Here, we will study the properties of two new fractional-type complex rheological models with piezoelectric properties: a basic fractional-type complex rheological Kelvin–Voigt–Faraday solid model, an ideal piezoelectric model material, and a basic fractional-type complex rheological Maxwell–Faraday model of a viscoelastic fluid, which is also an ideal piezoelectric model material.
5.1. New Basic Complex Rheological Kelvin–Voigt–Faraday Model of the Fractional Type with Piezoelectric Properties
The structural formula of the rheological complex Kelvin–Voigt–Faraday fractional-type model with piezoelectric propertied is .
The resulting normal stress at the ends of the basic rheological complex Kelvin–Voigt–Faraday model, consisting of the basic rheological complex Kelvin–Voigt fractional-type model in parallel with the Faraday ideal elastic and piezoelectric material element, is equal to the sum of the normal stresses of the elements connected in parallel (see
Figure 2a,
Figure 3a, and
Figure 4a,b).
In addition, the electric polarization voltage in Faraday’s ideal piezoelectric material element is equal to
Equation (6) is a differential constitutive relation of the fractional order and gives the relationship between the normal stress
of the fractional-type rheological basic complex Kelvin–Voight–Faraday model with piezoelectric properties and the axial dilatation
, or the axial dilatation rate of the fractional-type material,
. It is a constitutive relation of the fractional order, as follows:
The fractional-order differential constitutive relation given in Equation (8) for the rheological basic complex Kelvin–Voigt–Faraday model, a complex rheological material, is in the general form. By applying the Laplace transform
to the previous differential constitutive relation given in Equation (8), we obtain the Laplace transform
of the axial dilatation of that model as a differential function of the Laplace transform
of the normal stress, in the following form:
where
is the Laplace operator in the following form:
This Laplace operator
can be applied to the following expression:
Equation (9) is the product of two Laplace transforms: one Laplace transform,
, is of the normal stress
, and the other,
, which concluded that these two functions
and
are also in convolution with the axial dilatation function
. As such, the following equation can be derived:
The expression
is much smaller than one; that is,
. Therefore, we can expand the expression
to powers of the order and write it in the following form (see Reference [
27]):
Next, the inverse Laplace transform of the previous Equation (13) can be obtained in the following form:
Based on the property of the three functions, which are in convolution and for which it is true that
we can write the convolution integral in the following form:
Both Equations (15) and (16), i.e., the convolution integrals, are integral constitutive equations relating the axial dilatation and normal stress of the rheological basic complex fractional-type Kelvin–Voigt–Faraday model with piezoelectric polarization properties.
The Faraday element in the complex Kelvin–Voigt–Faraday model of the fractional type, due to deformation, axial dilatation
, and normal stress
, enters a state of polarization, with its axial dilatation
calculated using Equation (17), and the electric voltage stored in that element is expressed in the following form:
In the resting case of a rheological basic complex Kelvin–Voigt–Faraday fractional-type model in parallel with Faraday’s ideal elastic and piezoelectric model of the material at a very slow load change, when we can assume that the fractional-type axial dilatation rate is small and tends to zero (
), the material behaves as a basic ideal elastic material, and the normal stress of the material
is almost proportional to the dilatation
:
If the normal stress at the ends of a rheological basic complex Kelvin–Voigt–Faraday model, in the case where it is in parallel with the Faraday ideal piezoelectric material element, suddenly rises from zero to some finite value, , which remains constant in the following time interval, then we are interested in the behavior of this basic model of the complex material.
If we assume that the normal stress rises suddenly to some value and remains constant
, then the following can be determined:
In order to find the time dependence of the axial dilatation
of the rheological basic complex Kelvin–Voigt–Faraday fractional-type model, in parallel with Faraday’s ideal piezoelectric material element, we first apply the Laplace transform to the previous differential fractional-order constitutive equation and obtain the Laplace transform of axial dilatation
in the following form:
The solution for the axial dilatation
as a function of time for the basic complex Kelvin–Voigt–Faraday model, in parallel with Faraday’s element, when suddenly subjected to a constant normal stress and held at a constant normal stress
, is the inverse Laplace transform
of Equation (21):
Next, it is necessary to determine the approximate analytical expression for the time function as the inverse Laplace transform of the previous expression and transfer it into the time domain.
As such, it is necessary to develop the expression
in order by powers of
, which is a complex number, using the formula
(see References [
4,
5,
57,
63]). As such,
The inverse Laplace transform now gives an analytically approximate expression for the time-domain axial dilatation
for the basic complex rheological Kelvin–Voigt–Faraday fractional-type model with piezoelectric properties, when suddenly subjected to a constant normal stress
and kept under constant normal stress in the following form:
The dielectric displacement (shift)
, represented as
and
in the polarized Faraday element included in parallel in the basic complex rheological Kelvin–Voigt–Faraday fractional-type model with piezoelectric properties, can be expressed in the following form:
This model of a complex rheological Kelvin–Voigt–Faraday material of the fractional type with piezoelectric properties, with the parallel connection structure of the basic complex rheological Kelvin–Voight model of a solid elastoviscous fractional-type material in parallel connection with Faraday’s ideal piezoelectric material model, possesses the property of post-elasticity, when axial dilatation lags behind normal stress .
In
Figure 4c, the surface of the subsequent elasticity of that basic complex rheological Kelvin–Voigt–Faraday material of the fractional type with piezoelectric properties is presented in a coordinate system with coordinate axes: the elongation of axial dilatation
, the exponent
of fractional-order differentiation, in an interval greater than zero and less than or equal to one (
), and time.
