Cette thèse s'est concentrée sur l'étude de la rupture de matériaux à comportement fragile (céramiques monolithiques et microbétons). Ce travail fait l'objet de la thèse de Christophe DENOUAL (collaboration DGA/DCE/CTA et LMT-Cachan), dont la soutenance est prévue en 1998.
Endommagement et rupture de composites [0/90] 280
Liste des figures
173 Evolution de la longueur de glissement normalisée équivalente ïR/L en fonction de la longueur de glissement normalisée lR/z.
Liste des tableaux
Introduction
INTRODUCTION 5 niveau microscopique ou mésoscopique avant qu'elles n'entraînent une rupture macros-
Les fissurations et fractures multiples de matériaux au comportement quasi-fragile sont analysées à l'échelle microscopique et mésoscopique. Cela concerne particulièrement le comportement aux chocs des céramiques, la fissuration de la matrice et la rupture des fibres dans les composites à matrice fragile.
Chapitre 1
Comportement mécanique et
Loi de comportement et changement d'échelle
- Calcul de la densité d'énergie élastique récupérable
- Calcul de la densité d'énergie élastique irrécupérable
Cette déformation inélastique dépend de la variable d'endommagement D (ce n'est pas une variable interne supplémentaire). Le calcul de la densité d'énergie irréversible nécessite la connaissance du champ de contraintes résiduelles $- induit par le frottement.
Rupture d'un volume élémentaire endommagé et changement d'échelle
- Rupture fragile
- Rupture quasi-fragile
The damage variable, D, is a function of the crack density (ie, va2/S, where 242 is the crack size within a surface, S) and the Helmholtz free energy density, # and the velocity density of energy release, Y, are as follows [12]. The evolution of the mesoscopic damage is shown as a function of the evolution of the energy release rate density, Y - Y*(Do), in Fig.
Chapitre 2
Rupture de matériaux à comportement fragile
Dans le cas d'une rupture fragile, le comportement reste élastique jusqu'à la rupture, ce qui rend sa prédiction beaucoup plus difficile. Pour aborder la rupture de matériaux au comportement fragile, il s'agit de déterminer la probabilité de rupture en fonction du chargement (information mécanique) et de la population de défauts initiaux (information physique).
Probabilité de rupture d'un volume élémentaire
- Propagation instable de défauts
- Propagation stable de défauts
- Bornes de probabilité de rupture
- Isoprobabilité de rupture
- Applications à deux cas simples
- Hypothèses de travail
- Probabilité de rupture d'une structure
- Effet d'hétérogénéité des contraintes en fatigue
- Application à une structure
Ce raisonnement peut être valable quelle que soit la valeur de la probabilité de rupture du PFO. Dans le cas d'une propagation de défaut stable, l'expression de la probabilité de rupture d'une connexion est donnée par [HILD et ROUX, 19911. Dans le cas d'une propagation de défaut stable, l'expression de la probabilité de rupture d'une connexion est donnée par l'équation (2.26 ).
Procédures d'identification : études à différentes échelles
- Rupture du nitrure de silicium monolithique
L'identification de la répartition des défauts ne peut être comparée à celle constatée dans le matériau. COUB AGHA, 1996 ; YAACOUB AGHA et al., 19971 pour lequel l'identification de la répartition initiale des défauts est obtenue par l'analyse des limites d'endurance. La qualification de la méthode est réalisée à partir de mesures indépendantes de la répartition initiale des défauts par analyse systématique de 50 faciès de fracture.
TENSILE TEST SYSTEM
It enables a unified approach to failure probability taking into account the failure distribution and its evolution in the case of cyclic loading. Some specimens were tested with four strain gauges on the central section to verify stress alignment: the maximum difference between the four gauges is less than 1.3 x 104 (i.e. 7.4% of the tensile strain) and we conclude that the bending stresses are small (maximum about 11%) compared to tensile strains. As shown below, the dispersion of failure stress caused by bending deformations is small compared to the dispersion caused by initial faults.
MONOTONIC TENSILE TESTS ON SILICON NITRIDE
The initiation sites observed on the fracture surfaces of the specimens have defects whose size is between 20 and 250 pm (Fig. 3.1-2). Seventeen of the eighteen failures initiate within the sample volume (Fig. 4) rather than at the surface. This appears to be a back-test of the quality of the experimental setup that was designed for these tests: surface defects do not play a major role due to the homogeneity of the stress field induced by this setup.
