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    • 专利摘要:
    The invention relates to a method for estimating (100) the stress intensity factor (FIC) in a numerically modeled part, in the context of fatigue crack propagation modeling, comprising the following steps: - (E2 ): obtaining, from a numerical modeling of the part to be analyzed (20), a plurality of values simulated at different points of the part to be analyzed (20), (E3): determination of a converted value ( ΔKglobeff) of effective stress intensity amplitude corresponding to a straight edge of a plane crack and secondly of a length (a) converted from the crack corresponding to a straight-sided plane crack, said converted values being determined by equalization of the energy dissipated in the three-dimensional crack of the numerical modeling and the energy dissipated in the crack of a standard straight-sided crack model, - (E4): Converted value interpolation s of amplitude of effective stress intensity factor between two successive converted lengths (a), - (E5): Storage of converted values of effective stress intensity factor magnitude and interpolated.
    公开号:FR3052891A1
    申请号:FR1655705
    申请日:2016-06-20
    公开日:2017-12-22
    发明作者:Moura Pinho Raul Fernando De；Didier Jose Diego Soria
    申请人:SNECMA SAS；
    IPC主号:
    专利说明:

    Method for estimating the stress intensity factor and method for calculating the associated lifetime
    GENERAL TECHNICAL FIELD The invention relates to the analysis of crack propagation in mechanical parts. These parts are primarily intended for aircraft, but can be any mechanical component. The propagation is defined in a frame of propagation by fatigue, with a series of loading cycles.
    In particular, the invention relates to methods and systems for simulation, determination and interpolation of stress intensity factor (hereinafter FIC) of a digitally modeled part that is to be analyzed. The invention also relates to methods and systems for calculating the lifetime of the part to be analyzed.
    The FIC is broken down into three sizes, denoted Kl, KII and KIII, corresponding to the modes respectively of crack opening, plane shear and anti-plane shear.
    This description will be made for Kl but remains applicable to other quantities.
    Digital methods of crack propagation are efficient. For example, extended finite element (XFEM) or conformal cracking methods reliably predict crack propagation paths as well as calculate FICs along a crack front.
    However, the calculation of the crack propagation life is generally not integrated in the codes of trade or incompatible with the industrial requirements. This is why it is customary for the aeronautics industries to develop their own lifetime calculation code, the operation of which is based on specific post-treatments of the results resulting from numerical simulations of crack propagation.
    For a given crack dimension and at a given point of the crack front, the FIC is the magnitude characterizing the intensity of the loading in the vicinity of the crack tip. Depending on the magnitude of this magnitude during a fatigue loading cycle, the crack will spread more or less quickly. It is the role of numerical crack propagation simulations to provide FIC values along the crack front and for different crack lengths. From this list of values, the life calculation codes must be able to predict the crack propagation lifetime. Nevertheless, for reasons of computational capacity, the propagation steps, that is to say the iteration of the numerical simulations, are not taken too small, which means that not all the lengths of crack are simulated.
    In order to carry out the life calculation, the FICs must therefore be interpolated between the simulated propagation steps. It is therefore essential to have an efficient interpolation method so that the predicted lifetimes are relevant.
    STATE OF THE ART
    It is recalled that one places oneself within the framework of a fatigue crack modeling, that is to say a modeling of the evolution of a crack when several charge cycles are applied.
    The 3D cracking methods, generally obtained by finite elements, make it possible to obtain the FIC along a crack front at each propagation step as well as the geometrical information that characterize it (coordinates of the nodes of the crack front and faces and nodes associated with one of the faces of the crack). In practice, only the maximum Fie along the crack front for each propagation step (see Figure 1 the single point PI, P2, P3, P4 on each of the curved lines which represents each crack front Fl, F2, F3 , F4 at a different propagation pitch) is involved in the calculation of the service life. To use these FIC values, it is useful to match them with a corresponding crack length.
    Subsequently, the points between each tabulated value are obtained using a linear interpolation between the points of the tab.
    There are two major disadvantages to this interpolation method of the CIF.
    The first concerns the fact that the FIC entered in the form is the maximum FIC along the crack front for each propagation step. In other words, it is not necessarily the FIC corresponding to the same point of the front that is indicated (the points in Figure 1 relate to different locations of the crack front). With this method, we do not take into account the three-dimensional geometry of the crack and its evolution over time, which necessarily impacts the evolution of the FIC.
    The second drawback concerns the fact that this method does not make it possible to check whether the propagation step is sufficiently fine. It has been shown that if the propagation pitch is too coarse, the crack propagation lifetimes may no longer be conservative.
