Towards an Optimal Reliability of Metallic Structures during Random Loadings

Transcription

1 Towards an Optimal Reliability of Metallic Structures during Random Loadings Joseph Zarka 1) and Habib Karaouni 2) Laboratoire de Mécanique des Solides, Ecole Polytechnique, Palaiseau Cedex, France ABSTRACT Our objective in this paper is to show how to reach a real physical approach of the multiaxial random loadings which is very easy to perform and which allows the optimal reliability of a structure. The main idea of the approach is to find an Equivalence rule between two complex loadings relative to "Damage" which may be used as a "Quantification" or norm of any loading relative to one particular material. This rule must have the physical meaning of "damage" and allows for the construction of simple and practical cyclic radial loadings and tools for fatigue analysis or accelerated fatigue tests. A new framework for fatigue analysis and reliability of structures has been built which, based on a multi-scale analysis, allows the analyis to be reduced to the analysis on a 2-D window with a Characterized radial cyclic loading. Here, it is given one description for the smooth specimen and any general structure with any loading conditions. Simplified analysis of inelastic structures are done but the results are coupled with results from returns on real structures. As our proposed procedure is very simple and fast, it is possible to look after the optimal reliability of the structure. In this paper, only variable amplitudes and random multi-parametered loadings have been considered. In a following paper, we shall treat the incertainties on geometry, materials, initial state with Monte Carlo Simulations and the integration of the previous rules. 1) Directeur de Recherches au CNRS 2) Doctorant Institut Ligeron S.A, now Chief of Department, CADLM

2 CLASSICAL MATERIAL CHARACTERIZATIONS This paragraph is just here to explicit what we want to keep to characterize the materials and to give the definitions of the constants. On a smooth tensile specimen, it is easy to perform the following characterizations: The cyclic curves at ~± 1% of deformation are obtained (this is the best choice since the first tensile curve is a function of the initial state of the material, but it implies that the material is cyclically stable) eventually for different temperature,. usually, E 2 = Ee + Ep = 2 2 Σ 2Y + Σ 1/n' G 2K ' (1) where K ' and n ' are constant. The Wöhler curves are constructed during cyclic uniaxial loading, associated to various mean radial stresses and probabilities of failure: Usually, they are represented by the stress-life equation: ' Σ a = σ f N b + d (2) or, more practically, by the strain-life equation: E 2 = σ f (2N) b ' + ε Y f (2N) c (3) G In this equation, E is the strain amplitude, 2N is the number of cycles to failure with the 2 ' constitutive constants for the material: σ f is the fatigue strength coefficient, Y G is the Young ' modulus, ε f is the fatigue ductility coefficient, c is the fatigue ductility exponent, and b is the fatigue strength exponent, d is a constant term. Σ max = Σ a + Σ m, is the maximum or peak stress, Σ a is the stress amplitude, and, Σ m is the mean stress. Then, to take into account the mean stress, the endurance diagrams are constructed. A modified Goodman's diagram is usually employed for a given number of cycles, N, in the plane (Σ m,σ a ), a bounded domain is defined: Σ ' = Σ a a 1 Σ m σ (Goodman) (4) u During complex uniaxial loadings with variable amplitudes, different deterministic mathematical representations are based on peak-valley or range-mean matrix type, to define groups of constant amplitude cycles, each with a fixed load amplitude and mean value, such as cumulative exceeding curves (with the same reduction or counting of cycles) or sequential variable amplitude histories (usually the rainflow method of counting the cycles is preferred). During each of these groups of constant amplitude cycles some "damage" is induced. Various rules are proposed to define and cumulate the "damage", such as the linear ones by Palgreen-Miner: D i = N i (5) N Ri and failure when D i = 1 (6) (where N i is the number of cycles realized during the cyclic loading i for which the number of cycles to rupture is N Ri ) or non-linear ones such as those by Lemaitre-Chaboche. But the "damage" factor (scalar or tensor) is still the object of many researches; it is impossible to measure it during the whole loading path, even if, before failure, some slight changes on the elastic properties may be experimentally detected.

3 Moreover, the classical representations of the loading and the counting methods are purely based on mathematical aspects and ignore the particular mechanical behavior of the present materials in the structure. The probabilistic representation (with the power spectra density to measure the load amplitude intensity in the frequency range) is often employed during random (stationary gaussian) uniaxial loadings; it is only combined with the linear cumulation rule for damage. Several experts are thinking that non gaussian uniaxial loadings are more realistic and need special treatments. During cyclic multiaxial loading (several loading parameters inducing out-of-phase stresses), the critical plane approach seems to be reasonable (NISA/Endure, manual): γ max (1 + n Σ n max ) = (7) σ y (1+ν e ) σ f E Nb + n 2 (1+ν e) σ 2 f E σ 2b y N + (1+ν p )σ f N c + n 2 (1+ν p) σ f N b+c σ y where γ max is the maximum shear strain on the maximum shear strain amplitude plane, Σ max n is the maximum stress normal to the maximum shear strain amplitude plane with the supplementary constitutive constants for the material, ν is the elastic Poisson ratio, σ y is the elastic limit or yield stress, ν p is the plastic Poisson ratio (0.5 for metals), n is a material constant to correlate the tensile data to the torsion data. For non-cyclic multiaxial loadings, endurance criteria such that proposed by Dang Van are usually used; they have the form: C a + α P a + β P mean < b (8) where C a is the deviatoric stress amplitude, P a is the amplitude and P mean, the mean value of the pressure (first invariant of the stress tensor), α, β and b are some phenomenological constants of the material. The Dang Van's criterium is for example, written locally: Σ eq = τ (t) + a DV.p(t) b DV (9) where τ (t) is the local shear stress on a critical slip plane, p(t) is the local pressure and a DV, b DV are constants identified from endurance limits during alternate flexion and torsion tests. These local quantities may be expressed from the global ones by making some simplifications: C alt +a DV P(t) I E = Max n Max t b < 1 (10) DV where C alt is the alternate part of the tangential stress and P(t) is the global pressure. OBJECTIVES OF THE PAPER There are still many open problems: 1. what is the validity of these Fatigue/Endurance criteria? 2. what is the real physical meaning of this(ese) Damage parameter(s)? 3. how to extract or represent the random loading? 4. what is the real physical meaning the cumulation rules for Damage? Three classical approaches are used during reliability analysis of structures: i) Experimental approach where it is needed to have representative specimen and its loading and with usually also a prohibitive cost;

