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].

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.

Figure 1.
States of a deformable and polarizable continuum.

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 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

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

ηk,k=∂ηky∂yk=0E1

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

μly→+μey→+η→=0E2

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

P→=μly→−η→y→=μey→+η→η→≈μey→η→E3

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=njτjk, where nj 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.

ρu̇k=fkE+τmk,m+ρfkE4

ciE+εijkτjky→=0E5

where fkE is the electric force on the body and start from the dot product of polarization Pj with the electric field gradient Ek,i; Besides, ciE is called electric coupling, and it is the cross product of Pj with Ek; finally, fk is the external force on the body per unit mass.

Then, replacing the electric coupling ciE by the cross product between Pj and Ek, we can rewrite Eq. (5) as follow

εijkPjEky→+εijkτjky→=0E6

Factoring the permutation tensor εijk

εijkPjEk+τjk=0E7

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

τSjk=PjEk+τjkE8

The term PjEk is called the Maxwell stress tensor TEjk. On the other hand, multiplying Eq. (5) by εiqr is obtain

εiqrciE+εiqrεijkτjky→=0δjqδkrτjky→−δjrδkqτjky→=−εiqrciEτqry→−τrqy→=−εiqrciEE9

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

ddt(12ukukρdv+εinρdv)=μl(y→)Ek(y→)uk(y→)dv+μe(y→+η→)Ek(y→+η→)[ uk(y→)+η˙k ]dv+tkukds+ρfkukdvE10

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:

ddt12ukukρdv+εinρdv=ρdvddt12ukuk+εin+12ukuk+εinddtρdv=ρdvddt12ukuk+εin+12ukuk+εinddtρdvE11

The term ddtρdv represents the mass conservation. Therefore, this term is null in Eq. (11).

ddt12ukukρdv+εinρdv=ρdvddt12ukuk+εinE12

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

ddt12ukukρdv+εinρdv=ρdvukddtuk+ρdvddtεinE13

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

μly→Eky→uky→dv+μey→+η→Eky→+η→uky→+η̇kdv≈μly→Eky→uky→dv+μey→+η→Eky→+Ek,iy→ηiuky→+η̇kdvE14

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

=μly→Eky→uky→dv

+μey→+η→Eky→uky→+Eky→η̇k+Ek,iy→ηiuky→+Ek,iy→ηiη̇kdvE15

The second-order term ηiη̇k=ηiη̇i=12ddtηi2≈0 is zero, taking into account the infinitesimal displacement. Then, factorize Eky→uky→dv in Eq. (15), it takes a form:

=μly→+μey→+η→Eky→uky→dv+Eky→η̇k+Ek,iy→ηiuky→dvE16

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

=μey→+η→Eky→η̇kdv+μey→+η→Ek,iy→ηiuky→dvE17

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

=μey→+η→Eky→η̇kdv+PiEk,iy→uky→dvE18

In Eq. (18), the term μey→+η→Eky→η̇k is called the electric power of the body wE And the term PiEk,i is the electric force between the lattice and electronic volume elements. It was defined in Eq. (4) above.

μly→Eky→uky→dv+μey→+η→Eky→+η→uky→+η̇kdv=wEdv+fkEuky→dvE19

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:

tkukds= τmknmukds= τ1ky→+12dy1î1uky→+12dy1î1dy2dy3−τ1ky→−12dy1î1uky→−12dy1î1dy2dy3+τ2ky→+12dy2î2uky→+12dy2î2dy3dy1−τ2ky→−12dy2î2uky→−12dy2î2dy3dy1+τ3ky→+12dy3î3uky→+12dy3î3dy1dy2−τ3ky→−12dy3î3uky→−12dy3î3dy1dy2E20

Now, we apply Taylor’s expansion to Cauchy stress tensor τ1k and velocity uk. Next, the last result will be implemented in the components: two τ2k and three τ3k

τ1ky→+12dy1î1uky→+12dy1î1dy2dy3−τ1ky→−12dy1î1uky→−12dy1î1dy2dy3≈τ1ky→+12dy1τ1k,1y→uky→+12dy1uk,1y→dy2dy3−τ1ky→−12dy1τ1k,1y→uky→−12dy1uk,1y→dy2dy3E21

The products give

≈τ1kuk+τ1k12dy1uk,1+12dy1τ1k,1uk+14dy12τ1k,1uk,1dy2dy3−τ1kuk−τ1k12dy1uk,1−12dy1τ1k,1uk+14dy12τ1k,1uk,1dy2dy3E22

Adding similar terms:

≈τ1ky→dy1uk,1y→+dy1τ1k,1y→uky→dy2dy3E23

Factorizing dy1 that multiply to dy2 and dy3 it will transform into the volume element dv.

