Waves in Nonlocal Elastic Solid with Voids

Singh, Dilbag; Kaur, Gurwinderpal; Tomar, S. K.

doi:10.1007/s10659-016-9618-x

Cited by 109 publications

(55 citation statements)

References 28 publications

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“…The speed of the SV‐type wave remains the same as

c_{3} = \sqrt{c_{normalΘ}^{2} - ε^{2} ω^{2}}

, since this wave is independent of the thermal field. It is quite interesting to note that in a nonlocal elastic medium, the square of the speeds of the classical longitudinal wave (P‐wave) as well as the classical shear wave (SV‐wave) are frequency dependent (dispersive) and both are reduced by an amount equal to

ε^{2} ω^{2}

, a result earlier obtained by [13] in the relevant medium. The frequencies

ω = ω_{c_{j}} false(j = 1, 3 false)

act as the critical frequencies for the respective waves, beyond which the waves are no more propagating waves.…”

Section: Dispersion Relation and Its Solutionsmentioning

confidence: 62%

“…[42] and Sarkar et al. [43] as follows:

\begin{matrix} true \\ right μ \nabla^{2} \overset{u}{true} + (λ + μ) \overset{\nabla}{true} (\vec{\nabla} \cdot \vec{u}) - γ \overset{\nabla}{true} normalΘ = ρ (1 - ε^{2} \nabla^{2}) \overset{\vec{u}}{true}, \end{matrix}

\begin{matrix} true \\ right (() K^{*} + K_{normalΘ} \frac{\partial}{\partial t}) \nabla^{2} normalΘ = ρ C_{E} \overset{Θ}{true} + γ T_{0} \overset{\nabla}{true} \cdot \overset{\vec{u}}{true}, \end{matrix}

\begin{matrix} true \\ right (1 - ε^{2} \nabla^{2}) \overset{τ}{true} = {\vec{τ}}^{L} = μ (() \vec{\nabla} \vec{u} + \vec{\nabla} {\overset{u}{true}}^{T}) + (() λ \vec{\nabla} \cdot \vec{u} - γ Θ) δ_{i j} \overset{I}{true}, \end{matrix}

where

\nabla^{2} \equiv \partial^{2} / \partial x^{2} + \partial^{2} / \partial z^{2}

ε (= e_{0} a)

is the elastic nonlocal parameter [4, 13] having the dimension of length, a and e 0 , respectively are an internal characteristic length and a constant,

\overset{τ}{true}

is the stress tensor,

{\vec{τ}}^{L}

stands for stress tensor in local ...…”

Section: Governing Equations and Formulation Of The Problemmentioning

confidence: 99%

“…Khurana and Tomar [12] extended the theory of nonlocal elasticity to nonlocal theory of microstretch elasticity and then they investigated wave propagation in nonlocal microstretch solid. Sing [13] studied wave propagation in nonlocal elastic solid with voids.…”

Section: Introductionmentioning

confidence: 99%

“…They also applied these theories [16, 17] to study some one‐dimensional problems. Bachher and Sarkar [18] extended the nonlocal theory of elasticity with voids [13] to the nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer. They also applied this new theory to study the one‐dimensional transient response of a thermoelastic nonlocal infinite medium with voids.…”

Section: Introductionmentioning

confidence: 99%

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Waves in nonlocal thermoelastic solids of type III

Sarkar¹,

Bachher

Das³

et al. 2020

Z Angew Math Mech

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In this paper, the propagation of time-harmonic thermoelastic plane waves is studied in an infinite nonlocal elastic continuum. The type III Green-Naghdi model (with energy dissipation) of generalized thermoelasticity and the Eringen's nonlocal elasticity model are adopted to address this problem. We found two sets of the coupled longitudinal waves which are dispersive in nature and experience attenuation. In addition to the coupled waves, there also exists one independent vertically shear-type wave which is dispersive but experiences no attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear-type wave is found to face a critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. Reflection phenomenon of an incident coupled longitudinal waves from a rigid and thermally insulated boundary surface of a homogeneous and isotropic nonlocal thermoelastic half-space is investigated. Using these boundary conditions, the formulae for various reflection coefficients and their respective energy ratios are presented. For a particular model, various graphs are plotted to analyze the behavior of the phase speeds, reflection coefficients and their respective energy ratios.The amplitude ratios of the reflected waves and their respective energy ratios are determined analytically. For a particular model, the effect of elastic nonlocality parameter on the variations of phase speeds, attenuation coefficients, amplitude ratios and corresponding energy ratios of the reflected waves are presented graphically. Finally, analysis of the various results have been interpreted. K E Y W O R D Sdispersion, energy partition, green-nagdhi model, nonlocal, reflection

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“…The speed of the SV‐type wave remains the same as

c_{3} = \sqrt{c_{normalΘ}^{2} - ε^{2} ω^{2}}

ε^{2} ω^{2}

, a result earlier obtained by [13] in the relevant medium. The frequencies

ω = ω_{c_{j}} false(j = 1, 3 false)

act as the critical frequencies for the respective waves, beyond which the waves are no more propagating waves.…”

