Non-linear mechanical, electrical and thermal phenomena in piezoelectric crystals
Mechanical, electrical and thermal phenomena occurring in piezoelectric crystals were examined by non-linear approximation. For this purpose, use was made of the thermodynamic function of state, which describes an anisotropic body. Considered was the Gibbs function. The calculations included str...
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irk-123456789-1207162017-06-13T03:03:11Z Non-linear mechanical, electrical and thermal phenomena in piezoelectric crystals Warkusz, F. Linek, A. Mechanical, electrical and thermal phenomena occurring in piezoelectric crystals were examined by non-linear approximation. For this purpose, use was made of the thermodynamic function of state, which describes an anisotropic body. Considered was the Gibbs function. The calculations included strain tensor εij = f(σkl , En, T), induction vector Dm = f(σkl , En, T) and entropy S = f(σkl , En, T) as function of stress σkl , field strength En and temperature difference T. The equations obtained apply to anisotropic piezoelectric bodies provided that the “forces” σkl , En, T acting on the crystal are known. Механічні, електричні та термічні явища у п’єзоелектричних кристалах вивчаються у нелінійному наближенні. З цією метою використано термодинамічний потенціал, який описує анізотропне тіло. Розглянуто потенціал Гіббса. Розрахунки охоплюють тензор деформації εij = f(σkl , En, T), вектор індукції Dm = f(σkl , En, T) та ентропію S = f(σkl , En, T) як функцію механічного напруження σkl , величини поля En і різниці температур T. Отримано рівняння, які описують анізотропні п’єзоелектричні тіла, якщо відомі “сили” σkl , En, T, що діють на кристал. 2003 Article Non-linear mechanical, electrical and thermal phenomena in piezoelectric crystals / F. Warkusz, A. Linek // Condensed Matter Physics. — 2003. — Т. 6, № 2(34). — С. 333-345. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 65.40.-b, 77.65.-j, 77.65.Bn, 77.65.Ly, 05.70.Ce DOI:10.5488/CMP.6.2.333 http://dspace.nbuv.gov.ua/handle/123456789/120716 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
Mechanical, electrical and thermal phenomena occurring in piezoelectric
crystals were examined by non-linear approximation. For this purpose,
use was made of the thermodynamic function of state, which describes
an anisotropic body. Considered was the Gibbs function. The calculations
included strain tensor εij = f(σkl
, En, T), induction vector Dm =
f(σkl
, En, T) and entropy S = f(σkl
, En, T) as function of stress σkl
, field
strength En and temperature difference T. The equations obtained apply to
anisotropic piezoelectric bodies provided that the “forces” σkl
, En, T acting
on the crystal are known. |
| format |
Article |
| author |
Warkusz, F. Linek, A. |
| spellingShingle |
Warkusz, F. Linek, A. Non-linear mechanical, electrical and thermal phenomena in piezoelectric crystals Condensed Matter Physics |
| author_facet |
Warkusz, F. Linek, A. |
| author_sort |
Warkusz, F. |
| title |
Non-linear mechanical, electrical and thermal phenomena in piezoelectric crystals |
| title_short |
Non-linear mechanical, electrical and thermal phenomena in piezoelectric crystals |
| title_full |
Non-linear mechanical, electrical and thermal phenomena in piezoelectric crystals |
| title_fullStr |
Non-linear mechanical, electrical and thermal phenomena in piezoelectric crystals |
| title_full_unstemmed |
Non-linear mechanical, electrical and thermal phenomena in piezoelectric crystals |
| title_sort |
non-linear mechanical, electrical and thermal phenomena in piezoelectric crystals |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120716 |
| citation_txt |
Non-linear mechanical, electrical and
thermal phenomena in piezoelectric
crystals / F. Warkusz, A. Linek // Condensed Matter Physics. — 2003. — Т. 6, № 2(34). — С. 333-345. — Бібліогр.: 16 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT warkuszf nonlinearmechanicalelectricalandthermalphenomenainpiezoelectriccrystals AT lineka nonlinearmechanicalelectricalandthermalphenomenainpiezoelectriccrystals |
| first_indexed |
2025-07-08T18:27:28Z |
| last_indexed |
2025-07-08T18:27:28Z |
| _version_ |
1837104365954924544 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 2(34), pp. 333–345
Non-linear mechanical, electrical and
thermal phenomena in piezoelectric
crystals
F.Warkusz, A.Linek
Institute of Physics, Opole University, 45–052 Opole, Poland
Received August 25, 2002, in final form March 31, 2003
Mechanical, electrical and thermal phenomena occurring in piezoelectric
crystals were examined by non-linear approximation. For this purpose,
use was made of the thermodynamic function of state, which describes
an anisotropic body. Considered was the Gibbs function. The calcula-
tions included strain tensor εij = f(σkl, En, T ), induction vector Dm =
f(σkl, En, T ) and entropy S = f(σkl, En, T ) as function of stress σkl, field
strength En and temperature difference T . The equations obtained apply to
anisotropic piezoelectric bodies provided that the “forces” σkl, En, T acting
on the crystal are known.
