On the third-order bounds of the effective shear modulus of two-phase composites

2011 ◽  
Vol 43 (5) ◽  
pp. 269-275 ◽  
Author(s):  
X. Frank Xu
Keyword(s):  
Shear Modulus ◽  
Third Order ◽  
Two Phase ◽  
Effective Shear Modulus ◽  
The Third ◽  
Order Bounds

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%.

Probe size and 5th-order aberration

1987 ◽  
Vol 45 ◽  
pp. 134-135 ◽  
Author(s):  
Zhifeng Shao
Keyword(s):  
Electron Probe ◽  
Spherical Aberration ◽  
Wave Length ◽  
Cylindrical Symmetry ◽  
Electron Wave ◽  
Third Order ◽  
The Third ◽  
Probe Radius ◽  
Small Probe ◽  
Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

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.

Simulation of Two Phase Flow Dynamics in Flexible Riser Exit Geometries for Oil and Gas Applications

2018 ◽  
Vol 39 ◽  
pp. 1-13
Author(s):  
Ikpe E. Aniekan ◽  
Owunna Ikechukwu ◽  
Satope Paul
Keyword(s):  
Oil And Gas ◽  
Two Phase Flow ◽  
Flow Analysis ◽  
Computational Cost ◽  
Maximum Velocity ◽  
Oil Well ◽  
Phase Flow ◽  
Two Phase ◽  
The Third ◽  
Riser Pipe

Four different riser pipe exit configurations were modelled and the flow across them analysed using STAR CCM+ CFD codes. The analysis was limited to exit configurations because of the length to diameter ratio of riser pipes and the limitations of CFD codes available. Two phase flow analysis of the flow through each of the exit configurations was attempted. The various parameters required for detailed study of the flow were computed. The maximum velocity within the pipe in a two phase flow were determined to 3.42 m/s for an 8 (eight) inch riser pipe. After thorough analysis of the two phase flow regime in each of the individual exit configurations, the third and the fourth exit configurations were seen to have flow properties that ensures easy flow within the production system as well as ensure lower computational cost. Convergence (Iterations), total pressure, static pressure, velocity and pressure drop were used as criteria matrix for selecting ideal riser exit geometry, and the third exit geometry was adjudged the ideal exit geometry of all the geometries. The flow in the third riser exit configuration was modelled as a two phase flow. From the results of the two phase flow analysis, it was concluded that the third riser configuration be used in industrial applications to ensure free flow of crude oil and gas from the oil well during oil production.

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