Modeling of dynamic and spectral viscoelastic characteristics of composite materials

. When designing products made of composite materials intended for use in difficult conditions of inhomogeneous deformations and temperatures, it is important to take into account viscoelastic, including spectral and dynamic, properties of the binder and fillers. The article considers dynamic characteristics (complex modulus, complex malleability, their real and imaginary parts, loss angle tangent) and spectral characteristics of relaxation and creep and their dependence on each other. The above-mentioned characteristics were found for all known types of creep kernel and relaxation kernel. To find the spectral characteristics, one of the numerical methods of reversing the Laplace transformation was used - the method of quadrature formulas. Algorithms and computer programs have been compiled to implement this method


Introduction
When designing products made of composite materials intended for use in difficult conditions of inhomogeneous deformations and temperatures, it is important to take into account the viscoelastic properties of the binder and fillers [1].All viscoelastic characteristics, both static and dynamic, are expressed, ultimately, through relaxation and creep nuclei [2,17].
Relaxation and creep kernel reflect the specific properties of a particular material.There are many model variants of such cores proposed at different times scattered in the literature.All of them were systematized and brought together in the work [17], and a mutual connection between them was established there [17].An alternative and generalized approach is to determine the spectral characteristics of relaxation and creep -the density of the relaxation spectrum and the creep spectrum.It is necessary to link all the defining viscoelastic functions through spectral characteristics.It is also necessary to establish the relationship between the relaxation spectral density and the creep spectral density.
Thus, the density of relaxation spectra, for example, becomes the primary and main viscoelastic characteristic of the material, it is specific to each specific material and is subject to experimental determination in each specific case.In this work, a partial implementation of this program is proposed.
The function G t describes the change in stress over time with constant deformation.
This process is called stress relaxation.In this process, the tension decreases with time, that is, the function G t is decreasing.
The function J t describes the change in deformation over time at constant stress.This process is called creep deformation.In the process of creep at constant stress, the deformation increases, that is, the function J t is increasing.
In the theory of elasticity between an elastic module G and malleability J there is a simple connection [1][2][3][4] 1. GJ (2) However, in the theory of viscoelasticity, there is no such simple connection between the relaxation and creep functions.
Let's write down formulas (1) in the Laplace image space using the convolution theorem , .p pG p p p pJ p p (3) From formulas (3) follows 2 1. p G p J p (4) This expression defines the relationship between the images of relaxation and creep functions.
In the theory of the Laplace transform, the following limiting relations take place That is, relations of type (2) in the theory of viscoelasticity take place only in two limiting cases: when 0 t and when t

Dynamic characteristics of viscoelastic materials and the relationship between them
The most well-known dynamic characteristics are the complex module * G i and the complex malleability Where and g g are the images of the relaxation kernel and the creep kernel, respectively, in the Fourier space.
There is the following relationship between formulas (9) [14-16]: 1. G i J i (10) Complex modulus and complex malleability, as the name implies, have real and imaginary parts: J are real and imaginary parts of complex compliance, 1 .Pa By substituting formulas (11) into (10) Let's return to expressions (12) and write the second equation in the form Where the left side of the equality, by definition, is called the tangent of the loss angle Then it follows from ( 14) and ( 15)

Relaxation and creep spectra and their relation to relaxation and creep kernels
In [2][3], the relaxation function G t is defined as follows: Where H is the relaxation time distribution function (relaxation spectrum), 1 .

Pa s
Taking into account (6), the relaxation kernel t is expressed in terms of the relaxation spectrum H From this formula by substitution 1 we can get the expression Where 1 L is the inverse Laplace transform operator; 1 is the Dirac delta function, Then (20) takes a simpler form: In [2][3], the creep function J t is defined as Where j is the lag time distribution function, or creep spectrum,

Relation of relaxation and creep spectra to each otheR
According to (8), there is a mutual relationship between the relaxation kernel t and the creep kernel g t in the Laplace image space.Taking into account ( 18) and ( 23), it becomes obvious that the functions of the relaxation spectrum H and the creep spectrum j are also not independent.
Let's define formulas linking the spectra together.Let's start with the fact that from (18) we find the image of the relaxation kernel.Then, using (8), we find the image of the creep kernel.Applying the inverse Laplace transform operator

j
. By reverse substitution 1 we find the creep spectrum j , and then, taking into account ( 5) and ( 7), the final formula takes the form L and 2 L are operators of direct and inverse Laplace transformations, respectively, applied twice.Similarly, we can derive a formula expressing the relaxation spectrum through the creep spectrum:

Known types of relaxation and creep functionS
In [17], the known types of relaxation and creep kernels and their derivatives (Abel, Rabonov, Rzhanitsyn functions, etc.) were presented in the form of tables.Using these data, the dynamic and spectral characteristics of each of these kernels were further found.

