# Analytical and Numerical Investigation of Hardening Behavior of Porous Media

### Abstract

In this study, a comparative analysis is presented between a new proposed analytical model and numerical results for macroscopic behavior of porous media with isotropic hardening in its matrix. The macroscopic behavior of a sufficiently large representative volume element (RVE), with 200 identical spherical voids, was simulated numerically using finite element method and compared with elementary volume element that contains one void. The matrix of the porous material is considered as elasto-plastic with isotropic hardening obeys exponential law for isotropic hardening. A new parameter was added with exponential law for isotropic hardening to represent the new proposed analytical model for macroscopic isotropic porous hardening. The new added parameter *B* depended only on the porosity. The results of the new proposed analytical model were compared with numerical results for different types of cyclic loading. Very good agreements were found between the numerical results and the proposed analytical model.

### Downloads

### References

Balan, T. & Cazacu, O. 2013. Elastic–plastic ductile damage model based on strain-rate plastic potential. Mechanics Research Communications, 54(21-26.

Becker, R. & Needleman, A. 1986. Effect of yield surface curvature on necking and failure in porous plastic solids. Journal of applied mechanics, 53(3), pp 491-499.

Berisha, B., Hora, P., Wahlen, A. & Tong, L. 2010. A combined isotropic-kinematic hardening model for the simulation of warm forming and subsequent loading at room temperature. International journal of plasticity, 26(1), pp 126-140.

Besson, J. 2010. Continuum models of ductile fracture: a review. International Journal of Damage Mechanics, 19(1), pp 3-52.

Besson, J. & Guillemer-Neel, C. 2003. An extension of the Green and Gurson models to kinematic hardening. Mechanics of materials, 35(1-2), pp 1-18.

Besson, J., Moinereau, D. & Steglich, D. 2006. Local approach to fracture: Presses des MINES.

Buryachenko, V. A. 1996. The overall elastopIastic behavior of multiphase materials with isotropic components. Acta Mechanica, 119(1-4), pp 93-117.

Cao, J., Lee, W., Cheng, H. S., Seniw, M., Wang, H.-P. & Chung, K. 2009. Experimental and numerical investigation of combined isotropic-kinematic hardening behavior of sheet metals. International journal of plasticity, 25(5), pp 942-972.

Cardoso, R. P. & Yoon, J. W. 2009. Stress integration method for a nonlinear kinematic/isotropic hardening model and its characterization based on polycrystal plasticity. International journal of plasticity, 25(9), pp 1684-1710.

Carollo, V., Paggi, M. & Rossani, A. 2016. A two parameter elasto-plastic formulation for hardening pressure-dependent materials. Mechanics Research Communications, 77(1-4.

Chaaba, A. 2013. Reliability assessment by analytical calculation of the plastic collapse load of thin pressure vessels with strain hardening and large deformation. Thin-Walled Structures, 62(46-52.

Chaboche, J.-L. 1986. Time-independent constitutive theories for cyclic plasticity. International journal of plasticity, 2(2), pp 149-188.

Chaboche, J.-L. 2008. A review of some plasticity and viscoplasticity constitutive theories. International journal of plasticity, 24(10), pp 1642-1693.

Chaboche, J.-L., Kanouté, P. & Azzouz, F. 2012. Cyclic inelastic constitutive equations and their impact on the fatigue life predictions. International journal of plasticity, 35(44-66.

Chaboche, J. & Jung, O. 1997. Application of a kinematic hardening viscoplasticity model with thresholds to the residual stress relaxation. International journal of plasticity, 13(10), pp 785-807.

Chawla, N. & Deng, X. 2005. Microstructure and mechanical behavior of porous sintered steels. Materials Science and Engineering: A, 390(1-2), pp 98-112.

Chow, C. & Yang, X. 2003. Prediction of forming limit diagram with mixed anisotropic kinematic–isotropic hardening plastic constitutive model based on stress criteria. Journal of materials processing technology, 133(3), pp 304-310.

De Angelis, F. 2012. A comparative analysis of linear and nonlinear kinematic hardening rules in computational elastoplasticity. Technische Mechanik, 32(2-5), pp 164-173.