The rheological complex Kelvin–Voigt–Faraday fractional-type model, with piezoelectric properties has no internal degree of freedom.
5.2. New Basic Complex Rheological Maxwell–Faraday Model
We will now further study the properties of a new basic complex rheological model, a fractional-type model with piezoelectric properties that has a series connection structure of the basic complex rheological Maxwell model of a viscoelastic fluid material, a fractional type model, in series connection with a Faraday ideal elastic and piezoelectric model material. As a result, a new rheological basic complex Maxwell–Faraday model, a fractional-type model with piezoelectric properties, is obtained. See
Figure 2b and
Figure 3b.
Throughout the entire basic complex model with the Hooke ideal elastic element
, the Newton viscous element of the fractional type
, and the Faraday ideal elastic and piezoelectric material element, the normal stress
in all the regularly connected elements is the same, while the resulting fractional-type dilatation velocity is equal to the sum of the dilatation velocities of the elements
,
, and
in the order connection.
Figure 5b shows the component dilatations of these regularly connected elements of the complex rheological complex basic Maxwell–Faraday model of the fractional type.
Faraday’s ideal elastic and piezoelectric material model was added to a rheological basic complex Maxwell’s model of the fractional type, whose structural formula is , to create a new rheological basic complex Maxwell–Faraday model.
When using Hooke’s ideal elastic solid element
, Newton’s fractional-type viscous element
, and Faraday’s ideal elastic and piezoelectric material element, throughout the entire basic complex model
, the velocity of axial dilatation of the fractional type through the entire element is equal to the sum of the axial dilatation velocities of the elements,
,
, and
in a regular series connection, so in the sum of the axial dilatation rates, we use the dilatation rates of the fractional-type equation
. See
Figure 5a,b. In
Figure 5b, the decomposition of the axial dilatation of the new rheological basic complex Maxwell–Faraday model is presented and component dilatations are visible.
The normal stress is the same throughout the entirety of the new rheological basic complex Maxwell–Faraday model of the fractional type with piezoelectric properties, so we can determine the differential constitutive relations of each basic rheological element in this series connection; as such, the same normal stresses are a function of the resulting dilatation of the model (see
Figure 5b):
The differential constitutive relations between the normal stress
and velocity axial dilatation
of the fractional type (see
Figure 5b for an illustration of the idea of decomposition by dilatations) of each of the regularly connected elements of the complex structure of the Maxwell–Faraday model are determined by Equation (26), as shown previously.
For Hooke’s ideal elastic element, the normal stress is and is linearly related to the dilatation, and the velocity of fractional-type dilatation is a function of the normal stress of the fractional-type equation . For the new ideal viscous Newtonian element of the fractional-type fluid, the normal stress is differentially related by fractional order to the velocity of fractional-order axial dilatation . The velocity of fractional-type axial dilatation is expressed in terms of the normal stress . For the ideal elastic Faraday element with piezoelectric properties, the mechanical normal stress is linearly related to the axial dilatation and the electrical voltage due to the polarization of the Faraday element by mechanical strain, and also the dielectric displacements and are linearly related to the mechanical normal stress or dilatation. For this Faraday element, the velocity of fractional-type axial dilatation is a function of the velocity of fractional-type normal stress in the form .
The electric voltage can be expressed as and the dielectric displacement (shift) as , , and in the polarized Faraday’s element included in serial in the basic complex model.
Figure 5a presents the basic complex Maxwell–Faraday model and
Figure 5b presents the decomposition into component axial dilatations of rheological elements of the basic complex Maxwell–Faraday model of the fractional type with piezoelectric properties. This figure shows and explains how the differential constitutive relations of the fractional order are constructed.
The resulting velocity of axial dilatation
of the basic complex Maxwell–Faraday model of the serial structure is equal to the sum of the component fractional-type axial dilatation velocities of Newton’s fractional-type viscous element
, of Hooke’s ideal elastic element
, and of Faraday’s ideal elastic element with the property of piezoelectricity, and can be expressed in the following form:
As such,
and therefore
,
, and
, resulting in
. Therefore, the sum can be expressed as follows (see
Figure 5b):
Next, based on the previous analysis, it follows that this sum of fractional-type axial dilatations in the rheological basic complex Maxwell–Faraday model is of the following form (see
Figure 5b):
The last inhomogeneous ordinary differential Equation (29) of the fractional order is a differential constitutive relation of the fractional-order exponent , highlighting the axial dilatation and normal stress relationship of the rheological basic complex Maxwell–Faraday model of the fractional type with piezoelectric properties.
The last inhomogeneous differential constitutive equation, Equation (29), of fractional-order exponent
can be solved by applying the Laplace transform
, which gives the following:
Then, expanding in order by powers of the complex parameter, we obtain the following:
Using the property of the three functions being convolved, as well as the relationship between their Laplace transforms and the convolution integral, in a similar way as in [
5], we can produce the following convolution integral:
Equation (32), which is the convolution integral, represents an approximate analytical expression of the dependence of the normal stress in the basic complex Maxwell–Faraday model, as a function of the total axial dilatation over time, and is also the constitutive integral equation of that model.
If the material in the basic complex Maxwell–Faraday model is suddenly loaded to a certain value of normal stress , it will respond to an elastic deformation, , created instantaneously in Hooke’s ideal elastic element and Faraday’s piezoelectric element. This is because due to a sudden load added immediately at the beginning of the observation of the behavior of the material of the basic complex Maxwell–Faraday model, flow in the regularly connected modified Newton viscous element, in this case, in the viscous fluid, does not come to the fore. If we prevent the development of axial dilatation, assuming that the velocity of axial dilatation tends to zero, , then the normal stress is a function of time, which needs to be determined.
