Lattice-Misfit Stresses in a Circular Bi-Material Gallium-Nitride Assembly

2012 ◽  
Vol 80 (1) ◽  
Author(s):  
E. Suhir
Keyword(s):  
Gallium Nitride ◽  
Cross Sections ◽  
Circumferential Stress ◽  
Lattice Misfit ◽  
Theory Of Elasticity ◽  
Shearing Stress ◽  
Interfacial Shearing Stress ◽  
Stress Theory ◽  
Circular Configuration ◽  
Gan Film

A simple and physically meaningful analytical (“mathematical”) predictive model is developed using two-dimensional (plane-stress) theory-of-elasticity approach (TEA) for the evaluation of the effect of the circular configuration of the substrate (wafer) on the elastic lattice-misfit (mismatch) stresses (LMS) in a semiconductor and particularly in a gallium nitride (GaN) film grown on such a substrate. The addressed stresses include (1) the interfacial shearing stress supposedly responsible for the occurrence and growth of dislocations, for possible delaminations, and for the cohesive strength of the intermediate strain buffering material, if any, as well as (2) normal radial and circumferential (tangential) stresses acting in the film cross-sections and responsible for the short- and long-term strength (fracture toughness) of the film. The TEA results are compared with the formulas obtained using strength-of-materials approach (SMA). This approach considers, instead of the actual circular substrate, an elongated bi-material rectangular strip of unit width and of finite length equal to the wafer diameter. The numerical example is carried out, as an illustration, for a GaN film grown on a silicon carbide (SiC) substrate. It is concluded that the SMA model is acceptable for understanding the physics of the state of stress and for the prediction of the normal stresses in the major midportion of the assembly. The SMA model underestimates, however, the maximum interfacial shearing stress at the assembly periphery and, because of the very nature of the SMA, is unable to address the circumferential stress. The developed TEA model can be used, along with the author's earlier publications and the (traditional and routine) finite-element analyses (FEA), to assess the merits and shortcomings of a particular semiconductor crystal growth (SCG) technology, as far as the level of the expected LMS are concerned, before the actual experimentation and/or fabrication is decided upon and conducted.

Torsion of Noncylindrical Shafts of Circular Cross Section

1950 ◽  
Vol 17 (3) ◽  
pp. 275-282
Author(s):  
H. J. Reissner ◽  
G. J. Wennagel
Keyword(s):  
Cross Sections ◽  
Inverse Method ◽  
Direct Method ◽  
Circular Cross Section ◽  
Theory Of Elasticity ◽  
Elementary Functions ◽  
Single Term ◽  
Circumferential Displacement ◽  
Basic Differential Equation ◽  
Technical Interest

Abstract The theory of torsion of noncylindrical bodies of revolution, initiated by J. H. Michell and A. Föppl, is stated by a basic differential equation of the circumferential displacement and by a boundary condition of the shear stress along the generator surface. The solution of these two equations by the “direct” method of first assuming the boundary shape has not lent itself to closed solutions in terms of elementary functions, so that only approximation, infinite series, and experimental methods have been applied. A semi-inverse method analogous to Saint Venant’s semi-inverse method for cylindrical bodies has the disadvantage of the restriction to special boundary shapes but the advantage of exact solutions by means of elementary functions. By this method, bodies of conical, ellipsoidal, and hyperbolic boundary shapes have been obtained in a simple analysis. One class of integrals leading to other boundary shapes seems not to have been analyzed up to now, namely, the integrals in the form of a product of two functions of, respectively, axial (z) and radial (r) co-ordinates. A first suggestion of this possibility was given in Love’s treatise on the mathematical theory of elasticity. In the present paper, the classes of boundary shapes, displacements, and stress distributions are investigated analytically and numerically. The extent of the numerical investigation contains only the results of single-term integrals for full and hollow cross sections of technical interest. The detailed analysis of the boundary shapes, following from series integrals, presents essential mathematical obstacles. Overcoming these difficulties might lead to a multitude of solutions of interesting boundary shapes, and stress and strain distribution.

