Full Three-dimensional Elasticity Research Articles

Performance review of locking alleviation methods for continuum ANCF beam elements

The absolute nodal coordinate formulation (ANCF) is a nonlinear finite element approach proposed for the large deformation dynamics analysis of beam- and plate/shell-type structures. In the ANCF approach, elastic forces can be defined using three-dimensional elasticity-based continuum mechanics. This approach is often straightforward, and it makes it possible to use advanced material models in the ANCF framework. However, it has been pointed out in several studies that continuum ANCF-based elements with a full three-dimensional elasticity description can suffer from locking phenomena. In this study, a comparison between various combinations of locking alleviation techniques and their applicability to different ANCF beam variants is studied using numerical examples. Furthermore, the enhanced deformation gradient (EDG) technique, which has been proposed recently in finite element literature, is demonstrated for high-order ANCF beam elements. Based on the numerical tests, none of the currently available techniques are suitable for all types of ANCF elements. The paper also shows that the efficiency and accuracy of the techniques are case-dependent. For the ANCF beam element involving higher-order terms with respect to trapezoidal mode, however, the EDG-based techniques are preferable to reduce locking phenomena.

Simulations of microscopic propulsion of soft elastic bodies.

Using simulations that realistically model both hydrodynamic and elastic behavior, we study the motion of a microscopic, driven elastic sphere immersed in water. We first confirm the "jittery" relaxation recently predicted theoretically for an externally driven elastic sphere. The sphere is then divided in two and each section is driven internally with the two sections 180° out of phase. With periodic and perfectly symmetric driving, the elastic sphere spontaneously breaks symmetry and can attain macroscopic average swimming velocities to the right or left, the direction depending only on the initial state. With asymmetric driving the elastic sphere swims in one direction and the maximum speed is obtained with a 1/3:2/3 split. At high drive frequencies close to elastic resonances of the sphere, the motion can be quite efficient. At low drive frequencies the propulsion speed becomes independent of the elastic constants of the sphere and less efficient, but still substantial. Inertia is found to be an important driver of the behavior despite the small size of the spheres. As we model the full three-dimensional elasticity and compressible hydrodynamics, our simulations give not just qualitative indications but quantitative predictions for the motion.

Higher-order beam elements based on the absolute nodal coordinate formulation for three-dimensional elasticity

This study thoroughly examines various higher-order three and four-node beam elements for use in the absolute nodal coordinate formulation (ANCF). The paper carefully investigates which potential benefits and drawbacks the utilization of higher-order ANCF beam elements without in-slope vectors has in the case of the usage of full three-dimensional elasticity. When the elastic forces for shear-deformable ANCF beam elements are calculated using full three-dimensional elasticity—especially in the form of the St. Venant–Kirchhoff material law—Poisson locking severely deteriorates the accuracy of the numeric results. As shown in this paper, an existing approach to preventing this locking phenomenon for three-node beam elements can still produce unsatisfying results in load cases involving bidirectional bending. The results of this study show that enriching the polynomial basis used to approximate the beam kinematics provides a natural solution to this issue. As will be seen, these findings for three-node elements can also be extended to four-node elements. When using a sufficient approximation order in transverse directions, satisfying accuracy can be achieved both in conventional one-dimensional bending and in the above-mentioned bidirectional load case.

Gradient index lenses for flexural waves based on thickness variations

This work presents a method for the realization of gradient index devices for flexural waves in thin plates. Unlike recent approaches based on phononic crystals, the present approach is based on the thickness-dependence of the dispersion relation of flexural waves, which is used to create gradient index devices by means of local variations of the plate's thickness. Numerical simulations of known circularly symmetrical gradient index lenses have been performed. These simulations have been done using the multilayer multiple scattering method and the results prove their broadband efficiency and omnidirectional properties. Finally, finite element simulations employing the full three-dimensional elasticity equations also support the validity of the designed approach.

Thin-film evolution equation for a strained solid film on a deformable substrate: Numerical steady states

We consider the nonlinear behavior of the thin-film evolution equation for a strained solid film on a substrate. The evolution equation describes morphological changes to the film by surface diffusion in response to elastic energy, surface energy, and wetting energy. Due to the thin-film approximation, the elastic response of the film is determined analytically, resulting in a self-contained evolution equation which does not require separate numerical solution of the full three-dimensional elasticity problem. Using a pseudospectral predictor-corrector method we numerically determine the family of steady state solutions to this evolution equation which correspond to quantum dot and quantum ridge morphologies.

