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Elastic impact response of a nonlocal rectangular plate
Accepted Manuscript
Elastic impact response of a nonlocal rectangular plate Shahrokh Hosseini-Hashemi , Amin Sepahi-Boroujeni PII: DOI: Reference:
S0020-7683(17)30010-0 10.1016/j.ijsolstr.2017.01.010 SAS 9424
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
29 August 2016 13 December 2016 5 January 2017
Please cite this article as: Shahrokh Hosseini-Hashemi , Amin Sepahi-Boroujeni , Elastic impact response of a nonlocal rectangular plate, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.01.010
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ACCEPTED MANUSCRIPT Highlights
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The impact of a nonlocal plate with an indenter was theoretically studied. A conceptual contact law was utilized based on van-der Waals forces. Nonlocal parameter has significant effects on transient behavior of the plate. Despite macro elastic models, new model suggests that particle might stick to plate. Regardless of its distance to particle, plate remains under the effect of particle.
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Elastic impact response of a nonlocal rectangular plate Shahrokh Hosseini-Hashemi *, Amin Sepahi-Boroujeni Impact Research Laboratory, Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
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Abstract This paper concerns with the problem in which a spherical nano-particle impacts a rectangular orthotropic nano-plate. The plate is assumed thin, so that can be addressed by the Kirchhoff’s classical plate theory assumptions. The governing equations are derived using nonlocal theory of elasticity. The nano-particle is considered to collide with the plate
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transversely while the interactions are non-bonding van-der Waals forces. The contact force in terms of indenter-plate distance, i.e. contact law, is derived using a conceptual continuum model. Afterwards, the coupled equations of motion for both the plate and indenter together with the utilized contact law are solved and results are developed for the case of impact
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between a gold nano-particle and a single-layered graphene sheet with simply supported
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boundaries.
Keywords: Elastic; Impact; Orthotropic; Kirchhoff’s classical plate theory; Nonlocal theory;
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Contact law; Graphene sheet
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1. Introduction
Analyses of dynamics of nano-plates have been a major step in the development of
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NEMS, e.g. sensors and oscillators, devices that work based on the interaction of a nanostructure with external loads or objects. Several investigations like Lu et al. (2007), Pradhan and Phadikar (2009a), Jinbao et al. (2011), Jomehzadeh and Saidi (2011) and Jomehzadeh et al. (2012) conducted on nano-plate vibrations can be utilized to understand the vibrational behavior of such devices. However, in many practical applications, the response of plate
*
Corresponding author: E-mail address: [email protected] (Sh. Hosseini-Hashemi)
ACCEPTED MANUSCRIPT subjected to an external load or object must be taken under consideration. In this regard, investigations like Demiray and Eringen (1978) and He et al. (2012) have been done in the response of nano-plates to external harmonic or initial (i.e. impulse) force. Besides the above-mentioned fundamental investigations, studying the response of plate to an impact load, which is known as impact dynamics, is of vital importance. In spite of theoretical complications, this area due to wide applications has been considered as a major field in mechanics. In general, impact dynamics analyses include studying the collision of an
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external object (named as indenter or projectile) with a structure such as plate (named as target) to explain indenter and target motions as well as local displacements at the impact area. Depending on the type of problem, different models are employed in the analyses of impact phenomena, which may be categorized in three groups, Abrate (2001): 1- energybalance models, in which a quasi-static behavior is assumed for the structure; 2- spring-mass
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models, which analyzes the dynamic behavior of system using some simplified methods; and 3- complete models, in which dynamic behavior of structure is completely taken into account. At the macroscopic scale, various studies are done on the impact phenomena adoption the above-mentioned methods.
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Complete modeling of impact dynamics problems is based on the simultaneous solution of three equations including plate displacement, indenter motion, and contact force
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in terms of plate-particle distance. The latter relation is known as contact law. The first and the most famous contact law is known as the Hertz contact law. Some other theories, which
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have been developed and used in special cases, are described in Gao et al. (2014). Various contact laws have been utilized in complete modeling: Liu and Swaddiwudhipong (1997),
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Sburlati (2004), Vergani et al. (2011), Rossikhin and Shitikova (2011). Even though comprehensive studies have been conducted based on the macroscopic view, there is a gap in the literature regarding nano-scale investigations of the impact phenomenon. The necessity
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of applying non-local continuum theories in transient problems as well as the lack of appropriate solution methods account for the limited nano-scale studies. Another issue is to employ proper nano-scale contact laws. It should be kept in mind that all macroscopic contact laws, which are developed based on the field of stresses, energies or surface forces, are useless at nano-scale dimensions. Employing mathematical approximations, Hilder and Hill (2008a), Hilder and Hill (2008b), Hilder and Hill (2009) and Yue and James (2011) proposed approaches to investigate the non-bonding interactions with nano-tubes and graphenes. Chan and Hill (2010) employed mentioned method to illustrate the interaction of nano-particles
ACCEPTED MANUSCRIPT with graphene, which may be applied as a contact law in the impact problems. However, to the best of the authors' knowledge, the only published studies on the low-velocity impact on nano-plates/beams are Lee and Kwak (1993) and Seifoori and Liaghat (2013), which are classified as energy balanced methods (method 1). Employing complete modeling of impact, present study investigates the dynamics of a nano-plate imposed to the impact load of a nano-particle. The plate is assumed rectangular, orthotropic and thin, while the Kirchhoff’s classical plate theory (CPT) is employed together
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with non-local theory of elasticity to derive governing equations. An analytical approximation is utilized for the contact law of plate-indenter interaction, which is based on the LenardJones formula. This approach makes it possible to solve coupled equations of motion for the plate and indenter. As an example, the impact between a golden nano-particle and a single layered graphene sheet is numerically solved and the displacements of both plate and
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indenter together with contact forces are measured and analyzed in details. In addition, minimum number of mode shapes necessary for accurate answers are determined and some aspects of the problem such as the effects of impact velocity and non-local parameter on the dynamic behaviors of components are discussed for this case. As it will be explained,
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providing the possibility of employing expansion theorem to derive modal equations of motion, present method may be utilized to investigate impact response of plates in more
2.1
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2. Formulation
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general cases, such as impact response of thick plates or plates with large deformations.
Equations of motion
density of
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A thin orthotropic plate with a length of , width of , uniform thickness of , and is considered in this study. The origin of assigned
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system is located at one corner of the plate so that of undeformed mid-plane (i.e.,
).
axes are coincident with the edges
and
Poison's ratios in the corresponding directions, and A rigid spherical indenter with a radius of
and
Cartesian coordinate
are the moduli of elasticity and
represents shear modulus of elasticity.
