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2.3  Database of piezoelectric materials

m_piezo Dbval includes a number of material characteristics for piezoelectric materials. The properties are obtained from the datasheet of the material, but as we will illustrate, the data is not always sufficient to calculate all the material properties needed for the computations. Most of the information in the datasheet is generally related to the constitutive equations written in the form of (5). For PZT, PVDF, or piezoelectric composites based on PZT and PVDF, the general form of these matrices is:

(28)

For an orthotropic material, the compliance matrix [s^E] can be written as a function of the engineering constant E_i,ν_ij and G_ij as follows:

(29)

where z is aligned with the poling direction 3, and x,y with directions 1,2 respectively. Note that the matrix is symmetric so that:

(30)

A bulk piezoelectric ceramic exhibits transverse isotropic properties: the properties of the material are the same in the plane perpendicular to the poling direction. In this case, the compliance matrix reduces to:

(31)

and due to the symmetry we have:

(32)

where the subscript p refers to the in-plane properties. The matrix of piezoelectric coefficients is:

(33)

and the matrix of dielectric permittivities:

(34)

In order to use such a piezoelectric material in a 3D model, it is therefore necessary to have access to the 5 elastic constants E_p,E_z,ν_p,ν_zp and G_zp, 3 piezoelectric constants d₃₁,d₃₃, and d₁₅ and two dielectric constants є₁₁^T,є₃₃^T. Unfortunately, such constants are generally not given in that form, but can be calculated from the material properties found in the datasheet.
It is important to introduce the electromechanical coupling factors which are generally given in the datasheet and are a function of the elastic, piezoelectric and dielectric properties of the material. They measure the effectiveness of the conversion of mechanical energy into electrical energy (and vice-versa). There is one coupling factor for each piezoelectric mode:

(35)

In addition, coupling factors k_p for radial modes of thin discs, and k_t for thickness modes of arbitrary shaped thin plates are also commonly given in datasheet. k_p is related to k₃₁ through:

(36)

k_t is always lower than k₃₃ but there does not seem to be a simple explicit expression of k_t as a function of the material properties. The fact that k_t is lower than k₃₃ means that electrical energy conversion in the d₃₃-mode is less effective for a thin plate than for a rod. The definition of the coupling factors k₃₃ and k₁₅ also allows to write alternative expressions:

(37)

We illustrate the use of these different relationships to form the full set of mechanical, piezoelectric and dielectric properties for the material SONOX P502 from Ceramtec (http://www.ceramtec.com/) which is a soft piezoceramic. The properties found in the datasheet on matweb.com are given in Table 2.2 .

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Material property	value	unit
Piezoelectric properties
d33	440	10−12 m/V
d31	-185	10−12 m/V
d15	560	10−12 m/V
e33	16.7	C/m2=As/m2
g33	26.9  10−3	Vm/N
Permittivity
є33T	1850 є0	F/m
є33S	875 є0	F/m
є11T	1950 є0	F/m
є11S	1260 є0	F/m
Elastic properties
s11E	18.5   10−12	m2/N
s33E	20.7   10−12	m2/N
c33D	15.7   1010	N/m2
c55D	6.5   1010	N/m2
Coupling coefficients
k33	0.72
k15	0.74
k31	0.33
kp	0.62
kt	0.48
Density
ρ	7740	kg/m3

Table 2.2: Properties of SONOX P502 from the datasheet found on https://www.matweb.com (2013)

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E_p and E_z are computed directly from the definitions of s₁₁^E and s₃₃^E:

(38)

(39)

Knowing the value of s₁₁^E, d₃₁, є₃₃^T and k_p, s₁₂^E can be computed:

(40)

allowing to compute the value of ν_p:

(41)

and the value of G_p

(42)

From the value c₅₅^D and k₁₅, we compute

(43)

from which the the value of G_zp is computed:

(44)

The value of ν_zp cannot be calculated from the datasheet information. We therefore assume that, as for most PZT ceramics:

(45)

The value of ν_pz is calculated as:

(46)

The complete set of values is summarized in Table 2.3. These are the values used in m_piezo.

