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УДК 536.21
Теплопроводность в наноразмерных структурах
J. Sladek, V. Sladek, M. Repka
Институт строительства и архитектуры Словакской академии наук, Братислава, 84503, Словакия
Адекватное описание теплопередачи в наноразмерных структурах на основе классического закона Фурье представляется затруднительным. Предложена новая градиентная теория для описания таких структур с учетом размерного эффекта теплопроводности. Данная методика основана на учете вторых производных температуры в определяющем уравнении теплового потока высшего порядка в усовершенствованной модели континуума. Основные уравнения получены с помощью вариационного принципа. Общая двумерная краевая задача теплопроводности решается методом конечных элементов. Предложен смешанный метод конечных элементов с двумя непрерывными интерполяциями по C0 для температуры и температурных градиентов. Наложение ограничений на температуру и ее градиенты выполняется методом коллокации. В рамках параметрического подхода исследовано влияние внутреннего размера на распределение температуры.
Ключевые слова: наноразмерные структуры, градиентная теплопроводность, задача нестационарной теплопроводности, усовершенствованная модель континуума, смешанный метод конечных элементов
DOI 10.24412/1683-805X-2021-5-122-129
The heat conduction in nanosized structures
J. Sladek, V. Sladek, and M. Repka
Institute of Construction and Architecture, Slovak Academy of Sciences, Bratislava, 84503, Slovakia
Thermal transport cannot be well described by classical Fourier's law in nanosized structures. A novel gradient theory is developed in such structures adopting the size effect of heat conduction. This is achieved by considering the second derivatives of temperature in the constitutive equation for the high-order heat flux in an advanced continuum model. The variational principle is applied to derive the governing equations. The general two-dimensional boundary-value problem for heat conduction is analyzed by the finite element method (FEM). A mixed FEM with two independent C0 continuous interpolations for temperature and temperature gradients is developed. The constraints between temperature and its gradients are performed by the collocation approach. Parametric studies are given to evaluate the influence of the internal size on the temperature distribution.
Keywords: nanosized structures, gradient heat conduction, transient heat conduction problem, advanced continuum model, mixed FEM
1. Introduction
One can observe a rapid miniaturization of electronic devices since nanotechnologies bring a new quality. In the classical continuum mechanics the interaction of material microstructure is neglected and the results are size-independent. The size-dependent effects cannot be ignored in nanosized structures [1,
2]. The ability to release the produced heat in small electronic structures is reduced with respect to mac-rosized structures. Created hot spots in such structures can be leading to a damage. A new generation of nanosized structures requires a fundamental understanding of the heat transport. Nanoscale systems are very complex [3] and the first effort to introduce ad-
© Sladek J., Sladek V., Repka M., 2021
vanced continuum model with the scale effect is based on the nonlocal expression of the heat flux on the temperature gradients [4]. Majumdar [3] showed that in nanosized structures, heat transport represented by lattice vibrations or phonons can be analyzed as a radiative transfer problem. The nonlocal Fourier's law for the heat conduction of one-dimensional and two-dimensional thermal lattices is developed by Challa-mel et al. [5]. It is similar to the differential nonlocal model of Eringen for mechanical fields. This model was further extended to nonlocal thermoelasticity using Eringen's nonlocal theory [6]. Recently, Sarkar [7] has established a nonlocal heat conduction theory of generalized thermoelasticity. The generalized ther-moelasticity analyzed by the gradient elasticity and non-Fourier heat transfer is presented by Filopoulos et al. [8, 9].
Coupling of the Peierls-Boltzmann equation with density functional theory is a way for real modeling of thermal conductivity in such structures [10]. However, this modern ab initio descriptions of phonon thermal transport has extremely high requirements on computer memory in atomistic models. To reduce high computer memory requirements the multiscale approaches were developed, where atomistic and continuum subdomains are bridged. A multiscale hydrodynamic heat-transport model has been developed by Beardo et al. [11]. It is applicable to arbitrary geometries using finite-element methods. Results are compared with the experimental effective thermal conductivity of silicon thin films and periodic holey membranes for different sizes and temperatures. The model based on the hydrodynamic equation instead of an effective Fourier model can interpret successfully the current heat-transport experimental data in nanosized structures. A size-dependent thermoelastic model is established by Yu et al. [12] for higher order simple material by adopting both the size effect of heat conduction and elasticity with the aids of extended irreversible thermodynamics and generalized free energy. Recently, Yu and Deng [13] have derived governing equations for the gradienttype thermoelasticity with electron-lattice coupling mechanism for microscale metals. A novel gradient theory is developed in thermoelectricity, where the spatial size effect is considered for the heat transfer [14]. Stationary 2D boundary-value problems are analyzed there.
