Nonlinear Vibrations of Beams, Strings, Plates, and Membranes Without Initial Tension

2004 ◽  
Vol 71 (4) ◽  
pp. 551-559 ◽  
Author(s):  
Zhongping Bao ◽  
Subrata Mukherjee ◽  
Max Roman ◽  
Nadine Aubry
Keyword(s):  
Boundary Conditions ◽  
Young’S Modulus ◽  
Cross Sections ◽  
Nonlinear Vibrations ◽  
Young's Modulus ◽  
Thin Plates ◽  
Radius Of Gyration ◽  
Initial Tension ◽  
Various Boundary Conditions ◽  
Analytical Expressions

The subject of this paper is nonlinear vibrations of beams, strings (defined as beams with very thin uniform cross sections), plates and membranes (defined as very thin plates) without initial tension. Such problems are of great current interest in minute structures with some dimensions in the range of nanometers (nm) to micrometers (μm). A general discussion of these problems is followed by finite element method (FEM) analyses of beams and square plates with different boundary conditions. It is shown that the common practice of neglecting the bending stiffness of strings and membranes, while permissible in the presence of significant initial tension, is not appropriate in the case of nonlinear vibrations of such objects, with no initial tension, and with moderately large amplitude (of the order of the diameter of a string or the thickness of a plate). Approximate, but accurate analytical expressions are presented in this paper for the ratio of the nonlinear to the linear natural fundamental frequency of beams and plates, as functions of the ratio of amplitude to radius of gyration for beams, or the ratio of amplitude to thickness for square plates, for various boundary conditions. These expressions are independent of system parameters—the Young’s modulus, density, length, and radius of gyration for beams; the Young’s modulus, density, length of side, and thickness for square plates. (The plate formula exhibits explicit dependence on the Poisson’s ratio.) It is expected that these results will prove to be useful for the design of macro as well as micro and nano structures.

The Quasi-Static Analysis for Two-Dimensional Thin Plates

2011 ◽  
Vol 255-260 ◽  
pp. 166-169
Author(s):  
Li Chen ◽  
Yang Bai
Keyword(s):  
Boundary Conditions ◽  
Eigenfunction Expansion ◽  
Solution Space ◽  
Expansion Method ◽  
Fundamental Solutions ◽  
Thin Plates ◽  
Elastic Plates ◽  
Various Boundary Conditions ◽  
Eigenfunction Expansion Method ◽  
Concentration Phenomena

The eigenfunction expansion method is introduced into the numerical calculations of elastic plates. Based on the variational method, all the fundamental solutions of the governing equations are obtained directly. Using eigenfunction expansion method, various boundary conditions can be conveniently described by the combination of the eigenfunctions due to the completeness of the solution space. The coefficients of the combination are determined by the boundary conditions. In the numerical example, the stress concentration phenomena produced by the restriction of displacement conditions is discussed in detail.

A New Method of Measuring Young's Modulus for Flexible Thin Materials Using a Cantilever

2006 ◽  
Vol 3-4 ◽  
pp. 53-58 ◽  
Author(s):  
Atsumi Ohtsuki
Keyword(s):  
Young’S Modulus ◽  
Young's Modulus ◽  
Thin Sheet ◽  
New Method ◽  
Thin Plates ◽  
Flexible Materials ◽  
Steel Material ◽  
Fiber Optical ◽  
Deformation Behaviors ◽  
Fiber Materials

This paper describes a development of a new method (: Cantilever Method) to measure Young’s modulus of flexible materials. The method is based on a nonlinear deformation theory that takes into account large deformation behaviors. A set of testing devices was designed and machined. Measurements were carried out on two kinds of flexible materials (PVC: a high-polymer material and SWPA: a steel material). The modulus measured by this method is “Secant modulus”. The results of my evaluation confirm that the new method is suitable for flexible thin plates or rods. Based on the assessments made the method can be further applied to thin sheet and fiber materials (e.g., steel belt, glass fiber, carbon fiber, optical fiber, etc.).