4 ii) Probabilistic approach although only a very small number of cases, unknown cause of incidents, and final results which can not be verified; iii) Deterministic approach where it is needed to have exact representations of materials (plastic and fatigue properties), the exact geometry of structure and of its initial state of the structure, and the real loading paths on structures with the real «errors» during the simulation. Indeed, during a reliability analysis, i) we have to take into account the incertainties on geometry, material properties, loadings..., ii) we have no knowledge of the real initial state..., iii) and also, we can not control the errors during numerical simulations or measurements...! The objectives of this paper are to show how we propose to solve these various difficulties and how it may be possible to reach an optimal reliability of the structures although these several fuzzy aspects. REVIEW ON INELASTIC ANALYSIS OF STRUCTURES We are concerned by a general structure which is subjected to a general complex cyclic static or dynamical loading. It is well known that even if there is no collapse or ratchetting or excessive deformation, after a while, there will be shakedown and then failure of the structure by fatigue. We have proposed a very simple and practical framework for this analysis, (J. Zarka, J. Casier (1979)). LOADINGS ON THE STRUCTURE The structure has the volume Ω with the surface Ω. It is subjected to the following loading: on a part Uj Ω de Ω : given displacements U j d (t) on the complementary part Fi Ω de Ω : given surface loads F i d (t) in the volume Ω : specific load X d (t) in the volume Ω : initial strains E I (t). The full data U d (τ ), F d (τ ), X d (τ ), E I (τ ) for any τ [ 0, t] characterizes the loading path on the structure. PURELY ELASTIC RESPONSE At first the structure is assumed made of linear elastic materials. At any time t, the elastic response is given by : - the elastic displacement U el (t), - the associated strains E el (t) and - the elastic stresses S el (t) by using any tool ELAS which has the arguments: ELAS {Ω, Fi Ω, Uj Ω, X d, F i d, U j d, E I,M} (11) where M is the elastic coefficient matrix (M 1 = L) el (t) = L(E el (t) E I (t)) (12)

5 REAL RESPONSE There will be in the structure the fields: - U(t),E(t) Kinematically Admissible with U j d (t), on Uj W E(t) =1/2( U(t) + T U(t)) (13) - S(t), Statically Admissible with X d (t) in W and F j d (t), on Fi W S(t) = L(E(t) E I (t) E p (t)) (14) These fields may be splitted into one elastic part (previously analyzed) and one inelastic part: U(t) =U el (t) + U ine (t) (15) E(t) = E el (t) + E ine (t) (16) S(t) = S el (t) + R(t) (17) where U ine (t), E ine (t) = M R(t) + E p (t) K.A. with 0 sur Uj W, R(t) is S.A. with 0 in Ω and 0 on Fi W. These inelastic fields may be obtained, when the plastic strain filed is known, with: ELAS{ Ω, Fi Ω, Uj Ω,0,0,0,E p,m} (18) PARTICULAR CASE OF THE KINEMATIC HARDENING MATERIAL This material associated to the Mises criterium is classically defined by: f(s, y) = f(s y) =1/2 (S y) T 2 (S y) σ 0 (19) where S is the deviatoric part of the stress tensor and which defines in the stresses space the moving convex elastic surface C(y) = C 0 + y. (20) The internal parameters are here the plastic strain tensor; the transformed internal parameters y are linked to E p by: y =C E p E p =C -1 y (21) C is the work-hardening matrix. The normality flow rule is classically written symbolically: E.p (t) Ψ C(y) (S(t)) (22) i.e. external normals cone to C(y) at S. By using S S el + R the yield criterium may be rewritten f(s el + R C E p ) and then f(s el - Y) where Y = C E p R (23) is the structural transformed parameters field. It is obtained with the procedure ELAS (Ω, Fi Ω, Uj Ω, 0, 0, 0, E p, M) which means to U ine (t), E ine (t), R and then Y. The main point is that: since at any time S el is known independently of the previous load history, the Y is a locally known position C(S el ) = C 0 + S el!! Very often, it is possible to GUESS or to OBTAIN EXACTLY, the current position of the structural transformed parameters field Y, but it is, then, necessary to come back to the classical fields in order to characterize the eventual damage of the structure. So, if the field Y is assumed known, the same operator with the new arguments: ELAS (Ω, Fi Ω, Uj Ω, 0, 0, 0, Y/C, M' ) gives R and then E p.m ' is the inverse matrix of the modified elastic matrix L ' (L ' = M ' -1 ) given by: M ' = (M + C -1 dev) (24) The one to one relation in the structure between the 2 fields E p and Y is then elementary to expressed.

