Research Papers

Smooth Yield Surface Constitutive Modeling for Granular Materials

[+] Author and Article Information
Youssef Hammi

Center for Advanced Vehicular Systems,
Box 5405,
Mississippi State, MS 39762-5405
e-mail: yhammi@cavs.msstate.edu

Tonya W. Stone

Department of Mechanical Engineering,
Mississippi State University,
Box 9552,
Mississippi State, MS 39762-9552
e-mail: stone@me.msstate.edu

Bhasker Paliwal

Center for Advanced Vehicular Systems,
Box 5405,
Mississippi State, MS 39762-5405
e-mail: bhasker@cavs.msstate.edu

Mark F. Horstemeyer

Department of Mechanical Engineering,
Mississippi State University,
Box 9552,
Mississippi State, MS 39762-9552
e-mail: mfhorst@me.msstate.edu

Paul G. Allison

Department of Mechanical Engineering,
University of Alabama,
Box 870276,
Tuscaloosa, AL 35487-0276
e-mail: pallison@eng.ua.edu

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received June 13, 2016; final manuscript received October 14, 2016; published online November 14, 2016. Assoc. Editor: Curt Bronkhorst.

J. Eng. Mater. Technol 139(1), 011010 (Nov 14, 2016) (10 pages) Paper No: MATS-16-1181; doi: 10.1115/1.4034987 History: Received June 13, 2016; Revised October 14, 2016