, mass of
and initial velocity of
perpendicularly propelled toward the plate and strikes it at point
is
. According to the
CPT, non-zero resultant strains are: (1)
ACCEPTED MANUSCRIPT and according to Pradhan and Phadikar (2009a), the governing differential equation of motion is: (2) where
, is the mass momentum of inertia,
denotes displacement in
transversal distributed force (measured per unit area), and
direction,
are bending as:
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momentum resultants, which may be expressed as functions of local stresses of ∫
is
(3)
According to the non-local theory of elasticity, Hook's law may be formulated as Pradhan and
{
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Phadikar (2009a):
}
{
[
}
(4)
]
, is the 2-D Laplacian operator and
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in which
, is the non-local
parameter, introduced as a function of squared length with non-dimensional coefficient of then integrating along the thickness
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Based on Eqs. (1), multiplying Eq. (4) with
.
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, Eq. (4) can be rephrased as: }
{
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{
}
(5)
}
(6)
with flexural rigidities of plate defined as:
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{
}
{
Finally, pre-multiplying Eq. (2) with non-local operator
and employing Eq. (5),
non-local equation of motion for classical plate is derived as: ( 2.2
)
Orthogonality of mode shapes In the case of free vibrations, Eq. (7) may be reduced to:
(7)
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)
(8)
The mode shapes of vibrations, represented here by
(
), satisfy
equation of free vibrations. In Eq. (8), replacing displacement ( ) with the mode shape of , multiplying with arbitrary mode shape of
, then integrating over the area of plate ( )
result in:
∫
(
(
)
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∫ (9)
)
By employing partial integral properties to expand all terms of Eq. (8) and eliminating zero values on the boundaries of plate (i.e., clamped, simply supported, or free boundary
∫(
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conditions), the following symmetric form is obtained: )
(
)
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∫(
(10) )
in which the integrals are unaffected by the order of the subscripts of with
. In other
, multiplied with mode shape
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words, if Eq. (8) is rewritten by replacing
and
integrated over area, it will result in the same equation, but with coefficient of
and
instead of
∫
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can be derived as:
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in the left side. Accordingly, the orthogonality of mode shapes in a non-local classical plate
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∫
and in the case
(11) [
or
(
)
]
:
∫ (12) ∫
[
(
)
]
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3. Dynamic respose of plate-indenter system 3.1
Response to external load Expansion theorem is a strong tool to solve equations of motions in beams, plates, etc.
The theorem allows expanding functions over an orthogonal basis, an infinite sum, so that it will be possible to reduce the problem to simple modal equations. Providing orthogonal basis for the displacement (the orthogonality relations of basis, i.e. mode shapes has been derived in section 2.2 in the case of nonlocal CPT with different combination of boundary conditions),
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this theory may be utilized resulting in solvable modal equations. In the following, this theorem is employed to derive response of plate to external load. Due to Expansion theorem, displacement of
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displacements: ∑ in which
can be expanded as a series of modal
(13)
are defined as principal coordinates. Rewriting Eq. (7) using Eq. (13), pre-
multiplying Eq. (7) with arbitrary mode shape of
∫
∫
∫
[
(14) (
)
]
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∑
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∑
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can be rephrased as:
and integrating over the area of plate, it
Employing orthogonality properties of Eqs. (11) and (12), and after a few simplifications, the
(15)
is modal force and can be expressed as:
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where
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familiar form of modal equation is derived as follows:
∫
(16)
in which: ∫ represents mass portion of mode (
(17) ) in the response of plate.
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Response to impact load Impact force of
could be assumed as a concentrated force of
on the plate at point
imposed
, so it can be rewritten as: (18)
where
is the Dirac delta function. Combining Eq. (18) with Eq. (16) and employing partial
(
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integral properties, modal force can be reduced to: )
is defined here as the influence coefficient of impact force on vibration mode of (
(19) ).
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Eq. (15) is a simple ODE and can be solved using Laplace transform, resulting in: ∫
(20)
Now, the equation of motion for the indenter should be taken into account. According to the
∫ (21)
Deriving discrete equations
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3.3
, the following relations could be
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∫
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expressed for the indenter:
with a velocity of
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Newton's law, for displacement of
Solving equations (20) and (21) together with the equation of contact law will result in the
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dynamic response of plate to the impact force. It is worth mentioning that the contact law equation usually appears as a non-linear equation representing impact force in terms of indenter-plate distance. In the absence of an analytical solution for the set of these coupled equations, they may be solved employing a discrete numerical method, and by dividing the period of contact phenomenon into small increments of of
. For every increment, impact force
is assumed to be constant, which helps compute integrals of Eqs. (20) and (21). Assuming
zero initial conditions, i.e. (20) and (21) result in:
, and after some mathematical operations, Eqs.
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∑
∑
[ ∑
where index of
] (
(22)
∑
)
(23)
refers to the th increment. As a result, one can see that the time dependent
terms of plate response depend on the history of impact force during the contact procedure. In each interval, simplification of force to constant values leads to errors, which can be
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minimized by choosing adequately small time increments. The other reason for choosing very small time increments, as will be seen later, is the extremely transient nature of present nanoscale problem.
4. Contact law
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As mentioned before, interaction force between indenter and plate known as “contact law” depends on the indenter-plate distance. In the case of macro impact problems, various assumptions have led to several contact laws. These theories are based on geometries and material properties measured under the assumptions of continuum mechanics, which are not valid in nano-scale medium. Therefore, to present a reasonable solution for the impact
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problem in nano-sized objects, it seems inevitable to utilize a contact model based on nanoscale interaction forces.
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The majority of non-bonding interaction forces in neutral particles are classified as van-der Waals forces. These forces, depending on particles characteristics and distances,
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comprise attraction and repulsion effects. One of the most common and accurate models to
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state van-der Waals interaction is the well-known Lenard-Jones (L-J) equation: *
is L-J potential,
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in which
+
(24) is the distance between particles, and bold symbols
and
are
respectively energy and distance parameters, which depend on the physical features of particles. Herein, the utilized objects, including plate and indenter consist of several particles (i.e. atoms) interacting with each other. Discrete nature of structures makes it difficult to come up with a mathematical solution for the interaction force and potential. Consequently, it seems inevitable to employ nano-scale simulations such as molecular dynamics method. The major weakness of such methods is that, besides their significant complexity and timeconsuming computation processes, they do not present an analytical view of the problem.