Surface scattering analysis of phonon transport in the quantum limit using an elastic model

We have investigated the effect on phonon energy transport in mesoscopic systems and the reduction in the thermal conductance in the quantum limit due to phonon scattering by surface roughness, using full three-dimensional elasticity theory for an elastic beam with a rectangular cross section. At low frequencies, we find power laws for the scattering coefficients that are strongly mode dependent, and different from the ${\ensuremath{\omega}}^{2}$ dependence, deriving from Rayleigh scattering of scalar waves, that is often assumed. The scattering gives contributions to the reduction in thermal conductance with the same power laws. At higher frequencies, the scattering coefficients becomes large at the onset frequency of each mode due to the flat dispersion here. We use our results to attempt a quantitative understanding of the suppression of the thermal conductance from the universal value observed in the experiment.

Design analysis of composite laminate structures for light-weight armored vehicle by homogenization method

This project is in support of the US Army TACOM Composite Armored Vehicle (CAV) program and in collaboration with the University of Michigan to verify and build confidence on the homogenization analysis method of composite laminates for armored vehicles. The standard homogenization method is modified by applying the weighted material constants of the laminae of a laminate in order to reflect the nature of bending, when it is applied to form an equivalent homogenized laminate model with a single lamina. The weighting procedure is applied to the material constants through the thickness direction in which the periodic condition for the standard homogenization method is not assumed. This procedure makes appropriate change in the global stiffness of the two-dimensional homogenized single layer laminate in bending, while the unit cell model for the homogenization method is a three-dimensional. In this work, we shall describe a formulation of the weighted homogenization method together with several numerical examples to examine the validity of the present method with respect to the classical lamination theory, the first-order shear deformation theory, and the full three-dimensional elasticity.

Effective dimensional reduction algorithm for eigenvalue problems for thin elastic structures: a paradigm in three dimensions.

We present a methodology for the efficient numerical solution of eigenvalue problems of full three-dimensional elasticity for thin elastic structures, such as shells, plates and rods of arbitrary geometry, discretized by the finite element method. Such problems are solved by iterative methods, which, however, are known to suffer from slow convergence or even convergence failure, when the thickness is small. In this paper we show an effective way of resolving this difficulty by invoking a special preconditioning technique associated with the effective dimensional reduction algorithm (EDRA). As an example, we present an algorithm for computing the minimal eigenvalue of a thin elastic plate and we show both theoretically and numerically that it is robust with respect to both the thickness and discretization parameters, i.e. the convergence does not deteriorate with diminishing thickness or mesh refinement. This robustness is sine qua non for the efficient computation of large-scale eigenvalue problems for thin elastic structures.

Exact three-dimensional elasticity solutions for bending of moderately thick inhomogeneous and laminated strips under normal pressure

An exact three-dimensional solution is presented for the deformation and stress distribution in a semi-infinite strip clamped along its two edges, and subjected to uniform normal loading of the lateral surfaces. The strip is of constant moderate thickness and composed of anisotropic elastic material which is arbitrarily inhomogeneous in the through-thickness direction. The only material symmetry assumed is that of reflectional elastic symmetry in planes parallel to the mid-plane. The important special case of an anisotropic laminated plate is given by assuming piecewise-constant properties through the thickness.The general method of solution is to reformulate the full three-dimensional elasticity equations in a way that reduces the problem to solving a system of partial differential equations in the two in-plane independent variables only, and then obtaining asymptotic solutions in terms of an aspect ratio of the thickness divided by a typical in-plane length. In general, successive terms are expressed in terms of the approximate “classical laminate” solution. In the present problem, this solution is very simple and the expansion terminates after no more than three terms. This gives a closed-form analytical solution that is valid for any aspect ratio.

