Julian J. Rimoli

Effect of large deformation pre-loads on the wave properties of hexagonal lattices

We study linear wave propagation in nonlinear hexagonal lattices capable of undergoing large deformations, under different levels of pre-load. The lattices are composed of a set of masses connected by linear axial and angular springs, with the nonlinearity arising solely from geometric effects. By applying different levels of pre-load, the small amplitude linear wave propagation response can be varied from isotropic to highly directional. Analytical expressions for the stiffness of a unit cell in the deformed configuration are derived and they are used to analyze the dispersion surfaces and group velocity variation with pre-load. Numerical simulations on finite lattices demonstrate the validity of our unit cell predictions and illustrate the wave steering potential of our lattice.

Meshing strategies for the alleviation of mesh-induced effects in cohesive element models

One of the main approaches for modeling fracture and crack propagation in solid materials is adaptive insertion of cohesive elements, in which line-like (2D) or surface-like (3D) elements are inserted into the finite element mesh to model the nucleation and propagation of failure surfaces. In this approach, however, cracks are forced to propagate along element boundaries, following paths that in general require more energy per unit crack extension (greater driving forces) than those followed in the original continuum. This, in turn, leads to erroneous solutions. We illustrate how the introduction of a discretization produces mesh-induced anisotropy and mesh-induced toughness for problems involving brittle fracture. Subsequently, we quantify those effects through polar plots of the path deviation ratio for commonly adopted meshes. Finally, we propose to reduce those effects through a new type of mesh, which we term conjugate-directions mesh.

On the mesh dependency of cohesive zone models for crack propagation analysis

One of the main approaches for modeling fracture and crack propagation in solid mate- rials is adaptive insertion of cohesive elements, in which line-like (2D) or surface-like (3D) elements are inserted into thenite element mesh to model the nucleation and propagation of failure surfaces. In this approach, however, cracks are forced to propagate along element boundaries, following paths that in general require more energy per unit crack extension (greater driving forces) than those followed in the original continuum, which in turn leads to erroneous solutions. More specically, we illustrate how the introduction of a discretiza- tion produces two undesired effects: (i) mesh-induced anisotropy and (ii) mesh-induced toughness. Subsequently, we analyze those effects through polar plots of the path devia- tion ratio for 4k and K-means meshes. Finally, we propose a new type of mesh, termed conjugate-directions mesh, which signicantly reduces the undesired effects for meshes of practical size.

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

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.

A nonlocal finite difference scheme for simulation of wave propagation in 2D models with reduced numerical dispersion

The work deals with the reduction of numerical dispersion in simulations of wave propagation in solids. The phenomenon of numerical dispersion naturally results from time and spatial discretization present in a numerical model of mechanical continuum. Although discretization itself makes possible to model wave propagation in structures with complicated geometries and made of different materials, it inevitably causes simulation errors when improper time and length scales are chosen for the simulations domains. Therefore, by definition, any characteristic parameter for spatial and time resolution must create limitations on maximal wavenumber and frequency for a numerical model. It should be however noted that expected increase of the model quality and its functionality in terms of affordable wavenumbers, frequencies and speeds should not be achieved merely by denser mesh and reduced time integration step. The computational cost would be simply unacceptable. The authors present a nonlocal finite difference scheme with the coefficients calculated applying a Fourier series, which allows for considerable reduction of numerical dispersion. There are presented the results of analyses for 2D models, with isotropic and anisotropic materials, fulfilling the planar stress state. Reduced numerical dispersion is shown in the dispersion surfaces for longitudinal and shear waves propagating for different directions with respect to the mesh orientation and without dramatic increase of required number of nonlocal interactions. A case with the propagation of longitudinal wave in composite material is studied with given referential solution of the initial value problem for verification of the time-domain outcomes. The work gives a perspective of modeling of any type of real material dispersion according to measurements and with assumed accuracy.

Multiscale analysis of wave-damage interaction in two and three dimensional isotropic plates

In this paper a geometric multiscale finite element method (GMsFEM), recently developed by the authors, is applied to the analysis of wave propagation in damaged plates. The proposed methodology is based on the formulation of both two- and three-dimensional multi-node (or multiscale) elements capable of describing small defects without resorting to excessive mesh refinements. Each multiscale element is equipped with a local mesh that is used to compute the interpolation functions of the element itself and to resolve the local fluctuations of the solution near the defect. The computed shape functions guarantee the continuity of the solution between multiscale and conventional elements. This allows using an undistorted discretization in the uniform portion of the domain while limiting the use of multiscale elements only in the vicinity of the defects. In this article the method is applied to evaluate the reflection coefficients due to cracks of different size and orientation in an otherwise homogeneous plate. Also, numerical simulations of wave-damage interaction are used to compute the scattering coefficients associated to three-dimensional defects in isotropic plates.

Optical evaluation of the wave filtering properties of graded undulated lattices

We investigate and experimentally demonstrate the elastic wave filtering properties of graded undulated lattices. Square reticulates composed of curved beams are characterized by graded mechanical properties which result from the spatial modulation of the curvature parameter. Among such properties, the progressive formation of frequency bandgaps leads to strong wave attenuation over a broad frequency range. The experimental investigation of wave transmission and the detection of full wavefields effectively illustrate this behavior. Transmission measurements are conducted using a scanning laser Doppler vibrometer, while a dedicated digital image correlation procedure is implemented to capture in-plane wave motion at selected frequencies. The presented results illustrate the broadband attenuation characteristics resulting from spatial grading of the lattice curvature, whose in-depth investigation is enabled by the presented experimental procedures.

Modal-Based Finite Elements for Efficient Wave Propagation Analysis

This paper presents an extension to the Geometric Multi-Scale Finite Element Method (GMsFEM) developed by Casadei et al. to predict the dynamic response of heterogeneous materials and structures. The proposed approach introduces elements enriched by the natural modes over their own domain. When heterogeneities are present, the auxiliary fine-scale mesh from GMsFEM is used to calculate the modes numerically. The enrichment scheme is also chosen in such a way that it automatically satisfies continuity across boundaries. The computational efficiency of the method is compared to that of traditional finite element formulations through selected benchmark problems.Copyright

3D Model for Atomic Sputtering of Heterogeneous Ceramic Compounds

The erosion of the channel wall in Hall effect thru sters (HETs) limits the maximum HET operating life time. HET channel wall materials are often binary composites of boron nitride and silica. The heterogeneity of the material drive s the development of complex surface features and roughness during the erosion process. To aid the understanding of the erosion processes, a three dimensional model of the atomic sputtering of a heterogeneous material is developed. The model investigates, through a ray-tracing technique and empirical models for the erosion rate of each phase, the interaction between ion beams and the underlying grain-level microstructure. Simulated erosion histo ries and surface profiles are compared with empirical data collected from the eroded channel wall of the AFRL/UM P5 HET. Simulated surface features and expected surface roughnesses for moderate ion incidence angles of 30˚ and 45˚ resemble those found through scanning electron microscopy and optical profilometry of the P5 channel wall. Predic ted rms roughnesses for 30˚ incidence are on the order of 7 µm, and rms roughnesses measured on the channel wall are 6±2.5 µm. The composition of the channel wall surface is investig ated via X-ray photoelectron spectroscopy and is comparable to prior work, but the reduction in the presence of BN with erosion is not adequately captured by this model.