- Email: [email protected]
Transient thermal response of functionally graded piezoelectric laminates with an infinite row of parallel cracks normal to the bimaterial interface
Theoretical & Applied Mechanics Letters 9 (2019) 289-292
Contents lists available at ScienceDirect
Theoretical & Applied Mechanics Letters journal homepage: www.elsevier.com/locate/taml
Letter
Transient thermal response of functionally graded piezoelectric laminates with an infinite row of parallel cracks normal to the bimaterial interface Yoshiyuki Mabuchi, Sei Ueda* Department of Mechanical Engineering, Osaka Institute of Technology, Osaka, Japan
H I G H L I G H T S
• We focused on the transient thermal fracture problem of monomorph actuators using a functionally graded piezoelectric material strip (FGPM strip). • Using the Fourier transform technique, the electromechanical problem is reduced to a singular integral equation, which is solved numerically. • The stress intensity factors of the embedded crack are computed and presented as functions of the normalized time for the various values of the nonhomogeneous and geometric parameters. A R T I C L E I N F O
A B S T R A C T
Article history: Received 2 July 2019 Received in revised form 13 September 2019 Accepted 16 September 2019 Available online 30 September 2019
Keywords: Functionally graded piezoelectric material Fracture mechanics Stress intensity factor Elasticity
In this paper, the problem of a functionally graded piezoelectric material strip (FGPM strip) containing an infinite row of parallel cracks perpendicular to the interface between the FGPM strip and a homogeneous layer is analyzed under transient thermal loading condition. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the interface. Using the Fourier transforms, the electro-thermoelastic problem is reduced to a singular integral equation, which is solved numerically. The stress intensity factors are computed and presented as a function of the normalized time, the nonhomogeneous and geometric parameters. ©2019 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Recently, several kinds of piezoelectric actuators have been designed. Hall et al. [1] fabricated a mono-morph actuator made from semi-conductive piezoelectric ceramics. Therefore, the elucidation of fracture behavior of piezoelectric systems such as actuators is important. Moreover, the research on transient electro-thermo-elastic problem for functionally graded piezoelectric material (FGPM) laminate with a crack normal to the biomaterial interface has been reported [2]. On the other hand, cracks occur primarily from plural defects in material. The elucidation of interaction between cracks to affect fracture behavior is also important, because these cracks may lead to fracture of material [3].
* Corresponding author. E-mail address: [email protected] (S. Ueda).
In this paper, we theoretically analyzed the thermal fracture problem of mono-morph actuators using an FGPM strip with an infinite row of parallel cracks due to an instantaneous temperature change. The analytical laminate model of the mono-morph actuator consists of an FGPM strip and a homogeneous elastic layer. Material properties are exponentially dependent on the distance from the interface between the FGPM strip and the homogeneous elastic layer. Using the Fourier transform technique [4, 5], the electro-thermo-elastic problem is reduced to a singular integral equation, and the Gauss-Jacobi numerical integration formula was used in the numerical analysis of the singular integral equation [6]. The stress intensity factors of the embedded crack are computed and presented as functions of the normalized time for the various values of the nonhomogeneous and geometric parameters.
