Composites: Part A 33 (2002) 1115–1121
`
`www.elsevier.com/locate/compositesa
`
`Apparent coefficient of thermal expansion and residual stresses in
`multilayer capacitors
`
`Chun-Hway Hsueh*, Mattison K. Ferber
`
`Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6068, USA
`
`Received 11 August 2001; revised 20 April 2002; accepted 15 May 2002
`
`Abstract
`
`The thermal expansion behavior and residual stresses in multilayer capacitor (MLC) systems are analyzed in the present study. An MLC
`consists of a laminate of multiple alternating electrode layers and dielectric layers sandwiched between two ceramic cover layers. An
`analytical model is developed to derive simple closed-form solutions for the apparent coefficients of thermal expansion (CTEs) of the
`laminate. Plasticity of electrodes is included in the analysis. The predicted apparent CTEs are compared with measurements of some
`laminated ceramic composites. The effects of plasticity on apparent CTEs and residual stresses in MLC systems are discussed. q 2002
`Published by Elsevier Science Ltd.
`
`Keywords: A. Laminates; B. Residual/internal stress; B. Thermomechanical; C. Analytical modelling
`
`1. Introduction
`
`Multilayer capacitors (MLCs) typically consist of , 100
`alternate layers of electrodes and dielectric ceramics
`sandwiched between two ceramic cover layers [1 – 3].
`They are fabricated by screen-printing of electrode layers
`on dielectric layers and cosintering of the laminate.
`Conventionally, Ag – Pd is used as the electrode material
`and BaTiO3 is used as the dielectric ceramics. Due to the
`thermomechanical mismatch between the constituents, as-
`fabricated MLCs are generally subjected to residual
`stresses. The brittle dielectric layer may crack and/or the
`electrode/dielectric interface may debond due to the
`presence of these residual stresses. Such damage inevitably
`degrades the performance and reliability of the system.
`Also, because of the low yield strength of electrodes
`(, 70 MPa), plastic deformation can occur. The mechanical
`integrity of the system becomes a major concern, and
`various efforts have been devoted to analyze residual
`stresses, cracking, and interface debonding of the multilayer
`system [4 – 8]. While the classical lamination theory (CLT)
`provides a powerful tool to analyze the elastic behavior of
`laminated materials subjected to in-plane forces and
`moments, its solutions are often expressed in matrices,
`and computers are used to obtain the numerical results [9].
`* Corresponding author. Tel.: þ 1-865-576-6586; fax: þ 1-865-574-8445.
`
`E-mail address: hsuehc@ornl.gov (C.H. Hsueh).
`
`1359-835X/02/$ - see front matter q 2002 Published by Elsevier Science Ltd.
`PII: S 1 3 5 9 - 8 3 5 X ( 0 2 ) 0 0 0 5 4 - 4
`
`In MLCs, both the thermal strain and the plastic strain are
`present. We believe that these two strain components may
`also be added to CLT; however, we also believe that
`complexities in analyses based on CLT will be compounded
`in this case. To the best of the authors’ knowledge, there is
`no simple equation based on CLT to describe thermo-
`mechanical behavior of MLCs in the presence of plasticity.
`The purpose of the present study is to include plasticity
`of the electrodes and to derive simple closed-form solutions
`to describe both the thermal expansion behavior and
`residual stresses in MLCs. To achieve this, an analytical
`model other than CLT is developed to replace the laminate
`of electrodes and dielectric ceramics with an effective
`medium. The thermomechanical properties of the effective
`medium are related to the properties of both electrodes and
`dielectric ceramics. Then, closed-form analytical solutions
`of the residual stresses in the system are derived. Finally,
`comparisons are made between the present analytical
`solutions and existing measurements of apparent coeffi-
`cients of thermal expansion (CTEs) of some ceramic
`laminates, and the effects of plasticity on apparent CTEs
`and residual stresses in MLC systems are examined.
`
`2. Analyses
`
`An MLC system is shown schematically in Fig. 1(a),
`; and the total
`where the thicknesses of each cover layer is tc
`Exhibit 1031
`IPR2016-00636
`AVX Corporation
`
`000001
`
`

`

`1116
`
`C.-H. Hsueh, M.K. Ferber / Composites: Part A 33 (2002) 1115–1121
`
`; consists of
`
`infinite dimension in the x – y plane, the constituents in the
`system are subjected to biaxial stresses due to the
`thermomechanical mismatch between the constituents. To
`derive the thermomechanical properties of the effective
`medium, which consists of dielectric ceramics and electrodes
`; respectively, Fig. 1(c) is adopted for
`of thicknesses td and te
`modeling.