6 During monotonic radial loading, an ultimate state is reached and is constructed exactly. During cyclic radial loading, bounds are constructed with elementary rules. All these rules constitute the ZAC method. Even during contacts loadings such as successive rollings or impacts (which are not classical loadings) or dynamical loadings such as sismic loadings, it is possible to propose special simple rules. REVIEW ON THE ADVANCED INTELLIGENT DESIGN OF STRUCTURES (AIDS) AUTOMATIC LEARNING EXPERT SYSTEMS The engineers have to face very important problems in the design, the tests, the survey and the maintenance of their structures. These problems did not yet get full answers even from the best people in the world. Usually in these problems (such as no satisfactory constitutive modeling of materials, no real control of the accuracy of the numerical simulations, no real definition of the initial state and/or the effective loading of the structure ), there is no solution and the experts do not understand the problem in its whole. Moreover, the available data may be not statistically representative (i.e. are in limited number), and may be fuzzy, qualitative and/or missing in part. Other methods need to be used! Nowadays, automatic learning generators are able to automatically extract the rules from the raw examples base given by the expert. The experts know they do not know the full solution but they are able to build an examples base, for which the solution is known experimentally or numerically. Their main problem is to provide a good description of such an examples base. Basically such a automatic learning system includes five main functions : PREPARE: to transform the example files from user format (ASCII, Excel,...) into the own format of the system and to handle the discretization of the descriptors and the splitting of the initial data base into a training set and a test set. LEARN: to automatically extract a rules base from the training set according to the quality of available information (noise, sparseness of the training set,..) using Statistical, Symbolic, Numerical, Fuzzy logic, Neural Network or Genetic Algorithm learnings. TEST: to experimentally evaluate the quality of the extracted rule on the test set. INCLEAR: to allow the expert to visualize IN CLEAR the extracted rules with their format and to say what descriptors are the relievant pertinent descriptors which are kept. CONCLUDE: from the description of a new case, to deliver a conclusion based on the extracted rules. In all problems, it is necessary to consider one conclusion which may be a class or any continuous real number. Moreover, often, several conclusions may be considered together. The rules have to be automatically generated for each of these conclusions. Then an optional but fundamental sixth function, OPTIMIZE, based on genetic algorithms or other special optimization techniques (non convex function and non differentiable functions), may be used to solve the inverse problem i.e. when some conclusions and some descriptors have to belong to some given sets (or constraints) i.e the requirements, what are the possible solutions and in some particular cases what is the best solution if an objective function is given (cost, weight..)? Several automatic learning systems are now available. For example, at the Ecole Polytechnique (France), the LES program is based on symbolic learning and numerical

7 learning (authors M. Sebag and M. Terrien) but also the Wards systems Group at Frederick, (MD, USA), proposes the Neuroshell program wich is based on neural networks. These programs are all coupled with optimization add ons. GENERAL PRINCIPLE OF THE AIDS APPROACH Based on the automatic learning tools, we have defined a new approach, AIDS, where it is needed: i) To build a DATABASE of examples, to obtain some experimental, real or simulated results where the EXPERTS indicate all variables or descriptors which may take a part. This is, at first, done with some PRIMITIVE descriptors x, which are often in a different number and type for each example. Then, the data are transformed with the introduction of some INTELLIGENT descriptors XX, with the actual whole knowledge thanks to (but often insufficient) beautiful theories and models. These descriptors may be number, Boolean, string, name of file which give access to data bases, or treatments of curves, signals and images. The results or conclusions may be classes (good, not good...) or numbers (time, cost..). But for all examples, their number and their type are always the same, which is the only one way to allow the fusion of data coming from various tests or simulations. Usually, it is hoped to get ~50 to 500 examples in the data base with 10 to 1000 descriptors, 1 to 20 conclusions for each case. ii) To generate the RULES with any Automatic Learning Tool. Each conclusion is explained as function or set of rules of some among the input intelligent descriptors with a known reliability or accuracy. If this reliability is too low, that means that there is not enough data or we have bad or missing intelligent descriptors iii) To optimize at two levels (Inverse Problems). Considering the intelligent descriptors as independent; it is possible to get the OPTIMAL SOLUTION satisfying the special required properties and allowing the DISCOVERY OF NEW MECHANISMS, Considering the intelligent descriptors linked to primitive descriptors for a special family; it is possible to obtain the optimal solution that is technologically possible. So, not only a Practical Optimal Solution is obtained but also the Experts may learn the missing parts, may build models or theories based only on the retained intelligent descriptors and guided by the shapes of the rules or relationships. GENERAL DESCRIPTION OF MECHANICAL PROBLEMS Several tools were built to describe most of the mechanical problems. It is necessary to describe them in an "intelligent" way. MATERIAL DESCRIPTORS We use classical terms according the problem elastic coefficients, plastic properties, rupture and fatigue properties. All these descriptors are obtained on smooth specimen. FIELD DESCRIPTORS At any given time, any scalar field M varying between M min and M max may be characterized with: i) the interval [M min, M max ] is divided in n equal segments ii) the relative surfaces in the window for which the scalar M is included in the interval n, (S n /S 0 )

8 where S 0 is the surface of the domain. Very often, we assume that the stress field at a given time is represented by I1, its first invariant, J2, the second invariant of its deviatoric part and the norm of the gradient of this J2 (the problem being the fatigue life). It is also necessary to describe the evolution of the stress field with the time. Hablot (1990) introduced: - φ 1 = Sup t [M/Sup Ω (M)] envelop during the time of the relative M (included between 0 and 1). - φ 2 = Sup t [M] envelop along the time of the absolute value. - φ 3 = r r Sup t [ grad{(m)/sup Ω (M)} ] envelop of the norm of the gradient of the relative equivalent stress. This field shows the location of the maxima of the gradient during the evolution -φ 4 = r r Sup t [ grad(m) ] envelop of the norms of the absolute gradient. which expresses the most important gradients reached during the evolution, r r being a characteristic dimension. Many other quantities were also defined to represent curves, signals and pictures. In order to follow our "intelligent" approach for the fatigue analysis, we need to create a data base of examples/problems. Each case/problem is at first defined by its primitive description. There is no way, with such a description, to use the results obtained from one case. into another case/problem. The intelligent description has then to be performed. Since, we can not rely to the numerical (or experimental) results for the classical LOCAL dimensioning criteria, we proposed to perform only the cheapest elastic analysis or our simplified analysis of inelastic structures and to use only the simplest rules for damage criteria and cumulation of damage for materials but we shall couple them with the results from returns on real structures. INTELLIGENT REPRESENTATION OF ANY LOADING AT THE LEVEL OF THE MATERIAL HYPOTHESIS We shall assume two scales for the material characterization: i) the global macroscopical scale where the loading will drive to the asymptotic elastic shakedown behaviour ii) the local scale where there will be asymptoticcally plastic shakedown on some microscopical mechanisms for the local stresses equal to the global stresses (similar the the Sachs model). ONE PARAMETER RANDOM LOADING Equivalence rule Any one dimensional loading or block has: its center C f = (Σ max +Σ min )/2 and fluctuation F σ = (Σ max Σ min )/2 The local Cumulated plastic strain or local Dissipated energy during the real random loading is computed on the model representing the local behavior of the material such as, for example, the kinematic hardening behavior.

