Stress functions and stress-function spaces for 3-dimensional elastostatics and dynamics

2005 ◽  
pp. 492-495 ◽  
Author(s):  
Sitiro Minagawa
Keyword(s):  
Stress Function ◽  
Function Spaces ◽  
3 Dimensional ◽  
Stress Functions

scholarly journals Form-Finding of Shells Containing Both Tension and Compression Using the Airy Stress Function

2020 ◽  
Author(s):  
Masaaki Miki ◽  
Emil Adiels ◽  
William Baker ◽  
Toby Mitchell ◽  
Alexander Sehlstrom ◽  
...  
Keyword(s):  
Stress Function ◽  
Mixed Type ◽  
Airy Stress Function ◽  
Form Finding ◽  
Graphic Statics ◽  
Tension And Compression ◽  
Stress Functions ◽  
Complete Set ◽  
Pure Compression ◽  
Asymptotic Lines

Pure-compression shells have been the central topic in the form-finding of shells. This paper studies tension-compression mixed type shells by utilizing a NURBS-based isogeometric form-finding approach that analyzes Airy stress functions to expand the possible plan geometry. A complete set of smooth version graphic statics tools is provided to support the analyses. The method is validated using examples with known solutions, and a further example demonstrates the possible forms of shells that the proposed method permits. Additionally, a guideline to configure a proper set of boundary conditions is presented through the lens of asymptotic lines of the stress functions.

Analysis of Deep Beams

1951 ◽  
Vol 18 (2) ◽  
pp. 163-172
Author(s):  
H. D. Conway ◽  
L. Chow ◽  
G. W. Morgan
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Stress Distribution ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Normal Stress ◽  
Stress Function ◽  
Beam Theory ◽  
Deep Beam ◽  
Stress Functions

Abstract This paper presents a method of analyzing the stress distribution in a deep beam of finite length by superimposing two stress functions. The first stress function is chosen in the form of a trigonometric series which satisfies all but one of the boundary conditions—that of zero normal stress on the ends of the beam. The principle of least work is then used to obtain a second stress function giving the distribution of normal stress on the ends which is left by the first stress function. By superimposing the two solutions, all the boundary conditions are satisfied. Two particular cases of a given type of loading are solved in this way to investigate the stresses in a deep beam and their deviation from the ordinary beam theory. In addition, an approximate solution by the numerical method of finite difference is worked out for one of the two cases. Results from the two methods are compared and discussed. A method of obtaining an exact solution to the problem is given in an Appendix.

3D Thermal Stresses in Composite Laminates Under Steady-State Through-Thickness Thermal Conduction

2020 ◽  
Vol 12 (06) ◽  
pp. 2050065
Author(s):  
Yan Guo ◽  
Yanan Jiang ◽  
Ji Wang ◽  
Bin Huang
Keyword(s):  
Steady State ◽  
Plane Stress ◽  
Composite Laminates ◽  
Stress Function ◽  
Thermal Conduction ◽  
Thermal Stresses ◽  
Virtual Work ◽  
Ritz Method ◽  
Stress Fields ◽  
Stress Functions

In this study, 3D thermal stresses in composite laminates under steady-state through thickness thermal conduction are investigated by means of a stress function-based approach. One-dimensional thermal conduction is solved for composite laminate and the layerwise temperature distribution is calculated first. The principle of complementary virtual work is employed to develop the governing equations. Their solutions are obtained by using the stress function-based approach, where the stress functions are taken from the Lekhnitskii stress functions in terms of in-plane stress functions and out-of-plane stress functions. With the Rayleigh–Ritz method, the stress fields can be solved by first solving a standard eigenvalue problem. The proposed method is not merely computationally efficient and accurate. The stress fields also strictly satisfy the prescribed boundary conditions validated by the results of finite element method (FEM) results. Finally, some of the results will be given for discussion considering different layup stacking sequences, thermal conductivities and overall temperature differences. From the results, we find that the thermal conductivity greatly affects the stress distributions and peak values of stresses increase linearly for the present model. The proposed method can be used for predicting 3D thermal stresses in composite laminates when subjected to thermal loading.

A Development in the Use of Westergaard Stress Functions for the Solution of Griffith Crack Problems

1984 ◽  
Vol 8 (3) ◽  
pp. 142-145
Author(s):  
Zhang Heng ◽  
D. McCammond ◽  
B. Tabarrok
Keyword(s):  
Stress Function ◽  
Griffith Crack ◽  
Crack Problems ◽  
Stress Functions

Through some examples, it is shown that the use of Westergaard’s stress function in the solution of problems involving cracks in infinite plates may lead to erroneous results. The reasons are investigated in this paper and a procedure for avoiding such erroneous results is recommended.