Key words: piezoelectric crystals, thermodynamics, anisotropic bodies,
tensors
PACS: 65.40.-b, 77.65.-j, 77.65.Bn, 77.65.Ly, 05.70.Ce
1. Introduction
The present paper provides a thermodynamic description of elastic (mechanical),
electrical and thermal properties of crystals in a non-linear approximation. The
physical processes and phenomena that occur in a real crystal generally are of a
non-linear character, similarly to those in any other physical body which displays
a certain degree of nonlinearity. It is conventional to describe non-linear properties
(i.e., physical nonlinearity) in terms of higher-order material constants, and such
are the factors of proportionality incorporated in the Taylor series expansion of a
non-linear relation describing the given effect. In general, the mentioned constants
are tensors.
Nonlinearity manifests in the values of the second-order material constants being
the functions of the values of the applied forces which act onto the crystal, e.g. the
elasticity constants can be a function of the applied stress σ. Thus, the standard
Hooke equation may be written as follows:
ε = s(σ)σ, (1.1)
c© F.Warkusz, A.Linek 333
F.Warkusz, A.Linek
where ε denotes strain and s(σ) is an elasticity constant which varies according to
the stress σ applied. Hence, s(σ) takes the form
s(σ) = s1 + s2σ + s3σ
2 + . . . , (1.2)
where s1 is a second-order material constant (modulus of elasticity).
By linear approximation, equation (1.1) becomes as follows: ε = s1σ, where
s1 = ∂ε/∂σ.
Using non-linear approximation, we have
ε =
∂ε
∂σ
σ +
1
2!
∂2ε
∂σ2
σ2 +
1
3!
∂3ε
∂σ3
σ3 + . . . ,
where
s1 =
∂ε
∂σ
, s2 =
1
2!
∂2ε
∂σ2
, s3 =
1
3!
∂3ε
∂σ3
.
The coefficients s1, s2, s3, . . . of equation (1.2) can be regarded as material con-
stants which are proportional to the stresses (σ) of relevant powers. Hence, they are
material constants of second, third, fourth or higher order. Thus, the Hooke equation
for non-linear effects can be written as [1,2]
ε = s(σ)σ = s1σ + s2σ
2 + s3σ
3 + . . . or ε = (s1 + s2σ + s3σ
2 + . . .)σ.
With tensor notation for anisotropic bodies, the relation of
s = s1 + s2σ + s3σ
2 + . . .
takes the following form:
s = sijkl + sijklpqσpq + sijklpqrsσpqσrs + . . .
and the Hooke equation
ε = s(σ)σ = s1σ + s2σ
2 + s3σ
3 + . . .
becomes
εij = sijklσkl + sijklpqσpqσkl + sijklpqrsσpqσrsσkl + . . .
in tensor notation, where sijkl is a fourth-order tensor (second-order material con-
stant), sijklpq is a sixth-order tensor (third-order material constant), sijklpqrs is an
eight-order tensor (fourth-order material constant), and the coefficients i, j, k, l, p,
q, r, s take the values of 1, 2, 3. Summation signs have been omitted [1,3,4].
Higher-order material constants are formally derived from thermodynamic func-
tions. The procedure is similar to that for the linear case. If, for example, strain ε is
produced simultaneously by stress σ, by electric field E, and temperature variations
T , the equation of state for non-linear processes, which includes material constants
only up to the third order, can be written as
ε = s1σ + d1E + α1T + s2σ
2 + d2E
2 + α2T
2 + kσEσE + kσT σT + kETET,
334
Non-linear phenomena in piezoelectric crystals
where
s1 =
∂ε
∂σ
, d1 =
∂ε
∂E
, α1 =
∂ε
∂T
, s2 =
1
2
∂2ε
∂σ2
, d2 =
1
2
∂2ε
∂E2
,
α2 =
1
2
∂2ε
∂T 2
, kσE =
∂2ε
∂σ∂E
, kσT =
∂2ε
∂σ∂T
, kET =
∂2ε
∂E∂T
.
Apart from second-order material constants, s1, d1, α1, and third-order material
constants s2, d2, α2, there are also third-order “mixed” material constants kσE, kσT
and kET . The order of the material constant is defined by the order of the derivative
of the thermodynamic function and not by the order of the “force” acting onto the
crystal or by the order of the material constant tensor.
The problem will be presented more in detail in section 2.
2. Thermodynamic relations in crystals and Gibbs functions
According to the first law of thermodynamics, the total energy U of a body is the
sum of different energy types. In piezoelectric crystals, energy balance is primarily
accounted for by mechanical, electrical and thermal energy. The effect of magnetic
energy or gravitational energy on the phenomena occurring in piezoelectric crystals
may be neglected. In general, there are eight thermodynamic functions that can be
used to describe a piezoelectric phenomenon. The form of the function depends on
the choice of the independent variables, selected according to the conditions under
which the crystal is to be examined [2].