ACCEPTED MANUSCRIPT Some analytical averaging methods cover this problem, resulting in some mathematical formulation for interaction forces and potentials, Hilder and Hill (2008a), Hilder and Hill (2008b), Hilder and Hill (2009), Chan and Hill (2010), Yue and James (2011), Chan and Hill (2010) and Israelachvili (2011). The main idea of these methods is to divide nano-structured media into small elements. Each element represents the properties of its included particles. Next step will be to integrate over the domain of structure to have desired property. Afterwards, L-J force between plate and indenter is derived by employing the described
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method.
Fig. 1 shows a solid spherical indenter interacting with a thin plate. At first, indenter is assumed to be placed above the plate with a distance of , far enough from the plate edges. The interaction force was measured between the indenter and the region illustrated in Fig. 1. This region is considered as a circular zone of plate with a radius of
, which is concentric
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with the vertical projection of the indenter on the plate. As shown in Fig. 1, a small element of the circular region is assumed with respect to the the center of circle. Given that
polar coordinate system assigned on
is the number of particles per unit area of the plate, the
(25)
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number of atoms inside each element is:
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L-J potential of Eq. (24) can be rewritten for the system of plate element and indenter as: +
(26)
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*
L-J potential between indenter and circular region on the plate could be determined by
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integrating Eq. (26) over the circular domain: ∫
[
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∫
Eq. (27) shows clearly that by increasing
{
( ,
) }
{
(
quickly converges to its value in
example, if the radius of circular region on plate is chosen as
) }] (27) . For
, the first and the second
braces of Eq. (27) will have the values of 0.96000 and 0.99968, respectively. For
the
values will be 0.99000 and 0.99999. Therefore, it will not be an unreasonable assumption to approximate the potential with its value in *
+
, which results in: (28)
ACCEPTED MANUSCRIPT Vertical ( -direction) component of force applied to the indenter will be: )
(29)
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(
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Fig. 1, schematic illustration of spherical indenter interacts with a circular region of plate with a radius of
.
As an example, force-distance variations are plotted in Fig. 2 for a gold atom approaching a single-layer graphene sheet (SLGS). Gold-carbon L-J parameters are taken as
To measure
(30)
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Chan and Hill (2010):
, we consider that the SLGS is a hexagonal lattice and carbon atoms are placed
on the vertices. Since every atom is shared between three hexagons, we may allocate two
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atoms to every hexagonal region of the SLGS. The area of a hexagon with respect to the bond
will be
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length is calculated as √ √
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according to Chan and Hill (2010).
, and so the number of particles per unit area of the SLGS , where the bond length is
,
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Fig. 2, The variations of L-J contact force and potential vs. distance for a gold atom approaching a SLGS.
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5. Results and discussion
Dynamic response of the plate-indenter system can be derived by solving simultaneous equations of Eqs. (22), (23), and (29). As an instance, the impact phenomenon occurring between a nano-particle and a nano-plate is investigated. The plate is assumed to so its mass will be
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be a SLGS and the indenter is taken as a single gold atom with atomic mass of
and
The indenter strikes the nano-plate at its middle
y
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point with coordinates of
a
, and L-J parameters for the interaction
between atoms are taken as Eq. (30). According to Shen et al. (2010), assumed SLGS has the
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following features:
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(31)
in which, based on Shen et al. (2010), the nonlocal parameter is taken as e=0.05; and boundary conditions are assumed simply supported (SSSS), which imply the displacement function as Pradhan and Phadikar (2009a): ∑ where
∑
represents the
,
(32)
th mode of vibration. The number of vibrational modes (mn),
which should be measured through the solving process, determines the response precision. To guarantee that the higher modes of vibration are taken into account, the number of 900
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5.1
Dynamic response of the plate to the elastic impact Fig. 3-a illustrates plate displacement at the impact point as the nano-particle is
propelled with a velocity of 380 m/s. This figure also displays the position of indenter, which
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demonstrates that the particle almost hits the plate twice at t=1.6 ps and 16.7 ps. However, as this figure obviously shows, the indenter rebounds after reaching a specific distance from the plate. Eq. (29) can explain this phenomenon. Based on this equation, as the particle gets closer to the plate, the repulsion force approaches infinity. As a result, the indenter never hits the plate in practice. Moreover, the term describing long-range attraction force in Eq. (29)
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determines that as the particle recedes farther, it remains under the effect of the plate. This attraction force accounts for the return of particle toward the plate at t= 16.7 ps. Fig. 3-b shows the history of indenter-plate interaction force, indenter velocity, and relative velocity of indenter with respect to plate. As this figure implies, even though the interaction force is negligible between the two approach (impact) moments, the variations of indenter velocity
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reveals the accelerated motion of particle. In other words, the particle displays accelerated motion even while is far from the plate, which contrasts with the macro-scale observations.
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Finally, as the L-J forces disappear due to increase in the plate-particle distance, the motion of indenter continues with a constant velocity. This section of motion can be distinguished in
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Fig. 3-b where the interaction force and the velocity respectively approach zero and 74 m/s after t=16.7 ps. Comparison of the variations of interaction force and displacement indicates
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that the maximum interaction force corresponds to the closest distance between the plate and indenter. At these points, the velocity curve intersects the time axis, which shows a
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change in the direction of indenter motion. To reach a deeper understanding of this issue, Fig. 3 also presents the plate deflections
at various sequences. The trend of excited vibration modes can be obviously seen from this figure. At the beginning of excitation and prior to the stress wave propagation through the plate, the deflection locally occurs at the impact point. As the excitation continues, other vibrational modes are excited within the plate so that the vibration amplitude is a combination of several modes of vibration.
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Fig. 3, a- displacement, and; b- contact force, indenter velocity , and relative velocity of
5.2
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indenter with respect to plate vs. time, in the case of impact with initial velocity of 380 m/s
Effect of nonlocality
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To examine the effect of nonlocality, the particle is assumed to hit the plate with an
initial velocity of 300 m/s. Then, the problem is solved with (e=0.05) and without (e=0.00) nonlocal parameter. Under these conditions, Fig. 4 illustrates the displacements of both plate and particle at the impact point. As this figure indicates, employing the nonlocal theory leads to the absorption of particle into the plate so that after bouncing on the plate, the particle eventually forms an integrated system with the plate. On the other hand, the local theory results in the receding of particle into space. This study reveals the substantial effects of nonlocal parameter. As a result, the nonlocal theory plays a pivotal role particularly in the phenomena of the absorption of nano-particles into nano-plates. This conflict can be
ACCEPTED MANUSCRIPT explained by the reduction in the plate stiffness due to the effect of differential operator on the left side of Eq. (7). In other words, the local plate is stiffer than the nonlocal one. The relation of natural frequencies of SSSS plate also supports this claim. Natural frequencies of system can be expressed as Pradhan and Phadikar (2009b): (
)
[
(33)
]
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√
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Fig. 4, displacement of plate and indenter vs. time recorded based on nonlocal and local
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theories.
