Three-dimensional finite element analysis of stresses and energy density distributions around γ′ before coarsening loaded in the [1 1 0]-direction in Ni-based superalloy

doi:10.1016/S0921-5093(02)00901-2

Materials Science and Engineering: A

Volume 352, Issues 1–2, 15 July 2003, Pages 300-307

https://doi.org/10.1016/S0921-5093(02)00901-2 Get rights and content

Abstract

A three-dimensional (3D) anisotropic finite element (FE) analysis was employed to analyse the features about the elastic strain energy density, von Mises stress and hydrostatic pressure distributions in Ni-based superalloy single crystals when external loads applied along the [1 1 0]-direction. The asymmetry of the strain energy density, the von Mises stress and hydrostatic pressure could be used to predict the raft morphology for alloys exhibiting a positive or negative misfit. The reason was briefly discussed.

Introduction

One of the most striking characteristics of high-temperature creep deformation in Ni-based superalloy single crystals is the rapid directional coarsening of the cuboidal γ′ precipitates into preferentially orientated plates or rafts. The morphology of the rafts depends upon several parameters such as the direction of the creep stress and the sign of the misfit δ between the γ and γ′ phases. The understanding of the mechanisms responsible for raft formation and modeling of the process are of great interest, as this microstructural change is thought to have an influence on the creep resistance of superalloys used as turbine blades [1], [2]. Different experimental techniques [3], [4], [5], [6] and a number of finite element (FE) computations [2], [7], [8], [9], [10] have been applied for describing the directional coarsening of the γ′ precipitates in the case of a creep stress applied along the [0 0 1]-direction. However, except for the experimental observations conducted by Tien and Copley [11] and the dislocation model proposed by Buffiere and Ignat [12], most work did not describe the evolution of the raft morphology as a function of the creep stress direction. Tien and Copley studied the change in morphology of the γ′ precipitates due to stress annealing applied parallel to the [1 1 0] and [1 1 1] orientations. Experimental results indicate that if the external stress is along the [0 0 1]-direction, the rafted morphologies are: (1) plates aligned perpendicular to the stress axis and (2) plates or needles aligned parallel to the stress axis. If the stress axis is along [1 1 0], the rafted morphologies are also divided into two groups: (1) parallelepipeds aligned parallel to the [0 0 1]-direction and perpendicular to the [1 1 0] stress axis and (2) plates of γ′ with broad faces parallel to the {0 0 1}-planes containing the stress axis (see Fig. 1a and b).

Since in the case of a creep stress applied along the [0 0 1]-direction, the rafting mechanisms are rather well understood [7]; in this paper, only the creep stress applied along the [1 1 0]-direction is discussed. The aim of this study is to investigate how the elastic strain energy density, von Mises stress and hydrostatic pressure are affected by the external stress applied parallel to the [1 1 0]-orientation and the lattice misfit, in order to have a better understanding of the mechanism of rafting. Due to the high degree of elastic anisotropy ratios (see Table 1) of the precipitates and matrix, it is necessary to use the three-dimensional (3D) anisotropic FE method; the FE calculation is performed with the ANSYS software code.

Section snippets

Numerical models

The material model is shown in Fig. 2. The γ′ precipitates with the cuboidal shape are uniformly distributed in the γ matrix, and the volume fraction of the γ′ phase is ∼70%. Because the external load is along the [1 1 0]-direction, only the volume surrounded by the dashed lines (Fig. 2a) was investigated in a 3D FE model. Fig. 2b shows the 3D FE meshes. The global coordinate system is denoted as x, y, z. The crystal coordinate axes [1 0 0], [0 1 0], [0 0 1] are denoted as x′, y′, z′, respectively. The

Method for computation

After the FE calculation, the elastic strain energy density (G) of each element is obtained by $G= 12 (σ_{xx} ε_{xx} +σ_{yy} ε_{yy} +σ_{zz} ε_{zz} +σ_{xy} ε_{xy} +σ_{yz} ε_{yz} +σ_{zx} ε_{zx}).$ The von Mises stress σ_von is given by $σ_{von} ={12 [(σ_{xx} −σ_{yy})^{2} +(σ_{yy} −σ_{zz})^{2} +(σ_{zz} −σ_{xx})^{2} +6(σ_{xy}^{2} +σ_{yz}^{2} +σ_{zx}^{2})]}^{1/2} .$ The hydrostatic pressure σ_h is given by $σ_{h} = 13 (σ_{xx} +σ_{yy} +σ_{zz}).$

Results and discussion

In order to obtain the equivalent matrix channels for investigating the stress and energy distributions, it is necessary to specify a cutting plane, which is typical for the evaluation of the stress distribution with or without loading. In this work, the plane oo₁o₂o₃ is selected (see Fig. 2b). For simplicity, the matrix channel parallel to the loading direction is defined as parallel channel (PC), while the matrix channel that is 45° with respect to the loading direction is defined as declined

Conclusions

The distribution of elastic strain energy density, von Mises stress and hydrostatic pressure could be used to predict the rafting directions of γ′ in Ni-based superalloys when the external load is applied along the [1 1 0]-direction.

Under the external load, the difference in the hydrostatic pressure between PCs and DCs is smaller as compared with that of the elastic strain energy and von Mises stress.

Acknowledgements

This work was financially supported by the key projects of National Fundamental Research of China (G 19990650 and G 2000671).

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Three-dimensional finite element analysis of stresses and energy density distributions around γ′ before coarsening loaded in the [1 1 0]-direction in Ni-based superalloy

Abstract

Introduction

Section snippets

Numerical models

Method for computation

Results and discussion

Conclusions

Acknowledgements

Mater. Sci. Eng.

Acta Metall. Mater.

Scripta Metall. Mater.

Acta Mater.

Scripta Mater.

Acta Metall. Mater.

Acta Metall. Mater.

Acta Mater.