    There is also a last weakness to this approach relating to the geometry of the crack that must be considered, which can be complex (bifurcation for example). Indeed, the crack is an inherently three-dimensional construction and the problem of defining the "crack length" arises. This greatly limits the applicability of this method.
    There are theoretical methods for calculating crack propagation.
    For example, the weight functions method makes it possible to calculate FIC stress intensity factors from the stress field of the non-cracked structure and the crack geometry (references [1], [2] and [3]. ] for example - the references are given at the end of the description). Nevertheless, this method is limited to simple cases which makes it unusable as a rule.
    The so-called "perturbative" methods are more complex mathematically speaking, but allow to semi-analytically determine the cracking behavior as well as the stability of the front (see reference [4]). However, the majority of developments apply only under the assumption of at least semi-infinite medium.
    PRESENTATION OF THE INVENTION The invention thus proposes a method for estimating the stress intensity factor in a numerically modeled part, in the context of fatigue crack propagation modeling, comprising the following steps: - (E2 ): obtaining, from a numerical modeling of the part to be analyzed, a plurality of values simulated at different points of the part to be analyzed, said plurality of simulated values comprising, for different simulated propagation steps of a crack three-dimensional model that appears on the numerical modeling of the part, a set of simulated values of the stress intensity factor at associated points, the position of these points, and data relating to the cracked surface (coordinates of the nodes and faces of the elements three-dimensional defining one of the faces of the crack, the faces of the elements can be stored in the form of a table of conn ectivity), - (E3): determination, for each propagation step and for the set of values simulated for it, on the one hand, of a converted value of amplitude of the global equivalent effective stress intensity factor corresponding to that of a straight-edge plane crack and secondly of a converted length of the equivalent crack considered, said converted values being determined by equalization of the energy dissipated in the three-dimensional crack of the numerical modeling and the energy dissipated in the crack of a standard right-sided flat crack model, the energies being themselves determined according to the stress intensity factors, - (E4): Interpolation of converted factor factor amplitude values equivalent global effective stress intensity between two successive (a) converted crack lengths, - (E5): Storage of the converted intensity factor amplitude values overall equivalent effective stresses thus interpolated with the associated crack lengths.
    Thus, we refer to a global process "thermodynamically" equivalent to that of a straight-edge plane crack. A notion of length having a physical meaning is generated, which makes it possible to overcome the limitations mentioned above. The invention also relates to the following characteristics, taken alone or in combination: the interpolation step implements piecewise linear interpolation; the interpolation step implements an interpolation by minimizing the energy of the curvature, - the interpolation step comprises the following substeps: interpolation using linear interpolation. interpolation by interpolation by curvature energy minimization, the two interpolations being interchangeable, calculating at least one magnitude representative of a difference in amplitude values of the stress intensity factor between the two interpolations, said difference being calculated strictly between two values of lengths of the crack corresponding to two successive steps, comparison of said magnitude representative of the difference with a predetermined threshold, if the magnitude representative of the difference is greater than the threshold, generation of a calculation instruction setpoint from the numerical simulation of the part to be analyzed of the values of the stress intensity factor (FIC) along the new simulated front and the values of the position of the crack, between the two steps successive, - the method comprises prior to the obtaining step (E2), a calculation step (El) implementing the finite element mulation (or extended finite elements) at successive steps of the evolution of the crack in the part to be analyzed, said simulation being carried out by the processing unit, - the digital data recovered at obtaining (E2) data are obtained by means of finite element simulation or extended finite element simulation, - for the determination step (E3), the first subset of data makes it possible to know the effective amplitude of the rate of energy restitution AG by the following relations:
    E * = E in plane stresses
    in plane deformations
    Where K1, Kii and Km are, respectively, the FIC coefficients corresponding to the crack opening, the in-plane shear and the anti-plane shear modes, E is the Young's modulus, μ is the shear modulus, E * is the equivalent Young's modulus, v is the Poisson's ratio, the second subset of the data allows to know the increment of cracked surface per unit of dSurf crack front length, these formulations make it possible to obtain the value dissipated energy:
    wherein this dissipated energy is equalized with the dissipated energy of the crack of a standard straight-sided flat crack pattern, expressed as:
    Where the glob notation is relative to this standard model, in which, thanks to a modelization by a law of Paris with the parameters of crack closure of Elber, the link between dSurP '°' and
    is done
    Where C and n are Paris coefficients, in which, by equalizing the two dissipated energies, we obtain
    in which, by means of the relations between AGeff and AKeff, one determines and in which one determines Surf ^ '° ^ that is to say the length a. The invention also relates to a method for evaluating the lifetime of a numerically modeled part, in which a calculation of the lifetime of the part involving the converted factor amplitude values is carried out. equivalent global effective stress intensity interpolated according to an equivalent crack length according to a method as described above. The invention also relates to a system comprising a processing unit comprising calculation means and a memory, the unit being configured to implement an estimation method or a method of evaluating the propagation life of crack by fatigue as described previously. The invention also relates to a computer program product configured to be implemented by a system as described above, and comprising instructions for initiating the implementation of an estimation method or a duration evaluation method. of life as described above.
