An Analytical Study on Surface Energy Effect on Free Longitudinal Vibration of Cracked Nanorods
محورهای موضوعی : Applied Mechanics
H Shokrollahi
1
(Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran)
R Nazemnezhad
2
(School of Engineering, Damghan University, Damghan, Iran)
کلید واژه: Surface energy, Cracked nanorod, longitudinal vibration,
چکیده مقاله :
The present work analytically studies the free longitudinal vibration of nanorods in the presence of cracks based on the surface elasticity theory. To this end, governing equations of motion and corresponding boundary conditions are obtained using Hamilton’s principle. Due to considering the surface stress effect, as well as the surface density and the surface Lamé constants, the obtained governing equations of motion become non-homogeneous. The non-homogeneous governing equations are solved using appropriate analytical methods, and the natural frequencies are extracted. To have a comprehensive research, the effects of various parameters such as the length and radius of the nanorod, the crack severity, the crack position, the type of boundary condition, and the values of surface and bulk material properties on axial frequencies of the nanorod are investigated. Since this work considers the effects of all surface energy parameters, it can be claimed that it is a comprehensive study in this regard.
The present work analytically studies the free longitudinal vibration of nanorods in the presence of cracks based on the surface elasticity theory. To this end, governing equations of motion and corresponding boundary conditions are obtained using Hamilton’s principle. Due to considering the surface stress effect, as well as the surface density and the surface Lamé constants, the obtained governing equations of motion become non-homogeneous. The non-homogeneous governing equations are solved using appropriate analytical methods, and the natural frequencies are extracted. To have a comprehensive research, the effects of various parameters such as the length and radius of the nanorod, the crack severity, the crack position, the type of boundary condition, and the values of surface and bulk material properties on axial frequencies of the nanorod are investigated. Since this work considers the effects of all surface energy parameters, it can be claimed that it is a comprehensive study in this regard.
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An analytical study on surface energy effect on free longitudinal vibration of cracked nanorods
ABSTRACT
The present work analytically studies the free longitudinal vibration of nanorods in the presence of cracks based on the surface elasticity theory. To this end, governing equations of motion and corresponding boundary conditions are obtained using Hamilton’s principle. Due to considering the surface stress effect, as well as the surface density and the surface Lamé constants, the obtained governing equations of motion become non-homogeneous. The non-homogeneous governing equations are solved using appropriate analytical methods, and the natural frequencies are extracted. To have a comprehensive research, the effects of various parameters such as the length and radius of the nanorod, the crack severity, the crack position, the type of boundary condition, and the values of surface and bulk material properties on axial frequencies of the nanorod are investigated. Since this work considers the effects of all surface energy parameters, it can be claimed that it is a comprehensive study in this regard.
Keywords: Cracked nanorod; longitudinal vibration; surface energy.
1 INTRODUCTION
Defects may establish/create intentionally or unintentionally in structures. Defects are created intentionally when a structure is designed in such a way that one element of the structure collapses instead of the entire structure in critical situations like earthquakes [1]. Therefore, the source of this type of defect is an engineering design. On the other hand, the defects created unintentionally may appear due to issues such as material, manufacturing processes, and operation. This kind of defect is almost always destructive. Therefore, it should be detected before causing serious damage to structures. One of these destructive defects is the crack which is very common in structures. The crack can appear in structures with macro-, micro-, or nano-scale.
Cracked nanorods have gained significant attention in the field of nanotechnology due to their unique properties and potential applications. These nanorods, which are typically made of materials such as gold or silver, undergo controlled cracking to create gaps or breaks along their length. This deliberate manipulation of nanorod structures opens up a range of practical applications in various fields. The main practical applications of cracked nanorods include sensing and detection, catalysis, optics, electronics, and energy conversion [2, 3].
A literature survey on the investigation of the effects of the crack on the mechanical behaviors of nanoscale structures shows that this issue has been examined in various cases. Here, the related references are categorized according to the shape of the nano-structure, plate, beam, bar, and rod.
For plate-like nanostructures, Wang et al. [4] have simulated micro-crack healing in copper nano-plate during heating using the molecular dynamics method. In another study, the crack propagation process in a single-crystal aluminum plate with central cracks under tensile load has been simulated by the molecular dynamics method [5]. Hu et al. [6] have studied the surface energy effect on the fracture behavior of thin films with Mode-I and mode-II cracks.