`The in-plane strain in dielectric ceramics, 1
`d
`the elastic and the thermal strains, such that
`Þs
`þ a
`
`¼ ð1 2 n
`
`1
`
`d
`
`d
`
`DT
`
`d
`
`ð1Þ
`
`d
`Ed
`where E; n; and aare Young modulus, Poisson’s ratio, and
`the CTE, respectively, DT is the cooling temperature from
`the stress-free temperature, and the subscript, d, denotes
`dielectric ceramics.
`For the electrodes, the plastic strain is also involved. To
`analyze the plastic strain, the Von Mises yield criterion [10]
`is used. On the basis of this, it can be derived that plastic
`deformation occurs when the biaxial stress in the electrode,
`; reaches the yield strength, s
`: Furthermore, linear work
`y
`hardening is assumed, whereupon the plastic strain, 1p
`;
`ij
`satisfies the Prandtl – Reuss relation [10,11] such that
`¼ 3
`ds
`e
`Hs
`2
`e
`
`ð2Þ
`
`se
`
`d1p
`ij
`
`sij
`
`where H is the slope of the work-hardening curve, and sij is
`the deviatoric stress. In the present case, the deviatoric
`; are
`; sy
`; and sz
`stresses, sx
`¼ s
`¼ sy
`e
`3
`¼ 22s
`3
`
`sx
`
`sz
`
`e
`
`ð3aÞ
`ð3bÞ
`
`Combination of Eqs. (2), (3a), and (3b) yields
`¼ ds
`e
`2H
`
`d1p
`x
`
`y
`
`d1p
`z
`
`1p
`x
`
`1p
`z
`
`e
`
`y
`
`e
`
`y
`
`2H
`Þ
`2 s
`y
`
`e
`H
`
`¼ d1p
`¼ 2ds
`H
`With the boundary condition that plastic strains are zero
`¼ s
`; the plastic strains become
`when s
`e
`¼ s
`¼ 1p
`2 s
`y
`¼ 2ðs
`
`ð4aÞ
`ð4bÞ
`
`ð5aÞ
`ð5bÞ
`
`Superposing the plastic strain on the elastic and the thermal
`; is
`strains, the in-plane strain in electrodes, 1
`Þs
`þ a
`þ s
`ðfor s
`2 s
`y
`
`¼ ð1 2 n
`
`e
`
`Ee
`
`1
`
`e
`
`e
`
`e
`
`e
`
`2H
`
`DT
`
`e
`
`$ s
`y
`
`e
`
`Þ
`ð6Þ
`
`Fig. 1. Schematics showing (a) an MLC, (b) the MLC with the laminate of
`dielectric layers and electrode layers replaced by an effective medium, and
`(c) the effective medium consists of dielectrics and electrodes with
`; respectively.
`thicknesses td and te
`
`thicknesses of electrode layers and dielectric layers are te
`; respectively. The Cartesian coordinates, x; y; and z;
`and td
`are used, such that the z-axis is normal to the plane of the
`laminate. In the present analysis, the system is assumed to
`be infinite in the x – y plane, and no bending is induced by
`residual stresses because of the symmetric geometry. To
`analyze residual stresses in the system, the laminate of
`electrodes and dielectric ceramics are replaced by an
`þ te
`; such that the
`effective medium with a thickness td
`effective medium and cover layers are subjected to residual
`stresses, sp and s
`; respectively (see Fig. 1(b)). It is noted
`c
`that the solutions of sp and s
`c are contingent upon the
`determination of the thermomechanical properties of the
`effective medium. With the solution of sp; the residual
`; due
`
`stresses in electrodes and dielectric ceramics, se and s
`d
`to both the mismatch between electrodes and dielectric
`ceramics and the external constraint, sp; resulting from the
`presence of cover layers can be derived (see Fig. 1(c)).
`
`2.1. Thermomechanical properties of the effective medium
`
`Without constraints in the direction normal to the layers,
`the stress in the z-direction is zero in the system. With an
`
`where the subscript, e, denotes electrodes.