http://dx.doi.org/10.1016/j.taml.2019.06.009 2095-0349/© 2019 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
290
Y. Mabuchi and S. Ueda / Theoretical & Applied Mechanics Letters 9 (2019) 289-292
As shown in Fig. 1, suppose an FGPM strip with the thickness h1 containing an infinite row of parallel cracks of equal length 2c = b − a (0 < a < b < h1 ) being spaced at equal distance 2d perpendicular to the free boundaries bonded to an elastic layer with the thickness h2. The system of rectangular Cartesian coordinates (x, y, z) is introduced in the material in such a way that one of the crack is located along the z -axis, and the x -axis is parallel to the boundaries. The FGPM strip is poled in the z -direction and is in the plane strain conditions perpendicular to the y axis. It is assumed that initially the medium is at the uniform temperature TI and is suddenly subjected to a uniform temperature rise T0 H (t ) along the boundary z = h1, where H (t ) is the Heaviside step function and t denotes time. The temperature along the boundary z = h2 is maintained at TI . The crack faces remain thermally and electrically insulated. The material property parameters are taken to vary continuously along the z-direction inside the FGPM strip. The material properties of FGPM strip, such as the elastic stiffness constants c kl (z), piezoelectric constants e kl (z), dielectric constants εkk (z), stress -temperature coefficients λkk (z), coefficient of heat conduction κx (z), κz (z), and the pyroelectric constant p z (z), are one dimensionally dependent as [c kl (z) , e kl (z) , εkk (z)] = (c kl 0 , e kl 0 , εkk0 ) exp(βz), [ ] λkk (z) , p z (z) = (λkk0 , p z0 ) exp[(β + ω)z], [κx (z) , κz (z)] = (κx0 , κz0 ) exp(δz),
(1)
where β, ω, and δ are positive or negative constants, and the subscript 0 indicates the properties at the interface plane z = 0. For some materials, the thermal diffusivity τ0 indeed does not vary dramatically, then τ0 is assumed to be a constant. The material properties of the homogeneous elastic layer are the elastic stiffness constants c klE , stress-temperature coefficients λEkk , coefficient of heat conduction κE , and thermal diffusivity τE0 . The superscript E denotes the physical quantities of the homogeneous elastic layer. The crack problem may be solved using the superposition technique. In the problem considered here, since the heat conduction is one dimensional and straight crack does not obstruct the heat flow in this arrangement, determination of the temperature distribution and the resulting thermal stress would be quite straightforward and the related crack problem would be one of mode I. The lines of the centers of the cracks x = ±2nd (n = 0, 1, 2, ...) are the axes of symmetry of the configuration. We suppose that the crack is opened under the action of same distribution of the internal pressure σT0 (z, t ), where σT0 (z, t ) is the thermal stress induced by the time-dependent temperature change. The thermal stress σT0 (z, t ) has been already obtained in Ref. [2]. In the following, the subscripts x, y, z will be used to refer to the direction of coordinates. The governing equations for the electromechanical fields are given in previous paper [2]. The boundary conditions can be written as σxx (0, z, t ) = −σT0 (z, t ) , a < z < b, u x (0, z, t ) = 0, 0 ≤ z ≤ a, b ≤ z ≤ h 1 ,
(2)
} σzx (0, z, t ) = 0 , 0 ≤ z ≤ h1 , D x (0, z, t ) = 0
(3)
σEzx (0, z, t ) = 0 u xE (0, z, t ) = 0
} , −h 2 ≤ z ≤ 0,
(4)
σzx (x, h 1 , t ) = 0 σzz (x, h 1 , t ) = 0 , 0 ≤ x ≤ d , D z (x, h 1 , t ) = 0
(5)
σzx (x, 0, t ) = σEzx (x, 0, t ) σzz (x, 0, t ) = σEzz (x, 0, t ) D z (x, 0, t ) = 0 , 0 ≤ x ≤ d, u x (x, 0, t ) = u xE (x, 0, t ) u z (x, 0, t ) = u zE (x, 0, t )
(6)
σEzx (x, −h 2 , t ) = 0