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Material property	value	unit
Piezoelectric properties
d33	440	10−12 m/V
d31	-185	10−12 m/V
d15	560	10−12 m/V
Permittivity
є33T	1850 є0	F/m
є11T	1950 є0	F/m
Mechanical properties
Ep	54.05	GPa
Ez	48.31	GPa
Gzp	29.41	GPa
Gp	19.17	GPa
νp	0.4124
νzp	0.39
νpz	0.44
ρ	7740	kg/m3

Table 2.3: Properties of SONOX P502 to be used in 3D finite element models

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Note that there is some redundancy in the data from the datasheet, which allows to check for consistency. The two following coupling factors are computed from the data available and checked against the tabulated values.

(47)

(48)

The values are close to the values in Table 2.2. In addition, the value of g₃₃ is given by:

(49)

and corresponds exactly to the value tabulated. The value of e₃₃ can be computed using Equation (7), leading to:

(50)

where there is a difference of about 15% with the tabulated value of e₃₃= 16.7 C/m². Note however that this last value was found on matweb.com and is not given in the more recent datasheet on Ceramtec website (in 2023).
Using (37) to compute k₁₅ with the values from the datasheet, one gets:

(51)

which shows the non-consistency of the value of є₁₁^S in the datasheet. In fact, when computed using (8), one gets:

(52)

This illustrates the fact that it is difficult to obtain the full set of parameters needed for computation for piezoelectric materials, as there are often some inconsistencies amongst the data available from the manufacturers. What we believe is a "best compromise" was used in the material properties available in SDT.

From the input values in m_piezo  (Table 2.3), it is possible to compute the mechanical, piezoelectric and permittivity matrices used in the four different forms of the constitutive equations  (4),(5),(9),(10) using the relationships (6)-(8)) and (11)-(13).
The command p_piezo('TabDD',model) gives access to all the matrices based in the input values in m_piezo. This will be illustrated in section section 3.5.1.
As the mechanical properties of PZT are not strongly orthotropic, a simplification can be done by considering that the material is isotropic (for the mechanical and dielectric properties, not the piezoelectric properties). An isotropic version of SONOX P502 is included in m_piezo under the name of SONOX_P502_iso whose properties are given in Table 2.4.

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Material property	value	unit
Piezoelectric properties
d33	440	10−12 m/V
d31	-185	10−12 m/V
d15	560	10−12 m/V
Permittivity
єT	1850 є0	F/m
Mechanical properties
E	54	GPa
ν	0.41
ρ	7740	kg/m3

Table 2.4: Simplified material properties for SONOX P502 considering mechanical isotropy

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The second example is the PIC 255 PZT, also a soft piezoceramic, from PI ceramics. The properties found in the datasheet in the year 2013 are given in Table 2.5 (https://www.piceramic.com). Note that C₃₃^D was not given, therefore we estimated it from the value of PIC 155 given in the same datasheet, which is just slightly stiffer.

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Material property	value	unit
Piezoelectric properties
d33	400	10−12 m/V
d31	-180	10−12 m/V
d15	550	10−12 m/V
g31	-11.3  10−3	Vm/N
g33	25  10−3	Vm/N
Permittivity
є33T	1750 є0	F/m
є11T	1650 є0	F/m
Elastic properties
s11E	16.1   10−12	m2/N
s33E	20.7   10−12	m2/N
c33D	11   1010	N/m2
Coupling coefficients
k33	0.69
k15	0.66
k31	0.35
kp	0.62
kt	0.47
Density
ρ	7800	kg/m3

Table 2.5: Properties of PIC 255 from the datasheet (2013)

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E_p and E_z are computed directly from the definitions of s₁₁^E and s₃₃^E:

(53)

(54)

Knowing the value of s₁₁^E, d₃₁, є₃₃^T and k_p, s₁₂^E can be computed:

(55)

allowing to compute the value of ν_p:

(56)

and the value of G_p

(57)

The value of s₅₅^E can be computed as:

(58)

which leads to:

(59)

Again, the value of ν_zp cannot be calculated from the datasheet information. We cannot assume a value of 0.39 as previously, as it would lead to a non-physical value of ν_pz. As ν_p is in the range of 0.32 and ν_zp is typically slightly lower, we assume that :

(60)

The value of ν_pz is calculated as:

(61)

The complete set of values is summarized in Table 2.6. These are the values used in m_piezo. Note that there is some redundancy in the data from the datasheet, which allows to check for consistency. The two following coupling factors are computed from the data available and checked against the tabulated values.