When the normal stress rate
of the fractional type of the rheological basic complex Maxwell–Faraday model tends to zero,
, then the normal mechanical stress
tends to a value proportional to the axial dilatation rate of the fractional type,
:
In order to find the dependence of the normal stress
on time, when we keep the model of the material of the rheological basic complex Maxwell–Faraday model of the fractional type with piezoelectric properties at some constant velocity of fractional-type axial dilatation,
, the following can be determined:
Then, we apply the Laplace transformation to that fractional-order functional dependence outlined in Equation (34), and after applying the Laplace transformation to the previous differential constitutive relation in Equation (34) for the special case of the state of the model material, we obtain the following:
Then, by arranging the Laplace transform to the previous constitutive relation in Equation (35), we obtain
.
By solving the obtained relation using , the Laplace transform of the normal stress as a function of time in the rheological basic complex Maxwell–Faraday model can be given in Equation (36).
Next, it is necessary to determine an approximate analytical solution for the normal stress as a function of time, in a rheological basic complex Maxwell–Faraday model, as the inverse Laplace transformation of the previous Equation (36) and move from the complex domain of the Laplace transformation to the time domain.
To this end, it is necessary to develop the expression
in order of powers of
, which is a complex number, using the formula
(see References [
4,
5,
58,
63]). Therefore,
The inverse Laplace transform
of the previous expression
, given in Equation (36), now gives an approximate analytical solution for
in the time domain, in a basic complex Maxwell–Faraday model, in the form of a power order by degrees of time, in the following form:
The electric voltage polarization field of the Faraday element included serially in the rheological basic complex Maxwell–Faraday model of the fractional type with piezoelectric properties can be determined as follows:
And dielectric displacement is in the following form:
We can see from the previous solution (38), and from the graph of the relaxation surface of
, that, in that case, the normal stress
will asymptotically decrease and tend to zero, as shown in
Figure 5c. The occurrence of a decrease in normal stress
with the passage of time at a constant velocity of axial dilatation,
, is referred to as the normal stress relaxation of the material in the basic complex Maxwell–Faraday model. The studied material of the rheological basic complex fractional-type Maxwell–Faraday model with piezo-electric properties is a viscoelastic fluid of the fractional type with piezoelectric properties and can be used as a model for describing the properties of metals at very high temperatures and with piezoelectric properties.
The new rheological basic complex Maxwell–Faraday model has two internal degrees of freedom.
5.3. New Rheological Complex Lethersich–Faraday Model of a Viscoelastic Material
In this section, we study the new rheological complex Lethersich–Faraday model of an ideal material of the fractional type with piezoelectric properties in two variants: 1*—when Faraday’s ideal elastic material with polarization properties is connected in parallel with the rheological Kelvin–Voigt structure, as shown in
Figure 6b; and 2*—when Faraday’s ideal elastic material with polarization properties is connected in parallel with the rheological Newtonian viscous fluid element, as shown in
Figure 6c. In the first case, labeled 1*, the rheological complex Lethersich–Faraday model of an ideal fractional-type material with piezoelectric properties has the properties of a viscoelastic material and has the structural formula
. The second case, labeled 2*, is the rheological complex Lethersich–Faraday model of an ideal material, which has the properties of an elastoviscous solid material and has the structural formula
.
In
Figure 6, three new rheological complex Lethersich fractional-type models, in two variants, are visible. In
Figure 6a, the structure of the new Lethersich viscoelastic fluid material model is presented.
Figure 6b shows the structure of the new Lethersich–Faraday model of viscoelastic fluid with polarization properties, with Faraday’s element connected in parallel with the Kelvin–Voigt model.
Figure 6c shows the structure of the new Lethersich–Faraday-F elastoviscous solid model with polarization properties, with Faraday’s element connected in parallel with Newton’s viscous fluid element.
5.3.1. The Rheological Complex Linear Lethersich–Faraday Model of an Ideal Material with Piezoelectric Properties
This model contains the following parallel-connected elements: Hooke’s element of an ideal elastic material, Newton’s linear element of an ideal viscous fluid, and Faraday’s ideal elastic and piezoelectric element (which form, in parallel connection, the Kelvin–Voigt–Faraday linear model of an ideal material, with the quality of subsequent ecstaticity). They all form a sequential connection with the linear Newtonian element of an ideal viscous fluid. Let us now denote the specific axial deformations and axial dilatations of the Newtonian elements of an ideal viscous fluid in the rheological basic complex linear Kelvin–Voigt–Faraday model of an ideal solid elastoviscous material with piezoelectric properties by and . To establish the total axial dilatation of this new rheological linear Lethersich–Faraday model of an ideal material with piezoelectric properties, which is serially connected in the Kelvin–Voigt–Faraday model with the Newtonian linear element of an ideal viscous fluid, we calculate the sum of these axial dilatations: .
The constitutive relation between the normal stress at the points of the cross-sections of the test tube and the axial dilatations of the linear elements for the rheological complex Lethersich–Faraday linear model of an ideal material with piezoelectric properties is obtained from the relation of the sum of the component velocities of the axial dilatations in the form
. For Newton’s model of an ideal linear viscous fluid, the following applies:
Alternatively, for the rheological basic complex Kelvin–Voigt–Faraday ideal model of an ideal solid body with piezoelectric properties, the sum of the dilatations can be calculated as follows:
It follows that the axial dilatation as a function of time can be written in the following form:
in which
is the initial axial dilatation of the rheological basic complex Kelvin–Voigt–Faraday ideal linear model of an ideal solid body with piezoelectric properties.