} ,
0 ≤ x ≤ d,
(7)
σzx (d , z, t ) = 0 D x (d , z, t ) = 0 , ∂ u x (d , z, t ) = 0 ∂z
0 ≤ z ≤ h1 ,
(8)
σEzx (d , z, t ) = 0 , ∂ E u x (d , z, t ) = 0 ∂z
−h 2 ≤ z ≤ 0,
(9)
σEzz (x, −h 2 , t ) = 0
where u x (x, z, t ), u xE (x, z, t ), u z (x, z, t ), u zE (x, z, t ) are the displacement components, σxx (x, z, t ), σzz (x, z, t ), σEzz (x, z, t ), σzx (x, z, t ), σEzx (x, z, t ) are the stress components, D x (x, z, t ), D z (x, z, t ) are the electric displacement components, and the Eqs. (8) and (9) indicate the periodical boundary conditions. The general solutions of the governing equations for the FGPM strip are obtained by using the Fourier integral transform technique [4] u x (x, z, t ) =
6 ∫ i∑
π j =1
+
∞ −∞
6 ∑ ∞ ∑
( ) |s| a 1 j A 1 j exp |s| γ1 j x exp (−isz) ds s
( ) S n a 2 j n A 2 j n exp S n γ2 j n z sin (S n x) ,
j =1 n=1 6 1 ∑ u z (x, z, t ) = 2π j =1
+
∫
∞ −∞
6 ∑ ∞ ∑
( ) A 1 j exp |s| γ1 j x exp (−isz) ds
( ) S n A 2 j n exp S n γ2 j n z cos (S n x) ,
j =1 n=1 6 1 ∑ ϕ (x, z, t ) = 2π j =1
−
∫
6 ∑ ∞ ∑
∞ −∞
( ) b 1 j A 1 j exp |s| γ1 j x exp (−isz) ds
( ) S n b 2 j n A 2 j n exp S n γ2 j n z cos (S n x) ,
j =1 n=1
0 ≤ z ≤ h1 ,
(10) (
)
where A 1 j , A 2 j n j = 1 ∼ 6, n = 1, 2, ... are unknown functions to be ( ) solved, and S n = n π/d , γ1 j , γ2 j n, a1 j , a2 j n , b1 j , b2 j n j = 1 ∼ 6, n = 1, 2, ... are known functions. On the other hand, the general solutions of the governing equations for the elastic layer are
Y. Mabuchi and S. Ueda / Theoretical & Applied Mechanics Letters 9 (2019) 289-292 ∞ ∑
u (x, z, t ) = E x
[ Sn
n=1 ∞ ∑
u zE (x, z, t ) =
[ Sn
{B 1n + S n zB 2n } exp(−S n z) + {B 3n + S n zB 4n } exp(S n z)
]
z
sin(S n x),
{B 1n + (S n z + 3 − 4ν) B 2n } exp(−S n z)
h
−h2
0 < a < b < h1 ,
(12)
z→b
= Z ∞ exp(βb)(πc)1/2 Φ(1, t ),
0 < a < b < h1 ,
(13)
0.06 0.04 0.02
1/2
,
(14)
0.02
0.04
ℜ[Z ] ∞
σT0 (z, t ) , a < z < b,
0.08
0.1
0.08
(15)
In the integral equation, Mi (ξ, t ) (i = 1, 2, 3, 4) are known kernel functions obtained by using the boundary conditions. For the numerical calculations, the properties of cadmium selenide [7] are used as the properties of the FGPM strip at the plane z = 0. The normalized nonhomogeneous parameters βh1, ωh 1 and δh 1 are assumed to be βh 1 = ωh 1 = δh 1. The number of the terms for obtaining the highly accurate values of the infinite series and infinite integral solutions is depends on the geometric parameters. Then all numerical calculations are preformed to keep the relative errors smaller than 1.0 × 10−5. We consider the effect of the crack spacing d /h1 and the nonhomogeneous parameter βh1 on the stress intensity factors K IA and K IB for the Ti (titanium) elastic layer. Figure 2 shows the normalized stress intensity factors [K IA , K IB ]/λ110 |T0 |(πc)1/2 versus normalized time F = τ0 t /h 2 for d /h 1 = 0.5, 0.2 and 0.1 with c/h 1 = 0.1, h 2 /h 1 = 1.0, (a + b)/2h 1 = 0.5 and βh1 = 0.0 . Assume the top surface of the strip is heated from initial temperature TI to TI + T0(T0 > 0) suddenly. Note that the values of those intensity factors rise sharply at first, reach maximum values and then decrease and approach the static values with increasing F . Because the change of the thermal stress with F near the free boundary is more remarkable than that inside of
0.06 F
Fig. 2. The effect of the crack spacing d=h1 on the stress intensity factors K IA and K IB for T0 > 0
] 4 ∑ 1 + M i (ξ, z) dξ ξ − z i =1
π
Ti
: KIA : KIB 0
[KIA, KIB]/λ110|T0|(πc)1/2
(1 + u) (1 − u)
= exp(−βz)
T0 > 0.0 (a+b)/2h = 0.5 c/h1 = 0.1 βh1 = 0.0 h2/h1 = 1.0 d/h1 = 0.5 d/h1 = 0.2 d/h1 = 0.1
1/2
where ξ = (b − a)u/2 + (b + a)/2. The function G(ξ, t ) is the solution of the following singular integral equation obtained from the boundary conditions Eqs. (2)–(9).