    PRESENTATION OF THE FIGURES Other features, objects and advantages of the invention will emerge from the description which follows, which is purely illustrative and nonlimiting, and which should be read with reference to the appended drawings, in which: FIG. FIG. 2 illustrates a system for implementing the invention; FIG. 3 illustrates a part to be analyzed; FIG. 4 illustrates two methods and associated embodiments according to FIG. FIG. 5 illustrates points converted into equivalence with a straight-edge plane crack from a three-dimensional crack, according to a theory of energetic equivalence by thermodynamic analysis. FIG. 6 illustrates these same points with two interpolation methods: piecewise linear and by minimizing the curvature energy; FIG. 7 illustrates the evolution of the polynomial interpolation with the degree of the polynomial; FIG. 8 schematically illustrates a straight-sided flat crack, - FIGS. 9a to 9c illustrate ID-shaped functions resulting from a generalization of the finite elements and used to calculate the curvature energy; FIGS. 10 to 10c illustrate a change of representation to smooth metrics.
    DETAILED DESCRIPTION
    Referring to FIG. 2, a constraint intensity factor (KIC) value interpolation system K of a piece 20 to be analyzed, which is modeled numerically, is shown. As mentioned in the introduction, FIC K is broken down into three data K1, KII, KIII. The description will be given for Kl only.
    The part to be analyzed 20 is a part intended for aeronautics, for which the service life must be able to be estimated. The part 10 is typically a compressor or turbine fan blade, a motor disk, a flange, an engine attachment, a casing as illustrated in FIG. 3. This list is given for illustrative purposes, since, for the implementation of the process, the type of part does not intervene however.
    The system 10 comprises a data processing unit 12, for example a computer or a server, having calculation means 14 configured to implement a method which will be described in more detail below and, advantageously, having a memory 16. The computing means 14 may for example be a computer processor, microprocessor, microcontroller, etc.. The memory 16 may be for example a hard disk, a "flash" memory, or a delocalized storage space, of "cloud" type. The data processing unit 12 may also be adapted to implement numerical simulations, such as finite elements, of the part to be analyzed 20. The finite element simulation makes it possible to obtain, for each simulation step, data relating to said part and crack propagation.
    In particular, it is possible to obtain for each propagation step, the values of the FIC at associated points and the position of these points, generally along the crack front. The positions include for example the coordinates of the mesh nodes defining the crack front. It is possible to obtain data relating to the cracked surface: in addition to the coordinates of nodes, the faces of the three-dimensional elements defining one of the faces of the crack. These data are stored in the form of a connectivity table, conventionally known in finite elements. Other data, such as the mesh of the cracked surface as well as the front can also be obtained.
    With reference to FIG. 4, a method 100 of interpolation of the values of a FIC of a three-dimensional crack simulated by finite elements will be described.
    This method 100 is also advantageously used to improve the interpolation of the FIC. This will be described later.
    In a first step E1, a numerical simulation is performed on the part to be analyzed 10. This simulation makes it possible to obtain the aforementioned data.
    In a preferred embodiment, this simulation is done by finite elements or extended finite elements.
    This step E1 is performed either by the processing unit 12 of the system or by another system.
    In both cases, a step E2 of receiving said data (generated during step E1) by the processing unit 12 of the system is implemented. Specifically, for each simulated crack front, a first subset of data is defined that contains the FIC values at associated points on the crack front and their respective associated positions, and a second subset of data that contains the data relating to the cracked surface (the coordinates of the nodes defining the crack front as well as the surface mesh of one of the faces of the crack, for example the table of connectivity of the faces of the three-dimensional elements concerned) as they have been defined previously. Having a plurality of simulated fronts at different steps, thereby obtaining a plurality of first and second subsets of data.
    This is the first step of the actual interpolation process, since the simulation step E1 is not necessarily implemented expressly for subsequent interpolation.
    In step E3, the problem of three-dimensional crack geometry is reduced to a straight-edge plane crack problem. The straight-edge plane crack is a standard model known to those skilled in the art. For this, a thermodynamic equivalence based on an equality of energy dissipated between the two modelizations is carried out. The physical foundation is explained below.
