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Research Papers: Gas Turbines: Manufacturing, Materials, and Metallurgy

Application of a Smooth Approximation of the Schmid's Law to a Single Crystal Gas Turbine Blade

[+] Author and Article Information
Alessandro D. Ramaglia

Ansaldo Energia,
via N. Lorenzi 8,
Genova, Italy
e-mail: alessandro.ramaglia@aen.ansaldo.it

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the Journal of Engineering for Gas Turbines and Power. Manuscript received September 26, 2012; final manuscript received September 28, 2012; published online February 21, 2013. Editor: Dilip R. Ballal.

J. Eng. Gas Turbines Power 135(3), 032101 (Feb 21, 2013) (9 pages) Paper No: GTP-12-1379; doi: 10.1115/1.4007785 History: Received September 26, 2012; Revised September 28, 2012

In industrial practice the choice of the most suitable material model does not solely rely on the ability of the model in describing the intended phenomena. Most of the choice is often based on a trade-off between a great variety of factors. Robustness, cost, and time for the minimum testing campaign necessary to identify the model and preexisting standard practices are only a few of them. This is particularly true in the case of nonlinear structural analyses because of their intrinsic difficulties and the higher level of skills needed to carefully exploit their full potential. So, despite the great progress in this field, in certain cases it is desirable to use plasticity models that are rate independent and possess very simple hardening terms. This is for example the case in which long term creep can be an issue or when the designer may want to treat separately different phenomena contributing to inelastic deformation. If the material to be modeled is isotropic, commercial finite element (FE) packages are able to deal with such problems in almost every case. On the contrary for anisotropic materials like Ni-based superalloys cast as single crystals, the choice of the designer is more limited and despite the large amount of research literature on the subject, single crystal constitutive models remain quite difficult to handle, to implement into FE codes, to calibrate, and to validate. Such difficulties, coupled with the unavoidable approximations introduced by any model, often force the practice of using oversimplifications of the material behavior. In what follows this problem is addressed by showing how single crystal plasticity modeling can be reduced to the adoption of an anisotropic elastic behavior with a sort of von Mises yield surface.

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Figures

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Fig. 1

Yield surface corresponding to Schmid’s law and conceptual scheme used to represent it (normalized scale). In the case of the state of stress σ¯', the final yield surface obtained by taking into account cubic slip contains a smaller elastic domain with respect to the assumption of octahedral slip only. It is worth noting that the shape of this surface varies with the ratio between the yield stress along the two crystal directions 〈100〉 and 〈111〉.

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Fig. 2

Diagonal stress components do not activate cubic slip systems and yield is due only to octahedral systems

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Fig. 3

Variation of the maximum Schmid’s factor Mαd with direction: (a) octahedral slip only, (b) cube slip only, and (c) octahedral and cube slip

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Fig. 4

Dotted lines represent the actual yield surface as it is predicted by Schmid’s law for load case 1 of the previous paragraph. Solid lines are examples of the approximated yield surfaces fm for several values of the m parameter. As m increases, the actual yield surface given by the Schmid’s law is better and better approximated from the inside. (a) Case 1, state of stress σ¯' and (b) case 1, state of stress σ¯".

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Fig. 5

Tensile yield surface for various values of the ratio r=Rp〈100〉/Rp〈111〉: (a) octahedral slip only, r = 0.5, (b) combined octahedral and cubic slip for r = 0.77, and (c) octahedral and cubic slip for r = 1

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Fig. 6

(a) m = 10 and (b) m = 25: Effect of m on the approximation in the case of the general tensile load and r = 0.77. (c) Approximate yield surface for the case of octahedral slip only and m = 15.

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Fig. 7

Application of a postprocessing routine to a single crystal blade for (a) m = 25 and (b) m = 5 (normalized values)

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Fig. 8

Result of the explicit integration algorithm for various values of m (normalized material parameters)

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Fig. 9

Representation of the 18 slip systems on the unit fcc cell

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