Cement-Based Piezoelectricity Application: A Theoretical Approach

*Daniel A. Triana-Camacho, Jorge H. Quintero-Orozco and Jaime A. Perez-Taborda*

## **Abstract**

The linear theory of piezoelectricity has widely been used to evaluate the material constants of single crystals and ceramics, but what happens with amorphous structures that exhibit piezoelectric properties such as cement-based? In this chapter, we correlate the theoretical and experimental piezoelectric parameters for small deformations after compressive stress–strain, open circuit potential, and impedance spectroscopy on cement-based. Here, in detail, we introduce the theory of piezoelectricity for large deformations without including a functional for the energy; also, we show two generating equations in terms of a free energy's function for later it will be reduced to constitutional equations of piezoelectricity for infinitesimal deformations. Finally, here is shown piezoelectric and electrical parameters of gold nanoparticles mixed to cement paste: the axial elasticity parameter *Y* ¼ <sup>323</sup>*:*<sup>5</sup> � <sup>75</sup>*:*<sup>3</sup> *kN=m*<sup>2</sup> ½ �, the electroelastic parameter *<sup>γ</sup>* ¼ �20*:*<sup>5</sup> � <sup>6</sup>*:*<sup>9</sup> ½ � *mV=kN* , and dielectric constant *ε* ¼ ð Þ 939*:*6 � 82*:*9 *ε*<sup>0</sup> ½ � *F=m* , which have an interpretation as linear theory parameters *s D ijkl*, *gkij* and *ε<sup>T</sup> ik* discussed in the chapter.

**Keywords:** piezoelectricity, cement-based, nano-composites, constitutional equations, impedance spectroscopy

## **1. Introduction**

The direct piezoelectric effect creates an electric polarization on a continuum medium due to applied stress. The polarization can be macroscopic (effect over continuum medium) and nanoscopic and microscopy scales (effect over atoms, molecules, and electrical domains). Once the Curie brothers discovered the piezoelectric effect in 1880 [1], piezoelectricity investigations led to more data and constructed models based on crystallography to explain the electricity generation since electro-optics and thermodynamic. Voigt in 1894 proposed a piezoelectric parameter related to the strain of material; since the thermodynamic theory, he constructed a non-linear model and expressed the free energy of a piezoelectric crystal in terms of the electric field, strain, electric and elastic deformation potentials, temperature, pyroelectric and piezoelectric parameters [2]. Currently, we can see these constants in the constitutive equations of piezoelectricity. During 1956 and 1963, Toupin and Eringen used a variational formulation to construct a functional in terms of internal energy and derive the constitutive equations [3, 4]. Then, in 1971 Tiersten proposed to use the conservation equations of mass, electrical charge, linear momentum, angular momentum, and energy, adding a Legendre transformation to include a thermodynamic functional in terms of the free energy, achieving a reduction of the number of constitutive equations from 7 to 4 to facilitate theoretical calculations [5]. These constitutional equations and their linear approach gave support to the theoretical calculus of piezoelectric parameters of crystalline structures, e.g., zinc-blende [6, 7], zinc oxide [8, 9], and other crystals with similar symmetric of quartz [10, 11]. Finally, between 1991 and 2017, Yang has proposed modifications for the Legendre transformation of Tiersten, and he has included two models to describe the polarization in a deformable continuum medium [12, 13].

According to electrostatic theory, the macroscopic polarization *P* ! can be written in terms of electric charge distribution likewise with the electric field *E* ! induced into the continuum medium. Besides, the polarization starts with continuum medium deformation *S* ! for applied stress. Considering: (i) Uniqueness for the parameters that relate the polarization and the electric field, (ii) Stress produces equal deformations in each cell of crystal, (iii) The deformations lead dipole and quadrupole moments affecting the piezoelectric parameters directly. Based on the above considerations, the linear constitutive equations of piezoelectricity can be written, as Martin said [6].

displacement *η* respect to the lattice. It is produced by body deformation, as

*<sup>η</sup><sup>k</sup>*,*<sup>k</sup>* <sup>¼</sup> *<sup>∂</sup>ηk*ð Þ*<sup>y</sup> ∂yk*

Furthermore, if it is taken the two continuous mediums, the electric charge

We can show that the gradient of infinitesimal displacement Eq. (1) and the neutrality condition of electric charge density Eq. (2) are sufficient to explain the

! <sup>þ</sup> *<sup>η</sup>*

! þ *η* ! � �*<sup>η</sup>*

!� � <sup>þ</sup> *<sup>μ</sup><sup>e</sup> <sup>y</sup>*

density must be neutral to consider only the piezoelectric effect.

*Cement-Based Piezoelectricity Application: A Theoretical Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.95255*

*μ<sup>l</sup> y*

polarization in a deformable continuum

*States of a deformable and polarizable continuum.*

**Figure 1.**

**Figure 2.**

**145**

*P* ! <sup>¼</sup> *<sup>μ</sup><sup>l</sup> <sup>y</sup>*

*Volume elements of electronic and lattice continuum medium.*

!� � �*<sup>η</sup>*

! *y* ! h i � � <sup>¼</sup> *<sup>μ</sup><sup>e</sup> <sup>y</sup>*

¼ 0 (1)

! � � <sup>¼</sup> <sup>0</sup> (2)

! ≈*μ<sup>e</sup> y* !� �*<sup>η</sup>*

! (3)

described in **Figure 2**. From the assumption of the lattice and electronic continuum, the medium have equal volumes, some variation of infinitesimal displacement respect coordinates in the current state should be zero and can be written as

This chapter book is thought to be a working example that connects the piezoelectricity theory and experimental data of electromechanical and electrical properties. These data were obtained on cement paste mixed with gold nanoparticles.

## **2. Constitutional equations in detail and correlation with the piezoelectricity of cement-based composites**

In this section, we have selected Yang's differential approach to obtain the constitutional equations of piezoelectricity. The differential derivation shows the physics involved in the conservation laws differently. For example, it shows that the electric body couple and the Cauchy stress tensor are asymmetric. Also, it relates the local electric field with the electric interaction between the differential elements of the lattice continuum and the electronic continuum.

## **2.1 Conservation laws applied to a polarized continuum from differential approximation**

Regarding the study of the piezoelectric properties of cement paste, it is necessary to describe the material separately as two continua medium, as from the piezoelectric phenomenon, the crystal and their symmetry would have a lattice (positive charge) and an electronic component (negative charge) Those continua can be separated by mechanical stress. Their physical properties could change according to the coordinate systems or states. Therefore, the body will study in two states (reference state and current state), as is shown in **Figure 1**.

## *2.1.1 Electric charge conservation*

The conservation electric charge in the body takes importance with an infinitesimal displacement on the medium's current state to get polarization. For this reason, the phenomenon described in the above state is known as the two continuum medium model. The electronic continuum comes under an infinitesimal

## *Cement-Based Piezoelectricity Application: A Theoretical Approach DOI: http://dx.doi.org/10.5772/intechopen.95255*

**Figure 1.**

terms of internal energy and derive the constitutive equations [3, 4]. Then, in 1971 Tiersten proposed to use the conservation equations of mass, electrical charge, linear momentum, angular momentum, and energy, adding a Legendre transformation to include a thermodynamic functional in terms of the free energy, achieving a reduction of the number of constitutive equations from 7 to 4 to facilitate theoretical calculations [5]. These constitutional equations and their linear approach gave support to the theoretical calculus of piezoelectric parameters of crystalline structures, e.g., zinc-blende [6, 7], zinc oxide [8, 9], and other crystals with similar symmetric of quartz [10, 11]. Finally, between 1991 and 2017, Yang has proposed modifications for the Legendre transformation of Tiersten, and he has included two models to describe the polarization in a deformable continuum medium [12, 13].

*Cement Industry - Optimization, Characterization and Sustainable Application*

According to electrostatic theory, the macroscopic polarization *P*

in terms of electric charge distribution likewise with the electric field *E*

**2. Constitutional equations in detail and correlation with the**

In this section, we have selected Yang's differential approach to obtain the constitutional equations of piezoelectricity. The differential derivation shows the physics involved in the conservation laws differently. For example, it shows that the electric body couple and the Cauchy stress tensor are asymmetric. Also, it relates the local electric field with the electric interaction between the differential elements

**2.1 Conservation laws applied to a polarized continuum from differential**

sary to describe the material separately as two continua medium, as from the piezoelectric phenomenon, the crystal and their symmetry would have a lattice (positive charge) and an electronic component (negative charge) Those continua can be separated by mechanical stress. Their physical properties could change according to the coordinate systems or states. Therefore, the body will study in two

states (reference state and current state), as is shown in **Figure 1**.

Regarding the study of the piezoelectric properties of cement paste, it is neces-

The conservation electric charge in the body takes importance with an infinitesimal displacement on the medium's current state to get polarization. For this reason, the phenomenon described in the above state is known as the two continuum medium model. The electronic continuum comes under an infinitesimal

**piezoelectricity of cement-based composites**

of the lattice continuum and the electronic continuum.

deformation *S*

Martin said [6].

**approximation**

*2.1.1 Electric charge conservation*

**144**

!

the continuum medium. Besides, the polarization starts with continuum medium

that relate the polarization and the electric field, (ii) Stress produces equal deformations in each cell of crystal, (iii) The deformations lead dipole and quadrupole moments affecting the piezoelectric parameters directly. Based on the above considerations, the linear constitutive equations of piezoelectricity can be written, as

This chapter book is thought to be a working example that connects the piezoelectricity theory and experimental data of electromechanical and electrical properties. These data were obtained on cement paste mixed with gold nanoparticles.

for applied stress. Considering: (i) Uniqueness for the parameters

!

!

can be written

induced into

*States of a deformable and polarizable continuum.*

displacement *η* respect to the lattice. It is produced by body deformation, as described in **Figure 2**. From the assumption of the lattice and electronic continuum, the medium have equal volumes, some variation of infinitesimal displacement respect coordinates in the current state should be zero and can be written as

$$
\eta\_{k,k} = \frac{\partial \eta\_k(\mathbf{y})}{\partial \mathbf{y}\_k} = \mathbf{0} \tag{1}
$$

Furthermore, if it is taken the two continuous mediums, the electric charge density must be neutral to consider only the piezoelectric effect.

$$
\mu^l \left( \overrightarrow{\mathcal{Y}} \right) + \mu^\epsilon \left( \overrightarrow{\mathcal{Y}} + \overrightarrow{\eta} \right) = \mathbf{0} \tag{2}
$$

We can show that the gradient of infinitesimal displacement Eq. (1) and the neutrality condition of electric charge density Eq. (2) are sufficient to explain the polarization in a deformable continuum

$$\overrightarrow{P} = \mu^l \left(\overrightarrow{\mathcal{y}}\right) \left[ -\overrightarrow{\eta}\left(\overrightarrow{\mathcal{y}}\right) \right] = \mu^\epsilon \left(\overrightarrow{\mathcal{y}} + \overrightarrow{\eta}\right) \overrightarrow{\eta} \approx \mu^\epsilon \left(\overrightarrow{\mathcal{y}}\right) \overrightarrow{\eta} \tag{3}$$

**Figure 2.** *Volume elements of electronic and lattice continuum medium.*

#### *2.1.2 Energy conservation*

Once the body is deformed, electronic and lattice continua electric charges apply a quasi-static electric field. Tiersten et al. called it Maxwelliam electric field *Ek*. It is interacting on two continuum mediums producing an electrical force on each one. The other forces acting on the body are traction and body forces. The traction force *tk* per unit, the area is working on the surfaces of volume elements of the lattice (see **Figure 2**). Also, it can be written in terms of Cauchy stress tensor *τjk* as *tk* ¼ *n <sup>j</sup>τjk*, where *n <sup>j</sup>* is the normal vector. Moreover, body force refers to an external force acting on the body, for example, gravity. The previous three forces are necessary to get the conservation laws, including energy.

From the three forces above, it is possible to construct the Eqs. (4) and (5), linear and angular momentum conservation, respectively.

$$
\rho \dot{u}\_k = f\_k^E + \tau\_{mk,m} + \rho f\_k \tag{4}
$$

$$
\sigma\_i^E + \varepsilon\_{ijk}\tau\_{jk}\left(\stackrel{\rightarrow}{\mathcal{Y}}\right) = \mathbf{0} \tag{5}
$$

*d dt*

*d dt*

1 2

The term *<sup>d</sup>*

in Eq. (11).

expansion.

velocity,

form:

**147**

≈*μ<sup>l</sup> y*

<sup>¼</sup> *<sup>μ</sup><sup>l</sup> <sup>y</sup>*

<sup>þ</sup>*μ<sup>e</sup> <sup>y</sup>*

<sup>¼</sup> *<sup>μ</sup><sup>l</sup> <sup>y</sup>*

!� � <sup>þ</sup> *<sup>μ</sup><sup>e</sup> <sup>y</sup>*

1 2

total energy term is applied:

*ukukρdv* <sup>þ</sup> *<sup>ε</sup>inρdv* � � <sup>¼</sup> *<sup>ρ</sup>dv <sup>d</sup>*

*d dt*

> 1 2

!� �*uk <sup>y</sup>*

!� �*uk <sup>y</sup>*

!� �*dv*

!� �*uk <sup>y</sup>*

The second-order term *<sup>η</sup>iη*\_ *<sup>k</sup>* <sup>¼</sup> *<sup>η</sup>iη*\_*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>

infinitesimal displacement. Then, factorize *Ek y*

! þ *η* ! h i � � *Ek <sup>y</sup>*

*d dt*

*μ<sup>l</sup> y* !� �*Ek <sup>y</sup>*

!� �*Ek <sup>y</sup>*

!� �*Ek <sup>y</sup>*

! þ *η* ! � � *Ek <sup>y</sup>*

1 2

!� �*uk <sup>y</sup>*

*ukukρdv* <sup>þ</sup> *<sup>ε</sup>inρdv* � � <sup>¼</sup> *<sup>μ</sup><sup>l</sup> <sup>y</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.95255*

!� �*Ek <sup>y</sup>*

! þ *η* ! � �*Ek <sup>y</sup>*

<sup>þ</sup> *<sup>μ</sup><sup>e</sup> <sup>y</sup>*

*Cement-Based Piezoelectricity Application: A Theoretical Approach*

*dt*

<sup>¼</sup> *<sup>ρ</sup>dv <sup>d</sup> dt* 1 2

1 2

*ukukρdv* <sup>þ</sup> *<sup>ε</sup>inρdv* � � <sup>¼</sup> *<sup>ρ</sup>dv <sup>d</sup>*

*ukukρdv* <sup>þ</sup> *<sup>ε</sup>inρdv* � � <sup>¼</sup> *<sup>ρ</sup>dv uk*

!� �*dv* <sup>þ</sup> *<sup>μ</sup><sup>e</sup> <sup>y</sup>*

!� � <sup>þ</sup> *Ek <sup>y</sup>*

!� �*uk <sup>y</sup>*

!� �*dv* <sup>þ</sup> *<sup>μ</sup><sup>e</sup> <sup>y</sup>*

þ *ρ f <sup>k</sup>ukdv*

!� �*uk <sup>y</sup>*

The following steps from Eq. (10) conduct to develop conservation energy law. **In the first step**, we work on the **total energy term**. Then, the product rule to the

*ukuk* <sup>þ</sup> *<sup>ε</sup>in* � � <sup>þ</sup>

*ukuk* <sup>þ</sup> *<sup>ε</sup>in* � � <sup>þ</sup>

Then, here is writes the add of derivatives from kinetic and internal energies

**In the second step**, the **electric power term** will be developed by Taylor's

! þ *η* ! � �*Ek <sup>y</sup>*

From Eq. (14), it will solve the dot product between the electric field and

!� �*η*\_ *<sup>k</sup>* <sup>þ</sup> *Ek*,*<sup>i</sup> <sup>y</sup>*

!� �*dv* <sup>þ</sup> *Ek <sup>y</sup>*

2 *d dt η<sup>i</sup>*

h i*dv*

!� �*uk <sup>y</sup>*

! þ *η* ! � � *Ek <sup>y</sup>*

*dt*ð Þ *ρdv* represents the mass conservation. Therefore, this term is null

*dt*

*d*

1 2

*dt*ð Þþ *uk <sup>ρ</sup>dv <sup>d</sup>*

! þ *η* ! � � *uk <sup>y</sup>*

!� � <sup>þ</sup> *Ek*,*<sup>i</sup> <sup>y</sup>*

!� �*ηiuk <sup>y</sup>*

!� �*dv*

! þ *η* ! � � *uk <sup>y</sup>*

> 1 2

> 1 2

*ukuk* <sup>þ</sup> *<sup>ε</sup>in* � � *<sup>d</sup>*

*ukuk* <sup>þ</sup> *<sup>ε</sup>in* � � *<sup>d</sup>*

*dt*ð Þ *<sup>ρ</sup>dv*

*dt*ð Þ *<sup>ρ</sup>dv*

*dt <sup>ε</sup>in* � � (13)

!� � <sup>þ</sup> *<sup>η</sup>*\_*<sup>k</sup>* h i*dv*

!� �*ηiη*\_ *<sup>k</sup>*

(14)

(15)

(16)

*ukuk* <sup>þ</sup> *<sup>ε</sup>in* � � (12)

!� � <sup>þ</sup> *<sup>η</sup>*\_*<sup>k</sup>* h i*dv*

!� � <sup>þ</sup> *Ek*,*<sup>i</sup> <sup>y</sup>*

<sup>2</sup> ≈0 is zero, taking into account the

! h i � � *dv*

!� �*η*\_ *<sup>k</sup>* <sup>þ</sup> *Ek*,*<sup>i</sup> <sup>y</sup>*

!� �*dv* in Eq. (15), it takes a

!� �*ηiuk <sup>y</sup>*

!� �*η<sup>i</sup>* h i *uk <sup>y</sup>*

!� � <sup>þ</sup> *<sup>η</sup>*\_ *<sup>k</sup>* h i*dv* <sup>þ</sup> *tkukds*

(10)

(11)

where *f E <sup>k</sup>* is the electric force on the body and start from the dot product of polarization *P <sup>j</sup>* with the electric field gradient *Ek*,*<sup>i</sup>*; Besides, *c<sup>E</sup> <sup>i</sup>* is called electric coupling, and it is the cross product of *P <sup>j</sup>* with *Ek*; finally, *f <sup>k</sup>* is the external force on the body per unit mass.