In the present paper we confine ourselves to thermodynamics, that is, Gibbs
function G. Mathematical analysis enables us to define particular material constants,
as well as to establish many interesting relations between them, by linear and non-
linear approximation.
G = U − εijσij − EmDm − TS, (2.1)
where i, j, m = 1, 2, 3.
Let us consider the differential form dG, which describes the state of the crystal
following the application of three different “forces” – stress σij, electric field Em,
and temperature T . The forces acting onto the crystal are independent variables.
Thus, we have
dG = dU − εijdσij − σijdεij − EmdDm − DmdEm − SdT − TdS. (2.2)
By virtue of the first and second law of thermodynamics,
dU = σijdεij + EmdDm + TdS, (2.3)
where i, j, m = 1, 2, 3.
Substituting (2.3) into (2.2), we obtain
dG = −εijdσij − DmdEm − SdT. (2.4)
335
F.Warkusz, A.Linek
Strain εij, induction Dm and entropy S are functions of stress σkl, electric field
En and temperature T :
εij = f(σkl, En, T ), Dm = f(σkl, En, T ), S = f(σkl, En, T ).
Differentiating the function G with respect to individual independent variables,
and having determined the remaining values, we obtain
dG =
(
∂G
∂σij
)
E,T
dσij +
(
∂G
∂Em
)
σ,T
dEm +
(
∂G
∂T
)
σ,E
dT. (2.5)
Comparing the coefficients of (2.4) and (2.5), we can derive the relations that
describe particular quantities:
εij = −
∂G
∂σij
∣
∣
∣
∣
∣
E,T
, Dm = −
∂G
∂Em
∣
∣
∣
∣
∣
σ,T
, S = −
∂G
∂T
∣
∣
∣
∣
∣
σ,E
.
The indices in the lower part of the vertical line show the independent variable,
which takes a constant value during differentiation.
The function εij, Dm and S can be expanded into a Taylor series. All derivatives
incorporated in the Taylor’s series are the material constants.
2.1. Material constants derived from function εij
Second-order elasticity constant
∂εij
∂σkl
∣
∣
∣
∣
∣
E,T
=
∂
∂σkl
(
−
∂G
∂σij
)
= −
∂2G
∂σkl∂σij
∣
∣
∣
∣
∣
E,T
= sE,T
ijkl .
Second-order piezoelectric constant
∂εij
∂En
∣
∣
∣
∣
∣
σ,T
=
∂
∂En
(
−
∂G
∂σij
)
= −
∂2G
∂En∂σij
∣
∣
∣
∣
∣
T
= dT
ijn .
Second-order piezocalorific constant
∂εij
∂T
∣
∣
∣
∣
∣
σ,E
=
∂
∂T
(
−
∂G
∂σij
)
= −
∂2G
∂T∂σij
∣
∣
∣
∣
∣
E
= αE
ij .
Third-order elasticity constant
∂2εij
∂σkl∂σpq
∣
∣
∣
∣
∣
E,T
=
∂2
∂σkl∂σpq
(
−
∂G
∂σij
)
= −
∂3G
∂σkl∂σpq∂σij
∣
∣
∣
∣
∣
E,T
= sE,T
ijklpq .
Third-order piezoelectric constant
∂2εij
∂σkl∂En
∣
∣
∣
∣
∣
T
=
∂2
∂σkl∂En
(
−
∂G
∂σij
)
= −
∂3G
∂σkl∂En∂σij
∣
∣
∣
∣
∣
T
= dT
ijkln .
336
Non-linear phenomena in piezoelectric crystals
Third-order piezocalorific constant
∂2εij
∂σkl∂T
∣
∣
∣
∣
∣
E
=
∂2
∂σkl∂T
(
−
∂G
∂σij
)
= −
∂3G
∂σkl∂T∂σij
∣
∣
∣
∣
∣
E
= αE
ijkl .
Third-order electrostriction constant
∂2εij
∂En∂Et
∣
∣
∣
∣
∣
σ,T
=
∂2
∂En∂Et
(
−
∂G
∂σij
)
= −
∂3G
∂En∂Et∂σij
∣
∣
∣
∣
∣
T
= oT
ijnt .
Third-order constant
∂2εij
∂En∂T
∣
∣
∣
∣
∣
σ
=
∂2
∂En∂T
(
−
∂G
∂σij
)
= −
∂3
∂En∂T∂σij
= kijn .
Third-order thermal expansion constant
∂2εij
∂T∂T1
∣
∣
∣
∣
∣
σ,E
=
∂2
∂T∂T1
(
−
∂G
∂σij
)
= −
∂3G
∂T∂T1∂σij
∣
∣
∣
∣
∣
E
= rE
ij .
