Size Dependence of Residual Thermal Stresses in Micro Multilayer Ceramic Capacitors by Using Finite Element Unit Cell Model Including Strain Gradient Effect
The residual thermal stresses induced by the high-temperature sintering process in multilayer ceramic capacitors (MLCCs) are investigated by using a finite element unit cell model, in which the strain gradient effect is considered. The numerical results show that the residual thermal stresses depend...
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Інститут механіки ім. С.П. Тимошенка НАН України
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| Zitieren: | Size Dependence of Residual Thermal Stresses in Micro Multilayer Ceramic Capacitors by Using Finite Element Unit Cell Model Including Strain Gradient Effect / W.G. Jiang, C.A. Xiong, X.G. Wu // Прикладная механика. — 2013. — Т. 49, № 6. — С. 132-140. — Бібліогр.: 15 назв. — англ. |
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| author | Jiang, W.G. Xiong, C.A. Wu, X.G. |
| author_facet | Jiang, W.G. Xiong, C.A. Wu, X.G. |
| citation_txt | Size Dependence of Residual Thermal Stresses in Micro Multilayer Ceramic Capacitors by Using Finite Element Unit Cell Model Including Strain Gradient Effect / W.G. Jiang, C.A. Xiong, X.G. Wu // Прикладная механика. — 2013. — Т. 49, № 6. — С. 132-140. — Бібліогр.: 15 назв. — англ. |
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| container_title | Прикладная механика |
| description | The residual thermal stresses induced by the high-temperature sintering process in multilayer ceramic capacitors (MLCCs) are investigated by using a finite element unit cell model, in which the strain gradient effect is considered. The numerical results show that the residual thermal stresses depend on the lateral margin length, the thickness ratio of the dielectrics layer to the electrode layer, and the MLCC size. At a given thickness ratio, as the MLCC size is scaled down, the peak shear stress reduces significantly and the normal stresses along the length and thickness directions change slightly with the decrease in the ceramic layer thickness td as td >1 μm, but as td <1 μm, the normal stress components increase sharply with the increase in td. Thus, the residual thermal stresses induced by the sintering process exhibit strong size effects and, therefore, the strain gradient effect should be taken into account in the design and evaluation of MLCC devices.
Розглянуто задачу про залишкові температурні напруження в багатошарових керамічних конденсаторах, викликані високотемпературним процесом спікання. Застосовано модель одиничної чарунки в методі скінченних елементів, де враховано вплив градієнта деформацій. Отримані числові результати показують, що залишкові температурні напруження залежать від довжини поздовжнього краю, відношення товщини діелектричних шарів до товщини електродних шарів і розміру конденсатора. При заданому відношенні товщин, коли розмір конденсатора зменшується, то найвищі зсувні напруження зменшуються суттєво і нормальні напруження вздовж довжини та в напрямку зміни товщини змінюються незначно зі зменшенням товщини керамічного шару до величини, меншої 1 мікрона. Коли ця величина є більшою 1 мікрона, то нормальні напруження зростають різко зі збільшенням товщини керамічного шару. Таким чином, залишкові температурні напруження, викликані високотемпературним процесом спікання, виявляють сильний вплив розміру конденсатора і тому вплив градієнта деформації повинен братися до уваги в процесі створення і оцінювання роботи багатошарових керамічних конденсаторів.
