Three-dimensional elasticity problems for the prismatic bar

2006 ◽  
Vol 462 (2070) ◽  
pp. 1877-1896 ◽  
Author(s):  
J.R Barber
Keyword(s):  
Three Dimensional ◽  
Elastic Problem ◽  
Displacement Fields ◽  
Recursive Procedure ◽  
Axial Coordinate ◽  
Linear Elastic ◽  
Stress And Displacement Fields ◽  
Elasticity Problems ◽  
Finite Power ◽  
Three Dimensional Elasticity

A general solution is given to the three-dimensional linear elastic problem of a prismatic bar subjected to arbitrary tractions on its lateral surfaces, subject only to the restriction that they can be expanded as finite power series in the axial coordinate z . The solution is obtained by repeated differentiation of the tractions with respect to z , establishing a set of sub-problems . A recursive procedure is then developed for generating the solution to from that for . This procedure involves three steps: integration of the stress and displacement fields with respect to z , using an appropriate Papkovich–Neuber (P–N) representation; solution of two-dimensional in-plane and antiplane corrective problems for the tractions in that are independent of z ; and expression of these corrective solutions in P–N form. The method is illustrated by an example.

scholarly journals Stress and Strain Analysis of Functionally Graded Rectangular Plate with Exponentially Varying Properties

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Amin Hadi ◽  
Abbas Rastgoo ◽  
A. R. Daneshmehr ◽  
Farshad Ehsani
Keyword(s):  
Rectangular Plate ◽  
Three Dimensional ◽  
Strain Analysis ◽  
Functionally Graded ◽  
Stress And Strain ◽  
Governing Equations ◽  
Displacement Fields ◽  
Stress And Displacement Fields ◽  
Graded Material ◽  
Three Dimensional Elasticity

The bending of rectangular plate made of functionally graded material (FGM) is investigated by using three-dimensional elasticity theory. The governing equations obtained here are solved with static analysis considering the types of plates, which properties varying exponentially along direction. The value of Poisson’s ratio has been taken as a constant. The influence of different functionally graded variation on the stress and displacement fields was studied through a numerical example. The exact solution shows that the graded material properties have significant effects on the mechanical behavior of the plate.

The Uniform Flexure of an Incomplete Tore

1949 ◽  
Vol 2 (4) ◽  
pp. 469
Author(s):  
W Freiberger ◽  
RCT Smith
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Harmonic Functions ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Derivatives Of ◽  
Special Case

In this paper we discuss the flexure of an incomplete tore in the plane of its circular centre-line. We reduce the problem to the determination of two harmonic functions, subject to boundary conditions on the surface of the tore which involve the first two derivatives of the functions. We point out the relation of this solution to the general solution of three-dimensional elasticity problems. The special case of a narrow rectangular cross-section is solved exactly in Appendix II.

Analytical and 3D Finite Element Study of the Deflection of an Elastic Cantilever Bilayer Plate

2010 ◽  
Vol 78 (1) ◽  
Author(s):  
M. Chekchaki ◽  
V. Lazarus ◽  
J. Frelat
Keyword(s):  
Thin Films ◽  
Finite Element ◽  
Three Dimensional ◽  
Analytical Formula ◽  
Mechanical Stresses ◽  
Linear Elastic ◽  
Wide Range ◽  
Finite Element Calculations ◽  
Analytical Formulas ◽  
Three Dimensional Elasticity

The mechanical system considered is a bilayer cantilever plate. The substrate and the film are linear elastic. The film is subjected to isotropic uniform prestresses due for instance to volume variation associated with cooling, heating, or drying. This loading yields deflection of the plate. We recall Stoney’s analytical formula linking the total mechanical stresses to this deflection. We also derive a relationship between the prestresses and the deflection. We relax Stoney’s assumption of very thin films. The analytical formulas are derived by assuming that the stress and curvature states are uniform and biaxial. To quantify the validity of these assumptions, finite element calculations of the three-dimensional elasticity problem are performed for a wide range of plate geometries, Young’s and Poisson’s moduli. One purpose is to help any user of the formulas to estimate their accuracy. In particular, we show that for very thin films, both formulas written either on the total mechanical stresses or on the prestresses, are equivalent and accurate. The error associated with the misfit between our theorical study and numerical results are also presented. For thicker films, the observed deflection is satisfactorily reproduced by the expression involving the prestresses and not the total mechanical stresses.