Then, replacing the electric coupling *c<sup>E</sup> <sup>i</sup>* by the cross product between *P <sup>j</sup>* and *Ek*, we can rewrite Eq. (5) as follow

$$
\varepsilon\_{ijk} P\_j E\_k \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) + \varepsilon\_{ijk} \tau\_{jk} \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) = \mathbf{0} \tag{6}
$$

Factoring the permutation tensor *εijk*

$$
\varepsilon\_{ijk} \left[ P\_j E\_k + \tau\_{jk} \right] = \mathbf{0} \tag{7}
$$

The term in square brackets from Eq. (7) is a symmetry tensor that can be written as

$$
\pi^{S}\_{j\mathbb{k}} = P\_j E\_{\mathbb{k}} + \tau\_{j\mathbb{k}} \tag{8}
$$

The term *P jEk* is called the Maxwell stress tensor *T<sup>E</sup> jk*. On the other hand, multiplying Eq. (5) by *εiqr* is obtain

$$
\varepsilon\_{iqr}\mathfrak{c}\_i^E + \varepsilon\_{iqr}\mathfrak{e}\_{ijk}\mathfrak{r}\_{jk}\left(\stackrel{\rightarrow}{\mathcal{Y}}\right) = \mathbf{0}
$$

$$
\delta\_{iq}\delta\_{kr}\mathfrak{r}\_{jk}\left(\stackrel{\rightarrow}{\mathcal{Y}}\right) - \delta\_{jr}\delta\_{kq}\mathfrak{r}\_{jk}\left(\stackrel{\rightarrow}{\mathcal{Y}}\right) = -\varepsilon\_{iqr}\mathfrak{c}\_i^E \tag{9}
$$

$$
\mathfrak{r}\_{qr}\left(\stackrel{\rightarrow}{\mathcal{Y}}\right) - \mathfrak{r}\_{rq}\left(\stackrel{\rightarrow}{\mathcal{Y}}\right) = -\varepsilon\_{iqr}\mathfrak{c}\_i^E
$$

From Eq. (9) we can conclude that the Cauchy stress tensor (*tk* ¼ *n <sup>j</sup>τjk*) is asymmetry. Now, we get the total energy inside the continuum medium as a combination of kinetic and internal energy *εin* [13], both per mass unit. Here is performed the powers added due to the three forces above.

*Cement-Based Piezoelectricity Application: A Theoretical Approach DOI: http://dx.doi.org/10.5772/intechopen.95255*

*2.1.2 Energy conservation*

where *f*

written as

**146**

*E*

the body per unit mass.

we can rewrite Eq. (5) as follow

get the conservation laws, including energy.

Then, replacing the electric coupling *c<sup>E</sup>*

Factoring the permutation tensor *εijk*

multiplying Eq. (5) by *εiqr* is obtain

linear and angular momentum conservation, respectively.

*ρu*\_ *<sup>k</sup>* ¼ *f*

*Cement Industry - Optimization, Characterization and Sustainable Application*

*c E*

polarization *P <sup>j</sup>* with the electric field gradient *Ek*,*<sup>i</sup>*; Besides, *c<sup>E</sup>*

*εijkP jEk y*

!

*εijk P jEk* þ *τjk*

*τS*

The term *P jEk* is called the Maxwell stress tensor *T<sup>E</sup>*

*εiqrc E*

*τqr y* !

performed the powers added due to the three forces above.

!

*δjqδkrτjk y*

The term in square brackets from Eq. (7) is a symmetry tensor that can be

*<sup>i</sup>* þ *εiqrεijkτjk y*

� *τrq y* !

From Eq. (9) we can conclude that the Cauchy stress tensor (*tk* ¼ *n <sup>j</sup>τjk*) is asymmetry. Now, we get the total energy inside the continuum medium as a combination of kinetic and internal energy *εin* [13], both per mass unit. Here is

� *δjrδkqτjk y*

!

!

¼ �*εiqrc*

¼ 0

¼ �*εiqrc*

*E i*

*E i*

Once the body is deformed, electronic and lattice continua electric charges apply a quasi-static electric field. Tiersten et al. called it Maxwelliam electric field *Ek*. It is interacting on two continuum mediums producing an electrical force on each one. The other forces acting on the body are traction and body forces. The traction force *tk* per unit, the area is working on the surfaces of volume elements of the lattice (see **Figure 2**). Also, it can be written in terms of Cauchy stress tensor *τjk* as *tk* ¼ *n <sup>j</sup>τjk*, where *n <sup>j</sup>* is the normal vector. Moreover, body force refers to an external force acting on the body, for example, gravity. The previous three forces are necessary to

From the three forces above, it is possible to construct the Eqs. (4) and (5),

*<sup>k</sup>* þ *τmk*,*<sup>m</sup>* þ *ρ f <sup>k</sup>* (4)

¼ 0 (5)

*<sup>i</sup>* by the cross product between *P <sup>j</sup>* and *Ek*,

<sup>¼</sup> <sup>0</sup> (7)

*jk* ¼ *P jEk* þ *τjk* (8)

¼ 0 (6)

*jk*. On the other hand,

(9)

*<sup>i</sup>* is called electric

*E*

*<sup>i</sup>* þ *εijkτjk y*

!

*<sup>k</sup>* is the electric force on the body and start from the dot product of

þ *εijkτjk y*

!

coupling, and it is the cross product of *P <sup>j</sup>* with *Ek*; finally, *f <sup>k</sup>* is the external force on

$$\begin{split} \frac{d}{dt} \left( \frac{1}{2} u\_k u\_k \rho dv + \epsilon^{in} \rho dv \right) &= \mu^l \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) E\_k \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) u\_k \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) dv \\ &+ \mu^\ell \left( \stackrel{\rightarrow}{\mathcal{Y}} + \stackrel{\rightarrow}{\eta} \right) E\_k \left( \stackrel{\rightarrow}{\mathcal{Y}} + \stackrel{\rightarrow}{\eta} \right) \left[ u\_k \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) + \dot{\eta}\_k \right] dv + t\_k u\_k ds \\ &+ \rho f\_k u\_k dv \end{split} \tag{10}$$

The following steps from Eq. (10) conduct to develop conservation energy law. **In the first step**, we work on the **total energy term**. Then, the product rule to the total energy term is applied:

$$\begin{split} \frac{d}{dt} \left( \frac{1}{2} u\_k u\_k \rho d\upsilon + \epsilon^{in} \rho d\upsilon \right) &= \rho d\upsilon \frac{d}{dt} \left( \frac{1}{2} u\_k u\_k + \epsilon^{in} \right) + \left( \frac{1}{2} u\_k u\_k + \epsilon^{in} \right) \frac{d}{dt} (\rho d\upsilon) \\ &= \rho d\upsilon \frac{d}{dt} \left( \frac{1}{2} u\_k u\_k + \epsilon^{in} \right) + \left( \frac{1}{2} u\_k u\_k + \epsilon^{in} \right) \frac{d}{dt} (\rho d\upsilon) \end{split} \tag{11}$$

The term *<sup>d</sup> dt*ð Þ *ρdv* represents the mass conservation. Therefore, this term is null in Eq. (11).

$$\frac{d}{dt}\left(\frac{1}{2}u\_k u\_k \rho dv + \varepsilon^{in} \rho dv\right) = \rho dv \frac{d}{dt}\left(\frac{1}{2}u\_k u\_k + \varepsilon^{in}\right) \tag{12}$$

Then, here is writes the add of derivatives from kinetic and internal energies

$$\frac{d}{dt}\left(\frac{1}{2}\mu\_k\mu\_k\rho dv + \varepsilon^{in}\rho dv\right) = \rho dv \, u\_k \frac{d}{dt}(u\_k) + \rho dv \frac{d}{dt}\left(\varepsilon^{in}\right) \tag{13}$$

**In the second step**, the **electric power term** will be developed by Taylor's expansion.

$$\begin{aligned} \mu^l \begin{pmatrix} \vec{\boldsymbol{y}} \end{pmatrix} \boldsymbol{E}\_k \begin{pmatrix} \vec{\boldsymbol{y}} \end{pmatrix} \boldsymbol{u}\_k \begin{pmatrix} \vec{\boldsymbol{y}} \end{pmatrix} \boldsymbol{dv} + \mu^\epsilon \begin{pmatrix} \vec{\boldsymbol{y}} + \vec{\boldsymbol{\eta}} \end{pmatrix} \boldsymbol{E}\_k \begin{pmatrix} \vec{\boldsymbol{y}} + \vec{\boldsymbol{\eta}} \end{pmatrix} \left[ \boldsymbol{u}\_k \begin{pmatrix} \vec{\boldsymbol{y}} \end{pmatrix} + \boldsymbol{\dot{\eta}}\_k \right] \boldsymbol{dv} \\ \approx \mu^l \begin{pmatrix} \vec{\boldsymbol{y}} \end{pmatrix} \boldsymbol{E}\_k \begin{pmatrix} \vec{\boldsymbol{y}} \end{pmatrix} \boldsymbol{u}\_k \begin{pmatrix} \vec{\boldsymbol{y}} \end{pmatrix} \boldsymbol{dv} + \mu^\epsilon \begin{pmatrix} \vec{\boldsymbol{y}} \end{pmatrix} \boldsymbol{E}\_k \begin{pmatrix} \vec{\boldsymbol{y}} \end{pmatrix} + \boldsymbol{E}\_{k,i} \begin{pmatrix} \vec{\boldsymbol{y}} \end{pmatrix} \boldsymbol{\eta}\_i \end{aligned} \tag{14}$$

From Eq. (14), it will solve the dot product between the electric field and velocity,

$$\begin{aligned} \rho &= \mu^l \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \boldsymbol{E}\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \boldsymbol{u}\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) d\boldsymbol{v} \\ &+ \mu^r \left( \overrightarrow{\boldsymbol{\mathcal{y}}} + \overrightarrow{\boldsymbol{\eta}} \right) \left[ \boldsymbol{E}\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \boldsymbol{u}\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) + \boldsymbol{E}\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \boldsymbol{\dot{\eta}}\_k + \boldsymbol{E}\_{k,i} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \boldsymbol{\eta}\_i \boldsymbol{u}\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) + \boldsymbol{E}\_{k,i} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \boldsymbol{\eta}\_i \boldsymbol{\dot{\eta}}\_k \right] d\boldsymbol{v} \end{aligned} \tag{15}$$

The second-order term *<sup>η</sup>iη*\_ *<sup>k</sup>* <sup>¼</sup> *<sup>η</sup>iη*\_*<sup>i</sup>* <sup>¼</sup> <sup>1</sup> 2 *d dt η<sup>i</sup>* <sup>2</sup> ≈0 is zero, taking into account the infinitesimal displacement. Then, factorize *Ek y* !� �*uk <sup>y</sup>* !� �*dv* in Eq. (15), it takes a form:

$$= \left[ \mu^{l} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) + \mu^{\epsilon} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} + \overrightarrow{\boldsymbol{\eta}} \right) \right] E\_{k} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \boldsymbol{u}\_{k} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) d\boldsymbol{v} + \left[ E\_{k} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \dot{\boldsymbol{\eta}}\_{k} + E\_{k,i} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \boldsymbol{\eta}\_{i} \boldsymbol{u}\_{k} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \right] d\boldsymbol{v} \tag{16}$$

$$
\hat{\mu} = \mu^{\epsilon} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} + \overrightarrow{\boldsymbol{\eta}} \right) E\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \dot{\boldsymbol{\eta}}\_k d\boldsymbol{v} + \mu^{\epsilon} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} + \overrightarrow{\boldsymbol{\eta}} \right) E\_{k,i} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \boldsymbol{\eta}\_i \boldsymbol{u}\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) d\boldsymbol{v} \tag{17}
$$

$$
\hat{\mu} = \mu^{\xi} \left( \stackrel{\rightarrow}{\mathcal{Y}} + \stackrel{\rightarrow}{\eta} \right) E\_k \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) \dot{\eta}\_k dv + P\_i E\_{k,i} \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) u\_k \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) dv \tag{18}
$$

$$\mu^l \left( \overrightarrow{\boldsymbol{y}} \right) E\_k \left( \overrightarrow{\boldsymbol{y}} \right) \boldsymbol{u}\_k \left( \overrightarrow{\boldsymbol{y}} \right) d\boldsymbol{v} + \mu^c \left( \overrightarrow{\boldsymbol{y}} + \overrightarrow{\boldsymbol{\eta}} \right) E\_k \left( \overrightarrow{\boldsymbol{y}} + \overrightarrow{\boldsymbol{\eta}} \right) \left[ \boldsymbol{u}\_k \left( \overrightarrow{\boldsymbol{y}} \right) + \dot{\boldsymbol{\eta}}\_k \right] d\boldsymbol{v} = \boldsymbol{w}^E d\boldsymbol{v} + f\_k \, ^E \boldsymbol{u}\_k \left( \overrightarrow{\boldsymbol{y}} \right) d\boldsymbol{v} \tag{19}$$