A novel advanced continuum model for heat transfer with the size effect is developed for nano-sized structures. It is well known that nonlocal models lead to an explanation of the size effect. A spe-
cial treatment of the involved integral kernel in the nonlocal model makes the convolution integral formulation equivalent to the differential one [15]. In this paper, we extend the classical theory of heat conduction to the generalized nonlocal theory by introducing higher-order derivatives of temperature and higher-grade heat flux.
The higher-order derivatives occurred in governing equations require a powerful computational tool to solve general boundary-value problems. The finite element method (FEM) is often convenient in classical and gradient theories and it has been applied for many similar problems [16, 17]. The present approach is the first effort to develop a gradient theory for heat conduction problem. The second spatial derivative of temperature is considered in the constitutive equation of the higher-order heat flux in this model. The variational principle is applied to derive the finite element equation for a 2D heat conduction boundary-value problem. Due to the higher-order derivatives in governing equations, C1-continuous elements are needed to guarantee the continuity of the derivatives at element interfaces. It is a difficult task to develop C:-elements. Therefore, a mixed FEM formulation is developed here, where C0 continuous interpolation is independently applied to the temperature and its gradients. The constraints between the temperature and its gradients are satisfied at Gaussian integration points inside the elements [18, 19].
The computational method is verified on simple examples, where analytical solution is available in stationary case. The influence of the internal size of the material structure on the distribution of temperature is presented and discussed via numerical examples for stationary and transient problems.
2. A gradient theory for heat conduction problems
It is well known that thermal conductivity can be reduced significantly in nanostructures. If the size of a nanostructure is smaller than the phonon mean free-path, scattering occurs on the interfaces and thus thermal conductivity is reduced. Therefore, there is a need to develop a theory for heat conduction where size effects are considered. In classical Fourier heat conduction there is no size effect. Nonlocal heat transport discussed by Allen [4] could bring physical insights into size effects.
The constitutive relationships in the classical theory of thermoelectricity are given as
h = -j j, (1)
where the temperature differences are denoted by 9 = T- T0 with the reference temperature T0, and Kj is the thermal conductivity
The heat conduction equation in classical Fourier theory can be written as
X,,, + pc0 = w, (2)
where w is the volume density of the heat source, and p and c are the mass density and specific heat, respectively. An overdot denotes a time derivative.
The variation of the thermal potential has the following form
8y = -X, 80,,. +pc080. (3)
In the higher-grade theory, we incorporate the higher-order derivatives of temperature into the rate of thermal potential as
8V = -X, 80, +pc080- mlk 80.lk, (4)
where mlk is the higher-grade flux considered as ca-nonically conjugated field with 9,k. In linear theory
mrk =aki0, jl , (5)
where a,kjl is the higher-order thermal conductivity. To simplify the theory these material coefficients are proportional to the thermal conductivity with one length scale parameter l. Then, on can write a,kjl = -l 2SttKy, and
m„
( m- 1
11
m.
21
m12
V m22 J
= -l2
rk = -l 2 K0, Jkor
K11 0 0 0 1 f0 1 ,11
0 K22 0 0 0,21
0 0 K11 0 0,12
0 0 0 K22 J V0,22 j
= 12[G](n).
(6)
The energetic balance can be written as jSydV-SWe = 0,
(7)
where the rate of the work done by the external forces (or sources).
The variations of the temperature 9 being arbitrary except vanishing on the boundary portions where they are prescribed, i.e. S0l = 0.
110
The proposed generalization of the heat conduction leads to the modification on the governing equation
jSydV = J (-^.S0,; +pc0 S0- mlk S0,ik )dV
V V
= J [(Àii +pc0 )S0 + mk k S0,i ]dV
-J (n,X iS0 + nkm,k S0,i )dr
dV
= J [(Ài,i +pc0 - m,k ik )S0]dV
- J [ni - mikk)S0 + nkmikS0,i]dr
dV
= J [(X,, +pc0 - mlk )80]dV
V
-J (A80 + P8p)dT, (8)
dV
where n, being the components of the unit outward normal vector on the boundary dV.
