A continuum model of microheterogeneous media

Belov, P. A.; Lurie, S. A.

doi:10.1016/j.jappmathmech.2009.11.013

Cited by 14 publications

(10 citation statements)

References 7 publications

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“…Thus, change of the porosity volume fraction in media is ( − ϕ ) . The Lagrangian of considered model with surface effects could be written in the following form [18–20]:…”

Section: Theory Of Linear Elastic Materials With Voids and Surfacementioning

confidence: 99%

Bending problems in the theory of elastic materials with voids and surface effects

Lurie

Solyaev

Волков

et al. 2017

Mathematics and Mechanics of Solids

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In the present study, a comparison of pure, three-point, four-point and cantilever beam bending problems in the frame of the theory of elastic materials with voids (micro-dilatational elasticity) has been provided via analytical modelling and three-dimensional finite-element analysis. We consider the extended variant of the theory with surface effects using a variational approach. At first, we compare the known approximate semi-inverse analytical solution of the pure bending problem with a corresponding three-dimensional finite-element solution in the frame of micro-dilatational theory. It is demonstrated that in the numerical solution – unlike the analytical one – all boundary conditions are satisfied accurately, and there exist distortions of the cross-sections and lateral faces of the beam. The generalized analytical solution of the beam pure bending problem with surface effects is also established and compared with numerical simulations. The effective elastic properties of the beam with micro-dilatations are introduced by comparing its displacements and corresponding classical beam displacements. The influence of scale, coupling and surface parameters on the effective elastic moduli in different bending and simple tension tests is studied. It is shown that all considered types of bending experiments provide the determination of close values of the effective flexural modulus of the beams with different thickness. This means that it is possible to use any bending test with beams of different thickness for reliable identification of the material constants of the theory. It is also possible to use the simple analytical expression for an effective flexural modulus that follows from the analytical solution of the pure bending problem. It is also shown that stiffness of the beam with micro-dilatation in any bending tests should always be higher as compared with the stiffness of such a beam in simple tension. For thin beams, there are no scale effects, and its effective flexural modulus is equal to the material’s Young’s modulus without micro-dilatation. Possible rise of the negative size effects in the model with surface effects is also discussed.

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“…Thus, change of the porosity volume fraction in media is ( − ϕ ) . The Lagrangian of considered model with surface effects could be written in the following form [18–20]:…”

Section: Theory Of Linear Elastic Materials With Voids and Surfacementioning

confidence: 99%

Bending problems in the theory of elastic materials with voids and surface effects

Lurie

Solyaev

Волков

et al. 2017

Mathematics and Mechanics of Solids

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“…It has been argued some time, that gradient theories may be quite effective to describe phenomenologically the influence of underlying microstructures, and have been used for capturing scale effects in miniaturized components and devices [7][8][9]]. In connections with the present works it is noted that Lurie and co-workes [10][11][12][13][14] have employed a first-order unified gradient model of the medium with conserved dislocations to describe a spectrum of various surface phenomena and scale effects, and an applied interphase layer model was also proposed [15][16][17][18]. These applied gradient models contain essentially only one additional physical parameter to account for straingradient effects; more so, in particular, since they also provide a sufficient description of the cohesion field with adhesion interactions in the contact zone between different components.…”

mentioning

confidence: 60%

“…The gradient term i u is defined as n is the outwards unit normal to the surface. Note, that in common case strain energy density on the surface ( G E  ) can define more common adhesion properties [11,12,16,19]. For the model of Eq.…”

Section: Herementioning

confidence: 99%

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Gradient effects in fracture mechanics for nano-structured materials

Lurie

Belov

2014

Engineering Fracture Mechanics

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Abstract. Gradient elastic effects associated with the existence of an cohesion type interphase layer, within a simple and robust gradient model are discussed. Classical and gradient solutions within a simple and robust fracture model whose properties are described by the harmonic and Helmholtz equations, are compared. It is shown that using the gradient solution can be explain the various mechanisms of fracture for the microcrystalline structures in dependence from scale parameter, which can be defined in terms of standard characteristics of materials. We established the equation for the tensor of energy, within a gradient model and adhesion model. Conservation law for the tensor of energy and J-integral equation were formulated for the continual gradient adhesion model. IntroductionThe development of continuum media models accounting for various micro/nanostructures beyond the theory of classical elasticity appears to be crucial for the description of not only short-range interactions and cohesion forces, but also for the modeling of other size-dependent effects in the framework of generalized elasticity and plasticity theories. Such robust gradient were developed initially by Aifantis and co-workers in the early eighties [1,2], for gradient plasticity and in the early ninties [3,4] for gradient elasticity, respectively. Following the publication of Aifantis's initial models, various gradient theories appeared and applied to interface, shear banding dislocation and size effect and composite problems [5][6][7]. In the listed works the variants of gradient models for the description of cohesion scale-effect (without adhesive interactions) are developed. It has been argued some time, that gradient theories may be quite effective to describe phenomenologically the influence of underlying microstructures, and have been used for capturing scale effects in miniaturized components and devices [7][8][9]]. In connections with the present works it is noted that Lurie and co-workes [10][11][12][13][14] have employed a first-order unified gradient model of the medium with conserved dislocations to describe a spectrum of various surface phenomena and scale effects, and an applied interphase layer model was also proposed [15][16][17][18]. These applied gradient models contain essentially only one additional physical parameter to account for straingradient effects; more so, in particular, since they also provide a sufficient description of the cohesion field with adhesion interactions in the contact zone between different components. In this paper, using the gradient theory, we construct an analytical asymptotic solution for the model problem in the mechanics of fracture. We show that in contrast to the classical theory of gradient elasticity theory gives a non-singular solution for the crack. The novelty of the approach lies on the fact that the gradient theory allows us to construct asymptotic solutions of varying smoothness of the crack tip. In addition, we constructed an expression for the energy tensor and developed J in...

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“…The stress-strain state in structurally inhomogeneous is often described on the basis of the micropolar elasticity theory intensively developed now [1,[3][4][5][6][7][8][9]. The problem of constructing mathematical models of micropolar elastic thin plates, shells, and beams is relevant for advanced applications [10][11][12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Mathematical model of micropolar elastic thin plates and their strength and stiffness characteristics

Sarkisyan

2012

J Appl Mech Tech Phy

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A number of hypotheses were formulated using the properties of an asymptotic solution of boundary-value problems of the three-dimensional micropolar (moment asymmetric) theory of elasticity for areas with one geometrical parameter being substantially smaller than the other two (plates and shells). A general theory of bending deformation of micropolar elastic thin plates with independent fields of displacements and rotations is constructed. In the constructed model of a micropolar elastic plate, transverse shear strains are fully taken into account. A problem of determining the stress-strain state in bending deformation of micropolar elastic thin rectangular plates is considered. The numerical analysis reveals that plates made of a micropolar elastic material have high strength and stiffness characteristics.

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A continuum model of microheterogeneous media

Cited by 14 publications

References 7 publications

Bending problems in the theory of elastic materials with voids and surface effects

Bending problems in the theory of elastic materials with voids and surface effects

Gradient effects in fracture mechanics for nano-structured materials

Mathematical model of micropolar elastic thin plates and their strength and stiffness characteristics

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