$$\begin{split} \tau\_{1k} \left( \overrightarrow{\boldsymbol{y}} + \frac{1}{2} d\boldsymbol{y}\_1 \hat{\boldsymbol{1}}\_1 \right) \boldsymbol{u}\_k \left( \overrightarrow{\boldsymbol{y}} + \frac{1}{2} d\boldsymbol{y}\_1 \hat{\boldsymbol{1}}\_1 \right) d\boldsymbol{y}\_2 d\boldsymbol{y}\_3 - \tau\_{1k} \left( \overrightarrow{\boldsymbol{y}} - \frac{1}{2} d\boldsymbol{y}\_1 \hat{\boldsymbol{1}}\_1 \right) \boldsymbol{u}\_k \left( \overrightarrow{\boldsymbol{y}} - \frac{1}{2} d\boldsymbol{y}\_1 \hat{\boldsymbol{1}}\_1 \right) d\boldsymbol{y}\_2 d\boldsymbol{y}\_3 \\ \approx \left[ \tau\_{1k} \left( \overrightarrow{\boldsymbol{y}} \right) + \frac{1}{2} d\boldsymbol{y}\_1 \tau\_{1k,1} \left( \overrightarrow{\boldsymbol{y}}^\cdot \right) \right] \left[ \boldsymbol{u}\_k \left( \overrightarrow{\boldsymbol{y}} \right) + \frac{1}{2} d\boldsymbol{y}\_1 \boldsymbol{u}\_{k,1} \left( \overrightarrow{\boldsymbol{y}}^\cdot \right) \right] d\boldsymbol{y}\_2 d\boldsymbol{y}\_3 \\ - \left[ \tau\_{1k} \left( \overrightarrow{\boldsymbol{y}}^\cdot \right) - \frac{1}{2} d\boldsymbol{y}\_1 \tau\_{1k,1} \left( \overrightarrow{\boldsymbol{y}}^\cdot \right) \right] \left[ \boldsymbol{u}\_k \left( \overrightarrow{\boldsymbol{y}} \right) - \frac{1}{2} d\boldsymbol{y}\_1 \boldsymbol{u}\_{k,1} \left( \overrightarrow{\boldsymbol{y}}^\cdot \right) \right] d\boldsymbol{y}\_2 d\boldsymbol{y}\_3 \end{split} \tag{21}$$

$$\begin{aligned} \approx \left[ \tau\_{1k} u\_k + \tau\_{1k} \frac{1}{2} d\boldsymbol{\eta}\_1 \boldsymbol{u}\_{k,1} + \frac{1}{2} d\boldsymbol{\eta}\_1 \tau\_{1k,1} \boldsymbol{u}\_k + \frac{1}{4} d\boldsymbol{\eta}\_1 \,^2 \tau\_{1k,1} \boldsymbol{u}\_{k,1} \right] d\boldsymbol{\eta}\_2 d\boldsymbol{\eta}\_3 \\ - \left[ \tau\_{1k} \boldsymbol{u}\_k - \tau\_{1k} \frac{1}{2} d\boldsymbol{\eta}\_1 \boldsymbol{u}\_{k,1} - \frac{1}{2} d\boldsymbol{\eta}\_1 \tau\_{1k,1} \boldsymbol{u}\_k + \frac{1}{4} d\boldsymbol{\eta}\_1 \,^2 \tau\_{1k,1} \boldsymbol{u}\_{k,1} \right] d\boldsymbol{\eta}\_2 d\boldsymbol{\eta}\_3 \end{aligned} \tag{22}$$

$$
\approx \left[ \tau\_{1k} \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) dy\_1 u\_{k,1} \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) + dy\_1 \tau\_{1k,1} \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) u\_k \left( \stackrel{\rightarrow}{\mathcal{Y}} \right) \right] dy\_2 dy\_3 \tag{23}
$$

$$\approx \tau\_{1k} \left( \overrightarrow{\mathcal{Y}} \right) u\_{k,1} \left( \overrightarrow{\mathcal{Y}} \right) dv + \tau\_{1k,1} \left( \overrightarrow{\mathcal{Y}} \right) u\_k \left( \overrightarrow{\mathcal{Y}} \right) dv \tag{24}$$

$$\begin{split} t\_k u\_k ds &= \tau\_{1k} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) u\_{k,1} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) dv + \tau\_{1k,1} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) u\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) dv + \tau\_{2k} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) u\_{k,2} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) dv \\ &+ \tau\_{2k,2} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) u\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) dv + \tau\_{3k} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) u\_{k,3} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) dv + \tau\_{3k,3} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) u\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) dv \end{split} \tag{25}$$

$$
\tau\_k u\_k d\mathbf{s} = \tau\_{mk} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) u\_{k,m} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) d\boldsymbol{v} + \tau\_{mk,m} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) u\_k \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) d\boldsymbol{v} \tag{26}
$$

$$\begin{split} \rho \boldsymbol{dv} \, u\_k \frac{d}{dt} (\boldsymbol{u}\_k) + \rho \boldsymbol{dv} \frac{d}{dt} (\boldsymbol{\varepsilon}^{in}) &= \boldsymbol{w}^E \boldsymbol{d} \boldsymbol{v} + \boldsymbol{f}\_k \, ^E \boldsymbol{u}\_k \left( \stackrel{\cdot}{\boldsymbol{y}} \right) \boldsymbol{d} \boldsymbol{v} + \boldsymbol{\tau}\_{mk} \left( \stackrel{\cdot}{\boldsymbol{y}} \right) \boldsymbol{u}\_{k,m} \left( \stackrel{\cdot}{\boldsymbol{y}} \right) \boldsymbol{d} \boldsymbol{v} \\ &+ \boldsymbol{\tau}\_{mk,m} \left( \stackrel{\cdot}{\boldsymbol{y}} \right) \boldsymbol{u}\_k \left( \stackrel{\cdot}{\boldsymbol{y}} \right) \boldsymbol{d} \boldsymbol{v} + \rho \boldsymbol{f}\_k \boldsymbol{u}\_k \boldsymbol{d} \boldsymbol{v} \end{split} \tag{27}$$

$$
\left[\rho \frac{d}{dt}(u\_k) + f\_k^{\ E} + \tau\_{mk,m} \left(\overrightarrow{\boldsymbol{\mathcal{y}}}\right) + \rho f\_k\right] u\_k \left(\overrightarrow{\boldsymbol{\mathcal{y}}}\right) dv = \left\{\boldsymbol{w}^{E} - \rho \frac{d}{dt} \left[\boldsymbol{\varepsilon}^{in}\right] + \tau\_{mk} \left(\overrightarrow{\boldsymbol{\mathcal{y}}}\right) u\_{k,m} \left(\overrightarrow{\boldsymbol{\mathcal{y}}}\right)\right\} dv\tag{28}
$$

$$
\rho \dot{\varepsilon}^{\dot{m}} = \omega^E + \tau\_{mk} \left( \overrightarrow{\mathcal{Y}} \right) u\_{k,m} \left( \overrightarrow{\mathcal{Y}} \right) \tag{29}
$$

$$m^E = \mu^\varepsilon \left(\overrightarrow{\boldsymbol{\mathcal{Y}}} + \overrightarrow{\boldsymbol{\eta}}\right) E\_m\left(\overrightarrow{\boldsymbol{\mathcal{Y}}}\right) \frac{d\boldsymbol{\eta}\_m}{dt} = E\_m\left(\overrightarrow{\boldsymbol{\mathcal{Y}}}\right) \left\{ \frac{d}{dt} \left[\mu^\varepsilon \left(\overrightarrow{\boldsymbol{\mathcal{Y}}} + \overrightarrow{\boldsymbol{\eta}}\right) \boldsymbol{\eta}\_m\right] - \frac{d\mu^\varepsilon \left(\overrightarrow{\boldsymbol{\mathcal{Y}}} + \overrightarrow{\boldsymbol{\eta}}\right)}{dt} \boldsymbol{\eta}\_m \right\}$$

*Cement Industry - Optimization, Characterization and Sustainable Application*

$$
\mu w^E = E\_m \left( \overrightarrow{\mathcal{Y}} \right) \left\{ \dot{P}\_m - \dot{\mu}^\epsilon \left( \overrightarrow{\mathcal{Y}} + \overrightarrow{\eta} \right) \eta\_m \right\} \tag{30}
$$

polarization vector), and mechanical (Cauchy stress tensor) components, as is

! *uk*,*<sup>m</sup> <sup>y</sup>*

**2.2 Transformation of fundamental physical quantities in piezoelectricity to**

There are several reasons to consider two coordinate systems (reference and current state) for continuum. Firstly, it is not mathematically simple to describe the movement of each particle that compounds a continuum as seen on the gradient of velocity *uk*,*<sup>m</sup>* in Eq. (38); it is more appropriate to propose a coordinate system that describes the continuum in the reference system. The material behavior could be affected by the characteristics of the current state, too. For example, fluids and solids can change their mechanical behavior while changing the shape [14]. Hence, we refer to our study material (cement-based composites) whom we know the physical properties in the reference state *XL*. To explain the behavior of a material, we must include physical quantities respect the reference state *XL*: potential gradient *WK*, polarization *PL*, electric displacement D*L*, volume free charge

state, and relate the traction force with areas, both in the reference state. While the first Piola-Kirchhoff stress is connecting the traction force and electric force in the

To transform the electric field to a reference state, here will use follow:

The gradient of the potential *WK* is multiplying both sides by *XK*,*<sup>m</sup>*

*WKXK*,*<sup>m</sup>* ¼ *Emym*,*<sup>K</sup>XK*,*<sup>m</sup>* ¼ *E*<sup>m</sup>

*d*

*dt*ð Þ *WK XK*,*<sup>m</sup>* <sup>þ</sup> *WK*

This section will describe the **transformation of energy conservation from the current state to the reference state**, using Eq. (8), the symmetric tensor modifies

! � *<sup>E</sup>*\_ *mPm* (38)

*mk* in the current state to reference

*mkuk*,*<sup>m</sup>* � *PmEkuk*,*<sup>m</sup>* � *<sup>E</sup>*\_ *mPm* (39)

*WK* ¼ *Emym*,*<sup>K</sup>* (40)

*∂XK <sup>∂</sup>ym*

*Em* ¼ *WKXK*,*<sup>m</sup>* (42)

*d*

*dt*ð Þ *XK*,*<sup>m</sup>* (43)

*<sup>∂</sup>ym ∂XK* *KL* [15]. It raises

(41)

*ρχ*\_ ¼ *τmk y*

*Cement-Based Piezoelectricity Application: A Theoretical Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.95255*

density *ρ<sup>E</sup>*, mass density *ρ*<sup>0</sup> and the second Piola-Kirchhoff stress *T<sup>S</sup>*

by the transformation of symmetric tensor *τ<sup>S</sup>*

current state with regions in the reference state.

*2.2.1 Electric field and gradient of potential*

*ρχ*\_ <sup>¼</sup> *<sup>τ</sup><sup>S</sup>*

The derivative respect to time of *Em* becomes

*dt* ½ �¼ *WKXK*,*<sup>m</sup>*

The term *XK*,*<sup>m</sup>* from Eq. (43) is developed as follow

*<sup>E</sup>*\_ *<sup>m</sup>* <sup>¼</sup> *<sup>d</sup>*

shown in Eq. (38).

Eq. (38).

Therefore,

**151**

**the reference state**

Another form of electric charge conservation is *μ*\_ *<sup>e</sup> y* ! þ *η* ! � � <sup>þ</sup> *<sup>μ</sup><sup>e</sup> <sup>y</sup>* ! þ *η* ! � �*ui*,*<sup>i</sup>* <sup>¼</sup> 0, it will simplify the Eq. (30) to:

$$
\mu \boldsymbol{\nu}^{E} = E\_m \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \left\{ \dot{P}\_m + \mu^\epsilon \left( \overrightarrow{\boldsymbol{\mathcal{y}}} + \overrightarrow{\boldsymbol{\eta}} \right) \boldsymbol{u}\_{i,i} \boldsymbol{\eta}\_m \right\} \tag{31}
$$

The mass conservation *ρ*\_ þ *ρui*,*<sup>i</sup>* ¼ 0 has a similar mathematical structure as charge conservation. Therefore, the gradient of the speed *ui*,*<sup>i</sup>* in Eq. (31) was replaced

$$\boldsymbol{w}^{E} = E\_m \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \dot{\boldsymbol{P}}\_m + E\_m \left( \overrightarrow{\boldsymbol{\mathcal{y}}} \right) \boldsymbol{\mu}^{\ell} \left( \overrightarrow{\boldsymbol{\mathcal{y}}} + \overrightarrow{\boldsymbol{\eta}} \right) \frac{-\dot{\boldsymbol{\rho}}}{\rho} \eta\_m = E\_m \dot{\boldsymbol{P}}\_m - \frac{\dot{\boldsymbol{\rho}}}{\rho} E\_m \boldsymbol{P}\_m$$

$$\boldsymbol{w}^{E} = \frac{E\_m}{\rho} \left[ \rho \dot{\boldsymbol{P}}\_m - \dot{\boldsymbol{\rho}} \boldsymbol{P}\_m \right] \tag{32}$$

Eq. (32) has been used on Eq. (29)

$$
\rho \dot{\varepsilon}^{in} = \frac{E\_m}{\rho} \left[ \rho \dot{P}\_m - \dot{\rho} P\_m \right] + \tau\_{mk} \left( \overrightarrow{\mathcal{Y}} \right) u\_{k,m} \left( \overrightarrow{\mathcal{Y}} \right) \tag{33}
$$

With Legendre transformation showing in Eq. (34), Tiersten replaced the internal energy *εin* by free energy *χ* [5]. This transformation diminishes the number of constitutional equations. Besides, it offers a quantitative interpretation that can not get from the internal energy resulting in more useful for those who perform piezoelectricity experiments. After Section 2.3, we could see the *χ* will depend on the gradient of potential in the reference state and deformation tensor.

$$\chi = \epsilon^{in} - E\_m \frac{P\_m}{\rho} \tag{34}$$

Upon differentiating respect to time the Eq. (34).

$$
\dot{\chi} = \dot{\varepsilon}^{in} - \dot{E}\_m \frac{P\_m}{\rho} - E\_m \frac{\dot{P}\_m}{\rho} + E\_m \frac{P\_m}{\rho^2} \dot{\rho} \tag{35}
$$

Clear the term *ρε*\_ *in*

$$
\rho \dot{\varepsilon}^{in} = \rho \dot{\chi} + \dot{E}\_m P\_m + E\_m \dot{P}\_m - E\_m \frac{P\_m}{\rho} \dot{\rho} \tag{36}
$$

Using Eq. (36) on Eq. (29), we obtain:

$$\begin{aligned} \rho \dot{\chi} + \dot{E}\_m P\_m + E\_m \dot{P}\_m - E\_m \frac{P\_m}{\rho} \dot{\rho} &= \frac{E\_m}{\rho} \left[ \rho \dot{P}\_m - \dot{\rho} P\_m \right] + \tau\_{mk} \left( \overline{\dot{\mathcal{V}}} \right) u\_{k,m} \left( \overline{\dot{\mathcal{V}}} \right) \\ \rho \dot{\chi} + \dot{E}\_m P\_m + E\_m \dot{P}\_m - E\_m \dot{\rho} \frac{P\_m}{\rho} &= E\_m \dot{P}\_m - E\_m \dot{\rho} \frac{P\_m}{\rho} + \tau\_{mk} \left( \overline{\dot{\mathcal{V}}} \right) u\_{k,m} \left( \overline{\dot{\mathcal{V}}} \right) \end{aligned} \tag{37}$$

The similar terms *EmP*\_ *<sup>m</sup>* and �*Emρ*\_ *Pm <sup>ρ</sup>* in Eq. (37), are clear. Finally, we have rewritten **energy conservation** in terms of free energy, electrical (electric field and *Cement-Based Piezoelectricity Application: A Theoretical Approach DOI: http://dx.doi.org/10.5772/intechopen.95255*

*<sup>w</sup><sup>E</sup>* <sup>¼</sup> *Em <sup>y</sup>*

Another form of electric charge conservation is *μ*\_

*<sup>w</sup><sup>E</sup>* <sup>¼</sup> *Em <sup>y</sup>*

*<sup>P</sup>*\_ *<sup>m</sup>* <sup>þ</sup> *Em <sup>y</sup>*

it will simplify the Eq. (30) to:

*<sup>w</sup><sup>E</sup>* <sup>¼</sup> *Em <sup>y</sup>*

!� �

Eq. (32) has been used on Eq. (29)

*ρε*\_

*in* <sup>¼</sup> *Em ρ*

replaced

!� �

*Cement Industry - Optimization, Characterization and Sustainable Application*

!� �

!� � *μ<sup>e</sup> y* ! þ *η* ! � � �*ρ*\_

*<sup>w</sup><sup>E</sup>* <sup>¼</sup> *Em ρ*

*<sup>ρ</sup>P*\_ *<sup>m</sup>* � *<sup>ρ</sup>*\_*Pm* � � <sup>þ</sup> *<sup>τ</sup>mk <sup>y</sup>*

gradient of potential in the reference state and deformation tensor.