Symbols P and A are independent boundary densities conjugated with p = d9/dn, and 9, respectively, and given as P = nkn,mlk,
A = n, (X,-m,k,k)-IT + £ |^(xc)] 8(x-xc), (9)
dT c
1 = nk1 ,m,k (10) with A being the generalized heat flux, t, the Cartesian components of the unit tangent vector on dV,
and the jump at a corner on the oriented boundary
contour dV, defined as
[|(xc)] :=|(xc-0)-|(xc + 0). (11)
The independent field variables on the boundary (9, p) are energetically conjugated with (A, P). Thus, the rate of work of the external "forces" (A, P, Q) and body source w is given by
8We = Jw80dV - J A80dr- J P8pdT. (12)
V Ta Tp
Since the variation of temperature in (8) can be arbitrary we obtain the following governing equation from the energetic balance (7) and (12)
X ,, ,(x) - m,k, ,k(x) + Pc0 = w. (13)
From the variational formulation (8) we get also possible boundary conditions (b.c.): essential b.c.
0(x) = 0(x) on T0, T0CT, p(x) = p(x)on Tp, Tp cT,
natural b.c.
A(x) = A(x) on Tt, TauT0 = T, rAnrfl=i
P(x) = P(x) on T^, Tp urp = T, Tp o Tp = !
Considering the constitutive equation (6) and (1) the governing equation (13) can be rewritten into the form
(1 -12V2) X ,,(x) + pc0 = w. (16)
(14)
(15)
3. Finite element method formulation
The energetic balance (7) can be written as J (A,-80,. + mlk80,ik -pc080)dV
V
= J A80dF + J P8pdF - J w80dV (17)
ra fp v
in which an overbar is used for the prescribed boundary value. Applying the Gauss divergence theorem to the volume integral in Eq. (17), we obtain the governing equation Eq. (17), since variations of primary fields can be arbitrary. Symbols rA and rP denote parts of the boundary, where the flux and higher-grade flux are prescribed, respectively.
Vanishing variations of the primary fields have to be considered on corresponding parts of the boundary:
86lsv-Fa = ° 8Psv-rP = °
The presence of the second derivative of temperature in the constitutive equations increases the order of the PDE Eq. (19) and it requires C1 continuous interpolations of the temperature. The mixed FEM is developed here. C° continuous interpolation is applied independently for both temperature and temperature gradients, satisfying the constraint between these fields by collocation at selected internal points of the finite elements [19].
For a 2D element, the temperature in each element are expressed in terms of their nodal values and shape functions
0 = N 0 (£ ^2)q 0, (18)
where qe are the nodal temperature (Fig. 1).
The normal derivative of temperature, and temperature gradients are approximated according to Eq. (18) as
Fig. 1. Global Cartesian coordinates x1-x2-x3, curvilinear local coordinates for a 2D and 4-node quadrilateral finite element (color online)
p = (nldl + n2d2)0 = Bs(£^ ^2)q0,
0 = B0(£1, ^2)q0.
8 =
"0,1" "
_0,2 _ _5 2 _
(19)
Besides the approximation of temperature gradients 8 in Eq. (19), we need in the mixed FEM an independent approximation of 8:
8ln = A b (£1, £ 2)a, (2°)
where a is a vector composed of undetermined coefficients defined separately for each component of the temperature gradient.
The polynomial function matrix in Eq. (2°) for the 2D 4-node quadrilateral element is
A b (£1, £2) = [1 £1 £2 £&]. (21) The coincidence of the two independent approximations of the temperature gradients in Eqs. (19) and (2°) is assumed to be satisfied at Gauss quadrature points %c = (£f, £2), i.e.
