An effective method for calculating viscoelastic structural elements in mining using the Lanczos τ-method

Abstract

A deformed plate of variable thickness with an elliptical hole is considered. The evolution of the stress state and the applied load P are investigated for a material exhibiting creep properties, where the tension δ of the rods connecting two rigid attachments fixed along the hole contour varies with time. The mathematical model has been developed employing canonical polynomials and Chebyshev polynomials. The accuracy of the approximation is characterized by the residual function. Using the property of minimal deviation from zero of Chebyshev polynomials allows us to represent the residual function as a series, which ensures uniform convergence of the solution. A relaxation kernel in the form of Rabotnov’s fractional-exponential function is employed to capture the key behavioral features of the variable-thickness plate. This enables the solution of the corresponding viscoelastic problem based on the Volterra principle. The rheological parameters involved in the kernel show good agreement with experimental results. The solution of the problem obtained using the Lanczos τ-method approximates the desired solution better than the use of hypergeometric functions. Unlike standard methods, the Lanczos method minimizes the error over the entire interval.