≈τ1ky→uk,1y→dv+τ1k,1y→uky→dvE24

Reply the proceeding since Eq. (21) to Eq. (24) for the components τ2k and τ3k yield

tkukds=τ1k(y→)uk,1(y→)dv+τ1k,1(y→)uk(y→)dv+τ2k(y→)uk,2(y→)dv+τ2k,2(y→)uk(y→)dv+τ3k(y→)uk,3(y→)dv+τ3k,3(y→)uk(y→)dvE25

Each one of the components has information about two opposite faces of the volume element. Then, adding index m it reduces the Eq. (25) to:

tkukds=τmky→uk,my→dv+τmk,my→uky→dvE26

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

ρdv ukddt(uk)+ρdvddt(εin)=wEdv+fkEuk(y→)dv+τmk(y→)uk,m(y→)dv+τmk,m(y→)uk(y→)dv+ρfkukdvE27

Here is factoring the terms that contain uky→dv to the left side and the terms with the volume differential dv to the right side.

ρddtuk+fkE+τmk,my→+ρfkuky→dv=wE−ρddtεin+τmky→uk,my→dvE28

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.

ρε̇in=wE+τmky→uk,my→E29

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

wE=μey→+η→Emy→dηmdt=Emy→ddtμey→+η→ηm−dμey→+η→dtηm

wE=Emy→Ṗm−μ̇ey→+η→ηmE30

Another form of electric charge conservation is μ̇ey→+η→+μey→+η→ui,i=0, it will simplify the Eq. (30) to:

wE=Emy→Ṗm+μey→+η→ui,iηmE31

The mass conservation ρ̇+ρui,i=0 has a similar mathematical structure as charge conservation. Therefore, the gradient of the speed ui,i in Eq. (31) was replaced

wE=Emy→Ṗm+Emy→μey→+η→−ρ̇ρηm=EmṖm−ρ̇ρEmPm

wE=EmρρṖm−ρ̇PmE32

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

ρε̇in=EmρρṖm−ρ̇Pm+τmky→uk,my→E33

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.

χ=εin−EmPmρE34

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

χ̇=ε̇in−ĖmPmρ−EmṖmρ+EmPmρ2ρ̇E35

Clear the term ρε̇in

ρε̇in=ρχ̇+ĖmPm+EmṖm−EmPmρρ̇E36

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

ρχ̇+ĖmPm+EmṖm−EmPmρρ̇=EmρρṖm−ρ̇Pm+τmky→uk,my→ρχ̇+ĖmPm+EmṖm−Emρ̇Pmρ=EmṖm−Emρ̇Pmρ+τmky→uk,my→E37

The similar terms EmṖm and −Emρ̇Pmρ in Eq. (37), are clear. Finally, we have rewritten 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).

ρχ̇=τmky→uk,my→−ĖmPmE38

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,m 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 DL, volume free charge density ρE, mass density ρ0 and the second Piola-Kirchhoff stress TSKL [15]. It raises by the transformation of symmetric tensor τSmk 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).

ρχ̇=τSmkuk,m−PmEkuk,m−ĖmPmE39

2.2.1 Electric field and gradient of potential

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

WK=Emym,KE40

The gradient of the potential WK is multiplying both sides by XK,m

WKXK,m=Emym,KXK,m=Em∂ym∂XK∂XK∂ymE41

Therefore,

Em=WKXK,mE42

The derivative respect to time of Em becomes

Ėm=ddtWKXK,m=ddtWKXK,m+WKddtXK,mE43

The term XK,m from Eq. (43) is developed as follow

XK,m=δKLXL,m=∂yk∂XL∂XK∂ykXL,mE44

Derivative XK,m respect to time

ddtXK,m=ddt∂yk∂XL∂XK∂ykXL,m=ddtyk,LXK,kXL,mddtXK,m=ddtyk,LXK,kXL,m+ddtXK,kyk,LXL,m+ddtXL,myk,LXK,kE45

