Towards an Optimal Reliability of Metallic Structures during Random Loadings Joseph Zarka 1) and Habib Karaouni 2) Laboratoire de Mécanique des Solides, Ecole Polytechnique, 91128 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

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 p ∆E ∆E ∆E  ∆Σ   ∆Σ 1/n' = + = (1)  +  ' 2 2 2 2YG  2K  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: (2) Σa = σf' Nb + d or, more practically, by the strain-life equation: ∆E σ’f (3) = (2N)b + εf' (2N)c 2 YG ∆E In this equation, is the strain amplitude, 2N is the number of cycles to failure with the 2 constitutive constants for the material: σf' is the fatigue strength coefficient, YG is the Young modulus, εf' is the fatigue ductility coefficient, c is the fatigue ductility exponent, and b is the fatigue strength exponent, d is a constant term. Σmax = Σa + Σm , is the maximum or peak stress, Σa is the stress amplitude, and, Σm 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 (Σm ,Σa), a bounded domain is defined:  Σ  Σ'a = Σa 1 – m (Goodman) (4)  σu  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: N Di = i (5) NRi and failure when (6) ∑ Di = 1 (where Ni is the number of cycles realized during the cyclic loading i for which the number of cycles to rupture is NRi) 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.

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): Σ max γmax (1 + n n ) = (7) σy n n σ2 σ σ (1+νe) f Nb + (1+νe) f σy N2b + (1+νp)σf Nc + (1+νp) f Nb+c E 2 E 2 σy where γmax is the maximum shear strain on the maximum shear strain amplitude plane, Σnmax 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, σy is the elastic limit or yield stress, νp 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: Ca + α Pa + β Pmean < b (8) where Ca is the deviatoric stress amplitude, Pa is the amplitude and Pmean , 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: Σeq = τ (t) + aDV.p(t) ≤ bDV (9) where τ (t) is the local shear stress on a critical slip plane, p(t) is the local pressure and aDV , bDV 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 ||+a P(t)  IE = Max Max  alt DV (10)  2S (as there is local accommodation) εpc = ε0pc + 2∆N |∆εp| where ∆N = N – N0 and N0 is the number of cycles conducting to the initial cumulated plastic strain ε0pc ∆Σ = 2S + h ∆εp or in the plane (∆Σ , ∆N) the hyperbola (Figure 4) h (εpc – ε0pc ) ∆Σ = 2S + 2∆N Σ

∆Σ ∆Σeq

Cf

2S ∆N Figure 3: Equivalent cyclic loading

∆Ν

t

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 Cf and a range ∆Σeq higher or equal to the real fluctuation Fσ. 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 = 179000 Mpa ; h = 465 Mpa ; S =180 Mpa its endurance limit, Σa (Mpa) = 52492 N-0.505 + 214.8 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).

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 Cf = -102 Mpa ; Fσ = 399 Mpa ; εpc = 48.19 2 – Determination of the associated family of equivalent cyclic loadings (∆Σ, ∆N) 465 (48.19 – 0) ∆Σ = 360 + 2∆N 3 – Choice of one particular equivalent loading (Cf , ∆Σeq, ∆Neq) Cf = -102 ; ∆Σeq = 2* Fσ = 798 , ∆Neq = 25.66 4 – As Cf ≠ 0, a correction with the Goodman rule is done => (0 , ∆Σeq’, ∆Neq) C’f = 0 ; ∆Σeq’ = 345 , ∆Neq = 25.66 5 – Based on the analytical expression of Wöhler curve, we determine the number of cycles with constant amplitude (∆Nf) associated to ∆Σeq’ for this special steel Σa (Mpa) = 52492 N-0.505 + 214.8 (Wöhler curve) => ∆Nf = 144598 6 – At last, with our representation, fatigue initiation occurs after Nblock= ∆Nf / ∆Neq Nblock= 144598/25.66 = 5631 Table 1 : Number of blocks to rupture: measured, computed with Rainflow, and with our actual representation A Cf Measured Rainflow Actual ∆Σeq -4 110 440 817