0
Research Papers

A Coarse Model for the Multiaxial Elastic-Plastic Response of Ductile Porous Materials

[+] Author and Article Information
Andreas Schiffer

Department of Mechanical Engineering,
Khalifa University of Science and Technology,
Abu Dhabi 127788, UAE
e-mail: andreas.schiffer@ku.ac.ae

Panagiotis Zacharopoulos

Department of Aeronautical Engineering,
Imperial College London,
London SW7 2AZ, UK
e-mail: panagiotis.zacharopoulos08@imperial.ac.uk

Dennis Foo

Department of Aeronautical Engineering,
Imperial College London,
London SW7 2AZ, UK
e-mail: dennis.foo15@imperial.ac.uk

Vito L. Tagarielli

Department of Aeronautical Engineering,
Imperial College London,
London SW7 2AZ, UK
e-mail: v.tagarielli@imperial.ac.uk

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received November 10, 2018; final manuscript received April 4, 2019; published online May 13, 2019. Assoc. Editor: A. Amine Benzerga.

J. Appl. Mech 86(8), 081002 (May 13, 2019) (8 pages) Paper No: JAM-18-1638; doi: 10.1115/1.4043439 History: Received November 10, 2018; Accepted April 04, 2019

We propose a modeling strategy to predict the mechanical response of porous solids to imposed multiaxial strain histories. A coarse representation of the microstructure of a porous material is obtained by subdividing a volume element into cubic cells by a regular tessellation; some of these cells are modeled as a plastically incompressible elastic-plastic solid, representing the parent material, while the remaining cells, representing the pores, are treated as a weak and soft compressible solid displaying densification behavior at large compressive strains. The evolution of homogenized deviatoric and hydrostatic stress is explored for different porosities by finite element simulations. The predictions are found in good agreement with previously published numerical studies in which the microstructural geometry was explicitly modeled.

FIGURES IN THIS ARTICLE
<>
Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Aubertin, M., and Li, L., 2004, “A Porosity-Dependent Inelastic Criterion for Engineering Materials,” Int. J. Plast., 20(12), pp. 2179–2208. [CrossRef]
Nahshon, K., and Hutchinson, J. W., 2008, “Modification of the Gurson Model for Shear Failure,” Eur. J. Mech. A/Solids, 27(1), pp. 1–17. [CrossRef]
Cocks, A. C. F., 2001, “Constitutive Modelling of Powder Compaction and Sintering,” Prog. Mater. Sci., 46(3–4), pp. 201–229. [CrossRef]
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,” ASME J. Eng. Mater. Technol., 99(1), pp. 2–15. [CrossRef]
Perrin, G., and Leblond, J.-B., 2000, “Accelerated Void Growth in Porous Ductile Solids Containing Two Populations of Cavities,” Int. J. Plast., 16(1), pp. 91–120. [CrossRef]
Khan, A. S., and Zhang, H., 2000, “Mechanically Alloyed Nanocrystalline Iron and Copper Mixture: Behavior and Constitutive Modeling Over a Wide Range of Strain Rates,” Int. J. Plast., 16(12), pp. 1477–1492. [CrossRef]
Idiart, M. I., Danas, K., and Castañeda, P. P., 2006, “Second-Order Theory for Nonlinear Composites and Application to Isotropic Constituents,” C. R Mecanique, 334(10), pp. 575–581. [CrossRef]
Danas, K., and Aravas, N., 2012, “Numerical Modeling of Elastic-Plastic Porous Materials With Void Shape Effects at Finite Deformations,” Compos. Part B, 43(6), pp. 2544–2559. [CrossRef]
Danas, K., Idiart, M. I., and Castañeda, P. P., 2008, “A Homogenization-Based Constitutive Model for Isotropic Viscoplastic Porous Media,” Int. J. Solids Struct., 45(11–12), pp. 3392–3409. [CrossRef]
Danas, K., Idiart, M. I., and Castañeda, P. P., 2008, “A Homogenization-Based Constitutive Model for Two-Dimensional Viscoplastic Porous Media,” C. R Mecanique, 336(1–2), pp. 79–90. [CrossRef]
Shen, W. Q., Shao, J. F., Dormieux, L., and Kondo, D., 2012, “Approximate Criteria for Ductile Porous Materials Having a Green Type Matrix: Application to Double Porous Media,” Comput. Mater. Sci., 62, pp. 189–194. [CrossRef]
Bilger, N., Auslender, F., Bornert, M., Moulinec, H., and Zaoui, A., 2007, “Bounds and Estimates for the Effective Yield Surface of Porous Media With a Uniform or a Nonuniform Distribution of Voids,” Eur. J. Mech., 26(5), pp. 810–836. [CrossRef]
McElwain, D. L. S., Roberts, A. P., and Wilkins, A. H., 2006, “Yield Criterion of Porous Materials Subjected to Complex Stress States,” Acta Mater., 54(8), pp. 1995–2002. [CrossRef]
McElwain, D. L. S., Roberts, A. P., and Wilkins, A. H. 2006, Yield Functions for Porous Materials With Cubic Symmetry Using Different Definitions of Yield. Adv. Eng. Mater., 8(9), pp. 870–876. [CrossRef]
Öchsner, A., and Mishuris, G., 2009, “Modelling of the Multiaxial Elasto-Plastic Behaviour of Porous Metals With Internal Gas Pressure,” Finite Elem. Anal. Des., 45(2), pp. 104–112. [CrossRef]
Mbiakop, A., Constantinescu, A., and Danas, K., 2015, “On Void Shape Effects of Periodic Elasto-Plastic Materials Subjected to Cyclic Loading,” Eur. J. Mech. A/Solids, 49, pp. 481–499. [CrossRef]
Qidwai, M. A., Entchev, P. B., Lagoudas, D. C., and Degiorgi, V. G., 2001, “Modeling of the Thermomechanical Behavior of Porous Memory Alloys,” Int. J. Solids Struct., 38(48–49), pp. 8653–8671. [CrossRef]
Pastor, J., Francescato, P., Trillat, M., Loute, E., and Rousselier, G., 2004, “Ductile Failure of Cylindrically Porous Materials. Part II: Other Cases of Symmetry,” Eur. J. Mech. A/Solids, 23(2), pp. 191–201. [CrossRef]
Segurado, J., Parteder, E., Plankensteiner, A. F., and Böhm, H. J., 2002, “Micromechanical Studies of the Densification of Porous Molybdenum,” Mater. Sci. Eng. A, 333(1–2), pp. 270–278. [CrossRef]
Gărăjeu, M., Michel, J. C., and Suquet, P., 2000, “A Micromechanical Approach of Damage in Viscoplastic Materials by Evolution in Size, Shape and Distribution of Voids,” Comput. Methods Appl. Mech. Eng., 183(3–4), pp. 223–246. [CrossRef]
Kim, J., Gao, X., and Srivatsan, T. S., 2004, “Modeling of Void Growth in Ductile Solids: Effects of Stress Triaxiality and Initial Porosity,” Eng. Fract. Mech. 71(3), pp. 379–400. [CrossRef]
Siegkas, P., Petrinic, N., and Tagarielli, V. L., 2011, “The Compressive Response of a Titanium Foam at Low and High Strain Rates,” J. Mater. Sci., 46(8), pp. 2741–2747. [CrossRef]
Siegkas, P., Petrinic, N., and Tagarielli, V. L., 2014, “Modelling Stochastic Foam Geometries for FE Simulations Using 3D Voronoi Cells,” Proc. Mater. Sci., 4, pp. 221–226. [CrossRef]
Siegkas, P., Petrinic, N., and Tagarielli, V. L., 2016, “Measurements and Micro-Mechanical Modelling of the Response of Sintered Titanium Foams,” J. Mech. Behav. Biomed. Mater. 57, pp. 365–375. [CrossRef] [PubMed]
