopic tensors of strain rates E and rotation S 2. Other types of boundary conditions, such as Periodic requirements may be imposed. Those skilled in the art will choose these conditions according to the possibilities of the finite element calculation software available to them. Since the purpose of these calculations is not to determine the local microscopic values of the variables but to calculate the macroscopic curves, and only in relative terms from one test to another, the results depend little on the type of boundary conditions imposed. . For example, the only parameters of the aggregate model, namely the parameters of the crystal model, are identified on simple tensile tests in the 0 = 0, 45 and 90 directions using optimization software. The experimental data are the stress-strain curves 06H and the curves E33-e22 connecting the transverse deformations. The optimization software uses methods for identifying known parameters in themselves. The software controls the finite element calculation of the aggregate model on the same tests, and compares the simulated curves with the experimental curves. Based on the difference between the experimental results and the calculated results, the software calculates new values for the parameters, and then replicates this process until the differences between the experimental results and the simulated results are less than a value. defined in advance. The polycrystalline aggregate model is then used to generate curves 6a -En and 33 -E22 corresponding for example to simple tensile tests in directions 0 = 15, 30, 60 and 75. FIGS. 3a and 3b show a more refined 18x18x18 aggregate model than the simple 6x6x6 model of FIG. 2. In a manner known moreover, a geometric model of the grain morphology of the material is first established, such as that of the Figure 3a which comprises 447 grains. A regular mesh is then applied to the geometric model as shown in Figure 3b which has 18x18x18 = 5832 finite elements. The calculation points of each finite element, in the number of 2x2x2 = 8 or - 23 - 3x3x3 = 27 for example, are thus individually assigned to a crystalline orientation, that of the grain to which they belong. The boundary conditions are imposed on the 6 faces of the cube in the same way as for the simple aggregate of Figure 2. The calculation of the refined model of aggregate is much longer than that of the simple model, and its use will be reserved for situations in which grain morphology is essential and small deformations. In usual situations, macroscopic results differ little between simple and refined aggregates. The identification procedure of the reduced polycrystalline model is shown diagrammatically in FIG. 4. In a first step, called initialization, values are given to the parameters of the crystalline model and of the scale-change model, and if necessary to other models, which are part of the polycrystalline model. In this initialization step, several reduced textures are chosen, by combining known or determined texture components from the complete experimental texture of the material. The finite element calculation software is then able to calculate the response of the reduced polycrystalline model to all the mechanical stresses corresponding to the tests of the extended mechanical test base. In a second step, the results of these calculations are compared to the results of the tests by optimization software, in an iterative process which leads in a third step to the determination of the best reduced texture and reduced polycrystalline model parameters. In a fourth step, the model thus determined is used in the materials module of a finite element software, for the calculation of industrial objects. We will now illustrate this approach on the concrete example of a sheet of 2090-T3 cold-rolled aluminum alloy with a thickness of 1.6 mm. The crystallographic texture of this sheet is shown in FIG. 5 in the form of a discretized pole figure. In known manner, the pole associated with a direction in space defined by a unit vector n = (n ,, n2, n3) is the projection of the end of this vector on the plane X3 = 1 from the projection center (X1, X2, X3) = (0.0, u1). If n3 <0, it is the equivalent point (ûn ,, ûn2, ûn3) that is projected, so that all the poles are located in a circle of radius 2 in the projection plane. By convention, a homothety of center (0,0) and ratio 1/2 in the plane of projection reduces the radius of this circle to 1, and a rotation of -90 is applied so that the direction X1 of this plane is oriented downwards and the direction X2 to the right, as in FIG. 5 where (X1, X2) = (RD, TD).

