Out-of-Plane Elastic Waves in 2D Models of Solids: A Case Study for a Nonlocal Discretization Scheme with Reduced Numerical Dispersion

Abstract

The paper addresses the problem of numerical dispersion in simulations of wave propagation in solids. This characteristic of numerical models results from both spatial discretization and temporal discretization applied to carry out transient analyses. A denser mesh of degrees of freedom could be a straightforward solution to mitigate numerical dispersion, since it provides more advantageous relation between the model length scale and considered wavelengths. However, this approach also leads to higher computational effort. An alternative approach is the application of nonlocal discretization schemes, which employ a relatively sparse spatial distribution of nodes. Numerical analysis carried out to study the propagation of elastic waves in isotropic solid materials is demonstrated. Fourier-based nonlocal discretization for continuum mechanics is introduced for a two-dimensional model undergoing out-of-plane wave propagation. The results show gradual increase of the effectiveness of this approach while expanding the region of nonlocal interactions in the numerical model. A challenging case of high ratio between the model length scale and wavelength is investigated to present capability of the proposed approach. The elaborated discretization method also provides the perspective of accurate representation of any arbitrarily shaped dispersion relation based on physical properties of modelled materials.

Nonlocal elasticity in shape memory alloys modeled using peridynamics for solving dynamic problems

“…Taking into account Eq. (14), which defines the total strain in SMA, as well as Eqs. (13), (15) and (16), one may obtain…”

“…The plot of the strain rate is bounded by the allowed limits for isothermal phase transition, which is ±0.038% s −1 Fig. 16 Hysteretic response of the SMA model for the applied sinusoidal force of the propagating waves, especially in case of limited number of DOFs in the spatial domain being discretized [14]. Finally, even though the planned elaboration of the peridynamic approach will allow to go beyond the limit of isothermal phase transitions, the preliminary results of the simulations carried out for GFB's component are valuable due to the rigorous requirement regarding the temperature gradient in GFB that must be kept within specific limits of the order of several • C only [69].…”

“…Nonlocal elasticity leads to lower requirements for model mesh density with its spatial domain discretization. On the one hand, it helps to mitigate numerical dispersion, which emerges for crude meshes [11][12][13][14][15]. This behavior is observed when the distances between nodes in a numerical model are comparable with the wavelengths.…”

“…This behavior is observed when the distances between nodes in a numerical model are comparable with the wavelengths. Moreover, nonlocal elasticity, via long-range interactions, introduces the means for relatively easy formulation of arbitrary shapes for physical dispersion relationships [14,16].…”

On Nonlocal Modeling in Continuum Mechanics

“…Second, the phenomenon of elastic wave propagation in deformable solids can be handled more accurately via the nonlocal approach without the need for a further increase of mesh density (Eringen 1972). Having introduced nonlocal interactions into a modeled structure, one can conveniently form the shape of the dispersion curves and surfaces (Martowicz et al 2015a). Effectively, the model is considered as a periodic structure being investigated within the first Brillouin zone (Martowicz et al 2014a).…”