In this paper, the authors present an internal state variable (ISV) cap plasticity model to provide a physical representation of inelastic mechanical behaviors of granular materials under pressure and shear conditions. The formulation is dependent on several factors: nonlinear elasticity, yield limit, stress invariants, plastic flow, and ISV hardening laws to represent various mechanical states. Constitutive equations are established based on a modified Drucker–Prager cap plasticity model to describe the mechanical densification process. To avoid potential numerical difficulties, a transition yield surface function is introduced to smooth the intersection between the failure and cap surfaces for different shapes and octahedral profiles of the shear failure yield surface. The ISV model for the test case of a linear-shaped shear failure surface with Mises octahedral profile is implemented into a finite element code. Numerical simulations using a steel metal powder are presented to demonstrate the capabilities of the ISV cap plasticity model to represent densification of a steel powder during compaction. The formulation is general enough to also apply to other powder metals and geomaterials.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Piccolroaz, A. , Bigoni, D. , and Gajo, A. , 2006, “ An Elastoplastic Framework for Granular Materials Becoming Cohesive Through Mechanical Densification—Part I: Small Strain Formulation,” Eur. J. Mech. A. Solids, 25(2), pp. 334—357. [CrossRef]
Schwer, L. E. , and Murray, Y. D. , 1994, “ A Three-Invariant Smooth Cap Model With Mixed Hardening,” Int. J. Num. Anal. Methods Geomech., 18(10), pp. 657–688. [CrossRef]
Fossum, A. F. , and Brannon, R. M. , 2004, “ The Sandia Geomodel: Theory and User's Guide,” Sandia National Laboratories, Albuquerque, NM, Report No. SAND2004-3226 UC-405.
Foster, C. D. , Regueiro, R. A. , Fossum, A. F. , and Borja, R. I. , 2005, “ Implicit Numerical Integration of a Three-Invariant, Isotropic/Kinematic Hardening Cap Plasticity Model for Geomaterials,” Comput. Methods Appl. Mech. Eng., 194(50–51), pp. 5109–5138. [CrossRef]
La Ragione, L. , Prantil, V. C. , and Sharma, I. A. , 2008, “ Simplified Model for Inelastic Behavior of an Idealized Granular Material,” Int. J. Plast., 24(1), pp. 168–189. [CrossRef]
Chandler, H. W. , Sands, C. M. , Song, J. H. , Withers, P. J. , and McDonald, S. A. , 2008, “ A Plasticity Model for Powder Compaction Processes Incorporating Particle Deformation and Rearrangement,” Int. J. Solids Struct., 45(7–8), pp. 2056–2076. [CrossRef]
Yin, Z.-Y. , and Chang, C. S. , 2009, “ Non-Uniqueness of Critical State Line in Compression and Extension Conditions,” Int. J. Numer. Anal. Methods Geomech., 33(10), pp. 1315–1338. [CrossRef]
Chandler, H. W. , and Sands, C. M. , 2010, “ Including Friction in the Mathematics of Classical Plasticity,” Int. J. Numer. Anal. Methods Geomech., 34(1), pp. 53–72.
Zhu, Q. Z. , Shao, J. F. , and Mainguy, M. , 2010, “ A Micromechanics-Based Elastoplastic Damage Model for Granular Materials at Low Confining Pressure,” Int. J. Plast., 26(4), pp. 586–602. [CrossRef]
Kamrin, K. , 2010, “ Nonlinear Elasto-Plastic Model for Dense Granular Flow,” Int. J. Plast., 26(2), pp. 167–188. [CrossRef]
Motamedi, M. H. , and Foster, C. D. , 2015, “ An Improved Implicit Numerical Integration of a Non-Associated, Three-Invariant Cap Plasticity Model With Mixed Isotropic-Kinematic Hardening for Geomaterials,” Int. J. Numer. Anal. Methods Geomech., 39(17), pp. 1853–1883. [CrossRef]
Sandler, I. S. , 2005, “ Review of the Development of Cap Models for Geomaterials,” Shock Vib., 12(1), pp. 67–71. [CrossRef]
DorMohammadi, H. , and Khoei, A. R. , 2008, “ A Three-Invariant Cap Model with Isotropic-Kinematic Hardening Rule and Associated Plasticity for Granular Materials,” Int. J. Solids Struct., 45(2), pp. 631–656. [CrossRef]
Das, A. , Tengattini, A. , Nguyen, G. D. , Viggiani, G. , Hall, S. A. , and Einav, I. , 2014, “ A Thermomechanical Constitutive Model for Cemented Granular Materials With Quantifiable Internal Variables—Part II: Validation and Localization Analysis,” J. Mech. Phys. Solids., 70, pp. 382–405. [CrossRef]
Tengattini, A. , Das, A. , Nguyen, G. D. , Viggiani, G. , Hall, S. A. , and Einav, I. , 2014, “ A Thermomechanical Constitutive Model for Cemented Granular Materials With Quantifiable Internal Variables—Part I: Theory,” J. Mech. Phys. Solids., 70, pp. 281–296. [CrossRef]
Kohler, R. , and Hofstetter, G. , 2008, “ A Cap Model for Partially Saturated Soils,” Int. J. Numer. Anal. Methods Geomech., 32(8), pp. 981–1004. [CrossRef]
Han, L. H. , Elliott, J. A. , Bentham, A. C. , Mills, A. , Amidon, G. E. , and Hancock, B. C. , 2008, “ A Modified Drucker–Prager Cap Model for Die Compaction Simulation of Pharmaceutical Powders,” Int. J. Solids Struct., 45(10), pp. 3088–3106. [CrossRef]
Sinha, T. , Bharadwaj, R. , Curtis, J. S. , Hancock, B. C. , and Wassgren, C. , 2010, “ Finite Element Analysis of Pharmaceutical Tablet Compaction Using a Density Dependent Material Plasticity Model,” Powder Technol., 202(1–3), pp. 46–54. [CrossRef]
Diarra, H. , Mazel, V. , Boillon, A. , Rehault, L. , Busignies, V. , Bureau, S. , and Tchoreloff, P. , 2012, “ Finite Element Method (FEM) Modeling of the Powder Compaction of Cosmetic Products: Comparison Between Simulated and Experimental Results,” Powder Technol., 224, pp. 233–240. [CrossRef]