According to this equation, ignoring the nonlocal parameter of denominator leads to the rise of all natural frequencies, which implies increase in the stiffness of plate. The stiffer system is
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more resistant to the deflection caused by the contact force. Therefore, a minor portion of particle’s kinetic energy is passed to the plate Consequently as can e seen in the present
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example, the plate is unable to dissipate as much energy as that is required for trapping the
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particle and, in turn, the indenter escapes.
5.3
Effect of impact velocity The term of long-range attraction force, which appears in the function of contact force
in Eq. (7), requires that the indenter remain within the force field of the plate. Thus, it is expected that in a mild collision, the particle will get trapped within an equilibrium distance. To illustrate, Fig. 5-a depicts a stationary indenter lies above the plate without initial velocity. Because of long-range attraction force, the particle approaches to the plate with an accelerated motion and, after several fluctuations, is caught by the plate. In other words, the
ACCEPTED MANUSCRIPT indenter is coupled to the plate and remains within a specific distance from that, which depends on material and geometry of both the plate and the indenter. This phenomenon can be further clarified by observing the variations of contact force in Fig. 5-a. As this figure demonstrates, as the particle gets closer to the plate, the contact force varies from a negative value (attraction) to a positive maximum (repulsion). After each collision, this maximum decreases and approaches, but never reaches zero. The contact force eventually reduces while varies within a narrow range and leads to the synchronized movements of plate and
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particle.
As mentioned before, in the present numerical solution, the first 900 modes of vibration are taken into account for the plate (m=n=30 in Eq. (32)). However, to achieve an as precise as desirable solution, the appropriate number of modes of vibration should be determined. To study this issue, Fig. 6-a displays the energy spectrum of modes. This figure,
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which shows the energy of vibration of each mode vs. the frequency, obviously demonstrates that considering the first 50 modes will result in an acceptable, precise solution. It is worth mentioning that, as mentioned above, when the particle sticks to the plate, they form an integrated system that vibrates with some major specific frequencies such as
(see Fig.
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6). Excited modes and their amplitudes depend on materials, geometry of plate, mass of
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indenter, position of contact point, and impact velocity.
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Fig. 5, displacement of plate and indenter as well as contact force vs. time for indenter with an initial velocities of a-zero and b- 500 m/s. In the case of considering an initial velocity of 500 m/s for the indenter, substantially
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different behavior will be observed. As Fig. 5-b displays, the particle recedes after hitting the plate. According to this figure, the contact force varies from a negative value (attraction force)
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to a positive one (repulsion force) so that at the escapes moment, the contact force reaches its maximum. As the indenter gets farther, on the other hand, the repulsion force reduces to zero,
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turns to attraction force, and again approaches zero as the particle recedes into space. Generally, the dynamic response of plate consists of free vibrations with several vibration modes, which are excited after the impact phenomenon. As the spectrum plot of Fig. 6-b displays, the excitation mostly includes the first 20 ones. As the indenter is propelled toward the plate with an initial velocity of zero (Fig. 6-a), the major excited modes can be detected which means that the system vibrates with specific frequencies, while the initial velocity of indenter causes the excitation of a wide range of modes. This observation could be explained by the amount of energy transferred to the system after collision. Initial velocity of indenter
ACCEPTED MANUSCRIPT gives rise to the excitation of higher levels of energy each of which corresponds to a particular
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mode of vibration.
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Fig. 6, energy spectrum of vibration modes, , for indenter with an initial velocities of azero and b- 500 m/s .
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6. Conclusions
Present research work includes efforts to the dynamics study of the impact of a nano-
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particle with a nano-plate, using a complete approach. The plate was considered rectangular and orthotropic, conformed to the classical plate theory. Non-local theory was adopted to extract the motion relations. A theoretical contact law was utilized based on the non-bonding
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forces of Lenard-Jones, which made it possible to solve displacement equations simultaneously. These equations, including the displacements of plate and indenter, were numerically solved for a specific example in which a gold atom hits a single-layered graphene sheet (SLGS). Then, displacements of plate and indenter together with the initial velocity and contact force were analyzed. In addition, the minimum number of vibration modes required to reach a precise solution as well as other effective parameters such as nonlocal parameter and impact velocity were investigated. The introduced model provided an analytical approximation to the dynamic response of 2-D structures at nano-scale. Findings revealed the
ACCEPTED MANUSCRIPT pivotal role of the nonlocal parameter. Ignoring this parameter and, consequently, minor increase in the stiffness of model led to the impractical response of the system, while the indenter did not stick to the plate, but receded into space. This investigation also implies that in contrast to the macro-scaled contact models, the indenter approaches and sticks to the plate without initial velocity. It was also found that for a range of initial velocities the particle eventually sticks to the plate, although in macro-scale elastic impact, the indenter finally does recede into space. Present solution can be generalized in the dynamics analyses of single or describe the performance of sensors and actuators.
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Israelachvili, J.N., 2011. Intermolecular and Surface Forces: Revised Third Edition. Elsevier Science.
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Jinbao, W., Xiaoqiao, H., Kitipornchai, S., Hongwu, Z., 2011. Geometrical nonlinear free vibration of multi-layered graphene sheets. Journal of Physics D: Applied Physics 44, 135401. Jomehzadeh, E., Saidi, A.R., 2011. A Levy Type Solution for Free Vibration Analysis of a Nano-Plate Considering the Small Scale Effect. INTECH Open Access Publisher. Jomehzadeh, E., Saidi, A.R., Pugno, N.M., 2012. Large amplitude vibration of a bilayer graphene embedded in a nonlinear polymer matrix. Physica E: Low-dimensional Systems and Nanostructures 44, 1973-1982. Lee, D.I., Kwak, B.M., 1993. An analysis of low-velocity impact of spheres on elastic curved-shell structures. International Journal of Solids and Structures 30, 2879-2893. Liu, Z., Swaddiwudhipong, S., 1997. Response of Plate and Shell Structures due to Low Velocity Impact. Journal of Engineering Mechanics 123, 1230-1237. Lu, P., Zhang, P.Q., Lee, H.P., Wang, C.M., Reddy, J.N., 2007. Non-local elastic plate theories. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 463, 3225.