Dynamic and Spectral Characteristics of Creep and Relaxation Kernels
Using the data from the tables from [17] and the formula ( 9)-( 16), ( 21) and (24) the dynamic and spectral characteristics of the known creep kernels and relaxation kernels were obtained.
The results are shown in Tables 1-6.

Maxwell functions
Table 1 shows the dynamic and spectral characteristics of Maxwell kernels.
Kohlrausch functions Table 2 shows the dynamic and spectral characteristics of Kohlrausch kernels.
Table 2. Dynamic and spectral characteristics of Kohlrausch kernels.In Table 2 takes values 0 . When 0 , Kohlrausch's characteristics turn into Maxwell's characteristics.Abel functions Table 3 shows the dynamic and spectral characteristics of Abel kernels.4 shows the dynamic and spectral characteristics of the Rabotnov kernels.
Table 4. Dynamic and spectral characteristics of Rabotnov kernels.

Rzhanitsyn functions
Table 5 shows the dynamic and spectral characteristics of the Rzhanitsyn kernels.

№
Relaxation Creep In tables 1-6, the functions of the kernels are indicated under number 1, the spectra of the kernels are indicated under number 2, the complex module and complex compliance are indicated under number 3, respectively, and their real and imaginary parts are indicated under numbers 4-5.Number 6 indicates the tangent of the loss angle and its limit values at 0 and at , respectively.13), it is possible to obtain the following functions already in an analytical form: the Abel accumulation modulus and the Abel loss modulus, the Rabotnov accumulation modulus and the Rabotnov loss modulus, the real and imaginary parts of the complex malleability of Rzhanitsyn.

Numerical method for solving the problem
Relaxation and creep spectra, according to ( 21) and (24), are found through relaxation and creep kernels by reversing the Laplace transform.We will solve this problem using one of the numerical methods of reversing the Laplace transform -the method of quadrature formulas with equal coefficients [20].This method was described in detail in the previous work [17].

Examples of numerical solution of the problem
Computer programs were written that implement the method of quadrature formulas for finding spectra.Here are some calculations obtained using these programs.
Figures 1-6 show graphs obtained by the method of quadrature formulas.To construct the functions of the spectra, the following parameters were set: The green graphs correspond to the analytical functions (or in the form of an approximate finite series) of the spectra, while the red graphs correspond to the functions of the spectra obtained by the numerical method mentioned above.

Conclusions
After analyzing the results, we draw the following conclusions: 1) the graphs of the relaxation and creep spectra were obtained fairly accurately (the maximum error of calculations does not exceed 5% on average), despite the fact that the error is very noticeable in the initial time sections.Numerical finding of the functions of the Kohlrausch and Gavriljak-Negami yield spectra caused difficulties associated with a complex recursive formula in these kernels; 2) for all kernels (except for the kernels of Kohlrausch and Gavrilyak-Negami), their dynamic characteristics were obtained in the form of analytical functions, even though the functions of some kernels are represented as an approximate infinite series; 3) formulas defining the spectra through each other have not proved to be very effective in practice (with the exception of Maxwell spectra).It is possible that for greater efficiency, the direct and inverse Laplace transform in these formulas should be replaced by an approximate one (the integrand function should be written as an approximate infinite series).
, we get an expression similar to(20) from which we can get the function 2 1 -Negami 's characteristics turn into Rzhanitsyn's characteristics, and when 1 they turn into Maxwell's characteristics.

Figures 1 ,
Figures 1, 3 and 5 show graphs of the relaxation spectra of Abel, Rabotnov and Rzhanitsyn, respectively.Figures 2, 4 and 4 show graphs of the creep spectrum of Abel, Rabotnov and Rzhanitsyn, respectively.The green graphs correspond to the analytical functions (or in the form of an approximate finite series) of the spectra, while the red graphs correspond to the functions of the spectra obtained by the numerical method mentioned above.

Table 1 .
Dynamic and spectral characteristics of Maxwell kernels.

Table 3 .
Dynamic and spectral characteristics of Abel kernels.