  The coordinates of the pole are then X; = n2 / (1 + n3) and Xz = -n1 / 0 + n3). For a crystalline orientation defined by its microscopic local coordinate system (xi, x2, x3), the 3 poles {100}, {010} and {001} are the poles of the directions x ,, x2 and x3. These are the so-called "{100}" poles. In the case of a cubic face-centered crystallographic structure, the normals 10 at the sliding planes (1,1,1), (1,1,1,1), (-1,1,1) and (1,1,1) , -1) give the 4 poles called "{111}". A crystalline orientation can be defined equivalently by its poles {100} or by its poles {111}. In Figure 5, the discretized texture of the aluminum sheet 2090-T3 is defined by the poles {111}, each crystalline orientation corresponding to a group of 4 points. Experimental curves 611-e11 and e33 -E22 are available on aluminum sheet 2090-T3 every 15, i.e. for 0 = 0, 15, 30, 45, 60, 75 and 90. These curves were used first in the optimization software. Since only initial values of the Lankford ratio r, which is the ratio of the strain rates r = ε22 / ε33, were available, the curves e3, -E22 were "reconstructed" assuming the ratio r constant during the loading. . In reality, the ratio r evolves during loading, and it is one of the advantages of polycrystalline models to be able to represent this evolution. In order to determine the parameters of the "reduced polycrystalline model", it is necessary to provide the optimization software with initial values of these parameters. As for the texture parameters, different sets of initial parameters called "texture 1", "texture 2", "texture 3", ... are tried, as explained above. These sets combine the Etain, S, Cube, Cube45 texture components and additional components deduced from Figure 30 of Poles 5, for example (31.66, 54.04, 82.29) and (90, 33.40, 45). After a number of parameter identification calculations, an 8-grain texture of representation gives the best results. The optimized parameters of this texture are given in Table 1. Given the orthotropic symmetry, each triplet of angles of Euler corresponds to 4 grains of representation.

  Table 1. Texture parameters corresponding to FIG. 6. f 39.460 67.252 87.065 0.15334 79.146 40.124 23.825 0.09666 The results obtained with the identified model, curve 61, are compared with the tests, curve 62, in FIG. 6 which gives a representation of the evolution, every 15, of the Lankford ratio r as a function of the simple traction angle in the LT plane. To obtain a

  synthetic figure, we calculated the Lankford "secant" ratio between deformations 611 = 0.015 and 611 = 0.065, ie: r = 627611 = 0.065) - 622 (El 1 = 0.015) E 18) 33 16u = 0.065) - c33 \ sl l = 0.015) (q) This ratio is the slope of the curve connecting 622 and 633 if this curve is linear between 611 = 0.015 and 611 = 0.065 The reduced polycrystalline model

  identified is in fairly good agreement with the tests. The differences observed at 0 = 15, 30 and 45 are acceptable, the essential thing is to reproduce the two minimums at 0 and 75 and the two maximums at 45 and 90, because they are the ones who will determine for example the angular position and the amplitude of the stamping horns obtained during a stamping operation of a

  circular sheet to obtain a bucket, and it is these stamping horns that cause material losses and machining costs as well as irregularities in the thickness of the finished product. The reduced polycrystalline model identified seems ready for the calculation of industrial parts, as shown schematically in Figure 4.

  However, if one simulates with the model thus identified simple tensile tests for intermediate angles 0, one obtains the evolution, every 5, of the secant Lankford ratios shown in FIG. 7, where the curve 71 represents the results obtained. with the polycrystalline model identified, and curve 72 represents the results obtained with the tests.