Then, by differentiating in time the previous expression for the axial expansion in Equation (43) of the rheological basic Kelvin–Voigt–Faraday linear model of an ideal body with piezoelectric properties and adding the axial dilatation velocity
of the Newtonian linear element of an ideal viscous fluid to the axial dilatation velocity
of the rheological basic Lethersich–Faraday ideal linear material model, we obtain the following expression:
Equation (44) represents the differential constitutive relation of the connection between the velocity of axial dilatation and the normal stress of the new rheological complex Lethersich–Faraday linear model of an ideal viscoelastic fluid body with piezoelectric properties.
5.3.2. New Rheological Complex Lethersich–Faraday Model in the Form of Viscoelastic Fluid
The rheological complex fractional-type Lethersich–Faraday model of an ideal material with piezoelectric properties, as shown in
Figure 6b, contains the following parallel-connected elements: Hooke’s element of an ideal elastic material, Newton’s fractional-type element of an ideal viscous fluid, and Faraday’s ideal elastic piezoelectric element material (which make up, in parallel connection, a basic fractional-type Kelvin–Voigt–Faraday model of an ideal material with piezoelectric properties and with the quality of subsequent elastity). These elements are in series connection with Newton’s element of an ideal fractional-type viscous fluid.
Let us now denote the specific axial deformations and axial expansions of Newton’s elements of an ideal viscous fluid in the model presented in
Figure 6b and in the rheological complex Lethersich–Faraday model of an ideal fractional-type material with piezoelectric properties by
and
. And for the total axial dilatation
of this Lethersich–Faraday model, which is a series connection of the rheological basic complex Kelvin–Voigt–Faraday model of an ideal fractional-type solid body with piezoelectric properties and Newton’s element of an ideal viscous fractional-type fluid, we obtain the sum of these axial expansions and dilatations:
.
The differential constitutive relay connection between normal stress and axial dilatations for the rheological complex Lethersich–Faraday model of an ideal fractional-type material with piezoelectric properties is obtained from the relation of the sum of component axial velocities of axial dilatations in the form
, that is, the sum of the fractional-type axial dilatation velocities:
. For Newton’s element of an ideal viscous fractional-type fluid, the sum can be calculated as follows:
Meanwhile, for the rheological basic complex fractional-type Kelvin–Voigt–Faraday ideal model of an elastoviscous body with piezoelectric properties, the sum of the normal stresses can be calculated as follows:
From this, it follows that the rate of velocity of axial dilatation, of the fractional type, through the rheological basic complex Kelvin–Voigt–Faraday model of an ideal solid material of the fractional type with piezoelectric properties is the solution of an inhomogeneous differential equation of the fractional order, as follows:
Next, let us apply, as in the previous paragraph, the Laplace transform
to the previous inhomogeneous differential equation of the fractional order:
On the right-hand side, we have the product of two Laplace transforms of each function, which can be viewed as two functions, which are in convolution. So, we need to determine the inverse transforms of each of those functions individually: and .
The solution of the inhomogeneous differential equation of the fractional order, shown in Equation (47), which describes the axial dilatation through the rheological complex Kelvin–Voigt–Faraday model of an ideal material of the fractional type with piezoelectric properties, is a convolution integral that can be written as follows (see References [
64,
65,
66]):
in which
is the initial axial dilatation of the rheological complex Kelvin–Voigt–Faraday ideal model of an ideal visvoelastic fractional-type fluid body with piezoelectric properties.
Then, by differentiating in time the previous expression of axial dilatation shown in Equation (49) of the rheological basic Kelvin–Voigt–Faraday model of an ideal elastoviscous solid body, as well as adding the axial dilatation velocity
of the elementary Newtonian element of an ideal fractional-type viscous fluid, to obtain the axial dilatation velocity
of the rheological complex Lethersich–Faraday model of an ideal visclastic fluid material, we obtain the following expression:
From the structure, shown in
Figure 6b, of the rheological complex Lethersich–Faraday model of this ideal viscoelastic fluid material, which is formed by a series connection of the rheological basic Kelvin–Voigt–Faraday elastoviscous solid material and Newton’s element of an ideal viscous fluid of the fractional type, and from the properties of the models of these ideal materials that we have identified, the first being subsequent elasticity and the second being normal stress relaxation, it can be seen that the rheological complex Lethersich–Faraday material possesses both properties of subsequent elasticity and stress relaxation and that it is a more complex model of the material than its substructures.
The rheological complex Lethersich–Faraday model of an ideal viscoelastic fluid material belongs to the group of rheological viscoelastic creepers.
The new rheological complex Lethersich–Faraday model has one internal degree of freedom of mobility.
5.5. New Rheological Complex Jeffrey–Faraday Model of a Viscoelastic Fluid Material
In this section, we study the new rheological complex Jeffrey–Faraday model of an ideal fractional-type material with piezoelectric properties, in two variants: 1*—when Faraday’s ideal elastic material with polarization properties is connected in parallel with Hooke’s rheological ideal elastic element, as shown in
Figure 7b; and 2*—when Faraday’s ideal elastic material with polarization properties is connected in parallel with the Newton rheological elementary fractional-type viscous fluid element or the rheological basic complex Maxwell viscoelastic fluid model, as shown in
Figure 7c. The first case, labeled 1*, is the rheological fractional-type complex Jeffrey–Faraday model of an ideal material with piezoelectric properties, which has the properties of a viscoelastic fluid material and has the structural formula
. The second case, labeled 2*, is the rheological fractional-type complex Jeffrey–Faraday model of an ideal material with piezoelectric properties, which has the properties of an elastoviscous solid material and has the structural formula
or
.