    The first principle of thermodynamics, for a transformation of the infinitesimal crack front gives, expressed in energy density per unit length of crack front:
    The second principle of thermodynamics makes it possible to express the irreversibility of transformation:
    The dissipated energy density is always positive. The term dSurf represents the cracked surface increment per unit of crack front length during the infinitesimal transformation. This quantity is homogeneous to a length: it is a kind of increment of crack length in the thermodynamic sense of the term. It thus appears a "physical" magnitude that can be considered to perform an interpolation that has a physical meaning.
    The above elements make it possible to make the initial three-dimensional cracking problem equivalent to a planar plane straight-edge plane crack propagation problem. For more generality, we put ourselves in the case of the elasticity adapted, that is to say that the material has already been able to undergo a plastic deformation initially, but the crack propagation is done under a cyclic load of fatigue in elasticity.
    The temperature field may change over time but the temperature field does not vary spatially. Therefore, a fatigue cycle is associated with a temperature.
    A modeling hypothesis consists in considering that the law of propagation of crack is a law of Paris with correction of Elber (references [5], [6]), given by the following equation:
    with C (T) and n (T) are the coefficients of the law of Paris and a (T) and b (T) are the parameters of the Elber crack closure law, ΔΚ is the amplitude of the FIC, which makes it possible to get rid of the load ratio (that is to say of the ratio R = Kmin / Kmax) and 4 / ^ // is the effective amplitude of the FIC, which takes into account the closing effect of the crack. The link between ΔΚ and 4 / ^ // was established by Elber. The cracked dSurf surface increment per cycle is now related to the magnitude of the effective stress intensity factor
    There is a relation between K (or equivalence) and the rate of energy restitution G (or the amplitude of said effective rate AGgff by equivalence), given by the following relations:
    E * = E in plane stresses
    in plane strains where E is the Young's modulus, E * is the equivalent Young's modulus, μ is the shear modulus, v is the Poisson's ratio.
    The Fie is known at each point of the crack front and for each propagation step by means of the numerical simulation of E 1 and recovered in step E 2.
    By replacing the values by their amplitude in the previous relations, we obtain, by definition, a relation between the rate of restitution of energy and the amplitude of the factor of intensity of the effective stresses (s being the abscissa curvilinear along the crack front):
    The energy dissipated during the cracking process is obtained by the following relation (remaining in the assumption of infinitesimal transformation, thus considering only one loading cycle):
    The dSurf is known using the data relating to the cracked surface calculated by the numerical simulation in step E1 and retrieved in step E2.
    By postulating an equivalence of the 3D cracking problem with a straight-edge plane cracking problem, we can equalize the dissipated energies and the crack front lengths. Therefore, it is legitimate to write:
    The notation "glob" means that the data is specific to the modelization in plane cracking with right face.
    The plane straight-edge cracking also respects the law of propagation of Paris. We therefore obtain that
    By equalizing the two dissipated energies and rewriting the equation, we obtain the following relation
    Now dSurP '°' '/ dN = dSurP' ° '' (since dN = 1 here, since only one cycle is considered) is homogeneous to a crack length. Therefore, its integration over time between the first cycle and the last cycle provides an equivalent crack length SurP '° ^ denoted length a, in the thermodynamic sense. We obtain a length that has a physical meaning.
    The previous relationship established between
    and
    allows to inject the amplitude of the effective FIC in the previous equation.
    In this way, a relation is obtained between the amplitude of the global effective equivalent FIC in the plane-straight-edge crack modeling as a function of the length of the crack SurP ', that is to say the length a. These data can then be interpolated.
    This physical construction is valid in all generality. However, it requires knowing all the information of the problem for each loading cycle which is generally too expensive in terms of computation time and memory space for the safeguarding of all the results (at least a thousand cycles should to be simulated). Finite element calculations are often at fixed maximum propagation steps along the crack front, or preintegrate over a given number of cycles (10, 100 or 1000, for example). This amounts to de-refining the discretization in no propagation to reduce the calculation time.
    Therefore the law of propagation is replaced by:
    where coeff is the coefficient that allows the pre-integration of several cycles.
    This amounts to modifying the coefficients of the law of Paris C and n by the coefficients Ceq = C. coeff and Ugq = n.
    The link between dSurffg "* and dSurfs * ° * ^ is
    so the following:
    In addition, the term dSurffs) is replaced by the incremented surface between two simulated propagation steps dSurfgqis) (thus not infinitesimal).
    In conclusion, the above reasoning applies approximately by performing the previous replacements and the equivalent crack length is always deduced in the same way.
    Thus, in step E3, from the plurality of first and second subsets of data retrieved in step E2, that is to say the data of the FIC along the various cracks fronts simulated in FIG. different steps and data relating to the cracked surfaces, and by means of what has been explained above, the overall effective amplitude of the FIC and the length a of the associated straight-edge plane crack are determined. This determination is implemented by the processing unit 12.
