Theory of Filler Reenforcement in Natural and Synthetic Rubber. The Stresses in and about the Particles

1944 ◽  
Vol 17 (4) ◽  
pp. 865-874 ◽  
Author(s):  
John Rehner
Keyword(s):  
Elastic Constants ◽  
Surrounding Medium ◽  
Radial Distance ◽  
Theory Of Elasticity ◽  
Shear Stresses ◽  
Synthetic Rubber ◽  
Stress Components ◽  
Applied Tension ◽  
Infinitesimal Deformations ◽  
Range Of Values

Abstract An attempt is made to develop a general theory of filler reënforcement by determining the stresses occurring in and about a spherical particle imbedded in a rubberlike medium subjected to an applied tension. For a system containing a single particle rigidly attached to the adjacent medium, an application of the theory of elasticity shows that, for infinitesimal deformations, the stress components within the particle are independent of the radial distance from the origin, taken at the center of the particle. The stress components at a given point in the surrounding medium depend on the elastic constants both of the particle and of the medium, on the radius of the sphere, on the distance from the origin, and on the angle between the direction vector and the applied tension. Expressions are given for the average stresses in media containing many (independent) particles. Theoretical values of the bulk moduli of the synthetic rubbers considered in the treatment are derived from sound velocity data. Curves showing the spatial distribution of radial and shear stresses are presented for a range of values of elastic constants to be expected for different kinds of filler particles and rubberlike materials.

Axisymmetric Stress Analysis and Strength Evaluation of Bonded Shrink Fitted Joints of Circular Pipes Subjected to Internal Pressure and Tensile Loads

2000 ◽  
Author(s):  
Masahide Katsuo ◽  
Toshiyuki Sawa ◽  
Masahiro Yoneno
Keyword(s):  
Internal Pressure ◽  
Stress Analysis ◽  
Joint Strength ◽  
Tensile Load ◽  
Theory Of Elasticity ◽  
Shear Stresses ◽  
Stress Distributions ◽  
Strength Evaluation ◽  
Circular Pipes ◽  
Hollow Cylinders

Abstract This study deals with the stress analysis and the strength evaluation of a bonded shrink fitted joint of circular pipes subjected to an internal pressure and a tensile load. In the analysis, two pipes and the adhesive are replaced with finite hollow cylinders, and the stress distributions in the joint are analyzed by using the axisymmetric theory of elasticity. From the numerical calculations, the following results are obtained: (1) Both the compressive and shear stresses at the interface between the adherend and the adhesive increase as Young’s modulus of the adherend increases. (2) The stress becomes singular at the edges of the interfaces. (3) The joint strength can be evaluated using the compressive and shear stresses near the edge of the interface. In the experiments, bonded shrink fitted joints consisting of dissimilar circular pipes were manufactured, and rupture tests of the joints were carried out by applying an internal pressure, and a tensile load to the joints. From the results, the joint strength of the bonded shrink fitted joint was found to be greater than that of the shrink fitted joint. Furthermore, the numerical results are in fairly good agreement with the experimental ones.

scholarly journals The elastic equilibrium of isotropic plates and cylinders

1949 ◽  
Vol 195 (1043) ◽  
pp. 533-552 ◽  
Keyword(s):  
Fourier Series ◽  
General Solution ◽  
Power Series ◽  
Elastic Equilibrium ◽  
Shear Stresses ◽  
Stress Distributions ◽  
Isotropic Plates ◽  
First Approximation ◽  
Stress Components ◽  
Normal And Shear Stresses

A general solution of the elastic equations is obtained for problems of stress distributions in plates or cylinders when the bounding faces of the plates Z = ± h , or the flat ends of the cylinders, are free from applied normal and shear stresses. The solution is expressed either in the form of Fourier series in the co-ordinate Z , or in power series in Z , the coefficients of the series being certain functions of the x and y co-ordinates which are sufficient to satisfy boundary conditions over two bounding cylindrical surfaces normal to the planes Z = ± h . The form of the theory is greatly simplified by making use of complex combinations of stress components, and by using the complex variable z = x + iy . A first approximation to the part of the theory which deals with the bending of the plate yields a theory similar in character to that given recently by Reissner.