In
Figure 7, three new rheological complex Jeffrey fractional-type models, in two variants, are visible. In
Figure 7a, the structure of the new rheological complex Jeffrey fractional-type viscoelastic fluid material model is presented.
Figure 7b shows the structure of the new rheological complex Jeffrey–Faraday fractional-type model of a viscoelastic fluid with piezoelectric properties, with Faraday’s piezoelectric element connected in parallel with Hooke’s ideal elastic element.
Figure 7c shows the new Jeffrey–Faraday elastoviscous solid model, with Faraday’s element connected in parallel with Newton’s fractional-type viscous fluid element.
5.5.1. The Rheological Complex Linear Jeffrey–Faraday Model of an Ideal Material with Piezoelectric Properties
This model contains Hooke’s element of an ideal elastic material and Faraday’s ideal elastic and piezoelectric element connected in parallel, and they are in a sequential connection with the linear Newtonian element of an ideal viscous fluid and in parallel connection with another linear Newtonian element of an ideal viscous fluid. Let us now denote the specific axial deformations and axial dilatations of the linear Newtonian elements of an ideal viscous fluid and the rheological Hooke model of an ideal solid element material by and . And to obtain the total axial dilatation of this new rheological linear complex Jeffrey–Faraday model of an ideal viscoelastic model material with piezoelectric properties, we calculate the sum of these axial dilatations as follows: .
The rheological linear complex Jeffrey–Faraday model
of an ideal viscoelastic fluid material with piezoelectric properties, similar to that shown in
Figure 7b, contains a parallel connection of Newton’s linear element of an ideal viscous fluid and Maxwell’s model of an ideal viscoelastic material with a parallel connection to Hooke’s ideal elastic model and Faraday’s element of an ideal piezoelectric and ideal elastic material, for which there are individually constitutive relations between normal stress and axial dilatation, i.e., axial dilatation velocity, in the following form:
From here, by integrating Equation (57), the following can be determined:
Then, by adding
and
, we obtain the resulting normal stress
in the rheological linear complex Jeffrey–Faraday model
of an ideal viscoelastic fluid material with bolariyation under deformation, shaped by piezoelectric properties, in the following form:
This is measured over the velocity (rate) of axial dilatation as a function of time in the form of Equation (59).
5.5.2. New Rheological Complex Jeffrey–Faraday Model of a Viscoelastic Fluid Material
The new rheological complex Jeffrey–Faraday model
of an ideal fractional-type viscoelastic fluid material with piezoelectric properties, as shown in
Figure 7b, contains a parallel connection of a Newtonian element of an ideal fractional-type viscous fluid and a new basic complex Maxwell model of an ideal fractional-type viscoelastic material, for which Hooke’s ideal elastic element is connected in parallel to Faraday’s ideal piezoelectric and ideal elastic material elements, which are individually serially connected.
For the first elementary Newtonian element of an ideal fractional-type viscous fluid, the velocity of axial dilatation can be expressed in the following form:
For the second Newtonian element of an ideal fractional-type viscous fluid, in terms of the rate of velocity of fractional-type axial dilatation, the constitutive diferential relation of the element showing the diferential relation between normal stress and velocity rate of axial dilatation, of the fractional order, is in the following form:
For Hooke’s ideal elastic element connected in parallel in the model with Faraday’s ideal piezoelectric and ideal elastic material model, there are individually constitutive relations between normal stress and axial dilatation, i.e., the fractional-type dilatation rate of velocity, which take the following forms:
The rates of velocity of the dilatations of the fractional type of these two parallel-connected elements, Hooke’s ideal elastic element and Faraday’s elements of the ideal piezoelectric and ideal elastic material model, are equal, so
and the following equation can be formed:
The sum of the normal stresses in parallel elements, i.e., Hooke’s ideal elastic element and Faraday’s element of the ideal piezoelectric and ideal elastic material, can be expressed in the following form:
As such, we have determined
through the entire series connection of the structure of new complex Jeffrey–Faraday model
of an ideal viscoelastic fluid material, shown in
Figure 7b. The velocity of axial dilatation of Newton’s element of viscous fluid, connected in series with the parallel-connected Hooke’s ideal elastic element and Faraday’s ideal piezoelectric model and ideal elastic material model, can be expressed as follows:
Then, the sum of the fractional-type axial dilatation velocity rates of the two parallel elements, Hooke’s element and Faraday’s elastic material, and Newton’s element of an ideal viscous fluid, which is regularly related to them, can be expressed in the following form:
This value is equal to the axial dilatation velocity of the first Newtonian element of an ideal viscous fluid, which is connected in parallel with them, as shown in
Figure 7b:
We now determine the normal stress in the first Newtonian viscous fluid element, using the previous velocity of axial dilatation, which we determined using the following equation:
The normal voltage can be expressed in the following form:
Next, we can write a differential constitutive fractional-order relation for the complete rheological complex Jeffrey–Faraday model of an ideal fractional-type viscoelastic fluid material with piezoelectric properties, shown in
Figure 7b, which contains a serial connection of a Newtonian element of an ideal fractional-type viscous fluid and a rheological basic complex Maxwell–Faraday model of an ideal fractional-type viscoelastic material with piezoelectric properties, to which Hooke’s ideal elastic element is connected in parallel with the element of Faraday’s ideal piezoelectric and ideal elastic material model with coupled mechanical and electrical fields.