ACCEPTED MANUSCRIPT Pradhan, S.C., Phadikar, J.K., 2009a. Nonlocal elasticity theory for vibration of nanoplates. Journal of Sound and Vibration 325, 206-223. Pradhan, S.C., Phadikar, J.K., 2009b. Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models. Physics Letters A 373, 1062-1069. Rossikhin, Y., Shitikova, M., 2011. Dynamic response of a prestressed orthotropic plate impacted by a sphere, Proceedings of the 2nd European conference of Control, and Proceedings of the 2nd European conference on Mechanical Engineering. World Scientific and Engineering Academy and Society (WSEAS), Puerto De La Cruz, Tenerife, Spain, pp. 175-180. Sburlati, R., 2004. An exact solution for the impact law in thick elastic plates. International Journal of Solids and Structures 41, 2539-2550.
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Seifoori, S., Liaghat, G.H., 2013. Low velocity impact of a nanoparticle on nanobeams by using a nonlocal elasticity model and explicit finite element modeling. International Journal of Mechanical Sciences 69, 85-93. Shen, L., Shen, H.-S., Zhang, C.-L., 2010. Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Computational Materials Science 48, 680-685.
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Vergani, L., Guagliano, M., Khorshidi, K., 2011. 11th International Conference on the Mechanical Behavior of Materials (ICM11)Elasto-Plastic Response of Impacted Moderatly Thick Rectangular Plates with Different Boundary Conditions. Procedia Engineering 10, 1742-1747.
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Yue, C., James, M.H., 2011. Hydrogen storage inside graphene-oxide frameworks. Nanotechnology 22, 305403.
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Graphical Abstract
Elastic impact response of a nonlocal rectangular plate Shahrokh Hosseini-Hashemi , Amin Sepahi-Boroujeni PII: DOI: Reference:
S0020-7683(17)30010-0 10.1016/j.ijsolstr.2017.01.010 SAS 9424
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
29 August 2016 13 December 2016 5 January 2017
Please cite this article as: Shahrokh Hosseini-Hashemi , Amin Sepahi-Boroujeni , Elastic impact response of a nonlocal rectangular plate, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.01.010
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ACCEPTED MANUSCRIPT Highlights
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The impact of a nonlocal plate with an indenter was theoretically studied. A conceptual contact law was utilized based on van-der Waals forces. Nonlocal parameter has significant effects on transient behavior of the plate. Despite macro elastic models, new model suggests that particle might stick to plate. Regardless of its distance to particle, plate remains under the effect of particle.
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Elastic impact response of a nonlocal rectangular plate Shahrokh Hosseini-Hashemi *, Amin Sepahi-Boroujeni Impact Research Laboratory, Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
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Abstract This paper concerns with the problem in which a spherical nano-particle impacts a rectangular orthotropic nano-plate. The plate is assumed thin, so that can be addressed by the Kirchhoff’s classical plate theory assumptions. The governing equations are derived using nonlocal theory of elasticity. The nano-particle is considered to collide with the plate
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transversely while the interactions are non-bonding van-der Waals forces. The contact force in terms of indenter-plate distance, i.e. contact law, is derived using a conceptual continuum model. Afterwards, the coupled equations of motion for both the plate and indenter together with the utilized contact law are solved and results are developed for the case of impact
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between a gold nano-particle and a single-layered graphene sheet with simply supported
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boundaries.
Keywords: Elastic; Impact; Orthotropic; Kirchhoff’s classical plate theory; Nonlocal theory;
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Contact law; Graphene sheet
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1. Introduction
Analyses of dynamics of nano-plates have been a major step in the development of
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NEMS, e.g. sensors and oscillators, devices that work based on the interaction of a nanostructure with external loads or objects. Several investigations like Lu et al. (2007), Pradhan and Phadikar (2009a), Jinbao et al. (2011), Jomehzadeh and Saidi (2011) and Jomehzadeh et al. (2012) conducted on nano-plate vibrations can be utilized to understand the vibrational behavior of such devices. However, in many practical applications, the response of plate
*
Corresponding author: E-mail address: [email protected] (Sh. Hosseini-Hashemi)
ACCEPTED MANUSCRIPT subjected to an external load or object must be taken under consideration. In this regard, investigations like Demiray and Eringen (1978) and He et al. (2012) have been done in the response of nano-plates to external harmonic or initial (i.e. impulse) force. Besides the above-mentioned fundamental investigations, studying the response of plate to an impact load, which is known as impact dynamics, is of vital importance. In spite of theoretical complications, this area due to wide applications has been considered as a major field in mechanics. In general, impact dynamics analyses include studying the collision of an
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external object (named as indenter or projectile) with a structure such as plate (named as target) to explain indenter and target motions as well as local displacements at the impact area. Depending on the type of problem, different models are employed in the analyses of impact phenomena, which may be categorized in three groups, Abrate (2001): 1- energybalance models, in which a quasi-static behavior is assumed for the structure; 2- spring-mass
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models, which analyzes the dynamic behavior of system using some simplified methods; and 3- complete models, in which dynamic behavior of structure is completely taken into account. At the macroscopic scale, various studies are done on the impact phenomena adoption the above-mentioned methods.