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2013 ПРИКЛАДНАЯ МЕХАНИКА Том 49, № 6
132 ISSN0032–8243. Прикл. механика, 2013, 49, № 6
W . G . J i a n g , C . A . X i o n g , X . G . W u
SIZE DEPENDENCE OF RESIDUAL THERMAL STRESSES IN MICRO MULTI-
LAYER CERAMIC CAPACITORS BY USING FINITE ELEMENT UNIT CELL
MODEL INCLUDING STRAIN GRADIENT EFFECT
School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University,
Nanchang 330063, People’s Republic of China; E-mail: jiangwugui@gmail.com
Abstract. The residual thermal stresses induced by the high-temperature sintering proc-
ess in multilayer ceramic capacitors (MLCCs) are investigated by using a finite element unit
cell model, in which the strain gradient effect is considered. The numerical results show that
the residual thermal stresses depend on the lateral margin length, the thickness ratio of the
dielectrics layer to the electrode layer, and the MLCC size. At a given thickness ratio, as the
MLCC size is scaled down, the peak shear stress reduces significantly and the normal
stresses along the length and thickness directions change slightly with the decrease in the
ceramic layer thickness td as td >1 μm, but as td <1 μm, the normal stress components in-
crease sharply with the increase in td. Thus, the residual thermal stresses induced by the sin-
tering process exhibit strong size effects and, therefore, the strain gradient effect should be
taken into account in the design and evaluation of MLCC devices.
Key words: multilayer ceramic capacitor, residual stress, strain gradient plasticity, size
effect, unit cell model, finite element method.
1. Introduction.
As an important type of surface-mount devices, the multilayer ceramic capacitors
(MLCCs) have been widely used in a broad range of applications from household devices to
satellites because of their advantages of high capacitance density and small device size [1,
2]. A MLCC has a so-called “brick” structure consisting of dielectric ceramic layers (Ba-
TiO3) and internal metallic electrodes (Ni), which are laminated alternately with each other
and co-fired at a high temperature up to about 1200oC. The high-temperature sintering is an
important step in fabrication process of MLCCs. After sintering, the component is cooled
down to room temperature. Residual stress induced during the preparation processing could
not be completely relaxed, which may cause damage and failure of the systems [3, 4].
Saito and Chazono [4] investigated uniaxial compressive stress and electric field re-
sponses of X5R-type MLCCs, and their results showed that the capacitance changes of the
MLCCs have a considerable dependence on the stresses. As the number of dielectric layers
increases up to 500 and the dielectric layer thickness decreases to several microns, however,
it becomes more and more difficult to exactly characterize the detailed residual stress state
in MLCCs via experiments. Alternatively, the finite element method (FEM) is always effi-
ciently used to deal with these complicated problems [2, 5 – 8]. However, for a large num-
ber of the dielectric layers (e.g., 200 or even more) and a very small thickness of each layer
(e.g., 1 μm or even thinner), a full three-dimensional (3D) FEM simulation of the sintering
process of MLCCs requires a large number of discrete elements and a long computation
time. Jiang et al. [6] and Shin et al. [8] had simulated the residual thermal stress using a full
3D FE model. However, these full 3D FE models can’t capture the exact information at the
tip of the inner electrodes because coarse mesh densities were used there. For multilayer
133
devices, a widely used method is to construct a unit cell model with periodic boundary con-
ditions. Our previous theoretical [9] and numerical studies [6] have indicated that all the
magnitudes of residual stresses increase with the increase in the layer numbers up to about
200 layers, then approach to constants with the further increase of the layer number. There-
fore, for MLCCs which contain over 200 dielectric layers, a unit cell model is efficient for
estimating the residual thermal stresses in a MLCC under thermal loads.
Nowadays, as a development trend in MLCCs, miniaturization requires a single layer in
a MLCC become thinner and thinner. The metal electrode materials have been found to ex-
hibit a large plastic deformation near the electrode tips [6]. For a very small thickness of
each layer (e.g., 1 μm or even thinner), the classical finite element method does not include
intrinsic material lengths, and is doubtful for predicting the mechanical behaviour of the
MLCCs at the micron and sub-micron scales. The size-dependent material behaviour has
been observed in these scale regions [10], which have been owed to geometrically necessary
dislocations (GND) associated with non-uniform plastic deformation in small volumes [11].
Strain gradient plasticity theories have been developed based on the notion of GND. Also
based on the Taylor dislocation model [12], a lower-order, conventional theory of mecha-
nism-based strain gradient plasticity (CMSG) was developed by Huang et al. [13], which
does not involve higher-order stress nor additional boundary conditions. To our knowledge,
little work has been conducted on applying the CMSG to analyze the thermal-mechanical
problems.