For all those rheological elements of different material properties, we determined the differential constitutive relations individually, on the basis of which we now write that the resulting normal stress of this model is in the form of the sum of the component normal stresses in the parallel branches of the complex Jeffrey–Faraday model of an ideal viscoelastic fluid material, shown in
Figure 7b, in the following form:
The preceding constitutive relation shown in Equation (74) is in the form of an inhomogeneous differential relation of fractional-order exponent
, where
. From the previous relation, we form an ordinary inhomogeneous fractional differential equation of the following form:
It gives the relation between the dilatation
of Hooke’s element and the resulting normal stress
in the structure of the complex Jeffrey–Faraday model
of an ideal viscoelastic fluid material, shown in
Figure 7b, which can also be written in the following form:
From this, it can be determined that
because
in which
is the axial dilatation of the elements. Next, we need to determine the solutions of this inhomogeneous ordinary differential equation of the fractional order which is in the following form, adapted from Equation (77):
Alternatively, it can be transformed as follows:
Using Equation (77), the inhomogeneous differential equation of fractional-order exponent
is solved by applying the Laplace transform, then developing it in a degrees series and then returning it, with the inverse Laplace transform, as follows:
Then, the following can be determined:
At constant axial dilatation of the rheological complex Jeffrey–Faraday model of an ideal viscoelastic fluid material, relaxation of normal stress occurs.
The new rheological complex Jeffrey–Faraday model has one internal degree of freedom of mobility.
5.6. New Rheological Complex Jeffrey–Faraday Model as an Elastoviscous Solid Material
In this section, we study the second variant of the new rheological complex Jeffrey–Faraday-F model of an ideal elastoviscous fractional-type solid material with piezoelectric properties, when Faraday’s ideal elastic element with polarization properties is connected in parallel with the rheological fractional-type Newtonian viscous fluid element or the rheological basic complex Maxwell viscoelastic fluid model, as shown in
Figure 7c. As we wrote in this case, it is the rheological complex Jeffrey–Faraday model of a fractional-type ideal material with piezoelectric properties, which has the properties of an elastoviscous solid material and has the structural formula
or
.
The rheological complex Jeffrey–Faraday model of a fractional-type elastoviscous solid and piezoelectric material has the ability to polarize under deformation and loading.
This rheological model, the second rheological complex Jeffrey–Faraday-F model of an ideal elastoviscous solid material, belongs to a group of rheological elastoviscous oscillators.
The second new rheological complex Jeffrey–Faraday-F model has one internal degree of freedom of mobility at the point of regular connection in its substructure.
The constitutive relation of the second rheological complex Jeffrey–Faraday-F model of an elastoviscous solid material is easily determined by the sum of the component normal stresses of two elementary rheological elements connected in parallel in a basic complex model: the Newtonian ideal viscous fluid element and the Faraday element in one rheological basic complex Maxwell model of an elastoviscous fluid, as shown in
Figure 7c.
The normal stress through the entire structure of the rheological complex Jeffrey–Faraday-F model of an elastoviscous solid material is the resultant normal stress of the components:
The normal stress
at the points of the cross-section is the same throughout the entire modified Maxwell model
, so we can write the constitutive relations of each basic rheological element in this series connection; as such, the normal stresses
in the dilatation function are
The velocity rate of axial dilatation of the fractional-type rheological basic complex Maxwell model
, the fractional-type equal to the sum of the dilatation rates, the fractional-type Newtonian viscous element
, and Hooke’s ideal elastic element
can be expressed in the following form:
From this, it can be determined that
and therefore
and
. As such, it follows that a sum (see
Figure 7c) can be used for axial dilatation analysis:
This model has the property of subsequent elasticity, and the structure model of the rheological complex Jeffrey–Faraday-F model of an elastoviscous solid material in the substructure of the basic complex Maxwell model exhibits normal stress relaxation. The structure of the rheological complex Jeffrey–Faraday-F model of an elastoviscous solid material has the property of subsequent elasticity.
This rheological model, the second rheological complex Jeffrey–Faraday-F model of an ideal elastoviscous solid material, belongs to the group of rheological elastoviscous oscillators.
The second new rheological complex Jeffrey–Faraday model of a fractional-type elastoviscous material with piezoelectric properties has one internal degree of freedom of mobility at the point of regular connection in its substructure.
5.7. New Rheological Complex Burgers–Faraday Model of a Viscoelastic Fluid Material
In this section, we study two new rheological complex Burgers–Faraday
models of an ideal fractional-type material with piezoelectric properties: 1*—when Faraday’s ideal elastic element with polarization properties is connected in parallel with the rheological basic complex Kelvin–Voigt ideal elastoviscous fluid model, as shown in
Figure 8b; and 2*— when Faraday’s ideal elastic element with polarization properties is connected in parallel with the rheological basic complex Maxwell viscoelastic fluid model, as shown in
Figure 8c. The first case, labeled 1*, is the rheological complex Burgers–Faraday model of an ideal material, which has the properties of a viscoelastic fluid material and has the structural formula
,
.
The second case, labeled 2*, is the rheological complex Burgers–Faraday-F model of an ideal material, which has the properties of an elastoviscous solid material and has the structural formula
,
; see
Figure 8c.
In
Figure 8, the structures of three new rheological complex Burgers models, one Burgers model as well as two Burgers–Faraday models, are presented.
Figure 8a shows a new complex Burgers fractional-type viscoelastic fluid model.