9 TWO LOADINGS or blocks ARE SAID EQUIVALENT if they induce the SAME cumulative plastic strain (e pc ) or the SAME dissipated energy and they have the same center and almost the same fluctuation (Figure 1) ε pc1 ; C 1 f ; Fσ 1 ε pc2 ; C 2 f ; F σ 2 ε pc1 = ε pc2 C 1 f = C2 f F σ 2 ~ Fσ 1 Figure 1: Equivalence rule Example Let us consider that the local scale can be represented by the one- dimensional kinematic model α h Ke σ S Figure 2: One-dimensional kinematic model i) Equations At the local scale, we choose as a "measure" of any one-parametered variable amplitude loading, the cumulated plastic strain (or equivalently the dissipated energy) which is induced on a one-single-degree-of-freedom system representing the linear kinematic material. It is intuitive that even if the cumulated plastic strain IS not "the damage factor", they are both going towards the same direction. Moreover, if the mean stress and the stress amplitude are similar, we feel that the damage mechanisms will be similar too Then, two loadings are equivalent if they induce the same cumulated plastic strain and have the same center and roughly the same amplitude. This one single degree of freedom system is schematically represented by: σ α = Σ - y α y α = h α ε p = α ε pc = ε. p (u) du 0 where: K e, is the elastic modulus, h, is the hardening modulus, S, is the local elastic limit or the threshold of the slider (indeed the endurance limit of the material), σ α is the local stress at the level of the friction block, α is the internal parameter (here the glide of the slider), y α is the transformed internal parameter (indeed, the opposite of the residual load at the level of the slider), ε p is the actual plastic strain, ε pc is the cumulated plastic strain. t

10 Let us assume that we have the time history of one particular variable amplitude loading, we input it to the system and we compute the induced cumulated plastic strain. For this given cumulated plastic strain, we may built a particular family of "equivalent" cyclic loadings with constant amplitude (or range) and a number of cycles (similar to the Wöhler curve) ( Σ, N) from the initial cumulated plastic strain. We have Σ > 2S (as there is local accommodation) ε pc = ε 0 pc + 2 N ε p where N = N N 0 and N 0 is the number of cycles conducting to the initial cumulated plastic strain ε 0 pc Σ = 2S + h ε p or in the plane ( Σ, N) the hyperbola (Figure 4) Σ = 2S + h (ε pc ε0 pc ) 2 N Σ Σ C f Σ eq 2S N Figure 3: Equivalent cyclic loading t Ν Figure 4: Family of equivalent cyclic loading In order to insure that the physical phenomena are the same, we also select a mean stress equal to the center C f and a range Σ eq higher or equal to the real fluctuation F σ. A particular family of cyclic loadings with a constant amplitude and a number of cycles is so deduced to represent the real block ii) Application Tests on smooth specimen of Stainless Steel A304L at room temperature were done by V. Doquet at the LMS-X for EDF. (for this steel E = Mpa ; h = 465 Mpa ; S =180 Mpa its endurance limit, Σ a (Mpa) = N its Wöhler curve). The rainflow method was applied and the predicted number of blocks was computed. Often, the method was not conservative. A representative block loading was given (Figure 5). Figure 5: Reference Loading block from EDF The tests were done with various amplifying factors on this loading : A = 3.3 to 4. The block is repeated until fatigue occurs. The number of blocks has been measured (Table 1).

11 In our representation, we have to follow the Procedure: 1 Computation of the center, fluctuation and cumulated plastic strain for, for example, the block * 3.6 C f = -102 Mpa ; F σ = 399 Mpa ; ε pc = Determination of the associated family of equivalent cyclic loadings ( Σ, N) 465 ( ) Σ = N 3 Choice of one particular equivalent loading (C f, Σ eq, N eq ) C f = -102 ; Σ eq = 2* F σ = 798, N eq = As C f 0, a correction with the Goodman rule is done => (0, Σ eq, N eq ) C f = 0 ; Σ eq = 345, N eq = Based on the analytical expression of Wöhler curve, we determine the number of cycles with constant amplitude ( N f ) associated to Σ eq for this special steel Σ a (Mpa) = N (Wöhler curve) => N f = At last, with our representation, fatigue initiation occurs after N block = N f / N eq N block = /25.66 = 5631 Table 1 : Number of blocks to rupture: measured, computed with Rainflow, and with our actual representation A C f Σ eq Measured Rainflow Actual <N< It is worth while noting that our actual representation gives conservative results. MULTI-PARAMETER RANDOM STRESS PATH Behavior of a two parameter model To clarify our hypothesis and our representations, at first, let us consider at the microscopic scale, this special two-degrees of freedom inelastic behaviour similar to the two dimensional friction block. In the general Zarka's framework, we have to write the equations: y α. a. y α y α Figure 6: Two dimensional friction block Figure 7: Velocities fields in the transformed internal parameters space