The present study is aimed at investigating the influence of size scale on the residual
stresses induced by high-temperature processing, and thus figuring out what critical issues
would appear during further miniaturization of MLCCs for guiding the design of the
MLCCs. A two-dimensional (2D) axisymmetric unit cell FE model with periodic conditions
along the thickness direction is proposed to examine the influences of the lateral margin
length, the single layer thickness, as well as the capacitor size on the residual stresses in
MLCCs, in which the thermal-mechanical CMSG is included.
2. FE model.
2.1 Unit cell model for MLCCs under thermal load. A 2D axisymmetric unit cell
model was constructed using the commercial finite element program ABAQUS 6.9. A
MLCC structure consists of the active region, the housing margins and the lateral margins,
as shown in Fig 1, a.
Fig. 1. Schematic structure of a MLCC and (b) its unit cell model.
The geometry of a MLCC is symmetric with respect to the 2- and 3-axes but asymmet-
ric with respect to the 1-axis because the left and the right margins have alternating elec-
trodes (Fig 1, a). Franken et al. [5] and Prume et al. [7] presented a 2D plane-stress finite
134
element (FE) model to simulate the behavior of MLCCs and predict their reliability during
soldering and bending. Shin et al. [8] calculated the residual stresses in the active region of a
MLCC based on 2D plane-strain assumption. However, from our previous full 3D FE nu-
merical results [6] it can be shown that axisymmetric assumption is more reasonable than
plane stress or plane strain assumption. Therefore, for simplicity, it is assumed that a MLCC
is axisymmetric with respect to the thickness direction (i.e. 2-axis) of the MLCCs in our
simulations. A unit cell axisymmetric model with periodical boundary is built, as shown in
Fig. 1, b.
The initial temperature is set to 1250oC, a typical sintering temperature for BaTiO3
based MLCCs, and the final temperature is set to room temperature, 20oC. Axisymmetrical
8-node bilinear element using reduced integration (CAX8R) was adopted for capturing more
accurate stress fields. The deformation behavior of the dielectric ceramic layers was as-
sumed to be elastic, while the Ni electrode was considered to be an elastic-plastic material
including strain gradient effect.
2.2 Thermal-mechanical coupled constitutive law including strain gradient effect. The
shear flow stress is related to the dislocation density by [12]
b , (1)
where is the shear modulus, b is the magnitude of Burgers vector, and is an empirical
coefficient around 0,3. The dislocation density is composed of the density s for statis-
tically stored dislocations (SSDs), and the density G for geometrically necessary disloca-
tions (GND) [14],
s G , (2)
G is defined as [12]
p
G r
b
, (3)
where r is the Nye-factor to reflect the scalar measure of geometrically necessary disloca-
tions density, and r is around 1.90 for face-centered-cubic (FCC) polycrystals. p is effec-
tive plastic strain gradient, given by [13]
p pdt ; p p p1
4 ijk ijk ; p p p p
, , ,ijk ik j jk i ij k , (4)
where p
ij is plastic strain rate tensor.
The flow stress flow accounting for the nonuniform plastic deformation associated with
geometrically necessary dislocations, is defined as
2
p 2 2 2 p 2 p p
flow ref ref, ,f T M r b f T l , (5)
where
2
2 2
ref
l M r b
(6)
is the intrinsic material length in strain gradient plasticity, M = 3,06 and r =1,90 ref is a
reference stress in plasticity (e.g., yield stress). f is a non-dimensional function of plastic
135
strain P and temperature T. In this study, we assumed that the material parameters are time-
independent, so the stress-strain relation in uniaxial tension takes the form as,
flow 1
NP
Y
Y
E
, (7)
where E is the Young’s modulus, Y is the initial yield stress, and N is the plastic work
hardening exponent. In order to determine s , we examine uniaxial tension, which has a
vanishing plastic strain gradient 0P . s is determined as
2
ref
P
S
f
M b
. (8)
For an elastic-plastic material under thermal load, the strain rate ij is decomposed into
the elastic, plastic and thermal parts.