scholarly journals GEOMETRIC MODELING OF A SHAPE OF PARALLELOGRAM PLATES IN A PROBLEM OF FREE VIBRATIONS USING CONFORMAL RADII

2021 ◽  
pp. 143-159
Author(s):  
A.A. Chernyaev ◽  
Keyword(s):  
Cross Sections ◽  
Geometric Modeling ◽  
Interpolation Method ◽  
Geometric Characteristic ◽  
Free Vibrations ◽  
Theory Of Elasticity ◽  
Constant Thickness ◽  
Basic Frequency ◽  
Plate Vibrations ◽  
The Subject

The paper considers a method of geometric modeling applied when solving basic twodimensional problems of the theory of elasticity and structural mechanics, in particular the applied problems of engineering. The subject of this study is vibrations of thin elastic parallelogram plates of constant thickness. To determine a basic frequency of vibrations, the interpolation method based on the geometric characteristic of the shape of plates (membrane, cross sections of a rod) is proposed. This characteristic represents a ratio of interior and exterior conformal radii of the plate. As is known from the theory of conformal mappings, conformal radii are those obtained by mapping of a plate onto the interior and exterior of a unit disk. The paper presents basic terms, tables, and formulas related to the considered geometric method with a comparative analysis of the curve diagrams obtained using various interpolation formulas. The original computer program is also developed. The main advantage of the proposed method of determining the basic frequency of plate vibrations is a graphic representation of results that allows one to accurately determine the required solution on the graph among the other solutions corresponding to the considered case of parallelogram plates. Although there are many known approximate approaches, which are used to solve the considered problems, only geometric modeling technique based on the conformal radii ratio gives such an opportunity.

Phenomenological and Structural Aspects of the Mechanical Response of Arteries

2000 ◽  
Author(s):  
Ray W. Ogden ◽  
Christian A. J. Schulze-Bauer
Keyword(s):  
Residual Stresses ◽  
Mechanical Response ◽  
Physiological State ◽  
Circumferential Stress ◽  
Morphological Structure ◽  
Cylindrical Tube ◽  
Single Layer ◽  
Theory Of Elasticity ◽  
Starting Point ◽  
Circular Cylindrical Tube

Abstract In this paper we present some new data from extension-inflation tests on a human iliac artery and then, on the basis of the nonlinear theory of elasticity, we examine a possible model to represent this data. The model considers the artery initially as a thick-walled circular cylindrical tube which may consist of two or more concentric layers. In order to take some account of the architecture (morphological structure), each layer of the material is regarded as consisting of two families of mechanically equivalent helical fibers symmetrically disposed with respect to the cylinder axis. The resulting material properties are then orthotropic in each layer. General formulas for the pressure and the axial load in the symmetric inflation of an extended tube are obtained. The starting point is the unloaded circular cylindrical configuration, but (in general unknown) residual stresses are included in the formulation. The model is illustrated by specializing firstly to the case of a single layer so that the consequences of the hypothesis of uniform circumferential stress in the physiological state can be examined theoretically. This enables the required residual stresses to be calculated explicitly. Secondly, the equations are specialized for the membrane approximation in order to show how certain important characteristics of the experimental data can be replicated using a relatively simple anisotropic membrane model.