*in* � *<sup>E</sup>*\_ *<sup>m</sup>*

*Pm ρ*

*Pm*

*<sup>ρ</sup>*\_ <sup>¼</sup> *Em ρ*

Upon differentiating respect to time the Eq. (34).

*χ*\_ ¼ *ε*\_

*in*

*ρε*\_

Using Eq. (36) on Eq. (29), we obtain:

*ρχ*\_ <sup>þ</sup> *<sup>E</sup>*\_ *mPm* <sup>þ</sup> *EmP*\_ *<sup>m</sup>* � *Em*

*ρχ*\_ <sup>þ</sup> *<sup>E</sup>*\_ *mPm* <sup>þ</sup> *EmP*\_ *<sup>m</sup>* � *Emρ*\_

The similar terms *EmP*\_ *<sup>m</sup>* and �*Emρ*\_ *Pm*

Clear the term *ρε*\_

**150**

*<sup>P</sup>*\_ *<sup>m</sup>* � *<sup>μ</sup>*\_

*<sup>P</sup>*\_ *<sup>m</sup>* <sup>þ</sup> *<sup>μ</sup><sup>e</sup> <sup>y</sup>*

The mass conservation *ρ*\_ þ *ρui*,*<sup>i</sup>* ¼ 0 has a similar mathematical structure as charge conservation. Therefore, the gradient of the speed *ui*,*<sup>i</sup>* in Eq. (31) was

*<sup>e</sup> y* ! <sup>þ</sup> *<sup>η</sup>* ! � �

n o

! <sup>þ</sup> *<sup>η</sup>* ! � �

*ρ*

!� �

*<sup>ρ</sup>P*\_ *<sup>m</sup>* � *<sup>ρ</sup>*\_*Pm*

With Legendre transformation showing in Eq. (34), Tiersten replaced the internal energy *εin* by free energy *χ* [5]. This transformation diminishes the number of constitutional equations. Besides, it offers a quantitative interpretation that can not get from the internal energy resulting in more useful for those who perform piezoelectricity experiments. After Section 2.3, we could see the *χ* will depend on the

*<sup>χ</sup>* <sup>¼</sup> *<sup>ε</sup>in* � *Em*

*Pm <sup>ρ</sup>* � *Em*

*in* <sup>¼</sup> *ρχ*\_ <sup>þ</sup> *<sup>E</sup>*\_ *mPm* <sup>þ</sup> *EmP*\_ *<sup>m</sup>* � *Em*

*<sup>ρ</sup>* <sup>¼</sup> *EmP*\_ *<sup>m</sup>* � *Emρ*\_

rewritten **energy conservation** in terms of free energy, electrical (electric field and

*Pm ρ*

*P*\_ *m ρ* þ *Em*

*<sup>ρ</sup>P*\_ *<sup>m</sup>* � *<sup>ρ</sup>*\_*Pm* � � <sup>þ</sup> *<sup>τ</sup>mk <sup>y</sup>*

> *Pm ρ*

*Pm*

*Pm ρ*

þ *τmk y* !� �

*<sup>ρ</sup>* in Eq. (37), are clear. Finally, we have

!� �

*<sup>ρ</sup>*<sup>2</sup> *<sup>ρ</sup>*\_ (35)

*ρ*\_ (36)

*uk*,*<sup>m</sup> y* !� �

*uk*,*<sup>m</sup> y* !� �

n o

*ηm*

*ui*,*iη<sup>m</sup>*

*<sup>η</sup><sup>m</sup>* <sup>¼</sup> *EmP*\_ *<sup>m</sup>* � *<sup>ρ</sup>*\_

*uk*,*<sup>m</sup> y* !� �

� � (32)

<sup>þ</sup> *<sup>μ</sup><sup>e</sup> <sup>y</sup>*

*ρ EmPm*

! þ *η* ! � �

*<sup>e</sup> y* ! þ *η* ! � � (30)

*ui*,*<sup>i</sup>* ¼ 0,

(31)

(33)

(34)

(37)

polarization vector), and mechanical (Cauchy stress tensor) components, as is shown in Eq. (38).

$$
\rho \dot{\chi} = \tau\_{mk} \left( \overrightarrow{\dot{\chi}} \right) u\_{k,m} \left( \overrightarrow{\dot{\chi}} \right) - \dot{E}\_m P\_m \tag{38}
$$

## **2.2 Transformation of fundamental physical quantities in piezoelectricity to the reference state**

There are several reasons to consider two coordinate systems (reference and current state) for continuum. Firstly, it is not mathematically simple to describe the movement of each particle that compounds a continuum as seen on the gradient of velocity *uk*,*<sup>m</sup>* in Eq. (38); it is more appropriate to propose a coordinate system that describes the continuum in the reference system. The material behavior could be affected by the characteristics of the current state, too. For example, fluids and solids can change their mechanical behavior while changing the shape [14]. Hence, we refer to our study material (cement-based composites) whom we know the physical properties in the reference state *XL*. To explain the behavior of a material, we must include physical quantities respect the reference state *XL*: potential gradient *WK*, polarization *PL*, electric displacement D*L*, volume free charge density *ρ<sup>E</sup>*, mass density *ρ*<sup>0</sup> and the second Piola-Kirchhoff stress *T<sup>S</sup> KL* [15]. It raises by the transformation of symmetric tensor *τ<sup>S</sup> mk* in the current state to reference state, and relate the traction force with areas, both in the reference state. While the first Piola-Kirchhoff stress is connecting the traction force and electric force in the current state with regions in the reference state.

This section will describe the **transformation of energy conservation from the current state to the reference state**, using Eq. (8), the symmetric tensor modifies Eq. (38).

$$
\rho \dot{\chi} = \pi^S{}\_{mk} u\_{k,m} - P\_m E\_k u\_{k,m} - \dot{E}\_m P\_m \tag{39}
$$

#### *2.2.1 Electric field and gradient of potential*

To transform the electric field to a reference state, here will use follow:

$$\mathcal{W}\_K = E\_m \mathcal{y}\_{m,K} \tag{40}$$

The gradient of the potential *WK* is multiplying both sides by *XK*,*<sup>m</sup>*

$$\delta W\_{K} X\_{K,m} = E\_m \mathcal{Y}\_{m,K} X\_{K,m} = E\_m \frac{\partial \mathcal{Y}\_m}{\partial X\_K} \frac{\partial X\_K}{\partial \mathcal{Y}\_m} \tag{41}$$

Therefore,

$$E\_m = W\_K X\_{K,m} \tag{42}$$

The derivative respect to time of *Em* becomes

$$\dot{E}\_m = \frac{d}{dt} \left[ \mathcal{W}\_K \mathcal{X}\_{K,m} \right] = \frac{d}{dt} (\mathcal{W}\_K) \mathcal{X}\_{K,m} + \mathcal{W}\_K \frac{d}{dt} (\mathcal{X}\_{K,m}) \tag{43}$$

The term *XK*,*<sup>m</sup>* from Eq. (43) is developed as follow

*Cement Industry - Optimization, Characterization and Sustainable Application*

$$X\_{K,m} = \delta\_{KL} X\_{L,m} = \frac{\partial \mathcal{Y}\_k}{\partial X\_L} \frac{\partial X\_K}{\partial \mathcal{Y}\_k} X\_{L,m} \tag{44}$$

From Eq. (54) the term *PmEkuk*,*<sup>m</sup>* was removes to get

*Cement-Based Piezoelectricity Application: A Theoretical Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.95255*

Where *J* is the Jacobian, multiplying Eq. (56) by *J*

*ρχ*\_ <sup>¼</sup> *<sup>τ</sup><sup>S</sup>*

*ρχ*\_ <sup>¼</sup> *<sup>τ</sup><sup>S</sup>*

missing transform the symmetric Cauchy stress tensor *τ<sup>S</sup>*

*ρχ*\_ <sup>¼</sup> *<sup>τ</sup><sup>S</sup>*

*ρχ*\_ <sup>¼</sup> *<sup>τ</sup><sup>S</sup>*

*τS mk* ¼ *J* �1

*ym*,*Kyk*,*LT<sup>S</sup>*

*KL*ð Þ� *ωmk* þ *dmk J*

�1

*KLωmk* þ *J*

From Eq. (61), the product between symmetric tensor *T<sup>S</sup>*

*ρχ*\_ ¼ *J* �1

<sup>2</sup> ð Þ *uk*,*<sup>m</sup>* � *um*,*<sup>k</sup>* plus a symmetric tensor *dmk* <sup>¼</sup> <sup>1</sup>

*ym*,*Kyk*,*LT<sup>S</sup>*

*ym*,*Kyk*,*LT<sup>S</sup>*

*ym*,*<sup>L</sup>PL* ¼ *J*

�1

*Pm* ¼ *J* �1

*mkuk*,*<sup>m</sup>* � *J*

*mkuk*,*<sup>m</sup>* � *J*

*mkuk*,*<sup>m</sup>* � *J*

*mkuk*,*<sup>m</sup>* � *J*

Until now, in Eq. (59), we have obtained a partial transformation, and still

�1

�1 *PLW*\_ *<sup>K</sup>*

�1

*ym*,*Kyk*,*LT<sup>S</sup>*

*KLuk*,*<sup>m</sup>* � *J*

�1 *PKW*\_ *<sup>K</sup>*

*ym*,*Kyk*,*LT<sup>S</sup>*

The gradient of velocity *uk*,*<sup>m</sup>* can be separated on antisymmetric tensor *ωmk* ¼

�1 *PKW*\_ *<sup>K</sup>*

*J* �1

From Eq. (58) into Eq. (55) results in

*2.2.3 Second Piola-Kirchhoff stress*

The symmetric tensor *τ<sup>S</sup>*

*ρχ*\_ ¼ *J* �1

*ρχ*\_ ¼ *J* �1

tensor *ωmk* result be null

1

**153**

through a reverse transformation as follow:

Eq. (60) into Eq. (59) results in

*2.2.2 Polarization vector*

to the reference state.

To get

*ρχ*\_ <sup>¼</sup> *<sup>τ</sup><sup>S</sup>*

In this subsection, we will perform the transformation of the polarization vector

*mkuk*,*<sup>m</sup>* � *PmW*\_ *KXK*,*<sup>m</sup>* (55)

*PL* ¼ *JXL*,*iPi* (56)

*ym*,*<sup>L</sup>* we obtain

*ym*,*<sup>L</sup>PL* (58)

(59)

*KL*

(62)

*KL* and antisymmetric

*ym*,*<sup>L</sup>JXL*,*iPi* ¼ *δmiPi* (57)

�1

*ym*,*<sup>L</sup>PLW*\_ *KXK*,*<sup>m</sup>*

*PLW*\_ *<sup>K</sup>δKL*

*mk* is related with second Piola-Kirchhoff stress *T<sup>S</sup>*

�1

<sup>2</sup> ð Þ *uk*,*<sup>m</sup>* þ *um*,*<sup>k</sup>* .

*KLdmk* � *J*

�1 *PKW*\_ *<sup>K</sup>*

*<sup>∂</sup>ym ∂XL* *∂XK <sup>∂</sup>ym*

*mk*.

*KL* (60)

*PKW*\_ *<sup>K</sup>* (61)

Derivative *XK*,*<sup>m</sup>* respect to time

$$\frac{d}{dt}(X\_{K,m}) = \frac{d}{dt}\left(\frac{\partial \mathbf{y}\_k}{\partial X\_L}\frac{\partial X\_K}{\partial \mathbf{y}\_k}X\_{L,m}\right) = \frac{d}{dt}\left(\mathbf{y}\_{k,L}X\_{K,k}X\_{L,m}\right) \tag{45}$$

$$\frac{d}{dt}(X\_{K,m}) = \frac{d}{dt}\left(\mathbf{y}\_{k,L}\right)X\_{K,k}X\_{L,m} + \frac{d}{dt}(X\_{K,k})\mathbf{y}\_{k,L}X\_{L,m} + \frac{d}{dt}(X\_{L,m})\mathbf{y}\_{k,L}X\_{K,k}$$

Partial derivate of *y* and *X* are written in Leibniz notation.

$$\frac{d}{dt}(X\_{K,m}) = u\_{k,L}X\_{K,k}X\_{L,m} + \frac{d}{dt}(X\_{K,k})\frac{\partial \mathbf{y}\_k}{\partial \mathbf{X}\_L}\frac{\partial \mathbf{X}\_L}{\partial \mathbf{y}\_m} + \frac{d}{dt}(X\_{L,m})\frac{\partial \mathbf{y}\_k}{\partial \mathbf{X}\_L}\frac{\partial \mathbf{X}\_K}{\partial \mathbf{y}\_k} \tag{46}$$

The products of partial derivate are reduced to Kronecker delta.

$$\frac{d}{dt}(X\_{K,m}) = u\_{k,L}X\_{K,k}X\_{L,m} + \frac{d}{dt}(X\_{K,k})\delta\_{km} + \frac{d}{dt}(X\_{L,m})\delta\_{KL} \tag{47}$$

The index into *XK*,*<sup>k</sup>* and *XL*,*<sup>m</sup>* were exchanging due to commutation Kronecker deltas.

$$\frac{d}{dt}(X\_{K,m}) = u\_{k,L}X\_{K,k}X\_{L,m} + \frac{d}{dt}(X\_{K,m}) + \frac{d}{dt}(X\_{K,m})\tag{48}$$

In Eq. (48) was delete the term *<sup>d</sup> dt*ð Þ *XK*,*<sup>m</sup>* in both sides

$$0 = u\_{k,L} X\_{K,k} X\_{L,m} + \frac{d}{dt} (X\_{K,m}) \tag{49}$$

Clearing *<sup>d</sup> dt*ð Þ *XK*,*<sup>m</sup>* we obtain

$$\frac{d}{dt}(X\_{K,m}) = -u\_{k,L}X\_{K,k}X\_{L,m} \tag{50}$$

Substituting Eq. (50) into Eq. (43) becomes

$$\dot{E}\_m = \frac{d}{dt} [\mathcal{W}\_K X\_{K,m}] = \dot{\mathcal{W}}\_K X\_{K,m} - u\_{k,L} X\_{K,k} X\_{L,m} \mathcal{W}\_K \tag{51}$$

Then, we replace the Eq. (51) into Eq. (39).

$$
\rho \dot{\chi} = \pi^S\_{mk} u\_{k,m} - P\_m E\_k u\_{k,m} - P\_m \left( \dot{W}\_K X\_{K,m} - u\_{k,L} X\_{K,k} X\_{L,m} W\_K \right) \tag{52}
$$

The index *L* was changed by *k*, into *uk*,*<sup>L</sup>*, due to *XL*,*<sup>m</sup>*

$$\rho \dot{\chi} = \boldsymbol{\tau}^{\mathcal{S}}{}\_{mk} \boldsymbol{u}\_{k,m} - P\_m \boldsymbol{E}\_k \boldsymbol{u}\_{k,m} - P\_m \left( \dot{\mathcal{W}}\_K \mathbf{X}\_{K,m} - \boldsymbol{u}\_{k,m} \mathbf{X}\_{K,k} \boldsymbol{W}\_K \right) \tag{53}$$

In Eq. (53), the term *XK*,*kWK* is the electric field concerning the current state.

$$
\rho \dot{\chi} = \pi^S\_{mk} \mu\_{k,m} - P\_m E\_k \mu\_{k,m} - P\_m \dot{W}\_K X\_{K,m} + P\_m \mu\_{k,m} E\_k \tag{54}
$$

*Cement-Based Piezoelectricity Application: A Theoretical Approach DOI: http://dx.doi.org/10.5772/intechopen.95255*

From Eq. (54) the term *PmEkuk*,*<sup>m</sup>* was removes to get

$$
\rho \dot{\chi} = \pi^S{}\_{mk} u\_{k,m} - P\_m \dot{W}\_K X\_{K,m} \tag{55}
$$

## *2.2.2 Polarization vector*

*XK*,*<sup>m</sup>* <sup>¼</sup> *<sup>δ</sup>KLXL*,*<sup>m</sup>* <sup>¼</sup> *<sup>∂</sup>yk*

*Cement Industry - Optimization, Characterization and Sustainable Application*

Derivative *XK*,*<sup>m</sup>* respect to time

*dt*ð Þ¼ *XK*,*<sup>m</sup>*

*d dt yk*,*<sup>L</sup>* 

*dt*ð Þ¼ *XK*,*<sup>m</sup> uk*,*LXK*,*kXL*,*<sup>m</sup>* <sup>þ</sup>

*d*

*d*

In Eq. (48) was delete the term *<sup>d</sup>*

*dt*ð Þ *XK*,*<sup>m</sup>* we obtain

*<sup>E</sup>*\_ *<sup>m</sup>* <sup>¼</sup> *<sup>d</sup>*

*d dt*

*∂yk ∂XL*

*XK*,*kXL*,*<sup>m</sup>* þ

*dt*ð Þ¼ *XK*,*<sup>m</sup> uk*,*LXK*,*kXL*,*<sup>m</sup>* <sup>þ</sup>

*dt*ð Þ¼ *XK*,*<sup>m</sup> uk*,*LXK*,*kXL*,*<sup>m</sup>* <sup>þ</sup>

*d*

The index *L* was changed by *k*, into *uk*,*<sup>L</sup>*, due to *XL*,*<sup>m</sup>*

Substituting Eq. (50) into Eq. (43) becomes

Then, we replace the Eq. (51) into Eq. (39).