2.2. Material constants derived from function Dm
Second-order piezoelectric constant
∂Dm
∂σkl
∣
∣
∣
∣
∣
E,T
=
∂
∂σkl
(
−
∂G
∂Em
)
= −
∂2G
∂σkl∂Em
∣
∣
∣
∣
∣
T
= dT
klm .
Second-order permittivity constant
∂Dm
∂En
∣
∣
∣
∣
∣
σ,T
=
∂
∂En
(
−
∂G
∂Em
)
= −
∂2G
∂En∂Em
∣
∣
∣
∣
∣
σ,T
= εσ,T
mn .
Second-order electric heat constant
∂Dm
∂T
∣
∣
∣
∣
∣
σ,E
=
∂
∂T
(
−
∂G
∂Em
)
= −
∂2G
∂T∂Em
∣
∣
∣
∣
∣
σ
= pσ
m .
Third-order piezoelectric constant
∂2Dm
∂σkl∂σpq
∣
∣
∣
∣
∣
E,T
=
∂2
∂σkl∂σpq
(
−
∂G
∂Em
)
= −
∂3G
∂σkl∂σpq∂Em
∣
∣
∣
∣
∣
T
= dT
klpqm .
Third-order electrostriction constant
∂2Dm
∂σkl∂En
∣
∣
∣
∣
∣
T
=
∂2
∂σkl∂En
(
−
∂G
∂Em
)
= −
∂3G
∂σkl∂En∂Em
∣
∣
∣
∣
∣
T
= oT
klmn .
Third-order constant
∂2Dm
∂σkl∂T
∣
∣
∣
∣
∣
E
=
∂2
∂σkl∂T
(
−
∂G
∂Em
)
= −
∂3
∂σkl∂T∂Em
= kklm .
337
F.Warkusz, A.Linek
Third-order permittivity constant
∂2Dm
∂En∂Et
∣
∣
∣
∣
∣
σ,T
=
∂2
∂En∂Et
(
−
∂G
∂Em
)
= −
∂3G
∂En∂Et∂Em
∣
∣
∣
∣
∣
σ,T
= εσ,T
mnt .
Third-order electric heat constant
∂2Dm
∂En∂T
∣
∣
∣
∣
∣
σ
=
∂2
∂En∂T
(
−
∂G
∂Em
)
= −
∂3G
∂En∂T∂Em
∣
∣
∣
∣
∣
σ
= pσ
mn .
Third-order pyroelectric constant
∂2Dm
∂T1∂T
∣
∣
∣
∣
∣
σ,E
=
∂2
∂T1∂T
(
−
∂G
∂Em
)
= −
∂3G
∂T1∂T∂Em
∣
∣
∣
∣
∣
σ
= uσ
m .
2.3. Material constants derived from function S
Second-order piezocalorific constant
∂S
∂σkl
∣
∣
∣
∣
∣
E,T
=
∂
∂σkl
(
−
∂G
∂T
)
= −
∂2G
∂σkl∂T
∣
∣
∣
∣
∣
E
= αE
kl .
Second-order electric heat constant
∂S
∂En
∣
∣
∣
∣
∣
σ,T
=
∂
∂En
(
−
∂G
∂T
)
= −
∂2G
∂En∂T
∣
∣
∣
∣
∣
σ
= pσ
n .
Second-order thermal capacity constant
∂S
∂T
∣
∣
∣
∣
∣
σ,E
=
∂
∂T
(
−
∂G
∂T1
)
= −
∂2G
∂T∂T1
∣
∣
∣
∣
∣
σ,E
= ξσ,E .
Third-order piezocalorific constant
∂2S
∂σkl∂σpq
∣
∣
∣
∣
∣
E,T
=
∂2
∂σkl∂σpq
(
−
∂G
∂T
)
= −
∂3G
∂σkl∂σpq∂T
∣
∣
∣
∣
∣
E
= αE
klpq .
Third-order constant
∂2S
∂σkl∂En
∣
∣
∣
∣
∣
T
=
∂2
∂σkl∂En
(
−
∂G
∂T
)
= −
∂3
∂σkl∂En∂T
= kkln .
Third-order thermal expansion constant
∂2S
∂σkl∂T
∣
∣
∣
∣
∣
E
=
∂2
∂σkl∂T
(
−
∂G
∂T1
)
= −
∂3G
∂σkl∂T∂T1
∣
∣
∣
∣
∣
E
= rE
kl .
Third-order electric heat constant
∂2S
∂En∂Et
∣
∣
∣
∣
∣
σ,T
=
∂2
∂En∂Et
(
−
∂G
∂T
)
= −
∂3G
∂En∂Et∂T
∣
∣
∣
∣
∣
σ
= pσ
nt .
