Classical continuum theories restrict the response of the continuum stringently to local actions, thus these theories are not capable to explain some phenomena precisely where the length scales are often sufficiently short as in nanostructures where it is required to consider the small length scales. This paper within the framework of nonlocal elasticity is concerned with the study of wave-surface features in nonlocal elasticity for cubic crystals. The nonlocal Christoffel equation of wave motion is derived and dispersion relations are obtained. The present model predicts some notable features of the dispersion relations in cubic crystals in comparison with classical local model. By considering the wave and slowness surfaces in [100], [110], and [111] planes of cubic crystals a perceptible change is observed with nonlocality parameter. In nonlocal theory longitudinal and transverse waves, become dispersive and influenced by non-locality parameter, whereas theses waves are non-dispersive in its counterpart classical theory (local theory). It is found that phase and group wave velocities for longitudinal and transverse modes are influenced by the nonlocality parameter only when its value is greater than 0.001. Numerical calculation for crystals Silicon (Si), Aluminum (Al), Copper (Cu), Nickel (Ni), Gold (Au) are carried and found that velocities of longitudinal and transverse waves continuously decreases with increases of non-locality parameter. Polar diagram and wave’s surfaces for phase and group velocities (m/s) of longitudinal and transverse and slowness surfaces are represented graphically in nonlocal elasticity.
Keywords: Nonlocal,Wave surface,Slowness,nonlocality parameter,anisotropy factor,Nonlocal christoffel equation.
Received: 2 June 2020 / Revised: 6 July 2020 / Accepted:30 July 2020/ Published: 27 August 2020
This study originates a generalized nonlocal Christoffel equation in the nonlocal theory of elasticity. Influences of nonlocal parameter (e) on the wave’s spectrum, anisotropy factor, slowness surfaces in various directions are investigated. The results obtained are exhibited in the tabular forms and represented graphically considering cubic crystals.
In the conventional continuum mechanics, linear theory of elasticity is inherently size independent and predicts no dispersion and is valid only for small wave numbers. Elastic strain, the stress and the elastic strain energy of defects are singular at the imperfection line. Undoubtedly, if one makes use of classical elasticity within the imperfect region, then such unphysical singularities then the penalty has to be compensated. Because of such limitations, we need to consider the small length scales such as lattice spacing between individual atoms, grain size, the nonlocal elasticity theory pioneered by Edelen and Laws [1]; Edelen, et al. [2]; Eringen and Edelen [3]; Eringen [4] and Eringen [5]; Eringen [6] which state that the local position at a point is influenced by the action of all particles of the body. Edelen [7] published a treatise in which he gave a rigorous comprehensive analysis of the foundations of nonlocal theories.
In classical (local) elasticity, several researchers Miller and Musgrave [8]; Musgrave [9]; Farnell [10]; Brugger [11]; Musgrave [12]; Buchwald and Davis [13] and Mielnicki [14]in the past splendid introduction of the fundamental concepts is explained and studied the wave surfaces. Philip and Viswanathan [15] studied the behavior of the sections of the inverse velocity surfaces and found that a large number of cubic crystals exhibit cuspidal edges for the sections of energy surfaces along the (100), (110) and (111) directions. Narasimha and Viswanathan [16] studied elastic wave surfaces for the (111) plane of cubic crystals. Since all these studies are in accomplished in traditional classical continuum mechanics models, which are scale free or size-independent, and its application to extended wave limit according to the atomic theory is not capable to explicate the small nanoscale size effect Gurtin and Murdoch [17]; Gleiter [18]; Lim, et al. [19]. Consequently, properties which are associated with nanostructures like lattice spacing between atoms, grain size, surface stress, etc., must be taken into consideration in any of the classical continuum models to study the requirement of size-dependence and which is applicable to micro and nano structures. Recently, authors Khurana and Tomar [20]; Dilbag, et al. [21]; Kaur, et al. [22] studied waves problems microstretch solid, micropolar elastic solid half-space, and with voids in the context of nonlocal theory. Slowness is defined as the inverse of velocity and slowness surfaces, by means of Christoffel equation for the wave propagation in elastic media, displays many interesting features as in Buchwald and Davis [13]; Lin, et al. [23]; Fein and Smith [24]. Slowness surface has an vital physical significance as a succinct graphical representation of the variation of all types of velocity with respect to direction of the slowness vector and is used as a pictographic to explain. Slowness surface are two-dimensional entities in three-dimensional space. Studies of elastic waves in such simple and mostly isotropic systems are widely available in the books [25-30]. Verma [31] studied the thermoelastic slowness surfaces in anisotropic media with thermal relaxation in the local generalied thermoelasticity.
In the present work, due to the establishment of the nonlocal theory, the aspects of wave quantities required in constructing wave fields propagating elastic media are calculated as a function of the slowness vector or of its direction called the wave normal. Based on the nonlocal theory of elasticity by Eringen, analysis of some interesting wave-surface features are studied for an elastic materials of cubic symmetry. Longitudinal and transverse waves, become dispersive but non-attenuating and influenced by non-locality parameter in this nonlocal theory, whereas theses waves are non-dispersive in its counterpart classical continuum mechanics theory (local theory). Wave and slowness surfaces are studied in [100], [110], and [111] planes of cubic crystals. It is found that phase and group wave velocities for longitudinal and transverse waves are affected only when the magnitude non-locality parameter is greater than or equal to 0.001and decreases with increases of non-locality parameter. Phase and Group velocities (m/s) of longitudinal and transverse wave’s surfaces polar diagram of phase velocity (m/s) and slowness surfaces are also represented graphically in nonlocal elasticity materials for Silicon(Si), Aluminum (Al), Copper (Cu), Nickel (Ni), Gold (Au).
