scholarly journals Stress Singularities Resulting From Various Boundary Conditions in Angular Corners of Plates in Extension

1952 ◽  
Vol 19 (4) ◽  
pp. 526-528
Author(s):  
M. L. Williams
Keyword(s):  
Boundary Conditions ◽  
Poisson’S Ratio ◽  
Mixed Boundary Condition ◽  
Mixed Boundary ◽  
Poisson's Ratio ◽  
Vertex Angle ◽  
Stress Singularities ◽  
Stress Concentrations ◽  
Various Boundary Conditions ◽  
Free Case

Abstract As an analog to the bending case published in an earlier paper, the stress singularities in plates subjected to extension in their plane are discussed. Three sets of boundary conditions on the radial edges are investigated: free-free, clamped-clamped, and clamped-free. Providing the vertex angle is less than 180 degrees, it is found that unbounded stresses occur at the vertex only in the case of the mixed boundary condition with the strength of the singularity being somewhat stronger than for the similar bending case. For vertex angles between 180 and 360 degrees, all the cases considered may have stress singularities. In amplification of some work of Southwell, it is shown that there are certain analogies between the characteristic equations governing the stresses in extension and bending, respectively, if ν, Poisson’s ratio, is replaced by −ν. Finally, the free-free extensional plate behaves locally at the origin exactly the same as a clamped-clamped plate in bending, independent of Poisson’s ratio. In conclusion, it is noted that the free-free case analysis may be applied to stress concentrations in V-shaped notches.

Effect of poisson's ratio strains in adherends on stresses of an idealized lap joint

1973 ◽  
Vol 8 (2) ◽  
pp. 134-139 ◽  
Author(s):  
R D Adams ◽  
N A Peppiatt
Keyword(s):  
Shear Stress ◽  
Poisson’S Ratio ◽  
Longitudinal Shear ◽  
Poisson's Ratio ◽  
Shear Stresses ◽  
Lap Joint ◽  
Stress Concentrations ◽  
Maximum Value ◽  
Set Up ◽  
Transverse Stresses

Poisson's ratio strains in the adherends of a simple adhesive lap joint induce transverse stresses both in the adhesive and in the adherends. Two simultaneous second-order partial-differential equations were set up to describe the normal stresses along and across an adherend and were solved both by an approximate analytical method and a finite-difference technique: the two solutions agreed closely. The adhesive shear stresses can then be obtained by differentiating these solutions. The transverse shear stress has a maximum value for metals of about one-third of the maximum longitudinal shear stress, and this occurs at the corners of the lap, thus making the corners the most highly stressed parts of the adhesive. Bonding adherends of dissimilar stiffness was shown to produce greater stress concentrations in the adhesive than when similar adherends are used.

Buckling and Vibration of Circular Auxetic Plates

2014 ◽  
Vol 136 (2) ◽  
Author(s):  
Teik-Cheng Lim
Keyword(s):  
Poisson’S Ratio ◽  
Dynamic Properties ◽  
Elastic Stability ◽  
Buckling Load ◽  
Poisson's Ratio ◽  
Circular Plates ◽  
Critical Buckling Load ◽  
Various Boundary Conditions ◽  
Load Factors ◽  
Auxetic Materials

This paper evaluates the elastic stability and vibration characteristics of circular plates made from auxetic materials. By solving the general solutions for buckling and vibration of circular plates under various boundary conditions, the critical buckling load factors and fundamental frequencies of circular plates, within the scope of the first axisymmetric modes, were obtained for the entire range of Poisson's ratio for isotropic solids, i.e., from −1 to 0.5. Results for elastic stability reveal that as the Poisson's ratio of the plate becomes more negative, the critical bucking load gradually reduces. In the case of vibration, the decrease in Poisson's ratio not only decreases the fundamental frequency, but the decrease becomes very rapid as the Poisson's ratio approaches its lower limit. For both buckling and vibration, the plate's Poisson's ratio has no effect if the edge is fully clamped. The results obtained herein suggest that auxetic materials can be employed for attaining static and dynamic properties which are not common in plates made from conventional materials. Based on the exact results, empirical models were generated for design purposes so that both the critical buckling load factors and the frequency parameters can be conveniently obtained without calculating the Bessel functions.

