Higher order and scale-dependent micro-inertia effect on the longitudinal dispersion based on the modified couple stress theory
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Journal of Computational Design and Engineering, 2021, 8(1), 189–194
doi: 10.1093/jcde/qwaa070
Journal homepage: www.jcde.org
Advance Access Publication Date: 31 October 2020
RESEARCH ARTICLE
Higher order and scale-dependent micro-inertia effect
Downloaded from https://academic.oup.com/jcde/article/8/1/189/5948191 by guest on 11 September 2021
on the longitudinal dispersion based on the modified
couple stress theory
Delara Soltani1 , Majid Akbarzadeh Khorshidi2 and Hamid M. Sedighi 3,
*
1
Faculty of New Sciences and Technologies, University of Tehran, 13145-1384, Tehran, Iran; 2 Department of
Mechanical Engineering, Ferdowsi University of Mashhad, 9177948974, Mashhad, Iran and 3 Mechanical
Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, 61357-43337, Ahvaz,
Iran
*Corresponding author. E-mail: h.msedighi@scu.ac.ir http://orcid.org/0000-0002-3852-5473
Abstract
The conventional modified couple stress theory cannot model the correct behavior of the longitudinal dispersion and acts
the same as the classical theory in the face of such problems. In this paper, the micro-inertia-based couple stress theory is
used to triumph over this deficiency. The developed theory is imposed to tackle the longitudinal dispersion of aluminum
beams in two distinct scales. Convenient available experimental data obtained for a macro-scale aluminum rod and
aluminum crystals are utilized to determine the corresponding micro-inertia length scale parameters and show the
scale-dependent nature of this parameter for the first time. In addition, a higher order micro-rotation relation is employed
to describe the higher order micro-inertia effects. This relation leads to a developed equation of motion containing an
additional term compared with the first-order relation. The obtained results indicate that only higher order micro-inertia
effect that is proposed in this study for the first time is able to capture the highly nonlinear behavior of dispersion curves
(including an extremum/inflection point), which has experimentally been observed for phonons propagating in the
longitudinal direction in an aluminum crystal.
Keywords: couple stress; micro-inertia; dispersion; aluminum; higher order micro-rotations
1. Introduction micromorphic (Eringen & Suhubi, 1964), classical couple stress
(Toupin, 1962), and modified couple stress (Yang et al., 2002).
In classical continuum mechanics, a material particle in a de- The micro-rotation is an additional internal degree of freedom
formable continuum can only undergo translation. The modi- that should be defined as an independent vector. Nevertheless,
fied couple stress theory proposed by Yang, Chong, Lam, and it is usually related to the displacement vector by a meaning-
Tong (2002) considers rotations in addition to translations for ful relationship. In the conventional form of modified couple
each material particle. This non-classical continuum theory in- stress theory, the micro-rotations are defined the same as the
tends to exploit the micro-rotation concept proposed by Mindlin macro-rotations in continuum mechanics, while the real micro-
(1964) to describe the size-dependent behavior of structures rotation is different from macro-rotations. This assumption is
in micro-scale. The micro-rotation is the fundamental concept reasonable for a wide range of studies especially for static anal-
of several theories, i.e. micropolar (Cosserat & Cosserat, 1909), yses (Akbarzadeh Khorshidi, 2019, 2020a; Akbarzadeh Khorshidi
Received: 3 August 2020; Revised: 3 September 2020; Accepted: 1 October 2020
C The Author(s) 2020. Published by Oxford University Press on behalf of the Society for Computational Design and Engineering. This is an Open Access
article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits
unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
189
190 Higher order micro-inertia effect on the longitudinal dispersion
& Shariati, 2015, 2017, 2019; Akbarzadeh Khorshidi, Shariati, &
Emam, 2016; Akgoz & Civalek, 2011; Al-Basyouni et al., 2015;
Lam, Yang, Chong, Wang, & Tong, 2003; Malikan, 2017; Nateghi,
Salamat-talab, Rezapour, & Daneshian, 2012; Simsek & Reddy,
2013), or any studies in which the micro-rotation is negligible
compared with the macro-rotations. In fact, this effect can be
observed in some of dynamic analyses such as dispersion anal-
ysis especially in the crystalline solids, porous media, or any
kind of materials with microstructure. For a better explanation,
let’s consider the longitudinal dispersion where both classical
and couple stress-based continuum theories cannot justify the
nonlinear behavior of structures in the longitudinal dispersion
analysis.