Evaluation of Effective Elastic Mechanical Properties of Graphene Sheets

2012 ◽  
Author(s):  
Khalid I. Alzebdeh
Keyword(s):  
Boundary Conditions ◽  
Young’S Modulus ◽  
Graphene Sheet ◽  
Poisson’S Ratio ◽  
Elastic Moduli ◽  
Young's Modulus ◽  
Unit Cell ◽  
Poisson's Ratio ◽  
Graphene Sheets ◽  
Numerical Predictions

The mechanical behaviour of a single-layer nanostructured graphene sheet is investigated using an atomistic-based continuum model. This is achieved by equating the stored energy in a representative unit cell for a graphene sheet at atomistic scale to the strain energy of an equivalent continuum medium under prescribed boundary conditions. Proper displacement-controlled (essential) boundary conditions which generate a uniform strain field in the unit cell model are applied to calculate one elastic modulus at a time. Three atomistic finite element models are adopted with an assumption that force interactions among carbon atoms can be modeled by either spring-like or beam elements. Thus, elastic moduli for graphene structure are determined based on the proposed modeling approach. Then, effective Young’s modulus and Poisson’s ratio are extracted from the set of calculated elastic moduli. Results of Young’s modulus obtained by employing the different atomistic models show a good agreement with the published theoretical and numerical predictions. However, Poisson’s ratio exhibits sensitivity to the considered atomistic model. This observation is supported by a significant variation in estimates as can be found in the literature. Furthermore, isotropic behaviour of in-plane graphene sheets was validated based on current modeling.

Analysis of the Tightening Process of Bolted Joint With a Tensioner Using Spring Elements

1994 ◽  
Vol 116 (4) ◽  
pp. 443-448 ◽  
Author(s):  
T. Fukuoka
Keyword(s):  
Young’S Modulus ◽  
Young's Modulus ◽  
Clamping Force ◽  
Structural Members ◽  
Bolted Joint ◽  
Initial Tension ◽  
Elementary Method ◽  
The Relationship

Hydraulic bolt tensioners are frequently used to tighten critical structural members with accurately controlled clamping force. The ratio of desired clamping force to initial tension is the most important factor to be predicted in advance for given joint configurations, which is termed “effective tensile coefficient” here. In this paper, an elementary and extensive approach to estimate the coefficient is proposed using spring elements, and a simple and practical equation is presented to evaluate the magnitude of “effective tensile coefficient” in terms of five spring rates of each part consisting of the joint. The relationship between “effective tensile coefficient” and grip length is investigated, including the influences of Young’s modulus of fastened plate. The validity of the elementary method proposed here is ascertained by comparing the results to those by experiment and FEM.

Mechanics-based algorithms to determine the current state of a bridge using quasi-static loading and strain measurement

2018 ◽  
Vol 18 (5-6) ◽  
pp. 1874-1888 ◽  
Author(s):  
Pandi Pitchai ◽  
U Saravanan ◽  
Rupen Goswami
Keyword(s):  
Boundary Conditions ◽  
Young’S Modulus ◽  
Material Properties ◽  
High Performance ◽  
High Performance Concrete ◽  
Young's Modulus ◽  
Moving Load ◽  
Material Parameters ◽  
Current State ◽  
Slow Moving

Knowing the current state of a bridge is of interest for a variety of reasons. Some parameters that determine the current state of a bridge are the material properties and boundary conditions. Using strain measurements obtained from a slow-moving vehicle on a bridge, the boundary condition and material properties are determined through a mechanistic-based approach. Observing that the sign of the curvature would change at locations near the support when a load passes over a bridge with end rotational restraints, a methodology for determining the boundary conditions is proposed and validated. The linear elastic properties of the material that the bridge is made up of is determined from the strain measured at locations where the stress is independent of the material property. In this procedure, the structure is analyzed assuming some material properties and the stress at the measured point is determined. Then, the material parameters in the isotropic Hooke’s law are determined so that the stress estimated from the experimentally determined strains agrees with that obtained from the analysis with arbitrarily assumed material parameters. A prestressed high-performance concrete pi-shaped girder tested under a three-axle slow-moving load with strains measured at different locations is used to bring out the efficacy and appropriateness of the proposed methodologies. The mean value of Young’s modulus of the prestressed concrete bridge agrees well with the experimentally determined Young’s modulus.