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.

scholarly journals Nonlinear analysis of buckling and postbuckling for axially compressed functionally graded cylindrical panels with the Poisson's ratio varying smoothly along the thickness

2012 ◽  
Vol 34 (1) ◽  
pp. 27-44 ◽  
Author(s):  
Dao Van Dung ◽  
Le Kha Hoa
Keyword(s):  
Boundary Conditions ◽  
Poisson’S Ratio ◽  
Compressive Load ◽  
Cylindrical Panel ◽  
Functionally Graded ◽  
Poisson's Ratio ◽  
Postbuckling Behavior ◽  
Nonlinear Buckling ◽  
Dimensional Parameter ◽  
Classical Shell Theory

In this paper an approximate analytical solution to analyze the nonlinear buckling and postbuckling behavior of imperfect functionally graded panels with the Poisson's ratio also varying smoothly along the thickness is investigated. Based on the classical shell theory and von Karman's assumption of kinematic nonlinearity and applying Galerkin procedure, the equations for finding critical loads and load-deflection curves of cylindrical panel subjected to axial compressive load with two types boundary conditions, are given. Especially, the stiffness coefficients are analyzed in explicit form. Numerical results show various effects of the inhomogeneous parameter, dimensional parameter, boundary conditions on nonlinear stability of panel. An accuracy of present theoretical results is verified by the previous well-known results.

Post-Buckling of Rectangular Plates with Various Boundary Conditions

1970 ◽  
Vol 21 (2) ◽  
pp. 163-181 ◽  
Author(s):  
K. R. Rushton
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Relaxation Method ◽  
Rectangular Plates ◽  
Dynamic Relaxation ◽  
Various Boundary Conditions ◽  
Post Buckling ◽  
Dynamic Relaxation Method ◽  
Theoretical Results

SummaryThe Dynamic Relaxation method is used to analyse the post-buckling of flat rectangular plates. Extensive results are obtained for two types of problem in which the transverse edges are unloaded; in one case the transverse edges remain straight, in the other they are free to wave. Comparisons are made with alternative experimental and theoretical results. The potentiality of this approach to post-buckling problems is demonstrated by considering a variable thickness plate with mixed boundary conditions.

Classical Solution and Edge Effect in the Problem of Stability of an Axially Compressed Cylindrical Shell

2020 ◽  
Vol 22 (1) ◽  
pp. 48-54
Author(s):  
Guyk A. Manuylov ◽  
Sergey B. Kosytsyn ◽  
Maksim M. Begichev
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Critical Load ◽  
Classical Solution ◽  
Edge Effect ◽  
Poisson’S Ratio ◽  
Poisson's Ratio ◽  
Critical Stresses ◽  
Membrane Deformations ◽  
Initial Arrangement

The classical solution for critical stresses in the problem of stability of a circular longitudinally compressed cylindrical shell consists of two terms, reflecting the ability of the shell to resist buckling due to bending and membrane deformations. However, with usual boundary conditions the classical solution appears only with the absence of the Poisson expansion of a shell. With a non-zero Poisson's ratio, an axisymmetric edge effect presents. It reduces the critical load and causes the initial arrangement of its own forms to change as the load increases.

scholarly journals Basal Stress Concentrations Due to Abrupt Changes in Boundary Conditions: A Cause for High Till Concentration at the Bottom of a Glacier

1981 ◽  
Vol 2 ◽  
pp. 29-33 ◽  
Author(s):  
Kolumban Hutter ◽  
Vincent O.S. Olunloyo
Keyword(s):  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Viscosity Coefficient ◽  
Transition Line ◽  
Physical Processes ◽  
Stress Concentrations ◽  
Asymptotic Expressions ◽  
Abrupt Changes ◽  
Velocity Changes

The existence of cold patches at the base of a glacier suggests that the sliding law will depend on these patches, which will essentially affect the viscosity constant. In a poly thermal glacier, such as a glacier which is cold in its lower part and temperate in its upper part, basal boundary conditions change from no-slip to viscous sliding. It is anticipated that the viscosity constant of this sliding law will depend on the distance from the transition line between cold and temperate ice.The mixed boundary conditions, namely no-slip where the ice is cold and viscous sliding where it is temperate, induce large stresses and velocity changes close to the transition line. In fact, it is shown that, for a Newtonian fluid and all investigated discontinuities of boundary data, square-root singularities of the stresses will develop at the transition line. Asymptotic expressions for the basal stresses are derived. The explicit forms of these asymptotic expansions depend on the form of the spatial dependence of the sliding law and, furthermore, on the numerical values of the viscosity coefficient. It is, moreover, argued that the stress concentrations are sufficiently pronounced to account for the removal of basal rock especially in regions of high cleavage concentrations, the details again depending upon the sliding coefficients.No mathematical details of the problem solved are presented as attention is focused on the physical processes.

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