(62)

The values are very close to the values in Table 2.5. In addition, the value of g₃₃ and g₃₁ are given by:

(63)

and are also very close to the values tabulated.

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Material property	value	unit
Piezoelectric properties
d33	400	10−12 m/V
d31	-180	10−12 m/V
d15	550	10−12 m/V
Permittivity
є33T	1750 є0	F/m
є11T	1650 є0	F/m
Mechanical properties
Ep	62.11	GPa
Ez	48.31	GPa
Gzp	21.03	GPa
Gp	23.53	GPa
νp	0.3242
νzp	0.30
νpz	0.39
ρ	7800	kg/m3

Table 2.6: Properties of PIC 255 to be used in 3D finite element models from datasheet in 2013

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As shown in the derivations above, the datasheet for PZT material typically do not contain the full information to derive all the coefficients needed for computations, and some hypothesis need to be made. In addition, it is usual to have a variation of 10 % or more on these properties from batch to batch, and the datasheet are not updated for each batch. Note also that the properties are given at 20 ^∘C and are temperature dependant. The variations with temperature are rarely given in the datasheet. This may also account for inaccuracies in the computations.
The more recent datasheet found on PI Ceramics website (2023) leads to slightly different properties, and includes the value of C₃₃^D, which gives a more precise value for E_z. The new datasheet information is given in Table 2.7 and the resulting m_piezo  input parameters in Table 2.8. The updated properties are included in the PIC255b material in m_piezo. It is advised to use this updated material property, as the main difference is for the Young's modulus in the direction of poling. This parameter has an important impact on the longitudinal natural frequency of disks and rods.

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Material property	value	unit
Piezoelectric properties
d33	400	10−12 m/V
d31	-180	10−12 m/V
d15	550	10−12 m/V
g31	-11.8  10−3	Vm/N
g33	25  10−3	Vm/N
Permittivity
є33T	1800 є0	F/m
є11T	1750 є0	F/m
Elastic properties
s11E	16   10−12	m2/N
s33E	19   10−12	m2/N
c33D	15.4   1010	N/m2
Coupling coefficients
k33	0.69
k15	0.65
k31	0.35
kp	0.62
kt	0.47
Density
ρ	7800	kg/m3

Table 2.7: Properties of PIC 255b from the datasheet (2023)

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Material property	value	unit
Piezoelectric properties
d33	400	10−12 m/V
d31	-180	10−12 m/V
d15	550	10−12 m/V
Permittivity
є33T	1800 є0	F/m
є11T	1750 є0	F/m
Mechanical properties
Ep	62.5	GPa
Ez	52.63	GPa
Gzp	21.64	GPa
Gp	23.39	GPa
νp	0.3389
νzp	0.30
νpz	0.3562
ρ	7800	kg/m3

Table 2.8: Properties of PIC 255b to be used in 3D finite element models from datasheet in 2023

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In much the same way, the material properties of PIC 181 which is a hard piezoceramic from the same manufacturer have been updated from the PIC181 to the PIC181b properties in m_piezo.
Table 2.9 summarizes the different material properties available in SDT, and the year of the datasheet where the original data was found.

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Manufacturer	type	year	SDT name
Ceramtec	Sonox P502	2023	SONOX_P502
Ceramtec	Sonox P502 - simplified	2023	SONOX_P502_iso
PI Ceramics	PIC181	2013	PIC181
PI Ceramics	PIC181	2023	PIC181b
PI Ceramics	PIC255	2013	PIC255
PI Ceramics	PIC255	2023	PIC255b
Ferroperm	Pz21	2018	FerropermPz21
Ferroperm	Pz23	2018	FerropermPz23
Ferroperm	Pz24	2018	FerropermPz24
Ferroperm	Pz26	2018	FerropermPz26
Ferroperm	Pz27	2018	FerropermPz27
Ferroperm	Pz28	2018	FerropermPz28
Ferroperm	Pz29	2018	FerropermPz29
Ferroperm	Pz34	2018	FerropermPz34
Ferroperm	Pz46	2018	FerropermPz46
Noliac	NCE51	2012	Noliac.NCE51

Table 2.9: Piezoelectric materials available in SDT: manufacturer references and year, and SDT name

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