However, there are couple stress-based studies that en- Figure 1: Hierarchical nature of length scale parameter by changing the size of
sub-unit cells.
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gaged the length scale parameter to longitudinal wave prop-
agation using geometrical considerations (Guven, 2011) or
considered the micro-inertia to impose micro-structure effects sub-particles (Yang et al., 2002) that can have a particular degree
on the longitudinal dispersion relation (Akbarzadeh Khorshidi of freedom to generate additional motions (Akbarzadeh Khor-
& Soltani, 2020; Fathalilou, Sadeghi, & Rezazadeh, 2014; Geor- shidi & Soltani, 2020). In fact, this parameter has a fixed value
giadis & Velgaki, 2003; Goodarzi, Fotouhi, & Shodja, 2016; Shodja, until where the sub-particle is not required to change. As soon
Goodarzi, Delfani, & Haftbaradaran, 2015). Akbarzadeh Khor- as we go into a different scale, the sub-particle should alter and
shidi and Soltani (2020) developed the modified couple stress consequently the parameter changes (Fig. 1). Therefore, we can
theory based on the micro-inertia effects to apply a length mention that the micro-inertia length scale parameter is not
scale parameter in the longitudinal dynamic analyses especially size-dependent but it is scale-dependent.
for the longitudinal dispersion. They considered the first-order The present paper aims to utilize the micro-inertia effect
relation for the micro-rotation and examined the relation for coming from realistic micro-rotation concept (rather than the
carbon nano-structures (e.g. graphene, carbon nanotube, and conventional macro-rotation) to study the longitudinal disper-
graphite). The micro-rotation relation can be defined to take sion analysis in solids (for a particular material, aluminum). This
into account higher order internal motion (Georgiadis & Velgaki, study considers two different scales of aluminum and shows
2003). Considering the higher order effect leads to justifying the the scale-dependent nature of the micro-inertia length scale
highly nonlinear behavior of some materials in dispersion anal- parameter. This evaluation reveals that the length scale pa-
ysis (Yarnell, Warren, & Koenig, 1965). It should be mentioned rameters introduced in the micro-inertia-based modified couple
that there are numerous published papers that make use of stress theory are different when we examine two distinct points
the couple stress and strain gradient theories to solve disper- of view (macroscopic and crystalline). Moreover, it is shown that
sion equations and investigate the wave propagation in solids higher order micro-rotations are required to describe the non-
(Akbarzadeh Khorshidi, 2020b; Ghodrati, Yaghootian, Ghanbar linear falls in the dispersion curve that happened for some of
Zadeh, & Mohammad-Sedighi, 2018; Lim, Zhang, & Reddy, 2015; materials (i.e. longitudinal dispersion of aluminum crystals).