As (%c )a = B0 (%c )q0. (22)
Hence,
a = Ac )B0 (%c )q0 (23)
and one can derive the final expression for the independent approximation of 8 from Eqs. (23) and (2°) as
8ln = Ae (£1 £3)Lq0, (24)
where
L = Ac )B0 (% c ). The approximation on the derivatives of temperature gradients n is given as derivatives of the independent approximations for 8ln. One can write
n
ln
"51" 8in = "51"
_5 2 _ _5 2 _
A8 (£^ £2)a
= Ae(£1, £2)a = Ae(£1, £2)Lq0. (25) Substituting above given approximations into the functional (17), we obtain:
-J (8q 0 )T B T(% )[k ]B0 (£1, £2){q0}dV v
-J {8q 0}t LT A;T(% )l 2GA; (%)L{q 0 }dV
V
-J {8q0 }T N T(% )pcN0 (%){q 0 }dV = {8q 0}t
V
x J NTAdr + {8q0}t J BTPdF
-{8q0}T J NTwdV.
V
The variational statement Eq. (26) is valid for arbitrary {8qe}. Thus, the ordinary differential equations follow from Eq. (26):
-j (bTg )[k ]B 0 £i, ^+Lt A;t(^ )i 2GA; (^)L)
v
x {q0}dV -j N T(^)pcN0 ($ ){q 0 }dV
V
= j NTAdr + j BT^dr - j NTwdV. (27)
ra rp v
Suppose the time interval [0, T] is discretized uniformly into K subintervals; define tk=kAt (k = 0, 1, ..., K) where At = T/K is the time step. Let 0k=0(x, tk) be the solution restricted to time tk. Assuming constant approximation for time derivatives within the time interval [tk, tk+i] we get
0 (x, t) =
0k+1(x) -0k (x) At '
(28)
Substituting (28) into (27) the ordinary differential equations are transformed into the system of algebraic equations for k = 0, 1, ..., K:
-j bT(^ )[k ]B0 ^1, ^2)+Lt A;t(^)i 2GA;
V
X L{q0+1}dV-A jNT(^)pcN0($){q0+1}dV At v
= -A j NT(^ (^){q0 }dV At v
+ j NT Adr + j BT^dr - j NT wd V. (29)
4. Numerical results
To verify our FEM formulation and the corresponding computer code, we first consider simple boundary value problems, like infinite strip—1D problem and a hollow cylinder, where analytical solution for stationary case are available. The isotropic material properties for the heat conduction coefficient are considered, k = 1.6 W/K m. The specific heat coefficient corresponding to carbon and mass density are c = 600 J/kg K and p = 2300 kg/m3, respectively. The characteristic internal length parameter for the selected material structure is l=5 x 10-9 m.
4.1. An infinite strip
An analytical solution can be derived by assuming the following 1D stationary problem with boundary conditions (Fig. 2):
0(0) = T0 -T0 = 0,0(L) = 0l = TL -T0, (30) 0'(0) = 0 = 0'(L). ( )
In terms of the dimensionless coordinate xjL, the governing equation becomes
5* =d/dx*,
i 7 \2
K
1
7*2
V*20 = 0, V*2 = 5k 5
I
\Lj
Then, the heat flux is given as a, i = -k
*2
(31)
k^k-
f f l ^ 2 ^
1 - V*2 5
v v L, /
From Eq. (31), one can obtain the exact solution for temperature distribution
L ^ fL
0(x ) = Ax + B + Cch
—x l
j
)'(x*) = A + CL sh l
L * —x l
+ Dsh
v
L
+ D-ch
l
—x l
L
—x l
(32)
with
A =
D = I
L
B = /0L 1 + ch( Ll) =-C.
L sh( L/l)
Next, we analyze the finite strip numerically. This 1D problem can be solved numerically as a 2D problem with a large height of the strip H = 5L. The number of finite elements in the numerical analysis is 10 000. Various ratios of l/L are considered and numerical/analytical results are compared. One can observe, from Figs. 3 and 4, that there is perfect agreement on temperature and temperature gradients between both analytical solution and numerical FEM.
4.2. A hollow cylinder
An axially symmetric heat conduction problem, as shown in Fig. 5, is analyzed as the second example. To get analytical solution, it is convenient to utilize a
Fig. 2. An infinite strip with boundary conditions (color online)
Fig. 3. Comparison of the temperature distribution in the strip with various ratios l/L = 0.0 (1), 0.1 (2), 0.2 (3), 0.7 (4); lines and symbols correspond to exact and numerical results, respectively (color online)
Fig. 5. Geometry and boundary conditions of a hollow cylinder (color online)
polar coordinate system, where all the variables only depend on its radial coordinate r. Let us define L = r2 - r1 as the geometry length of the hollow cylinder. In order to get a parametric study for various values of the dimensionless parameter l/L, it is appropriate to use the dimensionless radial coordinate as
r
L L L r2 - r r2 - r
" n n '
_ L L _
If we fix r2 - r1 = x, then L = r2 - ri = (X-^ L e
1
x
x-1 x-1
L
r =
x-1
In the numerical calculation, we select x=3 and l/L e {0.1,0.2,0.7}, or equivalently we have L e {l/ 0.1, l/ 0.2, l/ 0.7}.
