Refinement of the Maxwell Formula for Fiber-Reinforced Composites

The effective properties of the fiber-reinforced composite materials with fibers of square cross-section are investigated. The novel formula for the effective coefficient of thermal conductivity refining the classical Maxwell formula (MF) is derived. The methods of asymptotic homogenization, boundary shape perturbation and Schwarz alternating process are applied. It is shown that the principal term of the asymptotic expansion of the refined formula in powers of small size of inclusions coincides with the classical MF. The corrections to the MF are obtained for different values of geometrical and physical properties of the constituents of the composite material. The analytical and numerical analyses are carried out and illustrated graphically. In particular, the derived refined formula and the MF are compared for the limiting values of the geometric dimensions and physical properties of the composite. It is shown that the refined formula is applicable for the inclusions with any conductivity in the entire range of the geometric sizes of inclusions, including the limiting cases of inclusions with zero thermal conductivity and maximally large inclusions.

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Refinement of the Maxwell formula for composite reinforced by circular cross-section fibers. Part I: using the Schwarz alternating method

Acta Mechanica ◽

10.1007/s00707-020-02788-3 ◽

2020 ◽

Vol 231 (12) ◽

pp. 4971-4990 ◽

Cited By ~ 1

Author(s):

Igor I. Andrianov ◽

Jan Awrejcewicz ◽

Galina A. Starushenko ◽

Vladimir A. Gabrinets

Keyword(s):

Cross Section ◽

Effective Properties ◽

Circular Cross Section ◽

Principal Term ◽

The Novel ◽

Asymptotic Results ◽

Asymptotic Homogenization ◽

Schwarz Alternating Method ◽

Reinforced Composite ◽

Coefficient Of Thermal Conductivity

Abstract The effective properties of the fiber-reinforced composite materials with fibers of circular cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For analytical solution of the periodically repeated cell problem, the Schwarz alternating process is employed. The principal term of the refined formula coincides with the classical Maxwell formula. On the other hand, the refined formula can be used far beyond the area of applicability of the Maxwell formula. It can be used for dilute and non-dilute composites. It is confirmed by comparison with known numerical and asymptotic results.

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Refinement of the Maxwell formula for a composite reinforced by circular cross-section fibres. Part II: using Padé approximants

Acta Mechanica ◽

10.1007/s00707-020-02789-2 ◽

2020 ◽

Vol 231 (12) ◽

pp. 5145-5157

Author(s):

Igor I. Andrianov ◽

Jan Awrejcewicz ◽

Galina A. Starushenko ◽

Vladimir A. Gabrinets

Keyword(s):

Composite Materials ◽

Cross Section ◽

Volume Fraction ◽

Effective Properties ◽

Circular Cross Section ◽

The Novel ◽

High Contrast ◽

Asymptotic Results ◽

Reinforced Composite ◽

Coefficient Of Thermal Conductivity

Abstract The effective properties of the fiber-reinforced composite materials with fibers of circle cross section are investigated. The novel estimation for the effective coefficient of thermal conductivity refining the classical Maxwell formula is derived. The method of asymptotic homogenization is used. For an analytical solution of the periodically repeated cell problem the Schwarz alternating process (SAP) was employed. Convergence of this method was proved by S. Mikhlin, S. Sobolev, V. Mityushev. Unfortunately, the rate of the convergence is often slow, especially for nondilute high-contrast composite materials. For improving this drawback we used Padé approximations for various forms of SAP solutions with the following additive matching of obtained expressions. As a result, the solutions in our paper are obtained in a fairly simple and convenient form. They can be used even for a volume fraction of inclusion very near the physically possible maximum value as well as for high-contrast composite constituents. The results are confirmed by comparison with known numerical and asymptotic results.

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Mathematical Model of the Effective Properties of a Fiber Reinforced Composite with a Linearly Graded Transition Zone

Tire Science and Technology ◽

10.2346/1.3519536 ◽

2010 ◽

Vol 38 (4) ◽

pp. 286-307

Author(s):

Carey F. Childers

Keyword(s):

Mathematical Model ◽

Transition Zone ◽

Effective Properties ◽

Local Stress ◽

Stress And Strain ◽

Fiber Reinforced ◽

Strain Fields ◽

Shear Moduli ◽

Fiber Reinforced Composite ◽

Reinforced Composite

Abstract Tires are fabricated using single ply fiber reinforced composite materials, which consist of a set of aligned stiff fibers of steel material embedded in a softer matrix of rubber material. The main goal is to develop a mathematical model to determine the local stress and strain fields for this isotropic fiber and matrix separated by a linearly graded transition zone. This model will then yield expressions for the internal stress and strain fields surrounding a single fiber. The fields will be obtained when radial, axial, and shear loads are applied. The composite is then homogenized to determine its effective mechanical properties—elastic moduli, Poisson ratios, and shear moduli. The model allows for analysis of how composites interact in order to design composites which gain full advantage of their properties.

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