  The large variations in this ratio, for example between 15 and 30, are not acceptable for finite element structure calculation. These variations are explained by the fact that the Lankford ratio is obtained by derivation, because it is a ratio of deformation velocities, it is therefore much more sensitive than the constraints to the changes of direction of the mechanical stress, in particular for reduced polycrystalline models. One of the originalities of the process is to recognize this difficulty and to extend the experimental base to remedy it. In the example of the 2090-T3 aluminum sheet, a simple interpolation between the values every 15 gave values every 5 shown in Fig. 8, curve 82 showing the results of the real and theoretical tests. But when the initial database is not as rich, the use of polycrystalline aggregate calculations, as shown in Figure 1, is essential to extend this base, because for example the minimum of 750 of the experimental curve of the Figure 6 can not be obtained by interpolation from values at 0, 45 and 90 only. The determination of the reduced texture and other parameters of the model was therefore taken up on the extended experimental basis. Optimizations with 8 or 10 grains of representation show that one of the components with 4 grains of representation tends to be eliminated, that is to say that its volume fraction tends towards 0, or that the poles regroup around the Cube component, which is the same since the Cube component is already present. The optimized reduced texture is reduced to a component with 4 grains of representation close to the texture component S and to the components with 1 grain of representation Cube and Cube45, that is 6 grains of representation in all. The suppression of the Cube45 representation grain does not degrade the results, so we arrive at an optimized texture with only 5 grains of representation defined by the parameters of Table 2. The other parameters of the model are those of equations (5) to (13) . Their optimized values are given in Tables 3 and 4. The elasticity parameters E and V and viscosity n and K are not optimized because the experimental data do not allow it. -27- Table 2. Texture parameters corresponding to Figures 8 and 9. ~ 2 f 68.3112 27.5545 55.7647 0.2013 0. 0. 0. 0.1947 Table 3. Parameters corresponding to Figures 8 and 9. E (MPa) vn K (MPa) 70000. 0.3 25. 20. Ro (MPa) Q (MPa) b C (MPa) D 84.9002 59.7546 5.1171 25269. 174.7983 Table 4. Parameters of the work hardening matrix corresponding to FIGS. 8 and 9. h1 h2 h3 h4 h5 h6 1.0000 0.3686 1.0568 0.6841 0.4495 0.4013 The results of the identified model are compared with the tests in Figures 8 and 9. In these figures the curves 81 and 91 represent the results obtained with the identified polycrystalline model and the curves 82 and 92 represent the results obtained with attempts. To obtain a synthetic figure, a deformation sil = 0.04 was chosen to represent the stress levels of figure 9. The experimental curves 82 and 92 and the curves of the model 81 and 91 are very close, the stress deviation does not exceed not 4%. It is important to note that it is not only the texture that gives the material its anisotropic behavior, but also the parameters of the work hardening matrix h1 to h6. It is essential to optimize the parameters of this matrix, whereas in the scientific literature they are often set a priori. On the other hand, the localization model of equations (8) to (10) is isotropic, which is not coherent with the rest of the polycrystalline model, which is anisotropic in nature. To improve the results of the reduced polycrystalline model, the isotropic localization model must be replaced by an anisotropic localization model, replacing the 2 scalar parameters C and D of the isotropic model with 2 order 4, C and D tensors. In the case of an initially orthotropic material, such as a rolled sheet, each of these 2 tensors is expressed as a function of 10 scalar parameters. In the general case, each of these 2 tensors is expressed as a function of 34 scalar parameters. If the Voigt notation is used for symmetric 2-order tensors: 611/612/323/312/322 331/323/331 // 633 _ - Q = [fi / 622/333/312/23/331 e (Eq 19) the scalar expressions C (Buf) and D / 3 of equations (8) and (10) ù = x = x become tensor expressions noted: C: (B-fi) - * C (B- / 3) and D: -> Da (Eq.20) where the second order C and D tensors have the following form in the orthotropic case:

  C13 C21 C23 C31 C32 C33 0 0 0 G = 0 0 0 C44 0 0 0 0 0 0 c55 0 0 0 0 0 0 C66

  The incompressibility of the plastic strain makes that the tensors C and D connect vectors such that +, 622 + f'33 = 0. It follows that the 12 parameters of equation (21) satisfy the 2 relations:

  C11 + C21 + C31 = Cl2 + C22 + C32 = Cl3 + C23 + C33 (Eq.22)

  There remain 10 parameters to be determined for each of the tensors C and D. For theoretical reasons, the parameters of the tensor C must be of the order of magnitude of the shear modulus of the material, u = El (] + v) / 2 . (Eq.21) - The optimization of the polycrystalline model reduced by means of the 10 + 10-2 = 18 additional anisotropy parameters should allow the curves 81 and 91, obtained with the reduced polycrystalline model determined by the method, to be brought closer together according to the invention, experimental curves 82 and 92. But the method is dependent on the models and software used, and for the example presented only an isotropic localization model was available. Predictions of the reduced polycrystalline model with the optimized parameters of Tables 2 to 4 were also calculated for simple tensile tests every 5 in the LS and TS planes of the plate. The curves obtained, not shown here, do not show sudden variations, which is satisfactory. The tensile tests in the LS and TS planes of a thin sheet (1.6 mm for the 2090-T3 aluminum sheet) are impossible to achieve. Aggregate calculations are needed to generate these experimental data. In the operations of stamping a thin sheet, the stresses are mainly in the plane LT of the sheet and only the properties of anisotropy in this plane are important. The example of the aluminum sheet 2090-T3 is therefore limited to the properties in the LT plane, but the method is in no way limited to this plane.