Sedighi & Bozorgmehri, 2016; Sedighi, Chan-Gizian, & Noghreha-
badi, 2014). However, the most accurate analysis might be repre-
2. The Modified Couple Stress Theory with
sented using both couple stress and micro-inertia effects espe-
Micro-Inertia
cially for some particular types of materials (Akbarzadeh Khor-
shidi & Soltani, 2020; Georgiadis & Velgaki, 2003). In addition, To derive the equation of motion of structures based on the
there are studies presenting an approach to account for the modified couple stress theory, the strain energy should be writ-
micro-inertia through a dynamic relaxation model (Rabczuk, Zi, ten. Also, the kinetic energy including the velocity of particles
Bordas, & Nguyen-Xuan, 2010; Rabczuk & Belytschko, 2007). The and the angular velocity of particles due to the micro-rotations
approach is also applicable for fracture and associated strength should be extracted. Then, using Hamilton’s principle and the
increase and dynamic fracture energy. Also, there are some first variation of total energy, the equation of motion will be de-
relevant papers that consider non-classical theories to study rived. The strain energy in a continuum based on the modified
the dispersion and vibration in micro/nano-structures (Civalek couple stress theory is defined as (Park & Gao, 2006)
& Demir, 2016; Civalek, Uzun, Yaylı, & Akgöz, 2020; Demir &
1
Civalek, 2013; Ebrahimi, Barati, & Civalek, 2020; Numanoğlu, U= (σi j εi j + mi j χi j )dV, i, j = 1, 2, 3 (1)
2
Akgöz, & Civalek, 2018).
Moreover, it has been put forward that the material length where ε, σ , χ and m are the strain tensor, Cauchy stress tensor,
scale parameter used in couple stress theories is not just a ma- symmetric curvature tensor and the deviatoric part of the couple
terial constant and can be a size-dependent parameter (Ak- stress tensor, respectively. The defined tensors can be expressed
barzadeh Khorshidi, 2018, 2020b). Recently, it is shown that the as (Akbarzadeh Khorshidi et al., 2016)
couple stress length scale parameter is size-dependent, while
1 1
the micro-inertia length scale parameter remains constant at a εi j = (∇u + (∇u)T ) = (ui, j + u j,i ) (2)
2 2
particular scale (Akbarzadeh Khorshidi & Soltani, 2020). In this
study, it is intended to represent that the micro-inertia length σi j = E tr(εii )δi j + 2μεi j (3)
scale parameter must choose in accordance with the consid-
1 1
ered scale, i.e. lattice size, grain size, etc. As we know, the mod- χi j = (∇θ + (∇θ)T ) = (θi, j + θ j,i ) (4)
2 2
ified couple stress theory assumes a continuum containing sev-
eral material particles so that each particle consists of several mi j = 2l 2 μχi j (5)
Journal of Computational Design and Engineering, 2021, 8(1), 189–194 191
where u is the displacement vector, θ is the macro-rotation 3. Longitudinal Dispersion with Micro-Inertia
vector (θ = (∇ × u)/2), E and μ are respectively Young’s and Effect
shear moduli, and l is a couple stress length scale parameter,
which is the square root of the ratio of the modulus of cur- In this study, Euler–Bernoulli beam model is employed to ana-
vature to the modulus of shear, expresses the effect of cou- lyze the longitudinal dispersion based on the modified couple
ple stress, and can be determined from bending or torsion stress theory including the micro-inertia effect. Therefore, the
tests. equation of motion can be expressed as (Akbarzadeh Khorshidi
The micro-inertia effect [the effect of real micro-rotation & Soltani, 2020)
(Akbarzadeh Khorshidi & Soltani, 2020; Mindlin, 1964)] should
c02 u − ü + lm
2
ü = 0 (8)
be considered in the dynamic analysis. In this study, we have
√
a rotational kinetic energy in addition to the conventional ki- where c0 = E /ρ is the classical phase velocity, u is the longi-
netic energy due to translation. The angular velocity of the tudinal displacement, and primes denote derivatives with re-
rotational kinetic energy comes from the micro-rotation of spect to x. Detail of derivation of Equation (8) can be found in
particles. Therefore, it is necessary to define the kinetic en- Appendix 1. Substituting a harmonic longitudinal wave prop-
√
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ergy as (Akbarzadeh Khorshidi & Soltani, 2020; Georgiadis & agating along the axial direction u = U exp[ik(x − ct)] (i = −1)
Velgaki, 2003) into Equation (8), the phase velocity of the wave can be obtained
as
1 2 1 2
T= ρ (u̇) dV + Im (ψ̇) dV (6)
2 2 1
c = c0 (9)
1 + lm
2 k2
where ψ is the micro-rotation vector (ψi j = ∂u j /∂ xi ; Mindlin,
1964), ρ is the density, Im denotes the mass moment of inertia where k = ω/c is the wavenumber and ω is the angular frequency.