e p th
ij ij ij ij . (9)
The elastic strain rate e
ij is obtained from the stress rate ij ,
'1
2 9
e kk
ij ij ijK
, (10)
where '
ij is deviatoric stress rate and K is bulk moduli. The plastic strain rate P
ij is propor-
tional to the deviatoric stress '
ij in the conventional J2-flow theory of plasticity,
P "3
2
p
ij ij
e
, (11)
where " "3 / 2e ij ij is the von Mises effective stress, and 2 / 3P P P
ij ij is the
equivalent plastic strain rate.
Thermal strain rate th due to thermal mismatch is determined by
th T , (12)
where T is the temperature change rate and is the coefficient of thermal expansion (CTE).
The above CMSG considering thermal-mechanical coupling is embedded into
ABAQUS by using the user subroutines UMAT and UEXPAN. The material properties used
in the present simulations are listed in Table 1.
Table 1. Material parameters used in the present simulations
Materials BaTiO3 Nickel
Young's modulus E (GPa) 108 210
Poisson's ratio 0,25 0,33
CTE (10-6/deg) 8 13,5
Yield strength Y (MPa) 690
Work hardening coefficient N 0,15
Burger vector b (Å) 2,4918
Intrinsic material length l (μm) 5,27
136
3. Results and Discussion.
3.1 Effect of the lateral margin length. A typical contour plot of the stress components
is shown in Fig. 2. For the MLCC, the distributions of the various stress components includ-
ing the strain gradient plasticity are shown in Fig. 2, which illustrates that ceramic materials
have a large deformation around the electrode. Because the singularity is unavoidable in the
interface corner between the electrode and the ceramics, the same mesh density is used in all
the simulations to possibly reduce the error due to the singularity during comparison. From
Fig. 2, a, it can be seen that the radial stress 11 in most of the electrode layers is tensile.
Similarly, the normal stress 11 along the length direction in the ceramic layer is mostly
compressive. Fig. 2, b, indicate that the normal stress 22 along the thickness direction and
shear stress 12 in most of the BaTiO3/Ni interface are continuous, but they change to be dis-
continuous severely near the electrode tips. In other words, the interface between the ceramic
layer and the electrode layers near the internal electrode tips is more likely to be torn.
Better understanding the effects of the lateral margin length on the residual stresses is
important for the design of MLCC structures [15], which has been investigated theoretically
by Jiang et al. [9]. In this section, we focus on the peak stresses near the internal electrode
tip using our proposed model. The length of the lateral margin Lm is varied from 10, 20, 30,
40 to 50 μm, where the dielectric layer thickness td is 1 μm, the electrode layer te is 1 μm,
and L=300 μm. It is found that with the increase in the lateral margin length Lm, all of the
stress components increase, because of the enhancing constraint near the electrode tip, as
shown in Fig. 3.
a
b
Fig. 2. (a) Contour plot of the stress components and (b) the shear stress contour near the
electrode tip for the case that td=2 μm and te=2 μm (scale coordinates factor x×1 and
y×10).
137
The capacitance of an MLCC is propor-
tional to the area of its active region. The
above results show that a higher capacitance
can be achieve by decreasing the lateral
margin length Lm, but bigger peak residual
stresses near the internal electrode tips will
also be caused synchronously. Therefore,
the optimal sizes of the lateral margin in
MLCCs should be determined by consider-
ing the balance of the capacitance and re-
sidual stresses.
3.2 Effect of the thickness ratio of the
dielectrics layer to the electrode layer. For
a representative electrode layer thickness
(te=1 μm), the change of the residual
stresses in MLCCs with respect to the
thickness of each dielectric ceramic layer in
the range of 0,3 – 3,0 μm were calculated,
as shown in Fig. 4, where L = 150 μm and
Lm=30 μm. From Fig. 4, a, it is seen that for
a specified thickness of the electrode layers,
in the majority of the ceramic layer of the
active region, the magnitudes of the residual
normal stress components along both length
and thickness directions increase signifi-
cantly with the decrease in the dielectric
layer thickness, while the shear stress is
almost zero and thickness-independent.