    For a given crack front, the simulated FIC data make it possible to know K (first subset of data) and using the preceding equations to calculate the data and the surface cracked surface dSurf ^ q (second subset of data) make it possible to calculate the length a. Due to the plurality of data corresponding to each edge, a plurality of pairs (a, ΔΚ ^ ° ^) is obtained. The equivalence obtained can also be applied to a straight-edge plane crack. The following relation shows that the system remains invariant:
    From this follows the fact that the cracked surfaces are the same, so the final system is identical to the initial problem. This proves the consistency of the method based on a thermodynamic analysis of the problem.
    FIG. 5 illustrates in the plane the representation of the pairs (a) The references F1, F2, F3, F4 are given with reference to FIG. 1, to signify to which front of the original crack the points of the graph correspond. This is a writing abuse.
    It is now recalled that the goal is to be able to determine Fie for crack states that have not been simulated numerically, that is to say for states that occur strictly between two successive steps. For this purpose, in a step E4, an interpolation of the function of the length a is performed. The interpolation is performed by the calculation means 14 of the processing unit 12. The interpolation can be performed according to several methods. We prefer interpolations that do not create information to the problem or add a minimum. For this, an IL linear interpolation or a minimization interpolation of the IMC curvature energy are suitable. Both interpolations can also be performed, as they may have different applications.
    Figure 6 illustrates these two interpolations. The minimization interpolation of the overall curvature energy can be seen as a method based on physical considerations specific to the cracking problem. Indeed, the scientific literature on the subject shows that if a crack propagates without connecting, then the evolution of the stress intensity factor with the crack extension can be represented by an infinitely differentiable function. This means that calculating the curvature of such a function makes sense.
    In addition, minimizing a curvature energy occurs in a few physical situations, for example in the physics of soap bubbles or more generally in the physics of membranes.
    Figure 7 illustrates a linear interpolation between three points. This example shows that it is possible to better and better interpolate with polynomials. The approximation will be all the better if the degree of the polynomial is high. It is also possible to demonstrate that the maximum difference between the linear interpolation and the polynomial approximation tends to zero when the degree of the polynomial tends to infinity. Thus, linear interpolation can be seen as the passage to the limit at infinity that one would apply to an infinitely differentiable function.
    This approach confers a "physical" character to piecewise linear interpolation rather than interpolation based on the minimization of curvature energy. To carry out an interpolation consists in supplementing the missing information with a principle which must at least respect the "physics" of the problem considered, that is to say not to introduce elements which would violate certain fundamental equations to which the problem obeys. It is for this reason that interpolation by minimizing curvature energy is a "physical" method because it does not contradict the theory of fracture mechanics. The method consisting of piecewise linear interpolations is not physical at first, but can be seen as the transition to the limit of a global polynomial interpolation method which would be physical and which by extensions would make the method of Piecewise linear interpolation almost "physical" as well.
    Another point of view concerning the linear interpolation method would be to see it as a method of minimizing the distance between points. This way of thinking then places it in the category of optimal methods. Linear interpolation minimizes the distance while the other minimizes the curvature energy under the constraints of the positions of the points to be connected. Minimizing distances can be seen as a way of connecting points without introducing additional information to the initial problem and this results in a continuous field. In the case of the second method, at least one additional information is introduced which is that the field is regular. In this case, the minimization of the curvature energy is an optimal method in the sense of minimizing the physical information to be introduced whereas the method of linear interpolation is optimal in the sense that the purely mathematical information is minimized.
    Finally, in a step E5, after the interpolation step, the interpolated data are stored, in order to be able to access it for other applications. The storage is typically in the memory 16. It is generally referred to as "form". The form is a table that gives the stress intensity factors according to the length of the crack.
    Consequently, the interpolation step E4 generates a file, in the form of a text format or a table containing the form, that is to say regrouping the converted data and the converted interpolated data. Step E5 consists in storing this file.
    Indeed, as indicated in the introduction, obtaining the form is generally used to calculate the lifetime of a part. Therefore, with reference to FIG. 4, a method 200 for determining the life of the part to be analyzed 20 is described.
    In a step E'I, the processing unit 12 receives the form calculated in step E4 and / or stored in step E5 of the previous method. This recovery step may simply consist of having access to the memory 16.
    In a step E'2, a method for calculating the lifetime of fatigue crack propagation of the part is implemented. Document [7] describes such a method.
    It is sufficient to use an existing code and replace the Fie form with that proposed in this invention to determine the crack propagation lifetime.
    This method 200 can be implemented by another system than that described above.