  The optimization of the "reduced polycrystalline model" on experimental data every 5 has shown the interest of a texture reduced to 5 grains of representation like that of Table 2. This information being now known, it is possible to determine again model parameters on experimental data every 15 only. The initial texture for optimization consists of the S and Cube texture components. Figure 10 shows the Lankford ratio r as a function of the simple pull angle in the LT plane. Curve 101 shows the results obtained experimentally and curve 102 shows the results obtained with the reduced polycrystalline model determined by the method according to the invention. The curves 101 and 102 show that the polycrystalline model obtained gives results overall as good as those of FIG. 8, although slightly different. The parameters of Tables 5 to 7 are different from the parameters of Tables 2 to 4, which shows that 2 different sets of parameters can give substantially equivalent results. The purpose of the method and in particular of its parameter optimization approach is not to give so-called physical values intrinsic to the parameters, but to obtain a model that best reproduces the experimental data. Table 5. Texture parameters corresponding to Figure 10. This, 0 ~ zf 69.2030 27.6300 54.7523 0.2067 0. 0. 0. 0.1731 Table 6. Parameters corresponding to Figure 10. E (MPa) vn K (MPa) 70000. 0.3 25. 20. Ro (MPa) Q (MPa) b C (MPa) D 83.1778 120.8708 2.9603 29403. 197.1514 10 Table 7. Parameters of the work hardening matrix corresponding to Figure 10. hl h2 h3 h4 h5 h6 1.0000 0.2861 0.8268 2.4772 0.1835 0.1138 With 5 grains of representation, initially those of the S and Cube texture components, the parameters were again determined with experimental data at 0, 15 and 90 only. The best optimization results of the parameters of the polycrystalline model are shown in FIGS. 11 and 12, presenting the results obtained experimentally 111 and 121 and the results obtained with the reduced polycrystalline model 112 and 122. The differences with the experimental data are clearly more important than in the previous figures. The shapes of the curves, due to the choice of the initial representation grains, do not change, but there is no guarantee that these shapes correspond to the studied material because the points at 0, 45 and 90 are too far apart to allow a reliable interpolation.5 - 31 The preceding figures give in a synthetic form the results of the process. In reality, the comparison of experimental data and model forecasts, for example in optimization software, is performed on the curves connecting the components of strain and strain tensors. FIGS. 13 and 14 show these curves for the 0 simple draw angles, curves 131 and 141, and 45, curves 132 and 142, and the model identified every 5 of FIGS. 8 and 9, Tables 2 to 4. 0 is representative of a small deviation with the experimental points shown in Figures 13 and 14, the angle 45 is representative of a larger gap. The pole pattern {111} of the model identified every 5, Table 2, is shown in Figure 15. The simplicity of this figure, compared to the experimental pole figure of Figure 5, illustrates the reduction in computing time of the reduced polycrystalline model compared to full textured polycrystalline models. The purpose of the method is not to accurately reproduce the experimental texture, however the poles of the identified reduced texture are well within the concentration zones of the experimental texture poles. The poles {111} of the reduced polycrystalline model identified every 5, Table 2, and the texture component S, are compared in Figure 16. The orientations are close, but the difference is sufficient to make the difference in the forecasts of the model. Indeed, if one optimizes on the basis of extensive tests (all 5), the parameters of the polycrystalline model with the Euler angles of the texture S imposed and not optimized (60, 32, 65), the voluminal fractions being they are optimized like the other parameters, the curves of FIG. 17 are obtained, the curve 171 corresponding to the results obtained experimentally and the curve 172 corresponding to the results obtained with the polycrystalline model according to the invention. The modification of the texture component S is essential to obtain good predictions with the reduced polycrystalline model. In the scientific literature, an isotropic texture background is added to the texture components to improve the predictions of polycrystalline models with small number of texture components at imposed Euler angles. Thanks to the method of determination of the process parameters, this isotropic bottom is obviously superfluous. In the example of alloy 2090-T3, the mechanical tests used for the determination of the parameters do not exceed 10% of strain (E11 = 0.1). The modeling of the evolution of the texture is not essential to determine the parameters of the reduced polycrystalline model. On the other hand, in many industrial applications, deformations exceed 20%. It is then necessary to model the evolution of the texture, as well as to use computation software in large deformations known elsewhere, as well in the methodology of determination of the parameters of the process as in the industrial calculations. In parameter optimization calculations, if the applied loading preserves the orthotropic symmetry of the reduced texture, for example a simple pull or an additional lamination in the L direction, some highly symmetrical representation grains such as the Cube representation grains { 0, 0, 0) or Cube45 {0, 45, 0} can be in a stable position and not evolve. This stability reduces the overall evolution of the texture, because these grains of representation have a large volume fraction. This is a peculiarity of the reduced polycrystalline model.