An exact three-dimensional solution for normal loading of inhomogeneous and laminated anisotropic elastic plates of moderate thickness

An exact three-dimensional solution is presented for the deformation and stress distribution in an elliptical plate under uniform normal loading of the lateral surfaces, and clamped along its edge. The plate is assumed to be of constant, moderate thickness and composed of anisotropic elastic material which is inhomogeneous in the through-thickness direction but symmetric about the mid-plane. The only material symmetry assumed is that of reflectional symmetry in planes parallel to the mid-plane. A transfer matrix method is used which, without making any further assumptions, gives the exact solution at each point in the plate in terms of the stress and displacement at the mid-plane. The two-dimensional differential equations governing these mid-plane values are found to be the same as those for an equivalent homogeneous plate whose constant elastic moduli are determined by appropriate through-thickness weighted averages of the inhomogeneous moduli. The solution of the two-dimensional problem is known for such a plate when subject to the specified surface and edge conditions, and yields a closed form analytical solution that satisfies all the governing equations and surface conditions of the full three-dimensional elasticity problem, with edge displacement conditions satisfied on the mid-plane. The important special case of an anisotropic laminated plate is given by assuming piecewise constant properties through the thickness.

Interactions of a spherical wave with a bilaminar plate composed of homogeneous and isotropic solid layers

The interaction of a spherical wave with a bilaminar plate of infinite lateral extent is considered theoretically. Each of the plate layers is assumed to be composed of a homogeneous and isotropic solid. Losses in the plate materials are modeled by using complex material properties. Both the source and the plate are immersed in an infinite, inviscid fluid. The solution is obtained by imposing appropriate boundary conditions on the full three-dimensional elasticity equations. Several cases involving a rubber-coated metal plate and a rubber-coated plastic plate are investigated numerically for a 1-kHz incident wave and for a 2-kHz incident wave. For the numerical study, the inviscid fluid is taken to be water. For vanishingly thin rubber coatings, the bilaminar plate results are found to approach the single-layer plate results [J. C. Piquette, ‘‘Spherical-wave scattering by a finite-thickness solid plate of infinite lateral extent, with some implications for panel measurements,’’ J. Acoust. Soc. Am. 83, 1284–1294 (1988)]. In the case of a single-layer plate, it was found that ‘‘overpressures’’ could be created. That is, the amplitudes of the reflected and/or transmitted waves on the acoustic axis, evaluated on either surface of the plate, can exceed that of the incident wave. In the present work, it is found that for the case of rubber-coated plastic there is a transition thickness of rubber through which the reflected and transmitted waves on the acoustic axis are enhanced. That is, the amplitudes of the reflected and the transmitted waves can be greater than those obtained from an uncoated plate. However, if the rubber thickness is increased still further, it is found that the amplitudes of the reflected and transmitted waves rapidly collapse, the overpressures associated with the uncoated plate disappear, and the coated-plate results evaluated on the acoustic axis approach normal-incidence plane-wave results. This wave-field collapse does not occur in the case of a rubber-coated metal. An important conclusion of this research is that the spherical-wave effects that may be introduced into panel measurements in which a metallic support plate has been used may be reduced, or even eliminated, by replacing the metallic support plate with a plastic support plate.

Spherical-wave scattering by a finite-thickness solid plate of infinite lateral extent, with some implications for panel measurements

The solution to the problem of the interaction of a spherical wave with a homogeneous and isotropic solid plate of infinite lateral extent, but finite thickness, is considered theoretically. Both the source and the plate are immersed in an infinite, inviscid fluid. Appropriate boundary conditions are imposed on the full three-dimensional elasticity equations. The solution is evaluated numerically for a variety of materials for a 1-kHz incident spherical wave and for a 5-kHz incident spherical wave. For the numerical study, the fluid medium is taken to be water. Under certain conditions, ‘‘overpressures’’ are predicted for both the reflected and transmitted fields (i.e., the amplitude of the reflected pressure and/or the transmitted pressure can exceed the maximum value of the amplitude of the incident pressure on the plate surface). These overpressures are consistent with the law of conservation of energy in the sense that, for a plate composed of lossless material, the total incident power is found to be equal to the sum of the total reflected power plus the total transmitted power. An important conclusion of this research is that the practice of attempting to reduce the influence of edge diffraction in panel tests by using samples of increasingly larger lateral extent may result in measurements that are substantially corrupted by wave-front curvature effects, particularly if the sample panel includes a steel backing plate.