New bounds on effective elastic moduli of two-component materials

1982 ◽  
Vol 380 (1779) ◽  
pp. 305-331 ◽  
Keyword(s):  
Shear Modulus ◽  
Bulk Modulus ◽  
Fourth Order ◽  
Geometrical Parameters ◽  
Effective Moduli ◽  
Third Order ◽  
Effective Shear Modulus ◽  
The Third ◽  
Two Component ◽  
Order Bounds

Based on the perturbation solution, we derive new bounds on the effective moduli of a two-component composite material which are exact up to fourth order in δ μ = μ 1 — μ 2 and δ K = K 1 — K 2 , where μ i and K i , i = 1, 2, are the shear and bulk modulus, respectively, of the phases. The bounds on the effective bulk modulus involve three microstructural parameters whereas eight parameters are needed in the bounds on the effective shear modulus. For engineering calculations, we recommend the third-order bounds on the effective shear modulus which require only two geometrical parameters. We show in detail how Hashin-Shtrikman’s bounds can be extended and how Walpole’s bounds can be improved using two inequalities on the two geometrical parameters that appear in the third-order bounds on the effective shear modulus. The third- and fourth-order bounds on the effective moduli are shown to be more restrictive than, or at worst, coincident with, existing bounds due to Hashin and Shtrikman, McCoy, Beran and Molyneux and Walpole.

New bounds on the effective thermal conductivity of N -phase materials

1982 ◽  
Vol 380 (1779) ◽  
pp. 333-348 ◽  
Keyword(s):  
Thermal Conductivity ◽  
Effective Thermal Conductivity ◽  
Fourth Order ◽  
Third Order ◽  
Two Phase ◽  
The Third ◽  
Component Material ◽  
Order Bounds ◽  
Correlation Parameters ◽  
Two Parameters

Based on the perturbation solution to the effective thermal conductivity problem for an N -component material, we derive new third- and fourth-order bounds for the effective thermal conductivity of the composite. The new third-order bounds are accurate to third order in | ϵ a — ϵ b |, where ϵ a is the thermal conductivity of phase a and require a total of ½ N ( N — 1) 2 third-order correlation parameters in their evaluation. For two-phase composites, these bounds require only one geometrical parameter and are identical to Beran’s bounds. For N ≽ 3, the third-order bounds use the same information as Beran’s bounds, but always more restrictive than Beran’s bounds. The new fourth-order bounds are accurate to fourth-order in | ϵ a - ϵ b | and require an additional ¼ N 2 (N — 1) 2 fourth-order correlation parameters. For N = 2, only two parameters are needed, and the fourth-order bounds are shown to be more restrictive than the third-order Beran’s bounds and the second-order Hashin-Shtrikman’s bounds. The applica­tion of the third-order bounds to a spherical cell material of Miller is illustrated.

scholarly journals Bounds on the conductivity of a random array of cylinders

1988 ◽  
Vol 417 (1852) ◽  
pp. 59-80 ◽  
Keyword(s):  
Fourth Order ◽  
Volume Fraction ◽  
Effective Conductivity ◽  
Basis Set ◽  
Circular Cylinders ◽  
Third Order ◽  
Entire Volume ◽  
The Third ◽  
Wide Range ◽  
Order Bounds

We consider the problem of determining rigorous third-order and fourth-order bounds on the effective conductivity σ e of a composite material composed of aligned, infinitely long, equisized, rigid, circular cylinders of conductivity σ 2 randomly distributed throughout a matrix of conductivity σ 1 . Both bounds involve the microstructural parameter ξ 2 which is an integral that depends upon S 3 , the three-point probability function of the composite (G. W. Milton, J. Mech. Phys. Solids 30, 177-191 (1982)). The key multidimensional integral ξ 2 is greatly simplified by expanding the orientation-dependent terms of its integrand in Chebyshev polynomials and using the orthogonality properties of this basis set. The resulting simplified expression is computed for an equilibrium distribution of rigid cylinders at selected ϕ 2 (cylinder volume fraction) values in the range 0 ≼ ϕ 2 ≼ 0.65. The physical significance of the parameter ξ 2 for general microstructures is briefly discussed. For a wide range of ϕ 2 and α = σ 2 /σ 1 , the third-order bounds significantly improve upon second-order bounds which only incorporate volume fraction information; the fourth-order bounds, in turn, are always more restrictive than the third-order bounds. The fourth-order bounds on σ e are found to be sharp enough to yield good estimates of σ e for a wide range of ϕ 2 , even when the phase conductivities differ by as much as two orders of magnitude. When the cylinders are perfectly conducting ( α = ∞), moreover, the fourth-order lower bound on σ e provides an excellent estimate of this quantity for the entire volume-fraction range studied here, i. e. up to a volume fraction of 65%.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

Exact equations for fluid and solid substitution

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. L21-L32 ◽  
Author(s):  
Nishank Saxena ◽  
Gary Mavko
Keyword(s):  
Shear Stress ◽  
Shear Modulus ◽  
Bulk Modulus ◽  
Fluid Substitution ◽  
Pore Shape ◽  
Effective Shear Modulus ◽  
Stiffness Measurement ◽  
Saturated Rock ◽  
Mean Stresses

We derived exact equations, elastic bulk and shear, for fluid and solid substitution in monomineralic isotropic rocks of arbitrary pore shape and suggested methods to obtain the required substitution parameters. We proved that the classical Gassmann’s bulk modulus equation for fluid-to-fluid substitution is exact for solid-to-solid substitution if compression-induced mean stresses (pressure) in initial and final pore solids are homogeneous and either the shear modulus of the substituted solid does not change or no shear stress is induced in pores. Moreover, when compression-induced mean stresses in initial and final pore solids are homogeneous, we evaluated exact generalizations of Gassmann’s bulk modulus equation, which depend on usually known parameters. For the effective shear modulus, we found general exactness conditions of Gassmann and other approximations. Using the new exact substitution equations, we interpreted that predicting solid-filled rock stiffness from a dry rock stiffness measurement requires more information (i.e., assumptions about the pore shape) compared to predicting the same from a fluid-saturated rock stiffness.

scholarly journals Asymptotic formulae for linear equations

1972 ◽  
Vol 13 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Asymptotic Behaviour ◽  
Fourth Order ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Fourth Order Equation ◽  
The Third ◽  
Asymptotic Formulae ◽  
Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

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 bounds on the conductivity of composites from symmetry considerations

1993 ◽  
Vol 443 (1918) ◽  
pp. 451-455 ◽  
Keyword(s):  
Upper Bound ◽  
Substantial Improvement ◽  
Higher Order ◽  
Geometric Parameters ◽  
Third Order ◽  
Symmetry Properties ◽  
Two Component ◽  
Order Bounds ◽  
Parameter Dependent ◽  
Independent Pair

It is well known that symmetry considerations can lead to improved bounds on, or even determine, the conductivity of two-component symmetric materials. The present work exploits symmetry properties to derive explicit higher-order bounds for three-component symmetric materials. The bounds contain geometric parame­ters. But even without any knowledge of these geometric parameters, substantial improvement on previous bounds is made. This is discussed in the context of equiaxed polycrystals. Results include a parameter-independent pair of bounds that for some polycrystals becomes third-order, and a parameter-dependent third-order upper bound that can be partially attained.

STANDARD NOMOGRAPHIC FORMS FOR EQUATIONS IN THREE VARIABLES

1935 ◽  
Vol 12 (1) ◽  
pp. 14-40 ◽  
Author(s):  
F. M. Wood
Keyword(s):  
Unit Circle ◽  
Fourth Order ◽  
Third Order ◽  
Rectangular Hyperbola ◽  
Practical Utility ◽  
The Third ◽  
Cubic Curve ◽  
Final Adjustment ◽  
Straight Lines ◽  
Suitable Location

Equations of the third and fourth nomographic order in three variables have been dealt with and classified. Equations of the third order may be reduced to one of two standard forms, α + β + γ = 0 and α + βγ = 0, which give alignment charts composed of three straight lines. Equations of the fourth order may also be reduced to one of two standard forms, resulting in charts composed of (a) two straight lines and a curve, or (b) two scales on a conic, and the third on another curve. Transformations of these four standard forms are given which permit of rapid and easy adjustment of the position and length of the scales for any given example, resulting in a chart of practical utility. Although the underlying theory has been studied by other writers, notably Soreau and Clark, it has possibly never appeared before in such a neat form. On this account, and also because of the standard transformations, it is felt that this article is of particular value.Standard forms have also been developed for third order equations leading to charts composed of two scales on a conic and a third straight scale, and in conclusion a third type of chart, in which all three scales appear on a single cubic curve, has been standardized. The practical value of the last type is questionable, but the conic charts are of use since we may arbitrarily choose the unit circle, or the rectangular hyperbola, for our conic scales. Final adjustment forms which permit suitable location of the scales in particular examples have been obtained in every case.

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.

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