scholarly journals Membrane analogy for multi-material bars under torsion

2019 ◽  
Vol 475 (2225) ◽  
pp. 20190124 ◽  
Author(s):  
Laura Galuppi ◽  
Gianni Royer-Carfagni
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Cross Sections ◽  
Three Dimensional ◽  
Element Model ◽  
Elastic Problem ◽  
Two Dimensional ◽  
Torsion Problem ◽  
Linear Elastic ◽  
Physical Apparatus

Prandtl's membrane analogy for the torsion problem of prismatic homogeneous bars is extended to multi-material cross sections. The linear elastic problem is governed by the same equations describing the deformation of an inflated membrane, differently tensioned in regions that correspond to the domains hosting different materials in the bar cross section, in a way proportional to the inverse of the material shear modulus. Multi-connected cross sections correspond to materials with vanishing stiffness inside the holes, implying infinite tension in the corresponding portions of the membrane. To define the interface constrains that allow to apply such a state of prestress to the membrane, a physical apparatus is proposed, which can be numerically modelled with a two-dimensional mesh implementable in commercial finite-element model codes. This approach presents noteworthy advantages with respect to the three-dimensional modelling of the twisted bar.

The method of caustics for the determination of normal loads acting on surfaces of elastic bodies

1980 ◽  
Vol 15 (1) ◽  
pp. 37-41 ◽  
Author(s):  
P S Theocaris ◽  
N I Ioakimidis
Keyword(s):  
Three Dimensional ◽  
Contact Problems ◽  
Two Dimensional ◽  
Concentrated Force ◽  
Elastic Bodies ◽  
Plate And Shell ◽  
Elasticity Problems ◽  
Three Dimensional Elasticity ◽  
Quantities Of Interest

The optical method of caustics constitutes an efficient experimental technique for the determination of quantities of interest in elasticity problems. Up to now, this method has been applied only to two-dimensional elasticity problems (including plate and shell problems). In this paper, the method of caustics is extended to the case of three-dimensional elasticity problems. The particular problems of a concentrated force and a uniformly distributed loading acting normally on a half-space (on a circular region) are treated in detail. Experimentally obtained caustics for the first of these problems were seen to be in satisfactory agreement with the corresponding theoretical forms. The treatment of various, more complicated, three-dimensional elasticity problems, including contact problems, by the method of caustics is also possible.

Three-dimensional deformations in a single-lap joint

1994 ◽  
Vol 29 (2) ◽  
pp. 137-145 ◽  
Author(s):  
M Y Tsai ◽  
J Morton
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Stress Distributions ◽  
Lap Joint ◽  
Displacement Fields ◽  
Element Analysis ◽  
Linear Elastic ◽  
Non Linear ◽  
Linear Effects ◽  
Peel Stress

The three-dimensional nature of the state of deformation in a single-lap test specimen is investigated in a linear elastic finite element analysis in which the boundary conditions account for the geometrically non-linear effects. The validity of the model is demonstrated by comparing the resulting displacement fields with those obtained from a moiré inteferometry experiment. The three-dimensional adherend and adhesive stress distributions are calculated and compared with those from a two-dimensional non-linear numerical analysis, Goland and Reissner's solution, and experimental measurements. The nature of the three-dimensional mechanics is described and discussed in detail. It is shown that three-dimensional regions exists in the specimen, where the adherend and adhesive stress distributions in the overlap near (and especially on) the free surface are quite different from those occurring in the interior. It is also shown that the adhesive peel stress is extremely sensitive to this three-dimensional effect, but the adhesive shear is not. It is also observed that the maximum value of the peel stress occurs at the end of the overlap in the central two-dimensional core region, rather than at the corners where the three-dimensional effects are found. The extent of three-dimensional regions is also quantified.

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