Micromechanics of Random Heterogeneous Materials: New Background, Opportunities and Prospects

2016 ◽  
Author(s):  
Valeriy A. Buryachenko
Keyword(s):  
Integral Equations ◽  
Heterogeneous Media ◽  
Effective Field ◽  
Local Fields ◽  
Homogeneous Field ◽  
Multi Scale ◽  
General Integral ◽  
Operator Form ◽  
Mechanics Of Heterogeneous Media ◽  
Improved Accuracy

One considers a linear heterogeneous media (e.g. filtration in porous media, composite materials, CMs, nanocomposites, peristatic CMs, and rough contacted surfaces). The idea of the effective field hypothesis (EFH, H1, see for references and details [1], [2]) dates back to Mossotti (1850) who pioneered the introduction of the effective field concept as a local homogeneous field acting on the inclusions and differing from the applied macroscopic one. It is proved that a concept of the EFH (even if this term is not mentioned) is a (first) background of all four groups of analytical methods in physics and mechanics of heterogeneous media (model methods, perturbation methods, self-consistent methods, and variational ones, see for refs. [1]). An operator form of the general integral equations (GIEs) is obtained which connects the driving fields and fluxes in a point being considered. Either the volume integral equations or boundary ones are used for these GIEs, new concept of the interface polarisation tensors are introduced. New GIEs present in fact the new (second) background (which does not use the EFH) of multi-scale analysis offering the opportunities for a fundamental jump in multiscale research of random heterogeneous media with drastically improved accuracy of local field estimations (with possible change of sign of predicted local fields).

New Background of Micromechanics of Random Heterogeneous Media: Opportunities and Prospects

2015 ◽  
Author(s):  
Valeriy A. Buryachenko
Keyword(s):  
Multiscale Analysis ◽  
Perturbation Methods ◽  
Heterogeneous Media ◽  
Effective Field ◽  
Local Fields ◽  
Homogeneous Field ◽  
General Integral ◽  
Operator Form ◽  
Mechanics Of Heterogeneous Media ◽  
Improved Accuracy

One considers a linear heterogeneous media (e.g. filtration in porous media, composite materials, CMs, nanocomposites, peristatic CMs, and rough contacted surfaces). The idea of the effective field hypothesis (EFH, H1, see for references and details [1], [2]) dates back to Mossotti (1850) who pioneered the introduction of the effective field concept as a local homogeneous field acting on the inclusions and differing from the applied macroscopic one. It is proved that a concept of the EFH (even if this term is not mentioned) is a (first) background of all four groups of analytical methods in physics and mechanics of heterogeneous media (model methods, perturbation methods, self-consistent methods, and variational ones, see for refs. [1]). An operator form of the general integral equations (GIEs) is obtained which connects the driving fields and fluxes in a point being considered. New GIEs present in fact the new (second) background (which does not use the EFH) of multiscale analysis offering the opportunities for a fundamental jump in multiscale research of random heterogeneous media with drastically improved accuracy of local field estimations (with possible change of sign of predicted local fields).

General Integral Equations and Bounds of the Effective Moduli of Random Structure Composites

2017 ◽  
Author(s):  
Valeriy A. Buryachenko
Keyword(s):  
Heterogeneous Media ◽  
Effective Field ◽  
Local Fields ◽  
Upper And Lower Bounds ◽  
Homogeneous Field ◽  
Random Structure ◽  
Variational Functional ◽  
General Integral ◽  
Mechanics Of Heterogeneous Media ◽  
The Difference

One considers a linear elastic random structure composite material (CM) with a homogeneous matrix. The idea of the effective field hypothesis (EFH, H1) dates back to Faraday, Poisson, Mossotti, Clausius, and Maxwell (1830–1870, see for references and details [1], [2]) who pioneered the introduction of the effective field concept as a local homogeneous field acting on the inclusions and differing from the applied macroscopic one. It is proved that a concept of the EFH (even if this term is not mentioned) is a (first) background of all four groups of analytical methods in physics and mechanics of heterogeneous media (model methods, perturbation methods, self-consistent methods, and variational ones, see for refs. [1]). New GIEs essentially define the new (second) background (which does not use the EFH) of multiscale analysis offering the opportunities for a fundamental jump in multiscale research of random heterogeneous media with drastically improved accuracy of local field estimations (with possible change of sign of predicted local fields). Estimates of the Hashin-Shtrikman (H-S) type are developed by extremizing of the classical variational functional involving either a classical GIE [1] or a new one. In the classical approach by Willis (1977), the H-S functional is extremized in the class of trial functions with a piece-wise constant polarisation tensors while in the current work we consider more general class of trial functions with a piece-wise constant effective fields. One demonstrates a better quality of proposed bounds, that is assessed from the difference between the upper and lower bounds for the concrete numerical examples.