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Complete modeling of impact dynamics problems is based on the simultaneous solution of three equations including plate displacement, indenter motion, and contact force
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in terms of plate-particle distance. The latter relation is known as contact law. The first and the most famous contact law is known as the Hertz contact law. Some other theories, which
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have been developed and used in special cases, are described in Gao et al. (2014). Various contact laws have been utilized in complete modeling: Liu and Swaddiwudhipong (1997),
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Sburlati (2004), Vergani et al. (2011), Rossikhin and Shitikova (2011). Even though comprehensive studies have been conducted based on the macroscopic view, there is a gap in the literature regarding nano-scale investigations of the impact phenomenon. The necessity
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of applying non-local continuum theories in transient problems as well as the lack of appropriate solution methods account for the limited nano-scale studies. Another issue is to employ proper nano-scale contact laws. It should be kept in mind that all macroscopic contact laws, which are developed based on the field of stresses, energies or surface forces, are useless at nano-scale dimensions. Employing mathematical approximations, Hilder and Hill (2008a), Hilder and Hill (2008b), Hilder and Hill (2009) and Yue and James (2011) proposed approaches to investigate the non-bonding interactions with nano-tubes and graphenes. Chan and Hill (2010) employed mentioned method to illustrate the interaction of nano-particles
ACCEPTED MANUSCRIPT with graphene, which may be applied as a contact law in the impact problems. However, to the best of the authors' knowledge, the only published studies on the low-velocity impact on nano-plates/beams are Lee and Kwak (1993) and Seifoori and Liaghat (2013), which are classified as energy balanced methods (method 1). Employing complete modeling of impact, present study investigates the dynamics of a nano-plate imposed to the impact load of a nano-particle. The plate is assumed rectangular, orthotropic and thin, while the Kirchhoff’s classical plate theory (CPT) is employed together
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with non-local theory of elasticity to derive governing equations. An analytical approximation is utilized for the contact law of plate-indenter interaction, which is based on the LenardJones formula. This approach makes it possible to solve coupled equations of motion for the plate and indenter. As an example, the impact between a golden nano-particle and a single layered graphene sheet is numerically solved and the displacements of both plate and
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indenter together with contact forces are measured and analyzed in details. In addition, minimum number of mode shapes necessary for accurate answers are determined and some aspects of the problem such as the effects of impact velocity and non-local parameter on the dynamic behaviors of components are discussed for this case. As it will be explained,
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providing the possibility of employing expansion theorem to derive modal equations of motion, present method may be utilized to investigate impact response of plates in more
2.1
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2. Formulation
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general cases, such as impact response of thick plates or plates with large deformations.
Equations of motion
density of
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A thin orthotropic plate with a length of , width of , uniform thickness of , and is considered in this study. The origin of assigned
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system is located at one corner of the plate so that of undeformed mid-plane (i.e.,
).
axes are coincident with the edges
and
Poison's ratios in the corresponding directions, and A rigid spherical indenter with a radius of
and
Cartesian coordinate
are the moduli of elasticity and
represents shear modulus of elasticity.
, mass of
and initial velocity of
perpendicularly propelled toward the plate and strikes it at point
is
. According to the
CPT, non-zero resultant strains are: (1)
ACCEPTED MANUSCRIPT and according to Pradhan and Phadikar (2009a), the governing differential equation of motion is: (2) where
, is the mass momentum of inertia,
denotes displacement in
transversal distributed force (measured per unit area), and
direction,
are bending as:
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momentum resultants, which may be expressed as functions of local stresses of ∫
is
(3)
According to the non-local theory of elasticity, Hook's law may be formulated as Pradhan and
{
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Phadikar (2009a):
}
{
[
}
(4)
]
, is the 2-D Laplacian operator and
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in which
, is the non-local
parameter, introduced as a function of squared length with non-dimensional coefficient of then integrating along the thickness
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Based on Eqs. (1), multiplying Eq. (4) with
.
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, Eq. (4) can be rephrased as: }
{
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{
}
(5)
}
(6)
with flexural rigidities of plate defined as:
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{
}
{
Finally, pre-multiplying Eq. (2) with non-local operator
and employing Eq. (5),
non-local equation of motion for classical plate is derived as: ( 2.2
)
Orthogonality of mode shapes In the case of free vibrations, Eq. (7) may be reduced to:
(7)
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)
(8)
The mode shapes of vibrations, represented here by
(
), satisfy
equation of free vibrations. In Eq. (8), replacing displacement ( ) with the mode shape of , multiplying with arbitrary mode shape of
, then integrating over the area of plate ( )
result in:
∫
(
(
)
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∫ (9)
)
By employing partial integral properties to expand all terms of Eq. (8) and eliminating zero values on the boundaries of plate (i.e., clamped, simply supported, or free boundary
∫(
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conditions), the following symmetric form is obtained: )
(
)
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∫(
(10) )
in which the integrals are unaffected by the order of the subscripts of with
. In other
, multiplied with mode shape
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words, if Eq. (8) is rewritten by replacing
and
integrated over area, it will result in the same equation, but with coefficient of
and
instead of
∫
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can be derived as:
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in the left side. Accordingly, the orthogonality of mode shapes in a non-local classical plate
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∫
and in the case
(11) [
or
(
)
]
:
∫ (12) ∫
[
(
)
]
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3. Dynamic respose of plate-indenter system 3.1
Response to external load Expansion theorem is a strong tool to solve equations of motions in beams, plates, etc.
The theorem allows expanding functions over an orthogonal basis, an infinite sum, so that it will be possible to reduce the problem to simple modal equations. Providing orthogonal basis for the displacement (the orthogonality relations of basis, i.e. mode shapes has been derived in section 2.2 in the case of nonlocal CPT with different combination of boundary conditions),
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this theory may be utilized resulting in solvable modal equations. In the following, this theorem is employed to derive response of plate to external load. Due to Expansion theorem, displacement of
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displacements: ∑ in which
can be expanded as a series of modal
(13)
are defined as principal coordinates. Rewriting Eq. (7) using Eq. (13), pre-
multiplying Eq. (7) with arbitrary mode shape of
∫
∫
∫
[
(14) (
)
]
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∑
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∑
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can be rephrased as:
and integrating over the area of plate, it
Employing orthogonality properties of Eqs. (11) and (12), and after a few simplifications, the
(15)
is modal force and can be expressed as:
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where
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familiar form of modal equation is derived as follows:
∫
(16)
in which: ∫ represents mass portion of mode (
(17) ) in the response of plate.
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Response to impact load Impact force of
could be assumed as a concentrated force of
on the plate at point
imposed
, so it can be rewritten as: (18)
where
is the Dirac delta function. Combining Eq. (18) with Eq. (16) and employing partial
(
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integral properties, modal force can be reduced to: )
is defined here as the influence coefficient of impact force on vibration mode of (
(19) ).
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Eq. (15) is a simple ODE and can be solved using Laplace transform, resulting in: ∫
(20)
Now, the equation of motion for the indenter should be taken into account. According to the
∫ (21)
Deriving discrete equations
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3.3
, the following relations could be
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∫
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expressed for the indenter:
with a velocity of
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Newton's law, for displacement of
Solving equations (20) and (21) together with the equation of contact law will result in the
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dynamic response of plate to the impact force. It is worth mentioning that the contact law equation usually appears as a non-linear equation representing impact force in terms of indenter-plate distance. In the absence of an analytical solution for the set of these coupled equations, they may be solved employing a discrete numerical method, and by dividing the period of contact phenomenon into small increments of of
. For every increment, impact force
is assumed to be constant, which helps compute integrals of Eqs. (20) and (21). Assuming
zero initial conditions, i.e. (20) and (21) result in:
, and after some mathematical operations, Eqs.