per unit volume related to the micro-rotation of particles, and By considering higher order micro-rotations, the equation of
a superimposed dot denotes a derivative with respect to time. motion of Euler–Bernoulli beams can be rewritten as
The main concept of considering micro-inertia into the modi-
fied couple stress theory is to consider several sub-particles into c02 u − ü + lm
2
ü − α 2 lm
4 (4)
ü = 0 (10)
each material particle [this concept has also been considered
Using the longitudinal wave function, the phase velocity can
in the original form of the modified couple stress theory (Yang
be acquired as
et al., 2002)]. We have Im = (ρd2 )/3 when our sub-particles
are cubes with edges of length 2d and the density of macro- 1
material and micro-material are the same (Akbarzadeh Khor- c = c0 (11)
1 + lm
2 k2 + α 2 l 4 k4
m
shidi & Soltani, 2020). The mass moment of inertia can be
represented by the micro-inertia length scale parameter lm This phase velocity equation is derived for the first time in
that can involve the micro-rotation effect in the longitudinal this study. This relation is able to describe the highly nonlinear
equation of motion [Im = ρ(lm )2 ]. For more information about dispersion curve (with a simultaneous rise and fall). This behav-
this type of modification on the modified couple stress the- ior is illustrated in the next section.
ory, Akbarzadeh Khorshidi and Soltani (2020) is required to
be read.
4. Results and Discussion
Using the variation of kinetic energy Equation (6), the general
form for the dynamic part of the equation of motion in terms of In this paper, two types of aluminum structure are used to
the displacement vector can be expressed as (Akbarzadeh Khor- study two different dispersion analyses in terms of the micro-
shidi & Soltani, 2020) inertia length scale parameter. First, the dispersion of longitu-
dinal elastic wave in a cylindrical aluminum rod is investigated
2 2
using Equation (9) and the appropriate length scale parameter
δT = 0 → ρ ü − lm ∇ ü = 0 (7)
is estimated according to the experimental data (Zemanek &
in which ∇ 2 ü = ∇(∇ · ü) − ∇ × (∇ × ü), and in Cartesian coordi- Rudnick, 1961). The material properties are E = 73.36 GPa and ρ =
nates (x, y, z): 2760 kg/m3 (Zemanek & Rudnick, 1961). The radius of the rod is
assumed as r = 12.7 mm (Zemanek & Rudnick, 1961; macro-scale
∇ 2 ü = ∇ 2 üx , ∇ 2 üy , ∇ 2 üz aluminum rod). In Fig. 2, the phase velocity ratio c/c0 is shown
versus the ratio of radius to wavelength r/λ (λ = 2π/k). The ap-
The higher order micro-inertia effect can be expressed by
propriate micro-inertia length scale parameter lm is estimated
describing the internal motions of sub-particles in greater de-
to be equal to 3.5 mm to be fit to the experimental results. Us-
tails (Georgiadis & Velgaki, 2003). In other words, the micro-
ing the estimated lm value, the unit cell (sub-unit cell defined
rotation can obey a higher order relation to consider the more
in Akbarzadeh Khorshidi & Soltani, 2020) size 2d can approxi-
accurate rate of deformation between the sub-particles rather √
mately be determined equal to 12 mm (lm = d/ 3). This size is a
than ψi j = ∂u j /∂ xi . For this case, the micro-rotation can be in-
reasonable value in accordance with the lattice structure of alu-
troduced as ψi j = ∂u j /∂ xi + x(∂ 2 u j /∂ xi2 ), where x denotes an
minum (in millimeter scale) proposed in Yan et al. (2015). In this
effective length inside the sub-particles or an equivalent ra-
figure, the classical phase velocity (solid black line) is related to
dius of particles in the higher order micro-rotations, so, it can
lm = 0 in Equation (9) that cannot model the real behavior. Note
be defined as a function of d or lm , x =αlm (α is a dimen-
that the conventional couple stress theories cannot consider the
sionless coefficient). This concept of using the higher order ef-
real nonlinear behavior of longitudinal wave dispersion as well,
fects was used for the strain gradient theory in De Domenico
because the additional parameters defined in the conventional
and Askes, 2018a,b. In this paper, the higher order micro-
form of these theories are not imported into the equation of mo-
rotation is employed for the longitudinal dispersion curves for
tion. In fact, the conventional form of modified couple stress
phonons propagating in aluminum crystals (De Domenico &
theory is the same as a classical theory for longitudinal wave
Askes, 2018b).