`
`
`Solutions of sd and se are subjected to the compatibility
`
`000002
`
`

`

`C.-H. Hsueh, M.K. Ferber / Composites: Part A 33 (2002) 1115–1121
`
`ð7aÞ
`ð7bÞ
`
`ÞDT
`
`ð8aÞ
`
`ð8bÞ
`
`and the equilibrium conditions; i.e.
`¼ 1
`þ te
`
`se
`
`e
`
`1
`
`d
`
`sd
`
`td
`
`þ te
`
`Þsp
`
`2 a
`d
`
`e
`
`d
`
`sy
`
`
`
`
`
`Combination of Eqs. (1), (6), (7a), and (7b) yields
`þ ða
`þ 1
`1 2 n
`sp 2
`
`e
`Ee
`2H
`2H
`þ 1
`þ 1 2 n
`1 2 n
`td
`e
`te
`Ee
`Ed
`2H
`
`þ te
`te
`
`td
`
`¼
`
`sd
`
`¼ ðtd
`
`
`ÞDT
`
`2 a
`d
`
`td
`
`þ te
`td
`
`¼
`
`se
`
`2 ða
`sp þ s
`1 2 n
`y
`d
`e
`2H
`Ed
`þ te
`þ 1
`1 2 n
`1 2 n
`e
`d
`td
`Ee
`Ed
`2H
`The resultant in-plane strain of the effective medium, 1p;
`can be obtained from 1
`d by substituting Eq. (8a) into Eq. (1),
`such that
`
`
`sy
`
`
`
`td
`
`þ te
`te
`
`1 2 n
`e
`Ee
`
`þ 1
`2H
`
`
`Þ 1 2 n
`
`e
`
`Ee
`
`d
`
`
`
`
`
`1117
`
`H ð11cÞ
`
`þ 1
`2H
`
`ð11dÞ
`
`
`
`
`ad
`
`þ 1
`2H
`þ 1
`2H
`
`e
`
`e
`
`Ee
`
`Ee
`
`
`
`H
`
`p ¼ te
`þ te
`td
`
`1 þ
`
`te
`
`þ
`ap ¼ a
`1 þ
`
`e
`
`te
`
`te
`
`d
`
`d
`
`tdEd
`ð1 2 n
`
`Þ 1 2 n
`tdEd
`ð1 2 n
`
`Þ 1 2 n
`tdEd
`ð1 2 n
`
`Due to the nature of biaxial stress, the elastic constants
`in the stress – strain relation always appear in a form of
`
`E=ð1 2 nÞ; which is defined as the biaxial modulus.
`
`in defining the elastic
`Hence, Eq. (11a) is sufficient
`constants of the effective medium.
`It should be noted that experimental determination of
`CTE of the laminate is generally obtained from the
`measurement of resultant strain of the laminate, which
`is subjected to a temperature change in the absence of
`external constraints. For convenience,
`this CTE is
`
`
`DT
`
`ad
`
`
`
`e
`
`þ 1
`2H
`
`e
`
`d
`
`Ee
`
`1p ¼
`
`sp 2
`1 þ
`
`te
`
`2H
`tdEd
`ð1 2 n
`
`d
`
`te
`e
`
`Ee
`
`
`þ a
`þ
`tdEd
`ð1 2 n
`
`þ 1
`Þ 1 2 n
`2H
`
`
`Þ 1 2 n
`
`
`ð9Þ
`
`Because the effective medium consists of thermo-elastic
`ceramics and thermo-elasto-plastic electrodes,
`it
`is
`logical to assume that the effective medium is thermo-
`elasto-plastic. Hence, similar
`to Eq.
`(6),
`the stress –
`strain relation for
`the effective medium can be
`formulated as
`
`1p ¼ ð1 2 npÞsp
`
`Ep
`
`þ sp 2 sp
`2Hp
`
`y
`
`þ apDT
`
`ð10Þ
`
`; Hp; and ap are the effective
`where Ep; np; sp
`y
`thermomechanical properties of the effective medium.
`Comparing Eqs.
`(10) with Eq.