Bier, W. , and Hartmann, S. , 2006, “ A Finite Strain Constitutive Model for Metal Powder Compaction Using a Unique and Convex Single Surface Yield Function,” Eur. J. Mech. A. Solids., 25(6), pp. 1009–1030. [CrossRef]
Heisserer, U. , Hartmann, S. , Düster, A. , Bier, W. , Yosibash, Z. , and Rank, E. , 2008, “ P-FEM for Finite Deformation Powder Compaction,” Comput. Methods Appl. Mech. Eng., 197(6–8), pp. 727–740. [CrossRef]
Drucker, D. C. , and Prager, W. , 1952, “ Soil Mechanics and Plastic Analysis or Limit Design,” Q. Appl. Math., 10(2), pp. 157–165.
Drucker, D. C. , 1953, “ Limit Analysis of Two and Three Dimensional Soil Mechanics Problems,” J. Mech. Phys. Solids., 1(4), pp. 217–226. [CrossRef]
Schofield, A. N. , and Wroth, C. P. , 1968, Critical State Soil Mechanics, McGraw Hill, Maidenhead, UK.
Drucker, D. C. , Gibson, R. E. , and Henkel, D. J. , 1957, “ Soil Mechanics and Work-Hardening Theories of Plasticity,” Trans. ASCE., 122, pp. 338–346.
Gurson, A. L. , 1977, “ Continuum Theory of Ductile Rupture by Void Nucleation and Growth—Part I: Yield Criteria and Bow Rules for Porous Ductile Media,” ASME J. Eng. Mater. Technol., 99(1), pp. 2–15. [CrossRef]
Shima, S. , and Oyane, M. , 1976, “ Plasticity Theory for Porous Metals,” Inter. J. Mech. Sci. 18(6), pp. 285–291. [CrossRef]
Fleck, N. A. , Kuhn, L. T. , and McMeeking, R. M. , 1992, “ Yielding of Metal Powder Bonded by Isolated Contacts,” J. Mech. Phys. Solids., 40(5), pp. 1139–1162. [CrossRef]
Fleck, N. A. , 1995, “ On the Cold Compaction of Powders,” J. Mech. Phys. Solids., 43(9), pp. 1409–1431. [CrossRef]
Coube, O. , and Riedel, H. , 2000, “ Numerical Simulation of Metal Powder Die Compaction With Special Consideration of Cracking,” Powder Metall., 43(2), pp. 123–131. [CrossRef]
Trasorras, J. , Krauss, T. M. , and Ferguson, B. L. , 1989, “ Modeling of Powder Compaction Using the Finite Element Method,” Adv. Powder Metall., 1, pp. 85–104.
Swan, C. C. , and Seo, Y. K. , 2000, “ A Smooth, Three-Surface Elasto-Plastic Cap Model: Rate Formulation, Integration Algorithm and Tangent Operators,” University of Iowa, Iowa City, IA.
Desai, C. S. , 1980, “ A General Basis for Yield, Failure, and Potential Functions in Plasticity,” Int. J. Numer. Anal. Methods Geomech., 4(4), pp. 361–375. [CrossRef]
Lade, P. V. , and Kim, M. K. , 1988, “ Single Hardening Constitutive Model for Frictional Materials—Part I: Yield Criterion and Plastic Work Contours,” Comput. Geotech., 6(1), pp. 13–29. [CrossRef]
Fossum, A. F. , and Brannon, R. M. , 2007, “ On a Viscoplastic Model for Rocks With Mechanism-Dependent Characteristic Times,” Acta Geotech., 1(2), pp. 89–106. [CrossRef]
DiMaggio, F. L. , and Sandler, I. S. , 1971 “ Material Models for Granular,” J. Eng. Mech., 97, pp. 935–950.
Gurson, A. L. , and McCabe, T. , 1992, “ Experimental Determination of Yield Functions for Compaction of Blended Metal Powders,” MPIF/APMI World Congress Powder Metall. Part. Mater., Vol. 1, pp. 21–26.
Lode, W. , 1926, “ Versuche über den Einfuss der mittleren Hauptspannung auf das Fliessen der Metalle Eisen Kupfer und Nickel. Z,” Agnew. Phys., 36, pp. 913–939.
Pelessone, D. , 1989, “ A Modified Formulation of the Cap Model,” Gulf Atomics, Technical Report No. GA-C 19579.
Sandler, I. S. , and Rubin, D. , 1979, “ An Algorithm and a Modular Subroutine for the Cap Model,” Int. J. Numer. Anal. Methods Geomech., 3(2), pp. 173–186. [CrossRef]
Pavier, E. , and Doremus, P. , 1999, “ Triaxial Characterisation of Iron Powder Behavior,” Powder Metall., 42(4), pp. 345–352. [CrossRef]
Launay, P. , and Gachon, H. , 1972, “ Strain and Ultimate Strength of Concrete Under Triaxial Stress,” Spec. Publ., 34, pp. 269–282.
Bigoni, D. , and Piccolroaz, A. , 2003, “ A New Yield Function for Geomaterials,” Constitutive Modelling and Analysis of Boundary Value Problems in Geotechnical Engineering, Napoli, Italy, Apr. 22–24, C. Viggiani , ed., Hevelius, Benevento, Italy, pp. 266–281.
ABAQUS, 2008, “ ABAQUS: Theory Manual 6.8,” Dassault Systèmes, Providence, RI.
Majzoobi, G. H. , and Jannesari, S. , 2015 “ Determination of the Constants of Cap Model for Compaction of Three Metal Powders,” Adv. Powder Technol., 26(3), pp. 928–934. [CrossRef]
MPIF, 2010, “ Method for Determination of Density of Compacted or Sintered Powder Metallurgy Products,” Metal Powder Industries Federation, Princeton, NJ, Standard No. 42.
ASTM, 2008, “ Standard Test Method for Splitting Tensile Strength of Intact Rock Core Specimens,” ASTM International, West Conshohocken, PA, Standard D3967-08.
ASTM, 2009, “ Standard Test Methods of Compression Testing of Metallic Materials at Room Temperature,” ASTM International, West Conshohocken, PA, Standard E9-09.
Birks, A. S. , Green, R. E. , and McIntire, P. , 1991, Ultrasonic Testing: Nondestructive Testing Handbook, Vol. 7, American Society for Nondestructive Testing Inc., Columbus, OH, pp. 398–402.
Sinka, I. C. , Cunningham, J. C. , and Zavaliangos, A. , 2001, “ Experimental Characterization and Numerical Simulation of Die Wall Friction in Pharmaceutical Powder Compaction,” PM2TEC 2001 International Conference on Powder Metallurgy & Particulate Materials, New Orleans, LA, Vol. 1, pp. 46–60.
Armstrong, S. , Aesoph, M. D. , and Gurson, A. L. , 1995, “ The Effects of Lubricant Content and Relative Powder Density on the Elastic, Yield and Failure Behavior of a Compacted Metal Powder,” Adv. Powder Metall. Part. Mater., 3, pp. 31–44.