For beam-like structures, there are considerable studies. The studies have considered the effect of the crack on the mechanical behavior of beam-like nanostructures based on various theories. Based on the nonlocal elasticity theory, free transverse vibration of Euler-Bernoulli [7-10] and Timoshenko [11-14] nanobeams, functionally graded nanobeams [15, 16], nanobeams with multiple cracks [17], and nanobeams embedded in the elastic medium [12, 15, 18] have been studied. In addition, Khorshidi et al. [19] have investigated buckling and postbuckling behaviors of cracked nanobeams made of single-crystalline nano-materials incorporating the beam's axial stretching via von Karman nonlinear theory. There are also some studies based on the surface elasticity theory. In this regard, free transverse vibration of thin and thick nanobeams with single and mixed-mode cracks are investigated by considering the surface energy effects [12, 20-23]. The other studies on the mechanical behaviors of nano-beams are based on the modified couple stress theory. Bending [24], buckling and postbuckling [25, 26], and free transverse vibration [27-29] of cracked nanobeams have been analyzed based on the modified couple stress theory.
For bar-like nanostructures in which the torsional behavior is desired, Loya et al. [30] and Rahmani et al. [31] have considered the effect of the crack on free torsional vibration of nanobeams using the nonlocal elasticity theory. In addition, free torsional vibration of cracked nanobeams incorporating the surface energy effect has been investigated by Nazemnezhad and Fahimi [32].
Finally, there are a few studies in which the axial or longitudinal behaviors of nanostructures in the presence of the crack are analyzed. In references [33, 34] free axial vibration of cracked nanorods has been investigated based on the nonlocal elasticity theory. Furthermore, the elastic medium effect as well as the crack effect on the axial vibration of nanorods is studied based on the nonlocal elasticity theory [35].
The above literature survey shows that the works considered the effect of the crack on mechanical behaviors of nano-scale structures have implemented the nonlocal elasticity theory, the modified couple stress theory, and the surface elasticity theory. Among the four types of mentioned nanostructures, three of them, plate-like, beam-like, and bar-like nanostructures, have been analyzed using the surface elasticity theory. Since the theories give different results for a specific problem, it is necessary to have a comprehensive insight into the various aspects of the problem for future research. The importance of the present study comes from two main aspects. The first aspect lies in the methodology employed to model cracks and mathematically illustrate their impact on dynamic systems. This encompasses both the formulation of equations of motion and the consideration of boundary conditions. It is widely acknowledged or predictable that cracks tend to decrease the natural frequencies of a system by inducing structural softening. However, the equations of motion have been derived in a manner that necessitates the development of a new solution method. This highlights the requirement for an alternative approach, which may not have been necessary or utilized in other similar problems. The second aspect pertains to the utilization of the theory of surface elasticity. The distinction between this particular theory and other proposed theories, such as non-local theory, lies in its omission of a tuning or matching parameter within the equations. In other proposed theories, a parameter is introduced to bring the theoretical results closer to the simulation or laboratory results by adjusting its value. Typically, this parameter's value is contingent upon various conditions, even within a specific problem. However, this is not the case with the theory of surface elasticity. Instead, it solely deals with a series of surface mechanical properties whose values are considered fixed for a specific material. Therefore, it is important to examine the behavior of the structure from the perspective of this theory. To this end, governing equation of motion and corresponding boundary conditions of cracked nanorods incorporating the surface energy effects are obtained using Hamilton’s principle. Due to considering the surface energy effect the obtained governing equation of motion becomes non-homogeneous. To extract the natural frequencies of the nanorod, firstly the non-homogeneous governing equation is converted to a homogeneous one using an appropriate change of variable, and then for clamped-clamped and clamped-free boundary conditions, the governing equation is solved using an analytical method. To conduct comprehensive research, the effects of various parameters such as the length and radius of the nanorod, the crack severity, the crack position, the type of boundary condition, and the values of surface and bulk material properties on the axial frequencies of nanorod are investigated.
2 PROBLEM FORMULATION
Consider a thin nanorod having length L (0 ≤ x ≤ L) and cross section of A, in a Cartesian coordinate system xyz, as shown in Fig. 1.
According to the simple rod theory, the components of displacement (u, v, and w) are as follows [36]
| (1) |
| (2) |
| (3) |
in which t is the time in sec. Having these displacements, assuming the rod is made from an isotropic material, the strains and stresses can be defined as
| (4) |
| (5) |
| (6) |
| (7) |
The Eqs. (4)-(7) are represented the strains and stresses related to the bulk material of the nanorod. If the surface energy effect is included in the analysis, the surface stress and strain components must be obtained. To this aim, the surface elasticity theory is proposed. In surface elasticity theory proposed by Gurtin and Murdoch [37], the relation between surface stress and strain can be expressed as
| (8) |
| (9) |
in which 
is residual surface stress related to no strain condition, 
is Kronecker delta, 
and 
are Lamé constants, 
are surface displacement components, and 
. Note that the positive and the negative signs are represented for upper and lower surfaces of the nanorod (for rectangular or quadrangular cross sections). Since the nanorod in this study has circular cross section, the positive and negative signs are disregarded.