Notes on problems in hexagonal aeolotropic materials

1951 ◽  
Vol 47 (2) ◽  
pp. 401-409 ◽  
Author(s):  
R. T. Shield
Keyword(s):  
General Solution ◽  
Stress Function ◽  
Harmonic Functions ◽  
Present Note ◽  
Three Dimensional ◽  
Stress Distributions ◽  
Axially Symmetric ◽  
Isotropic Materials ◽  
Stress Functions ◽  
Single Stress

Three-dimensional stress distributions in hexagonal aeolotropic materials have recently been considered by Elliott(1, 2), who obtained a general solution of the elastic equations of equilibrium in terms of two ‘harmonic’ functions, or, in the case of axially symmetric stress distributions, in terms of a single stress function. These stress functions are analogous to the stress functions employed to define stress systems in isotropic materials, and in the present note further problems in hexagonal aeolotropic media are solved, the method in each case being similar to that used for the corresponding problem in isotropic materials. Because of this similarity detailed explanations are unnecessary and only the essential steps in the working are given below.

Stress Function Interface and Boundary Conditions in Anisotropic Materials

1982 ◽  
Vol 49 (4) ◽  
pp. 787-791 ◽  
Author(s):  
E. E. Gdoutos ◽  
M. Kattis
Keyword(s):  
Boundary Conditions ◽  
Stress Function ◽  
Anisotropic Media ◽  
Anisotropic Materials ◽  
Function Approach ◽  
Airy Stress Function ◽  
Stress Functions ◽  
The Media

The stress and displacement continuity conditions for interfaces between two different anisotropic media were formulated in terms of the Airy stress functions of the media. It was shown that such formulation greatly facilitates the solution of the problems of composite anisotropic materials by the Airy stress function approach. Two examples were given to demonstrate the potentiality of the method.

scholarly journals Stress Intensity of Debonding for A Rigid Inclusion Near An Angle Dislocation

2011 ◽  
Vol 5 (1) ◽  
pp. 190-194
Author(s):  
Xianfeng Wang ◽  
Feng Xing ◽  
Norio Hasebe
Keyword(s):  
Stress Intensity ◽  
Rigid Inclusion ◽  
Stress Function ◽  
Hilbert Problem ◽  
Infinite Plate ◽  
Mapping Function ◽  
Function Technique ◽  
Rational Mapping ◽  
Stress Functions ◽  
Point Dislocation

The study of debonding is of importance in providing a good understanding of the bonded interfaces of dissimilar materials. The problem of debonding of an arbitrarily shaped rigid inclusion in an infinite plate with a point dislocation of thin plate bending is investigated in this paper. Herein, the point dislocation is defined with respect to the difference of the plate deflection angle. An analytical solution is obtained by using the complex stress function approach and the rational mapping function technique. In the derivation, the fundamental solutions of the stress boundary value problem are taken as the principal parts of the corresponding stress functions, and through analytical continuation, the problem of obtaining the complementary stress function is reduced to a Riemann-Hilbert problem. Without loss of generality, numerical results are calculated for a square rigid inclusion with a debonding. It is noted that the stress components are singular at the dislocation point, and a stress concentration can be found in the vicinity of the inclusion corner. We also obtain the stress intensity of a debonding in terms of the stress functions. It can be found that when a debonding starts from a corner of the inclusion and extends to another corner progressively, the stress intensity of the debonding increases monotonously; once the debonding extends over the corner points, the value of the stress intensity of the debonding gradually decreases. The relationships between the stress intensity of the debonding and the direction and position of the dislocation are also presented in this paper.

scholarly journals Displacements and Stress Functions of Straight Dislocations and Line Forces in Anisotropic Elasticity: A New Derivation and Its Relation to the Integral Formalism

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1721
Author(s):  
Markus Lazar
Keyword(s):  
Plane Strain ◽  
Stress Function ◽  
Function Fields ◽  
Three Dimensional ◽  
Anisotropic Elasticity ◽  
Green Tensor ◽  
Two Dimensional ◽  
Integral Formalism ◽  
Stress Functions ◽  
Generalized Plane Strain

The displacement and stress function fields of straight dislocations and lines forces are derived based on three-dimensional anisotropic incompatible elasticity. Using the two-dimensional anisotropic Green tensor of generalized plane strain, a Burgers-like formula for straight dislocations and body forces is derived and its relation to the solution of the displacement and stress function fields in the integral formalism is given. Moreover, the stress functions of a point force are calculated and the relation to the potential of a Dirac string is pointed out.

Discontinuous Maxwell–Rankine stress functions for space frames

2018 ◽  
Vol 33 (1) ◽  
pp. 35-47
Author(s):  
Allan McRobie ◽  
Chris Williams
Keyword(s):  
Stress Function ◽  
Three Dimensional ◽  
Shear Forces ◽  
Space Frames ◽  
Stress Functions

This article shows how bending and torsional moments in three-dimensional frames can be represented via a discontinuous Maxwell–Rankine stress function. The associated Rankine reciprocal contains polygonal faces whose areas represent forces. These faces are orthogonal to the member forces (which may include shear forces) and need not be orthogonal to the beams.

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