Numerical methods of Laplace transform inversion in the problem of determination of viscoelastic characteristics of composite materials

When designing products made of composite materials intended for operation in difficult conditions of inhomogeneous deformations and temperatures, it is important to consider the viscoelastic properties of the binder and fillers. The article analyzes the relationship between relaxation and creep characteristics. All known creep and relaxation nuclei in the literature are considered. The problem of transformation of creep characteristics into relaxation characteristics and Vice versa is discussed in detail. There is a simple relationship between the creep and relaxation functions in the Laplace image space. However, when returning to the space of the originals, in many cases there are great difficulties in reversing the Laplace transform. Two numerical methods for inverting the Laplace transform are considered: the use of the Fourier series in sine and the method of quadrature formulas. Algorithms and computer programs for realization of these methods are made. It is shown that the operating time of a computer program implementing the Fourier method by sine is almost 2 times less than the operating time of a computer program implementing the quadrature formula method. However, the first method is inferior to the latter method in accuracy of calculations: the relaxation functions and relaxation rates, it is advisable to find the first method, since the computational error is almost indiscernible, and the functions of creep and creep speed, the second way, because for most functions, the result obtained by the second method is much more accurate than the result obtained by the first method.


Introduction
For isotropic viscoelastic materials the defining relations between stress and strain tensors are written as follows where ( ) G t and ( ) J t are relaxation and creep functions respectively [1][2][3][4][5][6][7][8][9][10][11]. Function ( ) G t describes the change in stress over time under constant strain. This process is called stress relaxation [12]. Experiments shows that in this process the stress decreases with time, i.e. the function ( ) G t is decreasing and therefore ( ) 0 dG t dt < .
(2) J t describes the change in strain over time at constant stress. This process is called creep deformation [12]. Experiments shows that in the process of creep at constant stress, the deformation increases, i.e. the function ( ) J t is increasing and therefore ( ) 0 dJ t dt > . (3)
(4) In the theory of viscoelasticity, there is no such simple connection between relaxation and creep functions. We write the formulas (1) in the image space by Laplace, using the convolution theorem From these formulas follows In the Laplace transform theory, the following limit relations take place [2][3] This means that relations of type (4) in the theory of viscoelasticity take place only in two limiting cases: by 0 t → + and t → ∞ .

Analysis of relaxation and creep kernels
Conveniently, the relaxation function ( ) G t and creep function ( ) J t to represent in a dimensionless form. For this we denote where and fucntions t ψ ( ) and g t ( ) are dimensionless, function t ψ ( ) called the relaxation kernel and the function g t ( ) called the creep kernel. By virtue of (4) and (7)   In the Laplace image space between images of dimensionless functions t ψ ( ) and g t ( ) there is a relation of type (6), i.e.

Maxwell kernels
Maxwell kernels are the simplest of all known kernels in the literature. Table 1 shows Maxwell functions and their images.  Table 2 shows Abel functions and their images.    where 0 1 β < ≤ . Note that when 1 β = than Abel kernels transformed into Maxwell kernels. Table 3 shows Rabotnov functions and their images.
Note that when 1 β = than Rzhanitsyn kernels transformed into Maxwell kernels. Table 5 shows Kohlrausch functions and their images.

Kohlrausch kernels
Note that when 0 β = than Kohlrausch kernels transformed into Maxwell kernels. Table 6 shows Gavrillac-Negami functions and their images.
The absence of dimension in all the above functions is due to the parameter τ , i.e. the relaxation time. The relaxation time is the period of time during which the amplitude value of the disturbance in the unbalanced physical system decreases by a factor of e times ( e − the base of the natural logarithm) [11][12][13][14][15][16][17][18][19].

Numerical methods for inverting the Laplace transform
In most cases, finding the original function as an analytical function is impossible or, from a practical point of view, impractical. That is why approximate and numerical methods for inverting the Laplace transform have been developed. The next two methods will be considered [20].

Using Fourier series on sine (1st method)
The method is taken from [20], pp.52-54. Using it, the original function can be written as follows:

Method of quadrature formulas with equal coefficients (2nd method)
The method is taken from [20], pp.121-124. According to this method, the original function will take the following form: