Elastic plastic shakedown of 3D periodic heterogeneous media: a direct numerical approach

A direct approach for elastic-plastic analysis and shakedown is presented and its application to a two-dimensional rolling contact problem is demonstrated. The direct approach consists of an operator split technique, which transforms the elastic-plastic problem into a purely elastic problem and a residual problem with prescribed eigenstrains. The eigenstrains are determined using an incremental projection method which is valid for nonproportional loading and both elastic and plastic shakedown. The residual problem is solved analytically and also by using a finite element procedure which can be readily generalized to more difficult problems such as three-dimensional rolling point contact. The direct analysis employs linear-kinematic-hardening plastic behavior and thus either elastic or plastic shakedown is assured, however, the phenomenon of ratchetting which can lead to incremental collapse, cannot be treated within the present framework. Results are compared with full elastic-plastic finite element calculations and a step-by-step numerical scheme for elastic-plastic analysis. Good agreement between the methods is observed. Furthermore, the direct method results in substantial savings in computational effort over full elastic-plastic finite element calculations and is shown to be a straightforward and efficient method for obtaining the steady state (shakedown) solution in the analysis of rolling and/or sliding contact.

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Bounds on plastic strains for elastic plastic structures in plastic shakedown conditions

STRUCTURAL ENGINEERING AND MECHANICS ◽

10.12989/sem.2007.25.1.107 ◽

2007 ◽

Vol 25 (1) ◽

pp. 107-126 ◽

Cited By ~ 6

Author(s):

Francesco Giambanco ◽

Luigi Palizzolo ◽

Alessandra Caffarelli

Keyword(s):

Plastic Strains ◽

Elastic Plastic ◽

Plastic Shakedown

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Coda-wave decorrelation sensitivity kernels in 2-D elastic media: a numerical approach

Geophysical Journal International ◽

10.1093/gji/ggaa357 ◽

2020 ◽

Vol 223 (2) ◽

pp. 934-943

Author(s):

Alejandro Duran ◽

Thomas Planès ◽

Anne Obermann

Keyword(s):

Analytical Solution ◽

Numerical Simulations ◽

Heterogeneous Media ◽

Synthetic Data ◽

Numerical Approach ◽

Coda Waves ◽

Coda Wave ◽

Elastic Media ◽

S Wave ◽

Velocity Changes

SUMMARY Probabilistic sensitivity kernels based on the analytical solution of the diffusion and radiative transfer equations have been used to locate tiny changes detected in late arriving coda waves. These analytical kernels accurately describe the sensitivity of coda waves towards velocity changes located at a large distance from the sensors in the acoustic diffusive regime. They are also valid to describe the acoustic waveform distortions (decorrelations) induced by isotropically scattering perturbations. However, in elastic media, there is no analytical solution that describes the complex propagation of wave energy, including mode conversions, polarizations, etc. Here, we derive sensitivity kernels using numerical simulations of wave propagation in heterogeneous media in the acoustic and elastic regimes. We decompose the wavefield into P- and S-wave components at the perturbation location in order to construct separate P to P, S to S, P to S and S to P scattering sensitivity kernels. This allows us to describe the influence of P- and S-wave scattering perturbations separately. We test our approach using acoustic and elastic numerical simulations where localized scattering perturbations are introduced. We validate the numerical sensitivity kernels by comparing them with analytical kernel predictions and with measurements of coda decorrelations on the synthetic data.

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Lower Bound Methods in Elastic-Plastic Shakedown Analysis

ASME 2010 Pressure Vessels and Piping Conference: Volume 1 ◽

10.1115/pvp2010-26088 ◽

2010 ◽

Author(s):

Wolf Reinhardt ◽

Reza Adibi-Asl

Keyword(s):

Lower Bound ◽

Plastic Material ◽

Shakedown Analysis ◽

Elastic Plastic ◽

Perfectly Plastic ◽

Efficient Calculation ◽

Asme Code ◽

Elastic Shakedown ◽

Plastic Shakedown

Several methods were proposed in recent years that allow the efficient calculation of elastic and elastic-plastic shakedown limits. This paper establishes a uniform framework for such methods that are based on perfectly-plastic material behavour, and demonstrates the connection to Melan’s theorem of elastic shakedown. The paper discusses implications for simplified methods of establishing shakedown, such as those used in the ASME Code. The framework allows a clearer assessment of the limitations of such simplified approaches. Application examples are given.

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On modelling of thermodynamic interaction of particles suspended in two-dimensional medium

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.23.202104.444-460 ◽

2021 ◽

Vol 23 (4) ◽

pp. 444-460

Author(s):

Aleksei O. Syromyasov ◽

Yulia V. Ponkratova ◽

Tatyana V. Menshakova

Keyword(s):

Heterogeneous Media ◽

Domain Size ◽

Analytical Description ◽

Particle Method ◽

Numerical Approach ◽

Multipole Expansion ◽

Multipole Expansions ◽

Approximate Boundary Conditions ◽

Complicated Geometry ◽

Thermodynamic Processes

Analytical description of temperature distribution in a medium with foreign inclusions is difficult due to the complicated geometry of the problem, so asymptotic and numerical methods are usually used to model thermodynamic processes in heterogeneous media. To be convinced in convergence of these methods the authors consider model problem about two identical round particles in infinite planar medium with temperature gradient which is constant at infinity. Authors refine multipole expansion of the solution obtained earlier by continuing it up to higher powers of small parameter, that is nondimensional radius of thermodynamically interacting particles. Numerical approach to the problem using ANSYS software is described; in particular, appropriate choice of approximate boundary conditions is discussed. Authors ascertain that replacement of infinite medium by finite-sized domain is important source of error in FEM. To find domain boundaries in multiple inclusions’ problem the authors develop “fictituous particle” method; according to it the cloud of particles far from the center of the cloud acts approximately as a single equivalent particle of greater size and so may be replaced by it. Basing on particular quantitative data the dependence of domain size that provides acceptable accuracy on thermal conductivities of medium and of particles is explored. Authors establish series of numerical experiments confirming convergence of multipole expansions method and FEM as well; proximity of their results is illustrated, too.

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Three-Dimensional Elastic-Plastic Stress Analysis of Rolling Contact

Journal of Tribology ◽

10.1115/1.1491978 ◽

2002 ◽

Vol 124 (4) ◽

pp. 699-708 ◽

Cited By ~ 59

Author(s):

Yanyao Jiang ◽

Biqiang Xu ◽

Huseyin Sehitoglu

Keyword(s):

Finite Element ◽

Residual Stresses ◽

Stress Analysis ◽

Tangential Force ◽

Three Dimensional ◽

Rolling Contact ◽

Normal Surface ◽

Elastic Plastic ◽

Residual Strains ◽

Plastic Shakedown

Three-dimensional elastic-plastic rolling contact stress analysis was conducted incorporating elastic and plastic shakedown concepts. The Hertzian distribution was assumed for the normal surface contact load over a circular contact area. The tangential forces in both the rolling and lateral directions were considered and were assumed to be proportional to the Hertzian pressure. The elastic and plastic shakedown limits obtained for the three-dimensional contact problem revealed the role of both longitudinal and lateral shear traction on the shakedown results. An advanced cyclic plasticity model was implemented into a finite element code via the material subroutine. Finite element simulations were conducted to study the influences of the tangential surface forces in the two shear directions on residual stresses and residual strains. For all the cases simulated, the p0/k ratio (p0 is the maximum Hertzian pressure and k is the yield stress in shear) was 6.0. The Qx/P ratio, where Qx is the total tangential force on the contact surface in the rolling direction and P is the total normal surface pressure, ranged from 0 to 0.6. The Qy/P ratio (Qy is the total tangential force in the lateral direction) was either zero or 0.25. Residual stresses increase with increasing rolling passes but tend to stabilize. Residual strains also increase but the increase in residual strain per rolling pass (ratchetting rate) decays with rolling cycles. Residual stress levels can be as high as 2k when the Qx/P ratio is 0.6. Local accumulated shear strains can exceed 20 times the yield strain in shear after six rolling passes under extreme conditions. Comparisons of the two-dimensional and three-dimensional rolling contact results were provided to elucidate the differences in residual stresses and ratchetting strain predictions.