  To remedy this, it suffices in the optimization calculations to dissociate these representation grains, for example by replacing the representation grain Cube by 4 grains of representation {0, 0.1, 0}, {0, -0.1, 0}, {0.1, 0, 0) and {-0.1, 0, 0}, each having a volume fraction equal to 1/4 of the volume fraction of the initial Cube representation grain. These 4 grains of representation are not stable and will evolve in a divergent way preserving the orthotropic symmetry of the texture as the loading itself respects this symmetry. In finite element calculations of mechanical tests or industrial structures, this precaution is not necessary because no calculation point has a perfectly symmetrical loading, with the exception, for example, of simple tensile test specimens, which are generally not calculated by finite elements but considered as a single material point. By means of the process schematized in FIGS. 1 and 4, we have obtained a reduced polycrystalline model able to represent the behavior of a material. This model makes it possible to calculate the effect of mechanical stresses on an object, but for that it must be integrated in a model of computation by finite elements. With this model, it is then possible to calculate the evolution, between an instant 0 and an instant t, of the properties of the material of which the object is constituted for which the stresses and the deformations in the macroscopic state are studied. . The method which has just been described uses visco-plastic models, but the skilled person can use any model of anelastic behavior compatible with a polycrystalline model according to the invention and in particular, creep and fatigue behavior. creep. By way of example, the "reduced polycrystalline model" of the aluminum alloy 2090-T3 established according to the invention will now be used in a finite element calculation model of a punching operation of a thin sheet metal. . According to the diagram of FIG. 4, the materials module of a commercial finite element calculation software is used without making any modifications. This module comprises a conventional polycrystalline model for face-centered cubic crystal structure materials. The algorithm for integrating the behavior model at each calculation point is of the Runge-Kutta type, known elsewhere. More stable algorithms such as implicit algorithms can be used for the reduced polycrystalline model, given the small number of variables of this model, but in the software used these algorithms do not exist for the polycrystalline model. The finite element mesh of the punching operation of a sheet is shown in FIG. 18. For the sake of clarity, only a quarter of the device is shown. The sheet 181 0.6 m thick is fixed on a piece 182 having a circular hole of 24 mm in diameter and the punch 183 of 20 mm in diameter is pressed progressively to deform the sheet. The coefficient of friction between the pieces is taken equal to 0.2. In the finite element calculation, all 3 parts are modeled by 2460 elements and 3567 nodes. Sheet 181 has 4 elements finished in its thickness. Given the orthotropic symmetry of the material, it is possible to perform the calculation on the 1/4 of the pieces only, as shown in Figure 18. Nevertheless, as it is a calculation of 34 - demonstration, this possibility was not used. The finite elements are "c3d8" hexahedrons (6 faces and 8 nodes) in which the displacements are linear functions of the coordinates. The elements each comprise 8 calculation points, that is to say 8 points of integration of the model of behavior of the material. All these characteristics are conventional in finite element calculation software and known to those skilled in the art. The punching operation was calculated with the parameters of the reduced polycrystalline model of Tables 2, 3 and 4, between times t = 0 and t = 5000 seconds corresponding to a displacement of the punch equal to 5 mm as in FIG. The calculations were made with the large displacement and large deformation options known elsewhere. By cons, as the deformations do not exceed 20% in the sheet 181, the calculations were made without changing the texture. Figures 19, 20 and 21 show the stresses 611, 622 and 612. on the underside of the sheet at t = 5000 seconds. The stresses are maximum in the central part of the sheet, subjected to a state of stress close to biaxial tension 611 = 622. For comparison, the punching operation was also calculated with a macroscopic anisotropic model (Bron, F Besson, J., 2004. A yield function for anisotropic materials, Application to Aluminum Alloys, Int J. Plasticity 20, 937-963). The parameters of the Bron-Besson model were identified on the same experimental points 611-E11 and c33-c22 used to determine the parameters of the "reduced polycrystalline model", for the simple traction angles 0 = 0, 15, 30, 45, 60, 75 and 90 (all 15), see for example Figures 13 and 14. With a macroscopic model it is not necessary to extend the database of the experimental data every 5, because the nature of the equations ensures an interpolation sufficiently precise. The model thus identified describes very well the variations of the stress and the Lankford ratio with the angle O. This model is programmed with an implicit local integration algorithm. For comparison with the reduced polycrystalline model, it has also been used with the Runge-Kutta algorithm. Tables 8 and 9 summarize the calculations made with the same processor. The stability of the Runge-Kutta algorithm was verified by comparing two calculations with 20 and 200 time steps At = 100 and 10 seconds, which give very similar results. The implicit and Runge-Kutta algorithms give very similar results for the Bron-Besson model. With both algorithms, it was not possible to continue the calculations for the Bron-Besson model beyond t = 3490 seconds, because of the strong nonlinearities of the model. For the comparable calculations, it can be seen that the calculation times of the two models are of the same order of magnitude. The reduced polycrystalline model is a bit slower and consumes more time in local integration.