Figure 8b shows a new complex Burgers–Faraday viscoelastic fluid material model. In
Figure 8c, a new complex Burgers–Faraday-F elastoviscous solid material model is visible.
The second rheological complex Burgers–Faraday model
of an ideal viscoelastic fluid material with piezoelectric properties is shown in
Figure 8b, and represents a series connection of the rheological basic complex Maxwell viscoelastic fluid model and the rheological basic complex Kelvin–Voight–Faraday elastoviscous solid model (with a piezoelectric element connected in parallel with the Faraday element). Both rheological basic complex models are of the fractional type; one is formed of an ideal viscoelastic fluid and the second of elastoviscous solid materials.
The structural formula of the Burgers–Faraday viscoelastic fluid material is in the form .
The third rheological complex Burgers–Faraday model
of an ideal elastovisous solid material with piezoelectric properties is shown in
Figure 8c, and represents a series connection of the rheological basic complex Maxwell–Faraday elastoviscous solid model (with a piezoelectric element connected in parallel with the Faraday element) and the rheological basic complex Kelvin–Voight elastoviscous solid model. Both rheological basic complex models are of the fractional type, and both utilize ideal and different viscoelastic solid material models. The constitutive relations (equations of coupled mechanical normal stress and axial expansions) of this rheological complex material model are now easily determined using the following structural formula:
.
5.7.1. New Rheological Complex Burgers–Faraday Linear Model of a Linear Viscoelastic Fluid Material with Piezoelectric Properties
First, we will study the properties of the new rheological complex Burgers–Faraday linear model of a linear viscoelastic fluid material with piezoelectric properties, which has a structure similar and analogous to the structure of the model shown in
Figure 8b, in which the Newton elements are linear.
The total axial dilatation of the complex linear-type Burgers–Faraday viscoelastic fluid model
of an ideal material with piezoelectric properties is equal to the sum of the component axial dilatations
,
, and
, of the component substructure, as well as the basic complex Maxwell and Kelvin–Voigt–Faraday models (complex model with Faraday’s element connected in parallel). The mechanical normal stress in all of the cross-sections of the component substructures of the basic complex models is the same throughout the entire complex Burgers–Faraday model. Based on this analysis and these derived conclusions, we can determine the following:
Let us differentiate the first constitutive relation shown in Equation (89) from the previous system, and replace the terms from the third constitutive equation of the previous system given in Equations (90) and (91); based on that, we can write the following constitutive relation for the complex Burgers–Faraday linear model of an ideal viscoelastic fluid material:
Next, the last constitutive relation shown in Equation (92) can be differentiated once more in time to obtain the following:
Let us now multiply Equation (90) by
and Equation (94) by
and combine them with
for the component Kelvin–Voigt–Faraday complex model of an ideal material with piezoelectric properties so that we obtain the following relation:
Finally, we obtain the following differential relationship between the mechanical normal stress and the first and third derivatives by time of the axial dilatation in the following form:
5.7.2. New Rheological Complex Burgers–Faraday Model of a Fractional-Type Viscoelastic Fluid Material
The rheological complex Burgers–Faraday model
of an ideal viscoelastic fluid material is shown in
Figure 8b, and represents, as we explain, a series connection of the rheological basic complex Maxwell viscoelastic fluid material model and the rheological basic complex Kelvin–Voight–Faraday elastoviscous solid material model (with a Faraday piezoelectric element connected in parallel). Both rheological basic complex models are of the fractional type, and utilize ideal viscoelastic fluids as well as elastoviscous solid materials. The constitutive relations are differential equations of the fractional order of coupled mechanical normal stress and axial expansions. The structural formula of this rheological complex material model is expressed in the following form:
,
.
The constitutive relations (equations of relation between normal stress and axial dilatation) of this complex material model, a modified Burgers–Faraday model of an ideal fractional-type material with piezoelectric properties, contain derivatives of the fractional-order exponent , where , by time.
The total axial dilatation of the rheological complex Burgers–Faraday
ideal fractional-type viscoelastic fluid material model
with the piezoelectric properties of an ideal material is equal to the sum of the component axial dilatations
,
, and
of the component substructure, as well as the rheological complex basic models: the rheological basic complex Maxwell and the rheological basic complex Kelvin–Voigt–Faraday models (complex model with a parallel-connected Faraday ideal elastic element with piezoelectric properties). Mechanical normal stresses in all cross-sections of the component substructure of the basic complex models are equal throughout the entire rheological complex Burgers–Faraday viscoelastic fluid model:
. Based on this analysis and the derived conclusions, we can determine the following relations:
Let us differentiate the first constitutive relation using the differential operator of fractional-order exponent
,
, from the previous system, as shown in Equation (98), and replace the terms from the third constitutive formula in Equation (99) of the previous system; based on that, we could write the constitutive relation for the rheological complex Burgers–Faraday model
of an ideal viscoelastic fluid material in the following form:
or
As throughout the viscoelastic fluid model structure
, the normal stress is equal to
, the previous two equations of the fractional order can be expressed as follows:
These last two relations between normal stress
and axial dilatation
are coupled differential constitutive relations of the fractional-order exponent
of the structure model
of the ideal rheological viscoelastic fluid material model, which is shown in
Figure 8b.