Chen, Z., Wang, X., Giuliani, F., and Atkinson, A., 2015, “Microstructural Characteristics and Elastic Modulus of Porous Solids,” Acta Mater., 89, pp. 268–277. [CrossRef]
Panico, M., and Brinson, L. C., 2008, “Computational Modeling of Porous Shape Memory Alloys,” Int. J. Solids Struct., 45(21), pp. 5613–5626. [CrossRef]
Fritzen, F., Böhlke, T., and Schnack, E., 2009, “Periodic Three-Dimensional Mesh Generation for Crystalline Aggregates Based on Voronoi Tessellations,” Comput. Mech., 43(5), pp. 701–713. [CrossRef]
Fritzen, F., Forest, S., Bohlke, T., Kondo, D., and Kanit, T., 2012, “Computational Homogenization of Elastic-Plastic Porous Metals,” Int. J. Plast., 29, pp. 102–119. [CrossRef]
Shan, Z., and Gokhale, A. M., 2001, “Micromechanics of Complex Three-Dimensional Microstructures,” Acta Mater., 49(11), pp. 2001–2015. [CrossRef]
Shen, H., and Brinson, L. C., 2006, “A Numerical Investigation of the Effect of Boundary Conditions and Representative Volume Element Size for Porous Titanium,” J. Mech. Mater. Struct., 1(7), pp. 1179–1204. [CrossRef]
Shen, H., and Brinson, L. C., 2007, “Finite Element Modeling of Porous Titanium,” Int. J. Solids Struct., 44(1), pp. 320–335. [CrossRef]
Shen, H., Oppenheimer, S. M., Dunand, D. C., and Brinson, L. C., 2006, “Numerical Modeling of Pore Size and Distribution in Foamed Titanium,” Mech. Mater., 38(8–10), pp. 933–944. [CrossRef]
Bilger, N., Auslender, F., Bornert, M., Michel, J.-C., Moulinec, H., Suquet, P., and Zaoui, A., 2005, “Effect of a Nonuniform Distribution of Voids on the Plastic Response of Voided Materials: A Computational and Statistical Analysis,” Int. J. Solids Struct., 42(2), pp. 517–538. [CrossRef]
Maîtrejean, G., Terriault, P., and Brailovski, V., 2013, “Density Dependence of the Superelastic Behavior of Porous Shape Memory Alloys: Representative Volume Element and Scaling Relation Approaches,” Comput. Mater. Sci., 77, pp. 93–101. [CrossRef]
Degiorgi, V. G., and Qidwai, M. A., 2002, “A Computational Mesoscale Evaluation of Material Characteristics of Porous Shape Memory Alloys,” Smart Mater. Struct., 11(3), pp. 435–443. [CrossRef]
Simoneau, C., Terriault, P., Rivard, J., and Brailovski, V., 2014, “Modeling of Metallic Foam Morphology Using the Representative Volume Element Approach: Development and Experimental Validation,”, Int. J. Solids Struct., 51(21–22), pp. 3633–3641. [CrossRef]
Zacharopoulos, P., and Tagarielli, V. L., 2017, “Numerical Modelling of the Mechanical Response of a Sintered Titanium Foam,” Int. J. Solids Struct., 113, pp. 241–254. [CrossRef]
Deshpande, V. S., and Fleck, N. A., 2000. Isotropic Constitutive Models for Metallic Foams. J. Mech. Phys. Solids, 48(6–7):1253–1283. [CrossRef]
Dassault Systemes, abaqus Standard User's Manual, v6.12, Simulia Corp, Rhode Island, USA.
Kanit, T., Forest, S., Galliet, I., Mounoury, V., and Jeulin, D., 2003, “Determination of the Size of the Representative Volume Element for Random Composites: Statistical and Numerical Approach,” Int. J. Solids Struct., 40(13–14), pp. 3647–3679. [CrossRef]
Ostoja-Starzewski, M., 2011, “Stochastic Finite Elements: Where Is the Physics,” Theor. Appl. Mech., 38(4), pp. 379–396. [CrossRef]
Kurukuri, S., 2005, “Homogenisation of Damaged Concrete Meso-Structures Using Representative Volume Elements—Implementation and Application to SLang,” Master thesis, Bauhaus-University, Weimar.
Kassem, G. A., 2010, “Micromechanical Material Models for Polymer Composites Through Advanced Numerical Simulation Techniques,” Ph.D. thesis, RWTH Aachen University, Aachen, Germany.
Cheng, L., Danas, K., Constantinescu, A., and Kondo, D., 2017, “A Homogenization Model for Porous Ductile Solids Under Cyclic Loads Comprising a Matrix With Isotropic and Linear Kinematic Hardening,” Int. J. Solids Struct., 121, pp. 174–190. [CrossRef]
Danas, K., and Ponte Castaneda, P., 2012, Influence of the Lode Parameter and the Stress Triaxiality on the Failure of Elasto-Plastic Porous Materials. Int. J. Solids Struct., 49(11–12): 1325–1342. [CrossRef]
Kováčik, J., 1998, “The Tensile Behaviour of Porous Metals Made by Gasar Process,” Acta Mater., 46(15), pp. 5413–5422. [CrossRef]
Kováčik, J., 1999, “Correlation Between Young’s Modulus and Porosity in Porous Materials,” J. Mater. Sci. Lett., 18, pp. 1007–1010. [CrossRef]
Arezoo, S., Tagarielli, V. L., Siviour, C. R., and Petrinic, N., 2013, “Compressive Deformation of Rohacell Foams: Effects of Strain Rate and Temperature,” Int. J. Impact Eng., 51, pp. 50–57. [CrossRef]
Zhang, E., and Wang, B., 2005, “On the Compressive Behaviour of Sintered Porous Coppers With Low to Medium Porosities-Part I: Experimental Study,” Int. J. Mech. Sci., 47, pp. 744–756. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Nonperiodic microstructures of different porosities (1%, 5%, and 20% from left to right)