Partial derivate of *y* and *X* are written in Leibniz notation.

*d dt*ð Þ *XK*,*<sup>k</sup>*

The products of partial derivate are reduced to Kronecker delta.

The index into *XK*,*<sup>k</sup>* and *XL*,*<sup>m</sup>* were exchanging due to commutation

0 ¼ *uk*,*LXK*,*kXL*,*<sup>m</sup>* þ

*mkuk*,*<sup>m</sup>* � *PmEkuk*,*<sup>m</sup>* � *Pm <sup>W</sup>*\_ *KXK*,*<sup>m</sup>* � *uk*,*LXK*,*kXL*,*mWK*

*mkuk*,*<sup>m</sup>* � *PmEkuk*,*<sup>m</sup>* � *Pm <sup>W</sup>*\_ *KXK*,*<sup>m</sup>* � *uk*,*mXK*,*kWK*

In Eq. (53), the term *XK*,*kWK* is the electric field concerning the current state.

*∂XK ∂yk*

*XL*,*<sup>m</sup>*

*d*

*d*

*d*

*d*

Kronecker deltas.

Clearing *<sup>d</sup>*

*ρχ*\_ <sup>¼</sup> *<sup>τ</sup><sup>S</sup>*

**152**

*ρχ*\_ <sup>¼</sup> *<sup>τ</sup><sup>S</sup>*

*ρχ*\_ <sup>¼</sup> *<sup>τ</sup><sup>S</sup>*

*dt*ð Þ¼ *XK*,*<sup>m</sup>*

*∂XL*

¼ *d*

*dt*ð Þ *XK*,*<sup>k</sup> yk*,*<sup>L</sup>XL*,*<sup>m</sup>* <sup>þ</sup>

*∂yk ∂XL*

*d*

*∂XL <sup>∂</sup>ym* þ *d dt*ð Þ *XL*,*<sup>m</sup>*

*dt*ð Þ *XK*,*<sup>k</sup> <sup>δ</sup>km* <sup>þ</sup>

*dt*ð Þþ *XK*,*<sup>m</sup>*

*d*

*dt*ð Þ *XK*,*<sup>m</sup>* in both sides

*d*

*∂XK ∂yk*

*dt yk*,*<sup>L</sup>XK*,*kXL*,*<sup>m</sup>* 

*d*

*d*

*d*

*dt*ð Þ¼� *XK*,*<sup>m</sup> uk*,*LXK*,*kXL*,*<sup>m</sup>* (50)

(52)

(53)

*dt* ½ �¼ *WKXK*,*<sup>m</sup> <sup>W</sup>*\_ *KXK*,*<sup>m</sup>* � *uk*,*LXK*,*kXL*,*mWK* (51)

*mkuk*,*<sup>m</sup>* � *PmEkuk*,*<sup>m</sup>* � *PmW*\_ *KXK*,*<sup>m</sup>* <sup>þ</sup> *Pmuk*,*mEk* (54)

*dt*ð Þ *XL*,*<sup>m</sup> yk*,*<sup>L</sup>XK*,*<sup>k</sup>*

*∂yk ∂XL* *∂XK ∂yk*

*dt*ð Þ *XL*,*<sup>m</sup> <sup>δ</sup>KL* (47)

*dt*ð Þ *XK*,*<sup>m</sup>* (48)

*dt*ð Þ *XK*,*<sup>m</sup>* (49)

*XL*,*<sup>m</sup>* (44)

(45)

(46)

In this subsection, we will perform the transformation of the polarization vector to the reference state.

$$P\_L = J X\_{L,i} P\_i \tag{56}$$

Where *J* is the Jacobian, multiplying Eq. (56) by *J* �1 *ym*,*<sup>L</sup>* we obtain

$$J^{-1} \mathcal{Y}\_{m,L} P\_L = J^{-1} \mathcal{Y}\_{m,L} J X\_{L,i} P\_i = \delta\_{mi} P\_i \tag{57}$$

To get

$$P\_m = f^{-1} \mathcal{Y}\_{m,L} P\_L \tag{58}$$

From Eq. (58) into Eq. (55) results in

$$\begin{aligned} \rho \dot{\chi} &= \boldsymbol{\tau}^{\mathcal{S}}{}\_{mk} \boldsymbol{u}\_{k,m} - \boldsymbol{J}^{-1} \boldsymbol{\mathcal{y}}\_{m,L} \boldsymbol{P}\_L \dot{\boldsymbol{W}}\_K \boldsymbol{X}\_{K,m} \\ \rho \dot{\chi} &= \boldsymbol{\tau}^{\mathcal{S}}{}\_{mk} \boldsymbol{u}\_{k,m} - \boldsymbol{J}^{-1} \boldsymbol{P}\_L \dot{\boldsymbol{W}}\_K \frac{\partial \boldsymbol{\mathcal{y}}\_m}{\partial \boldsymbol{X}\_L} \frac{\partial \boldsymbol{X}\_K}{\partial \boldsymbol{\mathcal{y}}\_m} \\ \rho \dot{\chi} &= \boldsymbol{\tau}^{\mathcal{S}}{}\_{mk} \boldsymbol{u}\_{k,m} - \boldsymbol{J}^{-1} \boldsymbol{P}\_L \dot{\boldsymbol{W}}\_K \delta\_{KL} \\ \rho \dot{\chi} &= \boldsymbol{\tau}^{\mathcal{S}}{}\_{mk} \boldsymbol{u}\_{k,m} - \boldsymbol{J}^{-1} \boldsymbol{P}\_K \dot{\boldsymbol{W}}\_K \end{aligned} \tag{59}$$

Until now, in Eq. (59), we have obtained a partial transformation, and still missing transform the symmetric Cauchy stress tensor *τ<sup>S</sup> mk*.

#### *2.2.3 Second Piola-Kirchhoff stress*

The symmetric tensor *τ<sup>S</sup> mk* is related with second Piola-Kirchhoff stress *T<sup>S</sup> KL* through a reverse transformation as follow:

$$
\pi^{\mathcal{S}}{}\_{mk} = J^{-1} \mathcal{Y}\_{m,K} \mathcal{Y}\_{k,L} T^{\mathcal{S}}{}\_{KL} \tag{60}
$$

Eq. (60) into Eq. (59) results in

$$\rho \dot{\chi} = f^{-1} \mathcal{Y}\_{m,K} \mathcal{Y}\_{k,L} T^{\mathbb{S}}{}\_{KL} u\_{k,m} - f^{-1} P\_K \dot{W}\_K \tag{61}$$

The gradient of velocity *uk*,*<sup>m</sup>* can be separated on antisymmetric tensor *ωmk* ¼ 1 <sup>2</sup> ð Þ *uk*,*<sup>m</sup>* � *um*,*<sup>k</sup>* plus a symmetric tensor *dmk* <sup>¼</sup> <sup>1</sup> <sup>2</sup> ð Þ *uk*,*<sup>m</sup>* þ *um*,*<sup>k</sup>* .

$$\begin{aligned} \rho \dot{\chi} &= f^{-1} \boldsymbol{\mathcal{y}}\_{m,k} \boldsymbol{\mathcal{y}}\_{k,L} \boldsymbol{T}^{\operatorname{\mathbb{S}}} \operatorname{KL} (\boldsymbol{\alpha}\_{mk} + \boldsymbol{d}\_{mk}) - f^{-1} \boldsymbol{P}\_{K} \dot{\boldsymbol{W}}\_{K} \\\\ \rho \dot{\chi} &= f^{-1} \boldsymbol{\mathcal{y}}\_{m,k} \boldsymbol{\mathcal{y}}\_{k,L} \boldsymbol{T}^{\operatorname{\mathbb{S}}} \boldsymbol{\mathcal{C}}\_{K\mathcal{L}} \boldsymbol{\alpha}\_{mk} + f^{-1} \boldsymbol{\mathcal{y}}\_{m,k} \boldsymbol{\mathcal{y}}\_{k,L} \boldsymbol{T}^{\operatorname{\mathbb{S}}} \boldsymbol{d}\_{mk} - f^{-1} \boldsymbol{P}\_{K} \dot{\boldsymbol{W}}\_{K} \end{aligned} \tag{62}$$

From Eq. (61), the product between symmetric tensor *T<sup>S</sup> KL* and antisymmetric tensor *ωmk* result be null

*Cement Industry - Optimization, Characterization and Sustainable Application*

$$\rho \dot{\chi} = f^{-1} \mathcal{y}\_{m,K} \mathcal{y}\_{k,L} T^{\mathbb{S}}{}\_{KL} d\_{mk} - f^{-1} P\_K \dot{W}\_K \tag{63}$$

piezoelectricity's constitutional equations. Take into account Eq. (70), we can

*KL* <sup>¼</sup> *<sup>T</sup><sup>S</sup>*

Derivation respect to time the free energy into Eq. (71) as follow

*<sup>χ</sup>*\_ <sup>¼</sup> *<sup>∂</sup><sup>χ</sup> ∂EKL*

*<sup>E</sup>*\_ *KL* <sup>þ</sup> *<sup>ρ</sup>*<sup>0</sup> *<sup>∂</sup><sup>χ</sup>*

*∂WK*

*TS*

Substituting Eq. (72) into Eq. (70), we obtain

energy as a generating function, as shown in Eq. (75).

*cABCDEABECD* � *eABCWAEBC* � <sup>1</sup>

*cABCDEFGHEABECDEEFEGH* þ

*ABCDWAWBWCWD* þ … ,

odd electrolytic *d*, electrostrictive *b*, and electroelastic force even *a*.

*aABCDEFWAWBECDEEF* þ

*dABCDEWAEBCEDE* � <sup>1</sup>

*<sup>ρ</sup>*<sup>0</sup> *<sup>∂</sup><sup>χ</sup> ∂EKL*

resulting symmetric tensor *T<sup>S</sup>*

energy function with order three

*<sup>ρ</sup>*<sup>0</sup>*<sup>χ</sup>* <sup>¼</sup> <sup>1</sup> 2

> þ 1 2

þ 1 24

þ 1 4

� 1 <sup>24</sup> *<sup>χ</sup>*<sup>E</sup>

*2.3.1 The linear approach of piezoelectricity*

in Eq. (74) and Eq. (75) to obtain

**155**

*TS*

*Cement-Based Piezoelectricity Application: A Theoretical Approach*

*χ* ¼ *χ*ð Þ *EKL*,*WK*

*PK* ¼ *PK*ð Þ *EKL*,*WK*

*<sup>E</sup>*\_ *KL* <sup>þ</sup>

Both sides of Eq. (73) were compared to deduce two transformations, which

*KL* <sup>¼</sup> *<sup>ρ</sup>*<sup>0</sup> *<sup>∂</sup><sup>χ</sup>*

*PK* ¼ �*ρ*<sup>0</sup> *<sup>∂</sup><sup>χ</sup>*

The mathematical structure of the free energy function will define the order of constitutional equations. There are functions for the free energy of piezoelectric materials from order 1 to order 3 [15]. It means that piezoelectric material behavior depends on the free energy function and its parameters. Here is an example of free

> 2 *χE*

> > 1

1

The parameters are called elasticity *c*, piezoelectric *e*, electric permeability *χ<sup>E</sup>*,

We take on order one approach from Eq. (76) to free energy *χ*. Then, replacing it

*bABCDWAWBECD* � <sup>1</sup>

2

*ABWAWB* þ

<sup>6</sup> *dABCDEWAWBWCEDE*

1

6 *χE*

<sup>6</sup> *dABCDEFGWAEBCEDEEFG*

<sup>6</sup> *cABCDEFEABE*C*DEEF*

*ABCWAWBWC*

*∂EKL*

*∂WK*

*KL*ð Þ *EKL*, *WK*

*∂χ ∂WK*

*<sup>W</sup>*\_ *<sup>K</sup>*<sup>¼</sup> *<sup>T</sup><sup>S</sup>*

(71)

(74)

(75)

(76)

*W*\_ *<sup>K</sup>* (72)

*KLE*\_ *KL* � *PKW*\_ *<sup>K</sup>* (73)

*KL* and polarization *PK*. The transformations use free

propose the next dependence to the functions

*DOI: http://dx.doi.org/10.5772/intechopen.95255*

The term *ym*,*<sup>K</sup>yk*,*<sup>L</sup>dmk* will be solved as following

$$\mathcal{Y}\_{m,\mathcal{K}}\mathcal{Y}\_{k,\mathcal{L}}d\_{mk} = \mathcal{Y}\_{m,\mathcal{K}}\mathcal{Y}\_{k,\mathcal{L}}\frac{1}{2}(u\_{k,m} + u\_{m,k}) = \frac{1}{2}\left(u\_{k,m}\mathcal{Y}\_{m,\mathcal{K}}\mathcal{Y}\_{k,\mathcal{L}} + u\_{m,k}\mathcal{Y}\_{m,\mathcal{K}}\mathcal{Y}\_{k,\mathcal{L}}\right)$$

$$\mathcal{Y}\_{m,\mathcal{K}}\mathcal{Y}\_{k,\mathcal{L}}d\_{mk} = \frac{1}{2}\left(\frac{\partial u\_{k}}{\partial \mathcal{Y}\_{m}}\frac{\partial \mathcal{Y}\_{m}}{\partial \mathcal{X}\_{\mathcal{L}}}\frac{\partial \mathcal{Y}\_{k}}{\partial \mathcal{X}\_{\mathcal{L}}} + \frac{\partial u\_{m}}{\partial \mathcal{Y}\_{k}}\frac{\partial \mathcal{Y}\_{m}}{\partial \mathcal{X}\_{\mathcal{L}}}\frac{\partial \mathcal{Y}\_{k}}{\partial \mathcal{X}\_{\mathcal{L}}}\right) = \frac{1}{2}\left(\frac{\partial u\_{k}}{\partial \mathcal{X}\_{K}}\frac{\partial \mathcal{Y}\_{k}}{\partial \mathcal{X}\_{\mathcal{L}}} + \frac{\partial u\_{m}}{\partial \mathcal{X}\_{\mathcal{L}}}\frac{\partial \mathcal{Y}\_{m}}{\partial \mathcal{X}\_{K}}\right)$$

$$\mathcal{Y}\_{m,\mathcal{K}}\mathcal{Y}\_{k,\mathcal{L}}d\_{mk} = \frac{1}{2}\left(u\_{k,\mathcal{K}}\mathcal{Y}\_{k,\mathcal{L}} + u\_{m,\mathcal{L}}\mathcal{Y}\_{m,\mathcal{K}}\right) \tag{64}$$

We interchange the index *k* to *m* in *uk*,*K*.

