Wave Processes in Solids with Microstructure

Erofeyev, Vladimir I.

doi:10.1142/5157

Cited by 131 publications

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“…Similar equations were obtained using the homogenization of a periodically layered medium [14][15][16] or using strain gradient theories. 17 Here the dispersive term contains fourth-order space derivative of the displacement. Another generalization of the wave equation (1) …”

Section: Dispersive Wave Equationsmentioning

confidence: 99%

Two-Scale Microstructure Dynamics

Berezovski

Engelbrecht

2011

J. Multiscale Modelling

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Wave propagation in materials with embedded two different microstructures is considered. Each microstructure is characterized by its own length scale. The dual internal variables approach is adopted yielding in a Mindlin-type model including both microstructures. Equations of motion for microstructures are coupled with the balance of linear momentum for the macromotion, but not coupled with each other. Corresponding dispersion curves are provided and scale separation is pointed out.

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Section: Dispersive Wave Equationsmentioning

confidence: 99%

Two-Scale Microstructure Dynamics

Berezovski

Engelbrecht

2011

J. Multiscale Modelling

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“…Microstructured materials are characterized by the existence of intrinsic space-scales in matter, like the lattice period, the size of a grain or a crystallite, or the distance between the microcracks, etc., which introduce the scale-dependence into the governing equations (see, e.g., [3,5,15,22,23] and references therein). The scale-dependence involves dispersive as well as nonlinear effects.…”

Section: Introductionmentioning

confidence: 99%

“…In terms of material and geometrical parameters, constants in (5) (5) can be considered as an approximation of FSE (4) and is referred to as the hierarchical equation (HE) below. On the other hand, HE (5) is of Boussinesq type [1].…”

Section: Introductionmentioning

confidence: 99%

On the propagation of solitary waves in Mindlin-type microstructured solids

Tamm¹,

Salupere²

2010

Proc. Estonian Acad. Sci.

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The Mindlin-Engelbrecht-Pastrone model is applied to simulating 1D wave propagation in microstructured solids. The model takes into account the nonlinearity in micro-and macroscale. Numerical solutions are found for the full system of equations (FSE) and the hierarchical equation (HE). The latter is derived from the FSE by making use of the slaving principle. Analysis of results demonstrates good agreement between the solutions of the FSE and HE in the considered domain of parameters. For numerical integration the pseudospectral method is used.

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“…Other notations for the potential can also be used, for instance, those in Landau and Murnaghan nonlinear models. The relations between the constants in these models and the values of some constants can be found in [1]. These models can also be used to describe the behavior of materials with different resistances to tension and compression.…”

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confidence: 99%

“…This potential has a tensor-nonlinear term, which take into account the interaction between the second and third invariants of strains. The term αI 3 1 / √ I 2 can usually be omitted, because its influence is taken into account to a certain extent by the term γI 1 √…”

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confidence: 99%

Propagation of longitudinal and transverse waves in a multimodulus elastic medium

Baev

2009

J Appl Mech Tech Phy

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UDC 539.3:517.958 L. V. BaevA problem of propagation of longitudinal and transverse waves in a multimodulus elastic isotropic medium is considered. In the model used, the medium is described by a potential depending on three invariants of strains, which allows the influence of preliminary deformation of the medium on the longitudinal and transverse velocities to be taken into account. 1.In considering adiabatic processes of elastic deformation, it is usually postulated that internal energy depends on invariants of strain measures. In the general form, the tensor relation between the stresses and strains for a nonlinearly elastic isotropic medium can be written asHere K 0 , K 1 , and K 2 are the functions of three invariants of the strain tensor I 1 = ε mm , I 2 = ε mn ε nm , and I 3 = ε mn ε np ε pm (summation is performed over repeated indices from 1 to 3).If a potential W = W (I 1 , I 2 , I 3 ) exists, we obtainOne of the methods most frequently used in the classical elasticity theory is the choice of the potential in the form of the expansion of internal energy into a Taylor series with respect to the initial state with retaining a certain number of terms. If the second, third, and fourth powers of strains are retained, we have nine terms. The constants at the corresponding powers of strains are called the Lame constants of the second, third, and fourth order. Other notations for the potential can also be used, for instance, those in Landau and Murnaghan nonlinear models. The relations between the constants in these models and the values of some constants can be found in [1]. These models can also be used to describe the behavior of materials with different resistances to tension and compression. It is assumed that these moduli can become different, in particular, owing to internal damages of the material and appearance of cracks, which makes the elastic characteristics and the character of propagation of elastic waves in the material depend on the type of the stress state. The qualitative characteristic responsible for the type of the strain state is the third invariant of the strain tensor, while the first and second invariants characterize the changes in the volume and shape. The difference in resistances to tension and compression can be taken into account to a certain extent by introducing a term depending on ξ = I 1 √ I 2 into the potential. Potentials of this type were considered in [2,3]. The potential in [3] was chosen in the form W (ε i,j ) = 0.5[K + ϕ(ξ)]I 2 1 + [G + ψ(ξ)](I 2 − I 2 1 /3), where the functions ϕ and ψ are related as 3ξ 2 ϕ +2(3−ξ 2 )ψ = 0. In this case, however, experiments on tension and compression with measurements of longitudinal and transverse strains yield four relations for determining a smaller Lavrent'ev Institute

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Wave Processes in Solids with Microstructure

Cited by 131 publications

References 0 publications

Two-Scale Microstructure Dynamics

Two-Scale Microstructure Dynamics

On the propagation of solitary waves in Mindlin-type microstructured solids

Propagation of longitudinal and transverse waves in a multimodulus elastic medium

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