12 i) Local stresses S, is the applied 2-D stress vector, a are the internal parameters (the local components of the inelastic strains) in the two directions 1 and 2. S = 1 Σ, a = 1 2 α 2 The local stresses, s α,at the level of the inelastic mechanism are: σ α1 = A 11 Σ 1 + A 12 Σ 2 y α1 (25) σ α2 = A 21 Σ 1 + A 22 Σ 2 y α2 (26) or S α = α1 σ = A α S - y α α2 (27) where A α is the localization elastic matrix. and y α are the transformed internal parameters. ii) Transformed internal parameters They are linked to the internal parameters by: y α = y α1 y = B 11 B 12 α2 B 21 B 22 α 1 α = B a (28) 2 where B is a symmetric non-negative matrix. Two cases are possible : a) B is regular; there is a one-to-one relation between a and y α. It is just sufficient to determine y α to have the behaviour b) B is singular (at least of rank 1). The y α must belong to a linear subspace of compatibility here the line (L). It is necessary to follow y α and a at the same time. But in this paper, we shall assume only regular matrix iii) Evolution law Classical plastic evolution law is assumed. Yield condition (convexity) We may take the circle C 0 or (σ α1 ) 2 + (σ α2 ) 2 S 2 (29) C 0 is a fixed convex set, the circle, centered at S and the radius S in the plane (σ α1, σ α2 ). But in the transformed internal parameters plane it becomes the circle C yα (S) centered in A 11Σ 1 + A 12 Σ 2 A and the same radius S. 21 Σ 1 + A 22 Σ 2 Flow rule (normality) The flow rule is written in J.J. Moreau formalism:. a Ψ C0 (s α ) (30) That is to say ȧ is an external normal to the convex set C 0 in s α, or that ȧ belongs to the subdifferential to C 0 in s α. In the plane of the transformed internal parameters, the flow rule is written:. a - Ψ Cyα (y α ) et ẏ α - B Ψ Cyα (y α ) (31) i.e ȧ is an internal normal to C yα (S) in y α (see Figure 7).

13 iv) Deformations We consider small strains for which we have additivity of the elastic and inelastic (plastic) strains: e = e e + e p (32) e e and e p are the local elastic and plastic strains that are taken as e p = (A a ) T a (33) e e = M S (34) where M is the local elastic coefficient matrix. v) Particular loadings It is easy to integrate these equations during any kind of loading and with any initial condition y a0. Radial cyclic loadings The loading path is along a line in the applied stress plane; We may have only elastic or plastic shakedown behavior after some cycles. Although, the kind of behaviour is independant of the initial state, the final asymptotic state is in general function of this initial state. Elastic shakedown Elastic shakedown Plastic shakedown Plastic shakedown Figure 8: Radial and non radial loadings in the transformed internal parameters space

14 Non Radial cyclic loadings and variable amplitude loadings We may consider different cyclic loadings (circle, rectangle, diagonal, triangle random...) all of them with the same center and the same fluctuation. We may consider also any variable amplitudes loadings. It is easy to verify that all the loadings drive to plastic strains that are bounded by the circular cyclic loading with the same center and the same fluctuation. In the stresses plane In the transformed parameters plane In the plastic strains plane Figure 9: Non radial loadings are bounded by the circular loading GENERAL RULES FOR MULTI-PARAMETER LOADINGS The same concept as during one-parameter loadings are used: two loadings are "equivalent" relative to fatigue/damage if they produce the same cumulated plastic strain and have the same center and almost the same fluctuation (or slightly higher). By keeping the center and the fluctuation to be always the same, the range of microscopic plastic strains will be the same as it was noted in the two dimensional model (and we hope to keep the same unknown damage mechanisms). Then, we can choose a special equivalent radial cyclic loading in a given direction θ in the stress plane associated to a number of cycles.

15 These rules show that we have anisotropy in the model as it is seen if we generate a 2D random loading and applied it to the two dimensional model. The first step is the computation of the center, the fluctuation and the cumulated plastic strain of the random loading C f = 0 ; F σ = 1; ε pc = 12.2 The second step is the computation of the equivalent radial cyclic loading in the direction θ. The associated number of cycles is then deduced (Table 2). It is obvious that the directions are not equivalent. θ σ-var C f Σ F σ-var Figure 10 : Fluctuation in Σ-space Radial loading in the direction θ Table 2 : Number of cycles of the equivalent radial cyclic loading in different direction θ θ (degree) Nb cycles This model shows that the difference between the big-small cycles and small-big cycles is verified as the cumulated plastic strains are different during one parameter loading (Figure 11). ε pc = ε pc = Figure 11 : Sequence effects inducing different cumulated plastic strains The model shows at last that if we applied several "normal" or non gaussian one parameter loadings, we loose the "normality" when taking off the parts where there is no plastic strain increment and moreover, no equivalence between these signals. INTELLIGENT REPRESENTATION OF ANY LOADING AT THE LEVEL OF THE STRUCTURE One parameter random loadings The elastic analysis of the structure (with the macroscopic behaviour) gives: S el (t) = S el 0 + λ(t) Sel (35) As we are concerned by long life fatigue, necessarily, a macroscopic elastic shakedown state is reached. The stress field may be written as:

16 S(t) = R l + S el 0 + λ(t-t s) S el pour t t s (36) where R l is the stabilized residual stress field, S el 0 is the initial stress field, Sel is the direction of the loading, λ is the loading parameter and t s is the time when elastic shakedown is reached. When the plastic strain field at the limit state is important, it may be possible that the "damage" produced in the first block and the following ones are different. It will be necessary to distinguish these two first blocks and calculate for each one the equivalent loading. We propose the following procedure to build the equivalent radial cyclic loading at the level of the structure: An elastoplastic simulation of the structure is performed in order to reach the elastic shakedown state (our simple analysis). The stabilized residual stress field and the global plastic strain field are deduced. As we have only one loading parameter, the loading path in the loading parameters space is radial. The center C λ-var f and the fluctuation F λ-var of the real loading (Figure 12) are very easy to perform : C λ-var f = λ min + λ max 2 F λ-var = λ min - λ max (38) 2 then the elastic response in the stress space is also radial at each point (Figure 13) and may be written as: (37) C σ-var f = R l + S el 0 (39) (40) 2 Two points (in some critical zones) are selected with K 1 and K 2 as concentration factors for any component of S el. The center, fluctuation and cumulated plastic strain for these two points (with the microscopic behaviour) are computed. We deduce the family of equivalent cyclic loading at the level of the material and then transform each curve at the level of the structure as indicated in the figure: ( Σ i, N i ) ( λ/k i, N i ) ; i= 1, 2 (41) F σ-var = λ max S el - λ min S el The intersection of the two curves in the plane ( λ, N) gives the GLOBAL REPRESENTATION of the loading at the level of the structure (Figure 14). It is underlined that, for this equivalent loading there is only one amplitude and one number of cycles. Also, the representation can not be made purely mathematically (as it is done classically in the rainflow method and other ones) and it is strictly associated to the geometry of the structure.

17 λ min C f λ-var λ-var F λ max DS el λ F σ-var C f σ-var = R l +S Σ Figure 12: Loading parameters space λ Figure 13: Stress space Σ λ Σ eq 1 λ 1 = Σ eq 1 / K 1 λ 2S 1 K 1 K 2 Σ eq 2 Ν λ eq 1 2 λ 2 = Σ eq 2 / K 2 Ν f Ν Ν Figure 14 : Global representation of the loading at the level of the structure Multi-parameter random loadings For any loading in the space of the loading parameters, we may define its center C λ-var f and its fluctuation F λ-var. This center can be taken has the new origin; the elastic response of the structure is thus n S el (t) = S el 0 + λ i (t) S el i (42) i The center C σ-var f and the fluctuation F σ-var of the stress path and the local cumulated plastic strain or local dissipated energy during the special loading path is computed for two points in the structure. If we consider a radial loading in the direction γ in the loading parameters space (Figure 15), we may write this path as: λ i (t) = Λ i0 + Γ(t) Λ i (43) where Γ(t) is the new loading parameter varying within [-F λ-var, F λ-var ], Λ i are the components of the unit vector defining the radial direction in the λ-space, Λ i0 is equal to the center C λ-var f. It will induce a radial elastic response (Figure 16)

18 n where S el γ0 = Sel 0 + i=1 Λ i0 S el i S el γ and Sel (t) = Sel γ0 γ = (L i ) n i=1 + Γ(t) Sel γ (44) Λ i S el i S el γ F λ-var γ Cf λ-var C f σ-var λ Σ F σ-var Figure 15: Loading parameters space λ Figure 16: Stress space Σ Then, we follow the same procedure as for the one parameter loading to define ONE Equivalent radial cyclic loading in the γ direction ( Γ γ, N γ ). But any other direction with an associated number of cycles may be selected. INTELLIGENT DESCRIPTION OF ONE GENERAL FATIGUE CASE We know how to transform the general loading applied to the structure into a radial cyclic loading in a special direction and with a given number of cycles. Now, it is needed to transfer the primitive, passive descriptions of the geometry of structure, of its various material and of one block of its random loading parameters to an intelligent, active description such that the experimental results on one structure may be used to any other new one! Here, in any point of the structure, the Output Primitive descriptors or conclusions are: i) failure or no failure and, ii) when there is failure, the number of blocks of the loading. A multi-scale analysis is considered (Figure 17), for example for a nuclear vessel: i) at the global scale of the vessel with rather crude mesh; elastic or simplified inelastic analysis is performed due to the various loadings ii) then at the level of a substructure 3-D with a refined mesh (with the initial loading and the real loading, elastic, elastoplastic or simplified inelastic analysis are performed); the equivalent radial cyclic loading is characterized. iii) then at the scale of a 2-D detail (one slice by a plane of the substructure) with a much refined mesh ; iv) then a mobile window allows to focus at the different hot zones. The materials: base material, weld, and HAZ, are described from the material property database (smooth specimen data). The moving window is similar to a filter. Its size is such that the quantities which are needed become rather insensitive to the errors associated to the mesh size and distribution. v) the last level is within a region which is defined from one material within a window. These quantities are related to the description of the stress and the gradient of this stress field. Among them, there are:

19 i) the Maximum, Minimum and Average value of any field, such as I1 and J2, the Dang Van criterium...; ii) inside the window, volumes where any fatigue criterium is violated by a factor of 50%, 80% or 100%. The list of the relievant descriptors needed during fatigue analysis is given in Table 3. Figure 17: Multi-level Analysis GENERATION OF THE DESIGN RULES TO FATIGUE AVAILABLE EXPERIMENTAL DATA All the experimental data used in the study are coming from the tests which were performed at the Illinois University by K. S. Park. These tests were done in order to compare various design methods and to analyze the influence of some loading parameters and of the geometry on marine welded structures. Several structures, indeed several characteristical details were tested (Figure 18). They were selected among the classical ones in function of their relative importance. One of the main details is the detail N 20 which we selected for its simplicity and for the important informations given during the tests. Within all specimen, three materials were mechanically characterized: the base metal, the weld and the HAZ heat affected zone. DETAIL N 20 It is made of one central plate on which two lateral plates are welded. Moreover the real geometry of the weld were measured. Thirty five specimens with three different plate thickness were tested. Three thickness sizes of specimens were used in the experiments: 6.35mm, 12.7mm and 25.4mm for loading plates and 7.9mm, 15.9mm and 31.8mm for corresponding center plates. Three different materials were always considered: base material (for detail No.20, ASTM A- 36 steel), weld metal (with Shielded Metal Arc Welding method), heat affected zone.