COMPARISON WITH OTHER MONOTONIC TESTS
Compared to the pure stress case, the magnitude of the initiation faults is smaller on average (15 to 110 pn) [SI, and the average failure stress is higher (629 MPa instead of 526 MPa). Compared to the four-point bending case, the magnitude of the initiation faults is comparable on average [5] (90% were in the volume) and the average failure stress is higher (660 MPa instead of 629 MPa). Compared to the three-point bending case, the magnitude of the initiation faults is comparable on average [5] (70% were in the volume), the average failure stress is higher (721 MPa instead of 660 MPa).
EFFECTIVE VOLUME, Veff (mm3)
Contrary to Katamaya and Hattori [8], for this set of experiments, a correlation in terms of effective volume is not satisfactory. Also, a correlation in terms of effective surface is not satisfactory as it leads to a slope equal to -1117 with p2 equal to 0.983. They lead to a value of the shape parameter m, derived from an effective volume analysis (of the order of 25), which differs from the values observed experimentally (of the order of 9).
CYCLIC TENSILE TEST ON SILICON NITRIDE
The sets of specimens were made using different processing techniques: therefore the distribution of defects was not the same for the four sets of specimens. 7, the experimental failure strains are compared with the failure strains obtained from a calculation where the initial defects are assumed to be penny cracks loaded in mode 1. With this assumption, a Linear Elastic Fracture Mechanics approach shows that the critical intensity factor of smss KI , is again in the order of 3 5 m a &.
In the case of monotonic loading, a correlation can be obtained between the Weibull parameters and the parameters of the defect size distribution. If we assume that the initial flaws are modeled by cracks, the value of the critical flaw size, a, is related to the equivalent applied stress, o, by. Another way to prove this difference is to perform direct measurements of the error distribution.
PURE TENSION
Furthermore, it is possible to calculate the voltage drop for the same cumulative failure probability after N cycles. After N cycles we get the same cumulative failure probability by writing [IO]. where the constant A depends on the details of the error distribution. For example, in the case of the experiments in monotonic stress, a failure stress equal to 5 15 MPa corresponds to a cumulative failure probability equal to 0.25.
CONCLUSION
In the considered cases, the value of Su is 360 MPa (Y i 1) and C*N « 1, so the cumulative probability of failure during the cycles of cyclic loading does not develop much. In this study, it is shown that the failure properties of a ceramic structure depend not only on the volume of the structure and the profile of the stress field, but also on the distribution of defects. It is therefore important to determine these quantities in order to fully describe the failure conditions.
ACKNOWLEDGEMENTS
Fissuration sous-critique de ferrites manganèse-zinc
Analysis of the Failure of Ceramics Due to Subcritical
- Rupture par fatigue de fontes GS
- Introduction
- Reliability of structures containing flaws Statistical methods applied to predicting failure un-
- Identification from S-N curves
- Particularization
- Analysis of fatigue tests on austempered S.G
- Conclusions
The next step is to study the sensitivity of the error probability to the error size distribution. In the case of cyclic loading conditions, the stable microcrack propagation leads to the evolution of the flaw size distribution. It relates the expression of the cumulative initiation probability to the initial fiaw size distribution fo.
HIGH CYCLE FATIGUE BEHAVIOR OF SPHEROIDAL GRAPHITE CAST IRON
NOMENCLATURE
INTRODUCTION
In this paper, we will focus our attention on SG cast iron subjected to high cycle fatigue. In the first part of this paper, the crack propagation law is discussed with respect to the considered defects. The second part deals with an expression for the cumulative failure probability of a structure subjected to high cycle fatigue.
MICROCRACK PROPAGATION IN HIGH CYCLE FATIGUE
In the case of cyclic loading, a cyclic threshold stress can be defined as the lowest value of the stress level below which no failure occurs (ie the failure probability is equal to zero). Therefore, it is assumed that the evolution of the threshold voltage intensity factor Kth is only dependent on the initial error size i.a. The value of the function cp depends on the power n of the microcrack propagation law.
CUMULATIVE FAILURE PROBABILITY IN TENSION
The purpose of the next section is to derive an expression for the cumulative failure probability when the material experiences cyclic loading conditions for which the defects are randomly distributed within the structure and can grow stably. Since the propagation stage is neglected when Eq. 1 1 ) is used, this equation corresponds to a lower bound for the cumulative failure probability of the structure. In the next section, an identification procedure of the propagation law is proposed and applied to fatigue experiments performed on SG cast iron specimens.
