Anti-symmetric motion of a pre-stressed incompressible elastic layer near shear resonance

A two-dimensional model is derived for anti-symmetric motion in the vicinity of the shear resonance frequencies in a pre-stressed incompressible elastic plate. The method of asymptotic integration is used and a second-order solution, for infinitesimal displacement components and incremental pressure, is obtained in terms of the long-wave amplitude. The leading-order hyperbolic governing equation for the long-wave amplitude is observed to be not wave-like for certain pre-stressed states, with time and one of the in-plane spatial variables swapping roles. This phenomenon is shown to be intimately related to the possible existence of negative group velocity at low wave number, i.e. in the vicinity of shear resonance frequencies

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near cut-of frequencies

A two-dimensional model for extensional motion of a pre-stressed incompressible elastic layer near its cut-off frequencies is derived. Leading-order solutions for displacement and pressure are obtained in terms of the long wave amplitude by direct asymptotic integration. A governing equation, together with corrections for displacement and pressure, is derived from the second-order problem. A novel feature of this (two-dimensional) hyperbolic governing equation is that, for certain pre-stressed states, time and one of the two (in-plane) spatial variables can change roles. Although whenever this phenomenon occurs the equation still remains hyperbolic, it is clearly not wave-like. The second-order solution is completed by deriving a refined governing equation from the third-order problem. Asymptotic consistency, in the sense that the dispersion relation associated with the two-dimensional model concurs with the appropriate order expansion of the three-dimensional relation at each order, is verified. The model has particular application to stationary thickness vibration of, or transient response to high frequency shock loading in, thin walled bodies

A bending quasi-front generated by an instantaneous impulse loading at the edge of a semi-infinite pre-stressed incompressible elastic plate

A refined membrane-like theory is used to describe bending of a semi-infinite pre-stressed incompressible elastic plate subjected to an instantaneous impulse loading at the edge. A far-field solution for the quasi-front is obtained by using the method of matched asymptotic expansions. A leading-order hyperbolic membrane equation is used for an outer problem, whereas a refined (singularly perturbed) membrane equation of an inner problem describes a boundary layer, which smoothes a discontinuity predicted by the outer problem at the wave front. The inner problem is then reduced to one-dimensional by an appropriate choice of inner coordinates, motivated by the wave front geometry. Using the inherent symmetry of the outer problem, a solution for the quasi-front is derived that is valid in a vicinity of the tip of the wave front. Pre-stress is shown to affect geometry and type of the generated quasi-front; in addition to a classical receding quasi-front the pre-stressed plate can support propagation of an advancing quasi-front. Possible responses may even feature both types of quasi-front at the same time, which is illustrated by numerical examples. The case of a so-called narrow quasi-front, associated with a possible degeneration of contribution of singular perturbation terms to the governing equation, is studied qualitatively

Eigenvalue of a semi-infinite elastic strip

A semi-infinite elastic strip, subjected to traction free boundary conditions, is studied in the context of in-plane stationary vibrations. By using normal (Rayleigh–Lamb) mode expansion the problem of existence of the strip eigenmode is reformulated in terms of the linear dependence within infinite system of normal modes. The concept of Gram's determinant is used to introduce a generalized criterion of linear dependence, which is valid for infinite systems of modes and complex frequencies. Using this criterion, it is demonstrated numerically that in addition to the edge resonance for the Poisson ratio ν=0, there exists another value of ν≈0.22475 associated with an undamped resonance. This resonance is best explained physically by the orthogonality between the edge mode and the first Lamé mode. A semi-analytical proof for the existence of the edge resonance is then presented for both described cases with the help of the augmented scattering matrix formalism

Optimum structure to carry a uniform load between pinned supports: Exact analytical solution

This article is available open access through the publisher’s website at the link below. Copyright @ 2010 The Royal Society.Recent numerical evidence indicates that a parabolic funicular is not necessarily the optimal structural form to carry a uniform load between pinned supports. When the constituent material is capable of resisting equal limiting tensile and compressive stresses, a more efficient structure can be identified, comprising a central parabolic section and networks of truss bars emerging from the supports. In the current article, a precise geometry for this latter structure is identified, avoiding the inconsistencies that render the parabolic form non-optimal. Explicit analytical expressions for the geometry, stress and virtual-displacement fields within and above the structure are presented. Furthermore, a suitable displacement field below the structure is computed numerically and shown to satisfy the Michell–Hemp optimality criteria, hence formally establishing the global optimality of this new structural form

On a Lamb-type problem for a bi-axially pre-stressed incompressible elastic plate

