scholarly journals Higher order derivatives of bulk modulus of materials at infinite pressure

2016 ◽  
Vol 94 (8) ◽  
pp. 748-750 ◽  
Author(s):  
A. Dwivedi
Keyword(s):  
Bulk Modulus ◽  
Higher Order ◽  
Grüneisen Parameter ◽  
Third Order ◽  
Pressure Derivative ◽  
Basic Principles ◽  
Gruneisen Parameter ◽  
The Third ◽  
The Status ◽  
Derivatives Of

Pressure derivatives of bulk modulus of materials at infinite pressure or extreme compression have been studied using some basic principles of calculus. Expressions for higher order pressure derivatives at infinite pressure are obtained that are found to have the status of identities. A generalized formula is derived for the nth-order pressure derivative of bulk modulus in terms of the third-order Grüneisen parameter at infinite pressure.

scholarly journals Volume derivatives of the Grüneisen parameter in the limit of infinite pressure

2019 ◽  
Vol 97 (1) ◽  
pp. 114-116 ◽  
Author(s):  
A. Dwivedi
Keyword(s):  
Boundary Condition ◽  
Fourth Order ◽  
Fundamental Theorem ◽  
High Pressures ◽  
Grüneisen Parameter ◽  
Third Order ◽  
Gruneisen Parameter ◽  
The Third ◽  
Properties Of Materials ◽  
Derivatives Of

Expressions have been obtained for the volume derivatives of the Grüneisen parameter, which is directly related to the thermal and elastic properties of materials at high temperatures and high pressures. The higher order Grüneisen parameters are expressed in terms of the volume derivatives, and evaluated in the limit of infinite pressure. The results, that at extreme compression the third-order Grüneisen parameter remains finite and the fourth-order Grüneisen parameter tends to zero, have been used to derive a fundamental theorem according to which the volume derivatives of the Grüneisen parameter of different orders, all become zero in the limit of infinite pressure. However, the ratios of these derivatives remain finite at extreme compression. The formula due to Al’tshuler and used by Dorogokupets and Oganov for interpolating the Grüneisen parameter at intermediate compressions has been found to satisfy the boundary condition at infinite pressure obtained in the present study.

HIGHER ORDER THERMOELASTIC PROPERTIES OF THE EARTH LOWER MANTLE AND CORE

2010 ◽  
Vol 24 (09) ◽  
pp. 1187-1200 ◽  
Author(s):  
S. S. KUSHWAH ◽  
N. K. BHARDWAJ
Keyword(s):  
Bulk Modulus ◽  
Lower Mantle ◽  
Inner Core ◽  
Outer Core ◽  
Higher Order ◽  
Grüneisen Parameter ◽  
Free Volume Theory ◽  
Gruneisen Parameter ◽  
The Earth ◽  
Derivatives Of

We have used some of the most reliable high pressure equations of state (EOS) to determine the thermoelastic Grüneisen parameter and its higher order volume derivatives for the lower mantle, outer core and inner core of the Earth. The cross derivatives of bulk modulus with respect to pressure and temperature have also been obtained for the deep interior of the Earth using the results based on the modified free volume theory for the Grüneisen parameter. We have used five EOS viz. (a) modified Rydberg EOS, (b) modified Poirier–Tarantola EOS, (c) Hama–Suito EOS, (d) Stacey EOS, and (e) Kushwah EOS to determine pressure derivatives of bulk modulus. The results for thermoelastic parameters obtained in the present study show systematic variations with the increase in pressure.

Formulation of the third-order Grüneisen parameter at extreme compression

2012 ◽  
Vol 407 (12) ◽  
pp. 2082-2083 ◽  
Author(s):  
J. Shanker ◽  
K. Sunil ◽  
B.S. Sharma
Keyword(s):  
Grüneisen Parameter ◽  
Third Order ◽  
Gruneisen Parameter ◽  
The Third

Volume Dependence of Thermodynamic Properties for Solids at high Temperatures

2014 ◽  
Vol 69 (10-11) ◽  
pp. 532-538
Author(s):  
Guanglei Cui ◽  
Bai Fan ◽  
Zewen Zuo ◽  
Min Gu ◽  
Ruilan Yu
Keyword(s):  
Thermal Expansion ◽  
Sodium Chloride ◽  
Bulk Modulus ◽  
Grüneisen Parameter ◽  
Isothermal Bulk Modulus ◽  
Pressure Derivative ◽  
Gruneisen Parameter ◽  
Computing Model ◽  
Volume Dependence ◽  
Sodium And Potassium

AbstractA new computing model on the volume dependence of the product αKT of the thermal expansion coefficient α and the isothermal bulk modulus KT is proposed straightforward in this paper. Based on this revised formula, the volume dependence of Grüneisen parameter, entropy, Anderson-Grüneisen parameter, and first pressure derivative of bulk modulus, respectively, are thus investigated. The calculated results agree well with the previous work for magnesium oxide, sodium chloride, lithium, sodium, and potassium.

scholarly journals Third derivatives of the integrable part of an asteroid Hamiltonian

2007 ◽  
pp. 53-60 ◽  
Author(s):  
R. Pavlovic
Keyword(s):  
Third Order ◽  
Geometrical Condition ◽  
The Third ◽  
Derivatives Of ◽  
Quasi Convexity ◽  
Explicit Expressions

To apply the theorem of Nekhoroshev (1977) to asteroids, one first has to check whether a necessary geometrical condition is fulfilled: either convexity, or quasi-convexity, or only a 3-jet non-degeneracy. This requires computation of the derivatives of the integrable part of the corresponding Hamiltonian up to the third order over actions and a thorough analysis of their properties. In this paper we describe in detail the procedure of derivation and we give explicit expressions for the obtained derivatives. .

A SPECTRAL/hp NONLINEAR FINITE ELEMENT ANALYSIS OF HIGHER-ORDER BEAM THEORY WITH VISCOELASTICITY

2012 ◽  
Vol 04 (01) ◽  
pp. 1250010 ◽  
Author(s):  
V. P. VALLALA ◽  
G. S. PAYETTE ◽  
J. N. REDDY
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Virtual Work ◽  
Constitutive Relations ◽  
Beam Theory ◽  
Element Model ◽  
Higher Order ◽  
Time Step ◽  
Third Order ◽  
The Third

In this paper, a finite element model for efficient nonlinear analysis of the mechanical response of viscoelastic beams is presented. The principle of virtual work is utilized in conjunction with the third-order beam theory to develop displacement-based, weak-form Galerkin finite element model for both quasi-static and fully-transient analysis. The displacement field is assumed such that the third-order beam theory admits C0 Lagrange interpolation of all dependent variables and the constitutive equation can be that of an isotropic material. Also, higher-order interpolation functions of spectral/hp type are employed to efficiently eliminate numerical locking. The mechanical properties are considered to be linear viscoelastic while the beam may undergo von Kármán nonlinear geometric deformations. The constitutive equations are modeled using Prony exponential series with general n-parameter Kelvin chain as its mechanical analogy for quasi-static cases and a simple two-element Maxwell model for dynamic cases. The fully discretized finite element equations are obtained by approximating the convolution integrals from the viscous part of the constitutive relations using a trapezoidal rule. A two-point recurrence scheme is developed that uses the approximation of relaxation moduli with Prony series. This necessitates the data storage for only the last time step and not for the entire deformation history.

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