    Refinement of the error
    When all the information concerning the problem to be solved is available, the result of the interpolation will always be the same whatever the interpolation method used and which respects all the "physical" constraints of the problem. Typically, on the intervals II of FIG. 6, the interpolation appears intuitively as being of quality; on intervals 12, it appears intuitively as of lower quality.
    It is thus possible to see the discrepancies between two optimal interpolations called "physical and mathematical" as a means to identify a lack of information that would be useful to improve the interpolation or the degree of confidence in the interpolation.
    This consideration allows for the implementation of a relevance verification step, illustrated in Figure 6.
    In two interchangeable substeps E41 and E42 of the interpolation, piecewise linear interpolation and curvature energy interpolation are performed.
    Note that strictly between two values of length corresponding to two successive steps simulated, that is to say strictly in interpolated area, there are differences δ value between the two interpolations.
    In a substep E43, at least one of these differences δ is calculated, which is compared, in a substep E44, with a predetermined threshold value VSP.
    The predetermined threshold VSP is chosen according to the quality of the desired interpolation.
    It is also possible to compare a value of these differences for each zone between two successive steps, or to compare an average, or a maximum value, etc. We will therefore speak more generally of a magnitude δr representative of the difference. This magnitude at r indicates that there is a difference that could be quantified by the functions mentioned above.
    Finally, a comparison sub-step E45 between the representative magnitude δr and the VSP threshold is performed. If the representative magnitude δη is smaller than the VSP threshold, we can consider that the two interpolations are of quality and that the data, if they had been simulated, would be close to the two interpolation values.
    If the representative magnitude δη is greater than the threshold VSP, then it can be considered that there is too much uncertainty on the interpolation, and that obtaining new simulated values is useful or even necessary. For this purpose, a calculation instruction setpoint is generated from the numerical simulation, typically by finite elements (or extended finite elements) of the part to be analyzed, of at least one value of the FIC stress intensity factor. and the position of the crack front between the two successive steps. In other words, the processing unit 12 sends a setpoint to perform part of the step E1, with a reduced step for at least one simulation.
    We thus speak of refinement of the discretization in crack length. By way of example, a predetermined threshold value VSP of 2% may be suitable. The criterion depends on the specifications.
    We can for example decide that 5r = δ and that as soon as 6r / vc ax {AKf ^ ° ^ among linear interpolation and curvature)> 2% for a value of length a given, a re-calculation instruction of the simulation is generated.
    Finally, the storage step E5 may comprise the storage of the two interpolations and quantities representative of the differences 5r and / or the differences δ. APPENDIX 1: Example of interpolation by minimization of the curvature energy The interpolation by minimization of the curvature energy gives coherent physical predictions only in the context of a planar plane straight edge crack, such as illustrated in FIG. 8, where σ represents the stress and the length of the crack. However, the process described above makes it possible to thermodynamically consider any three-dimensional crack as a crack satisfying these properties.
    To perform this Interpolation, it is necessary to determine the curvature energy.
    We define the function f in the following way (x is the "virtual" crack length, called a before):
    The need to introduce "virtual" crack lengths is explained in APPENDIX 2
    The function f is written as a linear combination of polynomial functions by piece (of degree 5 to ensure a regularity C2 over the entire range of evolution of the crack length "virtual", it is a generalization of elements conventional finishes) reference given below:
    The first index indicates the number of the node on which the nodal value is taken.
    The second index indicates the nature of the nodal value: 1: Nodal value of the stress intensity factor (CIF)
    2: Nodal value of the derivative with respect to the FIC 3 z variable: Node value of the second derivative with respect to the FIC z-variable The one-dimensional element associated with these shape functions is defined in such a way that the first node is located at z = -1 and the second at z = 1. The curvilinear abscissa associated with the curve giving the evolution of the factor of stress intensity as functions of the crack length "virtual" is given by:
    Πη nnl-pra!
    Note that the degrees of freedom
    and
    are known. Optimization will focus on other degrees of freedom. From where we get (where Lx is the length of the virtual element):
    The tangent unit vector is obtained as follows:
    FIGS. 9a to 9c illustrate the values of the functions of shapes C11, C12, C13, C21, C22 and C23 and their derivative and second derivative. Note that these curves check their specific constraints (0 or 1) at the nodes.