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in IMA Journal of Applied Mathematics following peer review. The definitive publisher authenticated version J. D. KAPLUNOV AND A. V. PICHUGIN (2006). On a Lamb-type problem for a bi-axially pre-stressed incompressible, IMA Journal of Applied Mathematics. 71, 171−185. OUP, is available online at: http://dx.doi.org/10.1093/imamat/hxh097The far-field response of a bi-axially pre-stressed incompressible elastic plate, subjected to an instantaneous edge impulse loading, is studied using a refined long-wave low-frequency theory. The second-order correction introduced by the refined theory is demonstrated to smooth the discontinuity associated with one of the wave-fronts predicted by the leading order hyperbolic theory. The character of the so-called quasi-front is shown to depend greatly on both the material parameters and the pre-stress and may be either classical receding or advancing. Additionally, and in contrast to the analogous problem in linear isotropic elasticity, in a pre-stressed plate the dilatational quasi-front may propagate slower than the shear wave-front. This situation is demonstrated to lead to the formation of a head-wave quasi-front

On rational boundary conditions for higher-order long-wave models

Higher-order corrections to classical long-wave theories enable simple and efficient modelling of the onset of wave dispersion and size effects produced by underlying micro-structure. Since such models feature higher spatial derivatives, one needs to formulate additional boundary conditions when confined to bounded domains. There is a certain controversy associated with these boundary conditions, because it does not seem possible to justify their choice by purely physical considerations. In this paper an asymptotic model for onedimensional chain of particles is chosen as an exemplary higher-order theory. We demonstrate how the presence of higher-order derivative terms results in the existence of non-physical “extraneous” boundary layer-type solutions and argue that the additional boundary conditions should generally be formulated to eliminate the contribution of these boundary layers into the averaged solution. Several new methods of deriving additional boundary conditions are presented for essential boundary. The results are illustrated by numerical examples featuring comparisons with an exact solution for the finite chain

An asymptotic higher-order theory for rectangular beams

A direct asymptotic integration of the full threedimensional problem of elasticity is employed to derive a consistent governing equation for a beam with the rectangular cross-section. The governing equation is consistent in the sense that it has the same long-wave low-frequency behaviour as the exact solution of the original three-dimensional problem. Performance of the new beam equation is illustrated by comparing its predictions against the results of direct finite element computations. Limiting behaviours for beams with large (and small) aspect ratios, which can be established using classic plate theories, are recovered from the new governing equation to illustrate its consistency and also to illustrate the importance of using plate theories with the correctly refined boundary conditions. The implications for the correct choice of the shear correction factor in Timoshenko’s beam theory are also discusse

Optimum structure for a uniform load over multiple spans

This paper presents a new half-plane Michell structure that transmits a uniformly distributed load of infinite horizontal extent to a series of equally-spaced pinned supports. Full kinematic description of the structure is obtained for the case when the maximum allowable tensile stress is greater than or equal to the allowable compressive stress. Although formal proof of optimality of the solution presented is not yet available, the proposed analytical solution is supported by substantial numerical evidence, involving the solution of problems with in excess of 10 billion potential members. Furthermore, numerical solutions for various combinations of unequal allowable stresses suggest the existence of a family of related, simple, and practically relevant structures, which range in form from a Hemp-type arch with vertical hangers to a structure which strongly resembles a cable-stayed bridge

Large expert-curated database for benchmarking document similarity detection in biomedical literature search

Document recommendation systems for locating relevant literature have mostly relied on methods developed a decade ago. This is largely due to the lack of a large offline gold-standard benchmark of relevant documents that cover a variety of research fields such that newly developed literature search techniques can be compared, improved and translated into practice. To overcome this bottleneck, we have established the RElevant LIterature SearcH consortium consisting of more than 1500 scientists from 84 countries, who have collectively annotated the relevance of over 180 000 PubMed-listed articles with regard to their respective seed (input) article/s. The majority of annotations were contributed by highly experienced, original authors of the seed articles. The collected data cover 76% of all unique PubMed Medical Subject Headings descriptors. No systematic biases were observed across different experience levels, research fields or time spent on annotations. More importantly, annotations of the same document pairs contributed by different scientists were highly concordant. We further show that the three representative baseline methods used to generate recommended articles for evaluation (Okapi Best Matching 25, Term Frequency–Inverse Document Frequency and PubMed Related Articles) had similar overall performances. Additionally, we found that these methods each tend to produce distinct collections of recommended articles, suggesting that a hybrid method may be required to completely capture all relevant articles. The established database server located at https://relishdb.ict.griffith.edu.au is freely available for the downloading of annotation data and the blind testing of new methods. We expect that this benchmark will be useful for stimulating the development of new powerful techniques for title and title/abstract-based search engines for relevant articles in biomedical research