338
Non-linear phenomena in piezoelectric crystals
Third-order pyroelectric constant
∂2S
∂En∂T
∣
∣
∣
∣
∣
σ
=
∂2
∂En∂T
(
−
∂G
∂T1
)
= −
∂3G
∂En∂T∂T1
∣
∣
∣
∣
∣
σ
= uσ
n .
Third-order thermal capacity constant
∂2S
∂T∂T1
∣
∣
∣
∣
∣
σ,E
=
∂2
∂T∂T1
(
−
∂G
∂T2
)
= −
∂3G
∂T∂T1∂T2
∣
∣
∣
∣
∣
σ,E
= ρσ,E .
2.4. Equalities between material constants
∂Dm
∂σkl
∣
∣
∣
∣
∣
E,T
=
∂εkl
∂Em
∣
∣
∣
∣
∣
σ,T
= dT
klm ,
∂2S
∂σkl∂σpq
∣
∣
∣
∣
∣
E,T
=
∂2εkl
∂σpq∂T
∣
∣
∣
∣
∣
E
= αE
klpq ,
∂Dm
∂σkl∂σpq
∣
∣
∣
∣
∣
E,T
=
∂2εkl
∂σpq∂Em
∣
∣
∣
∣
∣
T
= dT
klmpq ,
∂2S
∂En∂T
∣
∣
∣
∣
∣
σ
=
∂Dn
∂T∂T1
∣
∣
∣
∣
∣
σ,E
= uσ
n ,
∂S
∂En
∣
∣
∣
∣
∣
σ,T
=
∂Dn
∂T
∣
∣
∣
∣
∣
σ,E
= pσ
n ,
∂2S
∂σkl∂T
∣
∣
∣
∣
∣
E
=
∂2εkl
∂T∂T1
∣
∣
∣
∣
∣
σ,E
= rE
kl ,
∂2S
∂En∂Et
∣
∣
∣
∣
∣
σ,T
=
∂Dn
∂Et∂T
∣
∣
∣
∣
∣
σ
= pσ
nt ,
∂Dm
∂σkl∂En
∣
∣
∣
∣
∣
T
=
∂2εkl
∂En∂Em
∣
∣
∣
∣
∣
σ,T
= oT
klmn ,
∂S
∂σkl
∣
∣
∣
∣
∣
E,T
=
∂εkl
∂T
∣
∣
∣
∣
∣
σ,E
= αE
kl ,
∂2εij
∂En∂T
∣
∣
∣
∣
∣
E
=
∂2Dn
∂σij∂T
∣
∣
∣
∣
∣
E
=
∂2S
∂σij∂En
∣
∣
∣
∣
∣
T
= kijn .
With the material constants (tensors) defined above, we can rewrite function εij,
Dm and S from Taylor series into form:
εij = (sE,T
ijkl + 1
2
sE,T
ijklpqσpq + 1
2
dT
ijklnEn + 1
2
αE
ijklT )σkl
+ (dT
ijn + 1
2
dT
klijnσkl + 1
2
oT
ijntEt + 1
2
kijnT )En
+ (αE
ij + 1
2
αE
ijklσkl + 1
2
kijnEn + 1
2
rE
ijT1)T, (2.6)
Dm = (dT
mkl + 1
2
dT
mklpqσpq + 1
2
kmklT + 1
2
oT
klmnEn)σkl
+ (εT,σ
nm + 1
2
εT,σ
nmtEt + 1
2
oT
nmklσkl + 1
2
pσ
nmT )En
+ (pσ
m + 1
2
kmklσkl + 1
2
pσ
mnEn + 1
2
uσ
mT1)T, (2.7)
S = (αE
kl + 1
2
αE
klpqσpq + 1
2
kklnEn + 1
2
rE
klT )σkl
+ (pσ
n + 1
2
knklσkl + 1
2
pσ
ntEt + 1
2
uσ
nT )En
+ (ξE,σ + 1
2
rE
klσkl + 1
2
uσ
nEn + ρE,σT1)T. (2.8)
339
F.Warkusz, A.Linek
3. Linear and non-linear effects of the equations of state
Considering, for example, the equation of state derived from the Gibbs function
G for non-linear effects, we can easily obtain the equations of state for linear effects
by neglecting higher-order tensors. Thus, by virtue of (2.6), (2.7) and (2.8) we have
εij = sE,T
ijklσkl + dT
ijnEn + αE
ijT,
Dm = dT
mklσkl + εT,σ
nmEn + pσ
mT,
S = αE
klσkl + pσ
nEn + ξE,σT (3.1)
which are valid for linear effects [7,8,9].
By virtue of the symmetry of some tensors (tensor components in matrix tables),
the relations of (3.1) can be presented in matrix form.
Hooke’s law is often expressed in its contracted notation. Then, the equivalence
between the components of the compliance fourth-rank tensor sijkl and the compo-
nents of the 6 × 6 matrix s is shown to be:
sijkl = smn , m, n = 1, 2, 3,
2sijkl = smn , m = 1, 2, 3, n = 4, 5, 6,
4sijkl = smn , m, n = 4, 5, 6
in which the following contraction rule is applied for replacing a pair of indices by a
single contracted index: 11 → 1, 22 → 2, 33 → 3, (23, 32) → 4, (13, 31) → 5, (12,
21) → 6.
