•Discussing the limitations of Werkmeister’s criteria and the simplification adopted in European Standard.•Defining a “representative cycle number” to distinguish the post-compaction and second ...cyclic compression periods.•Proposing more reasonable and versatile new criteria to characterize the shakedown ranges of granular materials.
The Werkmeister’s criteria are widely utilized for characterizing the permanent axial strain behaviour of granular materials. These criteria are based on shakedown theory and are adopted in the European Standards. In this paper, the limitations of Werkmeister’s criteria and the simplification adopted in European Standard are discussed. To overcome these shortcomings, new criteria on characterizing the shakedown ranges of granular materials are proposed. The mathematical expression and physical meanings of the proposed criteria are presented and discussed in detail. It is noteworthy mentioning that, based on the new criteria, the characterizing of shakedown range only assesses the whole development trend of permanent axial strain occurred during the secondary cyclic compression period. Firstly, the performances of the new criteria and Werkmeister’s criteria are compared using the test data from existing literature. It can be concluded that the new criteria are capable of giving quite satisfactory prediction. Furthermore, in this study, several cyclic triaxial tests with step-wisely increasing cyclic deviatoric stress were conducted to verify the new criteria. The test results show a transitional range, which indicates the onset of the disruption of external loading to the original stable structure of granular materials.
The shakedown map is one of the key prediction models used in railway engineering to evaluate rolling contact fatigue (RCF) damages of wheels and rails. The objective of this work is to construct the ...response diagram of RCF and optimize the classical shakedown map based on the rolling-sliding tests. A method based on rolling-sliding tests for constructing the response diagram was presented. Firstly, the traction coefficient (μ) and load factor (P0/ke) were taken as the X-axis coordinate and Y-axis coordinate of the coordinate system of the classical shakedown map, respectively. Then, test parameters (P0 and μ) were designed and obtained using the various X and Y coordinate in four different regions of shakedown map (i.e., different plots of μ versus P0/ke). After that, for various test parameters, the rolling-sliding tests were carried out using a twin-disc testing apparatus under dry condition to investigate the RCF damages of U75V rail steel. The results showed that three types of damage states could be observed after rolling-sliding tests under different test parameters: (I) neither plastic flow of materials nor fatigue cracks, (II) plastic flow, and (III) fatigue cracks with plastic flow. Thus, the response diagram consists of the above three damage regions. Furthermore, with the increase of P0 and μ, all the depth of plastic flow, the length of fatigue cracks and the wear rates were increased. Moreover, materials were work hardened in the damage states of plastic flow and fatigue cracks. The hardness increment was increased with P0 and μ. In addition, the boundary of ratchetting region (i.e., shakedown limit) in the classical shakedown map for partial slip was optimized based on the RCF damages and the new response diagram. The shape of optimized shakedown limit curve was same as that of the classical one, whereas the position of the optimized curve was decreased.
•Three types of damage states could be observed under various P0 and μ conditions.•Depth of plastic flow, length of cracks and wear rates increased with P0 and μ.•Response diagram of RCF consists of three damage regions.•RCF crack was ratchetting effect and plastic flow was plastic shakedown response.•The position of optimized shakedown limit curve was decreased.
Rail materials with different shear yield strengths can exhibit different rolling contact fatigue (RCF) damage response during cyclic contact loading. The objective of this work is to achieve the ...actual shakedown limits in the shakedown map for rail materials with different shear yield strengths through RCF simulation tests of wheel-rail, and to investigate the relationship between the shear yield strength (ke) and the shakedown limit. Rolling-sliding tests were performed to investigate the RCF damage states of three types of rail materials with different shear yield strengths. Then, according to the differences of RCF damage states for each rail material and the method of nonlinear curve fitting, the actual shakedown limits for three types of rail materials were obtained. The actual shakedown limits for the three types of rail materials were lower than the original one in the classical shakedown map. With an increase in the shear yield strength (ke) of the rail material, the actual shakedown limit is reduced. Moreover, a relationship emerged which allowed a simple method to be proposed to obtain the actual shakedown limit based on the shear yield strength. The changing process and differences in the impact of different shear stress levels on the microstructural damage transformation of rail materials from the perspective of contact mechanics have been discussed. The behavior of severe accumulation of residual stresses during cyclic loading and more likely initiation of microcracks are the main reasons why rail materials with a higher shear yield strength (ke) under great contact loads are more prone to developing RCF crack damages. That is, the shakedown limit for the material with a higher shear yield strength (ke) is much lower. In view of the application of the shakedown map in the evaluation and prediction for RCF damage of rail materials with different shear yield strengths, different shakedown limits should be applied to different rail materials when analyzing their likely performance.
•The deformation depth and average crack length of damaged rail material decrease as the ke increases.•With the increase in ke, the hardness increment increases.•As the ke increases, the actual shakedown limit reduces.•A method to obtain the actual shakedown limit based on the ke was proposed.•The higher ke rail material shows a severe piling up of residual stresses during cyclic loading.
A modified numerical procedure for the shakedown analysis of structures under dual cyclic loadings, based on the Abdalla method, is proposed in this paper. Based on the proposed numerical procedure, ...the shakedown analysis of the thick cylindrical vessels with crossholes (TCVCs) under cyclic internal pressure and cyclic thermal loading was carried out. The effects of material parameters (elastic modulus and thermal expansion coefficient) and crosshole radius on the elastic shakedown limit of TCVCs are discussed and, finally, normalized and formularized. Furthermore, the obtained shakedown limit boundary formulation is compared with FEA results and is verified to evaluate the shakedown behavior of TCVCs under cyclic internal pressure and cyclic thermal loading.
