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.

scholarly journals Self-Consistent Calculation of Particle-Hole Diagrams on the Matsubara Frequency: Flex Approximation

1997 ◽  
Vol 08 (05) ◽  
pp. 1145-1158
Author(s):  
J. J. Rodríguez-Núñez ◽  
S. Schafroth
Keyword(s):  
Hubbard Model ◽  
Imaginary Part ◽  
Chemical Potential ◽  
Order Approximation ◽  
The Self ◽  
Green Functions ◽  
Matsubara Frequency ◽  
Self Energy ◽  
Peak Structure ◽  
Self Consistent

We implement the numerical method of summing Green function diagrams on the Matsubara frequency axis for the fluctuation exchange (FLEX) approximation. Our method has previously been applied to the attractive Hubbard model for low density. Here we apply our numerical algorithm to the Hubbard model close to half filling (ρ =0.40), and for T/t = 0.03, in order to study the dynamics of one- and two-particle Green functions. For the values of the chosen parameters we see the formation of three branches which we associate with the two-peak structure in the imaginary part of the self-energy. From the imaginary part of the self-energy we conclude that our system is a Fermi liquid (for the temperature investigated here), since Im Σ( k , ω) ≈ w2 around the chemical potential. We have compared our fully self-consistent FLEX solutions with a lower order approximation where the internal Green functions are approximated by free Green functions. These two approches, i.e., the fully self-consistent and the non-self-consistent ones give different results for the parameters considered here. However, they have similar global results for small densities.

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.

Variational estimates for the creep behaviour of polycrystals

1995 ◽  
Vol 448 (1932) ◽  
pp. 121-142 ◽  
Keyword(s):  
Heterogeneous Media ◽  
Creep Behaviour ◽  
The Self ◽  
Polycrystalline Materials ◽  
Variational Procedure ◽  
Constitutive Behaviour ◽  
Effective Potentials ◽  
Self Consistent ◽  
Geometric Arrangements

A variational procedure is developed for estimating the effective constitutive behaviour of polycrystalline materials undergoing high-temperature creep. The procedure is based on a new variational principle allowing the determination of the effective potential function of a given nonlinear polycrystal in terms of the corre­sponding potential for a linear comparison polycrystal with an identical geometric arrangements of its constituent single-crystal grains. As such, it constitutes an extension, to locally anisotropic behaviour, of the variational procedure devel­oped by Ponte Castañeda (1991) for nonlinear heterogeneous media with locally isotropic behaviour. By way of an example, the procedure is applied to the de­termination of bounds of the Hashin-Shtrikman type for the effective potentials of statistically isotropic nonlinear polycrystals. The bounds are computed for the special class of untextured FCC polycrystals with isotropic pure power-law viscous behaviour, first considered by Hutchinson (1976), in the context of a calculation of the self-consistent type. The new bounds are found to be more restrictive than the corresponding classical Taylor-Bishop-Hill bounds, and also more re­strictive, if only slightly so, than related bounds of the Hashin-Shtrikman type by Dendievel et al . (1991). The new procedure has the advantage over the self-consistent procedure of Hutchinson (1976) that it may be applied, without any essential complications, to aggregates of crystals with slip systems exhibiting dif­ferent creep rules - with, for example, different power exponents - and to general loading conditions. However, the distinctive feature of the new variational proce­dure is that it may be used in conjunction with other types of known bounds and estimates for linear polycrystals to generate corresponding bounds and estimates for nonlinear polycrystals.

scholarly journals Crystal theory of metals: calculation of the elastic constants

1942 ◽  
Vol 180 (983) ◽  
pp. 451-476 ◽  
Keyword(s):  
Potential Energy ◽  
Perturbation Method ◽  
Elastic Constants ◽  
Equations Of Motion ◽  
The Self ◽  
Consistent Field ◽  
Self Consistent ◽  
Self Consistent Field ◽  
Set Up ◽  
Approximate Equations

The approximate equations of motion for the electrons in a cyclic lattice of a metal are set up with the help of the self-consistent field. The displacements of the ions are then considered as perturbations of the motion of the electrons. The change of the boundary is compensated by a co-ordinate transformation. The change of the potential energy of the lattice due to a homogeneous deformation is calculated by the perturbation method. The calculated values of the elastic constants are found to be in satisfactory agreement with the observed values.

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