Furthermore, the full tensor suffixes of the stresses σ and strains ε are contracted
according to the scheme:
σ11 = σ1 , σ22 = σ2 , σ33 = σ3 ,
(σ23, σ32) = σ4 , (σ13, σ31) = σ5 , (σ12, σ21) = σ6 ,
ε11 = ε1 , ε22 = ε2 , ε33 = ε3 ,
(2ε23, 2ε32) = ε4 , (2ε13, 2ε31) = ε5 , (2ε12, 2ε21) = ε6
and
dijk = dmn , m, n = 1, 2, 3,
2dijk = dmn , m = 1, 2, 3, n = 4, 5, 6,
α11 = α1 , α22 = α2 , α33 = α3 ,
(α23, α32) = α4, (α13, α31) = α5, (α12, α21) = α6 .
The relations of (3.1) constitute the starting set of the equations of state, which
describe the environment, and they are widely used to solve the problems dealt with
in piezoelectricity. Basic effects are described by the material constants (tensors)
sE,T
ijkl , εT,σ
nm, ξE,σ that occur at the diagonal of the set of equations, whereas conjugate
340
Non-linear phenomena in piezoelectric crystals
effects are defined by the remaining constants. On rewriting the material constants
of (3.1) in a matrix form, we obtain a symmetrical matrix of linear effects:
εij
Dm
S
=
sE,T
ijkl dT
ijn αE
ij
dT
mkl εT,σ
nm pσ
m
αE
kl pσ
n ξE,σ
σkl
En
T
. (3.2)
The equivalence between the components of the compliance sixth-rank tensor sijklpq
and the components of the 6 × 21 matrix is shown to be:
Sijklpq = smno , m = 1, 2, 3, n = o = 1, 2, 3,
2sijklpq = smno , m = 1, 2, 3, n 6= o, n = 1, 2, o = 2, 3,
4sijklpq = smno , m = 1, 2, 3, n = 1, 2, 3, o = 4, 5, 6, n = o = 4, 5, 6,
8sijklpq = smno , m = 1, 2, 3, n = 4, 5, o = 5, 6,
2sijklpq = smno , m = 4, 5, 6, n = o = 1, 2, 3,
4sijklpq = smno , m = 4, 5, 6, n 6= o, n = 1, 2, o = 2, 3,
8sijklpq = smno , m = 4, 5, 6, n = 1, 2, 3, o = 4, 5, 6, n = o = 4, 5, 6,
16sijklpq = smno , m = 4, 5, 6, n = 4, 5, o = 5, 6.
The equation
εij =
1
2
sET
ijklpqσklσpq , i, j, k, l, p, q = 1, 2, 3
can be written in the form
εm =
1
2
smnoσnσo , m, n, o = 1, 2, 3, . . . .
Considering equations (2.6), (2.7) and (2.8) we obtain in a non-linear approximation:
εij = sijklσkl + 1
2
sijklpqσklσpq + . . . ,
Dm = dmklσkl + 1
2
dmklpqσklσpq + . . . ,
S = αklσkl + 1
2
αklpqσklσpq + . . .
if En = 0, ∆T = 0,
εij = dnijEn + 1
2
oijntEnEt + . . . ,
Dm = εnmEn + 1
2
εnmtEnEt + . . . ,
S = pnEn + 1
2
pntEnEt + . . .
if σkl = 0, ∆T = 0,
εij = αijT + 1
2
rijTT + . . . ,
Dm = pmT + 1
2
umTT + . . . ,
S = ξT + 1
2
ρTT + . . .
if σkl = 0, En = 0.
An analogical matrix equation can be derived for non-linear effects (equations
(2.6), (2.7) and (2.8)):
εij
Dm
S
=
A D E
D B F
E F C
σkl
En
T
, (3.3)
341
F.Warkusz, A.Linek
where the basic effects are given by the following constants:
A = sE,T
ijkl + 1
2
sE,T
ijklpqσpq + 1
2
dT
ijklnEn + 1
2
αE
ijklT,
B = εT,σ
nmt + 1
2
εT,σ
nmtEn + 1
2
oT
nmklσkl + 1
2
pσ
nmT,
C = ξE,σ + 1
2
ρE,σT + 1
2
uσ
nEn + 1
2
rE
klσkl
and the conjugate effects are described by
D = dT
ijn + 1
2
dT
klijnσkl + 1
2
oT
ijntEt + 1
2
kijnT,
E = αE
ij + 1
2
αE
ijklσkl + 1
2
rE
ijT + 1
2
kijnEn ,
F = pσ
m + 1
2
pσ
mnEn + 1
2
kmklσkl + 1
2
uσ
mT.
