## 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 = 323.5 ± 75.3 kN / m 2 , the electroelastic parameter γ = − 20.5 ± 6.9 mV / kN , and dielectric constant ε = 939.6 ± 82.9 ε 0 F / m , which have an interpretation as linear theory parameters s ijkl D , g kij and ε ik T 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

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 displacement

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

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

#### 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

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

where

Then, replacing the electric coupling

Factoring the permutation tensor

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

The term

From Eq. (9) we can conclude that the Cauchy stress tensor (

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:

The term

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 expansion.

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

The second-order term

Replacing Eq. (2) in Eq. (16), we obtain:

With Eq. (3), Eq. (17) takes the form:

In Eq. (18), the term

**In the third step**, we will solve the **traction force** in terms of Cauchy stress tensor. Then, the power due to the traction force takes the form:

Now, we apply Taylor’s expansion to Cauchy stress tensor

The products give

Adding similar terms:

Factorizing

Reply the proceeding since Eq. (21) to Eq. (24) for the components

Each one of the components has information about two opposite faces of the volume element. Then, adding index

From the results of Eq. (13), Eq. (19), and Eq. (26) into Eq. (10), we obtain

Here is factoring the terms that contain

From Eq. (28), the term in the square bracket is null by Eq. (4). Then, we obtain Eq. (29) for energy conservation that depends on internal energy.

Remember from Eq. (19) that electric power can be written as:

Another form of electric charge conservation is

The mass conservation

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

With Legendre transformation showing in Eq. (34), Tiersten replaced the internal energy

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

Clear the term

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

The similar terms **energy conservation** in terms of free energy, electrical (electric field and polarization vector), and mechanical (Cauchy stress tensor) components, as is shown in Eq. (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

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).

#### 2.2.1 Electric field and gradient of potential

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

The gradient of the potential

Therefore,

The derivative respect to time of

The term

Derivative

Partial derivate of

The products of partial derivate are reduced to Kronecker delta.

The index into

In Eq. (48) was delete the term

Clearing

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

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

The index

In Eq. (53), the term

From Eq. (54) the term

#### 2.2.2 Polarization vector

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

Where

To get

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

Until now, in Eq. (59), we have obtained a partial transformation, and still missing transform the symmetric Cauchy stress tensor

#### 2.2.3 Second Piola-Kirchhoff stress

The symmetric tensor

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

The gradient of velocity

From Eq. (61), the product between symmetric tensor

The term

We interchange the index

With

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

Factoring the inverse of Jacobian, we get

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

Using mass transformation to the reference state

### 2.3 Constitutional equations from free energy

The conservation laws are valid for any piezoelectric material, including cement-based 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 piezoelectricity’s constitutional equations. Take into account Eq. (70), we can propose the next dependence to the functions

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

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

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

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

The parameters are called elasticity

#### 2.3.1 The linear approach of piezoelectricity

We take on order one approach from Eq. (76) to free energy

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

Here is considering symmetry to parameters elastic

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

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

Solving

Factoring

where the term

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).

The electromechanical properties are defined by piezoelectric charge

### 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

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 .

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

The axial piezoelectric parameter:

For a total deformation

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 can perform a transformation to get a real part of the capacitance

The geometry of copper electrodes (an approximation to parallel plates) is related to capacitance. Therefore, we can calculate the dielectric parameter

where

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

Where

### 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

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 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

## Acknowledgments

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

In the current state, the continuum has a volume

The capital letter in the index is for the reference state

The velocity of the continuum is denoted by lower case letter

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