$$\begin{split} \mathcal{Y}\_{m,\mathbb{K}}\mathcal{Y}\_{k,\mathbb{L}}d\_{mk} &= \frac{1}{2} \left( u\_{m,\mathbb{K}}\mathcal{Y}\_{k,\mathbb{L}} + u\_{m,\mathbb{L}}\mathcal{Y}\_{m,\mathbb{K}} \right) = \frac{1}{2} \left( \dot{\mathcal{Y}}\_{m,\mathbb{K}}\mathcal{Y}\_{k,\mathbb{L}} + \dot{\mathcal{Y}}\_{m,\mathbb{L}}\mathcal{Y}\_{m,\mathbb{K}} \right) \\ \mathcal{Y}\_{m,\mathbb{K}}\mathcal{Y}\_{k,\mathbb{L}}d\_{mk} &= \frac{1}{2} \frac{d}{dt} \left( \mathcal{Y}\_{m,\mathbb{K}}\mathcal{Y}\_{k,\mathbb{L}} \right) = \frac{d}{dt} \left[ \frac{1}{2} \left( \mathcal{Y}\_{m,\mathbb{K}}\mathcal{Y}\_{k,\mathbb{L}} - \delta\_{\mathbb{K}\mathbb{L}} \right) \right] \end{split} \tag{65}$$

With *<sup>m</sup>* <sup>¼</sup> *<sup>k</sup>* the term <sup>1</sup> <sup>2</sup> *ym*,*<sup>K</sup>yk*,*<sup>L</sup>* � *<sup>δ</sup>KL* is known as the finite strain tensor *EKL* in the reference state, and Ę whit an uppercase index will represent the electric field vector in the reference state. Then, we reduce *ym*,*<sup>K</sup>yk*,*<sup>L</sup>dmk* to:

$$
\mathcal{Y}\_{m,\mathcal{K}} \mathcal{Y}\_{k,\mathcal{L}} d\_{mk} = \frac{d}{dt} (E\_{\mathcal{K}}) = \dot{E}\_{\mathcal{K}} \tag{66}
$$

Substituting Eq. (66) into Eq. (63), we obtain

$$
\rho \dot{\chi} = \int^{-1} T^{\mathcal{S}}{}\_{\text{KL}} \dot{E}\_{\text{KL}} - \int^{-1} P\_K \dot{W}\_K \tag{67}
$$

Factoring the inverse of Jacobian, we get

$$\rho \dot{\chi} = f^{-1} \left( T^{\mathbb{S}}{}\_{\text{KL}} \dot{E}\_{\text{KL}} - P\_K \dot{W}\_K \right) \tag{68}$$

Multiplying both sides into Eq. (68) by the Jacobian gives

$$J\rho\dot{\chi} = \mathcal{J}^{-1}(T^{\mathcal{S}}{}\_{\text{KL}}\dot{E}\_{\text{KL}} - P\_K\dot{W}\_K) \tag{69}$$

Using mass transformation to the reference state *<sup>ρ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>ρ</sup><sup>J</sup>* into Eq. (69), we get a new equation for energy conservation in terms of physical quantities in the reference state. Symmetric tensor *T<sup>S</sup> KL*, strain tensor *EKL*, polarization *PK* and gradient of potential *WK*.

$$
\rho^0 \dot{\chi} = T^{\mathbb{S}}{}\_{\text{KL}} \dot{\mathcal{E}}\_{\text{KL}} - P\_K \dot{\mathcal{W}}\_K \tag{70}
$$

#### **2.3 Constitutional equations from free energy**

The conservation laws are valid for any piezoelectric material, including cementbased composites. However, a specific material's piezoelectric properties are determined by a set of functions that describes free energy, symmetric tensor, and polarization. Once we replace these functions into Eq. (70), we will get the

*ρχ*\_ ¼ *J* �1

The term *ym*,*<sup>K</sup>yk*,*<sup>L</sup>dmk* will be solved as following

1 2

> *∂yk ∂XL* þ *∂um ∂yk*

*ym*,*<sup>K</sup>yk*,*<sup>L</sup>dmk* <sup>¼</sup> <sup>1</sup>

*um*,*Kyk*,*<sup>L</sup>* þ *um*,*Lym*,*<sup>K</sup>* 

> *dt ym*,*<sup>K</sup>yk*,*<sup>L</sup>*

<sup>2</sup> *ym*,*<sup>K</sup>yk*,*<sup>L</sup>* � *δKL* 

*ym*,*<sup>K</sup>yk*,*<sup>L</sup>dmk* <sup>¼</sup> *<sup>d</sup>*

�<sup>1</sup> *T<sup>S</sup>*

vector in the reference state. Then, we reduce *ym*,*<sup>K</sup>yk*,*<sup>L</sup>dmk* to:

*ρχ*\_ ¼ *J* �1 *TS*

*ρχ*\_ ¼ *J*

Multiplying both sides into Eq. (68) by the Jacobian gives

*<sup>J</sup>ρχ*\_ <sup>¼</sup> *JJ*�<sup>1</sup> *<sup>T</sup><sup>S</sup>*

*<sup>ρ</sup>*<sup>0</sup>*χ*\_ <sup>¼</sup> *<sup>T</sup><sup>S</sup>*

*<sup>∂</sup>ym ∂XK*

*ym*,*<sup>K</sup>yk*,*<sup>L</sup>dmk* ¼ *ym*,*<sup>K</sup>yk*,*<sup>L</sup>*

2

*∂uk <sup>∂</sup>ym*

We interchange the index *k* to *m* in *uk*,*K*.

2 *d*

Substituting Eq. (66) into Eq. (63), we obtain

Factoring the inverse of Jacobian, we get

**2.3 Constitutional equations from free energy**

2

*ym*,*<sup>K</sup>yk*,*<sup>L</sup>dmk* <sup>¼</sup> <sup>1</sup>

*ym*,*<sup>K</sup>yk*,*<sup>L</sup>dmk* <sup>¼</sup> <sup>1</sup>

*ym*,*<sup>K</sup>yk*,*<sup>L</sup>dmk* <sup>¼</sup> <sup>1</sup>

With *<sup>m</sup>* <sup>¼</sup> *<sup>k</sup>* the term <sup>1</sup>

ence state. Symmetric tensor *T<sup>S</sup>*

of potential *WK*.

**154**

*ym*,*Kyk*,*LT<sup>S</sup>*

*Cement Industry - Optimization, Characterization and Sustainable Application*

ð Þ¼ *uk*,*<sup>m</sup>* þ *um*,*<sup>k</sup>*

*KLdmk* � *J*

1

*∂yk ∂XL*

<sup>2</sup> *uk*,*Kyk*,*<sup>L</sup>* <sup>þ</sup> *um*,*Lym*,*<sup>K</sup>* 

¼ 1

¼ *d dt* 1

the reference state, and Ę whit an uppercase index will represent the electric field

*KLE*\_ *KL* � *<sup>J</sup>*

�1

*KLE*\_ *KL* � *PKW*\_ *<sup>K</sup>*

*KLE*\_ *KL* � *PKW*\_ *<sup>K</sup>*

Using mass transformation to the reference state *<sup>ρ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>ρ</sup><sup>J</sup>* into Eq. (69), we get a new equation for energy conservation in terms of physical quantities in the refer-

The conservation laws are valid for any piezoelectric material, including cementbased composites. However, a specific material's piezoelectric properties are determined by a set of functions that describes free energy, symmetric tensor, and polarization. Once we replace these functions into Eq. (70), we will get the

*<sup>∂</sup>ym ∂XK* �1

¼ 1 2

*PKW*\_ *<sup>K</sup>* (63)

<sup>2</sup> *uk*,*mym*,*<sup>K</sup>yk*,*<sup>L</sup>* <sup>þ</sup> *um*,*kym*,*<sup>K</sup>yk*,*<sup>L</sup>* 

> *∂yk ∂XL* þ *∂um ∂XL*

*<sup>∂</sup>ym ∂XK*

(64)

*∂uk ∂XK*

<sup>2</sup> *<sup>y</sup>*\_*<sup>m</sup>*,*<sup>K</sup>yk*,*<sup>L</sup>* <sup>þ</sup> *<sup>y</sup>*\_*<sup>m</sup>*,*<sup>L</sup>y*m,*<sup>K</sup>* 

(65)

is known as the finite strain tensor *EKL* in

*dt*ð Þ¼ *EKL <sup>E</sup>*\_ *KL* (66)

(68)

(69)

*KL*, strain tensor *EKL*, polarization *PK* and gradient

*KLE*\_ *KL* � *PKW*\_ *<sup>K</sup>* (70)

*PKW*\_ *<sup>K</sup>* (67)

<sup>2</sup> *ym*,*<sup>K</sup>yk*,*<sup>L</sup>* � *<sup>δ</sup>KL*

piezoelectricity's constitutional equations. Take into account Eq. (70), we can propose the next dependence to the functions

$$\begin{aligned} \chi &= \chi(\mathbf{E}\_{\rm KL}, \mathbf{W}\_K) \\ T^{\rm S}\_{\rm KL} &= T^{\rm S}\_{\rm KL}(\mathbf{E}\_{\rm KL}, \mathbf{W}\_K) \\ P\_K &= P\_K(\mathbf{E}\_{\rm KL}, \mathbf{W}\_K) \end{aligned} \tag{71}$$

Derivation respect to time the free energy into Eq. (71) as follow

$$
\dot{\chi} = \frac{\partial \chi}{\partial E\_{KL}} \dot{E}\_{KL} + \frac{\partial \chi}{\partial W\_K} \dot{W}\_K \tag{72}
$$

Substituting Eq. (72) into Eq. (70), we obtain

$$
\rho^0 \frac{\partial \chi}{\partial \mathcal{E}\_{\rm KL}} \dot{\mathcal{E}}\_{\rm KL} + \rho^0 \frac{\partial \chi}{\partial \mathcal{W}\_{\rm K}} \dot{\mathcal{W}}\_{\rm K} = T^S{}\_{\rm KL} \dot{\mathcal{E}}\_{\rm KL} - P\_K \dot{\mathcal{W}}\_K \tag{73}
$$

Both sides of Eq. (73) were compared to deduce two transformations, which resulting symmetric tensor *T<sup>S</sup> KL* and polarization *PK*. The transformations use free energy as a generating function, as shown in Eq. (75).

$$T^{\mathbb{S}}{}\_{KL} = \rho^0 \frac{\partial \mathbb{X}}{\partial \mathbb{E}\_{KL}} \tag{74}$$

$$P\_K = -\rho^0 \frac{\partial \chi}{\partial W\_K} \tag{75}$$

The mathematical structure of the free energy function will define the order of constitutional equations. There are functions for the free energy of piezoelectric materials from order 1 to order 3 [15]. It means that piezoelectric material behavior depends on the free energy function and its parameters. Here is an example of free energy function with order three

$$\begin{aligned} \rho^0 \chi &= \frac{1}{2} c\_{ABCD} E\_{AB} E\_{CD} - e\_{ABC} W\_A E\_{BC} - \frac{1}{2} \chi^E\_{AB} W\_A W\_B + \frac{1}{6} c\_{ABCDEF} E\_{AB} E\_{CD} E\_{EF} \\ &+ \frac{1}{2} d\_{ABCDEF} W\_A E\_{BC} E\_{DE} - \frac{1}{2} b\_{ABCD} W\_A W\_B E\_{CD} - \frac{1}{6} \chi^E\_{ABC} W\_A W\_B W\_C \\ &+ \frac{1}{24} c\_{ABCDEFG} E\_{AB} E\_{CD} E\_{EF} E\_{GH} + \frac{1}{6} d\_{ABCDEFG} W\_A E\_{BC} E\_{DE} E\_{FG} \\ &+ \frac{1}{4} a\_{ABCDEF} W\_A W\_B E\_{CD} E\_{EF} + \frac{1}{6} d\_{ABCDEF} W\_A W\_B W\_C E\_{DE} \\ &- \frac{1}{24} \chi^E\_{ABCD} W\_A W\_B W\_C W\_D + \dots, \end{aligned} \tag{76}$$

The parameters are called elasticity *c*, piezoelectric *e*, electric permeability *χ<sup>E</sup>*, odd electrolytic *d*, electrostrictive *b*, and electroelastic force even *a*.

#### *2.3.1 The linear approach of piezoelectricity*

We take on order one approach from Eq. (76) to free energy *χ*. Then, replacing it in Eq. (74) and Eq. (75) to obtain

*Cement Industry - Optimization, Characterization and Sustainable Application*

$$\mathcal{T}^{\mathcal{S}}\_{AB} = \frac{\partial}{\partial \mathcal{E}\_{AB}} \left( \frac{1}{2} \varepsilon\_{ABCD} E\_{AB} E\_{CD} - e\_{ABC} W\_A E\_{BC} - \frac{1}{2} \chi^{E}\_{AB} W\_A W\_B \right) \tag{77}$$

*Di* <sup>¼</sup> *diklSkl* <sup>þ</sup> *<sup>ε</sup><sup>T</sup>*

The electromechanical properties are defined by piezoelectric charge *dikl* and voltage *gkij* constants. Unlike parameters *cijkl* and *eikl* These new piezoelectric constants are taken out directly from experiments, as shown in the next section.

**2.4 Electromechanical and electrical properties of cement-based composites**

zoelectric and mechanical properties [16] due to increased deformable crystal structures. Zeolites, oxides, and carbon nanotubes are the most used cement-based composites to improve these properties [17]. Chen et al. also report some piezoelectric parameters of cement-based composites such as piezoelectric charge *d*33, voltage *g*33*:* And the coupling factor *Kt*. As was mentioned in the previous section, these piezoelectric parameters come from linear piezoelectricity theory. However, the crystalline structure of Calcium Silicate Hydrate (C-S-H) that compose the cement is a complex system described by linear theory. It could also be combined with statistical physics and mean-field homogenization theory tools to get the macroscale properties [18]. Here are show piezoelectric and electrical parameters of gold nanoparticles mixed to cement paste, which we hope to lead to our system's

constitutional equations.

particle sizes are shown in **Figure 4**.

*Scheme of nanoparticle physical synthesis by laser ablation.*

shown in **Figure 3**.

**Figure 3.**

**157**

Incorporating piezoelectric nanocomposites into cement paste improves its pie-

Next, we introduce a brief description of the gold nanoparticles' physical synthesis [19, 20]. They are produced by laser ablation at 532 nm. A gold plate at 99.9999% purity is put inside a beaker filled with 50 mL of ultrapure water. Then, the pulse laser spot with an energy of 30 mJ beats the gold plate by 10 minutes, as

At the time, the gold nanoparticles were brought to be characterized by dynam-

agglomerated. These measures are required because the gold nanoparticles directly affect the piezoelectric properties of cement cylinders. Some results of gold nano-

Also, the gold nanoparticles in water must be mixed quickly with the cement. The ratio of water/cement used was 0.47 mL/g. Then, the admixture was poured

ical light scattering (DLS). If not done quickly, the gold nanoparticles were

into cylindrical molds that contained copper wires as follows in **Figure 5**.