192 Higher order micro-inertia effect on the longitudinal dispersion
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Figure 4: Limit behavior of first-order and higher order micro-inertia effects in
the longitudinal dispersion curves.
Figure 2: Comparison of experimental and calculated longitudinal phase velocity is approximately 4 Å). In fact, the predicted unit cell size and
ratios in a macro-scale cylindrical rod. the lattice size are in the same scale. In this figure, the classical
dispersion curve (solid black line) refers to the classical contin-
uum mechanics or the conventional form of the couple stress
theories where lm = 0. The classical curve is linear and can-
not describe the nonlinear behavior of the experimental points.
Therefore, the micro-inertia-based modified couple stress the-
ory should be used to model the nonlinearity. But this time, un-
like the macro-scale aluminum rod (Fig. 2), the micro-inertia-
based modified couple stress theory cannot capture the real
behavior of the wave propagating in aluminum crystals (dash-
dotted green line). Interestingly, the higher order micro-inertia
effect can predict much better results to capture the highly non-
linear behavior (dashed red line).
In fact, only the higher order theory can visualize the angular
frequency drop as can be seen in Fig. 4. This figure shows the lon-
gitudinal dispersion curves for higher wavenumbers and is pre-
sented to express the point that the first-order dispersion curve
(without higher order micro-rotations) does not meet any drop
and tends to a limit value after rising, while the higher order
curve tends to decrease after the peak and shows the inflection
point properly. This trend takes place due to the presence of pa-
rameter α that applies the higher order effects to the equations.
Figure 3: The angular frequency of aluminum crystals in the longitudinal mode.
In this case, this parameter is presumed to be equal to 1 ( x =lm)
to achieve the best response. Also, this parameter can be 0 to ig-
propagation. Thus, it is necessary to add micro-inertia effect (lm ) nore the higher order effect. The influence of this parameter on
to detect the real longitudinal dispersion curve. the angular frequency is demonstrated in Fig. 5. The results rep-
On the other hand, the longitudinal dispersion of the alu- resent that the level of nonlinearity increases as α increases. In
√
minum nanobeam model (in the scale of aluminum crystals) this figure, α = 2 3 (solid purple line) denotes the case that the
is investigated and the results are compared with the experi- effective length x is equal to the length of sub-particle edge 2d.
mental dispersion curve for phonons propagating in the lon- The results indicate that the scale of the micro-inertia length
gitudinal crystallographic direction in an aluminum crystal at scale parameters should be the same as the scale of studied
300◦ K (De Domenico & Askes, 2018) (the material properties structures. In other words, this parameter is a scale-dependent
are the same as what we have used for the cylindrical rod). parameter and has a connection with the lattice structure of
Figure 3 illustrates the angular frequency versus the dimension- materials.