  The skilled person will interpret these figures. The essential result is that the reduced polycrystalline model according to the method leads to computation times sufficiently short for 3D calculations of complex industrial operations on a single processor, such as coin forming calculations.

  Table 8. Punching calculations performed with the reduced polycrystalline model. Model algorithm t (s) Force (N) t calculation (s) including local PR RK 20 no 2000 1161.58 15749 9503 PR RK 200 no 2000 1151.50 14993 4091 PR RK 330 step 3300 2130.18 30359 10241 PR RK 500 step 5000 3741.52 529491 23315 Table 9. Punching calculations made with the model Bron-Besson Model algorithm t (s) Force (N) t calculation (s) of which local BB imp 115 not 2000 1550.18 11892 5330 BB RK 200 not 2000 1551.81 10571 550 BB RK 330 step 3300 3073.09 18378 947 BB imp 716 step 3405 3218. 54 60234 17655 The force-displacement curves of the punch for the two models are compared in FIG. 22, the curve 221 corresponding to the results obtained with the Bron-Besson model and the curve 222 with the results. obtained with the polycrystalline model according to the invention. At the same force applied to the punch, the displacement of the punch and therefore the deformations of the sheet are smaller with the Bron-Besson model. Yet the parameters of both models were identified on the same experimental data. But these data are exclusively data in simple traction. But Figures 19 to 21 show that the central part of the sheet is in a state close to the biaxial traction 611 = 622. The models have not been identified for this type of loading. The calculation of a biaxial tensile test with the two models shows that the deformations are weaker with the Bron-Besson model.

  The various states of loading of a sheet in its plane are represented in the space {611, 622, 612} in FIG. 23. The plastic flow surface of Tresca was used to visualize these states, although does not correspond to the actual behavior of the material. It appears clearly in FIG. 23 that the biaxial traction TB {60, co, 0} is not an interpolation but a strong extrapolation of the experimental data in simple traction corresponding to the ellipse passing the points TSO {6o, 0, 0} and TS45 {6o / 2, 60/2, (Yo / 2} (and TS90 {0, 60, 0} hidden by the surface of Tresca.) It is therefore not surprising that the results of both models are different from each other, and different from the actual punching curve which is not represented here, because these two models are not capable of extrapolation outside the domain of identification of their parameters, which is reduced to the ellipse TSO -TS45 These results confirm the need to generate an extended test basis according to the scheme of Figure 1. This extended base must absolutely cover the loadings that will be suffered by the material in the industrial applications concerned, with regard to the constraints, the deformations, the deformation rates and, if appropriate, the temperature. For example, the computations of polycrystalline aggregates according to the method according to the invention make it possible to generate a biaxial tensile test TB and all the intermediate states between the points TB and TSO, TB and TS45 and TB and TS90 of FIG. 23. However, it is preferable to have an experimental biaxial test, and to use the aggregate calculations only to generate the intermediate states (interpolation). This biaxial test may be a biaxial tensile test or a test producing a biaxial state of the stresses in a portion of the test specimen, such as a notch tensile test, punching test or swelling of a sheet, or a compression test of a cylindrical disk. If the industrial part also undergoes stress states close to simple shear, point CS {0, 0, ao / 2} in FIG. 23, it will also be necessary to extend the experimental base in this direction by means of polycrystalline aggregate calculations and preferably perform shear tests to interpolate rather than extrapolate from the actual experimental base.