Bending analysis of sandwich plates with composite face sheets and compliance functionally graded syntactic foam core

2016 ◽  
Vol 230 (20) ◽  
pp. 3606-3630 ◽  
Author(s):  
Mohammad Javad Lashkari ◽  
Omid Rahmani
Keyword(s):  
Plate Theory ◽  
Theory Of Elasticity ◽  
Functionally Graded ◽  
Shear Stresses ◽  
Rectangular Plates ◽  
Governing Equations ◽  
Positive Role ◽  
Virtual Displacement ◽  
The Core ◽  
Composite Face

In this paper, the problem of a rectangular plate with functionally graded soft core and composite face sheets is considered using high order sandwich plate theory. This theory applies no assumptions on the displacement and stress fields in the core. Face sheets were treated using classical theory and core was exposed to the theory of elasticity. Governing equations and boundary conditions are derived using principle of virtual displacement and the governing equations are based on eight primary variables including six displacements and two shear stresses. This solution is able to present localized displacements and stresses in places where concentrated loads are exerted to the structure since the displacements in the core can take a nonlinear form which could not be seen in the previous theories such as classical and higher order shear theories. This theory is suitable for rectangular plates under all types of loadings distributed or concentrated which can be different on upper and lower face sheets at the same point. The results were compared with the published literature using theory of elasticity and showed good agreement confirming the accuracy of the present theory. Subsequently, the solution for the core with functionally graded material is presented and effectively indicates positive role of functionally graded core.

Realistic Modeling of Edge Effect Stresses in Bimaterial Elements

1990 ◽  
Vol 112 (1) ◽  
pp. 16-23 ◽  
Author(s):  
J. W. Eischen ◽  
C. Chung ◽  
J. H. Kim
Keyword(s):  
Thermal Loading ◽  
Thermal Stresses ◽  
Theory Of Elasticity ◽  
Shear Stresses ◽  
Strength Of Materials ◽  
Approximate Methods ◽  
Interfacial Stresses ◽  
Distribution Of Stress ◽  
Laminated Structures ◽  
Peak Value

A classic paper by Timoshenko in 1925 dealt with thermal stresses in bimetal thermostats and has been widely used for designing laminated structures, and in contemporary studies of stresses in electronic devices. Timoshenko’s analysis, which is based on strength of materials theory, is unable to predict the distribution of the interfacial shear and normal stresses known to exist based on more sophisticated analyses involving the theory of elasticity (Bogy (1970) and Hess (1969)). Suhir (1986) has recently provided a very insightful approximate method whereby these interfacial stresses are estimated by simple closed-form formulas. The purpose of the present paper is to compare three independent methods of predicting the interfacial normal and shear stresses in bimaterial strips subjected to thermal loading. These are: 1.) Theory of elasticity via an eigenfunction expansion approach proposed by Hess, 2.) Extended strength of materials theory proposed by Suhir, 3.) Finite element stress analysis. Two material configurations which figure prominently in the electronics area have been studied. These are the molydeneum/aluminum and aluminum/silicon material systems. It has been discovered that when the two layers are nearly the same thickness, the approximate methods adequately predict the peak values of the interfacial stresses but err in a fundamental manner in the prediction of the distribution of stress. This may not be of concern to designers who are interested mainly in maximum stress alone. However, it has been shown that if one layer is relatively thin compared to the other, the approximate methods have difficulty in predicting both the peak value of stress and its associated distribution.

A note on the finite plane-strain equations for isotropic incompressible materials

1955 ◽  
Vol 51 (2) ◽  
pp. 363-367 ◽  
Author(s):  
J. E. Adkins
Keyword(s):  
Differential Equations ◽  
Plane Strain ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Theory Of Elasticity ◽  
Finite Plane ◽  
Incompressible Materials ◽  
Stress Components ◽  
Strain Invariants

For elastic deformations beyond the range of the classical infinitesimal theory of elasticity, the governing differential equations are non-linear in form, and orthodox methods of solution are not usually applicable. Simplifying features appear, however, when a restriction is imposed either upon the form of the deformation, or upon the form of strain-energy function employed to define the elastic properties of the material. Thus in the problems of torsion and flexure considered by Rivlin (4, 5, 6) it is possible to avoid introducing partial differential equations into the analysis, while in the theory of finite plane strain developed by Adkins, Green and Shield (1) the reduction in the number of dependent and independent variables involved introduces some measure of simplicity. Some further simplification is achieved when the strain-energy function can be considered as a linear function of the strain invariants as postulated by Mooney(2) for incompressible materials. In the present paper the plane-strain equations for a Mooney material are reduced to symmetrical forms which do not involve the stress components, and some special solutions of these equations are derived.