    We can now calculate the unit vector tangent to the curve K ,. The formulas given previously lead to:
    There remains only the calculation of the curvature. For this, the derivative of tx with respect to the curvilinear abscissa is calculated:
    To determine the derivative terms with respect to the curvilinear abscissa, the differential transition formulas from the curvilinear coordinate to the virtual coordinate are expressed:
    This matrix equation is invertible, which makes it possible to obtain:
    We notice that Diffx "^ = Diffs ^, which makes it possible to determine
    and
    easily. Finally, we find:
    With:
    The curvature is deduced automatically:
    The Diffx matrix must now be expiicted. For this, the following standard form of 2x2 matrix inversion is used:
    with det (A) = a. d - b. vs
    So we have :
    The function we are trying to minimize is the following (if the maiiiage consists of n-1 element):
    A Newtonian algorithm is then used preferentially to determine the parameters that minimize this function. It is thus necessary to know the Hessian gradient vectors and matrices (with respect to the unknown variabies) of the quantities involved in the optimization probeme.
    This algorithm can be implemented by the processing unit 12. APPENDIX 2: Variable metrics method
    Figure 10a illustrates the points of the front at different propagation steps. Note that the distance between two same points of the front is not constant between two steps, which means that the length scale associated with each element is different. We are talking about different metrics.
    Having elements with different metrics (therefore discontinuous at the point of intersection) while imposing a regularity connection C2 between the elements leads to numerical instabilities.
    Another point of view is that the elements do not have the same contributions to the functional (because of their different dimensions) while they all carry the same amount of information. Whatever the point of view, this can induce a numerical bias that must be treated advantageously.
    To solve this problem, it suffices to place oneself in a virtual representation space in which the interpolation elements all have the same dimensions. For example, if the discrete crack lengths are χ e {0,1; 0.2; 0.4; 0.6; 0.8}, we will take ^ virtual ^ (Ij 2; 3; 4; 5; 6; 7; 8; 9; 10; 11).
    To go from the virtual representation to the real representation, we use a bijection between the two marks (see Figure 10b).
    We use the same interpolation functions as before. The correspondence between the two spaces of representation is done by imposing the degrees of freedom // ^ and each node. Then, to have a metric that evolves in a "soft" way between each element, the process presented previously is implemented. This makes it possible to obtain a smooth correspondence between the two representation spaces as represented in FIG. 10c. References: [1] Bueckner HGZ (1970), "A novel principle for the computation of stress intensity factors", Angew, Math. Mech, Vol. 50, p. 529-546.
    [2] Rice J.R. (1972), "Some remarks on elastic crack-tip stress fields", Int. J. Solids and Structures, 8, 751-758.
    [3] Rice J.R. (1989), "Weight function theory for three-dimensional elastic crack analysis". Fracture Mechanics: Perspectives and Directions (Twentieth Symposium), ASTM STP 1020, R. P. Wei and R. P. Gangloff, Eds., American Society for Testing Materials, Philadelphia, pp. 29-57.
    [4] Lazarus V. (1997), "Some three-dimensional problems of the mechanics of brittle fracture". Thesis University Paris 6.
    [5] W. Elber. "The significance of fatigue crack closure". ASTM STP, 486: 230-242, 1971.
    [6] PC. Paris, F. Erdogan, 1963, "A critical analysis of crack propagation laws". J Basic Eng 85, pp 528-534.
    [7] "Nasgro - Fracture Mechanics and Fatigue Crack Growth Analysis Software - Reference Manual", Version 6.2, 2011.
    权利要求:
    Claims (10)
    [1" id="c-fr-0001]
    claims
    A method for estimating (100) the stress intensity factor (FIC) in a numerically modeled part, in the context of fatigue crack propagation modeling, comprising the following steps: - (E2): obtaining from a numerical modeling of the part to be analyzed (20), a plurality of values simulated at different points of the part to be analyzed (20), said plurality of simulated values comprising, for different simulated propagation steps d a three-dimensional crack that appears on the numerical modeling of the part, a set of simulated values of the stress intensity factor (FIC K) at associated points, the position of these points, and data relating to the cracked surface of the crack, - (E3): determination, for each propagation step and for the set of values simulated for it, of a part of a converted value (ΔΚ ^^^ * ') of amplitude of the factor of intensity of co equivalent global effective stress corresponding to that of a straight-edge plane crack and secondly of a length (a) converted from the equivalent crack considered, said converted values being determined by equalization of the energy dissipated in the three-dimensional crack numerical modeling and dissipated energy in the crack of a standard right-sided flat crack model, the energies being themselves determined according to the stress intensity factors, - (E4): Interpolation of values Equivalent global effective stress intensity magnitude factor converts between two successive converted (a) crack lengths, - (E5): Storage of equivalent global effective stress intensity factor converted magnitude values thus interpolated with the associated crack lengths.
    [2" id="c-fr-0002]
    The estimation method according to claim 1, wherein the interpolation step (E4) implements piecewise linear interpolation.
    [3" id="c-fr-0003]
    3. Estimation method according to claim 1, wherein the interpolation step (E4) implements interpolation by minimizing the curvature energy.
