Frantisek Vales

Dispersion of elastic waves in the contact-impact problem of a long cylinder

Numerical dispersion of two-dimensional finite elements was studied. The outcome of the dispersion study was verified by the numerical and analytical solutions to the longitudinal impact of two long cylindrical bars. In accordance with the results of the dispersion analysis it was demonstrated that the quadratic elements showed better accuracy than the linear ones.

Analytical solution of transient in-plane vibration of a thin viscoelastic disc and its multi-precision evaluation

The analytical solution of transient in-plane vibration of a thin viscoelastic disc caused by a radial pressure load is presented. Using the multi-precision implementation of FFT based algorithm, the inverse Laplace transform is carried out and the spatio-temporal distributions of displacement components are obtained. The efficiency and the accuracy of the process of analytical solution evaluation are discussed in detail and the consequences in context of the application of results are mentioned. Finally, the results of numerical simulation are presented. They are used for the verification of the correctness of the derived analytical formulae and their evaluation and for the determination of numerical model capabilities.

Numerical Laplace inversion in problems of elastodynamics: Comparison of four algorithms

Abstract The objective of this work is to find a suitable algorithm for numerical Laplace inversion which could be used for effective and precise solution of elastodynamic problems. For this purpose, the capabilities of four algorithms are studied using three transforms resulted from analytical solutions of longitudinal waves in a thin rod, flexural waves in a thin beam and plane waves in a strip. In particular, the Gaver–Stehfest algorithm, the Gaver–Wynn’s rho algorithm, the Fixed-Talbot algorithm and the FFT algorithm combined with Wynn’s epsilon accelerator are tested. The codes written in Maplexa016 employing multi-precision computations are presented for each method. Given the results obtained, the last mentioned algorithm proves to be the best. It is most efficient and it gives results of reasonable accuracy nearly for all tested times ranging from 3 × 10 − 7 s to 3u202f×u202f10 3 u2009s.

PROPAGATION OF NON-STATIONARY WAVES IN VISCOELASTIC STRIP COMPOSED OF ORTHOTROPIC LAYERS

The analytical solution for transient waves caused by transverse impact on an infinite layered strip with free-free boundaries is presented in this work. The strip is composed of two horizontal layers of different heights. The materials of both layers are assumed to be linear viscoelastic and orthotropic. The case of special orthotropy is assumed for simplicity. The dissipative behaviour of each layer is modelled by the discrete model of standard linear viscoelastic solid in Zener configuration. The solving procedure used in this work follows the methods applied in previously published works dealing with the problems of a viscoelastic orthotropic strip and the symmetric case of a layered strip. The system of four linear partial integro-differential equations describing the non-stationary state of plane stress in the strip is solved by means of integral transform method. Concretely, the Laplace transform in time domain and the Fourier transform in spatial domain are applied. As a results of this procedure, the final formulas for displacement components in both layers are derived in Laplace domain. These transforms contain eight spectra of Fourier integrals which can be found as the solution of the system of eight complex equations arising from boundary conditions of the problem. When the spectra are known, the resulting formulas for the Laplace transforms of displacement components are obtained. The presented new formulation of non-symmetric problem enables to study the wave phenomena in general two-layered solids of strip-like geometry and it is fundamental for solving a problem with arbitrary number of layers. 1322 Available online at www.eccomasproceedia.org Eccomas Proceedia COMPDYN (2017) 1322-1329

PROBLEM OF NON-STATIONARY WAVES IN VISCOELASTIC ORTHOTROPIC STRIP SOLVED USING ANALYTICAL METHOD

The transient response of an infinite orthotropic strip subjected to a transverse load is investigated using an analytical method in this work. The material properties of the strip are described by the discrete model of standard linear viscoelastic solid. In this study, the case of special orthotropy is assumed, i.e., the principal material and geometric axes of the strip are coincident. Once the final system of equations describing the plane-stress problem solved is derived, the integral transform method is used to obtain the Laplace transforms of displacement and velocity components. The inversion of the resulted formulae back to time domain is carried out by the help of numerical inverse Laplace transform. In particular, an algorithm based on the FFT and Wynn’s epsilon accelerator was used for this purpose. In the last part of this work, the analytical results obtained for selected orthotropic material are compared to those resulted from numerical simulation performed in the finite element code MSC.Marc. The comparison made showed good agreement between analytical and numerical results and proved their correctness. Finally, the efficiency of both approaches is discussed.

Analytical Solution of In-Plane Response of a Thin Viscoelastic Disc Under Impact Load

This paper concerns the analytical solution of the in-plane response of a thin viscoelastic disc to a dynamic load applied to its rim. The exact analytical relations for the Laplace transforms of radial and circumferential displacements are derived in terms of Bessel functions for the case of radial and torsional loads defined by even and odd functions of angular variable, respectively. The numerical evaluation of the analytical solution is then made for the case of an impulse radial load and transient wave phenomena are studied in the disc. With respect to the complexity of presented formulae, the multi-precision implementation of FFT based numerical algorithm for the inverse Laplace transform is used. The obtained analytical results are then compared to the results of numerical simulation performed in the finite element system MSC.Marc. The presented analytical solution can be used as a benchmark solution for the testing of numerical methods.

Symmetry preserving algorithm for a dynamic contact-impact problem

In the finite element method, the contact constraints can be introduced either before or after the fi- nite element discretization has been performed, leading to the so-called pre-discretization or postdiscretization techniques [1]. In the paper [2] we focused on the pre-discretization approach, showing this technique to lead naturally to the use of surface integration points as contactors. It was shown that the proposed method preserved the symmetry of the algorithmic approximation with respect to contact boundaries. On the outcome there was nothing like a master or slave definition of contact surface.