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∑
∑
[ ∑
where index of
] (
(22)
∑
)
(23)
refers to the th increment. As a result, one can see that the time dependent
terms of plate response depend on the history of impact force during the contact procedure. In each interval, simplification of force to constant values leads to errors, which can be
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minimized by choosing adequately small time increments. The other reason for choosing very small time increments, as will be seen later, is the extremely transient nature of present nanoscale problem.
4. Contact law
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As mentioned before, interaction force between indenter and plate known as “contact law” depends on the indenter-plate distance. In the case of macro impact problems, various assumptions have led to several contact laws. These theories are based on geometries and material properties measured under the assumptions of continuum mechanics, which are not valid in nano-scale medium. Therefore, to present a reasonable solution for the impact
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problem in nano-sized objects, it seems inevitable to utilize a contact model based on nanoscale interaction forces.
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The majority of non-bonding interaction forces in neutral particles are classified as van-der Waals forces. These forces, depending on particles characteristics and distances,
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comprise attraction and repulsion effects. One of the most common and accurate models to
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state van-der Waals interaction is the well-known Lenard-Jones (L-J) equation: *
is L-J potential,
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in which
+
(24) is the distance between particles, and bold symbols
and
are
respectively energy and distance parameters, which depend on the physical features of particles. Herein, the utilized objects, including plate and indenter consist of several particles (i.e. atoms) interacting with each other. Discrete nature of structures makes it difficult to come up with a mathematical solution for the interaction force and potential. Consequently, it seems inevitable to employ nano-scale simulations such as molecular dynamics method. The major weakness of such methods is that, besides their significant complexity and timeconsuming computation processes, they do not present an analytical view of the problem.
ACCEPTED MANUSCRIPT Some analytical averaging methods cover this problem, resulting in some mathematical formulation for interaction forces and potentials, Hilder and Hill (2008a), Hilder and Hill (2008b), Hilder and Hill (2009), Chan and Hill (2010), Yue and James (2011), Chan and Hill (2010) and Israelachvili (2011). The main idea of these methods is to divide nano-structured media into small elements. Each element represents the properties of its included particles. Next step will be to integrate over the domain of structure to have desired property. Afterwards, L-J force between plate and indenter is derived by employing the described
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method.
Fig. 1 shows a solid spherical indenter interacting with a thin plate. At first, indenter is assumed to be placed above the plate with a distance of , far enough from the plate edges. The interaction force was measured between the indenter and the region illustrated in Fig. 1. This region is considered as a circular zone of plate with a radius of
, which is concentric
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with the vertical projection of the indenter on the plate. As shown in Fig. 1, a small element of the circular region is assumed with respect to the the center of circle. Given that
polar coordinate system assigned on
is the number of particles per unit area of the plate, the
(25)
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number of atoms inside each element is:
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L-J potential of Eq. (24) can be rewritten for the system of plate element and indenter as: +
(26)
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*
L-J potential between indenter and circular region on the plate could be determined by
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integrating Eq. (26) over the circular domain: ∫
[
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∫
Eq. (27) shows clearly that by increasing
{
( ,
) }
{
(
quickly converges to its value in
example, if the radius of circular region on plate is chosen as
) }] (27) . For
, the first and the second
braces of Eq. (27) will have the values of 0.96000 and 0.99968, respectively. For
the
values will be 0.99000 and 0.99999. Therefore, it will not be an unreasonable assumption to approximate the potential with its value in *
+
, which results in: (28)
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(29)
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(
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Fig. 1, schematic illustration of spherical indenter interacts with a circular region of plate with a radius of
.
As an example, force-distance variations are plotted in Fig. 2 for a gold atom approaching a single-layer graphene sheet (SLGS). Gold-carbon L-J parameters are taken as
To measure
(30)
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Chan and Hill (2010):
, we consider that the SLGS is a hexagonal lattice and carbon atoms are placed
on the vertices. Since every atom is shared between three hexagons, we may allocate two
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atoms to every hexagonal region of the SLGS. The area of a hexagon with respect to the bond
will be
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length is calculated as √ √
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according to Chan and Hill (2010).
, and so the number of particles per unit area of the SLGS , where the bond length is
,
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Fig. 2, The variations of L-J contact force and potential vs. distance for a gold atom approaching a SLGS.
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5. Results and discussion
Dynamic response of the plate-indenter system can be derived by solving simultaneous equations of Eqs. (22), (23), and (29). As an instance, the impact phenomenon occurring between a nano-particle and a nano-plate is investigated. The plate is assumed to so its mass will be
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be a SLGS and the indenter is taken as a single gold atom with atomic mass of
and
The indenter strikes the nano-plate at its middle
y
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point with coordinates of
a
, and L-J parameters for the interaction
between atoms are taken as Eq. (30). According to Shen et al. (2010), assumed SLGS has the
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following features:
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(31)
in which, based on Shen et al. (2010), the nonlocal parameter is taken as e=0.05; and boundary conditions are assumed simply supported (SSSS), which imply the displacement function as Pradhan and Phadikar (2009a): ∑ where
∑
represents the
,
(32)
th mode of vibration. The number of vibrational modes (mn),
which should be measured through the solving process, determines the response precision. To guarantee that the higher modes of vibration are taken into account, the number of 900
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5.1
Dynamic response of the plate to the elastic impact Fig. 3-a illustrates plate displacement at the impact point as the nano-particle is
propelled with a velocity of 380 m/s. This figure also displays the position of indenter, which
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demonstrates that the particle almost hits the plate twice at t=1.6 ps and 16.7 ps. However, as this figure obviously shows, the indenter rebounds after reaching a specific distance from the plate. Eq. (29) can explain this phenomenon. Based on this equation, as the particle gets closer to the plate, the repulsion force approaches infinity. As a result, the indenter never hits the plate in practice. Moreover, the term describing long-range attraction force in Eq. (29)
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determines that as the particle recedes farther, it remains under the effect of the plate. This attraction force accounts for the return of particle toward the plate at t= 16.7 ps. Fig. 3-b shows the history of indenter-plate interaction force, indenter velocity, and relative velocity of indenter with respect to plate. As this figure implies, even though the interaction force is negligible between the two approach (impact) moments, the variations of indenter velocity
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reveals the accelerated motion of particle. In other words, the particle displays accelerated motion even while is far from the plate, which contrasts with the macro-scale observations.