    [4" id="c-fr-0004]
    4. Estimation method according to claim 1, wherein the interpolation step (E4) comprises the following sub-steps: - (E41) interpolation using a linear interpolation, - (E42) interpolation to using interpolation by minimizing the curvature energy, the two interpolations being interchangeable, - (E43) calculating at least one magnitude representative of a difference (5r) of amplitude values of the factor of Equivalent global effective stress intensity (FIC) between the two interpolations, said difference being calculated strictly between two length values of the crack (a) corresponding to two successive steps, - (E44) comparison of said magnitude representative of the difference (5r ) with a predetermined threshold (VSP), - (E45) if the magnitude representative of the difference (5r) is greater than the threshold (VSP), generation of a calculation instruction instruction from the numerical simulation of the part at a nalysing (10) values of the stress intensity factor (FIC) and the values of the position of the crack, between the two successive steps.
    [5" id="c-fr-0005]
    5. Estimation method according to any one of the preceding claims, comprising, prior to the obtaining step (E2), a calculation step (E1) implementing the finite element simulation at successive steps of the evolution of the crack in the part to be analyzed (20), said simulation being carried out by the processing unit (12).
    [6" id="c-fr-0006]
    An estimation method according to any one of the preceding claims, wherein the digital data retrieved at the obtaining step (E2) of the data is obtained by a finite element or extended finite element simulation.
    [7" id="c-fr-0007]
    7. Estimation method according to any one of the preceding claims, wherein, for the determination step (E3), the first subset of data makes it possible to know the effective amplitude of the energy release rate AG. by the following relations and their equivalence in matters of amolitude:

    E * = E in plane stresses f * = £ / (! - υ ^) in plane deformations

    Where Kl, Kii and Km are, respectively, the FIC coefficients corresponding to the modes of crack opening, plane shear and anti-plane shear, E is the Young's modulus, E * is the Young's modulus equivalent, μ is the shear modulus, v is the Poisson's ratio, the second subset of data allows to know the increment of cracked surface dSurf, these formulations make it possible to obtain the value of the energy dissipated:

    wherein this dissipated energy is equalized with the dissipated energy of the crack of a standard straight-sided flat crack pattern, expressed as:

    Where the glob notation is relative to this standard model. in which, thanks to a modelization by a law of Paris with the closing parameters of Rssure of Elber, the link between dSurf ^ '"" ^ and

    3St performed

    where C and n are Paris coefficients, in which, by equalising the two dissipated energies, we obtain

    in which, by means of the relations between AGeff and AKeff, one determines and in which one determines SurP '°' ', that is to say the length a.
    [8" id="c-fr-0008]
    8. A method of evaluating (200) the lifetime of a digitally modeled part (20), in which a lifetime calculation of the part (20) involving the converted values of a part (20) is performed. interpolated effective stress intensity factor amplitude according to a method according to one of claims 1 to 7.
    [9" id="c-fr-0009]
    9. System comprising a processing unit (12) comprising calculation means (14) and a memory (16), the unit being configured to implement an estimation method according to any one of claims 1 to 7 , or a lifetime evaluation method according to claim 8.
    [10" id="c-fr-0010]
    A computer program product configured to be implemented by a system according to claim 9, and including instructions for causing the implementation of an estimation method according to any one of claims 1 to 7, or a lifetime evaluation method according to claim 8.
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    优先权:
    申请号 | 申请日 | 专利标题
    FR1655705|2016-06-20|
    FR1655705A|FR3052891B1|2016-06-20|2016-06-20|METHOD OF ESTIMATING THE STRESS INTENSITY FACTOR AND METHOD OF CALCULATING THE LIFETIME OF THE ASSOCIATED|FR1655705A| FR3052891B1|2016-06-20|2016-06-20|METHOD OF ESTIMATING THE STRESS INTENSITY FACTOR AND METHOD OF CALCULATING THE LIFETIME OF THE ASSOCIATED|
    EP17742480.1A| EP3472736B1|2016-06-20|2017-06-20|Method for estimating stress intensity factors and method for calculating associated service life|
    RU2019101373A| RU2748411C2|2016-06-20|2017-06-20|Computer-implemented method for estimating the service life of a cracked part and a system for estimating the service life of a part|
    CN201780044396.1A| CN109478210A|2016-06-20|2017-06-20|Method for estimating the method for stress intensity factor and for calculating the service life used in connection with|
    PCT/FR2017/051633| WO2017220923A1|2016-06-20|2017-06-20|Method for estimating stress intensity factors and method for calculating associated service life|
    US16/311,320| US20190197211A1|2016-06-20|2017-06-20|Method for estimating stress intensity factors and method for calculating associated service life|
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