a
0.08
0
where Z ∞ is known constant and the function Φ(u, t ) are given by Φ(u, t )
TI
K IB = lim+ [2π(z − b)]1/2 σxx (0, z, t )
[
x
Fig. 1. Geometry of the crack problem in a functionally graded piezoelectric laminate
z→a
= − Z ∞ exp(βa)(πc)1/2 Φ(−1, t ),
G(ξ, t )
Elastic
K IA = lim− [2π(a − z)]1/2 σxx (0, z, t )
b
0
)
[KIA, KIB]/λ110|T0|(πc)1/2
(
∫
a
(11)
where B j n j = 1 ∼ 4, n = 1, 2... are the unknown functions to be solved, and ν denotes the Poisson's ratio. Substituting the displacements and electric potential solutions Eqs. (10) and (11) into the constitutive equations, one can obtain the stresses and electric displacement components. The stress intensity factors K IA at z = a and K IB at z = b may be evaluated as
G(ξ, t ) =
FGPM
b
2c
· cos(S n x), −h 2 ≤ z ≤ 0,
h1 TI + T0H (t)
2d
]
− {B 3n + (S n z − 3 + 4ν) B 4n } exp(S n z)
n=1
291
T0 > 0.0 (a+b)/2h = 0.5 c/h1 = 0.1 d/h1 = 0.2 h2/h1 = 1.0 β/h1 = 1.0 β/h1 = 0.0 β/h1 =1.0
0.06 0.04 0.02
Ti
: KIA : KIB
0 0
0.02
0.04
0.06
0.08
0.1
F
Fig. 3. The effect of the nonhomogeneous parameter ¯h1 on the stress intensity factors K IA and K IB for T0 > 0
the FGPM layer [2], the curves of K IB are later to reach stable values than those of K IA. Under heating thermal load, the stress intensity factors decrease monotonically with decreasing d /h1. Figure 3 shows the effect of βh1 on the time dependencies of the normalized stress intensity factors [K IA , K IB ]/λ110 |T0 |(πc)1/2 for βh 1 = 1.0 , 0.0 and −1.0 with c/h 1 = 0.1, h 2 /h 1 = 1.0, (a + b)/2h 1 = 0.5 and d /h1 = 0.2. In this case, the decrease of βh1 is beneficial for decreasing the maximum values of the stress intensity factors.
References [1] A. Hall, M. Allahverdi, E.K. Akdogan, et al., Piezoelectric/electrostrictive multimaterial PMN-PT monomorph actuators, Journal of the European Ceramic Society 25 (2005) 2991–2997.
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Y. Mabuchi and S. Ueda / Theoretical & Applied Mechanics Letters 9 (2019) 289-292
[2] S. Ueda, M. Okada, Y. Nakaue, Transient thermal response of a
Theory of Elasticity, John Wiley & Sons, Inc., New York (1969).
functionally graded piezoelectric laminate with a crack normal to the bimaterial interface, Journal of Thermal Stresses 41 (2018) 98–118. [3] S. Ueda, Y. Ashida, Infinite row of parallel cracks in a functionally graded piezoelectric material strip under mechanical and transient thermal loading conditions, Journal of Thermal Stresses 32 (2009) 1103–1125. [4] I.N. Sneddon, M. Lowengrub, Crack Problems in the Classical
[5] F. Erdogan, B.H. Wu, Crack problems in FGM layers under thermal stresses, Journal of Thermal Stresses 19 (1996) 237–265.