Recognizing an Eringen-type nonlocal differential model [5] the stress may be associated with the displacement in the analogous case of nonlocal elasticity. The integral constitutive relations can be represented in a linear differential form as an Eringen type differential model for the nonlocal elastic media as:
In the nonlocal theory of elasticity, elasto-dynamical Equation 7 describing the inertial forces can be written with (the displacement) as
Equation 16 are the strain tensor represents the relation in terms of displacement parameters.
Using (15), (16) in (13) we have
The roots of Equation 17 can be studied and represented graphically as three unique velocity surfaces in the nonlocal elasticity for any cubic material. It is convenient to examine and study the waves and slowness surfaces in [100], [110], and [111] planes of cubic crystals in nonlocal elasticity, intersection of these surfaces with the three principal orthogonal planes can be investigated in nonlocal elasticity. Relationships describing the wave velocity surfaces in the context of nonlocal elasticity for each plane can be derived using (17).
3.1. Propagation along a Cube Face
For propagation in the plane of a cube face (001), the above equations are simplified by
These curves exhibit the maximal symmetry of the cubic system - the stiffness tensor has the same form for all cubic classes, and so is invariant for the symmetry operations in the holosymmetric class. The velocity of the quasi-transverse wave has extrema in the [100] and [110] directions, given by
3.2. Propagation in a Diagonal Plane
In order to study the wave propagation in the (111) plane, it is convenient to transform to a new set of axes
non-attenuating and are influenced by non-locality parameter, while theses waves are non-dispersive and non-attenuating in its counterpart local theory of elasticity. It can be seen that the transverse wave in nonlocal elastic solid travels slower than that of longitudinal wave even in the presence non-locality parameter likewise as in case of classical continuum mechanics.
3.3. Wave Surface along a Cube Edge
If we consider the problem of progressive waves propagation along the [001] edge of a cubic crystal, we take
As regards the polarization of the waves, Equation 25 demonstrates that in this case one wave corresponds to corresponds to a pure transverse wave. The remaining two waves are polarized in the plane (001) on mutually perpendicular directions, one being quasi-transverse, and the other quasi-longitudinal. It is interesting to note that the directions of polarization of the last two waves are influenced by the nonlocality parameter.
3.5. Special Case
which are wave velocities of longitudinal and transverse waves in the classical continuum mechanics theory, (local theory) which becomes non-dispersive non-attenuating.
The phase vectors depict the direction of the phase velocity whereas group vectors depict the direction of the
eigenvalues corresponding to a longitudinal wave, and two transverse waves in the same manner as in local case. The largest eigenvalue of Equation 17 corresponds to the longitudinal wave propagation an is uniquely defined, because the velocity of the longitudinal wave is always greater than those of the transverse. Therefore the slowness sheet is the innermost one and is away from the other two which are coincident (in isotropic case) is a function of non local parameter. Further the polarization vector of the longitudinal wave is tangent to the wave front normal and the polarization vectors of the transverse waves are normal vector of the longitudinal wave, with the three eigenvectors forming an orthogonal system.
The numerical computation is carried out over cubic materials. Physical data of the substances that crystallize in the cubic system has only three independent stiffness constants are given in Table 1.
From the Equation 21 the velocity of the quasi-transverse wave has extrema in the [100] and [110] directions, in which propagation direction [100]; Polarization[100] (Longitudinal) and (100) plane (Transverse) velocity is given by in Table 2. From the table values for crystals Silicon(Si), Aluminum (Al), Copper (Cu), Nickel (Ni), Gold (Au) found that velocities of longitudinal and transverse waves continuously decreases with increases of non-locality parameter when it is greater than 0.001 and the no change is observed when non local parameter is less than 0.001.
In Table 3,propagation direction [110]; polarization[100] (longitudinal) and [110] plane (transverse) and
Figure-1. Wave surfaces of quasi-longitudinal, quasi-transverse and transverse for silicon for in the absence of nonlocal parameter.
Figure-2. Wave surfaces of quasi-longitudinal, quasi-transverse and transverse for silicon for nonlocal parameter .
Figure-3. Wave surfaces of quasi-longitudinal, quasi-transverse and transverse for silicon for nonlocal parameter .
Figure-4. Slowness surfaces of quasi-longitudinal, quasi-transverse and transverse for silicon in the absence of nonlocal parameter.
Figure-5. Slowness surfaces of quasi-longitudinal, quasi-transverse and transverse for silicon for nonlocal parameter
Figure-6. Slowness surfaces of quasi-longitudinal, quasi-transverse and transverse for silicon for nonlocal parameter
Various features of the slowness or wave surface have been well acknowledged and qualitatively understood in classical (local) elastic solids, a very few study has been accomplished in nonlocal elasticity. Analytical scheme to determine the wave surface in general is based on the Christoffel equation [33] has been undertaken in this article we have derived the nonlocal Christoffel equation for anisotropic material, and then specializing it to the material of cubic symmetry. On the basis of this equation the following conclusions are drawn:
Propagation along a cube face transverse wave polarized along OX3, with velocity
Funding: This study received no specific financial support. |
Competing Interests: The author declares that there are no conflicts of interests regarding the publication of this paper. |
Acknowledgement: Author is thankful to the Professor S.K. Tomar of Panjab University, Chandigarh under his guidance work was initiated when the author was visiting the Department of Mathematics, Center for Advanced Study in Mathematics, Panjab University, Chandigarh under the visiting scientists scheme of University Grants Commission, New Delhi, INDIA. |
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