The Torsionless Bending of a Hollow Beam by a Transverse Load

1937 ◽  
Vol 4 (1) ◽  
pp. A25-A30
Author(s):  
W. L. Schwalbe
Keyword(s):  
Cross Sections ◽  
Equilateral Triangle ◽  
Bending Moment ◽  
Longitudinal Section ◽  
Transverse Load ◽  
Shearing Stress ◽  
Numerical Work ◽  
Hollow Beam ◽  
Transverse Loads ◽  
Hollow Beams

Abstract The author discusses the bending of hollow beams when subjected to transverse loads, and points out that shearing stresses and strains in the cross sections are necessary, and a particular longitudinal section remains plane only if the resultant of the shearing stress, and hence the plane of the applied bending moment, possesses a particular location. The author determines the location of this resultant shearing stress by applying a method based on St. Venant’s theory. Applications of the method are made to two hollow sections. One of the sections is that of an equilateral triangle which serves as a measure of accuracy for the numerical work presented by the author, since the location of the resultant of the shearing stresses is known by symmetry.

scholarly journals The torsion of a semi-infinite elastic solid by an elliptical stamp

1968 ◽  
Vol 9 (1) ◽  
pp. 36-45
Author(s):  
Mumtaz K. Kassir
Keyword(s):  
Field Equation ◽  
Classical Theory ◽  
Cross Sections ◽  
Plane Surface ◽  
Boundary Surface ◽  
Elastic Solid ◽  
Theory Of Elasticity ◽  
Circular Area

The problem of determining, within the limits of the classical theory of elasticity, the displacements and stresses in the interior of a semi-infinite solid (z ≧ 0) when a part of the boundary surface (z = 0) is forced to rotate through a given angle ω about an axis which is normal to the undeformed plane surface of the solid, has been discussed by several authors [7, 8, 9, 1, 11, and others]. All of this work is concerned with rotating a circular area of the boundary surface and the field equation to be solved is, essentially, J. H. Mitchell's equation for the torsion of bars of varying circular cross-sections.

Effect of Couple-Stresses on Force Transfer Between Embedded Microfibers

1967 ◽  
Vol 1 (2) ◽  
pp. 174-187 ◽  
Author(s):  
M.A. Sadowsky ◽  
Y.C. Hsu ◽  
M.A. Hussain
Keyword(s):  
Stress Field ◽  
Couple Stress Theory ◽  
Physical Phenomenon ◽  
Lattice Parameter ◽  
Theory Of Elasticity ◽  
Crystal Lattice Parameter ◽  
Tangential Displacement ◽  
Stress Theory ◽  
Couple Stresses ◽  
Force Transfer

The present investigation is a continuation of the Weitsman problem, reference 1. The investigation is concerned with the nature of the stress field which develops in the presence of couple- stresses in composite material formed of microfibers embedded in a filler substance. The case is considered when one row consists of an infinitely long microfiber, while the second row is composed of two semi- infinite microfibers. An exact solution, based on a two-dimensional theory of elastic ity with inclusion of couple-stresses is obtained for the stress field which is activated by giving the two semi-infinite microfibers a tangential displacement of separation. The couple-stress theory gives a better representation of the actual physical phenomenon than the classical theory of elasticity. This is especially true and essentially important for bodies having a dimension comparable to the crystal lattice parameter of the material. This is believed to be the case with an elastic microfilm in bond with microfibers. The investigation led to results which are at wide variance with the Weitsman solution near x = 0 and the asymptotic solution shows the same trend as reported previously in reference num bered 2 i.e., bonding layers are dominated by boundary layer behavior throughout their entire thickness.

scholarly journals Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

2017 ◽  
Vol 4 (1) ◽  
pp. 119-133 ◽  
Author(s):  
V.V. Zozulya
Keyword(s):  
Fourier Series ◽  
Couple Stress ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Couple Stress Theory ◽  
Order Theory ◽  
Theory Of Elasticity ◽  
High Order ◽  
Stress Theory ◽  
Curved Rods

AbstractNew models for plane curved rods based on linear couple stress theory of elasticity have been developed.2-D theory is developed from general 2-D equations of linear couple stress elasticity using a special curvilinear system of coordinates related to the middle line of the rod as well as special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and rotation along with body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate.Thereby, all equations of elasticity including Hooke’s law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. Timoshenko’s and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear couple stress theory of elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scales when taking into account couple stress and rotation effects.

Sign in / Sign up

Export Citation Format

Share Document