Multi-Scale Finite Element Simulation of Progressive Damage in Composite Structures

2006 ◽  
Author(s):  
Qing Yu ◽  
Jer-Fang Wu
Keyword(s):  
Composite Structures ◽  
Heterogeneous Media ◽  
Spatial Scales ◽  
Offshore Structures ◽  
Local Fields ◽  
Homogenization Theory ◽  
Patch Repair ◽  
Progressive Damage ◽  
Multi Scale ◽  
Macro Scale

A methodology for analyzing progressive damage accumulation on multiple spatial scales (micro- and macro-scale) in composite materials is presented in this paper. Idealization (homogenization) of heterogeneous media and evolution of damage on micro- and macro-scales are considered simultaneously at each incremental analysis step. The classical mathematical homogenization theory is extended to account for damage effects on distinct spatial scales through the introduction of an asymptotic expansion of damage parameter (or damage tensor in general). Local solutions on micro-scale provide the homogenized material properties that a global structure behaves on the macro-scales. The responses in the local fields, i.e. microscopic phases, can be reconstructed through the scale linking relations along with the global responses as input. The application of this multi-scale simulation method to composite patch repair for offshore structures is demonstrated by numerical examples.

On a General Formula for the Transition Temperature of Superconductors

1974 ◽  
Vol 29 (3) ◽  
pp. 445-451 ◽  
Author(s):  
W. Kessel
Keyword(s):  
Transition Temperature ◽  
Integral Equations ◽  
General Formula ◽  
Transition Point ◽  
Energy Gap ◽  
Eliashberg Equations ◽  
Operator Form ◽  
Method Of Solution ◽  
Theory Of Superconductivity ◽  
Phonon Properties

A method of solution of the Eliashberg equations in the theory of superconductivity is derived which uses the fact that near the transition point the energy gap is small compared to the energies over which the electron-phonon properties vary appreciably. On this basis the Eliashberg equations are converted into linear inhomogeneous integral equations. Their solution is given in operator form and provides a general formula for the transition temperature

A Boundary Element Method for Plane Anisotropic Elastic Media

1990 ◽  
Vol 57 (3) ◽  
pp. 600-606 ◽  
Author(s):  
Kyu J. Lee ◽  
A. K. Mal
Keyword(s):  
Integral Equations ◽  
Complex Plane ◽  
Singular Integral Equations ◽  
Boundary Curve ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Numerical Formulation ◽  
Complex Linear ◽  
Anisotropic Elastic ◽  
Improved Accuracy

The general problem of plane anisotropic elastostatics is formulated in terms of a system of singular integral equations with Cauchy kernels by means of the classical stress function approach. The integral equations are represented over the image of the boundary in the complex plane and a numerical scheme is developed for their solution. The boundary curve is discretized and suitable polynomial approximations of the unknown functions in terms of the complex variable are introduced. This reduces the equations to a set of complex linear algebraic equations which can be inverted to yield the stresses in a straightforward manner. The major difference between the present technique and the previous ones is in the numerical formulation. The integral equations are discretized in the complex plane and not in terms of real variables which depend on arc length, resulting in improved accuracy in presence of strong boundary curvature.

Thermal Expansion of Three-Phase Composite Materials

1989 ◽  
Vol 56 (2) ◽  
pp. 418-422 ◽  
Author(s):  
George J. Dvorak ◽  
Tungyang Chen
Keyword(s):  
Thermal Expansion ◽  
Heterogeneous Media ◽  
Transversely Isotropic ◽  
Local Fields ◽  
Stress Fields ◽  
Cylindrical Shape ◽  
Thermal Expansion Coefficients ◽  
Decomposition Procedure ◽  
Transverse Geometry ◽  
Expansion Coefficients

Exact expressions are found for overall thermal expansion coefficients of a composite medium consisting of three perfectly-bonded transversely isotropic phases of cylindrical shape and arbitrary transverse geometry. The results show that macroscopic thermal expansion coefficients depend only on the thermoelastic constants and volume fractions of the phases, and on the overall compliance. The derivation is based on a decomposition procedure which indicates that spatially uniform elastic strain fields can be created in certain heterogeneous media by superposition of uniform phase thermal strains with local strains caused by piecewise uniform stress fields, which are in equilibrium with prescribed surface tractions. The procedure also allows evaluation of thermal stress fields in the aggregate in terms of known local fields caused by axisymmetric overall stresses. Finally, averages of local fields are found with the help of known mechanical stress and strain concentration factors.

Sign in / Sign up

Export Citation Format

Share Document