Failure of micron scale Single Crystal Silicon bars due to torsion developed by MEMS micro instruments

1998 ◽  
Vol 518 ◽  
Author(s):  
Taher Saif ◽  
N. C. MacDonald
Keyword(s):  
Single Crystal ◽  
Single Crystal Silicon ◽  
Theory Of Elasticity ◽  
Shear Stresses ◽  
Electrostatic Actuator ◽  
Crystal Silicon ◽  
Cross Sectional ◽  
Lever Arm ◽  
Spring Constants

AbstractWe present an experimental study on a single crystal silicon (SCS) bar subjected to pure torsion using MEMS micro instruments. The bar is in the form of a pillar, anchored at one end to the silicon substrate. It is attached to a lever arm at the other end. The pillar has a minimum cross sectional area at its mid height. The cross section coincides with the (100) plane of SCS. Torsion is generated by applying two equal forces on the lever arm on either side of the pillar. Two micro instruments apply the forces. Each consists of an electrostatic actuator and a component that calibrates it. The actuator generates high force (≈ 200 µN at 50 V) and is capable of developing large displacements (≈ 10 μm). Calibration involves determination of the force generated by the actuator at an applied voltage, as well as the linear and higher order spring constants of its springs. Each microinstrument is thus calibrated independently.With the application of forces by the two micro instruments, a torque is generated which twists the pillar. The angle of twist at different applied voltages are recorded using an angular scale. The corresponding torques are determined from the calibration parameters of the actuators. Torque is applied until the pillar fractures. Two such sample pillars, samples 1 and 2, are tested. There cross sectional areas are 1 and 2.25 µm2. We find that both the pillars behave linearly until failure. The stresses prior to fracture are evaluated based on anisotropic theory of elasticity. Samples 1 and 2 fail at shear stresses of 5.6 and 2.6 GPa respectively. The fracture surfaces seem to coincide with the (111) plane of SCS.

scholarly journals The influence of surface effects in the problem of theory of elasticity for a circular hole in a half-plane

2020 ◽  
pp. 250-259
Author(s):  
D. V Gandilyan
Keyword(s):  
Fourier Series ◽  
Stress Concentration ◽  
Half Plane ◽  
Surface Stress ◽  
Surface Elasticity ◽  
Surface Effects ◽  
Theory Of Elasticity ◽  
Plane Boundary ◽  
Problem Solution ◽  
Stress Components

Surface effects are important for modeling structures, such as nanofilms, nanoporous materials, and other nanoscale constructions. In the current study, we consider the problem of the theory of elasticity - the problem of a half-plane containing a circular hole, stretched by constant stresses applied at infinity, and take into account surface effects such as surface elasticity and surface stresses. The problem solution has been obtained by expanding the Fourier series with the variables written in the bipolar coordinate system (which simplifies the problem solution because one of the coordinates becomes a constant on the hole contour), where the stress components are expressed through a bi-harmonic stress function. The parametric coefficients involved in the solution, namely in the Fourier series, are determined in order to satisfy the boundary conditions on the hole contour. To solve the problem, in addition to the equations of the theory of elasticity, the equations of surface elasticity were used, in particular, by applying the generalized Young-Laplace’s law and the Shuttleworth’s law; the surface stress on the hole contour has been calculated directly. Using recurrence relations for the stress components at the boundary, stress concentration values have been obtained. The resulting expressions can be considered as a generalized solution of the problem in case of the classical elasticity. The stress concentrations are compared for the cases with and without taking into account surface effects at various points on the hole contour. The contribution caused by the surface effects depending on the relative distance between the hole and the half-plane boundary is studied. It is shown that despite a quite simple geometry, owing to the fairly small distance between the hole and the half-plane boundary, the stress concentration with and without taking into account the surface stress are significantly different from each other, due to the significant contribution of surface effects.