`(9),
`the effective
`properties can be obtained, such that
`
`
`
`
`Þ 1 2 n
`tdEd
`ð1 2 n
`Ee
`Þð1 2 n
`ðtd
`þ te
`teEe
`
`d
`
`þ 1
`2H
`
`Þ
`
`e
`
`e
`
`¼ 1 þ
`
`te
`
`Ep
`1 2 np
`
`¼ te
`þ te
`td
`
`sy
`
`sp
`y
`
`defined as the apparent CTE, aa: Hence, the apparent
`CTE is
`
`ð12Þ
`
`ð13Þ
`
`ad
`
`
`
`aa ¼ 1p
`DT
`
`ðat sp ¼ 0Þ
`
`Combination of Eqs. (9) and (12) yields
`
`þ
`
`2s
`y
`2HDT
`
`aa ¼
`
`e
`
`þ a
`1 þ
`
`te
`
`e
`
`d
`
`e
`
`Ee
`
`d
`
`
`þ 1
`Þ 1 2 n
`tdEd
`ð1 2 n
`
`
`te
`Ee
`2H
`þ 1
`Þ 1 2 n
`tdEd
`ð1 2 n
`2H
`
`ð11aÞ
`
`ð11bÞ
`
`The apparent CTE given by Eq. (13) is in the x – y
`plane. To derive the apparent CTE normal to the layer,
`;
`aa
`the strain in the z-direction in the absence of
`z
`external constraints is required. Following the similar
`those for deriving aa and
`analytical procedures as
`including the plastic strain, 1p
`z (Eq. (5b)), in electrodes,
`the apparent CTE normal to the layer is
`
`ð14Þ
`
`ae
`
`þ te
`þ te
`
`ad
`
`td
`
`þ td
`
`
`ða
`
`e
`
`Þ
`
`2 a
`d
`
`þ 1
`H
`
`
`þ 2n
`2 2n
`d
`e
`HDT
`Ed
`Ee
`Þ
`ð1 2 n
`þ 1
`þ te
`tdEd
`2H
`
`sy
`
`
`
`Þ
`
`d
`
`d
`
`ð1 2 n
`þ te
`tdEd
`1 2 n
`e
`Ee
`
`nd
`
`þ td
`
`1 2 2n
`e
`Ee
`
`
`
`aa
`z
`
`¼ te
`þ te
`td
`
`000003
`
`

`

`C.-H. Hsueh, M.K. Ferber / Composites: Part A 33 (2002) 1115–1121
`
`nd
`
`
`
`2
`
`¼
`
`aa
`z
`
`td
`
`þ te
`Ee
`1 2 n
`d
`tdEd
`
`
`ða
`2
`Ed
`þ 1 2 n
`teEe
`
`ð19Þ
`
`ae
`
`þ te
`þ te
`
`ad
`
`td
`
`þ td
`
`Þ
`
`2 a
`e
`
`d
`
`e
`
`ne
`
`It can be seen that the apparent CTEs in a laminate are
`anisotropic; i.e. aa – aa
`: In the special case of Ed
`¼ n
`z
`; the apparent CTEs become
`þ te
`aa ¼ aa
`¼ a
`¼ td
`þ te
`td
`¼ n
`Þ
`
`ae
`
`ad
`
`e
`
`z
`
`rom
`
`nd
`
`ðfor Ed
`
`¼ Ee and n
`
`d
`
`e
`
`¼ Ee and
`
`ð20Þ
`
`Hence, when the constituents of the laminate have the same
`elastic constants, the apparent CTEs of the laminate can be
`obtained using rule-of-mixtures.
`
`Þ and
`Ignoring the difference in Poisson’s ratios ðn
`d and n
`e
`; it can be obtained from Eqs. (18) – (20)
`. Ed
`assuming Ee
`; and aa ,
`
`that aa . arom and aa
`
`. a
`, arom when a
`d
`e
`: Hence, aa shifts from
`om and aa
`
`. arom when a
`, a
`d
`om in the direction of the CTE of the harder layer, and aa
`z
`shifts from a
`rom in the direction of the CTE of the softer
`layer.
