Stepwise Loading of Half-Spaces in Elliptical Contact

1996 ◽  
Vol 63 (3) ◽  
pp. 766-773 ◽  
Author(s):  
J. Ja¨ger
Keyword(s):  
Stress Distribution ◽  
Analytical Solution ◽  
Plasticity Theory ◽  
Oblique Loading ◽  
Tangential Forces ◽  
Elliptical Contact ◽  
Numerical Discretization ◽  
Force Displacement ◽  
Discretized Problem ◽  
Stepwise Loading

A new analytical solution for Hertzian surfaces in contact under stepwise oblique loading is presented in this paper. It is written as a superposition of so-called Catta-neo-Mindlin functions, which represent the tangential stress distribution for constant normal and monotonically increasing tangential forces. The size of the stick zone is determined by cones in the force space, which are also known as yield cones. It has a simpler form than the differential force-displacement relations of plasticity theory and the numerical discretization of the influence integrals. A short computer algorithm is proposed and compared with a numerical solution of the discretized problem.

scholarly journals A Non-Singular Analytical Technique for Reinforced Non-Circular Holes in Orthotropic Laminate

2020 ◽  
Vol 25 (1) ◽  
pp. 92-105
Author(s):  
Pradeep Mohan ◽  
R. Ramesh Kumar
Keyword(s):  
Stress Distribution ◽  
Analytical Solution ◽  
Power Series ◽  
Orthotropic Shell ◽  
Infinite Plate ◽  
Stress Singularity ◽  
Power Series Solution ◽  
Element Analysis ◽  
Full Field ◽  
Analytical Technique

AbstractThe intricacy in Lekhnitskii’s available single power series solution for stress distribution around hole edge for both circular and noncircular holes represented by a hole shape parameter ε is decoupled by introducing a new technique. Unknown coefficients in the power series in ε are solved by an iterative technique. Full field stress distribution is obtained by following an available method on Fourier solution. The present analytical solution for reinforced square hole in an orthotropic infinite plate is derived by completely eliminating stress singularity that depends on the concept of stress ratio. The region of validity of the present analytical solution on reinforcement area is arrived at based on a comparison with the finite element analysis. The present study will also be useful for deriving analytical solution for orthotropic shell with reinforced noncircular holes.

An Anisotropic Generalization of the Bridgman Analysis of Tensile Necking

1983 ◽  
Vol 105 (4) ◽  
pp. 264-267 ◽  
Author(s):  
M. A. Eisenberg ◽  
C. F. Yen
Keyword(s):  
Stress Distribution ◽  
Analytical Solution ◽  
Plastic Flow ◽  
Approximate Analytical Solution ◽  
Parameter Family ◽  
Isotropic Case ◽  
Anisotropic Plastic

Tensile necking in anisotropic bars is analyzed in the spirit of P. W. Bridgman’s treatment of the isotropic case. Anisotropic plastic flow causes an initially axisymmetric bar to develop an elliptical neck. Using physical approximations analogous to Bridgman’s, an approximate analytical solution for the stress distribution is obtained. The solution is shown to be asymptotically correct in two important limiting cases: (a) the fully developed anisotropic neck, and (b) the isotropic limit. In the latter case it is shown that the solution is a member of a one-parameter family of solutions, which includes the Bridgman and the Davidenkov and Spiridonova solutions.

Analytical Solution of a Borehole Problem Using Strain Gradient Plasticity

2002 ◽  
Vol 124 (3) ◽  
pp. 365-370 ◽  
Author(s):  
X.-L. Gao
Keyword(s):  
Analytical Solution ◽  
Strain Gradient ◽  
Plastic Region ◽  
Yield Condition ◽  
Plasticity Theory ◽  
Strain Gradient Plasticity ◽  
Spatial Gradient ◽  
Gradient Plasticity ◽  
Present Solution ◽  
Classical Plasticity

An analytical solution is presented for the borehole problem of an elasto-plastic plane strain body containing a traction-free circular hole and subjected to uniform far field stress. A strain gradient plasticity theory is used to describe the constitutive behavior of the material undergoing plastic deformations, whereas the generalized Hooke’s law is invoked to represent the material response in the elastic region. This gradient plasticity theory introduces a higher-order spatial gradient of the effective plastic strain into the yield condition to account for the nonlocal interactions among material points, while leaving other relations in classical plasticity unaltered. The solution gives explicit expressions for the stress, strain, and displacement components. The hole radius enters these expressions not only in nondimensional forms but also with its own dimensional identity, unlike classical plasticity-based solutions. As a result, the current solution can capture the size effect in a quantitative manner. The classical plasticity-based solution of the borehole problem is obtained as a special case of the present solution. Numerical results for the plastic region radius and the stress concentration factor are provided to illustrate the application and significance of the newly derived solution.

