Propagation of waves in an incompressible rotating transversely isotropic nonlocal elastic solid

Abstract

In this paper, the nonlocal elasticity theory is applied to study the propagation of plane wave and Rayleigh-type surface wave in an incompressible, rotating and transversely isotropic material. The governing equations of motion for an incompressible, rotating, transversely isotropic and nonlocal elastic medium are specialized for a plane. The medium is assumed rotating about an axis perpendicular to the plane. The transverse isotropy axis is taken perpendicular to the surface. The specialized governing equations are first applied to derive a velocity equation for homogeneous plane wave. The specialized governing equations along with traction free boundary conditions are also applied to derive the secular equation governing the wave speed of Rayleigh wave. The speeds of plane wave and Rayleigh wave are computed and illustrated graphically to observe the effects of nonlocality, rotation, anisotropy, frequency and propagation direction. It is noticed from the theory and numerical results that the speeds of both plane wave and Rayleigh wave decrease sharply with an increase in nonlocal parameter or rotation parameter. The speeds of plane wave and Rayleigh wave increase logarithmically with anisotropy material parameter. The feasible ranges of nonlocality, rotation or anisotropy parameters for the existence of plane wave or Rayleigh surface wave are determined for a given wave speed when the values of other parameters are fixed.

Expressions of nonlocal quantities and application to Stoneley waves in weakly nonlocal orthotropic elastic half-spaces

“…Moreover, the authors want to state that these expressions correspond to the weakly nonlocal model of elasticity, not to the Eringen's nonlocal elasticity theory. These expressions will be obtained in the next section by solving directly the differential equation ( 1) along with the extra condition (3).…”

“…where = le 0 (e 0 is the material constant, l is the atomic spacing) and ∇ = [ ∂ ∂x 1 , ∂ ∂x 2 , ∂ ∂x 3 ] T is the gradient operator; the researchers have carried out a large number of investigations of harmonic plane waves propagating in various nonlocal elastic media, including nonlocal purely elastic media [2,3], nonlocal thermoelastic media [4][5][6], nonlocal piezoelastic media [7][8][9], nonlocal micropolar elastic media [10][11][12][13][14][15], nonlocal porous elastic media [16][17][18], and nonlocal elastic solids with voids [19][20][21][22][23]. These works investigated the propagation of harmonic plane waves in infinite nonlocal continuum solids [2,4,6,7,10,14,16,17,19,21], the reflection of harmonic plane waves from free boundaries of nonlocal elastic half-spaces [3,4,6,10,13,15,16,19], the reflection and transmission of harmonic plane waves through plane interfaces of two nonlocal elastic half-spaces [8,9,12], the propagation characteristics of Rayleigh waves…”

“…where ϵ = l e 0 ( e 0 is the material constant, l is the atomic spacing) and ∇ = [ ∂ ∂ x 1 , ∂ ∂ x 2 , ∂ ∂ x 3 ] T is the gradient operator; the researchers have carried out a large number of investigations of harmonic plane waves propagating in various nonlocal elastic media, including nonlocal purely elastic media [2, 3], nonlocal thermoelastic media [4–6], nonlocal piezoelastic media [7–9], nonlocal micropolar elastic media [10–15], nonlocal porous elastic media [16–18], and nonlocal elastic solids with voids [19–23]. These works investigated the propagation of harmonic plane waves in infinite nonlocal continuum solids [2, 4, 6, 7,…”

On the well-posedness of Eringen’s non-local elasticity for harmonic plane wave problems

“…(b) Application of Eringen's method to problems of harmonic plane waves Following Eringen's method, researchers have carried out a large number of investigations of harmonic plane waves propagating in various non-local elastic media. They are non-local purely elastic media [11,12], non-local visco-elastic media [13], non-local diffusive elastic media [14], nonlocal thermo-elastic media [15][16][17][18][19][20], non-local piezo-elastic media [21][22][23][24], non-local micropolar elastic media [25][26][27][28][29][30][31][32], non-local porous elastic media [33][34][35][36][37] and non-local elastic solids with voids [38][39][40][41][42]. These works investigated the propagation of harmonic plane waves in infinite non-local continuum solids [11,15,17,21,25,29,33,34,38,40], the reflection of harmonic plane waves from free-boundaries of non-local elastic half-spaces [12,15,…”

“…These works investigated the propagation of harmonic plane waves in infinite non-local continuum solids [11,15,17,21,25,29,33,34,38,40], the reflection of harmonic plane waves from free-boundaries of non-local elastic half-spaces [12,15,17,25,28,30,33,38], the reflection and transmission of harmonic plane waves through plane interfaces of two non-local elastic halfspaces [22,24,27,31,36]. They also considered the propagation characteristics of Rayleigh waves [5,[12][13][14]16,[18][19][20]23,26,32,35,37,42], Lamb waves [33,41] and Love waves [39] in non-local elastic media.…”