As we can see, the matrix of non-linear effects is also a symmetrical one.
4. Summary
Using the Gibbs function (G), the equations describing the physical quantities εij,
Dm, S have been derived for the non-linear effects of piezoelectric bodies. Relevant
equations are included in section 3. It has been demonstrated that the relations
between the material constants, which are tensors of zeroth to sixth order, can be
written in non-linear approximation in the form of material equations where the
elements of the matrices are the sums of appropriate matrices which represent the
tensors (relation of (3.3)). The relations of (3.3) were used to derive the linear effects
(relations of (3.2)).
The equations describing linear and non-linear phenomena for particular crystal-
lographic point groups reduce because of monocrystal symmetry [5,6,12,13,14,15,16].
From the rewritten relations of (2.6, 2.7, 2.8) in the matrix notation (table 1)
we can easily identify the interrelations of the tensor components, especially for
separate physical effects. Further simplifications and decays of some components of
higher order tensors may be obtained for appropriate point groups (crystal classes).
Table 1. Matrices of the coefficients (tensor components) in non-linear expansion.
σn σnσo Enσo σnT En EnEt EnT T T 2
εm
smn
6 × 6
smno
6 × 21
dmno
6 × 15
αmnl
6 × 6
dnm
6 × 3
omnt
6 × 6
kmn
6 × 3
αm
6 × 1
rm
6 × 1
Dm
dmn
3 × 6
dnmo
3 × 21
oomn
3 × 15
knm
3 × 6
εmn
3 × 3
εmnt
3 × 6
pnm
3 × 3
pm
3 × 1
um
3 × 1
S
αn
1 × 6
αno
1 × 21
kon
1 × 15
rn
1 × 6
pn
1 × 3
pnt
1 × 6
un
1 × 3
ξ
1 × 1
ρ
1 × 1
(6 + 3 + 1) × (6 + 21 + 15 + 6 + 3 + 6 + 3 + 1 + 1) = 10 × 62
342
Non-linear phenomena in piezoelectric crystals
In 1952, with the aim to interpret the properties of the tetragonal antiferroelectric
crystal ND4D2PO4, Mason [10,11] made use of the elastic Gibbs function G1 :
G1 = U − εijσij − TS
in its differential form:
dG1 = −εijdσkl + EndDn − SdT.
The equations and formulae describing εij, En and S, which have been derived by
Mason [10], depend on σkl, Dn and T . So it is not surprising that they differ from
the equations presented in this paper (equations (2.6), (2.7) and (2.8)) both in linear
and non-linear approximations. In our calculations we used a Gibbs function which
had the form of G:
G = U − εijσkl − EnDn − TS
and a differential form of
dG = −εijdσkl − DndEn − SdT.
The equations obtained for εij, Dn and S depend on σkl, En and T . There is a
definite sense in which our results differ from those achieved by Mason. For example,
according to Mason (function G1) [10] we have the following relations for the linear
effects:
εij =
∂εij
∂σkl
∣
∣
∣
∣
∣
D,T
σkl +
∂εij
∂Dm
∣
∣
∣
∣
∣
σ,T
Dm +
∂εij
∂T
∣
∣
∣
∣
∣
σ,D
T,
En =
∂En
∂σkl
∣
∣
∣
∣
∣
D,T
σkl +
∂En
∂Dm
∣
∣
∣
∣
∣
σ,T
Dm +
∂En
∂T
∣
∣
∣
∣
∣
σ,D
T,
S =
∂S
∂σkl
∣
∣
∣
∣
∣
D,T
σkl +
∂S
∂Dm
∣
∣
∣
∣
∣
σ,T
Dm +
∂S
∂T
∣
∣
∣
∣
∣
σ,D
T.
Our results (based on function G) take the form
εij =
∂εij
∂σkl
∣
∣
∣
∣
∣
E,T
σkl +
∂εij
∂En
∣
∣
∣
∣
∣
σ,T
En +
∂εij
∂T
∣
∣
∣
∣
∣
σ,E
T,
Dm =
∂Dm
∂σkl
∣
∣
∣
∣
∣
E,T
σkl +
∂Dm
∂En
∣
∣
∣
∣
∣
σ,T
En +
∂Dm
∂T
∣
∣
∣
∣
∣
σ,E
T,
S =
∂S
∂σkl
∣
∣
∣
∣
∣
E,T
σkl +
∂S
∂En
∣
∣
∣
∣
∣
σ,T
En +
∂S
∂T
∣
∣
∣
∣
∣
σ,E
T.
Relevant differences have been underlined.
