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Dynamical boundary value problem for a viscoelastic half-space with cut

N. Shavlakadze, N. Odishelidze, B. Pachulia, F. Criado-Aldeanueva

TL;DR

This work addresses the dynamical boundary-value problem for a viscoelastic half-space with a strip cut, under harmonic shear, by reducing it to a singular integral equation of the first kind. The authors apply Fourier transform techniques and contour integration to obtain a one-dimensional integral equation for the jump in boundary displacement, then solve it with a Chebyshev polynomial expansion, yielding an infinite system that is shown to be quasi-completely regular in $l_2$. They prove existence and convergence of the reduced problem, deriving a rate of $O(N^{-3/2})$ for truncated solutions, and provide explicit expressions for the stress-intensity factors $K_I+iK_{II}$. Numerical experiments demonstrate rapid convergence and reveal how the material parameters $G$ and $G_0$ influence the intensity factor, validating the method and its practical utility for viscoelastic crack-like contact problems.

Abstract

The dynamical boundary value problem for viscoelastic half-space with cut in the form of a strip is considered. The problem is reduced to the singular integral equation of first kind. Using the method of orthogonal polynomials, the integral equation is reduced to an infinite system of linear algebraic equations. The quasi-completely regularity of the obtained system is proved and the reduction method for approximate solution is developed.

Dynamical boundary value problem for a viscoelastic half-space with cut

TL;DR

This work addresses the dynamical boundary-value problem for a viscoelastic half-space with a strip cut, under harmonic shear, by reducing it to a singular integral equation of the first kind. The authors apply Fourier transform techniques and contour integration to obtain a one-dimensional integral equation for the jump in boundary displacement, then solve it with a Chebyshev polynomial expansion, yielding an infinite system that is shown to be quasi-completely regular in . They prove existence and convergence of the reduced problem, deriving a rate of for truncated solutions, and provide explicit expressions for the stress-intensity factors . Numerical experiments demonstrate rapid convergence and reveal how the material parameters and influence the intensity factor, validating the method and its practical utility for viscoelastic crack-like contact problems.

Abstract

The dynamical boundary value problem for viscoelastic half-space with cut in the form of a strip is considered. The problem is reduced to the singular integral equation of first kind. Using the method of orthogonal polynomials, the integral equation is reduced to an infinite system of linear algebraic equations. The quasi-completely regularity of the obtained system is proved and the reduction method for approximate solution is developed.
Paper Structure (5 sections, 1 theorem, 42 equations, 1 figure)

This paper contains 5 sections, 1 theorem, 42 equations, 1 figure.

Table of Contents

  1. Statement of the problem
  2. Reduction of the problem to the integral equation
  3. Reduction of integral equation (\ref{['eq:1.9']}) to the infinite system of linear algebraic equations
  4. Discussion and Numerical Results
  5. Conclusion

Key Result

Theorem 3.1

The infinite system of linear algebraic equations (eq:1.11) is quasi-completely regular in the space $l_2$. Consequently, the integral equation (eq:1.9) has the unique solution in the form (eq:1.10). The intensity factor of stress $\tau_{yz}(0,y,t)$ at the end of cut is calculated using the formula

Figures (1)

  • Figure 1: Sketch of the posed problem

Theorems & Definitions (1)

  • Theorem 3.1