Section: Dispersion Relation and Its Solutionsmentioning

confidence: 62%

“…[42] and Sarkar et al. [43] as follows:

\begin{matrix} true \\ right μ \nabla^{2} \overset{u}{true} + (λ + μ) \overset{\nabla}{true} (\vec{\nabla} \cdot \vec{u}) - γ \overset{\nabla}{true} normalΘ = ρ (1 - ε^{2} \nabla^{2}) \overset{\vec{u}}{true}, \end{matrix}

\begin{matrix} true \\ right (() K^{*} + K_{normalΘ} \frac{\partial}{\partial t}) \nabla^{2} normalΘ = ρ C_{E} \overset{Θ}{true} + γ T_{0} \overset{\nabla}{true} \cdot \overset{\vec{u}}{true}, \end{matrix}

\begin{matrix} true \\ right (1 - ε^{2} \nabla^{2}) \overset{τ}{true} = {\vec{τ}}^{L} = μ (() \vec{\nabla} \vec{u} + \vec{\nabla} {\overset{u}{true}}^{T}) + (() λ \vec{\nabla} \cdot \vec{u} - γ Θ) δ_{i j} \overset{I}{true}, \end{matrix}

where

\nabla^{2} \equiv \partial^{2} / \partial x^{2} + \partial^{2} / \partial z^{2}

ε (= e_{0} a)

is the elastic nonlocal parameter [4, 13] having the dimension of length, a and e 0 , respectively are an internal characteristic length and a constant,

\overset{τ}{true}

is the stress tensor,

{\vec{τ}}^{L}

stands for stress tensor in local ...…”

Section: Governing Equations and Formulation Of The Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Waves in nonlocal thermoelastic solids of type III

Sarkar¹,

Bachher

Das³

et al. 2020

Z Angew Math Mech

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show abstract

“…Waves in nonlocal elastic solid with voids has been studied by Sing et al. [49]. They also discussed the reflection phenomena of P‐wave from a stress‐free solid half‐space.…”

Section: Introductionmentioning

confidence: 99%

Thermoelastic plane waves under the modified Green–Lindsay model with two‐temperature formulation

Sarkar

Mondal²

2020

Z Angew Math Mech

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The present study is concerned with the reflection and propagation of thermoelastic plane waves from the stress‐free and thermally insulated surface of a homogeneous, isotropic thermally conducting elastic half‐space in the frame of modified Green–Lindsay theory of generalized thermoelasticity with two‐temperature (2TMGL). The thermoelastic coupling effect creates two types of coupled longitudinal waves which are dispersive as well as exhibit attenuation. Different from the thermoelastic coupling effect, there also exists one independent vertically shear‐type (SV‐type) wave. In contrast to the classical Green–Lindsay (GL) and Lord–Shulman (LS) theories of generalized thermoelasticity, the SV‐type wave is not only dispersive in nature but also experiences attenuation. Analytical expressions for the amplitude ratios and their respective energy ratios for reflected thermoelastic waves are determined when a coupled longitudinal wave is made incident on the free surface. The paper concludes with the numerical results on the phase speeds, amplitude ratios and their respective energy ratios for specific parameter choices. Various graphs have been plotted to analyze the behavior of these quantities. The characteristics of employing the 2TMGL model are discussed by comparing the numerical results obtained for the present model with those obtained in case of the two‐temperature Green–Lindsay (2TGL) and two‐temperature Lord–Shulman (2TLS) models.

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Reflection of plane waves in generalized thermoelasticity of type III with nonlocal effect

Das¹,

Sarkar

2019

Math Methods in App Sciences

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The generalized thermoelasticity theory based upon the Green and Naghdi model III of thermoelasticity as well as the Eringen's nonlocal elasticity model is used to study the propagation of harmonic plane waves in a nonlocal thermoelastic medium. We found two sets of coupled longitudinal waves, which are dispersive in nature and experience attenuation. In addition to the coupled waves, there also exists one independent vertically shear-type wave, which is dispersive but experiences no attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear-type wave is found to face a critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. The problem of reflection of the thermoelastic waves at the stress-free insulated and isothermal boundary of a homogeneous, isotropic nonlocal thermoelastic half-space has also been investigated. The formulae for various reflection coefficients and their respective energy ratios are determined in various cases. For a particular material, the effects of the angular frequency and the elastic nonlocal parameter have been shown on phase speeds and the attenuation coefficients of the propagating waves. The effect of the elastic nonlocality on the reflection coefficients and the energy ratios has been observed and depicted graphically. Finally, analysis of the various results has been interpreted. KEYWORDS dispersion, energy partition, nonlocal, thermoelasticity, phase speeds and attenuation, reflection MSC CLASSIFICATION 74J10; 73B18; 73D15; 35Q74 Math Meth Appl Sci. 2020;43:1313-1336.wileyonlinelibrary.com/journal/mma

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Waves in Nonlocal Elastic Solid with Voids

Cited by 109 publications

References 28 publications

Waves in nonlocal thermoelastic solids of type III

Waves in nonlocal thermoelastic solids of type III

Thermoelastic plane waves under the modified Green–Lindsay model with two‐temperature formulation

Reflection of plane waves in generalized thermoelasticity of type III with nonlocal effect

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