Considering the physical properties of the crystals, the symmetries of the tensor
components and the order of differentiation of the thermodynamic function of state
as a total differential, we can use a more concise matrix notation, e.g:
343
F.Warkusz, A.Linek
sijkl → smn (m, n = 1, . . . , 6), fourth-order tensor, second-order physical
constant;
sijklpq → smno (m, n, o = 1, . . . , 6), sixth-order tensor, third-order physical
constant;
dijkln → dmno (m, n = 1, . . . , 6, o = 1, 2, 3), fifth-order tensor, third-order
physical constant;
dnij → dnm (n = 1, 2, 3, m = 1, . . . , 6), third-order tensor, second-order
physical constant;
oijnt → omnt (m = 1, . . . , 6, n, t = 1, 2, 3), fourth-order tensor, third-
order physical constant;
αij → αm (m = 1, . . . , 6), second-order tensor, second-order physical
constant, etc.
During transition from tensor to matrix notation we neglected the coefficients
that were different for relevant components (those for sijkl and sijklpq are listed in
section 3).
Equally concise are the matrices of the coefficients (tensor components) in non-
linear expansion. All of the coefficients (tensor components) incorporated in equa-
tions (2.6), (2.7) and (2.8), which occur on the right-hand side of table 1, are pre-
sented in the form of matrix 10 × 62:
(6 + 3 + 1) × (6 + 21 + 15 + 6 + 3 + 6 + 3 + 1 + 1) = 10 × 62.
For linear effects we have matrix 10 × 10:
(6 + 3 + 1) × (6 + 3 + 1) = 10 × 10.
Matrix notation is clear and very convenient, and in its external form it is similar for
all (eight) thermodynamic functions, which have been used to describe the physical
quantities for piezoelectric crystals. Making use of the data reported in the litera-
ture [1,3,5,13,14,15,16], we can establish the number of independent components of
particular tensors for each point group, as well as define the relations between some
elements of these matrices.
References
1. Nye J.F. Physical Properties of Crystals. Oxford University Press, 1985.
2. Soluch W. // Wstȩp do Piezoelektroniki. Wydawnictwo Komunikacji i La̧czności,
Warszawa, 1980 (in Polish).
3. Juretschke H.J. Crystal Physics. W.A. Benjamin, INC. Massachusetts, 1974.
4. Weber H.-J., Balashova E.V., Kizhaev S.A. // J. Phys.: Condens. Matter, 2000, vol. 12,
p. 1485.
5. Fumi F.G., Ripamonti C. // Acta Cryst., 1980, vol. A36, p. 535.
6. Abrahams S.C. // Acta Cryst., 1994, vol. A50, p. 658.
344
Non-linear phenomena in piezoelectric crystals
7. Briss R.R. Symmetry and Magnetism. Amsterdam, North-Holland Publishing Co.,
1966.
8. Wooster W.A. Tensors and Group Theory for the Physical Properties of Crystals.
1973.
9. Warkusz F. // Zeszyty Naukowe UO, Fizyka, 1997, vol. 27, p. 195 (in Polish).
10. Mason W.P. // Phys. Rev., 1952, vol. 88, p. 480.
11. Mason W.P. Crystal Physics of Interaction Processes. New York, London, Academic
Press, 1966.
12. Parton V.Z., Kudryavtsev B.A. Electromagnetoelasticity of Piezoelectronics and Elec-
trically Conductive Solids. Moscow, Nauka Publ., 1988 (in Russian); Electromagne-
toelasticity: Piezoelectrics and Electrically Conductive Solids. Taylor & Francis, 1988
(English translation).
13. Fumi F.G. // Il Nuovo Cimento, 1952, vol. 9, p. 739.
14. Fieshi R., Fumi F.G. // Il Nuovo Cimento, 1953, vol. 10, p. 865.
15. Fumi F.G., Ripamonti C. // Acta Cryst., 1980, vol. A36, p. 551.
16. Fumi F.G. // Acta Cryst. 1997, vol. A53, p. 101.
Нелінійні механічні, електричні та термічні явища в
п’єзоелектричних кристалах
Ф.Варкуш, А.Лінек
Інститут фізики, університет м. Ополє, 45–052 Ополє, Польща
Отримано 25 серпня 2002 р., в остаточному вигляді –
31 березня 2003 р.
Механічні, електричні та термічні явища у п’єзоелектричних криста-
лах вивчаються у нелінійному наближенні. З цією метою використа-
но термодинамічний потенціал, який описує анізотропне тіло. Роз-
глянуто потенціал Гіббса. Розрахунки охоплюють тензор деформа-
ції εij = f(σkl, En, T ), вектор індукції Dm = f(σkl, En, T ) та ентропію
S = f(σkl, En, T ) як функцію механічного напруження σkl, величини
поля En і різниці температур T . Отримано рівняння, які описують
анізотропні п’єзоелектричні тіла, якщо відомі “сили” σkl, En, T , що
діють на кристал.
Ключові слова: п’єзоелектричні кристали, термодинаміка,
анізотропні тіла, тензори
PACS: 65.40.-b, 77.65.-j, 77.65.Bn, 77.65.Ly, 05.70.Ce
345
346
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