Self‐consistent phonon calculation of the elastic constants of the β phase of solid N2

1976 ◽  
Vol 64 (12) ◽  
pp. 5121-5125 ◽  
Author(s):  
Victor V. Goldman ◽  
Michael L. Klein
Keyword(s):  
Elastic Constants ◽  
Β Phase ◽  
Self Consistent ◽  
Phonon Calculation ◽  
Solid N2

Self‐consistent phonon calculation of the β phase and the α‐β transition in solid nitrogen

1974 ◽  
Vol 61 (4) ◽  
pp. 1411-1414 ◽  
Author(s):  
J. C. Raich ◽  
N. S. Gillis ◽  
T. R. Koehler
Keyword(s):  
Β Phase ◽  
Solid Nitrogen ◽  
Self Consistent ◽  
Phonon Calculation

Elastic constants of parahydrogen determined by Brillouin scattering

1978 ◽  
Vol 56 (11) ◽  
pp. 1494-1501 ◽  
Author(s):  
P. J. Thomas ◽  
S. C. Rand ◽  
B. P. Stoicheff
Keyword(s):  
Elastic Constants ◽  
Elastic Anisotropy ◽  
Brillouin Scattering ◽  
Direct Determination ◽  
Experimental Measurements ◽  
Frequency Shifts ◽  
Self Consistent ◽  
Good Agreement ◽  
Phonon Calculation

Five single crystals of parahydrogen were grown near the triple point, and their Brillouin spectra examined in known crystal directions. An analysis of the measured frequency shifts of longitudinal and transverse components led to the direct determination of the adiabatic elastic constants at 13.2 K: C11 = 3.34 ± 0.05, C13 = 0.56 ± 0.03, C33 = 4.08 ± 0.06, C44 = 1.04 ± 0.03 kbar. The fifth elastic constant C12 = 1.30 ± 0.05 kbar was evaluated from the relation C11 + C12 = C13 + C33. Values of the following constants were calculated: elastic anisotropy, A = 1.02 ± 0.06, adiabatic bulk modulus, Bs = 1.73 ± 0.04 kbar, and Debye temperature, θD = 111.2 K. All of these values were compared with earlier experimental measurements. The elastic constants were found to be in good agreement with a recent self-consistent phonon calculation by Goldman.

Constitutive Analysis of the High-Temperature Deformation Behavior of Single Phase α-Ti and α+β Ti-6Al-4V Alloy

2007 ◽  
Vol 539-543 ◽  
pp. 3607-3612 ◽  
Author(s):  
Jeoung Han Kim ◽  
Jong Taek Yeom ◽  
Nho Kwang Park ◽  
Chong Soo Lee
Keyword(s):  
High Temperature ◽  
Flow Stress ◽  
Deformation Behavior ◽  
Single Phase ◽  
Volume Fraction ◽  
High Temperature Deformation ◽  
Temperature Deformation ◽  
Β Phase ◽  
Α Phase ◽  
Self Consistent

The high-temperature deformation behavior of the single-phase α (Ti-7.0Al-1.5V) and α + β (Ti-6Al-4V) alloy were determined and compared within the framework of self-consistent scheme at various temperature ranges. For this purpose, isothermal hot compression tests were conducted at temperatures between 650°C ~ 950°C to determine the effect of α/β phase volume fraction on average flow stress under hot-working condition. The flow behavior of α phase was estimated from the compression test results of single-phase α alloy whose chemical composition is close to that of α phase of Ti-6Al-4V alloy. On the other hand, the flow stress of β phase in Ti-6Al-4V was predicted by using self-consistent method. The flow stress of α phase was higher than that of β phase above 750°C, while the β phase revealed higher flow stress than α phase at 650°C. Also, at temperature above 750°C, the predicted strain rate of β phase was higher than that of α phase. It was found that the relative strength between α and β phase significantly varied with temperature.

Composition dependence of the elastic constants of β-phase and (α+β)-phase PdHx

Ultrasonics ◽  
2010 ◽  
Vol 50 (2) ◽  
pp. 155-160 ◽  
Author(s):  
D.J. Safarik ◽  
R.B. Schwarz ◽  
S.N. Paglieri ◽  
R.L. Quintana ◽  
D.G. Tuggle ◽  
...  
Keyword(s):  
Elastic Constants ◽  
Composition Dependence ◽  
Β Phase

Implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for any crystal symmetry