Fig 4, b shows that near the electrode tip,
the peak normal stresses along the length
and thickness directions decrease gradually,
but oppositely, the shear stress increases
sharply.
Considering these factors comprehen-
sively, we can draw a conclusion that for a
specified thickness of electrode layers, de-
creasing the ceramic layer thickness (i.e.,
decreasing the thickness ratio of the ceramic
layer to the electrode layer) may cause the
increase in the residual thermal stresses in
the MLCC during the sintering process,
which is against the further miniaturization
of the MLCCs.
3.3 Effect of size-scale. Our previous
[6] and present results suggest that the re-
sidual stress concentration is closely related
to the relative thickness of the dielectric
ceramic layer with respect to the electrode
layer, and therefore a possible approach to
decrease the ceramic layer thickness td
without the failure of MLCC devices is to decrease the electrode layer thickness te simulta-
neously. If the thickness ratio of the ceramic layer to the electrode layer is kept to be a con-
stant (i.e., the thicknesses of the electrode layer and ceramic layer decrease simultaneously),
the change trend of the residual thermal stresses induced by the sintering process with the
decrease in the single layer thickness, is not clear until now.
Fig. 3. Variations of the peak residual stresses
near the electrode tip with respect to the lat-
eral margin length.
a
b
Fig. 4. (a) Variations of the residual stresses
in the active region of the dielectric ceramic
layer and (b) variations of the peak stresses
near the electrode tip as a function of the
thickness ratio in case of te=1 μm.
138
Table 2. Geometric parameters for investigating the size effect
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6
td (μm )/te (μm) 10/10 5/5 2/2 1/1 0.5/0.5 0.1/0.1
L/2 (μm) 1500 750 300 150 75 15
Lm (μm) 300 150 60 30 15 3
In this section, the thicknesses of the ceramic and electrode layers reduce to micron and
submicron scale, where the MLCC size is scaled down accordingly, as depicted in Table 2.
The same thickness ratio of the dielectrics layer to the electrode layer, 1,0, is kept in all
cases in our simulations. From Fig. 5, it can be seen that near the internal electrode tip, if
without CMSG being considered, the peak residual thermal stresses are size independent,
while if with CMSG being considered, the peak stresses are almost size- independent above
2 µm and exhibit a strong size effect below 1µm. Fig. 5 clearly demonstrates that for a given
thickness ratio, as the MLCC size is scaled down, the peak shear stress reduces significantly
and the normal stresses along the length and thickness directions change slightly with the
decrease in the single layer thickness td as td >1 μm, but as td <1 μm, the normal stress com-
ponents increase sharply with the further decrease in td. In other words, the reduction of the
single layer thickness is benefit to the relaxation of the residual stress when the single layer
thickness td is above 1 μm, but when td is below 1 μm, the excessive rapid increase in the
residual thermal normal stresses along the length and thickness directions could cause the
cracking of the brittle dielectric ceramic layers near the electrode tip. The numerical results
based on the conventional FEM can not present this size effect on residual thermal stresses
induced by the sintering process in micro MLCCs.
4. Conclusions.
The residual thermal stresses induced by the high-temperature sintering process in mul-
tilayer ceramic capacitors (MLCCs) were investigated by using a finite element unit cell
model, in which the CMSG thermal-mechanical coupling is considered. The unit cell model
with periodic boundary conditions along the thickness direction was proposed to represent
the multilayer structure effect. The effects of the lateral margin length, the thickness ratio of
the dielectrics layer to the electrode layer, as well as the size-scale dependence on the resid-
ual thermal stresses were all examined. Some conclusions can be drawn as follow:
1. The normal stress along the thickness direction and the shear stress in most of the ce-
ramics/electrode interface are continuous, but they change to be discontinuous severely near
Fig. 5. Comparison of the residual thermal stresses obtained between by the present model
and by the classical FE results with respect to the single layer thickness, where the MLCC
size is scaled down.