Residual Stress in an Autofrettaged Tube Taking Bauschinger Effect as a Function of the Prior Plastic Strain

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Xiaoping Huang ◽  
Torgeir Moan
Keyword(s):  
Residual Stress ◽  
Stress Distribution ◽  
Analytical Solution ◽  
Plastic Strain ◽  
Young’S Modulus ◽  
Bauschinger Effect ◽  
Young's Modulus ◽  
Residual Stress Distribution ◽  
Present Solution ◽  
Stress Dependent

Autofrettage is a practical method for increasing the elastic carrying capacity and the fatigue life of thick-walled cylinders such as cannon and high-pressure tubular reactor. Many analytical and numerical solutions for determining the residual stress distribution in an autofrettaged tube have been reported. It is still difficult to model the Bauchinger effect, which is dependent on the prior plasticity in an analytical solution. The reduced Young’s modulus during unloading affects residual stress distribution. However, until now this effect has not been considered in any analytical model. In this paper, an autofrettage analytical solution considering Young’s modulus and the reverse yield stress dependent on the prior plasticity, based on the actual tensile-compressive curve of the material and the von Mises yield criterion, has been proposed. New model incorporates the Bauschinger effect factor and the unloading modulus variation as a function of prior plastic strain, and hence of the radius. Thereafter it assumes a fixed nonlinear unloading profile. The comparison of predicted residual stress distribution by the present solution with that of fixed unloading curve model, and test results shows that the present solution gives accurate prediction of residual stress distribution of an autofrettaged tube. This analytical procedure for the cylinder permits an excellent representation of various pressure vessel steels.

Elliptical Contact Area Between Elastic Bodies Bounded by High Order Surfaces

2005 ◽  
Author(s):  
Emanuel N. Diaconescu
Keyword(s):  
Analytical Solution ◽  
Pressure Distribution ◽  
Fourth Order ◽  
Contact Area ◽  
High Order ◽  
Normal Displacement ◽  
Hertz Theory ◽  
Elastic Bodies ◽  
Surface Normal ◽  
Elliptical Contact

Hertz theory fails when contacting surfaces are expressed by a sum of even polynomials of higher powers than two. An alternative analytical solution implies the knowledge of contact area. In the case of elliptical domains, there are some published proposals for the correlation between pressure distribution and surface normal displacement. This paper identifies the class of high order surfaces which lead to elliptical contact domains and solves a contact between fourth order surfaces.

scholarly journals Analytical solution and numerical verification for the pressure-relief method of a circular tunnel

2018 ◽  
Vol 38 (3) ◽  
pp. 338-351
Author(s):  
Shunchuan Wu ◽  
Miaofei Xu ◽  
Yongtao Gao ◽  
Shihuai Zhang ◽  
Fan Chen
Keyword(s):  
Stress Distribution ◽  
Analytical Solution ◽  
High Stress ◽  
Mapping Function ◽  
Mapping Method ◽  
Numerical Verification ◽  
Fast Method ◽  
Circular Tunnel ◽  
Variable Theory ◽  
Releasing Effect

This paper presents an elastic analytical solution to a circular tunnel with releasing slots at high stress areas near the hole by using a conformal mapping method and the complex variable theory. Compared to the original stress distribution around the circular hole, the releasing effect on elastic stresses is evaluated. After grooving slots, low stress area is generated where the high stress concentration is located. This is agreeable with what was predicted by the finite difference FLAC2D. Besides, displacements are obtained along the periphery of the released hole and are in accordance with those of FLAC2D. In addition to the intersection of the mapping contour, the influences of the sampling points distribution, series number in mapping function, and slot shape are discussed. It is inevitable that the mapping accuracies for the slot and the circle cannot be satisfied at the same time The mapping effect on the circle has to be considered primarily since the stress distribution around the circle is much more significant than the tunnel stability. The analytical solution can be available and fast method of estimating the releasing effect of the application on the tunnel without rock parameters.