scholarly journals Atomic Simulation of Nanoindentation on the Regular Wrinkled Graphene Sheet

Materials ◽  
2020 ◽  
Vol 13 (5) ◽  
pp. 1127
Author(s):  
Ruonan Wang ◽  
Haosheng Pang ◽  
Minglin Li ◽  
Lianfeng Lai
Keyword(s):  
Mechanical Properties ◽  
Boundary Condition ◽  
Boundary Conditions ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Md Simulations ◽  
Force Microscopy ◽  
Atomic Simulation ◽  
Atomic Force ◽  
Surface Wrinkling

Surface landscapes have vague impact on the mechanical properties of graphene. In this paper, single-layered graphene sheets (SLGS) with regular wrinkles were first constructed by applying shear deformation using molecular dynamics (MD) simulations and then indented to extract their mechanical properties. The influence of the boundary condition of SLGS were considered. The wrinkle features and wrinkle formation processes of SLGS were found to be significantly related to the boundary conditions as well as the applied shear displacement and velocity. The wrinkling amplitude and degree of wrinkling increased with the increase in the applied shear displacements, and the trends of wrinkling wavelengths changed with the different boundary conditions. With the fixed boundary condition, the degree of graphene wrinkling was only affected when the velocity was greater than a certain value. The effect of wrinkles on the mechanical characterization of SLGS by atomic force microscopy (AFM) nanoindentation was finally investigated. The regular surface wrinkling of SLGS was found to weaken the Young’s modulus of graphene. The Young’s modulus of graphene deteriorates with the increase in the degree of regular wrinkling.

Three-dimensional elasticity solution of simply supported functionally graded rectangular plates with internal elastic line supports

2009 ◽  
Vol 44 (4) ◽  
pp. 249-261 ◽  
Author(s):  
Y P Xu ◽  
D Zhou
Keyword(s):  
Boundary Conditions ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Three Dimensional ◽  
Lower Surface ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Elastic Line ◽  
Simply Supported ◽  
Three Dimensional Elasticity

This paper studies the stress and displacement distributions of simply supported functionally graded rectangular plates with internal elastic line supports. The Young's modulus is graded through the thickness following the exponential law and the Poisson's ratio is kept constant. On the basis of three-dimensional elasticity theory, the solutions of displacements and stresses of the plate under static loads, which exactly satisfy the governing differential equations and the simply supported boundary conditions at four edges of the plate, are analytically derived. The reaction forces of the internal elastic line supports are regarded as the unknown external forces acting on the lower surface of the plate. The unknown coefficients in the solutions are then determined by the boundary conditions on the upper and lower surfaces of the plate. Convergence and comparison studies demonstrate the correctness and effectiveness of the proposed method. The effect of variations in Young's modulus on the displacements and stresses of rectangular plates and the effect of internal elastic line supports on the mechanical properties of plates are investigated.

Stability of Thin Gold Nano Wires

2009 ◽  
Vol 4 ◽  
pp. 65-77 ◽  
Author(s):  
Veena Verma ◽  
Keya Dharamvir
Keyword(s):  
Shear Modulus ◽  
Young’S Modulus ◽  
Cross Sections ◽  
Young's Modulus ◽  
Poisson Ratio ◽  
Gold Nanowires ◽  
Tube Radius ◽  
Nano Wires ◽  
Gupta Potential ◽  
Bulk Gold

Various gold nanowires with very small cross-sections (few atoms) have been studied using the Gupta potential. Gold nanowire icosa structure is found to be most stable among structures studied. The values of cohesive energy, Young’s modulus and shear modulus have been computed and all the values (except poisson ratio) are more than that of bulk gold. Another striking observation about gold nanostructures is that the Young’s modulus increases with tube radius whereas shear modulus decreases.

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