[6] F. Erdogan, G.D. Gupta, T.S. Cook, Methods of Analysis and Solution of Crack Problems. (Edited by G.C.Sih), Noordhoff, Leyden (1972), pp. 368. [7] F. Ashida, T.R. Tauchert, Transient response of a piezothermoelastic circular disk under axisymmetric heating, Acta Mechanica 128 (1998) 1–14.
Contents lists available at ScienceDirect
Theoretical & Applied Mechanics Letters journal homepage: www.elsevier.com/locate/taml
Letter
Transient thermal response of functionally graded piezoelectric laminates with an infinite row of parallel cracks normal to the bimaterial interface Yoshiyuki Mabuchi, Sei Ueda* Department of Mechanical Engineering, Osaka Institute of Technology, Osaka, Japan
H I G H L I G H T S
• We focused on the transient thermal fracture problem of monomorph actuators using a functionally graded piezoelectric material strip (FGPM strip). • Using the Fourier transform technique, the electromechanical problem is reduced to a singular integral equation, which is solved numerically. • The stress intensity factors of the embedded crack are computed and presented as functions of the normalized time for the various values of the nonhomogeneous and geometric parameters. A R T I C L E I N F O
A B S T R A C T
Article history: Received 2 July 2019 Received in revised form 13 September 2019 Accepted 16 September 2019 Available online 30 September 2019
Keywords: Functionally graded piezoelectric material Fracture mechanics Stress intensity factor Elasticity
In this paper, the problem of a functionally graded piezoelectric material strip (FGPM strip) containing an infinite row of parallel cracks perpendicular to the interface between the FGPM strip and a homogeneous layer is analyzed under transient thermal loading condition. The crack faces are supposed to be completely insulated. Material properties are assumed to be exponentially dependent on the distance from the interface. Using the Fourier transforms, the electro-thermoelastic problem is reduced to a singular integral equation, which is solved numerically. The stress intensity factors are computed and presented as a function of the normalized time, the nonhomogeneous and geometric parameters. ©2019 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Recently, several kinds of piezoelectric actuators have been designed. Hall et al. [1] fabricated a mono-morph actuator made from semi-conductive piezoelectric ceramics. Therefore, the elucidation of fracture behavior of piezoelectric systems such as actuators is important. Moreover, the research on transient electro-thermo-elastic problem for functionally graded piezoelectric material (FGPM) laminate with a crack normal to the biomaterial interface has been reported [2]. On the other hand, cracks occur primarily from plural defects in material. The elucidation of interaction between cracks to affect fracture behavior is also important, because these cracks may lead to fracture of material [3].
* Corresponding author. E-mail address: [email protected] (S. Ueda).
In this paper, we theoretically analyzed the thermal fracture problem of mono-morph actuators using an FGPM strip with an infinite row of parallel cracks due to an instantaneous temperature change. The analytical laminate model of the mono-morph actuator consists of an FGPM strip and a homogeneous elastic layer. Material properties are exponentially dependent on the distance from the interface between the FGPM strip and the homogeneous elastic layer. Using the Fourier transform technique [4, 5], the electro-thermo-elastic problem is reduced to a singular integral equation, and the Gauss-Jacobi numerical integration formula was used in the numerical analysis of the singular integral equation [6]. The stress intensity factors of the embedded crack are computed and presented as functions of the normalized time for the various values of the nonhomogeneous and geometric parameters.