*Sij* ¼ *s D*

*Cement-Based Piezoelectricity Application: A Theoretical Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.95255*

*ikEk* (88)

*ijklTkl* þ *gkijDk* (89)

$$P\_A = -\frac{\partial}{\partial W\_A} \left( \frac{1}{2} c\_{ABCD} E\_{AB} E\_{CD} - e\_{ABC} W\_A E\_{BC} - \frac{1}{2} \chi^E\_{\ \ AB} W\_A W\_B \right) \tag{78}$$

The approximation is possible if we consider an infinitesimal deformation, weak electric field, and low amplitude displacements around the reference state. Hence, it approaches require a nomenclature exchange for physical quantities. Thus, second Piola-Kirchhoff stress will be replaced by infinitesimal Cauchy stress tensor *TS KL* ! *Tij*; finite strain tensor will be exchanged by infinitesimal strain tensor *EKL* ! *Skl*; potential gradient, polarization, and displacement electric vector are similar either reference or current state: *WK* ! *Ek*, *PL* ! *Pi*, and D*<sup>L</sup>* ! *Di*. Then, Eqs. (77) and (78) follow:

$$T\_{\vec{\eta}} = \frac{\partial}{\partial \mathbf{S}\_{\vec{\eta}}} \left( \frac{\mathbf{1}}{2} c\_{\vec{\eta}kl} \mathbf{S}\_{\vec{\eta}} \mathbf{S}\_{kl} - e\_{\vec{\eta}k} E\_i \mathbf{S}\_{jk} - \frac{\mathbf{1}}{2} \boldsymbol{\chi}^E {}\_{\vec{\eta}} E\_i E\_j \right) \tag{79}$$

$$P\_i = -\frac{\partial}{\partial E\_i} \left( \frac{1}{2} c\_{ijkl} \mathbf{S}\_{ij} \mathbf{S}\_{kl} - c\_{ijk} E\_i \mathbf{S}\_{jk} - \frac{1}{2} \chi^E{}\_{ij} E\_i E\_j \right) \tag{80}$$

Here is considering symmetry to parameters elastic *cijkl*, piezoelectric *ekij*, and electric *χik* when they have odd permutations. Differentiating the Eq. (79) and Eq. (80), we obtain

$$T\_{\vec{\+}\vec{\}} = \mathcal{c}\_{\vec{\+}\vec{\&}l} \mathbf{S}\_{kl} - \mathcal{e}\_{k\vec{\+}\vec{\!}} E\_k \tag{81}$$

$$P\_i = e\_{ikl} \mathbf{S}\_{kl} + \chi^E\_{\ ik} E\_k \tag{82}$$

The polarization can be written in terms of electric displacement vector too.

$$P\_i = D\_i - \varepsilon\_0 E\_i \tag{83}$$

From Eq. (83) into Eq. (82) gives

$$
\varepsilon D\_i - \varepsilon\_0 E\_i = \varepsilon\_{ikl} \mathbb{S}\_{kl} + \chi^E\_{\ ik} E\_k \tag{84}
$$

Solving *Di*,

$$D\_i = \varepsilon\_{ikl}\mathbb{S}\_{kl} + \varepsilon\_0 E\_i + \chi^E\_{\ ik} E\_k = \varepsilon\_{ikl}\mathbb{S}\_{kl} + \varepsilon\_0 \delta\_{ik} E\_i + \chi^E\_{\ ik} E\_k \tag{85}$$

Factoring *Ek*,

$$D\_i = \varepsilon\_{ikl} \mathbf{S}\_{kl} + \left(\varepsilon\_0 \delta\_{ik} + \boldsymbol{\chi}\_{\ \ ik}^E\right) E\_k \tag{86}$$

where the term *<sup>ε</sup>*0*δik* <sup>þ</sup> *<sup>χ</sup><sup>E</sup> ik* is defined as dielectric constant *εik*. Finally, we have the linear constitutional equation for the electric displacement vector.

$$D\_i = \varepsilon\_{ikl} \mathbf{S}\_{kl} + \varepsilon\_{ik} E\_k \tag{87}$$

We have seen several forms to present the linear constitutional equations in piezoelectricity. Next, we include another form of constitutional equations shown in the IEEE standard for piezoelectricity. It can be obtained inverting the matrix formed by Eq. (81) and Eq. (82).

*Cement-Based Piezoelectricity Application: A Theoretical Approach DOI: http://dx.doi.org/10.5772/intechopen.95255*

*TS*

*TS*

*AB* <sup>¼</sup> *<sup>∂</sup> ∂EAB*

*PA* ¼ � *<sup>∂</sup>*

Eqs. (77) and (78) follow:

Eq. (80), we obtain

Solving *Di*,

Factoring *Ek*,

**156**

where the term *<sup>ε</sup>*0*δik* <sup>þ</sup> *<sup>χ</sup><sup>E</sup>*

formed by Eq. (81) and Eq. (82).

*∂WA*

1 2

1 2

*Tij* <sup>¼</sup> *<sup>∂</sup> ∂Sij*

*Pi* ¼ � *<sup>∂</sup> ∂Ei*

From Eq. (83) into Eq. (82) gives

*Di* <sup>¼</sup> *eiklSkl* <sup>þ</sup> *<sup>ε</sup>*0*Ei* <sup>þ</sup> *<sup>χ</sup><sup>E</sup>*

1 2

> 1 2

*cABCDEABECD* � *eABCWAEBC* � <sup>1</sup>

*Cement Industry - Optimization, Characterization and Sustainable Application*

*cABCDEABECD* � *eABCWAEBC* � <sup>1</sup>

*KL* ! *Tij*; finite strain tensor will be exchanged by infinitesimal strain tensor *EKL* ! *Skl*; potential gradient, polarization, and displacement electric vector are similar either reference or current state: *WK* ! *Ek*, *PL* ! *Pi*, and D*<sup>L</sup>* ! *Di*. Then,

*cijklSijSkl* � *eijkEiSjk* � <sup>1</sup>

*cijklSijSk*<sup>l</sup> � *eijkEiSjk* � <sup>1</sup>

Here is considering symmetry to parameters elastic *cijkl*, piezoelectric *ekij*, and electric *χik* when they have odd permutations. Differentiating the Eq. (79) and

*Pi* <sup>¼</sup> *eiklSkl* <sup>þ</sup> *<sup>χ</sup><sup>E</sup>*

*Di* � *<sup>ε</sup>*0*Ei* <sup>¼</sup> *eiklSkl* <sup>þ</sup> *<sup>χ</sup><sup>E</sup>*

*Di* <sup>¼</sup> *eiklSkl* <sup>þ</sup> *<sup>ε</sup>*0*δik* <sup>þ</sup> *<sup>χ</sup><sup>E</sup>*

We have seen several forms to present the linear constitutional equations in piezoelectricity. Next, we include another form of constitutional equations shown in the IEEE standard for piezoelectricity. It can be obtained inverting the matrix

the linear constitutional equation for the electric displacement vector.

The polarization can be written in terms of electric displacement vector too.

The approximation is possible if we consider an infinitesimal deformation, weak electric field, and low amplitude displacements around the reference state. Hence, it approaches require a nomenclature exchange for physical quantities. Thus, second Piola-Kirchhoff stress will be replaced by infinitesimal Cauchy stress tensor

2 *χE*

2 *χE*

2 *χE ijEiE <sup>j</sup>*

2 *χE ijEiE <sup>j</sup>*

*Tij* ¼ *cijklSkl* � *ekijEk* (81)

*Pi* ¼ *Di* � *ε*0*Ei* (83)

*ikEk* <sup>¼</sup> *eiklSkl* <sup>þ</sup> *<sup>ε</sup>*0*δikEi* <sup>þ</sup> *<sup>χ</sup><sup>E</sup>*

*ik*

*ik* is defined as dielectric constant *εik*. Finally, we have

*Di* ¼ *eiklSkl* þ *εikEk* (87)

*Ek* (86)

*ikEk* (82)

*ikEk* (84)

*ikEk* (85)

*ABWAWB*

*ABWAWB*

(77)

(78)

(79)

(80)

$$D\_i = d\_{ikl} \mathbf{S}\_{kl} + \varepsilon\_{ik}^T E\_k \tag{88}$$

$$\mathbf{S}\_{ij} = \mathbf{s}\_{ijkl}^{D} T\_{kl} + \mathbf{g}\_{kij} D\_{k} \tag{89}$$

The electromechanical properties are defined by piezoelectric charge *dikl* and voltage *gkij* constants. Unlike parameters *cijkl* and *eikl* These new piezoelectric constants are taken out directly from experiments, as shown in the next section.

#### **2.4 Electromechanical and electrical properties of cement-based composites**

Incorporating piezoelectric nanocomposites into cement paste improves its piezoelectric and mechanical properties [16] due to increased deformable crystal structures. Zeolites, oxides, and carbon nanotubes are the most used cement-based composites to improve these properties [17]. Chen et al. also report some piezoelectric parameters of cement-based composites such as piezoelectric charge *d*33, voltage *g*33*:* And the coupling factor *Kt*. As was mentioned in the previous section, these piezoelectric parameters come from linear piezoelectricity theory. However, the crystalline structure of Calcium Silicate Hydrate (C-S-H) that compose the cement is a complex system described by linear theory. It could also be combined with statistical physics and mean-field homogenization theory tools to get the macroscale properties [18]. Here are show piezoelectric and electrical parameters of gold nanoparticles mixed to cement paste, which we hope to lead to our system's constitutional equations.

Next, we introduce a brief description of the gold nanoparticles' physical synthesis [19, 20]. They are produced by laser ablation at 532 nm. A gold plate at 99.9999% purity is put inside a beaker filled with 50 mL of ultrapure water. Then, the pulse laser spot with an energy of 30 mJ beats the gold plate by 10 minutes, as shown in **Figure 3**.

At the time, the gold nanoparticles were brought to be characterized by dynamical light scattering (DLS). If not done quickly, the gold nanoparticles were agglomerated. These measures are required because the gold nanoparticles directly affect the piezoelectric properties of cement cylinders. Some results of gold nanoparticle sizes are shown in **Figure 4**.

Also, the gold nanoparticles in water must be mixed quickly with the cement. The ratio of water/cement used was 0.47 mL/g. Then, the admixture was poured into cylindrical molds that contained copper wires as follows in **Figure 5**.

**Figure 3.** *Scheme of nanoparticle physical synthesis by laser ablation.*

**Figure 4.** *The particle size distribution of gold nanoparticles suspended in water to concentration 442 ppm.*

The cement cylinders were dried one day. Then it leaves curing for 28 days and finally to thermal treatment one day more. After 14 days, electromechanical measurements were performed, as shown in **Figure 6**.

Electromechanical measurements consist of two measurements performed in parallel: the cement cylinders under compressive strength test in the axial direction, open circuit potential (OCP) measurements in the electrodes of cement cylinders. From mechanical and electrical data, we calculated an electroelastic parameter with units ½ � *mV=kN* , it has the same interpretation of piezoelectric parameter *e* in linear theory. From **Figure 7**, an example of voltage-force curves for identically cement samples with gold nanoparticles is shown. We did get from the above measurements the axial elasticity parameter:

$$Y = \text{323.5} \pm \text{75.3} \ \left[ kN/m^2 \right] \tag{90}$$

can perform a transformation to get a real part of the capacitance *C*<sup>0</sup>

*OCP-force curves from cement cylinders with gold nanoparticles concentrated to 658 ppm.*

*C*0

ð Þ¼ *<sup>ω</sup>* <sup>1</sup>

The geometry of copper electrodes (an approximation to parallel plates) is related to capacitance. Therefore, we can calculate the dielectric parameter *ε* since 1 MHz; this parameter is a real number that depends on the frequency and

*ε ω*ð Þ*ε*<sup>0</sup> <sup>¼</sup> *<sup>d</sup>* <sup>∗</sup>*C*<sup>0</sup>

where *ε*<sup>0</sup> is the electric permittivity of free space, *A* is the transversal section,

frequency dependence as follow

*Experimental setup of electromechanical measurements.*

*Cement-Based Piezoelectricity Application: A Theoretical Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.95255*

and *d* is the thickness between electrodes.

is given by

**159**

**Figure 7.**

**Figure 6.**

. It has

*<sup>ω</sup>Z*<sup>00</sup> (92)

*<sup>A</sup>* (93)

The axial piezoelectric parameter:

$$\chi = -20.5 \pm 6.9 \,\mathrm{[mV/kN]}.\tag{91}$$

For a total deformation *S* ¼ 0*:*57 � 0*:*09 ½ � *mm* in the axial direction.

The electrical properties of cement cylinders were obtained from the imaginary part of impedance; an example of these curves in **Figure 8**. From impedance data

*Cement-Based Piezoelectricity Application: A Theoretical Approach DOI: http://dx.doi.org/10.5772/intechopen.95255*

**Figure 6.** *Experimental setup of electromechanical measurements.*

**Figure 7.** *OCP-force curves from cement cylinders with gold nanoparticles concentrated to 658 ppm.*

can perform a transformation to get a real part of the capacitance *C*<sup>0</sup> . It has frequency dependence as follow

$$C'(\omega) = \frac{1}{aZ''} \tag{92}$$

The geometry of copper electrodes (an approximation to parallel plates) is related to capacitance. Therefore, we can calculate the dielectric parameter *ε* since 1 MHz; this parameter is a real number that depends on the frequency and is given by

$$
\varepsilon(\alpha)\varepsilon\_0 = \frac{d \ast C}{A} \tag{93}
$$

where *ε*<sup>0</sup> is the electric permittivity of free space, *A* is the transversal section, and *d* is the thickness between electrodes.

The cement cylinders were dried one day. Then it leaves curing for 28 days and finally to thermal treatment one day more. After 14 days, electromechanical

*The particle size distribution of gold nanoparticles suspended in water to concentration 442 ppm.*

*Cement Industry - Optimization, Characterization and Sustainable Application*

Electromechanical measurements consist of two measurements performed in parallel: the cement cylinders under compressive strength test in the axial direction, open circuit potential (OCP) measurements in the electrodes of cement cylinders. From mechanical and electrical data, we calculated an electroelastic parameter with units ½ � *mV=kN* , it has the same interpretation of piezoelectric parameter *e* in linear theory. From **Figure 7**, an example of voltage-force curves for identically cement samples with gold nanoparticles is shown. We did get from the above measure-

For a total deformation *S* ¼ 0*:*57 � 0*:*09 ½ � *mm* in the axial direction.

The electrical properties of cement cylinders were obtained from the imaginary part of impedance; an example of these curves in **Figure 8**. From impedance data

*<sup>Y</sup>* <sup>¼</sup> <sup>323</sup>*:*<sup>5</sup> � <sup>75</sup>*:*<sup>3</sup> *kN=m*<sup>2</sup> (90)

*γ* ¼ �20*:*5 � 6*:*9 ½ � *mV=kN :* (91)

measurements were performed, as shown in **Figure 6**.

ments the axial elasticity parameter:

*Molds and dimensions of cement cylinders.*

**Figure 4.**

**Figure 5.**

**158**

The axial piezoelectric parameter:

From the data in **Figure 8** and Eq. (92) and Eq. (93), we obtain the dielectric constant:

$$
\varepsilon = (\mathfrak{B}\mathfrak{B}\mathfrak{A} \pm \mathfrak{B}\mathfrak{A}\mathfrak{B})\varepsilon\_0 \tag{94}
$$

information from deep within the concrete. The experimental control of the NPs embedded within the cement paste's dispersions and piezoresistive responses is essential to have a good signal-to-noise ratio within the sensing. Knowing the coupling between the electromechanical equations from a theoretical approach is

*The image shows a network of IoT sensors based on cement-based composites piezoresistivity as an active part of*

This chapter proposed a mathematical physicist construction of the linear theory

*D*

*ijkl*, *gkij* and *ε<sup>T</sup>*

*ik*

of piezoelectricity since classical movement laws and the conservation of their physical quantities (mass, charge, linear momentum, angular momentum, and energy) over time. This construction takes parts of Eringen, Tiersten, and Yang's research without including the variational formulation or energy functional to deduce the constitutional equations. We have also presented some results of piezoelectric and dielectric constants obtained for cement mixed to gold nanoparticles. We got the axial elasticity parameter *<sup>Y</sup>* <sup>¼</sup> <sup>323</sup>*:*<sup>5</sup> � <sup>75</sup>*:*<sup>3</sup> *kN=m*<sup>2</sup> ½ �, the electroelastic

another crucial factor in making viable these technological solutions.

*Cement-Based Piezoelectricity Application: A Theoretical Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.95255*

parameter *γ* ¼ �20*:*5 � 6*:*9 [mV/kN], and dielectric constant *ε* ¼ ð Þ 939*:*6 � 82*:*9 *ε*0½ � *F=m* , which can be compared with parameters *s*

Engineering of the Universidad de los Andes Colombia.

authorship, and/or publication of this book chapter.

respectively presents into constitutional equations discussed in the chapter.

We would like to thank the Vice-rector for research in project N 2676 of the Universidad Industrial de Santander, the CIMBIOS research group for the ablation laser system (Universidad Industrial de Santander), and the CA Perez-Lopez for his support in the editing of images of the Department of Electrical and Electronic

The authors declared no potential conflicts of interest concerning the research,

**3. Conclusions**

**Figure 9.**

*smart construction.*

**Acknowledgements**

**Conflict of interest**

**161**

Where *ε*<sup>0</sup> has unit ½ � *F=m* . The piezoelectric and electrical properties of cement paste mixed with gold nanoparticles exhibit reproducibility and linearity of the piezoelectric parameter.

#### **2.5 Future studies and remarks**

The Piezoelectric parameters are an initial point to beginning a new connection with piezoelectricity theory by inverse modeling and constructing new free energy functions and constitutional equations. To catch out with researchers in this scope, we suggest thinking about the next research questions; how is the piezoelectric parameter presented related to the piezoelectric parameter formulated by linear theory for piezoelectricity? Is the free energy function of order one sufficient to describe cement paste's piezoelectric with gold nanoparticles? How to develop a new function for free energy that models cement paste's piezoelectric behavior of cement paste with gold nanoparticles?

