Methods for Numerical Prediction of Relaxation and Reduction Properties of Polymer Textile Materials
- PDF / 214,973 Bytes
- 6 Pages / 594 x 792 pts Page_size
- 78 Downloads / 197 Views
Fibre Chemistry, Vol. 52, No. 3, September, 2020 (Russian Original No. 3, May-June, 2020)
METHODS FOR NUMERICAL PREDICTION OF RELAXATION AND REDUCTION PROPERTIES OF POLYMER TEXTILE MATERIALS N. V. Pereborova, A. G. Makarov, I. M. Egorov, and V. I. Vagner
UDC 539.434:677.494
Methods of numerical prediction of relaxation and reduction properties of polymer textile materials are considered. Numerical prediction is performed based on integration of the governing Boltsmann– Volterra equations applied to relaxation-reduction processes of the materials under investigation.
Mathematical simulation of functional–consumption relaxation and reduction processes of materials in the textile industry and in light industry based on which their relaxation and reduction properties may be predicted dynamically should be performed on the basis of the governing Boltsmann–Volterra equation [1–3] with different integral kernels: t
σ t = E 0 ε t − (E 0 − E ∞ ) ⋅ ε θ ⋅ ϕ'ε,t −θ dθ,
∫
(1)
0
where t is time; σt – voltage; εt – deformation; E∞ – modulus of viscoelasticity; E0 – modulus of elasticity; and ϕ’ε, t – relaxation kernel. One of the following integral probability distribution functions [4–7] should be selected as the integral kernel: – probability integral (which characterizes a normal distribution):
1
ϕt =
2π
⋅
an ⋅ln( t / τ) − z2 / 2
∫e
dz ,
(2)
−∞
– hyperbolic tangent: −1
ϕt =
−A 1⎡ ⎛ A t ⎞⎤ ⎡ ⎛ t ⎞ ⎤ + = + ⎢ ⎢1 th ⎜ ln ⎟⎥ 1 ⎜ ⎟ ⎥ . τ ⎠⎦ ⎢⎣ ⎝ τ ⎠ ⎥⎦ 2⎣ ⎝2
(3)
– Kohlrausch function, which does not possess a centrally symmetric graph, unlike the probability integral and the hyperbolic tangent, but possesses a relatively simple form:
ϕt = 1 − e −(t / τ) , k
(4)
– normalized arctangent logarithm, which characterizes the Cauchy probability distribution, ϕt =
1 2
+
⎛ 1⎞ arctg ⎜ ⎟ . ⎜b ⎟ π ⎝ n⎠ 1
(5)
Here an, A, k, and bn are the parameters of the intensity of the relaxation process, which characterizes the rate of the process, and t/τ is relative time. The existence of several mathematical models in which different integrand functions are used is justified and St. Petersburg State University of Industrial Technologies and Design; e-mail: [email protected]. Translated from Khimicheskie Volokna, No. 3, pp. 21–24, May–June, 2020.
154
0015-0541/20/5203-0154© 2020 Springer Science+Business Media LLC
makes it possible to obtain the results of relaxation-reduction predictions that are independent of each other. The predictable relaxation-reduction characteristics obtained by averaging the characteristics determined with the use of different mathematical models possess a higher degree of reliability than are characteristics which are determined with the use of a single mathematical model [8–10]. In order to predict a relaxation-reduction process applied to materials used in the textile industry and in light industry we will use the governing equation (1) with integrand kernel ϕ’ε, t in the form of the derivative of the normalized arctangent logarithm function (5) which corresponds to the Cauchy probability distr
Data Loading...