http://dx.doi.org/10.1016/j.taml.2019.06.009 2095-0349/© 2019 The Authors. Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
290
Y. Mabuchi and S. Ueda / Theoretical & Applied Mechanics Letters 9 (2019) 289-292
As shown in Fig. 1, suppose an FGPM strip with the thickness h1 containing an infinite row of parallel cracks of equal length 2c = b − a (0 < a < b < h1 ) being spaced at equal distance 2d perpendicular to the free boundaries bonded to an elastic layer with the thickness h2. The system of rectangular Cartesian coordinates (x, y, z) is introduced in the material in such a way that one of the crack is located along the z -axis, and the x -axis is parallel to the boundaries. The FGPM strip is poled in the z -direction and is in the plane strain conditions perpendicular to the y axis. It is assumed that initially the medium is at the uniform temperature TI and is suddenly subjected to a uniform temperature rise T0 H (t ) along the boundary z = h1, where H (t ) is the Heaviside step function and t denotes time. The temperature along the boundary z = h2 is maintained at TI . The crack faces remain thermally and electrically insulated. The material property parameters are taken to vary continuously along the z-direction inside the FGPM strip. The material properties of FGPM strip, such as the elastic stiffness constants c kl (z), piezoelectric constants e kl (z), dielectric constants εkk (z), stress -temperature coefficients λkk (z), coefficient of heat conduction κx (z), κz (z), and the pyroelectric constant p z (z), are one dimensionally dependent as [c kl (z) , e kl (z) , εkk (z)] = (c kl 0 , e kl 0 , εkk0 ) exp(βz), [ ] λkk (z) , p z (z) = (λkk0 , p z0 ) exp[(β + ω)z], [κx (z) , κz (z)] = (κx0 , κz0 ) exp(δz),
(1)
where β, ω, and δ are positive or negative constants, and the subscript 0 indicates the properties at the interface plane z = 0. For some materials, the thermal diffusivity τ0 indeed does not vary dramatically, then τ0 is assumed to be a constant. The material properties of the homogeneous elastic layer are the elastic stiffness constants c klE , stress-temperature coefficients λEkk , coefficient of heat conduction κE , and thermal diffusivity τE0 . The superscript E denotes the physical quantities of the homogeneous elastic layer. The crack problem may be solved using the superposition technique. In the problem considered here, since the heat conduction is one dimensional and straight crack does not obstruct the heat flow in this arrangement, determination of the temperature distribution and the resulting thermal stress would be quite straightforward and the related crack problem would be one of mode I. The lines of the centers of the cracks x = ±2nd (n = 0, 1, 2, ...) are the axes of symmetry of the configuration. We suppose that the crack is opened under the action of same distribution of the internal pressure σT0 (z, t ), where σT0 (z, t ) is the thermal stress induced by the time-dependent temperature change. The thermal stress σT0 (z, t ) has been already obtained in Ref. [2]. In the following, the subscripts x, y, z will be used to refer to the direction of coordinates. The governing equations for the electromechanical fields are given in previous paper [2]. The boundary conditions can be written as σxx (0, z, t ) = −σT0 (z, t ) , a < z < b, u x (0, z, t ) = 0, 0 ≤ z ≤ a, b ≤ z ≤ h 1 ,
(2)
} σzx (0, z, t ) = 0 , 0 ≤ z ≤ h1 , D x (0, z, t ) = 0
(3)
σEzx (0, z, t ) = 0 u xE (0, z, t ) = 0
} , −h 2 ≤ z ≤ 0,
(4)
σzx (x, h 1 , t ) = 0 σzz (x, h 1 , t ) = 0 , 0 ≤ x ≤ d , D z (x, h 1 , t ) = 0
(5)
σzx (x, 0, t ) = σEzx (x, 0, t ) σzz (x, 0, t ) = σEzz (x, 0, t ) D z (x, 0, t ) = 0 , 0 ≤ x ≤ d, u x (x, 0, t ) = u xE (x, 0, t ) u z (x, 0, t ) = u zE (x, 0, t )
(6)
σEzx (x, −h 2 , t ) = 0
} ,
0 ≤ x ≤ d,
(7)
σzx (d , z, t ) = 0 D x (d , z, t ) = 0 , ∂ u x (d , z, t ) = 0 ∂z
0 ≤ z ≤ h1 ,
(8)
σEzx (d , z, t ) = 0 , ∂ E u x (d , z, t ) = 0 ∂z
−h 2 ≤ z ≤ 0,
(9)
σEzz (x, −h 2 , t ) = 0
where u x (x, z, t ), u xE (x, z, t ), u z (x, z, t ), u zE (x, z, t ) are the displacement components, σxx (x, z, t ), σzz (x, z, t ), σEzz (x, z, t ), σzx (x, z, t ), σEzx (x, z, t ) are the stress components, D x (x, z, t ), D z (x, z, t ) are the electric displacement components, and the Eqs. (8) and (9) indicate the periodical boundary conditions. The general solutions of the governing equations for the FGPM strip are obtained by using the Fourier integral transform technique [4] u x (x, z, t ) =
6 ∫ i∑
π j =1
+
∞ −∞
6 ∑ ∞ ∑
( ) |s| a 1 j A 1 j exp |s| γ1 j x exp (−isz) ds s
( ) S n a 2 j n A 2 j n exp S n γ2 j n z sin (S n x) ,
j =1 n=1 6 1 ∑ u z (x, z, t ) = 2π j =1
+
∫
∞ −∞
6 ∑ ∞ ∑
( ) A 1 j exp |s| γ1 j x exp (−isz) ds
( ) S n A 2 j n exp S n γ2 j n z cos (S n x) ,
j =1 n=1 6 1 ∑ ϕ (x, z, t ) = 2π j =1
−
∫
6 ∑ ∞ ∑
∞ −∞
( ) b 1 j A 1 j exp |s| γ1 j x exp (−isz) ds
( ) S n b 2 j n A 2 j n exp S n γ2 j n z cos (S n x) ,
j =1 n=1
0 ≤ z ≤ h1 ,
(10) (
)
where A 1 j , A 2 j n j = 1 ∼ 6, n = 1, 2, ... are unknown functions to be ( ) solved, and S n = n π/d , γ1 j , γ2 j n, a1 j , a2 j n , b1 j , b2 j n j = 1 ∼ 6, n = 1, 2, ... are known functions. On the other hand, the general solutions of the governing equations for the elastic layer are
Y. Mabuchi and S. Ueda / Theoretical & Applied Mechanics Letters 9 (2019) 289-292 ∞ ∑
u (x, z, t ) = E x
[ Sn
n=1 ∞ ∑
u zE (x, z, t ) =
[ Sn
{B 1n + S n zB 2n } exp(−S n z) + {B 3n + S n zB 4n } exp(S n z)
]
z
sin(S n x),
{B 1n + (S n z + 3 − 4ν) B 2n } exp(−S n z)
h
−h2
0 < a < b < h1 ,
(12)
z→b
= Z ∞ exp(βb)(πc)1/2 Φ(1, t ),
0 < a < b < h1 ,
(13)
0.06 0.04 0.02
1/2
,
(14)
0.02
0.04
ℜ[Z ] ∞
σT0 (z, t ) , a < z < b,
0.08
0.1
0.08
(15)
In the integral equation, Mi (ξ, t ) (i = 1, 2, 3, 4) are known kernel functions obtained by using the boundary conditions. For the numerical calculations, the properties of cadmium selenide [7] are used as the properties of the FGPM strip at the plane z = 0. The normalized nonhomogeneous parameters βh1, ωh 1 and δh 1 are assumed to be βh 1 = ωh 1 = δh 1. The number of the terms for obtaining the highly accurate values of the infinite series and infinite integral solutions is depends on the geometric parameters. Then all numerical calculations are preformed to keep the