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Elastic-Plastic Shakedown Evaluation Using ASME Code Section III and Section VIII Methods

ASME 2010 Pressure Vessels and Piping Conference: Volume 3 ◽

10.1115/pvp2010-25669 ◽

2010 ◽

Author(s):

David P. Molitoris ◽

John V. Gregg ◽

Edward E. Heald ◽

David H. Roarty ◽

Benjamin E. Heald

Keyword(s):

Elastic Analysis ◽

Acceptance Criteria ◽

Elastic Plastic ◽

Division 1 ◽

Plastic Analysis ◽

Asme Code ◽

Section Viii ◽

American Society ◽

Plastic Shakedown ◽

Division 2

Section III, Division 1 and Section VIII, Division 2 of the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel (B&PV) Code provide procedures for demonstrating shakedown using elastic-plastic analysis. While these procedures may be used in place of elastic analysis procedures, they are typically employed after the elastic analysis and simplified elastic-plastic analysis limits have been exceeded. In using the Section III, Division 1 and Section VIII, Division 2 procedures for elastic-plastic shakedown analyses, three concerns are raised. First, the Section III, Division 1 procedure is vague, which can result in inconsistent results between analysts. Second, the acceptance criteria contained in both procedures are vague, which can also result in inconsistent results between analysts. Lastly, differences in the procedures and acceptance criteria can result in demonstration of component elastic-plastic shakedown under Section III, Division 1 but not under Section VIII, Division 2. The authors presume that the ASME Code intends to provide similar design and analysis conclusions, which may not be a correct assumption. To demonstrate these concerns, a nozzle benchmark design subject to a representative thermal and pressure transient was evaluated using the two Code elastic-plastic shakedown procedures. Shakedown was successfully demonstrated using the Section III, Division 1 procedure. However, shakedown could not be demonstrated using the Section VIII, Division 2 procedure. The conflicting results seem to indicate that, for the nozzle design evaluated, the Section VIII, Division 2 procedure is considerably more conservative than the Section III, Division 1 procedure. To further assess the conservative nature of the Section VIII, Division 2 procedure, the nozzle benchmark design was evaluated using the same thermal transient, but without a pressure load. While shakedown was technically not observed using the Section VIII, Division 2 acceptance criteria, engineering judgment concluded that shakedown was demonstrated. Based on the results of all the evaluations, recommendations for modifications to both procedures were presented for consideration.

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The Incremental Strain Growth of Elastic-Plastic Bodies Subjected to High Levels of Cyclic Thermal Loading

Journal of Applied Mechanics ◽

10.1115/1.3167659 ◽

1984 ◽

Vol 51 (3) ◽

pp. 470-474 ◽

Cited By ~ 10

Author(s):

A. R. S. Ponter ◽

A. C. F. Cocks

Keyword(s):

Mechanical Loading ◽

Yield Surface ◽

Thermal Loading ◽

Local Curvature ◽

Cyclic Thermal Loading ◽

Elastic Plastic ◽

Plastic Shakedown ◽

Incremental Strain

A linearized method of analysis proposed in an accompanying paper [1] is used to obtain the ratchet rate for two types of thermal loading problems where parts of the structure experience reversed plastic straining. For structures that can shakedown plasticially it is found that for a given increment of load beyond the plastic shakedown boundary, the rate of ratchet increases with increasing level of thermal loading. When a structure is unable to shakedown plastically it ratchets at low mechanical loading as the result of a localized mechanism that involves some reversed plasticity. It is shown that the ratchet rate in such situations can be substantial but its value is very dependent on the local curvature of the yield and not the accuracy of the yield surface itself.

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