Micromechanics of Random Structure Thermoperistatic Composites

2016 ◽  
Author(s):  
Valeriy A. Buryachenko
Keyword(s):  
Elastic Moduli ◽  
Effective Properties ◽  
Local Stress ◽  
Local Fields ◽  
Random Structure ◽  
Influence Functions ◽  
Effective Elastic Moduli ◽  
Displacement Fields ◽  
General Integral ◽  
Inclusion Phase

In contrast to the classical local and nonlocal theories, the peridynamic equation of motion introduced by Silling (J. Mech. Phys. Solids 2000; 48: 175–209) is free of any spatial derivatives of displacement. The new general integral equations (GIE) connecting the displacement fields in the point being considered and the surrounding points of random structure composite materials (CMs) is proposed. For statistically homogeneous thermoperistatic media subjected to homogeneous volumetric boundary loading, one proved that the effective behaviour of this media is governing by conventional effective constitutive equation which is intrinsic to the local thermoelasticity theory. It was made by the most exploitation of the popular tools and concepts used in conventional thermoelasticity of CMs and adapted to thermoperistatics. A generalization of the Hills equality to peri-static composites is proved. The classical representations of effective elastic moduli through the mechanical influence functions for elastic CMs are generalized to the case of peristatics, and the energetic definition of effective elastic moduli is proposed. The general results establishing the links between the effective properties (effective elastic moduli, effective thermal expansion) and the corresponding mechanical and transformation influence functions are obtained by the use of the decomposition of local fields into load and residual fields. Effective properties of thermoperistatic CM are expressed through the introduced local stress polarization tensor averaged over the extended inclusion phase. This similarity opens a way for straightforward expansion of analytical micromechanics tools for locally elastic CMs to the new area of random structure peri-dynamic CMs. Detailed numerical examples for 1D case are considered.

Micromechanical Background of Random Structure Thermoperistatic Composites

2015 ◽  
Author(s):  
Valeriy A. Buryachenko
Keyword(s):  
Elastic Moduli ◽  
Effective Properties ◽  
Local Fields ◽  
Random Structure ◽  
Effective Elastic Moduli ◽  
Displacement Fields ◽  
General Integral ◽  
Spatial Derivatives ◽  
Effective Thermal Expansion ◽  
Derivatives Of

In contrast to the classical local and nonlocal theories, the peridynamic equation of motion introduced by Silling (J. Mech. Phys. Solids 2000; 48: 175–209) is free of any spatial derivatives of displacement. The new general integral equations (GIE) connecting the displacement fields in the point being considered and the surrounding points of random structure composite materials (CMs) is proposed. For statistically homogemneous thermoperistatic media subjected to homogeneous volumetric boundary loading, one proved that the effective behaviour of this media is governing by conventional effective constitutive equation which is intrinsic to the local thermoelasticity theory. It was made by the most exploitation of the popular tools and concepts used in conventional thermoelasticity of CMs and adapted to thermoperistatics. The general results establishing the links between the effective properties (effective elastic moduli, effective thermal expansion) and the corresponding mechanical and transformation influence functions are obtained by the use of decomposition of local fields into the load and residual fields similarly to the locally elastic CMs. This similarity opens a way for straightforward expansion of analytical micromechanics tools for locally elastic CMs to the new area of random structure peridynamic CMs. Detailed numerical examples for 1D case are considered.

Variational principles and generalized Hill’s bounds in micromechanics of linear peridynamic random structure composites

2019 ◽  
Vol 25 (3) ◽  
pp. 682-704 ◽  
Author(s):  
Valeriy A Buryachenko
Keyword(s):  
Elastic Moduli ◽  
Variational Principles ◽  
Virtual Work ◽  
Static Problem ◽  
Material Behavior ◽  
Reciprocal Theorem ◽  
Random Structure ◽  
Effective Elastic Moduli ◽  
Peridynamic Model ◽  
Non Local

We consider a static problem for statistically homogeneous matrix linear peridynamic composite materials (CMs). The basic feature of the peridynamic model considered is a continuum description of a material behavior as the integrated non-local force interactions between infinitesimal particles. In contrast to these classical local and non-local theories, the peridynamic equation of motion introduced by Silling ( J Mech Phys Solids 2000; 48: 175–209) is free of any spatial derivatives of displacement. Estimation of effective moduli of peridynamic CMs is performed by generalization of some methods used in locally elastic micromechanics. Namely, the admissible displacement and force fields are defined. The theorem of work and energy, Betti’s reciprocal theorem, and the theorem of virtual work are proved. Principles of minimum of both potential energy and complimentary energy are generalized. The strain energy bounds are estimated for both the displacement and force homogeneous volumetric boundary conditions. The classical representations of effective elastic moduli through the mechanical influence functions for elastic CM are generalized to the case of peridynamics, and the energetic definition of effective elastic moduli is proposed. Generalized Hill’s bounds on the effective elastic moduli of peridynamic random structure composites are obtained. In contrast to the classical Hill’s bounds, in the new bounds, comparable scales of the inclusion size and horizon are taken into account that lead to dependance of the bounds on both the size and shape of the inclusions. The numerical examples are considered for the 1D case.

scholarly journals On the respect to the Hashin-Shtrikman bounds of some analytical methods applying to porous media for estimating elastic moduli

2021 ◽  
Vol 15 (2) ◽  
pp. 14-25
Author(s):  
N. Nguyen ◽  
N.Q Tran ◽  
B.A Tran ◽  
Q.H Do
Keyword(s):  
Porous Media ◽  
Elastic Moduli ◽  
Effective Properties ◽  
Minimum Energy ◽  
Three Dimensions ◽  
Effective Elastic Moduli ◽  
The Finite Element Method ◽  
Energy Principle ◽  
Minimum Energy Principle ◽  
Recent Method