`Using the thermomechanical properties of Al2O3,
`BaZrO3, SnO2, and CaTiO3 listed in Table 1 and the
`volume fractions of constituents in Al2O3/BaZrO3, Al2O3/
`SnO2, and Al2O3/CaTiO3 laminates listed in Table 2, the
`calculated aa and aa
`z from Eqs. (18) and (19) are shown in
`Table 2. The apparent CTEs of the laminates were measured
`during heating [12] and are also shown in Table 2 for
`comparison. Good agreement is obtained between measured
`and predicted apparent CTEs for Al2O3/BaZrO3 and Al2O3/
`the
`SnO2 laminates. For the Al2O3/CaTiO3 laminate,
`predicted aa is significantly different from the measurement.
`However, it is noted that cracks normal to the layers were
`observed in CaTiO3 layers which, in turn, results in a lower
`in-plane CTE for CaTiO3 layers during heating [12]. As a
`result, the measured aa is lower than the prediction for the
`Al2O3/CaTiO3 laminate.
`
`z
`
`z
`
`e
`
`ar
`
`ar
`
`3.2. Effects of plasticity on CTEs and residual stresses in
`MLCs
`
`The material properties of CR-1206 and AR-1206 MLCs
`(KEMET Electronics Corp., Ft. Inn, SC 29644-0849) are
`used in the present study to examine the essential trends of
`
`Table 1
`Thermomechanical properties of constituents in laminated ceramic
`composites
`
`Materials
`
`Young’s modulus,
`E (Gpa)
`
`Poisson’s
`ratio
`
`CTE
`
`( £ 10
`26/8C)
`
`Al2O3
`BaZrO3
`SnO2
`CaTiO3
`
`380
`220
`253
`280
`
`0.26
`0.25
`0.293
`0.25
`
`9.09
`6.26
`5.8
`13.4
`
`1118
`
`2.2. Residual stresses in MLC
`
`The residual stresses in dielectric ceramics and electro-
`; given by Eqs. (8a) and (8b) are contingent
`
`des, sd and s
`e
`upon the determination of sp: With the thermomechanical
`properties of the effective medium defined by Eqs. (11a) –
`(11d), the residual stresses in the effective medium and
`cover layers, sp and s
`c (see Fig. 1(b)), can be derived using
`the compatibility and the equilibrium conditions; i.e.
`ð1 2 npÞsp
`Þs
`þ sp 2 sp
`þ a
`Ep
`2Hp
`þ te
`
`þ apDT ¼ ð1 2 n
`
`c
`
`Ec
`
`c
`
`c
`
`DT
`ð15aÞ
`ð15bÞ
`
`y
`
`¼ 0
`
`sc
`
`Þsp þ 2tc
`
`ðtd
`
`c
`
`c
`
`ð16Þ
`
`ð17Þ
`
`Solution of Eqs. (15a) and (15b) yields
`ÞDT
`2 ðap 2 a
`sp
`y
`2Hp
`Þ
`Þð1 2 n
`þ te
`þ ðtd
`þ 1
`1 2 np
`Ep
`2Hp
`2tcEc
`¼ 2ðtd
`Þsp
`þ te
`2tc
`With the solution of sp; the solutions of residual stresses in
`; s
`the MLC (i.e. s
`
`d and se) are complete. It should be
`c
`noted that the solutions derived in Sections 2.1 and 2.2 are
`$
`based on the condition that electrode layers yield (i.e. s
`Þ: When s
`e
`; the plastic strain component shall be
`, s
`y
`e
`deleted from the above analyses, and this can be achieved by
`letting H ! 1 in the above analyses.
`
`sp ¼
`
`sc
`
`sy
`
`3. Results
`
`The CTEs of some laminated ceramic composites have
`been measured recently [12]. The present analytical
`solutions of the apparent CTEs (Eqs. (13) and (14)) are
`compared with those measurements. Then, the effects of
`plasticity on the apparent CTEs and residual stresses in
`MLCs are presented.