  The method according to the invention can be used to determine the impact of the method of manufacturing an object comprising a cold-forming deformation forming operation, such as stamping, and in particular to determine the mechanical properties of the object in each of its points. Indeed, the impact of a shaping operation by plastic deformation on the mechanical properties depends on the deformation actually performed and this deformation generally depends on the point considered. The mechanical properties after shaping are therefore not, in general, identical in all points of the object. It is then possible to use the evaluation of the distribution of the mechanical properties of the object obtained by the method according to the invention, to determine the service behavior of this object. The advantage of the process according to the invention is that this evaluation of the distribution of the mechanical properties of an object can be made under very diverse conditions from a limited number of mechanical tests. Indeed, to evaluate the mechanical properties of objects made of the same material, but made by shaping in various ways, it suffices to carry out the mechanical tests necessary for the construction of the polycrystalline model according to the invention, then to apply the method according to the invention to each of the particular cases envisaged.

  By way of example, the method according to the invention can be used to evaluate the impact behavior of a piece of motor vehicle obtained by stamping or bending or by any other forming process inducing plastic deformations. For this, using a calculation method such as a finite element calculation method, incorporating the reduced polycrystalline model representative of the material, the shaping operation is simulated and the properties determined. mechanics of the piece in each of its points. From this result, and using a suitable calculation method known to those skilled in the art, it determines, for example, the effects on the parts of a shock resulting from a collision. The method according to the invention can also be used to determine the behavior of a part consisting of a material that can exhibit a transformation of a sentence, that is to say a change in crystallographic structure under the effect of the thermomechanical stresses that it undergoes under the conditions of service or shaping considered. The reduced polycrystalline model whose parameters are determined by the method according to the invention is essentially a numerical model which can be implemented by any appropriate means, in particular by computer by integrating it into a known finite element calculation code. in himself. This model can be used in the same way in any method of calculating structural deformation and, in particular, in finite difference computation methods or in methods that involve only one point of the studied object such as is the case in the modeling of flat products rolling. The person skilled in the art knows how to make the necessary adaptations which relate only to the integration of the model into the calculation method. The process is independent of the polycrystalline model and the finite element and optimization software used. The process according to the invention has been described in detail for applications with anisotropy, with the example of rolled aluminum. But it is absolutely general and can be applied equally well to all the situations badly modeled by the macroscopic models, for example to the cyclic stresses, to the solicitations over long durations, type creep or relaxation, etc. The application examples have been described for materials having a face-centered cubic crystallographic structure. But the invention is not limited to these examples and one skilled in the art will be able to apply it to other crystalline structures and to various textures. It also applies to mixed structures made of crystals of different natures. For that, it suffices to take into account grains of representation of which a part corresponds to a first type of crystal and another part corresponds to a second type of crystal. In summary, the method according to the invention makes it possible: to extend an experimental database relating to a material by interpolation and if necessary extrapolation from a limited number of suitably chosen mechanical tests, using if necessary to polycrystalline aggregate calculations, - determine a polycrystalline behavior model to represent the behavior of the solid material, adjust the parameters of the behavior model on the extended basis of experimental data, identify the effect of the mechanical and possibly thermal stresses to which the object is subjected during the period of time considered, and calculate the effect of the stresses at each point of the object by using a calculation method incorporating the model of behavior of the material. The method can be applied to any type of material whose behavior can be represented by a model of the polycrystalline type, and in particular to any metallic material.

The stresses to which the object is subjected, which may be forces or displacements distributed over space and time and which may be supplemented by thermal stresses, may be determined by any means and in particular by measurement recordings. .

The calculation methods used by the method are numerical computation methods implemented by computer with the aid of adapted software. Of course, the invention is not limited to the examples just described.

