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A constitutive model for the mechanical behavior at high temperatures of superalloys with a high volume fraction of the ? ? phase is derived from an analysis of the main deformation mechanisms. The model assumes periodically distributed cubic ? ? particles and an homogeneous slip distribution perpendicular to the slip planes inside the matrix channels and the particles. It accounts for octahedral and cubic slip in the ? channels, recovery by climb of loops around the precipitates until complete annihilation, shear of both matrix and precipitates by complete matrix dislocations and partial slip reversal during unloading. The back-stresses of the constitutive law are identified with the long-range internal stresses in the microstructure which are explicitly calculated by Fourier series for a simplified distribution of the plastic strains in the microstructure. In particular the three types of channels are distinguished. The internal variables of the model are the plastic strains produced by each mechanism, the number of mobile loops in the ? channels and the scalar dislocation density in the ?/? ? interfaces for each slip system.
On the Connection between Continuum Crystal Plasticity and the Mechanics of Discrete Dislocations
(2003)
The calculation of the glide force on a dislocation in finite elasticity requires a suitable description of the variation of the deformation due to dislocation motion. This is achieved in this paper by extending the linear Somigliana-type dislocation model to finite strains. The core is modeled by a strip of finite width where the displacement jump continuously decreases with the distance to the dislocation tip. In the case of finite transformations, this distance can be defined in the deformed or in the undeformed state. By defining it in the undeformed natural state of the crystal, we show that the work dissipated by friction in the core region only depends on the dislocation motion. Using the principle of virtual work, the equation of the dislocation motion and several alternative expressions for the glide force are derived. The obtained force may be interpreted as a generalization of Eshelby's configurational forces for a noncoherent singular surface. In the linear case, the glide force of Peach and Koehler as well as the well-known expression of the self-force due to De Wit and Koehler are retrieved.
A stochastic model for fatigue short crack growth is presented. It takes into account the interaction between the crack-tip plastic zone and grain boundaries. The process is Markovian. It is completely described by the crack length and the size of the plastic zone. The integro-differential equation giving the evolution of the transition probability distribution is derived.
Simulations of cavitation processes on a grain boundary under creep conditions have been carried out, taking into account nucleation, growth, coalescence and sintering of multiple cavities. Cavity growth rates have been calculated through a resolution procedure based on the use of holomorphic complex functions. Three dimensionless parameters have been identified that are responsible for the cavitation development. Parameter studies have been carried out in order to characterise the mechanisms that are responsible for the cavitation development. These studies have been used to develop a physically motivated, simplified model in order to describe the cavitation development of the simulations in terms of global state variables. The simplified model is able to reproduce the cavitation development for all considered combinations of the dimensionless parameters and allows the description of the thickening behaviour of a cavitating grain boundary.
The present work focuses on the cubic slip phenomenon in single crystal superalloys with a high fraction of γ' precipitates at high temperature.
The macroscopically apparent cubic slip mechanism is known to significantly reduce the tensile and creep strength of <1 1 1> oriented specimens. However, recent results obtained by 3D dislocation dynamics (Vattré et al., 2009) suggest a new interpretation of the so-called pseudo-cubic slip related to the nature of the dislocation network formation at the interfaces. These results are taken into account in a micromechanical model to improve the estimation of the strain hardening anisotropy. In accordance with the discrete simulations, it is shown that a key role is played by the combination of the activated octahedral slip systems as a function of the crystal orientation. In the <1 1 1> case, the contribution of the activated systems to kinematical hardening compensate, whereas strain hardening remains high for the <0 0 1> case. The experimental dependence of plastic flow on the orientation is also explained at the dislocation-scale mechanisms.
Results are presented for the alloy CMSX-4 in several orientations.
An extension of the constitutive viscoplastic model of Meric and Cailletaud is presented. The new model
accounts for coupling of plasticity and viscoplasticity to describe the rate dependence at low and moderate
temperatures. The model contains two boundaries: an elastic one and a viscoplastic one. Between the
boundaries, the only contribution to yielding is the rate dependent viscoplastic mechanism. Once the viscoplastic
boundary is reached, an additional rate independent flow mechanism becomes active. However, the extended
model is not able to predict properly both creep and long-term relaxation tests simultaneously. Therefore, a
deformation-induced softening is assumed, which is supposed to mainly affect static recovery.
The model has been calibrated with the mechanical tests on a single crystal nickel-basis superalloy. The uniaxial
tests have been carried out in the temperature interval 600°C – 1100°C in [001], [011] and [111] specimens.
The predicted creep, relaxation and non-isothermal cyclic tests exhibit reasonable agreement with the
experimental observations.
Crack growth tests have been performed at 950 °C with Single Edge Notch specimens of the Ni-based single crystal superalloy PWA1483. In particular, several orientations and frequencies have been investigated, thus allowing the assessment of the influence of these parameters on the crack growth rate. In addition, oxidation experiments have been carried out to characterize the kinetics of the outer oxide scale growth at the same temperature.
On the other side, crack growth has been simulated with the Finite Element program ABAQUS in real test conditions by the node release technique. The nodes are released according to the measured crack growth rate.
The simulation results are compared with the test results on the basis of the computed Crack Tip Opening Displacement (CTOD). For this purpose, the crack is propagated until a stabilized value of the CTOD is obtained. This is usually the case when the crack has crossed the initial plastic zone. The procedure provides an evaluation of the effects of cycle frequency, crystal orientation, plasticity and oxide induced crack closure.