One-dimensional Structural Elements Research Articles

Optimizing the SERS Performance of 3D Substrates through Tunable 3D Plasmonic Coupling toward Label-Free Liver Cancer Cell Classification

Three-dimensional (3D) plasmonic nanostructures are emerging as excellent surface-enhanced Raman spectroscopy (SERS) substrates for chemical and biomedical applications. However, the correlation of 3D (including both in-plane and out-of-plane) plasmonic coupling with the SERS properties to deepen the understanding of 3D SERS substrates remains a challenge. Here, we perform correlation studies of 3D plasmonic coupling and SERS properties of the 3D hierarchical SERS substrates by tuning the multiscale structural elements. The effects of zero-dimensional (0D; the size of the building blocks), one-dimensional (1D; the thickness of the 3D substrates), and two-dimensional (2D; the composition of individual monolayers) structural elements on 3D plasmonic coupling are studied by performing UV-vis-near-infrared (NIR) spectroscopy and measuring SERS performance. It shows that both the extinction spectra and SERS enhancement are tuned at the 3D structural level. It is demonstrated that the plasmonic resonance wavelength (PRW) stemming from the 3D plasmonic coupling correlates with the SERS averaged surface enhancement factor (ASEF) and is improved by more than tenfold at the optimum 3D nanostructure. The optimized substrate is used to quantitatively analyze two small biological molecules. Moreover, as a proof-of-concept study, the substrate is first applied to differentiate between living liver normal and cancer cells with a high prediction accuracy through the spectral features of the cell membranes and the metabolites secreted outside the cells. We expect that the tuning of plasmonic coupling at the 3D level can open up new routes to design high-performance SERS substrates for wide applications.

Perturbation approach to Eringen’s local/non-local constitutive equation with applications to 1-D structures

Eringen’s two-phase local/non-local constitutive equation is preferred over its full non-local counterpart due to mathematical simplifications it provides. Then again, an integro-differential equation must be solved, which requires rigorous examination of the existence of an exact solution in certain forms. For this purpose, some additional constraints are attained to strain field for the sake of an exact solution which may be in contrast with the balance equations. It is the aim of this study to look for possible approximated solutions in series by a perturbation approach. Indeed, we find that response of structures with non-local constitutive relation may be approximated by a set of local elasticity problems, the existence and uniqueness of which are ensured. The present approach does not require any more conditions than physical boundary conditions, such as constitutive boundary conditions. It is applied to simple one-dimensional structural elements, and numerical evidence on possible convergence of the series expansion is provided. Some structural problems of bars and beams, which may be the simplified models of nanostructures in modern engineering applications, are discussed and solutions to them are given in closed-form.

Dynamic center of elasticity concept for one-dimensional structural elements

The center of elasticity concept for one-dimensional structural elements is most often understood in the context of a point in the cross-section lying between the flexural and torsional centers of the section. However, the elastic center as a fundamental property of the element itself may be discussed in terms of the displacements and loads applied to a particular point. The elastic center is the point at which an applied force produces pure translation, and an applied moment produces pure rotation about the same axis. This concept is extended to the dynamic regime by including both elastic and inertial properties, leading to a dynamic center of elasticity which is valid across a range of frequencies. Three methods are proposed to locate the dynamic elastic center of a simple lumped parameter system, revealing that the elastic center location varies greatly, even moving off to infinity at certain critical frequencies. Each method is also applied to a continuously deformable cantilevered Euler-Bernoulli beam, producing rich dynamic behavior. The three methods are shown to yield equivalent results for both systems. Finally, the results are compared with dynamic stiffness and modal analysis to obtain physical insight into the dynamic behavior of a system's elastic center.

Coupled Transverse and Axial Vibrations Including Warping Effect in Asymmetric Short Beams

AbstractTo model the displacement field of asymmetric one-dimensional structural elements, in which a transverse-bending motion couples with axial motion, a formulation with five displacement compo...

Homogenization of heterogeneous masonry beams

This study presents a two-scale model to describe the out-of-plane masonry response. One-dimensional (1D) structural elements, like masonry columns or strips of long wall characterized by the periodic repetition of bricks and mortar arranged in stack bond, are considered. A damage-friction plasticity law is adopted to model the mortar joint constitutive response, while the bricks are assumed as linear elastic. A 1D beam formulation is introduced at both the structural and micromechanical scale, linking the two levels by means of a kinematic map. This expresses the microscopic beam strains in the masonry unit cell (UC) as function of the macroscopic generalized strains. The kinematic field in the UC is completed by adding an unknown periodic fluctuation term. A nonlinear homogenization procedure is developed, proposing a semi-analytical solution for the micromechanical problem, based on the fiber discretization of the mortar joints. A force-based beam-column finite element procedure is adopted at the structural scale and the solution algorithm for the element state determination is illustrated in details. Some numerical applications, showing the UC constitutive response and the behavior of masonry structural elements, are finally presented.

An FEM-based method to evaluate and optimize vibration power flow through a beam-to-plate connection

Power flow is an important measure in vibratory energy propagation path analysis. For one-dimensional structural elements, i.e., bars or beams, power flow is obtained using a relationship involving degrees of freedom (DOF) of interest, and the respective internal forces. Several published works on vibration analysis involving the finite element method (FEM) and its optimization describe measures of interest only in terms of DOF (such as displacement and mean square velocity). However, in scenarios involving the coupling of different structures, these measures are inadequate. For example, minimizing only dynamical displacements at specific points in a structure could result in the internal forces at those points becoming unacceptably large. In such cases, minimizing power flow is preferable over minimizing displacements. In this manuscript, using FEM and a gradient-based optimization method, the authors propose a technique to evaluate and optimize vibratory power flow specifically in cases involving beam–plate coupling. The total power flow at the connection point is defined as a function of the global displacement vector, and it is evaluated for a given frequency by harmonic analysis; the relevant sensitivities are obtained using elementary matrices and the adjoint method. Geometrical parameters of the beam are used as design variables. A description of the methodology and two examples of its application to beam–plate structures are presented.