139
the electrode tips. In other words, the interface between the ceramic layer and the electrode
layers is more likely to be torn near the internal electrode tips.
2. A higher capacitance can be achieve by decreasing the lateral margin length Lm, but
bigger peak residual stresses near the internal electrode tips will also be caused synchro-
nously. Therefore, the optimal sizes of the lateral margin in MLCCs should be determined
by considering the balance of the capacitance and residual stresses.
3. For a specified thickness of electrode layers, decreasing the ceramic layer thickness
(i.e., decreasing the thickness ratio of the ceramic layer to the electrode layer) may cause the
increase in the residual thermal stresses in the MLCC during the sintering process, which is
against the further miniaturization.
4. For a given thickness ratio, as the MLCC size is scaled down, the peak shear stress re-
duces significantly and the normal stresses along the length and thickness directions change
slightly with the decrease in the single layer thickness td as td >1 μm, but as td <1 μm, the nor-
mal stress components increase sharply with the continued decrease in td. In other words, the
reduction of the single layer thickness is benefit to the relaxation of the residual stress when
the single layer thickness td is above 1 μm, but when td is below 1 μm, the excessive rapid in-
crease in the residual thermal normal stresses along the length and thickness directions could
cause the cracking of the brittle dielectric layers near the electrode tip.
5. The residual thermal stresses induced by the sintering process exhibit strong size ef-
fects and, therefore, strain gradient effect should be taken into account in the design and
evaluation of MLCC devices.
ACKNOWLEDGMENTS.
This work was supported by the National Science Foundation of China (10902048), the
Foundation of Jiangxi Educational Committee (GJJ08229) and the Natural Science Founda-
tion of Jiangxi Province (2008GZW0010). The first author would appreciate the excellent
advice from Dr. S. Qu.
Р Е ЗЮМ Е . Розглянуто задачу про залишкові температурні напруження в багатошарових ке-
рамічних конденсаторах, викликані високотемпературним процесом спікання. Застосовано модель
одиничної чарунки в методі скінченних елементів, де враховано вплив градієнта деформацій. Отри-
мані числові результати показують, що залишкові температурні напруження залежать від довжини
поздовжнього краю, відношення товщини діелектричних шарів до товщини електродних шарів і
розміру конденсатора. При заданому відношенні товщин, коли розмір конденсатора зменшується, то
найвищі зсувні напруження зменшуються суттєво і нормальні напруження вздовж довжини та в на-
прямку зміни товщини змінюються незначно зі зменшенням товщини керамічного шару до величини,
меншої 1 мікрона. Коли ця величина є більшою 1 мікрона, то нормальні напруження зростають різко
зі збільшенням товщини керамічного шару. Таким чином, залишкові температурні напруження, ви-
кликані високотемпературним процесом спікання, виявляють сильний вплив розміру конденсатора і
тому вплив градієнта деформації повинен братися до уваги в процесі створення і оцінювання роботи
багатошарових керамічних конденсаторів.