Doghri, I. & Friebel, C. 2005. Effective elasto-plastic properties of inclusion-reinforced composites. Study of shape, orientation and cyclic response. Mechanics of materials, 37(1), pp 45-68.

Doghri, I. & Ouaar, A. 2003. Homogenization of two-phase elasto-plastic composite materials and structures: study of tangent operators, cyclic plasticity and numerical algorithms. International journal of solids and structures, 40(7), pp 1681-1712.

Dung, N. L. 1992. Three-dimensional void growth in plastic materials. Mechanics Research Communications, 19(3), pp 227-235.

Fritzen, F., Forest, S., Böhlke, T., Kondo, D. & Kanit, T. 2012. Computational homogenization of elasto-plastic porous metals. International journal of plasticity, 29(102-119.

Gurson, A. L. 1977. Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules for porous ductile media. Journal of engineering materials and technology, 99(1), pp 2-15.

Hashemi, E. & Farshi, B. 2011. Cyclic Loading of Beams Based on Kinematic Hardening Models: A Finite Element Approach. International Journal of Modeling and Optimization, 1(3), pp 210.

Jiang, T., Shao, J.-F. & Xu, W. 2011. A micromechanical analysis of elastoplastic behavior of porous materials. Mechanics Research Communications, 38(6), pp 437-442.

Jin, C., Niu, J., He, S. & Fu, C. 2008. Modeling thermal cycling induced micro-damage in aluminum welds: An extension of Gurson void nucleation model. Computational Materials Science, 43(4), pp 1165-1171.

Khan, A. S. & Jackson, K. M. 1999. On the evolution of isotropic and kinematic hardening with finite plastic deformation Part I: compression/tension loading of OFHC copper cylinders. International journal of plasticity, 15(12), pp 1265-1275.

Khdir, Y.-K., Kanit, T., Zaïri, F. & Naït-Abdelaziz, M. 2014. Computational homogenization of plastic porous media with two populations of voids. Materials Science and Engineering: A, 597(324-330.

Khdir, Y.-K., Kanit, T., Zaïri, F. & Naït-Abdelaziz, M. 2015. A computational homogenization of random porous media: Effect of void shape and void content on the overall yield surface. European Journal of Mechanics-A/Solids, 49(137-145.

Khdir, Y., Kanit, T., Zaïri, F. & Nait-Abdelaziz, M. 2013. Computational homogenization of elastic–plastic composites. International journal of solids and structures, 50(18), pp 2829-2835.

Khoei, A. & Azami, A. 2005. A single cone-cap plasticity with an isotropic hardening rule for powder materials. International Journal of Mechanical Sciences, 47(1), pp 94-109.

Kossa, A. & Szabó, L. 2009. Exact integration of the von Mises elastoplasticity model with combined linear isotropic-kinematic hardening. International journal of plasticity, 25(6), pp 1083-1106.

Kuna, M. & Sun, D. 1996. Three-dimensional cell model analyses of void growth in ductile materials. International Journal of fracture, 81(3), pp 235-258.

Lademo, O.-G., Hopperstad, O. S. & Langseth, M. 1999. An evaluation of yield criteria and flow rules for aluminium alloys. International journal of plasticity, 15(2), pp 191-208.

Leblond, J.-B., Perrin, G. & Devaux, J. 1995. An improved Gurson-type model for hardenable ductile metals. European journal of mechanics. A. Solids, 14(4), pp 499-527.

Leu, S.-Y. & Li, R.-S. 2012. Exact solutions of sequential limit analysis of pressurized cylinders with combined hardening based on a generalized Hölder inequality: Formulation and validation. International Journal of Mechanical Sciences, 64(1), pp 47-53.

Mahmoudi, A., Pezeshki-Najafabadi, S. & Badnava, H. 2011. Parameter determination of Chaboche kinematic hardening model using a multi objective Genetic Algorithm. Computational Materials Science, 50(3), pp 1114-1122.

Mear, M. & Hutchinson, J. 1985. Influence of yield surface curvature on flow localization in dilatant plasticity. Mechanics of materials, 4(3-4), pp 395-407.