`
`3.1. Comparison with measurements of apparent CTEs
`
`The CTEs have been measured for the following
`laminated ceramic composites: Al2O3/BaZrO3, Al2O3/-
`SnO2, and Al2O3/CaTiO3 [12]. The solutions for an elastic
`system can be obtained from the present solutions by
`deleting the plastic strain. In the absence of plasticity, the
`effective CTE and the apparent CTE are identical. Letting
`; described by
`the apparent CTEs, aa and aa
`z
`Eqs. (13) and (14) can be reduced to
`þ teEe
`1 2 n
`e
`þ teEe
`1 2 n
`e
`
`ae
`
`H ! 1;
`
`ad
`
`tdEd
`1 2 n
`d
`tdEd
`1 2 n
`d
`
`aa ¼
`
`ð18Þ
`
`000004
`
`

`

`C.-H. Hsueh, M.K. Ferber / Composites: Part A 33 (2002) 1115–1121
`
`1119
`
`Table 2
`Comparison of measured and calculated apparent CTEs of laminated ceramic composites
`
`( £ 1026/8C)
`
`
`
`a
`
`za
`
`aa ( £ 1026/8C)
`
`
`
`Laminate
`
`Al2O3 vol. fraction
`
`Al2O3/BaZrO3
`Al2O3/SnO2
`Al2O3/CaTiO3
`
`0.7
`0.69
`0.75
`
`8.3
`7.87
`7.19
`
`8.53
`8.31
`9.93
`
`8.33
`7.33
`9.22
`
`8.06
`7.81
`10.3
`
`Measured
`
`Predicted
`
`Measured
`
`Predicted
`
`effects of plasticity on CTEs and residual stresses in MLCs.
`CR-1206 consists of 263 layers of 3 mm thick BaTiO3
`dielectrics and 264 layers of 2 mm thick Ag – Pd electrodes
`sandwiched between two 85 mm thick cover layers. This
`¼ 85 mm; td
`¼ 789 mm; and te
`¼ 528 mm
`corresponds to tc
`in Fig. 1. The cover layers and dielectrics are of the same
`¼ Ed
`¼ 91 GPa;
`material. The elastic constants are [13]: Ec
`¼ a
`¼ 0:35; a
`¼ 8 £
`¼ 110 GPa; n
`¼ n
`¼ 0:33; n
`Ee
`¼ 11 £ 10
`d
`e
`c
`d
`26=8C: The yield strength of Ag – Pd
`26=8C; a
`10
`e
`; is 74 MPa. The work hardening coefficient,
`electrodes, s
`; such that H ¼ 11 GPa [13].
`y
`H, is taken to be 10% of Ee
`the apparent CTEs of the
`Using the above properties,
`dielectrics/electrodes laminate and residual stresses in CR-
`1206 as functions of the cooling temperature, DT; are shown
`in Fig. 2(a) and (b), respectively. When the cooling
`
`c
`
`; and (b) residual stresses, sp; s
`; s
`Fig. 2. (a) Apparent CTEs, aa and aa
`z
`d
`c
`; in CR-1206 MLC as functions of the cooling temperature, DT:
`and s
`e
`
`;
`
`temperature is less than , 245 8C, electrodes remain elastic
`and apparent CTEs remain constant; however, they are not
`isotropic (i.e. aa – aa
`z ). Using rule-of-mixtures, the calcu-
`rom is 9.2 £ 10
`26/8C. Because Ee
`;
`lated a
`. a
`. Ed and a
`d
`e
`
`aa . arom and aa
`, a
`rom (see Fig. 2(a)) which are in
`z
`agreement with the discussion in Section 3.1. When the
`cooling temperature is greater than , 245 8C, electrodes
`yield and aa decreases while aa
`z increases as the cooling
`temperature increases because of the contribution of the
`plastic deformation in electrodes. It
`is noted that
`the
`apparent CTE versus DT curve in Fig. 2(a) shows a
`discontinuity when the electrode yields. This is due to the
`constraint of cover layers imposed on the laminate. Without
`
`cover layers, the electrode yields at DT ¼ 2268 8C (i.e. at
`
`the intersection of the extension of the constant apparent
`CTE and the apparent CTE curve when the electrode
`yields). However, in the presence of constraints of cover
`layers, the electrode yields at a smaller cooling temperature
`
`(DT ¼ 2245 8C in this case). Fig. 2(b) shows that the
`constraint on the laminate, sp; due to the presence of cover
`layers is relatively small during the cooling process. This is
`because cover layers are relatively thin compared to the
`laminate. Also, because dielectrics and cover layers are of
`the same material, the residual stresses in dielectrics and
`¼ s
`cover layers are of the same (i.e. s
`d). While dielectrics
`c
`and cover layers are subjected to biaxial compression,
`electrodes are subjected to biaxial tension. The magnitudes
`of residual stresses increase linearly with the cooling
`temperature. After electrodes yield,
`the magnitudes of
`residual stresses increase at a slower rate with the cooling
`temperature.