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From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 06.09.2010 Утверждена в печать 26.06.2013
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| id | nasplib_isofts_kiev_ua-123456789-87922 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0032-8243 |
| language | Russian |
| last_indexed | 2025-12-07T18:08:32Z |
| publishDate | 2013 |
| publisher | Інститут механіки ім. С.П. Тимошенка НАН України |
| record_format | dspace |
| spelling | Jiang, W.G. Xiong, C.A. Wu, X.G. 2015-10-29T19:24:11Z 2015-10-29T19:24:11Z 2013 Size Dependence of Residual Thermal Stresses in Micro Multilayer Ceramic Capacitors by Using Finite Element Unit Cell Model Including Strain Gradient Effect / W.G. Jiang, C.A. Xiong, X.G. Wu // Прикладная механика. — 2013. — Т. 49, № 6. — С. 132-140. — Бібліогр.: 15 назв. — англ. 0032-8243 https://nasplib.isofts.kiev.ua/handle/123456789/87922 The residual thermal stresses induced by the high-temperature sintering process in multilayer ceramic capacitors (MLCCs) are investigated by using a finite element unit cell model, in which the strain gradient effect is considered. The numerical results show that the residual thermal stresses depend on the lateral margin length, the thickness ratio of the dielectrics layer to the electrode layer, and the MLCC size. At a given thickness ratio, as the MLCC size is scaled down, the peak shear stress reduces significantly and the normal stresses along the length and thickness directions change slightly with the decrease in the ceramic layer thickness td as td >1 μm, but as td <1 μm, the normal stress components increase sharply with the increase in td. Thus, the residual thermal stresses induced by the sintering process exhibit strong size effects and, therefore, the strain gradient effect should be taken into account in the design and evaluation of MLCC devices. Розглянуто задачу про залишкові температурні напруження в багатошарових керамічних конденсаторах, викликані високотемпературним процесом спікання. Застосовано модель одиничної чарунки в методі скінченних елементів, де враховано вплив градієнта деформацій. Отримані числові результати показують, що залишкові температурні напруження залежать від довжини поздовжнього краю, відношення товщини діелектричних шарів до товщини електродних шарів і розміру конденсатора. При заданому відношенні товщин, коли розмір конденсатора зменшується, то найвищі зсувні напруження зменшуються суттєво і нормальні напруження вздовж довжини та в напрямку зміни товщини змінюються незначно зі зменшенням товщини керамічного шару до величини, меншої 1 мікрона. Коли ця величина є більшою 1 мікрона, то нормальні напруження зростають різко зі збільшенням товщини керамічного шару. Таким чином, залишкові температурні напруження, викликані високотемпературним процесом спікання, виявляють сильний вплив розміру конденсатора і тому вплив градієнта деформації повинен братися до уваги в процесі створення і оцінювання роботи багатошарових керамічних конденсаторів. This work was supported by the National Science Foundation of China (10902048), the Foundation of Jiangxi Educational Committee (GJJ08229) and the Natural Science Foundation of Jiangxi Province (2008GZW0010). The first author would appreciate the excellent advice from Dr. S. Qu. ru Інститут механіки ім. С.П. Тимошенка НАН України Прикладная механика Size Dependence of Residual Thermal Stresses in Micro Multilayer Ceramic Capacitors by Using Finite Element Unit Cell Model Including Strain Gradient Effect Article published earlier |
| spellingShingle | Size Dependence of Residual Thermal Stresses in Micro Multilayer Ceramic Capacitors by Using Finite Element Unit Cell Model Including Strain Gradient Effect Jiang, W.G. Xiong, C.A. Wu, X.G. |
| title | Size Dependence of Residual Thermal Stresses in Micro Multilayer Ceramic Capacitors by Using Finite Element Unit Cell Model Including Strain Gradient Effect |
| title_full | Size Dependence of Residual Thermal Stresses in Micro Multilayer Ceramic Capacitors by Using Finite Element Unit Cell Model Including Strain Gradient Effect |
| title_fullStr | Size Dependence of Residual Thermal Stresses in Micro Multilayer Ceramic Capacitors by Using Finite Element Unit Cell Model Including Strain Gradient Effect |
| title_full_unstemmed | Size Dependence of Residual Thermal Stresses in Micro Multilayer Ceramic Capacitors by Using Finite Element Unit Cell Model Including Strain Gradient Effect |
| title_short | Size Dependence of Residual Thermal Stresses in Micro Multilayer Ceramic Capacitors by Using Finite Element Unit Cell Model Including Strain Gradient Effect |
| title_sort | size dependence of residual thermal stresses in micro multilayer ceramic capacitors by using finite element unit cell model including strain gradient effect |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/87922 |
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