Mosler, J. 2010. Variationally consistent modeling of finite strain plasticity theory with non-linear kinematic hardening. Computer Methods in Applied Mechanics and Engineering, 199(45-48), pp 2753-2764.

Rezaiee-Pajand, M. & Sinaie, S. 2009. On the calibration of the Chaboche hardening model and a modified hardening rule for uniaxial ratcheting prediction. International journal of solids and structures, 46(16), pp 3009-3017.

Ristinmaa, M. 1995. Cyclic plasticity model using one yield surface only. International journal of plasticity, 11(2), pp 163-181.

Ristinmaa, M. 1997. Void growth in cyclic loaded porous plastic solid. Mechanics of materials, 26(4), pp 227-245.

Rousselier, G., Barlat, F. & Yoon, J. W. 2010. A novel approach for anisotropic hardening modeling. Part II: Anisotropic hardening in proportional and non-proportional loadings, application to initially isotropic material. International journal of plasticity, 26(7), pp 1029-1049.

Samrout, H., El Abdi, R. & Chaboche, J. 1997. Model for 28CrMoV5-8 steel undergoing thermomechanical cyclic loadings. International journal of solids and structures, 34(35-36), pp 4547-4556.

Seifert, T. & Schmidt, I. 2008. Line‐search methods in general return mapping algorithms with application to porous plasticity. International journal for numerical methods in engineering, 73(10), pp 1468-1495.

Seifert, T. & Schmidt, I. 2009. Plastic yielding in cyclically loaded porous materials. International journal of plasticity, 25(12), pp 2435-2453.

Siad, L., Liu, W. K. & Benabbes, A. 2009. Explicit numerical study of softening in porous ductile solids. Mechanics Research Communications, 36(2), pp 236-245.

Simo, J. C. & Hughes, T. J. 2006. Computational inelasticity: Springer Science & Business Media.

Sorić, J., Tonković, Z. & Krätzig, W. 2000. A new formulation of numerical algorithms for modelling of elastoplastic cyclic response of shell-like structures. Computers & Structures, 78(1-3), pp 161-168.

Steglich, D., Pirondi, A., Bonora, N. & Brocks, W. 2005. Micromechanical modelling of cyclic plasticity incorporating damage. International journal of solids and structures, 42(2), pp 337-351.

Taherizadeh, A., Ghaei, A., Green, D. E. & Altenhof, W. J. 2009. Finite element simulation of springback for a channel draw process with drawbead using different hardening models. International Journal of Mechanical Sciences, 51(4), pp 314-325.

Tvergaard, V. 1982. On localization in ductile materials containing spherical voids. International Journal of fracture, 18(4), pp 237-252.

Tvergaard, V. & Needleman, A. 1984. Analysis of the cup-cone fracture in a round tensile bar. Acta metallurgica, 32(1), pp 157-169.

Verleene, A., Dubar, L., Dubois, A., Dubar, M. & Oudin, J. 2002. Hardening behaviour law versus rigid perfectly plastic law: application to a cold forging tool steel. International journal of plasticity, 18(8), pp 997-1011.

Verma, R. K., Kuwabara, T., Chung, K. & Haldar, A. 2011. Experimental evaluation and constitutive modeling of non-proportional deformation for asymmetric steels. International journal of plasticity, 27(1), pp 82-101.

Voyiadjis, G. Z. & Song, C. R. 2002. Multi-scale non-local approach for geomaterials. Mechanics Research Communications, 29(2-3), pp 121-129.

Zhang, Z., Zhuang, Z., Gao, Y., Liu, Z. & Nie, J. 2011. Cyclic plastic behavior analysis based on the micromorphic mixed hardening plasticity model. Computational Materials Science, 50(3), pp 1136-1144.

Zhao, K. & Lee, J. 2001. Material properties of aluminum alloy for accurate draw-bend simulation. Journal of engineering materials and technology, 123(3), pp 287-292.

*Polytechnic Journal*,

*9*(2), 1-10. https://doi.org/10.25156/ptj.v9n2y2019.pp1-10

Copyright (c) 2019 Younis Kh. Khdir

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Authors who publish with this journal agree to the following terms:

1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License [CC BY-NC-ND 4.0] that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.

2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.

3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).