Relationship between interface stresses and the orientation of each grain boundary with respect to the tensile axis.

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Thermally-activated nucleation of cavities

Intergranular cavity nucleation is provided from the formation of a stable cluster of vacancies. The surface of cavities is constituted by a lenticular segment shape [77] (explain in details subsection ?? in Chapter ??). This cavity geometry requires both a fairly quick superficial diffusion, in order to keep the constant curvature of cavities during growth, and an isotropic superficial energy. The creep nucleation mechanism was initially proposed by Greenwood et al. [78]. This mechanism assumes that the cavities nucleate preferentially by condensation of the vacancies under the effect of the normal local stress acting on the grain boundaries. Finally, Raj and Ashby [77] showed that this cavity nucleation mechanism occurs more quickly if the normal stress to the grain boundary is high and the surface energy of the cavity is low. Then, they deduce that it is easier to generate cavities at the multiple junction joints where the local stresses are higher (Figs. 1.13a and 1.13b). From the thermodynamic point of view, Raj and Ashby [77] proposed an energy barrier based on the variation of the Gibbs free energy, G.
G is given by [77]:
G = −nV + freeSfree − interfaceSinterface (1.22)
Three terms contribute to the energy variation:
1) the work induced by the application of a remote tensile stress on an elastic medium containing a cavity of volume V.
2) the energy to be supplied for the creation of the cavity free surface, Sfree.
3) the loss of the energy of the grain boundary due to the reduction of its surface, consumed by the cavity growth, Sinterface.

Coupling between diffusion and viscoplasticity

In diffusion growth models, the creep deformation of the surrounding grains is neglected. The atoms coming from the cavities deposit all along grain boundaries. In fact, the vacancies necessary for cavity growth, diffuse only from the area near to the cavities. In this case the diffusion distance, is quite short. This will increase the cavity growth rate. This problem was studied by several authors [95, 96, 98– 100]. These models assume that the deformation away from the cavities takes place by grain creep deformation. A schematic presentation of the coupling of grain boundary diffusion and viscoplastic deformation is illustrated in Fig. 1.16.

Measurement of the reduction in fracture area and observations of the fractured Specimens

The FEG-SEM observations of fracture surfaces (Fig. ??), show two different types of fracture surface, either necking fracture (Fig. ??) or intergranular fracture (Fig. ??).
The necking fracture surface is characterized by voids and dimples (Fig. ??). Dimples are generally observed on the ductile fracture surface [57]. On the contrary, grain boundaries can be clearly observed on the intergranular fracture surfaces (Fig. ??).
The measured reduction in area at fracture (Z): Z = S0 − Sf S0 · 100% (2.4).
with S0 the original area and Sf the area after fracture, Z varies between 10% and 70% (Fig. ??). Generally, necking fracture, short term tests, is accompanied by a higher area reduction (Z% > 40%) and high failure strain in comparison to intergranular fracture. For intergranular fracture, long term tests, the reduction in fracture surface areas is lower than 30% based on our FEG-SEM observations of the fracture surfaces as shown in Fig. ??.
Figure 2.3: FEG-SEM observations of the fracture surfaces of the HN 823 alloy for two different test conditions. (a) necking fracture 550C, 280MPa, 332h, Z%=42%; (b) intergranular fracture 500C, 310MPa, 5082h, Z%=10%. minimum strain rate leads to a high area reduction. On the contrary, a low creep rate leads to a low area reduction. A high strain rate plateau is observed, with an area reduction of about 55%. A drastic decrease is observed between 10−8 and 10−7s−1. Finally, the area reduction seems to decrease much slower down to 10−10s−1 (Fig. ??). A value of almost 15% is reached. No clear temperature effect can be observed.
According to our area reduction measurements and fracture surface observations, it can be concluded that a high minimum creep rate leads to necking fracture, and a low minimum creep rate leads to intergranular fracture.

Observations of the longitudinal sections

The observations of the polished longitudinal sections by FEG-SEM are focused on precipitates, cavities and short cracks. In all specimens, two types of precipitates are observed chromium carbides, M23C6, and titanium carbonitrides, Ti(C,N). The M23C6 precipitates, generally Cr23C6 precipitates, appear at grain boundaries and triple junctions. The Ti(C,N) particles are generally rectangular precipitates observed in the matrix. The affinity of titanium to carbon is higher than the Cr one. Thus, Ti(C,N) precipitates are more stable than M23C6 precipitates. But, in comparison to the Ti(C,N) carbonitrides, the M23C6 precipitates are produced preferentially at in service temperatures ranging from 500 to 550C [? ].
For a few carbonitrides, short cracks ( 1μm) can be observed (as shown in Fig. ??) at the interfaces with the metallic matrix. But the crack sizes remain always similar to the precipitate size. And neither propagation nor coalescence is observed. Consequently, such micro cracks are not considered in the creep damage modeling presented in section ??.
Intergranular cavities can be observed in specimens which failed due to intergranular damage, but not observed in the ones fractured by necking. Intergranular cavities are initiated at the interfaces of M23C6 precipitates and matrix as shown in Fig. ??. Then, cavities grow along grain boundaries and triple points.
This is why long term creep damage is also called intergranular cavitation damage. Furthermore, growth and coalescence of cavities lead to intergranular cracks. Cracks propagate along the grain boundaries perpendicularly to the tensile axis as shown in Fig. ??, which agrees with many observations reported in literature.