`The AR-1206 MLC consists of 94 layers of 9 mm thick
`BaTiO3 dielectrics and 95 layers of 2 mm thick Ag – Pd
`electrodes sandwiched between two 85 mm thick cover
`layers. The corresponding apparent CTEs of the dielec-
`trics/electrodes laminate and residual stresses in AR-1206
`as functions of the cooling temperature, DT, are shown in
`Fig. 3(a) and (b), respectively. The MLCs are normally
`sintered at 1320 8C, and DT ¼, 21300 8C when they
`are cooled to room temperature. At DT ¼ 21300 8C; s
`d in
`CR-1206 is more compressive than in AR-1206 by 46 MPa
`(see Figs. 2(b) and 3(b)). Using four-point bending tests,
`the strengths of CR-1206 and AR-1206 are 250 ^ 25 and
`220 ^ 26 MPa, respectively. Hence, the higher strength of
`CR-1206 as compared to AR-1206 could be attributed to the
`higher residual compressive stress in CR-1206’s dielectric.
`
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`
`

`

`1120
`
`C.-H. Hsueh, M.K. Ferber / Composites: Part A 33 (2002) 1115–1121
`
`theory, an analytical model is developed to analyze the
`thermal expansion behavior and residual stresses in MLCs.
`The laminate is replaced by an effective medium, and its
`effective thermomechanical properties are derived. While
`residual stresses in cover layers can be obtained from the
`thermomechanical mismatch between the effective medium
`and cover layers (Fig. 1(b)), residual stresses in electrodes
`and dielectrics can be obtained from both their mismatch
`and the constraints on the effective medium, sp; imposed by
`cover layers (Fig. 1(c)). The predicted apparent CTEs of a
`laminate are compared to the measurements of some
`laminated ceramic composites, and good agreements are
`obtained. It is noted that the apparent CTEs in a laminate is
`anisotropic because of the geometry. While the apparent in-
`plane CTE, aa; shifts from the CTE calculated based on
`; toward the harder layer’s CTE, the
`rule-of-mixtures, a
`rom
`; shifts from a
`apparent out of plane CTE, aa
`rom toward the
`z
`softer layer’s CTE. The effects of plasticity on apparent
`CTEs in laminates and residual stresses in MLCs are
`examined. In the presence of plasticity, aa and aa
`z are
`modified (Fig. 2(a)), and residual stresses increase at a
`slower rate during cooling (Fig. 2(b)).
`
`Acknowledgements
`
`The authors thank Dr P.F. Becher, Dr R.D. Ott, and
`Dr M.J. Lance for reviewing the manuscript. Research is
`jointly sponsored by the US Department of Energy, Division
`of Materials Sciences and Engineering, Office of Basic
`Energy Sciences, and Electronic Ceramics Program under
`contract DE-AC05-00OR22725 with UT-Battelle.
`
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`; and (b) residual stresses, sp; s
`; s
`Fig. 3. (a) Apparent CTEs, aa and aa
`z
`d
`c
`; in AR-1206 MLC as functions of the cooling temperature, DT:
`and s
`e
`
`;
`
`The differences in measured strength (i.e. 30 MPa) and
`calculated s
`d (i.e. 46 MPa) are of the same order of
`magnitude.
`It should be noted that the temperature dependence of
`materials properties is not considered in the present
`calculations. In the presence of such temperature depen-
`dence, the solutions can be obtained by using the elastic –
`plastic properties at
`the measuring temperature and
`replacing the thermal strain aDT by an integral of the
`CTE over the temperature change range in the present
`analytical solutions. It should also be noted that, dielectrics
`in-between electrodes could be poled when an electric field
`is applied. In this case, the elastic constants of dielectrics
`will be modified and the strain due to domain switch shall be
`superposed on the thermal strain.
`
`4. Conclusions
`
`An MLC consists of a laminate of electrode layers and
`dielectric layers sandwiched between two ceramic cover
`layers (Fig. 1(a)). While both dielectric layers and cover
`layers are thermo-elastic, electrode layers are thermo-
`elasto-plastic. As an alternative to the classical lamination
`
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`

`C.-H. Hsueh, M.K. Ferber / Composites: Part A 33 (2002) 1115–1121
`
`1121
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`
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`