2019 ◽  
Vol 34 (2) ◽  
pp. 103-109
Author(s):  
Arnold C. Vermeulen ◽  
Christopher M. Kube ◽  
Nicholas Norberg
Keyword(s):  
Elastic Constants ◽  
Ultrasonic Waves ◽  
The Self ◽  
Crystal Symmetry ◽  
Intermediate Step ◽  
X Rays ◽  
X Ray ◽  
Eshelby Inclusion ◽  
Eshelby Model ◽  
Self Consistent

In this paper, we will report about the implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for general, triclinic crystal symmetry. With applying appropriate symmetry relations, the point groups of higher crystal symmetries are covered as well. This simplifies the implementation effort to cover the calculations for any crystal symmetry. In the literature, several models can be found to estimate the polycrystalline elastic properties from single crystal elastic constants. In general, this is an intermediate step toward the calculation of the polycrystalline response to different techniques using X-rays, neutrons, or ultrasonic waves. In the case of X-ray residual stress analysis, the final goal is the calculation of X-ray Elastic constants. Contrary to the models of Reuss, Voigt, and Hill, the Kröner–Eshelby model has the benefit that, because of the implementation of the Eshelby inclusion model, it can be expanded to cover more complicated systems that exhibit multiple phases, inclusions or pores and that these can be optionally combined with a polycrystalline matrix that is anisotropic, i.e., contains texture. We will discuss a recent theoretical development where the approaches of calculating bounds of Reuss and Voigt, the tighter bounds of Hashin–Shtrikman and Dederichs–Zeller are brought together in one unifying model that converges to the self-consistent solution of Kröner–Eshelby. For the implementation of the Kröner–Eshelby model the well-known Voigt notation is adopted. The 4-rank tensor operations have been rewritten into 2-rank matrix operations. The practical difficulties of the Voigt notation, as usually concealed in the scientific literature, will be discussed. Last, we will show a practical X-ray example in which the various models are applied and compared.

Self-consistent calculations of elastic constants of polycrystalline graphite

Carbon ◽  
1976 ◽  
Vol 14 (4) ◽  
pp. 185-189 ◽  
Author(s):  
R.E. Smith ◽  
G.B. Spence ◽  
J.E. Guberntist ◽  
J.A. Krumhansl
Keyword(s):  
Elastic Constants ◽  
Polycrystalline Graphite ◽  
Self Consistent

scholarly journals Determining Ti-17 β-Phase Single-Crystal Elasticity Constants through X-Ray Diffraction and Inverse Scale Transition Model

2011 ◽  
Vol 681 ◽  
pp. 97-102 ◽  
Author(s):  
Sylvain Fréour ◽  
Emmanuel Lacoste ◽  
Manuel François ◽  
Ronald Guillén
Keyword(s):  
Mechanical Load ◽  
Cubic Phase ◽  
X Ray Diffraction ◽  
X Ray ◽  
Β Phase ◽  
Consistent Model ◽  
Self Consistent ◽  
Diffraction Lattice ◽  
Two Phases ◽  
Multi Phase

The scope of this work is the determination of single-crystals elastic constants (SEC) from X-ray diffraction lattice strains measurements performed on multi-phase polycrystals submitted to mechanical load through a bending device. An explicit three scales inverse self-consistent model is developed in order to express the SEC of a cubic phase, embedded in a multi-phase polycrystal, as a function of its X-ray Elasticity Constants. Finally, it is applied to a two-phases (α+β) titanium based alloy (Ti-17), in order to estimate Ti-17 β-phase unknown SEC. The purpose of the present work is to account the proper microstructure of the material. In particular, the morphologic texture of Ti-17 a-phase, i.e. the relative disorientation of the needle-shaped grains constituting this phase, is considered owing to the so-called Generalized Self-Consistent model.

Nonlinear Elastic Properties of Micro-Heterogeneous Media

1994 ◽  
Vol 116 (3) ◽  
pp. 325-330 ◽  
Author(s):  
E. Kro¨ner
Keyword(s):  
Statistical Methods ◽  
Elastic Constants ◽  
Linear Elasticity ◽  
Heterogeneous Media ◽  
Nonlinear Problems ◽  
The Self ◽  
Green Functions ◽  
First Approximation ◽  
Self Consistent ◽  
Nonlinear Elastic

Utilizing statistical methods known from linear elasticity it is shown how effective 3rd (and higher) order elastic constants (TOEC) of micro-heterogeneous media can be calculated. Emphasis is put on the self consistent scheme. The ensemble average of the fluctuating TOEC yields a 0th approximation to the rigorous selfconsistent moduli. A first approximation is also given in closed form. The insight that the well-established statistical methods of the linear theory, which uses Green functions, are applicable also to nonlinear problems is considered as the main result of this paper.

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