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Modeling of intergranular fracture

Dyson observed that cavities do not nucleate at the same time but continuously one after another [74]. Cavity nucleation starts rather early and continues over the whole creep test [93, 185, 240]. Riedel [51] proposed a model to predict long term creep lifetimes. The Riedel model assumes that, under creep conditions, cavities nucleate continuously as shown by Dyson [74] and then grow by grain boundary vacancy diffusion. The nucleation and growth of intergranular cavities cause coalescence of cavities, and then microcracks form. The fracture of specimens takes place as the area fraction of cavities and short cracks reaches a critical value, !f [51]. So that, the values of the different input parameters such as the cavity nucleation rate, ˙N0, the grain boundary vacancy self diffusion coefficient, Dgb/v, and the critical damage, !f , should be evaluated. Their ranges of variation should be evaluated too, in order to assess the sensitivity of the predictions with respect to the uncertainties in the input parameter values.

Cavity geometry parameter

Particular cavity shapes (Fig. ??) could reduce the nucleation energy barrier, which leads to earlier cavity nucleation at grain boundaries but not in the crystals [51]. In addition, voids along grain boundaries can display different shapes depending if they nucleate at grain boundaries, triple points, quadruple points or at the interfaces with precipitates located at grain boundaries [77]. In the following, only grain boundary cavity nucleation is taken into account in agreement with our observations. In Fig. ??, the following notations are used:
– b is the average normal stress acting on the grain boundary facet, in this study, b is considered as applied stress, .
– L is the half distance between two cavities.
– rb is the half-length of penny shape cavity (Fig. ??).
– r is the radius of the corresponding spherical cavity (broken line) which has the same volume as the lenticular one ( < 2 ) of half-length rb (Fig. ??).
The cavity geometry is defined by the half-length of the lenticular cavity, rb, and the angle . This angle can be calculated by considering the balance of the surface tension. Eq. ??, links the grain boundary surface energy, gb, and the cavity free surface energy, s [51]. cos = gb 2 s.

Table of contents :

1 Introduction 
1.1 Materials under study
1.1.1 Chemical composition
1.1.2 Effect of the main chemical elements
1.1.3 Grain size
1.1.4 Secondary phases
1.1.5 Microstructure evolution at high temperature
1.2 Creep background
1.2.1 Creep deformation
1.2.2 Phenomenological viscoplasticity laws
1.3 Creep deformation mechanisms
1.3.1 Diffusion creep
1.3.2 Grain Boundary Sliding
1.3.3 Dislocation creep
1.3.4 Deformation map
1.4 Damage mechanisms
1.4.1 Necking
1.4.2 Intergranular fracture
1.4.3 Physically-based lifetime prediction
1.5 Conclusion and summary of the manuscript
2 Modeling of creep cavity nucleation 
2.1 Introduction
2.2 Experimental background and results
2.2.1 Material under study
2.2.2 Microscopic observations
2.3 Macroscopic and crystalline constitutive laws
2.3.1 Macroscopic isotropic creep flow rules
2.3.2 Crystal constitutive laws
2.4 Interfacial stress field calculations
2.4.1 Influence of the particle elasticity constants
2.4.2 Influence of the random orientations of the two neighbor grains
2.4.3 Time evolution of the normal stress fields
2.4.4 Influence of temperature and remote stress
2.4.5 Relationship between interface stresses and the orientation of each grain boundary with respect to the tensile axis.
2.5 Interface fracture
2.5.1 The stress criterion
2.5.2 First prediction of cavity nucleation rate
2.6 Discussion
2.6.1 Local interfacial stress
2.6.2 The fracture criterion
2.6.3 Evaluation of the Dyson law prefactor
2.7 Conclusion
3 Effect of the particle geometry 
3.1 Experimental observations
3.2 Interfacial stress field calculations
3.3 Precipitate shape factor effect
3.3.1 The Eshelby theory
3.3.2 Finite Element calculations
3.4 Precipitate sharp tip effect
3.4.1 Precipitate symmetric tip
3.4.2 Precipitate asymmetric tip
3.5 Discussion of the modeling assumptions
3.5.1 2D-3D comparison
3.5.2 Evolution of the average inclusion stresses during straining
3.5.3 Influence of the lattice rotation
3.6 Summary and conclusion
4 Lifetime prediction of 316L(N) 
4.1 Introduction
4.2 Creep damage mechanisms
4.2.1 Necking
4.2.2 Intergranular damage
4.2.3 Thermally-activated nucleation of stable vacancy nuclei .
4.2.4 Interface fracture
4.3 stress concentrators
4.3.1 Slip bands
4.3.2 Grain boundary sliding
4.3.3 Intergranular inclusion embedded in metallic grains
4.4 Long term lifetime prediction
4.4.1 Final evaluation of the Dyson law prefactor 0
4.4.2 lifetime predictions in 316 SSs
4.5 Discussion and conclusion
4.5.1 Cavity nucleation model
4.5.2 Evaluation of cavity nucleation rate
4.5.3 Lifetime predictions
4.5.4 Comparison of the long term creep resistance in Incoloy 800, 316L(N) and Grade 91 steel
5 Conclusions, work in progress and perspectives 
5.1 Conclusions
5.1.1 Experimental investigation of damage mechanisms
5.1.2 Finite element calculations
5.1.3 Enhanced prediction of creep lifetimes
5.2 Work in progress
5.3 Perspectives
5.3.1 Local stress concentration
5.3.2 Intergranular Diffusion
5.3.3 Precipitation
A Uncertainty in 0 
B Interface normal stress values 
C Cohesive law: HINTE

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