20 Table 3 : Relievant Intelligent Descriptors during Fatigue Analysis Descriptors: stress distribution (72) MOY t(i1 moy) Ω MOY t(i1 max) Ω MOY t(i1 min) Ω MOY t(i1 lay1) Ω MOY t(i1 lay2) Ω MOY t(i1 lay3) Ω MOY t(i1 lay4) Ω MOY t(i1 lay5) Ω MOY t(j2 moy) Ω MOY t(j2 max) Ω MOY t(j2 min) Ω MOY t(j2 lay1) Ω MOY t(j2 lay2) Ω MOY t(j2 lay3) Ω MOY t(j2 lay4) Ω MOY t(j2 lay5) Ω MOY t( grad J 2 moy) Ω MOY t( grad J 2 max) Ω MOY t( grad J 2 min) Ω MOY t( grad J 2 lay1) Ω MOY t( grad J 2 lay2) Ω MOY t( grad J 2 lay3) Ω MOY t( grad J 2 lay4) Ω MOY t( grad J 2 lay5) Ω MAX t(j2 moy) Ω MAX t (J2 max) Ω MAX t (J2 min) Ω MAX t(i1 lay1) Ω MAX t (I1 lay2) Ω MAX t (I1 lay3) Ω MAX t (I1 lay4) Ω MAX t (I1 lay5) Ω MAX t(i1 moy) Ω MAX t (I1 max) Ω MAX t (I1 min) Ω MAX t(j2 lay1) Ω MAX t (J2 lay2) Ω MAX t (J2 lay3) Ω MAX t (J2 lay4) Ω MAX t (J2 lay5) Ω MAX t( grad J 2 moy) Ω MAX t( grad J 2 max) Ω MAX t ( grad J 2 min) Ω MAX t( gradj 2 lay1) Ω MAX t( gradj 2 lay2) Ω MAX t( gradj 2 lay3) Ω MAX t( gradj 2 lay4) Ω MAX t( gradj 2 lay5) Ω MIN t(i1 moy) Ω MIN t(i1 max) Ω MIN t(i1 min) Ω MIN t(i1 lay1) Ω MIN t (I1 lay2) Ω MIN t (I1 lay3) Ω MIN t (I1 lay4) Ω MIN t (I1 lay5) Ω MIN t(j2 moy) Ω MIN t(j2 max) Ω MIN t(j2 min) Ω MIN t(j2 lay1) Ω MIN t (J2 lay2) Ω MIN t (J2 lay3) Ω MIN t (J2 lay4) Ω MIN t (J2 lay5) Ω MIN t( grad J 2 moy) Ω MIN t( grad J 2 max) Ω MIN t( grad J 2 min) Ω MIN t( grad J 2 lay1) Ω MIN t ( grad J 2 lay2) Ω MIN t ( grad J 2 lay3) Ω MIN t ( grad J 2 lay4) Ω MIN t ( grad J 2 lay5) Ω Descriptors: fatigue (12) Descriptors: material (11) (E DV moy ) Ω (E DV max ) Ω (E DV min ) Ω (N S-N moy ) Ω (N S-N max ) Ω (N S-N min ) Ω (ε pc moy ) Ω (ε pc max ) Ω (ε pc min ) Ω Young Modulus Poisson ratio Macroscopic elastic limit Macroscopic workhardening modulus Cyclic hardening Exponent n Cyclic Strength Coefficient K Ultimate Strength σ u Cyclic Yield Strength σ y Fatigue Strength Coefficient σ f ' Fatigue Strength Exponent b Fatigue Ductility Exponent c Y G ν σ y H

21 . Figure 18: Classical welded details (from S.K PARK) Figure 19: 3-D detail No.20 Figure 20: 2-D detail No.20 Figure 21: Detail n 20 with its various hot points

22 Table 4 : Review of the experimental results (from S.K PARK) The experimental set up in the tension-compression machine, implies that the grips induce important initial stresses which were measured thanks to four strain gages located in the vicinity of the weld on the lateral plates. During each test, a fluctuating tensile loading was applied: a mean stress level varying 0 and Mpa with a random amplitude block (these stresses result from a long analysis on real naval structure). In principle, such a block was applied several times until one crack is initiated or the maximum number of 1500 times (which represents 125 years on the real structure). During the tests, special cares on the critical zones were taken, as these zones are the most sensitive to the crack initiation damage (Table 4). For each sample, we have thus the geometry, the real loading, the place where cracks appear (but unhappily, the initial stresses due to the weld were not measured). MODELLING AND MESHING WITH ALGOR Script files for the design system SD3 from Algor were written to allow the systematical input of the geometry and the mesh of the specimen according the given experimental data. It was taken into account that 3 materials had to be differentiated. The Heat Affected Zone was not perfectly described in the report. It was guessed according the other geometrical informations. The system Algor proposes several mesh generators. For a 2D structure, it uses Supergen. In order to take into account the different materials and to have a thinner mesh around the sensitive points, it is necessary to give some more informations.