In view of the dimensionless radial coordinate r* = r/L, the governing equation becomes
1 -( L
1
1 ^
ov +—dt
= 0,
where dv = d/dr
The general solution is given as 0( r ) = A ln( r/L) + B + CI0 ( r/l ) + ( r/l ),
ar0(r) = ^-1 + C l-Il(r/l) - D l-Kl(rll), r I l
where In(z) and Kn(z) are modified Bessel functions of the first kind.
Taking into account the boundary conditions
0(ri) = 01, 0(r2) = 02, 0(ri) = 0,0'(r2) = 0 we can find the integration constants
A = (0i - 02)[ln(ri/r2) + C(Io(rjl) - Io(rJl))
+ D ( K o(rJ l ) - K o(r2/1 ))]-i, B = 0i - [ln(r^L) + CIo(rJL) + DKo(rjL)]A, C = CA, D = IDA,
where
C = 1 Q
-r2ll) - -l)
D=1
Q
- Ix(r2l l ) -1 I1(rj I )
'1 2 Q = ^1(r1/1) lx(r2l l) - rj l) l1(rx/1).
Figures 6, 7 show the radial distribution of the temperature and its radial gradient in classical as well
Fig. 4. Comparison of the temperature gradient distribution in the strip with various ratios l/L = 0.0 (1), 0.1 (2), 0.2 (3), 0.7 (4); lines and symbols correspond to exact and numerical results, respectively (color online)
Fig. 6. Temperature distribution in the hollow cylinder with various ratios l/L = 0.0 (1), 0.1 (2), 0.2 (3), 0.7 (4); lines and symbols correspond to exact and numerical results, respectively (color online)
--------------------*----------
>> /
\ P
\ / —*— 1
S — e— 2
- 3
-tzuu ...................
0.5 1.0 1.5
rIL
Fig. 7. Temperature gradient distribution in the hollow cylinder with various ratios l/L = 0.0 (1), 0.1 (2), 0.2 (3), 0.7 (4); lines and symbols correspond to exact and numerical results, respectively (color online)
as higher-grade theory of heat conduction. The number of finite elements used in the numerical analysis is 26 000 and perfect coincidence is observed between the numerical and analytical results. The influence of the microlength scale parameter on results is increasing with approaching the ratio l/L to one.
5. Conclusions
A novel gradient theory is developed for transient heat conduction problems in nanosized structures. The second order derivatives of temperature in the constitutive equation for the high-order heat flux is considered in the advanced continuum model. The experimentally observed size effect in nanosized structures is included in that model. The variational principle is applied to derive the governing equations. The mixed FEM with the C0 continuous interpolation for temperature and temperature gradients is developed for 2D problems, since the higher-order derivatives in the gradient theory cannot be approximated by standard FEM approach. Numerical results for an infinite strip and hollow cylinder showed that the variation of temperature along the thickness is affected significantly by the internal size parameter l/L (i.e. the ratio of the microlength scale parameter and the characteristic length of the structure).
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Funding
The authors acknowledge the support by the Slovak Science and Technology Assistance Agency re-
gistered under number APVV-18-0004 and VEGA-
2/0061/20.
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Received 15.05.2021, revised 06.07.2021, accepted 06.07.2021
Сведения об авторах
Jan Sladek, DrSc, Prof., Head of Department of Mechanics, Institute of Construction and Architecture, Slovak Academy of Sciences, Slovakia, jan.sladek@savba.sk
Vladimir Sladek, DrSc, Prof., Senior Researcher, Institute of Construction and Architecture, Slovak Academy of Sciences, Slovakia, vladimir.sladek@savba.sk
Miroslav Repka, PhD Researcher, Institute of Construction and Architecture, Slovak Academy of Sciences, Slovakia, miroslav.repka@savba.sk