To repeat, of the ideal viscoelastic fluid material with piezoelectric properties, with the fractional-derivative-order exponent , where , represents the regular serial connection of the Maxwell and Kelvin–Voight–Faraday models, both rheological complex fractional-type models of an ideal material with piezoelectric properties, one utilizing a viscoelastic fluid and the second an elastoviscous solid, and in regular sequential conection gives the final results for the viscoelastic fluid material model. And under certain conditions, of the ideal viscoelastic fluid material with piezoelectric properties, with fractional-derivative-order exponent , where , has properties of subsequent elasticity and/or relaxation of normal stress in its substructure.
From this system, expressed in Equations (102) and (103), of differential constitutive relations of the fractional-order exponent
, where
, we can eliminate the velocity of dilatation of this fractional type and obtain only one differential constitutive relation highlighting the connection between the normal stress and axial dilatation of the structure of the rheological complex Burgers–Faraday model
of an ideal fractional-type viscoelastic fluid material with piezoelectric properties in the domain of the Laplace transform, as follows:
Next, let us integrate the second equation, Equation (105), into the first equation, Equation (104), to obtain one relation:
And next, by applying the inverse Laplace transform in Equation (106), it is possible to obtain the convolutional integral and the time function of the normal stress from the axial dilatation of the structure for the rheological complex Burgers–Faraday model of an ideal fractional-type material with piezoelectric properties.
This rheological model, the second rheological complex Burgers–Faraday-F model of ideal viscoelastic fluid material, belongs to a group of rheological viscoelastic creepers.
The second new rheological complex Burgers–Faraday model of a fractional-type viscoelastic fluid material with piezoelectric properties has two internal degrees of freedom of mobility at the points of regular connection in its substructure.
5.8. Second New Rheological Complex Burgers–Faraday-F Model of an Elastoviscous Solid Material
Next, we will analyze the constitutive relations of the substructure of the second new complex Burgers–Faraday-F model of an elastoviscous solid material, when a Faraday element is connected in parallel to the substructure of a basic complex Maxwell model.
Along the entire substructure of the complex Maxwell model, the normal stress is the same, and the velocity of axial dilatation in individual regularly connected elements can be expressed as follows: in the Hooke element, it is
, i.e.,
, while in the Newton element of a fractional-type fluid, it is
. The total velocity of axial dilatation in the substructure of the Maxwell viscoelastic material can now be expressed in the following form:
If we apply the Laplace transform to the previous differential constitutive relations expressed in Equations (107) and (119) of the fractional-order exponent
, where
, we obtain the following:
We obtained the previous relationship shown in Equation (110) as the relationship between the Laplace transform of the axial dilatation in the Maxwell model and the Laplace transform of the normal stress in that model, which is a substructure of the complex Burgers–Faraday-F model.
The fractional-type axial dilatation velocity
in a parallel-connected Faraday element with Maxwell’s model is expressed in the following form:
Next, using the previous differential constitutive relationship demonstrated in Equation (111), of the fractional-order exponent
, where
, it can be determined that
, in a substructure consisting of a Faraday element connected in parallel with a Maxwell model, which together form the substructure of the complex Burgers–Faraday-F model, is equal to
If we apply the Laplace transform to the previous constitutive relation of the fractional order, we obtain the following:
We obtained the previous expression as the relationship between the Laplace transform of the normal stress in a Faraday element with piezoelectric properties, connected in parallel with the rheological basic complex Maxwell model of the fractional type, and the Laplace transform of the normal stress in the Maxwell model, which is a substructure of the complex Burgers–Faraday-F model.
The total normal stress
in the Maxwell–Faraday model, which contains a parallel-connected Faraday element, which is a substructure of the complex Burgers–Faraday-F model, can be expressed as follows:
If we apply the Laplace transform to the previous constitutive relationship expressed in Equation (114), using the fractional-order exponent
, where
, we obtain the following:
We obtained the previous expression as the relationship between the Laplace transform of in the basic complex Maxwell–Faraday elastoviscous solid model, which contains a parallel-connected Faraday element, which is a substructure of the complex Burgers–Faraday-F model, and the Laplace transform of in the basic Maxwell–Faraday model.
Next, we use the knowledge that the substructure of the complex Burgers–Faraday-F elastoviscous solid material consists of the order of two substructures—the structure of the basic complex Maxwell–Faraday elastoviscous solid material model and the substructure consisting of the solid material—and we write the following differential constitutive relations of fractional-order exponent
, where
:
Next, we establish the following system, including the differential constitutive relations of the fractional-order exponent
, where
, of the complex Burgers–Faraday-F elastoviscous solid material model in the following form:
These differential constitutive relations of fractional-order exponent , where , as shown in Equations (119)–(124), give the relationships of axial dilatation and normal stress for the complex Burgers–Faraday-F elastoviscous solid material, as well as the normal stresses and axial dilatations in the substructures of the basic complex Kelvin–Voigt elastoviscous solid material and the basic complex Maxwell–Faraday elastoviscous solid material.
If we apply the Laplace transform to the previous two differential constitutive relations of fractional-order exponent
, where
, as shown in Equations (129)–(124), we obtain the following relations:
These differential constitutive relations of the fractional-order exponent , where , give the connections between the Laplace transformations of the normal stress and the Laplace transformations of the axial dilatation of the complex Burgers–Faraday-F elastoviscous solid material, and the axial dilatations , namely , and the normal stresses , namely , in the substructures of the basic complex Kelvin–Voigt elastoviscous solid material and the basic complex Maxwell–Faraday elastoviscous solid material.
This new model, the second complex Burgers–Faraday-F model of an ideal elastoviscous solid material, belongs to a group of rheological elastoviscous oscillators.
The second new complex Burgers–Faraday-F model, has two internal degrees of freedom of mobility at the points of regular connections in its substructures.