Missouri University of Science and Technology Scholars' Mine Materials Science and Engineering Faculty Research & Creative Works Materials Science and Engineering 01 Jul 2018 Repeated Loading Model for Elastic-Plastic Contact of Geomaterial Jian Wang Qimin Li Changwei Yang Caizhi Zhou Missouri University of Science and Technology, [email protected] Follow this and additional works at: https://scholarsmine.mst.edu/matsci_eng_facwork Part of the Materials Science and Engineering Commons Recommended Citation J. Wang et al., "Repeated Loading Model for Elastic-Plastic Contact of Geomaterial," Advances in Mechanical Engineering, vol. 10, no. 7, SAGE Publications Inc., Jul 2018. The definitive version is available at https://doi.org/10.1177/1687814018788778 This Article - Journal is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Materials Science and Engineering Faculty Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. Research Article Advances in Mechanical Engineering 2018, Vol. 10(7) 1–15 Ó The Author(s) 2018 Repeated loading model for elastic– DOI: 10.1177/1687814018788778 plastic contact of geomaterial journals.sagepub.com/home/ade Jian Wang1,2,3 ,QiminLi4, Changwei Yang1 and Caizhi Zhou5 Abstract A new nonlinear hysteretic model with considering the loading, unloading, and reloading processes is developed based on Drucker–Prager yield criterion and finite-element analysis. This model can be used for multiple repeated elastic– plastic normal direction contact problems between two identical spherical geomaterials. After examining the influence of material properties, strain hardening, and loading histories, we found that the hysteretic phenomena (represented by residual displacement and plastic work) become weak after the first cycle, and the subsequent cycles step into elastic shakedown state eventually. A critical number of cycles can be used to estimate the state of ratchetting, plastic shake- down, as well as elastic shakedown. It also found that the subsequent curves will be stiffer than the previous ones, espe- cially when the yield strength is high and ratchetting effect is not strong. This new model can be used for a wide range of geomaterials under different loading levels, and it can also be extended to describe the constitutive behavior of spheres under earthquake as well as aftershocks. Keywords Contact mechanics, elastic–plastic contact, cyclic loading–unloading, Drucker–Prager criterion, elastic shakedown Date received: 28 February 2018; accepted: 20 June 2018 Handling Editor: Shun-Peng Zhu Introduction than other types of models with dampers for the energy dissipation. The contact zone is under locally plastic Interaction between two spheres is the most fundamen- tal problems for discrete geological material simulation, among which repeated loading and unloading is espe- cially important for dynamic action, such as compacted 1School of Civil Engineering, Southwest Jiaotong University, Chengdu, discrete rock mass or soil under seismic force. China 2MOE Key Laboratory of High-Speed Railway Engineering, Southwest Usually, very simple linear models of contact, like Jiaotong University, Chengdu, China linear viscoelastic models, are used in Discrete Element 3National Engineering Laboratory for Technology of Geological Disaster Methods, which account for more than 60% of all Prevention in Land Transportation, Southwest Jiaotong University, cases.1 The main disadvantage of non-hysteretic models Chengdu, China 4 is their artificial assumption of force–displacement School of Water Resources and Environment, China University of Geosciences, Beijing, Beijing, China behavior, which would not guarantee their accuracy to 5Department of Materials Science and Engineering, Missouri University of describe the overall macroscope response of particle sys- Science and Technology, Rolla, MO, USA tems under repeated loading in elastic–plastic regime. With regard to the interaction without cohesion, it Corresponding author: Changwei Yang, School of Civil Engineering, Southwest Jiaotong has been realized that hysteretic models consisted of University, Chengdu 610031, China. springs and sliders would be more physically intuitive Email: [email protected] Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage). 2 Advances in Mechanical Engineering deformation, rather than the artificial damper subsequent load–unload process after the initial loading– behavior.2 unloading cycle would be fully elastic. They proposed an Although many contact models of the hysteretic unloading model of the power law relation with noncon- type have been developed for corresponding materials stant exponent, which was also been adopted by Song separately,3–7 relatively few attentions have been paid and Komvopoulos.11 Jackson et al.13 found high residual to the unloading and reloading responses of elastic– stresses remained after an elastic–plastic hemispherical plastic deformation especially, which is essential for contact being unloaded. Kadin et al.14 presented the cyclic loading simulation between spheres. numerical results of two successive loading and unload- The main difficulties for cyclic loading model with ing cycles on the same model as Etsion et al.12 and hysteretic law result from the definition of stiffness inferred that the secondary plastic flow might reach a coefficients and assessment of the evolution of stress steady state and the successive loading–unloading cycles state during the unloading and reloading processes. become fully elastic from the evolution of equivalent von Literatures concerned include the elastic–plastic inden- Mises stress distribution. Later, Zait et al.15 extended the tation of rigid sphere to compliant base and interference unloading contact model of Etsion et al.12 including stick between two flexible spheres, which could be mutually contact condition. emulated substantially. All the above-mentioned results are mainly fit for Generally speaking, the understanding of unloading metallic material following the von Mises yield criterion process goes through elastic assumption to elastic– (i.e. Maxwell–Huber–Hencky–von Mises theory), which plastic verification. Assuming the unloading behavior are not suitable for more complicated geo materials with purely elastic, Johnson8 proposed the analytical model hydrostatic pressure dependence. Besides, the reloading of unloading for elastic–plastic indentation. Later, the model based on reliable FEA results has not been put model had been modified by Li and Gu9 based on forward, which is crucial for cyclic loading problems. the superposition of Steuermann solutions instead of Although Sadd et al.16 developed a nonlinear hys- the original contact surface curvature. That modifica- teretic contact law including loading, unloading, and tion made it more closely fit with the numerical results reloading stages for dry cohesionless granular media 10 with large plastic deformation. Yan and Li studied the simulation based on photo-elastic experiment, the inter- deformation of a rigid sphere under cyclic indentation. particle contact behavior did not exceed the elastic In their study, an elastic–perfectly plastic half-space was regime substantially. adopted and they found that the deformation during Kostek17 presented a nonlinear mathematical model unloading from a contact beyond the elastic limit was describing the hysteresis of dry contact of rough planar perfectly elastic and the reloading curve was the same as surfaces loaded in the normal direction, which is hard the unloading curve. The material of the half-space used to transform to spherical contact. by them represented a typical steel material, with high Therefore, the main purpose of this article is to develop elastic modulus and yield stress. However, it is still unloading and reloading models for geomaterial under unclear whether the unloading and reloading response cyclic loading and with elastic–plastic deformation. Also, would be different for other materials with low elastic the influence of material properties and loading histories modulus and yield stress. on the evolution of elastic–plastic deformation will be 11 Song and Komvopoulos examined the unloading examined. This article is structured as follows. FEA model behavior of indented elastic–plastic materials as well as and yield criterion are described in sections ‘‘Numerical the evolution of plasticity due to four consecutive inden- model’’ and ‘‘Material constitutive model,’’ respectively. tation cycles by finite-element analysis (FEA) and found Schematic contact model is presented in section ‘‘Contact the occurrence of plastic deformation during unloading model,’’ and loading model of the first cycle, unloading was attributed to the effect of the residual stresses in the model, and reloading model are discussed in sections elastic material surrounded by the plastic regime. Also, ‘‘Loading model,’’‘‘Unloading model,’’ and ‘‘Reloading they deduced