A reduced micromorphic model for multiscale materials and its applications in wave propagation

Shaat, Mohamed

doi:10.1016/j.compstruct.2018.06.057

Cited by 40 publications

(41 citation statements)

References 43 publications

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“…Because the classical theories of continuum mechanics do not have the ability to represent the nanoscale phenomena, the reduced micromorphic model (RMM) 23 is introduced to study such phenomena at the micro-scale level. The RMM introduces the micro-strain tensor as an unknown measure, besides the displacement components.…”

Section: The Rmm In Cartesian Coordinatesmentioning

confidence: 99%

“…ij ij ijk is the free energy function in terms of internal variables, t ij is the micro-stress tensor, ij τ can be defined as the residual stress and m ijk is a higher order micro-stress tensor. According to 23,24 , the free energy is taken in the form where λ m and μ m are the elastic moduli of the microstructure, λ and μ are the elastic moduli of the confined material between two particles, λ c and μ c are two elastic moduli accounting for the coupling between the micro-strain and the macro-strain, 1  and 2  are length scale parameters. Such a medium is composed of deformed molecules and have twelve degree of freedom: three translational, three rotational and six micro-deformations 25 and 26 .…”

Section: The Rmm In Cartesian Coordinatesmentioning

confidence: 99%

“…(27) may be found in series form as in (30). Applying the boundary conditions (28), we find that the first boundary condition is automatically satisfied while the second one gives numerical Results Table 1 shows the values for the physical parameters 23 for an epoxy matrix and the inclusions materials. The following figures are obtained depending on the numerical values for materials parameters listed in Tables (1) and (2).…”

Section: Half-space Involving Spherical Symmetrymentioning

confidence: 99%

“…Also, the types of waves are acoustic waves, optic waves, and standing waves for the relaxed micromorphic model. Shaat 23 reduced the number of the material parameters to 8, the coupling between materials parameters are 8 dependent + 0 independent. Three types of waves are obtained.…”

mentioning

confidence: 99%

“…into Eq. (13) and neglecting the dynamic effects, we get the field equations in static equilibrium as follows:(23)(24)(25) into Eq. (26) to obtain the field equations of a hemisphere made of elastic phononic material in spherical coordinate as follows:…”

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Reduced micromorphic model in orthogonal curvilinear coordinates and its application to a metamaterial hemisphere

Dhaba

2020

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The reduced micromorphic model (RMM) is used to study the effect of the applied force on a hemisphere made of phononic crystals that belongs to the metamaterials group. The strain tensor, the micro-strain tensor and the coupling between them are the kinematic relations used to measure the deformation and micro-deformation of the representative volume element of these materials. The free energy function, the constitutive relations, the field equations, and the boundary conditions are presented firstly in the Cartesian coordinate. Then, the orthogonal curvilinear coordinates are introduced as a general coordinate to describe the physical quantities included in the RMM. The spherical coordinates are deduced as a special case from the curvilinear coordinates to study the deformation and microdeformation for the hemisphere. The kinematic relations and the governing equations of the model are considered to changing with the radius of the hemisphere only. The analytical solutions of the field equations are also obtained by using the Frobenius series satisfying the given boundary conditions and consequently the value of the physical constants of the problem is determined. Numerical applications for the obtained results are introduced with discussion. The results showed that the displacement has a greater effect rather than the micro-strain, when it is measured relative to the classical physical quantities while the micro-strain has a greater effect rather than the displacement, when it is measured relative to the nanoscale physical quantities.Metamaterials are fabricated engineering materials with periodic internal structures. These materials have special properties that do not exist in other natural materials. They can be designed to prohibit the propagation of elastic waves in the bandgap frequency range effectively. This property has many potential applications in the vibration and noise reduction areas. Besides this property, there are many applications for metamaterials in the fields of technology, industrial engineering, telecommunication equipment, optical filters, medical devices and mobile communication systems and others.The main idea of constructing metamaterials is attributed to Sir J.C. Bose in 1898. He suggested the idea of the existence of artificial materials by carrying the microwave experiment on twisted structures 1,2 . Winston E. Kock 3,4 , used the optical radio waves properties to develop new type of antennas made of metamaterial. V. G. Veselago 5 , introduced a theoretical model that predicts the propagation of electromagnetic waves in left-handed materials. During the last few decades, more attention was given to metamaterials. This is because of their important applications in recent industries. Theoretical formulation and mechanical applications can be found in refs. [6][7][8][9] .The idea of building metamaterials was extended to include elastic and acoustic waves by Ding et al. 10 , where the authors proposed metamaterials with simultaneously negative bulk modulus and mass density. Wu et al. 1...

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Section: The Rmm In Cartesian Coordinatesmentioning

confidence: 99%

Section: The Rmm In Cartesian Coordinatesmentioning

confidence: 99%

Section: Half-space Involving Spherical Symmetrymentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Reduced micromorphic model in orthogonal curvilinear coordinates and its application to a metamaterial hemisphere

Dhaba

2020

Sci Rep

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An approximate analytical solution for a dynamical simple shear deformation problem of an infinite strip including flexoelectric and micro‐inertial effects by Laplace transform

El‐Dhaba,

Ghaleb

2024

Z Angew Math Mech

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We study the flexoelectric effect induced in an infinite strip of a dielectric material, due to an applied shear deformation at the upper surface, while the lower surface is fixed. The model incorporates dynamic spontaneous polarization, flexocoupling deformation, and microinertia effects to evaluate these effects combined. The variation of the Hamiltonian functional in the medium, and of the external forces yields the governing equations and the boundary conditions. There are many types of applied boundary conditions besides the initial conditions, where the initial conditions are imposed due to the dynamics of many effects in the mathematical modelling formulation. In addition to that, the boundary conditions are divided into two groups, where the first group contains mechanical boundary conditions such as traction boundary conditions and displacement boundary conditions for the classical treatment as in the theory of elasticity while for the strain gradient theory of elasticity, conditions for higher‐order traction vector should be imposed and cannot be neglected. The proposed technique depends on applying Laplace's transform to the field equations to obtain the characteristic equation, the roots of which are expressed in terms of the ‐transform parameter and other parameters related to the elastic and electric properties of the material. Those roots which produce a bounded solution are then approximated using Taylor's expansion, then the boundary conditions are applied to complete the solution. This technique enables to obtain the solution in terms of simple analytical functions that are easily inverted from the ‐domain to the time domain. This technique is applied to a simple shear deformation problem to study the dynamical flexoelectric effect in dielectric materials due to shear displacement and external electric potential. A suitable material, such as Strontium Titanate material, , is used for numerical simulation, and the results are analyzed, discussed, and represented graphically.

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Nonlocal Approaches to the Dynamics of Metamaterials

Failla

Ghavanloo

2021

Springer Tracts in Mechanical Engineering

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A reduced micromorphic model for multiscale materials and its applications in wave propagation

Cited by 40 publications

References 43 publications

Reduced micromorphic model in orthogonal curvilinear coordinates and its application to a metamaterial hemisphere

Reduced micromorphic model in orthogonal curvilinear coordinates and its application to a metamaterial hemisphere

An approximate analytical solution for a dynamical simple shear deformation problem of an infinite strip including flexoelectric and micro‐inertial effects by Laplace transform

Nonlocal Approaches to the Dynamics of Metamaterials

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