In this work, some popular analytic formulas such as Maxwell (MA), Mori-Tanaka approximation (MTA), and a recent method, named the Polarization approximation (PA) will be applied to estimate the elastic moduli for some porous media. These approximations are simple and robust but can be lack reliability in many cases. The Hashin-Shtrikman (H-S) bounds do not supply an exact value but a range that has been admitted by researchers in material science. Meanwhile, the effective properties by unit cell method using the finite element method (FEM) are considered accurate. Different shapes of void inclusions in two or three dimensions are employed to investigate. Results generated by H-S bounds and FEM will be utilized as references. The comparison suggests that the method constructed from the minimum energy principle PA can give a better estimation in some cases. The discussion gives out some remarks which are helpful for the evaluation of effective elastic moduli. Keywords: Maxwell approximation; polarization approximation; Mori-Tanaka approximation; effective elastic moduli; porous medium.

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.

Modeling of the Effective Elastic Properties of Multifunctional Carbon Nanocomposites Due to Agglomeration of Straight Circular Carbon Nanotubes in a Polymer Matrix

2013 ◽  
Vol 81 (2) ◽  
Author(s):  
Chetan Shivaputra Jarali ◽  
Somaraddi R. Basavaraddi ◽  
Björn Kiefer ◽  
Sharanabasava C. Pilli ◽  
Y. Charles Lu
Keyword(s):  
Carbon Nanotubes ◽  
Elastic Properties ◽  
Elastic Moduli ◽  
Volume Fraction ◽  
Effective Properties ◽  
Transverse Isotropy ◽  
Effective Elastic Moduli ◽  
Effective Elastic Properties ◽  
The Matrix ◽  
Nanotube Composites

In the present study, the effective elastic properties of multifunctional carbon nanotube composites are derived due to the agglomeration of straight circular carbon nanotubes dispersed in soft polymer matrices. The agglomeration of CNTs is common due to the size of nanotubes, which is at nanoscales. Furthermore, it has been proved that straight circular CNTs provide higher stiffness and elastic properties than any other shape of the nanofibers. Therefore, in the present study, the agglomeration effect on the effective elastic moduli of the CNT polymer nanocomposites is investigated when circular CNTs are aligned straight as well as distributed randomly in the matrix. The Mori–Tanaka micromechanics theory is adopted to newly derive the expressions for the effective elastic moduli of the CNT composites including the effect of agglomeration. In this direction, analytical expressions are developed to establish the volume fraction relationships for the agglomeration regions with high, and dilute CNT concentrations. The volume of the matrix in which there may not be any CNTs due to agglomeration is also included in the present formulation. The agglomeration volume fractions are subsequently adopted to develop the effective relations of the composites for transverse isotropy and isotropic straight CNTs. The validation of the modeling technique is assessed with results reported, and the variations in the effective properties for high and dilute agglomeration concentrations are investigated.

Generalized Method of Effective Field in Linear Peridynamic Micromechanics of Random Structure Composites

2020 ◽  
Author(s):  
Valeriy A. Buryachenko
Keyword(s):  
Effective Field ◽  
Random Structure ◽  
Entire Space ◽  
Displacement Fields ◽  
Inclusion Phase ◽  
Self Consistent ◽  
The Media ◽  
Spatial Derivatives ◽  
Similarity And Difference ◽  
Derivatives Of

Abstract A statistically homogeneous random matrix medium with the bond-based peridynamic properties of constituents is considered. For the media subjected to remote homogeneous volumetric boundary loading, one proved that the effective behavior of this media is governing by conventional effective constitutive equation which is the same as for the local elasticity theory. The average is performed over the surface of the extended inclusion phase rather than over an entire space. Any spatial derivatives of displacement fields are not required. The basic hypotheses of locally elastic micromechanics are generalized to their peri-static counterparts. In particular, in the generalized method of effective field proposed, the effective field is evaluated from self-consistent estimations by the use of closing of a corresponding integral equation in the framework of the quasi-crystalline approximation. In so doing, the classical effective field hypothesis is relaxed, and the hypothesis of the ellipsoidal symmetry of the random structure of CMs is not used. One demonstrates some similarity and difference with respect to other methods (the dilute approximation and Mori-Tanaka approach) proposed before in peridynamic micromechanics of CMs. Comparative numerical analyses of these methods are performed for 1D case.

Method of Fundamental Solution in Thermoelasticity of Random Structure Matrix Composites

2018 ◽  
Author(s):  
Valeriy A. Buryachenko
Keyword(s):  
Fundamental Solution ◽  
Effective Properties ◽  
Effective Field ◽  
Stress And Strain ◽  
Random Structure ◽  
Infinite Matrix ◽  
Matrix Composites ◽  
Strain Fields ◽  
General Integral ◽  
Single Inclusion

One considers linear thermoelastic composite media, which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non-canonical (i.e. non-ellipsoidal) shape. The representations of the effective properties (effective moduli, thermal expansion, and stored energy) are expressed through the statistical averages of the interface polarization tensors (generalizing the initial concepts, see e.g. [1] and [2]) introduced apparently for the first time. The new general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced in terms of boundary interface integrals estimated by the method of fundamental solution for a single inclusion inside the infinite matrix. This enables one to reconsider basic concepts of micromechanics such as effective field hypothesis, quasi-crystalline approximation, and the hypothesis of ellipsoidal symmetry. The results of this reconsideration are quantitatively estimated for some modeled composite reinforced by aligned homogeneous heterogeneities of non canonical shape. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.

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