Claims (23)

  1. A method for determining a polycrystalline model for representing the behavior of a solid material subjected to stresses, said solid material being represented by a plurality of so-called representation grains, grouping grains of crystalline orientations close to said solid material said polycrystalline model using: a crystalline model representing the behavior of a representation grain, a texture representative of the texture of said material, said texture comprising at least one orientation and a volume fraction for each of said representation grains; said method comprising at least one iteration of the following operations: determination, by simulation by means of said polycrystalline model of a stress on said material, of said theoretical behavior of said material; comparing said theoretical behavior with a behavior, said real, representative of said material obtained from a base of mechanical tests; readjustment of said polycrystalline model according to said comparison, said readjustment comprising a modification of the number of grains of representation, or of a parameter of said texture. 25   2. Method according to any one of the preceding claims, characterized in that it further comprises an initialization step, said step comprising: an initialization of the number of representation grains used by the polycrystalline model, an initialization of the texture, said initialization comprising a choice of at least one texture or at least one combination of textures, and an initialization of at least one parameter of the crystalline model.   3. Method according to any one of the preceding claims, characterized in that it comprises a determination of the number of grains of representation representing the solid material.   4. Method according to any one of the preceding claims, characterized in that the number of grains of representation is less than or equal to twenty. 10   5. Method according to any one of the preceding claims, characterized in that it further comprises a determination of the texture most representative of the texture of said material.   6. Method according to claim 5, characterized in that it comprises a determination of an orientation and / or a volume fraction of at least one grain of representation.   7. Method according to any of the preceding claims, characterized in that it further comprises a determination of the crystalline model representing the behavior of a grain of representation.   8. Method according to any one of the preceding claims, characterized in that it further comprises a determination of a scale-change model, said scale-change model comprising a homogenization model for the passage. from the scale of the grain of representation, called the microscopic scale, to the scale of all the grains of representation, called the macroscopic scale, and a model of location for the passage from the macroscopic scale to the microscopic scale. 30   9. A method according to claim 8, characterized in that the localization model is an anisotropic model involving tensors of order 4.5-.   10. Method according to any one of the preceding claims, characterized in that the crystalline model is a model of microscopic behavior such as twinning.   11. Method according to any one of the preceding claims, characterized in that the behavior of a representation grain is determined from a plurality of sliding systems associated with the solid material.   12. Method according to any one of the preceding claims, characterized in that it further comprises a determination of a pattern of evolution of the texture, said model comprising a calculation of a rotation of the crystal lattice of each grain of representation with respect to a predetermined mark.   13. Method according to any one of the preceding claims, characterized in that the mechanical test base is a test base, said extended, obtained by enriching a base of real mechanical tests with at least one mechanical test. , said theoretical, obtained from said real mechanical tests.   14. The method of claim 13, characterized in that at least one theoretical mechanical test is obtained by interpolation or extrapolation of the actual mechanical tests.   15. Method according to any one of claims 13 or 14, characterized in that at least one theoretical mechanical test is obtained, by simulation of a mechanical stress on a polycrystalline grain aggregate, modeled in finite elements, and whose texture is representative of said solid material, the behavior of each of said grains being represented by a predetermined crystalline model.   16. Method according to claim 15, characterized in that it further comprises a determination of the crystalline model, said determination comprising at least one iteration of the following operations: determination, by simulation on the polycrystalline aggregate, of the so-called simulated behavior of said material ; comparing said simulated behavior with actual mechanical tests; readjustment, according to said comparison, of said crystal model representing the behavior of each of said grains of the polycrystalline aggregate.   17. The polycrystalline model as defined in any one of claims 1 to 16.   18. Use of the polycrystalline model according to claim 17 for determining the effects of a thermomechanical stress on a piece made of a solid material.   19. Use according to claim 18, characterized in that it comprises a determination of deformations and / or stresses generated by the thermomechanical stress at a plurality of points of the part by means of a finite element model of said part .   20. Use according to any one of claims 18 or 19, characterized in that the piece of solid material is a metal part.   21. Use according to any one of claims 18 to 20, characterized in that the thermomechanical stress comprises a stress associated with a plastic deformation forming operation such as stamping, drawing, cutting, drilling, rolling or forging.- 45 -   22. Use according to any one of claims 18 to 21, characterized in that the thermomechanical stress comprises a load generated by the use in service of the metal part.   23. Use according to claim 22, characterized in that the effect of a stress generated by the use in service of the part comprises a phenomenon of creep and / or fatigue.