Solution of general integral equations of micromechanics of heterogeneous materials

Buryachenko, Valeriy A.

doi:10.1016/j.ijsolstr.2014.06.008

Cited by 28 publications

(51 citation statements)

References 85 publications

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“…were S denotes the unit sphere and d m denotes a differential solid angle on S in the direction of any unit vector m. Equation ( 18) was proved in [21] for the 3D case whereas the case d = 2 can be justified in a similar manner. Equation ( 18) at d = 1 can also be reduced to (17) by the variable exchange x − z → r, x + y → s. Indeed, the origin-centered unit 1D "sphere" is the set {−1, 1}, which has a measure of 2. Then the integrand of ( 18) is reduced to the integrand of ( 17) by the use of the equality (7).…”

Section: C(xx)mentioning

confidence: 99%

“…The approximative character of the variational bounds and their restricted opportunities (for the formula manipulations) of the variational methods in comparison with the different self-consistent methods is explained by a fundamental reason. Indeed, the self-consistent methods in linear statics of mechanics of CMs were exploited in a wide class of micromechanical problems for CMs with non-ellipsoidal, coated, and continuously inhomogeneous heterogeneities with possible non-ideal interface as well as for different non-local problems (inhomogeneous remote loading, functionally graded materials, clustered materials, bounded media, nanocomposites, non-local constitutive law, e.g., peridynamics; see, e.g., [16,17] for references). Analyses of so much diverse (in the sense of microtopology and constitutive laws) problems are essentially simplified due to linear dependence of the effective properties on average fields (strains and stresses) inside the inclusions.…”

Section: Introductionmentioning

confidence: 99%

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

Buryachenko¹

2019

Mathematics and Mechanics of Solids

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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.

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Section: C(xx)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Buryachenko¹

2019

Mathematics and Mechanics of Solids

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“…In particular, the residual local fields were presented in terms of certain transformation influence functions with the establishment of their relation to the analogous mechanical influence functions (50) and (51) for the special case of piecewise uniform eigenfields. As an analogy between the aforementioned theories and their peristatic version, we will assume that the nonlocal operatorL is presented in the form (6) and (72), and for the phase…”

Section: Transformation Influence Functionsmentioning

confidence: 99%

Effective properties of thermoperistatic random structure composites: Some background principles

Buryachenko¹

2016

Mathematics and Mechanics of Solids

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The basic feature of the peridynamic model considered here is a continuum description of a material's behavior as the integrated nonlocal force interactions between infinitesimal particles. In contrast to 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. A theory of thermoelastic composite materials (CMs) with nonlocal thermoperistatic properties of multiphase constituents of arbitrary geometry is analyzed for statistically homogeneous CMs subjected to homogeneous loading. A generalization of the Hill's equality to peristatic 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. A detected similarity of results for both the peristatic and locally elastic composites is explained fundamentally, as the methods used for obtaining the results widely exploit the Hill's condition and the self-adjointness of the stress operator. However, the representation of effective properties for composites with both the local thermoelastic and nonlocal thermoperistatic properties do not always coincide. Therefore, the representation of effective eigenfields through mechanical influence functions generalizing Levin's representation does not in general hold for thermoperistatic CMs; this is demonstrated for a one-dimensional numerical example.

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“…In recent years, integral equations have been encountered in variety applied sciences, such as mathematics, engineering, thermodynamic, molecular properties, electromagnetics, Stokes flow, heat and mass transfer, and micromechanics [1][2][3][4][5][16][17][18][19][20][21]. Most of the time, integral equations and their different types are too hard to be solved analytically.…”

Section: Introductionmentioning

confidence: 99%

A New Numerical Method for Solving Functional Delay Integral Equations With Variable Bounds by Using Generalized Mott Polynomials

Kürkçü¹

2018

Anadolu University Journal of Science and Technology-a Applied Sciences and Engineering

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In this study, the delay integral equations with variable bounds are considered and their approximate solutions are obtained by using a new numerical method based on matrices, collocation points and the generalized Mott polynomials including a parameter-. An error analysis technique consisting of the residual function is performed. The numerical examples are applied to illustrate the practicability and usability of the method. The behavior of the solutions is monitored in terms of the parameter-. The accuracy of the method is scrutinized for different values of N and also the numerical results are discussed in figures and tables.

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Solution of general integral equations of micromechanics of heterogeneous materials

Cited by 28 publications

References 85 publications

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

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

Effective properties of thermoperistatic random structure composites: Some background principles

A New Numerical Method for Solving Functional Delay Integral Equations With Variable Bounds by Using Generalized Mott Polynomials

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