In this chapter, we have intended to contribute to the theory of piezoelectricity for large deformations without including an energy function. **Figure 9** shows a possible use around IoT as intelligent sensing of devices based on cement-based composites' piezoresistivity. Without reaching into depth in the technical and engineering aspect that smart construction, active sensing system entails; we highlight how the Eqs. (88) and (89) that relate the electromechanical properties and that are defined by piezoelectric charge *dikl* and voltage *gkij* constants are present as indicators to improve the detection resolution in large structures with large deformations.

The sensors analyze the deformations, temperature, relative humidity, and other critical parameters of the concrete in real-time. This data is captured via wireless communication (WAN/BLE) and deployed on a secure and scalable platform (Cloud) capable of collecting data to facilitate remote decision making with

#### **Figure 8.**

*The imaginary part of electrical impedance represented in a Bode plot was performed on two cement cylinders with gold nanoparticles concentrated to 658 ppm.*

**Figure 9.**

From the data in **Figure 8** and Eq. (92) and Eq. (93), we obtain the dielectric

*Cement Industry - Optimization, Characterization and Sustainable Application*

Where *ε*<sup>0</sup> has unit ½ � *F=m* . The piezoelectric and electrical properties of cement paste mixed with gold nanoparticles exhibit reproducibility and linearity of the

The Piezoelectric parameters are an initial point to beginning a new connection with piezoelectricity theory by inverse modeling and constructing new free energy functions and constitutional equations. To catch out with researchers in this scope, we suggest thinking about the next research questions; how is the piezoelectric parameter presented related to the piezoelectric parameter formulated by linear theory for piezoelectricity? Is the free energy function of order one sufficient to describe cement paste's piezoelectric with gold nanoparticles? How to develop a new function for free energy that models cement paste's piezoelectric behavior of

In this chapter, we have intended to contribute to the theory of piezoelectricity for large deformations without including an energy function. **Figure 9** shows a possible use around IoT as intelligent sensing of devices based on cement-based composites' piezoresistivity. Without reaching into depth in the technical and engineering aspect that smart construction, active sensing system entails; we highlight how the Eqs. (88) and (89) that relate the electromechanical properties and that are defined by piezoelectric charge *dikl* and voltage *gkij* constants are present as indicators to improve the detection resolution in large structures with large deformations. The sensors analyze the deformations, temperature, relative humidity, and other critical parameters of the concrete in real-time. This data is captured via wireless communication (WAN/BLE) and deployed on a secure and scalable platform (Cloud) capable of collecting data to facilitate remote decision making with

*The imaginary part of electrical impedance represented in a Bode plot was performed on two cement cylinders*

*ε* ¼ ð Þ 939*:*6 � 82*:*9 *ε*<sup>0</sup> (94)

constant:

**Figure 8.**

**160**

*with gold nanoparticles concentrated to 658 ppm.*

piezoelectric parameter.

**2.5 Future studies and remarks**

cement paste with gold nanoparticles?

*The image shows a network of IoT sensors based on cement-based composites piezoresistivity as an active part of smart construction.*

information from deep within the concrete. The experimental control of the NPs embedded within the cement paste's dispersions and piezoresistive responses is essential to have a good signal-to-noise ratio within the sensing. Knowing the coupling between the electromechanical equations from a theoretical approach is another crucial factor in making viable these technological solutions.

## **3. Conclusions**

This chapter proposed a mathematical physicist construction of the linear theory of piezoelectricity since classical movement laws and the conservation of their physical quantities (mass, charge, linear momentum, angular momentum, and energy) over time. This construction takes parts of Eringen, Tiersten, and Yang's research without including the variational formulation or energy functional to deduce the constitutional equations. We have also presented some results of piezoelectric and dielectric constants obtained for cement mixed to gold nanoparticles. We got the axial elasticity parameter *<sup>Y</sup>* <sup>¼</sup> <sup>323</sup>*:*<sup>5</sup> � <sup>75</sup>*:*<sup>3</sup> *kN=m*<sup>2</sup> ½ �, the electroelastic parameter *γ* ¼ �20*:*5 � 6*:*9 [mV/kN], and dielectric constant *ε* ¼ ð Þ 939*:*6 � 82*:*9 *ε*0½ � *F=m* , which can be compared with parameters *s D ijkl*, *gkij* and *ε<sup>T</sup> ik* respectively presents into constitutional equations discussed in the chapter.

## **Acknowledgements**

We would like to thank the Vice-rector for research in project N 2676 of the Universidad Industrial de Santander, the CIMBIOS research group for the ablation laser system (Universidad Industrial de Santander), and the CA Perez-Lopez for his support in the editing of images of the Department of Electrical and Electronic Engineering of the Universidad de los Andes Colombia.

## **Conflict of interest**

The authors declared no potential conflicts of interest concerning the research, authorship, and/or publication of this book chapter.

## **Appendices and nomenclature**

In the reference state, the continuum has a volume *V*, and mass density *ρ*0.

**References**

4020-4670-4

24900192

[1] Katzir, S. The beginnings of piezoelectricity: A study in mundane physics. Springer Science & Business Media; 2007. 273 p. DOI: 10.1007/978-1-

dielectrische influenz und

[2] Voigt, W. Piezo-und pyroelectricität,

*DOI: http://dx.doi.org/10.5772/intechopen.95255*

*Cement-Based Piezoelectricity Application: A Theoretical Approach*

[10] Bechmann, R. Elastic and

Physical Review, 1958;110(1): 1060-1061. DOI: 10.1103/PhysRev.

110.1060

4899-6453-3

90048-K

piezoelectric constants of alpha-quartz.

[11] Tiersten, H. F. Linear piezoelectric plate vibrations: Elements of the linear theory of piezoelectricity and the vibrations piezoelectric plates. Springer, Boston, MA. 2013. DOI: 10.1007/978-1-

[12] Yang, J. On the derivation of electric body force, couple and power in an electroelastic body. Acta Mechanica Solida Sinica. 2015;28(6):613-617.

[13] Yang, J. Differential derivation of momentum and energy equations in electroelasticity. Acta Mechanica Solida

[14] Abeyaratne, R., & Knowles, J. K. A continuum model of a thermoelastic solid capable of undergoing phase transitions. Journal of the Mechanics and Physics of Solids. 1993;41(3): 541-571. DOI: 10.1016/0022-5096(93)

[16] Chen, J., Qiu, Q., Han, Y., & Lau, D. Piezoelectric materials for sustainable building structures: Fundamentals and applications. Renewable and Sustainable Energy Reviews. 2019;101:14-25. DOI:

[17] Paul, S. C., van Rooyen, A. S., van Zijl, Gideon P. A. G., & Petrik, L. F. Properties of cement-based composites using nanoparticles: A comprehensive review. Construction and Building Materials. 2018;189:1019-1034. DOI: 10.1016/j.conbuildmat.2018.09.062

Sinica. 2017;30(1), 21-26.

[15] Tiersten, H. F. Non-linear electroelastic equations cubic in the small field variables. The Journal of the Acoustical Society of America. 1975;57 (3):660-666. DOI:10.1121/1.380490

10.1016/j.rser.2018.09.038

[3] Eringen, A. C. On the foundations of electroelastostatics. 1963;1(1):127-153. DOI: 10.1016/0020-7225(63)90028-4

[4] Toupin, R. A. The elastic dielectric. Journal of Rational Mechanics and Analysis. 1956;5(6):849-915. Retrieved from http://www.jstor.org/stable/

[5] Tiersten, H. F. On the non-linear equations of thermo-electroelasticity. International Journal of Engineering Science. 1972;9(7):587-604. DOI: 10.1016/0020-7225(71)90062-0

[6] Martin, R. M. Piezoelectricity. Physical Review B. 1972;5(4):1607-1613.

[7] Casamento, J., Chang, C. S., Shao, Y. T., Wright, J., Muller, D. A., Xing, H., & Jena, D. Structural and piezoelectric properties of ultra-thin ScxAl1 xN films grown on GaN by molecular beam epitaxy. Applied Physics Letters. 2020; 117(11):112101. DOI: 10.1063/5.0013943

[8] Auld, B. A. Acoustic fields and waves in solids Рипол Классик. John Wiley &

[9] Ma, J., Ren, J., Jia, Y., Wu, Z., Chen, L., Haugen, N. O., ... & Liu, Y. High efficiency bi-harvesting light/vibration energy using piezoelectric zinc oxide nanorods for dye decomposition. Nano Energy. 2019;62:376-383. DOI: 10.1016/

Sons, New York. 1973.

j.nanoen.2019.05.058

**163**

DOI: 10.1103/PhysRevB.5.1607

electrostriction bei krystallen ohne symmetriecentrum. Annalen Der Physik. 1985;291(8):701-731. DOI: 10.1002/andp.18952910812

In the current state, the continuum has a volume *v*, mass density *ρ*, electronic charge density *μ<sup>e</sup>* and lattice charge density *μ<sup>l</sup>* . Besides, In the current state with infinitesimal displacement *η*, the electronic charge does not change its volume.

The capital letter in the index is for the reference state *XK* And the lowercase letters to the current state *yi* . Also, the index in the physics quantities can denote a vector. For example *XK*, *yi* , *ui*; or a tensor, for example *EKL*, *τjk*. Another form to present a vector quantity is the right-pointing arrow *y* !.

The velocity of the continuum is denoted by lower case letter *u*, and just makes sense in the current state.

The partial derivate is denoted by comma separation in the indexes. For example *yi*,*<sup>i</sup>* .

## **Author details**

Daniel A. Triana-Camacho<sup>1</sup> , Jorge H. Quintero-Orozco<sup>1</sup> and Jaime A. Perez-Taborda<sup>2</sup> \*

1 CIMBIOS Group, Physics Department, Universidad Industrial de Santander, Bucaramanga, Colombia

2 Department of Electrical and Electronic Engineering, Universidad de los Andes, Bogotá, Colombia

\*Address all correspondence to: jaimeandres@protonmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Cement-Based Piezoelectricity Application: A Theoretical Approach DOI: http://dx.doi.org/10.5772/intechopen.95255*

## **References**

**Appendices and nomenclature**

letters to the current state *yi*

vector. For example *XK*, *yi*

sense in the current state.

.

For example *yi*,*<sup>i</sup>*

**Author details**

Daniel A. Triana-Camacho<sup>1</sup>

Bucaramanga, Colombia

Bogotá, Colombia

**162**

and Jaime A. Perez-Taborda<sup>2</sup>

provided the original work is properly cited.

charge density *μ<sup>e</sup>* and lattice charge density *μ<sup>l</sup>*

present a vector quantity is the right-pointing arrow *y*

In the reference state, the continuum has a volume *V*, and mass density *ρ*0. In the current state, the continuum has a volume *v*, mass density *ρ*, electronic

*Cement Industry - Optimization, Characterization and Sustainable Application*

infinitesimal displacement *η*, the electronic charge does not change its volume. The capital letter in the index is for the reference state *XK* And the lowercase

The partial derivate is denoted by comma separation in the indexes.

The velocity of the continuum is denoted by lower case letter *u*, and just makes

, Jorge H. Quintero-Orozco<sup>1</sup>

1 CIMBIOS Group, Physics Department, Universidad Industrial de Santander,

2 Department of Electrical and Electronic Engineering, Universidad de los Andes,

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*

\*Address all correspondence to: jaimeandres@protonmail.com

. Besides, In the current state with

. Also, the index in the physics quantities can denote a

, *ui*; or a tensor, for example *EKL*, *τjk*. Another form to

!.

[1] Katzir, S. The beginnings of piezoelectricity: A study in mundane physics. Springer Science & Business Media; 2007. 273 p. DOI: 10.1007/978-1- 4020-4670-4

[2] Voigt, W. Piezo-und pyroelectricität, dielectrische influenz und electrostriction bei krystallen ohne symmetriecentrum. Annalen Der Physik. 1985;291(8):701-731. DOI: 10.1002/andp.18952910812

[3] Eringen, A. C. On the foundations of electroelastostatics. 1963;1(1):127-153. DOI: 10.1016/0020-7225(63)90028-4

[4] Toupin, R. A. The elastic dielectric. Journal of Rational Mechanics and Analysis. 1956;5(6):849-915. Retrieved from http://www.jstor.org/stable/ 24900192

[5] Tiersten, H. F. On the non-linear equations of thermo-electroelasticity. International Journal of Engineering Science. 1972;9(7):587-604. DOI: 10.1016/0020-7225(71)90062-0

[6] Martin, R. M. Piezoelectricity. Physical Review B. 1972;5(4):1607-1613. DOI: 10.1103/PhysRevB.5.1607

[7] Casamento, J., Chang, C. S., Shao, Y. T., Wright, J., Muller, D. A., Xing, H., & Jena, D. Structural and piezoelectric properties of ultra-thin ScxAl1 xN films grown on GaN by molecular beam epitaxy. Applied Physics Letters. 2020; 117(11):112101. DOI: 10.1063/5.0013943

[8] Auld, B. A. Acoustic fields and waves in solids Рипол Классик. John Wiley & Sons, New York. 1973.

[9] Ma, J., Ren, J., Jia, Y., Wu, Z., Chen, L., Haugen, N. O., ... & Liu, Y. High efficiency bi-harvesting light/vibration energy using piezoelectric zinc oxide nanorods for dye decomposition. Nano Energy. 2019;62:376-383. DOI: 10.1016/ j.nanoen.2019.05.058

[10] Bechmann, R. Elastic and piezoelectric constants of alpha-quartz. Physical Review, 1958;110(1): 1060-1061. DOI: 10.1103/PhysRev. 110.1060

[11] Tiersten, H. F. Linear piezoelectric plate vibrations: Elements of the linear theory of piezoelectricity and the vibrations piezoelectric plates. Springer, Boston, MA. 2013. DOI: 10.1007/978-1- 4899-6453-3

[12] Yang, J. On the derivation of electric body force, couple and power in an electroelastic body. Acta Mechanica Solida Sinica. 2015;28(6):613-617.

[13] Yang, J. Differential derivation of momentum and energy equations in electroelasticity. Acta Mechanica Solida Sinica. 2017;30(1), 21-26.

[14] Abeyaratne, R., & Knowles, J. K. A continuum model of a thermoelastic solid capable of undergoing phase transitions. Journal of the Mechanics and Physics of Solids. 1993;41(3): 541-571. DOI: 10.1016/0022-5096(93) 90048-K

[15] Tiersten, H. F. Non-linear electroelastic equations cubic in the small field variables. The Journal of the Acoustical Society of America. 1975;57 (3):660-666. DOI:10.1121/1.380490

[16] Chen, J., Qiu, Q., Han, Y., & Lau, D. Piezoelectric materials for sustainable building structures: Fundamentals and applications. Renewable and Sustainable Energy Reviews. 2019;101:14-25. DOI: 10.1016/j.rser.2018.09.038

[17] Paul, S. C., van Rooyen, A. S., van Zijl, Gideon P. A. G., & Petrik, L. F. Properties of cement-based composites using nanoparticles: A comprehensive review. Construction and Building Materials. 2018;189:1019-1034. DOI: 10.1016/j.conbuildmat.2018.09.062

Section 4

Sustainable Applications of

Cement

**165**

[18] Qomi, M. J. A., Ulm, F. J., & Pellenq, R. J. M. Physical origins of thermal properties of cement paste. Physical Review Applied. 2015;3(6): 064010. DOI: 10.1103/ PhysRevApplied.3.064010

[19] Yan, Z., & Chrisey, D. B. Pulsed laser ablation in liquid for micro-/ nanostructure generation. Journal of Photochemistry and Photobiology C: Photochemistry Reviews. 2012;13(3): 204-223. DOI:10.1016/ j.jphotochemrev.2012.04.004

[20] Huang, H., Lai, J., Lu, J., & Li, Z. Pulsed laser ablation of bulk target and particle products in liquid for nanomaterial fabrication. AIP Advances. 2019;9(1):015307. DOI: 10.1063/ 1.5082695

Section 4