The use of ‘C’-shaped specimens to investigate the interlaminar fracture of woven composite laminates

1995 ◽  
Vol 30 (2) ◽  
pp. 129-134 ◽  
Author(s):  
G Hognestad ◽  
D A Hills
Keyword(s):  
Stress Distribution ◽  
Analytical Solution ◽  
Composite Laminates ◽  
Epoxy Composite ◽  
Woven Composite ◽  
Interlaminar Fracture ◽  
Profound Influence ◽  
Woven Composite Laminates

‘C’-shaped test specimens have been used to find the interlaminar strength of woven carbon/epoxy composite laminates in tension and in combined tension with minor transverse tractions. An analytical solution for the stress distribution within the “C” specimen, of arbitrary lay-up, is developed to permit a rigorous interpretation of the results, which confirm that the presence of relatively minor through-thickness stresses will have a profound influence on the strength of the component.

A Theoretical Model for the Normal Contact Force of Two Elastoplastic Ellipsoidal Bodies

2020 ◽  
Vol 88 (3) ◽  
Author(s):  
Verena Becker ◽  
Marc Kamlah
Keyword(s):  
Theoretical Model ◽  
Contact Force ◽  
General Description ◽  
Element Analysis ◽  
Normal Contact ◽  
Normal Contact Force ◽  
Static Contact ◽  
Elliptical Contact ◽  
Individual Grains ◽  
Force Displacement

Abstract To model the mechanical behavior of granular materials, a reliable description of the material properties is indispensable. Individual grains are usually not perfectly spherical. In batteries, for instance, lithium nickel manganese cobalt oxide (NMC) is a frequently used material, consisting out of particles with possibly ellipsoidal like shapes. As particles may plastically deform under increasing stresses, the paper presents a theoretical model for the normal contact force of elastoplastic ellipsoidal bodies for the use in the context of mechanical discrete element method (DEM). The model can be considered as extension of the elastic, elastic-plastic, fully plastic Thornton model by using a more general description to incorporate elliptical contact areas. The focus is on a normal contact force description as continuous function of time for all regimes, elastic, elastoplastic, and fully plastic loading, as well as unloading from elastoplastic loading, while the evolution of the plastic contact area is not considered here. All underlying formulae to describe the force-displacement relationship for the static contact problem are derived, partly based on finite element analysis (FEA). To verify the new model, FEAs are performed and their results compared with the model predictions.

Comparison of stress distribution in a three unit cement retained implant supported prosthesis with different implant abutment connections: A photoelastic analysis

2020 ◽  
Vol 12 (1) ◽  
pp. 35-43
Author(s):  
Dr Poojya R ◽  
Dr Darakshan Nazir ◽  
Dr Shruthi C S
Keyword(s):  
Stress Distribution ◽  
Axial Loading ◽  
The Body ◽  
Partial Denture ◽  
Fixed Partial Denture ◽  
Group I ◽  
Significant Difference ◽  
Implant Abutment ◽  
Oblique Loading ◽  
Group Ii

Aim: With the emphasis on success of implant supported prosthesis, and health of the surrounding tissues that are related to accuracy, and fit between the implant components, stability at implant abutment interface is of prime importance. The aim of this study is to evaluate and compare the stress distribution in three unit cement retained implant supported fixed partial denture with different implant abutment connections through photo elasticity. Materials and methods: Two photo elastic resin models were fabricated of standard dimensions (44mmx22mmx10mm). Group I sample: Three unit cement retained implant supported fixed partial denture with Internal implant abutment connection (Internal hexagonal connection) (Paltop Advanced, Keystone Dental Company, US)Group II sample: Three unit cement retained implant supported fixed partial denture with conical Morse taper connection (1.5 degree Morse taper) (Paltop Conical Active, Keystone Dental Company, US). Three unit cement retained implant supported fixed partial denture simulated missing mandibular first molar. Axial and oblique loads of 100N were placed on each implant and pontic area for 10 sec. Ten tests were done for each group. The stress values around the implants were derived from the colored fringe patterns obtained through polariscope, which were photographed after load applications from which values were derived. Results: Under axial loading, there was statistically significant difference between internal hexagonal connection and Morse taper connection in three unit implant supported prosthesis. Stresses were more in Group II sample with Morse taper connection. Under oblique loading, there was no statistically significant difference between Group I and Group II samples. Conclusion: Within the limitations of this in vitro study, it can be concluded that Internal hexagonal connection showed less stresses as compared to Morse taper connection in a three unit cement retained implant supported prosthesis. Stresses were concentrated more in apical area under axial loading; while under oblique loading stresses were seen on the side of application of force on the body of the implant and on the apical region. However, stresses were uniformly distributed in both groups I and group II samples. In both groups stresses under oblique loading were more than axial loading, but that was not statistically significant.

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