23 ELASTIC ANALYSIS WITH NISA FROM EMRC It is necessary to edit the nisa file within the preprocessor DISPLAY, to define the materials, the boundary conditions. It is also necessary to introduce at first the initial stresses due to the grips and the misalignment. As underlined in the report, these stresses may be important although the care brought by the technicians. It is, at last, necessary to define the loading on the specimen. For elastic only some unit loadings are introduced and then combined. For the elastoplastic analysis, the full time history of the loading and the representative radial cyclic loading are introduced. With NISA, four types of analysis were performed: i) elastic analysis ii) simplified elastoplastic analysis with ZAC method with the representative radial cyclic loading (loading2) iii) classical incremental analysis with the representative radial cyclic loading (loading2) iv) classical incremental analysis with the real variable loading (loading1) Without giving details, we obtained the results for the 8 sensitive zones (IJP and TOE) and also for some 4 other randomly chosen zones within the specimen. The size of the zones or windows has been taken as a fundamental parameter of the study, 5 sizes were retained (.1 mm,.2mm,.4mm,.8 mm and 1. mm). The result of these analysis consists into the deduction of the initial stress field due to the grips and the applied stress field due to the random traction for each of the 12 regions and for each of the 3 differentiated materials. P m 1 2 P 3 Figure 22: Variable Loading1 Figure 23: Cyclic Loading2 FATIGUE ANALYSIS From these results of the various analysis, it is possible to use the post processor ENDURE from EMRC. Only the crack initiation option of the program was used: the number of applied blocks necessary to induce fatigue in any point of the structure. This last file was used to produce one of the intelligent descriptors, the mean value of the life taken on all the nodes within a zone and for one material. We also compute, the Dang-Van criteria which is often used in fatigue and which is expressed in any node of the mesh: E DV = Max t τ max(t) + α p(t) - β β - α p(t) (45) where τ max (t) is the local shear stress on a critical slip plane, p(t) is the local pressure and α, β are constants identified from endurance limits during alternate flexion and torsion tests

24 COMPILATION OF DATA AND AUTOMATIC LEARNING Several details were tested. For each size of the window, several text files are induced. All these text files are merged. The observed conclusions are added. The data base has been so created. In the Excel files name1.xls, name2.xls, name4.xls and name8.xls, name10.xls the results of all the analysis corresponding to a window size of.1 mm,.2 mm,.4 mm,.8 mm and 1.0 mm are given. We recall that in our approach, each subregion i.e. one material in one region, is considered as an independent case/example for which we have its own intelligent descriptors and its two conclusions Failure or No failure and when there is failure, the number of cycles or number of blocks. Table 5 : Extract of one data base Des_Ela8.xls Classes Conclusions Descriptors Material Descriptors stress distribution example Nbcyc C1 C2 k MOY t(i1 moy) Ω MOY t(j2 moy) Ω MOY t( grad J 2 moy) Ω MOY t(i1 max) Ω d20h12_ela8_r1_m d20h4_ela8_r6_m d20q4_ela8_r3_m d20h8_ela8_r5_m d20q3_ela8_r5_m d20c3_ela8_r5_m d20h5_ela8_r8_m d20q1_ela8_r1_m d20q7_ela8_r7_m d20h9_ela8_r2_m d20c2_ela8_r8_m d20h12_ela8_r1_m We used L.E.S from the Ecole Polytechnique and Neuroshell from Wards systems to extract automatically the rules. Only the learning of Failure or No Failure has been done for the moment. We also used the Probabilistic Neural Networks (PNN) which are able to train on sparse data sets and to separate data into a specified number of output categories. From the results, the best window size may be taken equal to.8 mm during the elastic analysis. Moreover, the simplified and elastoplastic analysis give almost the same reliability than the elastic one. Then only elastic analysis for any new structure with a window size of.8 mm have to be taken. These rules have to be considered as generalized fatigue criterium. CONCLUSIONS Although the rules were generated on the detail n 20, they can now be used to analyze any other structure subjected to any complex multi-parameter and variable loadings. However, more experimental tests are still necessary to qualify fully our approach

25 ACKNOWLEDGMENTS This paper was written in part during the regular visits of the first author to UCSD's Center of Excellence for Advanced Materials, under ONR contract (R. Barsoum, Coordinator) to the University of California San Diego and for the other part by the second author under the sponsorship of Ligeron S.A (A. Azarian, Coordinator). REFERENCES Dang-Van, K. and Cailletaud, G. and Flavenot, J.F. and Ledouaron, A. and. Lieurade, H.P (1984), Critère d amorçage en fatigue à grand nombre de cycles sous sollicitations multiaxiales, Amorçage des fissures sous sollicitations complexes, Journées internationales de printemps, p , Paris. Karaouni, H. (2001), Nouveaux outils pour la conception fiable des structures, Thèse de doctorat, Ecole Polytechnique. Khil, D.P. (1997), Fatigue strength of stainless stell weldments in air, NSWCCD-65-TR- 2000/04, february Morel, F. (2000), Fatigue multiaxiale sous chargement d amplitude variable, Thèse de Doctorat, Université de Poitiers. Robert, J.L.(1992), Contribution à l étude de la fatigue multiaxiale sous sollicitations périodiques ou aléatoires, Thèse de l INSA de Lyon. Taheri, S. and Doquet, V. (2000), Le non conservatisme de la méthode rainflow en déformation contrôlée, EDF, HI-74/00/006/A. Zarka, J. and Casier, J. (1979), "Elastic Plastic Response of a Structure to a Cyclic loading: Practical Rules", Mechanics Today, Vol 6, S. Nemat-Nasser editor. Zarka, J. Frelat, J. Inglebert, G.and Navidi, P. (1990), A New Approach in Inelastic Analysis of Structures distributed free by CADLM. Zarka, J. and Navidi, P. (2002) Practical Analysis of Inelastic Materials and Structures, Vol 2: Intelligent Optimal Design, to appear; manuscript distributed on a free CD by CADLM Zarka, J. and Karaouni, H (2002), New rules to be included in codes and standard to represent multiaxial variable amplitudes loadings on structures, ICONE Zarka, J. and Karaouni, H (2002), Fatigue during multi-axial radom laodings, ICONE