Grahic Jump Location
Fig. 1

The ISV cap model showing the yield surface in the meridional (q, p) plane

Grahic Jump Location
Fig. 2

Hyperbolic-shaped shear failure yield surface smoothly connected to a cap surface in the meridional (q, p) plane

Grahic Jump Location
Fig. 3

Exponent-shaped shear failure yield surface smoothly connected to cap surface in the meridional (q, p) plane

Grahic Jump Location
Fig. 4

Three-dimensional representation of the linear-, hyperbolic- and exponent-shaped shear yield surfaces smoothly connected to the cap surface with a Mises, Gudehus, William–Warnke, and Mohr–Coulomb octahedral profiles

Grahic Jump Location
Fig. 5

Yield surface profile in the octahedral plane

Grahic Jump Location
Fig. 6

Octahedral shear failure envelopes plotted at allowable values of strength ratio (a) with no smoothing and (b) with smoothing parameter, γ=0.995

Grahic Jump Location
Fig. 7

Density–pressure compressibility curves associated with cap hardening for FC-0205 steel powders from isostatic compaction tests

Grahic Jump Location
Fig. 8

Failure stress versus green density for FC-0205 steel cylindrical samples

Grahic Jump Location
Fig. 9

Material cohesion (d) versus green density for FC-0205 steel cylindrical samples

Grahic Jump Location
Fig. 10

Interparticle friction (tan β) versus green density for FC-0205 steel cylindrical samples

Grahic Jump Location
Fig. 11

Comparison of the elastic Young's modulus E as function of the porosity for the FC-0205 steel powder with the AS1000 metal powders (data from Ref. [51])

Grahic Jump Location
Fig. 12

(a) Hoop strain contours on the cylindrical die after simulated compaction of a FC-0205 copper steel powder and (b) strain gauges located on cylindrical die to measure hoop strain during compaction of FC-0205 powder

Grahic Jump Location
Fig. 13

Comparison of measured and finite element hoop strains for a 6.35 cm FC-0205 copper steel cylindrical sample

Grahic Jump Location
Fig. 14

Calibrated cap eccentricity versus green density for FC-0205 steel powders

Grahic Jump Location
Fig. 15

Procedure to build pq-plots from different experiments

Grahic Jump Location
Fig. 16

Isodensity curves in the p-q meridional plane representing the evolution of the ISV cap model with respect to relative density (% theoretical) for FC-0205 steel powders



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