scholarly journals Local Elastic Constants For Epoxy-Nanotube Composites From Molecular Dynamics Simulation

2007 ◽  
Vol 1056 ◽  
Author(s):  
S. J. V. Frankland ◽  
T. S. Gates
Keyword(s):  
Molecular Dynamics ◽  
Molecular Dynamics Simulation ◽  
Numerical Analysis ◽  
Elastic Constants ◽  
Radial Distance ◽  
Dynamics Simulation ◽  
Statistical Uncertainty ◽  
Nanotube Composites

ABSTRACTA method from molecular dynamics simulation is developed for determining local elastic constants of an epoxy/nanotube composite. The local values of C11, C33, K12, and K13 elastic constants are calculated for an epoxy/nanotube composite as a function of radial distance from the nanotube. While the results possess a significant amount of statistical uncertainty resulting from both the numerical analysis and the molecular fluctuations during the simulation, the following observations can be made. If the size of the region around the nanotube is increased from shells of 1Å to 6Å in thickness, then the scatter in the data reduces enough to observe trends. All the elastic constants determined are at a minimum 20Å from the center of the nanotube. The C11, C33, and K12 follow similar trends as a function of radial distance from the nanotube. The K13 decreases greater distances from the nanotube and becomes negative which may be a symptom of the statistical averaging.

Constitutive Equations for Graphene Based on Anisotropic Theory

2011 ◽  
Vol 314-316 ◽  
pp. 1268-1272
Author(s):  
Shu Hao Ban ◽  
Ai Ping Hu ◽  
Xue Dong Jiang ◽  
Xiao Yan Li
Keyword(s):  
Elastic Constants ◽  
Material Properties ◽  
Constitutive Equations ◽  
Single Layer ◽  
Plane Stress State ◽  
Single Layer Graphene ◽  
Layer Graphene ◽  
Simple Tension ◽  
Stress Components ◽  
Shear Strains

The constitutive equations, describing the relations between stresses and strains in the single layer graphene, were obtained based on the anisotropic theory. The equations showed that (1) there are 3 nonzero stress components in graphene for it is in the plane stress state; (2) there are 4 elastic constants independent in the constitutive equations describing the material properties of graphene; and (3) the normal stress merely results in normal strains if graphene in simple tension in primary directions but generates the normal strains and shear strains if in non-primary directions. In addition, the Hamilton method determining the 4 elastic constants in constitutive equations of graphene was introduced.

scholarly journals The equivalent modulus of elasticity of soil mediums for designing shallow foundations

2021 ◽  
Author(s):  
Lysandros Pantelidis
Keyword(s):  
Elastic Constants ◽  
Poisson’S Ratio ◽  
Theory Of Elasticity ◽  
Shallow Foundations ◽  
Poisson's Ratio ◽  
Deformation Pattern ◽  
Lateral Expansion ◽  
Principle Of Superposition ◽  
Soil Medium ◽  
Equivalent Modulus

Abstract As known, in a Winkler type of analysis the soil medium underneath the foundation is violently replaced by a row of parallel springs having constant ks. For the effective calculation of the latter, which is called the modulus of subgrade reaction, the two elastic constants of the soil (the elastic modulus, E and the Poisson’s ratio, ν) must be known. Although for homogenous soils this generally seems not to be a problem, the same does not stand for stratified mediums or mediums with linearly increasing modulus with depth. In such an analysis, the proper pair of elastic constant values of soil should be selected. This refers to a Poisson’s ratio value equal to zero corresponding to the deformation pattern of springs (compression with no lateral expansion) and the respective modulus. In the present paper a method for calculating the equivalent elastic constants for the above mentioned mediums is proposed based on the theory of elasticity combining the principle of superposition. Various cases are considered, since the equivalent modulus, Eeq, depends on the rigidity and the shape of the footing. As shown, the derived Eeq values not only return reliable settlement results, but also settlement profiles that are similar to those corresponding to the original soil mediums.

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