Mass Matrix Templates: General Description and 1D Examples

This article is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The main focus is on adjusting the mass matrix. Two well known discretization methods, described in FEM textbooks since the late 1960s, lead to diagonally lumped and consistent mass matrices, respectively. Those models are sufficient to cover many engineering applications but for some problems they fall short. The gap can be filled with a more general approach that relies on the use of templates. These are algebraic forms that carry free parameters. Templates have the virtue of producing a set of mass matrices that satisfy certain a priori constraint conditions such as symmetry, nonnegativity, invariance and momentum conservation. In particular, the diagonally lumped and consistent versions can be obtained as instances. Availability of free parameters, however, allows the mass matrix to be customized to special needs, such as high precision vibration frequencies or minimally dispersive wave propagation. An attractive feature of templates for FEM programming is that only one element implementation as module with free parameters is needed, and need not be recoded when the application problem class changes. The paper provides a general overview of the topic, and illustrates it with one-dimensional structural elements: bars and beams.

Construction of Customized Mass-Stiffness Pairs Using Templates

This paper is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The exposition focuses on adjusting the mass matrix while a separately provided stiffness matrix is kept fixed. Two well known kinetic-energy discretization methods described in finite-element method (FEM) textbooks since the mid-1960s lead to diagonally lumped and consistent mass matrices, respectively. These two models are sufficient to cover many engineering applications. Occasionally, however, they fall short. The gap can be filled with a more general approach that relies on the use of templates. These are algebraic forms that carry free parameters. This approach is discussed in this paper using one-dimensional structural elements as examples. Templates have the virtue of producing a set of mass matrices that satisfy certain a priori constraint conditions such as symmetry, nonnegativity, invariance, and momentum conservation. In particula...

Maximum likelihood estimation of structural wave components from noisy data.

In this paper, a general framework is developed for determining the underlying parameters of general signal models through the application of maximum likelihood estimation theory for functions whose variables separate. This method extends previous work in sinusoidal and exponential estimation to include models with other functional bases, such as exponential functions with nonconstant amplitudes and Bessel functions. Nonuniform spatial sampling is also possible with this technique. The maximum likelihood method is applied to the identification of wave components along one-dimensional structural elements. Results are given which demonstrate the viability and accuracy of the technique estimating exponential and Bessel function model parameters from noisy simulation data.

Spatially Convolving Wave Propagation Sensors for Structural Control-Part I: Analytical Development for One-Dimensional Structures

A novel method for sensing actual disturbance information which can propagate along one-dimensional structural waveguides is developed. The motivation for this work arises from the inability to realize most active wave control designs using output information from physical measurements such as deflection, slope, curvature and internal shear force. Often the use of actual physical measurements leads to instability and suboptimal performance of certain active wave control loops. Thus, there is a desire to develop wave-propagation filters which can extract magnitude and direction of structural disturbances along dispersive and non-dispersive members. In this work a distributed wave sensing scheme is reported which exploits the characterization of 1-dimensional structural motion in terms of traveling waves. This form of the solution is then combined with distributed sensors to convolve future and past measurements into a single temporal signal to overcome the problem of causality and spatial aliasing inherent in discrete sensor approaches. Thus, by imposing specific shapes to spatially convolving sensors, it is possible to combine the output of these sensors with point measurements to observe directional propagating wave components. Because this method requires some approximation of the spatial domain, there will be errors due to spatial discretization and truncation. This work addresses these issues and presents analytical results for one-dimensional structural elements.

Weakly decaying energy separation and uniqueness of motions of an elastic-plastic oscillator with work-hardening

A useful and natural first step in the analysis of motions of heavy objects supported by a variety of structural members is to ignore many of the geometrical characteristics of the supporting members. For example, the motion of a block suspended from a helical spring usually is studied first by regarding the spring as a one-dimensional structural element whose change in length determines the force that it exerts on the block. When the force varies linearly with change in length, the resulting spring-mass system is the familiar harmonic oscillator. If structural members undergo significant permanent deformation, it is standard practice to employ one-dimensional structural elements which, unlike the idealized spring, are capable of exhibiting hysteresis. The elastic-perfectly plastic element provides an example of this type in which there is an upper bound for the magnitude of the force that the element can provide. Below this upper “yield force”, the element behaves elastically, as would a spring. When the upper bound would be surpassed through further elastic behavior, the element continues to deform but with no further increase in the force it provides. The resulting large permanent deformations might destabilize a structure consisting of elastic-plastic elements, but this tendency sometimes can be offset by the ability of the elements to dissipate mechanical energy as they undergo permanent deformations.

Minimum Weight Design for Structural Eigenvalue Problems by Optimal Control Theory*, †

ABSTRACT The application of optimal control theory to minimum weight design of continuous one-dimensional structural elements subject to eigenvalue constraints is discussed. If not only the value of an eigenvalue is prescribed but also its position in the sequence of the ordered eigenvalues—for example, the critical buckling load of a column—the corresponding optimal control problem is shown to include necessarily all eigenvalues. Considering the unspecified eigenvalues as free parameters, necessary conditions for minimum weight design are derived. These conditions are compared with those obtained by use of variational methods. Attention is focused on the special case of multimodal solutions.