relative errors smaller than 1.0 × 10−5. We consider the effect of the crack spacing d /h1 and the nonhomogeneous parameter βh1 on the stress intensity factors K IA and K IB for the Ti (titanium) elastic layer. Figure 2 shows the normalized stress intensity factors [K IA , K IB ]/λ110 |T0 |(πc)1/2 versus normalized time F = τ0 t /h 2 for d /h 1 = 0.5, 0.2 and 0.1 with c/h 1 = 0.1, h 2 /h 1 = 1.0, (a + b)/2h 1 = 0.5 and βh1 = 0.0 . Assume the top surface of the strip is heated from initial temperature TI to TI + T0(T0 > 0) suddenly. Note that the values of those intensity factors rise sharply at first, reach maximum values and then decrease and approach the static values with increasing F . Because the change of the thermal stress with F near the free boundary is more remarkable than that inside of
0.06 F
Fig. 2. The effect of the crack spacing d=h1 on the stress intensity factors K IA and K IB for T0 > 0
] 4 ∑ 1 + M i (ξ, z) dξ ξ − z i =1
π
Ti
: KIA : KIB 0
[KIA, KIB]/λ110|T0|(πc)1/2
(1 + u) (1 − u)
= exp(−βz)
T0 > 0.0 (a+b)/2h = 0.5 c/h1 = 0.1 βh1 = 0.0 h2/h1 = 1.0 d/h1 = 0.5 d/h1 = 0.2 d/h1 = 0.1
1/2
where ξ = (b − a)u/2 + (b + a)/2. The function G(ξ, t ) is the solution of the following singular integral equation obtained from the boundary conditions Eqs. (2)–(9).
a
0.08
0
where Z ∞ is known constant and the function Φ(u, t ) are given by Φ(u, t )
TI
K IB = lim+ [2π(z − b)]1/2 σxx (0, z, t )
[
x
Fig. 1. Geometry of the crack problem in a functionally graded piezoelectric laminate
z→a
= − Z ∞ exp(βa)(πc)1/2 Φ(−1, t ),
G(ξ, t )
Elastic
K IA = lim− [2π(a − z)]1/2 σxx (0, z, t )
b
0
)
[KIA, KIB]/λ110|T0|(πc)1/2
(
∫
a
(11)
where B j n j = 1 ∼ 4, n = 1, 2... are the unknown functions to be solved, and ν denotes the Poisson's ratio. Substituting the displacements and electric potential solutions Eqs. (10) and (11) into the constitutive equations, one can obtain the stresses and electric displacement components. The stress intensity factors K IA at z = a and K IB at z = b may be evaluated as
G(ξ, t ) =
FGPM
b
2c
· cos(S n x), −h 2 ≤ z ≤ 0,
h1 TI + T0H (t)
2d
]
− {B 3n + (S n z − 3 + 4ν) B 4n } exp(S n z)
n=1
291
T0 > 0.0 (a+b)/2h = 0.5 c/h1 = 0.1 d/h1 = 0.2 h2/h1 = 1.0 β/h1 = 1.0 β/h1 = 0.0 β/h1 =1.0
0.06 0.04 0.02
Ti
: KIA : KIB
0 0
0.02
0.04
0.06
0.08
0.1
F
Fig. 3. The effect of the nonhomogeneous parameter ¯h1 on the stress intensity factors K IA and K IB for T0 > 0
the FGPM layer [2], the curves of K IB are later to reach stable values than those of K IA. Under heating thermal load, the stress intensity factors decrease monotonically with decreasing d /h1. Figure 3 shows the effect of βh1 on the time dependencies of the normalized stress intensity factors [K IA , K IB ]/λ110 |T0 |(πc)1/2 for βh 1 = 1.0 , 0.0 and −1.0 with c/h 1 = 0.1, h 2 /h 1 = 1.0, (a + b)/2h 1 = 0.5 and d /h1 = 0.2. In this case, the decrease of βh1 is beneficial for decreasing the maximum values of the stress intensity factors.
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Theory of Elasticity, John Wiley & Sons, Inc., New York (1969).
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