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

ddtXK,m=uk,LXK,kXL,m+ddtXK,k∂yk∂XL∂XL∂ym+ddtXL,m∂yk∂XL∂XK∂ykE46

The products of partial derivate are reduced to Kronecker delta.

ddtXK,m=uk,LXK,kXL,m+ddtXK,kδkm+ddtXL,mδKLE47

The index into XK,k and XL,m were exchanging due to commutation Kronecker deltas.

ddtXK,m=uk,LXK,kXL,m+ddtXK,m+ddtXK,mE48

In Eq. (48) was delete the term ddtXK,m in both sides

0=uk,LXK,kXL,m+ddtXK,mE49

Clearing ddtXK,m we obtain

ddtXK,m=−uk,LXK,kXL,mE50

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

Ėm=ddtWKXK,m=ẆKXK,m−uk,LXK,kXL,mWKE51

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

ρχ̇=τSmkuk,m−PmEkuk,m−PmẆKXK,m−uk,LXK,kXL,mWKE52

The index L was changed by k, into uk,L, due to XL,m

ρχ̇=τSmkuk,m−PmEkuk,m−PmẆKXK,m−uk,mXK,kWKE53

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

ρχ̇=τSmkuk,m−PmEkuk,m−PmẆKXK,m+Pmuk,mEkE54

From Eq. (54) the term PmEkuk,m was removes to get

ρχ̇=τSmkuk,m−PmẆKXK,mE55

2.2.2 Polarization vector

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

PL=JXL,iPiE56

Where J is the Jacobian, multiplying Eq. (56) by J−1ym,L we obtain

J−1ym,LPL=J−1ym,LJXL,iPi=δmiPiE57

To get

Pm=J−1ym,LPLE58

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

ρχ̇=τSmkuk,m−J−1ym,LPLẆKXK,mρχ̇=τSmkuk,m−J−1PLẆK∂ym∂XL∂XK∂ymρχ̇=τSmkuk,m−J−1PLẆKδKLρχ̇=τSmkuk,m−J−1PKẆKE59

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

2.2.3 Second Piola-Kirchhoff stress

The symmetric tensor τSmk is related with second Piola-Kirchhoff stress TSKL through a reverse transformation as follow:

τSmk=J−1ym,Kyk,LTSKLE60

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

ρχ̇=J−1ym,Kyk,LTSKLuk,m−J−1PKẆKE61

The gradient of velocity uk,m can be separated on antisymmetric tensor ωmk=12uk,m−um,k plus a symmetric tensor dmk=12uk,m+um,k.

ρχ̇=J−1ym,Kyk,LTSKLωmk+dmk−J−1PKẆKρχ̇=J−1ym,Kyk,LTSKLωmk+J−1ym,Kyk,LTSKLdmk−J−1PKẆKE62

From Eq. (61), the product between symmetric tensor TSKL and antisymmetric tensor ωmk result be null

ρχ̇=J−1ym,Kyk,LTSKLdmk−J−1PKẆKE63

The term ym,Kyk,Ldmk will be solved as following

ym,Kyk,Ldmk=ym,Kyk,L12uk,m+um,k=12uk,mym,Kyk,L+um,kym,Kyk,Lym,Kyk,Ldmk=12∂uk∂ym∂ym∂XK∂yk∂XL+∂um∂yk∂ym∂XK∂yk∂XL=12∂uk∂XK∂yk∂XL+∂um∂XL∂ym∂XKym,Kyk,Ldmk=12uk,Kyk,L+um,Lym,KE64

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

ym,Kyk,Ldmk=12um,Kyk,L+um,Lym,K=12ẏm,Kyk,L+ẏm,Lym,Kym,Kyk,Ldmk=12ddtym,Kyk,L=ddt12ym,Kyk,L−δKLE65

With m=k the term 12ym,Kyk,L−δKL 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,Kyk,Ldmk to:

ym,Kyk,Ldmk=ddtEKL=ĖKLE66

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

ρχ̇=J−1TSKLĖKL−J−1PKẆKE67

Factoring the inverse of Jacobian, we get

ρχ̇=J−1TSKLĖKL−PKẆKE68

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

Jρχ̇=JJ−1TSKLĖKL−PKẆKE69

Using mass transformation to the reference state ρ0=ρJ into Eq. (69), we get a new equation for energy conservation in terms of physical quantities in the reference state. Symmetric tensor TSKL, strain tensor EKL, polarization PK and gradient of potential WK.