Using Eqs. (1)-(3), surface stresses effective to longitudinal vibration of nanorod are obtained as
| (10) |
| (11) |
In order to arrive to the governing equation and the boundary conditions, the bulk and surface stresses and strains must be used in Hamilton’s principle defined by Eq. (12),
| (12) |
The variation of kinetic energy of nanorod take into account the effect of kinetic energy of surface density, can be written as
| (13) |
in which ρ and ρs are bulk and surface density, respectively, and A and S are surface and periphery of nanorod, respectively. Substituting Eqs. (4) and (6) in forming the potential energy relation, the variation of potential energy can be expressed as
| (14) |
Substituting Eqs. (13) and (14) into Eq. (12), and integrating the resulted equation by part, the governing equation and the corresponding boundary conditions of cracked nanorods incorporating the surface energy effects are obtained as follows,
| (15) |
| (16) |
Assume that in the crack location (x = LC) an equivalent linear spring K connecting the two segments of the nanorod, then for each segments of the nanorod, i.e. 0 ≤ x < LC and LC < x ≤ L, the Eqs. (15) and (16) must be applied. Implementing Eq. (15) leads to following equations,
| (17) |
| (18) |
where 
and 
.
In crack location, x = LC, following continuity conditions must be satisfied,
| (19) |
| (20) |
Moreover, end conditions of nanorod for clamped-clamped and clamped-free nanorods are as Eq. (21) and (22) respectively,
| (21) |
| (22) |
Eqs. (19) and (22) imply that the relation of boundary condition at the free end of the nanorod as well as one of the relations of the continuity conditions are inhomogeneous. Therefore, in order to solve the governing equations of motion, these relations must be homogenized at first.
It is worth mentioning here that it has not been reported in literature that homogeneous relations of boundary conditions and/or equations of motion are changed to inhomogeneous ones by considering the surface energy effects on various mechanical behaviors of nanosized structures. Therefore, the present study reports this issue for the first time.
In order to study the vibration characteristics of the cracked nanorod, Eqs. (17)-(20) along with Eq. (21) and (22) must be solved for the clamped-clamped and the clamped-free conditions, respectively. The method of solution of obtained equations is presented in next two subsections.
2.1 Clamped-clamped cracked nanorod
As mentioned before, the first step in solving the governing equations of motion is homogenization of Eqs. (19) and (22). At first, the equations and boundary conditions must be homogenized. To this aim it is supposed that
| (23) |
| (24) |
Substituting u1(x,t) and u2(x,t) into Eqs. (17)-(21), the related equations for u1cc(x) and u2cc(x) are obtained as follows
| (25) |
| (26) |
| (27) |
| (28) |
| (29) |
Solving Eqs. (25)-(29) leads to
| (30) |
| (31) |
Now, substituting Eqs. (30) and (31) into Eqs. (23) and (24) and using u1(x,t) and u2(x,t) in Eqs. (17)-(21) leads to following equations,
| (32) |
| (33) |
| (34) |
| (35) |
| (36) |
where 
. Assuming harmonic displacements as
| (37) |
| (38) |
Eqs. (32)-(36) can be rewritten as
| (39) |
| (40) |
| (41) |
| (42) |
| (43) |
The solutions of Eqs. (39) and (40) are
| (44) |
| (45) |
where 
.
Application of Eqs. (41)-(43) leads to B1 = 0 and following system of equations,
| (46) |
To have a nontrivial solution, determinant of coefficient matrix must set to be zero. The resulted equation is as follows
| (47) |
Solving Eq. (47) numerically, using MATLAB software, the natural frequencies of the clamped-clamped cracked nanorod are obtained.
2.2 Clamped-free cracked nanorod
Similar to section 2.1, suppose that 
and 
. Substituting u1(x,t) and u2(x,t) into Eqs. (17)-(20) and (22), the related equations for u1cf(x) and u2cf(x) are obtained as follows
| (48) |
| (49) |
| (50) |
| (51) |
| (52) |
Solving Eqs. (48)-(52) leads to,
| (30) |
| (31) |
Now, substituting Eqs. (53) and (54) into and and using u1(x,t) and u2(x,t) in Eqs. (17)-(20) and (22) leads to following equations,
| (55) |
| (56) |
| (57) |
| (58) |
| (59) |
Assuming harmonic displacements as 
and 
, Eqs. (55)-(59) can be rewritten as
| (60) |
| (61) |
| (62) |
| (63) |
| (64) |
The solution of Eqs. (60) and (61) are 
and 
where . Application of Eqs. (62)-(64) leads to B1 = 0 and following system of equations,