The expression for the cumulative failure probability then depends only on the initial failure distribution and the value of the cyclic limit stress. For a given value of b (2. e., a given probability of failure), the development of the number of cycles to failure mainly depends on the law of crack growth. Therefore, the analysis of the constant probability of failure allows the identification of the parameters of the crack growth law if the distribution of the size of the defect is known (ie, different values of the constant b have been identified).
CONCLUSIONS
This result shows that the function g accounts for the influence of the load ratio for different cumulative failure probabilities. A simplified expression for the cumulative failure probability is derived for cyclic failure in the case of tension. These results demonstrate that the cumulative failure probability expression proposed herein is capable of modeling fatigue data obtained on SG cast iron.
ACKNOWLEDGMENTS
122 CHAPTER 2: RUPTURE DE MATÉRIAUX FRAGILES Table 2: Material parameters modeling high cycle fatigue of SG cast iron. The threshold stress intensity factor, which depends on the current size of the defect, is considered as a material constant for the described SG cast iron. Experimental data on SG cast iron in tension are analyzed within this framework for experiments with a load ratio R = 0.1.
Perspectives
Les développements récents des outils informatiques permettent désormais d'envisager l'utilisation d'outils de simulation numérique pour modéliser le processus de coulée. La modélisation de ce processus permet de prédire l'état géométrique et métallurgique de la pièce coulée. Un programme complet de simulation du processus de coulée doit également inclure des critères d'apparition de défauts, notamment lors de la solidification.
Endommagement et rupture de
Endommagement de céramiques sous charge- ment dynamique
- Potentiel d'état - couplage d'état
An edge-on impact configuration can be used to analyze damage state and evolution by observing the crack front pattern and velocity. This impact test can be used to identify damage mechanisms through a comparison of experimental and simulated values of the crack front velocity. Second, it is assumed that the influence of stress propagation in the thickness of the edge-on impact is weak (i.e. the tile is in equilibrium.
TI"&
Fragmentation de céramiques sous impact
A probabilistic approach for
- Probabilistic approach
- Applications
- Lois d'évolution, identification et applications
The space-time graph is composed of the union of obscured zones in which no faults can arise and the complementary zone in which defects can arise. An equivalent form for equation (3) can be found in Jeulin (1985). The variable P1 can be split into an infinite number of events, defined by the probability of finding a new defect at t during a time step dt in an interaction zone 2fn) (T - t). Using equations (4) and (7), the evolution law of the damage variable D can be written as.
3 but are clearly present in the last frame of the same shot (not shown in this paper, see Ref. To compare the evolution of radial and hoop stresses, a typical result is given in Fig. It can be observed that the radial stress reaches a significant value before any significant hoop stress evolution.
A PROBABILISTIC MODEL FOR THE DYNAMIC FRAGMENTATION OF BRITTLE SOLIDS
HILD1 and C. DENOUAL2
- Fissuration matricielle de composites sous chargement monotone
- Modèle de comportement de composites unidirectionnels
La deuxième variable, d, est obtenue à partir de l'expression de la densité d'énergie irréversible qi (voir paragraphe l.l.2.iii). En particulier, la mesure de la variable interne d implique la quantification de la densité énergétique. Le calcul de la variable d de dommage d’interface est obtenu à partir du calcul de la densité d’énergie irrécupérable (cf.
Ce résultat montre à nouveau que la longueur de glissement représentative IR peut être approchée au premier ordre par ZR, mais qu'elle diverge à mesure qu'elle s'approche de la saturation (Figure 3.8).
- Applications à des composites unidirectionnels
- INTRODUCTION
- PHYSICAL MODEL
- CHAPITRE 3 : ENDOMMAGEMENT DE COMPOSITES
- The Unit Cell
- Interrogation of First Loading Response
Afin d’étudier l’évolution de la longueur de glissement représentative IR, par souci de simplicité nous utiliserons une méthode numérique. La limitation de la méthode est le nombre d'éléments (de l'ordre de IO5 dans notre cas). 3.9 - Développement de la longueur de glissement équivalente normalisée ïR/z en fonction de la longueur de glissement normalisée ERIE.