ρ0χ̇=TSKLĖKL−PKẆKE70

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

χ=χEKLWKTSKL=TSKLEKLWKPK=PKEKLWKE71

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

χ̇=∂χ∂EKLĖKL+∂χ∂WKẆKE72

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

ρ0∂χ∂EKLĖKL+ρ0∂χ∂WKẆK=TSKLĖKL−PKẆKE73

Both sides of Eq. (73) were compared to deduce two transformations, which resulting symmetric tensor TSKL and polarization PK. The transformations use free energy as a generating function, as shown in Eq. (75).

TSKL=ρ0∂χ∂EKLE74

PK=−ρ0∂χ∂WKE75

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

ρ0χ= 12cABCDEABECD−eABCWAEBC−12χEABWAWB+16cABCDEFEABECDEEF+12dABCDEWAEBCEDE−12bABCDWAWBECD−16χEABCWAWBWC+124cABCDEFGHEABECDEEFEGH+16dABCDEFGWAEBCEDEEFG+14aABCDEFWAWBECDEEF+16dABCDEWAWBWCEDE−124χEABCDWAWBWCWD+…,E76

The parameters are called elasticity c, piezoelectric e, electric permeability χE, 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

TSAB=∂∂EAB12cABCDEABECD−eABCWAEBC−12χEABWAWBE77

PA=−∂∂WA12cABCDEABECD−eABCWAEBC−12χEABWAWBE78

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 TSKL→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 DL→Di. Then, Eqs. (77) and (78) follow:

Tij=∂∂Sij12cijklSijSkl−eijkEiSjk−12χEijEiEjE79

Pi=−∂∂Ei12cijklSijSkl−eijkEiSjk−12χEijEiEjE80

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

Tij=cijklSkl−ekijEkE81

Pi=eiklSkl+χEikEkE82

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

Pi=Di−ε0EiE83

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

Di−ε0Ei=eiklSkl+χEikEkE84

Solving Di,

Di=eiklSkl+ε0Ei+χEikEk=eiklSkl+ε0δikEi+χEikEkE85

Factoring Ek,

Di=eiklSkl+ε0δik+χEikEkE86

where the term ε0δik+χEik is defined as dielectric constant εik. Finally, we have the linear constitutional equation for the electric displacement vector.

Di=eiklSkl+εikEkE87

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).

Di=diklSkl+εikTEkE88

Sij=sijklDTkl+gkijDkE89

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 d33, voltage g33. 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.

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

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.

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

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 5.
Molds and dimensions of cement cylinders.

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.

Figure 6.
Experimental setup of electromechanical measurements.

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 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:

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

Y=323.5±75.3kN/m2E90

The axial piezoelectric parameter:

γ=−20.5±6.9mV/kN.E91

For a total deformation S=0.57±0.09mm 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 can perform a transformation to get a real part of the capacitance C′. It has frequency dependence as follow

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.

C′ω=1ωZ′′E92

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

εωε0=d∗C′AE93

where ε0 is the electric permittivity of free space, A is the transversal section, and d is the thickness between electrodes.

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

ε=939.6±82.9ε0E94

Where ε0 has unit 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.

Figure 9.
The image shows a network of IoT sensors based on cement-based composites piezoresistivity as an active part of smart construction.

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 Y=323.5±75.3kN/m2, the electroelastic parameter γ=−20.5±6.9 [mV/kN], and dielectric constant ε=939.6±82.9ε0F/m, which can be compared with parameters sijklD, gkij and εikT respectively presents into constitutional equations discussed in the chapter.

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 V, and mass density ρ0.

In the current state, the continuum has a volume v, mass density ρ, electronic charge density μe and lattice charge density μl. 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,i.

James Leung

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