Grahic Jump Location
Fig. 2

Plastic constitutive response of the fictitious pore material; prescribed history of the equivalent Deshpande–Fleck stress as a function of the corresponding equivalent plastic strain

Grahic Jump Location
Fig. 3

(a) Example of SVEs at increasing NCELL, for the choice NCELL = NFE, (b) sensitivity of compressive and tensile flow stress at a strain of 0.01 to the parameter NCELL, and (c) sensitivity of the macroscopic compressive or tensile stress at the first yield to NCELL. All error bars indicate the range of the outputs.

Grahic Jump Location
Fig. 4

(a) Examples of SVEs at increasing NFE, for the choice NCELL = 125, (b) sensitivity of compressive and tensile flow stress at a strain of 0.01 to the parameter NFE, (c) sensitivity of macroscopic compressive or tensile stress at the first yield to NFE, and (d) sensitivity of the compressive strength to NFE/NCELL

Grahic Jump Location
Fig. 5

(a) Time histories of von Mises stress for loading cases 1–9 and (b) time histories of von Mises stress for loading cases 10–18

Grahic Jump Location
Fig. 6

(a) Time histories of hydrostatic stress for loading cases 1–9 and (b) time histories of hydrostatic stress for loading cases 10–18

Grahic Jump Location
Fig. 7

Yield surface and its evolution for a 30% porous solid. Squares correspond to an equivalent von Mises strain of 0.01 and circles correspond to an equivalent von Mises strain of 0.04.

Grahic Jump Location
Fig. 8

Predicted initial yield surfaces at different porosity. Results by Fritzen et al. [28] are fitted by the modified GTN model [28] and compared with the predictions from the current study. Here, σF denotes the yield stress of the parent material.

Grahic Jump Location
Fig. 9

Compressive Young’s modulus and flow stress at 0.01 strain as a function of relative density

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In