When exposed to cyclic quasi-static loading, elastic bodies in contact may develop a favourable condition where slip ceases after a few cycles, an occurrence commonly known as frictional shakedown. ...If the amplitude of the cyclic load is greater than a so-called shakedown limit, shakedown cannot occur. In this review paper, the validity of shakedown theorems in the context of conforming contacts with à la Coulomb friction is first discussed. Then, an optimisation method for determining the shakedown limit of elastic discrete three-dimensional systems is reviewed. Finally, an incremental Gauss–Seidel algorithm, extended to three-dimensional systems, is here illustrated in details for the first time. The algorithm allows us to describe the transient response of normal-tangential coupled systems under a given cyclic loading scenario, and to determine their possible shakedown depending on the initial conditions. An example concerning a discrete conforming contact problem, where either coupling or uncoupling conditions can be imposed, is illustrated.
This article presents an elastoplastic continuum topology optimization method for shakedown under the SIMP‐based framework for the first time, aiming to maximize shakedown load‐carrying capacity ...under variable repeated loading. In contrast to most elastoplastic topology optimizations which have path‐dependent properties, the shakedown topology optimization only needs loading vertices of the loading domain rather than any complete loading history. Based on Melan's lower bound theorem, the gradient‐based topology optimization framework synthesizing the shakedown analysis and sensitivity analysis is developed to maximize the shakedown multiplier. Path‐independent sensitivity related to the specified density interpolation model is derived analytically for the first time in conjunction with the adjoint method. Besides, a primal‐dual algorithm of shakedown analysis is implemented for computing the shakedown load limit of structures and corresponding adjoint variables needed in the sensitivity analysis. In addition to the density interpolation scheme of material Young's module, the yield strength is also interpolated to overcome numerical difficulties. Several numerical examples demonstrate the effectiveness of the proposed method, which indicates the shakedown load‐carrying performance enhancements.
•Initiate the study of the fatigue behavior of the classic double network hydrogels.•Damage accumulates over thousands of cycles for an uncut sample.•Crack extends over thousands of cycles for a cut ...sample.•Determine a fatigue threshold and compare with the Lake-Thomas model.
The discovery of tough hydrogels of many chemical compositions, and their emerging applications in medicine, clothing, and engineering, has raised a fundamental question: How do hydrogels behave under many cycles of stretch? This paper initiates the study of the fatigue behavior of the classic PAMPS/PAAM double network hydrogels discovered by Gong and her co-workers 25. We reproduce the hydrogels, and prepare samples of two types, with or without a crack cut before the test. When an uncut sample is subject to cyclic stretches, internal damage accumulates over thousands of cycles until a steady state is reached. When a cut sample is subject to cyclic stretches, the crack extends cycle by cycle if the amplitude of stretch is above a certain value. A threshold of energy release rate exists, below which the crack remains stationary as the sample is cycled. We find a threshold around 400 J/m2 for hydrogels containing PAAM networks of a low density of crosslinkers, and around 200 J/m2 for hydrogels containing PAAM networks of a high density of crosslinkers. The experimental findings are compared to the Lake-Thomas model adapted to the double-network hydrogels.
This paper establishes a semi-analytical lower-bound dynamic shakedown method to investigate the shakedown of the pavement on saturated subgrade based on dynamic Biot's consolidation theory, ...considering a rolling-sliding contact between vehicle tire and pavement. This analysis has demonstrated that properties of the saturated subgrade generally have negative effects on the shakedown limit of the pavement system under high-speed traffic loadings. In general, the classical single-phased shakedown theorem overestimates the shakedown limit of the pavement on the soft ground of low permeability, particularly under high-speed traffic loadings. The shakedown limit of the pavement on a saturated subgrade gradually reduces with decreasing permeability of the subgrade. The sliding-induced friction significantly affects the shakedown limit and the location of critical shakedown failure. The permeability of the saturated subgrade has considerable effects on the shakedown limit for the case of low tire-pavement friction. The shakedown limit of the pavement on a saturated subgrade and a single-phased subgrade shows a considerable difference at the optimum modulus ratio, at which the critical shakedown failure tends to shift from the saturated subgrade to the pavement layer. The proposed method will be more desirable to evaluate the performance of the pavement in the saturated soft ground.
Shape memory alloys (SMAs) offer interesting perspectives in various fields such as aeronautics, robotics, biomedical sciences, or structural engineering. The distinctive properties of those ...materials stem from a solid/solid phase transformation occurring at a microscopic level. Modeling the rather complex behavior of SMAs is a topic of active research. Lately, SMA models coupling phase-transformation with permanent inelasticity have been proposed to capture degradation effects which are frequently observed experimentally for cyclic loadings — a phenomenon referred to as functional fatigue. In this paper, the classical static and kinematic shakedown of plasticity theory are extended to such material models. Those results give conditions for the energy dissipation to remain bounded, which is beneficial for the fatigue life. Analytical shakedown limits are obtained for a 3-bar truss example and compared with numerical results from step-by-step simulations. We consider the problem of a nitinol stent submitted to cyclic pressure and mixed pressure–bending as an application, showing how the approach presented can be combined with finite-element analysis to study shakedown of complex 3D structures.
•Rigorous extensions of Melan’s and Koiter’s theorems are presented•Easy-to-use safety coefficients are introduced for the shakedown design of structures•Analytical shakedown limits are compared with numerical step-by-step simulations•A shakedown map is obtained for nitinol stents under cyclic pressure–bending
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