less wavenumber that can be defined as k(2d)/2π. As defined in
Section 2, 2d denotes the side of the unit cube. The appropriate
5. Conclusion
micro-inertia length scale parameter is estimated based on the
experimental data (lm = 0.57 Å). The sub-unit cell/sub-particle This study utilizes the micro-inertia effect to describe the real
size defined in Akbarzadeh Khorshidi and Soltani (2020) can be behavior of longitudinal waves propagating into the materials
determined using the value estimated for the parameter (2d = 2 based on the new version of the modified couple stress the-
Å). This size matches the lattice constant in the crystal lattice ory proposed by Akbarzadeh Khorshidi and Soltani (2020). This
of aluminum nanostructure (the lattice constant of aluminum developed formulation considers the realistic material particle
Journal of Computational Design and Engineering, 2021, 8(1), 189–194 193
Akbarzadeh Khorshidi, M. (2020a). Length scale parameter of
single trabecula in cancellous bone. Biomechanics and Model-
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fect in modified couple stress theory: Dispersion analysis of
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Figure 5: The effect of higher order parameter α on the longitudinal dispersive flexibility constants for a multi-spring model: A solution for
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to define a new length scale parameter that is imported into the (2016). Postbuckling of functionally graded nanobeams
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that this additional length scale parameter is scale dependent. Akbarzadeh Khorshidi, M., & Soltani, D. (2020). Nanostructure-
In addition, the available experimental results reveal that the dependent dispersion of carbon nano-structures: New in-
longitudinal dispersion curve of some materials (or some special sights into the modified couple stress theory. Mathematical
cases) is highly nonlinear so that the conventional non-classical Methods in the Applied Sciences, 1–17. https://doi.org/10.1002/
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ran University of Ahvaz for its financial support (Grant No.
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Conflict of Interest dient and inertia gradient beam theories for the simulation
of flexural wave dispersion in carbon nanotubes. Composites
None declared.
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Appendix 1
Mathematical Modelling, 48, 196–207.
Mindlin, R. D. (1964). Micro-structure in linear elasticity. Archive To derive Equation (8), we should start with the strain and kinetic
for Rational Mechanics and Analysis, 16, 51–78. energies shown in Equations (1) and (6). For 1D motion (longitu-
Nateghi, A., Salamat-talab, M., Rezapour, J., & Daneshian, B. dinal wave motion), we have u = {u, 0, 0}, ε = {u , 0, 0}, σ = Eu
(2012). Size dependent buckling analysis of functionally and χ = {0, 0, 0}. Therefore, the strain and kinetic energies are
graded micro beams based on modified couple stress theory. reduced to
Applied Mathematical Modelling, 36(10), 4971–4987. 1
E u dV
2
U= (A.1)
Numanoğlu, H. M., Akgöz, B., & Civalek, O. (2018). On dynamic 2
analysis of nanorods, International Journal of Engineering Sci-
ence, 130, 33–50.
1 1
Park, S. K., & Gao, X. L. (2006). Bernoulli–Euler beam model based T= ρ u̇2 dV + Imu̇2 dV (A.2)
2 2
on a modified couple stress theory. Journal of Micromechanics
By applying Hamilton’s principle δ(T−U) = 0, we have:
and Microengineering, 16, 2355–2359.
Rabczuk, T., & Belytschko, T. (2007). A three-dimensional 1 1 1
δT − δU = ρδ u̇2 dV + Imδ u̇2 dV − E δu2 dV
large deformation meshfree method for arbitrary evolving 2 2 2
cracks. Computer Methods in Applied Mechanics and Engineering,
196(29–30), 2777–2799. = − ρ üδu dV + Imü δu dV + E u δu dV
Rabczuk, T., Zi, G., Bordas, S., & Nguyen-Xuan, H. (2010). A sim-
ple and robust three-dimensional cracking-particle method = (E u − ρ ü + Imü ) δu dV =0 (A.3)
without enrichment. Computer Methods in Applied Mechanics
and Engineering, 199(37–40), 2437–2455. From the last part of Equation (A.3), the equation of motion
Sedighi, H. M., & Bozorgmehri, A. (2016). Dynamic instabil- can be deduced as
ity analysis of doubly clamped cylindrical nanowires in
E u − ρ ü + Imü = 0 (A.4)
the presence of Casimir attraction and surface effects us-
√
ing modified couple stress theory. Acta Mechanica, 227, Using c0 = E /ρ and Im = ρ(lm ) , Equation (A.4) is simplified
2
1575–1591. as Equation (8).You can also read