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Finally, as the L-J forces disappear due to increase in the plate-particle distance, the motion of indenter continues with a constant velocity. This section of motion can be distinguished in
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Fig. 3-b where the interaction force and the velocity respectively approach zero and 74 m/s after t=16.7 ps. Comparison of the variations of interaction force and displacement indicates
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that the maximum interaction force corresponds to the closest distance between the plate and indenter. At these points, the velocity curve intersects the time axis, which shows a
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change in the direction of indenter motion. To reach a deeper understanding of this issue, Fig. 3 also presents the plate deflections
at various sequences. The trend of excited vibration modes can be obviously seen from this figure. At the beginning of excitation and prior to the stress wave propagation through the plate, the deflection locally occurs at the impact point. As the excitation continues, other vibrational modes are excited within the plate so that the vibration amplitude is a combination of several modes of vibration.
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Fig. 3, a- displacement, and; b- contact force, indenter velocity , and relative velocity of
5.2
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indenter with respect to plate vs. time, in the case of impact with initial velocity of 380 m/s
Effect of nonlocality
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To examine the effect of nonlocality, the particle is assumed to hit the plate with an
initial velocity of 300 m/s. Then, the problem is solved with (e=0.05) and without (e=0.00) nonlocal parameter. Under these conditions, Fig. 4 illustrates the displacements of both plate and particle at the impact point. As this figure indicates, employing the nonlocal theory leads to the absorption of particle into the plate so that after bouncing on the plate, the particle eventually forms an integrated system with the plate. On the other hand, the local theory results in the receding of particle into space. This study reveals the substantial effects of nonlocal parameter. As a result, the nonlocal theory plays a pivotal role particularly in the phenomena of the absorption of nano-particles into nano-plates. This conflict can be
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)
[
(33)
]
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√
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Fig. 4, displacement of plate and indenter vs. time recorded based on nonlocal and local
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theories.
According to this equation, ignoring the nonlocal parameter of denominator leads to the rise of all natural frequencies, which implies increase in the stiffness of plate. The stiffer system is
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more resistant to the deflection caused by the contact force. Therefore, a minor portion of particle’s kinetic energy is passed to the plate Consequently as can e seen in the present
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example, the plate is unable to dissipate as much energy as that is required for trapping the
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particle and, in turn, the indenter escapes.
5.3
Effect of impact velocity The term of long-range attraction force, which appears in the function of contact force
in Eq. (7), requires that the indenter remain within the force field of the plate. Thus, it is expected that in a mild collision, the particle will get trapped within an equilibrium distance. To illustrate, Fig. 5-a depicts a stationary indenter lies above the plate without initial velocity. Because of long-range attraction force, the particle approaches to the plate with an accelerated motion and, after several fluctuations, is caught by the plate. In other words, the
ACCEPTED MANUSCRIPT indenter is coupled to the plate and remains within a specific distance from that, which depends on material and geometry of both the plate and the indenter. This phenomenon can be further clarified by observing the variations of contact force in Fig. 5-a. As this figure demonstrates, as the particle gets closer to the plate, the contact force varies from a negative value (attraction) to a positive maximum (repulsion). After each collision, this maximum decreases and approaches, but never reaches zero. The contact force eventually reduces while varies within a narrow range and leads to the synchronized movements of plate and
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particle.
As mentioned before, in the present numerical solution, the first 900 modes of vibration are taken into account for the plate (m=n=30 in Eq. (32)). However, to achieve an as precise as desirable solution, the appropriate number of modes of vibration should be determined. To study this issue, Fig. 6-a displays the energy spectrum of modes. This figure,
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which shows the energy of vibration of each mode vs. the frequency, obviously demonstrates that considering the first 50 modes will result in an acceptable, precise solution. It is worth mentioning that, as mentioned above, when the particle sticks to the plate, they form an integrated system that vibrates with some major specific frequencies such as
(see Fig.
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6). Excited modes and their amplitudes depend on materials, geometry of plate, mass of
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indenter, position of contact point, and impact velocity.
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Fig. 5, displacement of plate and indenter as well as contact force vs. time for indenter with an initial velocities of a-zero and b- 500 m/s. In the case of considering an initial velocity of 500 m/s for the indenter, substantially
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different behavior will be observed. As Fig. 5-b displays, the particle recedes after hitting the plate. According to this figure, the contact force varies from a negative value (attraction force)
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to a positive one (repulsion force) so that at the escapes moment, the contact force reaches its maximum. As the indenter gets farther, on the other hand, the repulsion force reduces to zero,
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turns to attraction force, and again approaches zero as the particle recedes into space. Generally, the dynamic response of plate consists of free vibrations with several vibration modes, which are excited after the impact phenomenon. As the spectrum plot of Fig. 6-b displays, the excitation mostly includes the first 20 ones. As the indenter is propelled toward the plate with an initial velocity of zero (Fig. 6-a), the major excited modes can be detected which means that the system vibrates with specific frequencies, while the initial velocity of indenter causes the excitation of a wide range of modes. This observation could be explained by the amount of energy transferred to the system after collision. Initial velocity of indenter
ACCEPTED MANUSCRIPT gives rise to the excitation of higher levels of energy each of which corresponds to a particular
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mode of vibration.
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Fig. 6, energy spectrum of vibration modes, , for indenter with an initial velocities of azero and b- 500 m/s .
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6. Conclusions
Present research work includes efforts to the dynamics study of the impact of a nano-
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particle with a nano-plate, using a complete approach. The plate was considered rectangular and orthotropic, conformed to the classical plate theory. Non-local theory was adopted to extract the motion relations. A theoretical contact law was utilized based on the non-bonding
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forces of Lenard-Jones, which made it possible to solve displacement equations simultaneously. These equations, including the displacements of plate and indenter, were numerically solved for a specific example in which a gold atom hits a single-layered graphene sheet (SLGS). Then, displacements of plate and indenter together with the initial velocity and contact force were analyzed. In addition, the minimum number of vibration modes required to reach a precise solution as well as other effective parameters such as nonlocal parameter and impact velocity were investigated. The introduced model provided an analytical approximation to the dynamic response of 2-D structures at nano-scale. Findings revealed the
ACCEPTED MANUSCRIPT pivotal role of the nonlocal parameter. Ignoring this parameter and, consequently, minor increase in the stiffness of model led to the impractical response of the system, while the indenter did not stick to the plate, but receded into space. This investigation also implies that in contrast to the macro-scaled contact models, the indenter approaches and sticks to the plate without initial velocity. It was also found that for a range of initial velocities the particle eventually sticks to the plate, although in macro-scale elastic impact, the indenter finally does recede into space. Present solution can be generalized in the dynamics analyses of single or describe the performance of sensors and actuators.
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