dissipative materials, ilyushin's postulate and...

Joe GoddardUniversity of California, San Diego

Department of Aerospace and Mechanical Engineering

INTERNATIONAL SYMPOSIUM ON PLASTICITY 2006Halifax, Nova Scotia, Canada

July 17-22, 2006

DISSIPATIVE MATERIALS, ILYUSHIN'S POSTULATE AND HYPOPLASTICITY

Overview & Summary• Objectives are to:

explore specific forms of a general continuum model proposed over 20 years ago* to describe “curious” rheological effects in nominally Stokesian suspensions,

provide a continuum framework for further micromechanical modeling and experiment on the viscoplasticity of suspensions and granular media, and

consider implications for the modeling of more general elastoplastic bodies.

*Adv. Coll. Interface Sci. 17, 241,1982; JNNFM,14,141, 1984 (J. Fluid Mech., 2006).

Review of the Rheologyof

Stokesian Suspensions

(Generalized Einstein Problem)

Standard model of Stokesian suspensions*

*Note the analogy to elasticity of solid composites.

Viscometric flows(simple-shear) of“simple fluids”

2

13


2

13

τ


2

13N1

τ


2

13N1 N2

τ

Rheological “curiosities” in sphere suspensions

Viscosity recovery on shear reversalGadala-Maria and Acrivos, J.Rheol. 24,799,1980



Anisotropic structure in simple shear



Anisotropic structure in simple shearHusband & Gadala-Maria, J.Rheol. 31, 95,1987




Linear normal stress in simple shear




Linear normal stress in simple shearZarraga, I. et al., J.Rheol. 44,




Linear normal stress in simple shearZarraga, I. et al., J.Rheol. 44, 185, 2000, anticipated in JDG 1982.




Linear normal stress in simple shearZarraga, I. et al., J.Rheol. 44, 185, 2000, anticipated in JDG 1982.

≡ κ

Partial reversal and recovery of shear and normal forceson reversal of steady shearing*

*Kolli, G. et al. J. Rheol., 46, 321, 2002

(normalized torque and axial thrust in torsional split-ring apparatus)




• As discussed below, torque and thrust




• As discussed below, torque and thrust should abruptly change sign but not




• As discussed below, torque and thrust should abruptly change sign but not value for Stokesian suspensions.

Proposed continuum model* :1.) Strictly dissipative material with memory

2.) but no characteristic time

*JDG, Adv. Coll. Interface Sci. 17, 241,1982 & JNNFM,14,141,1984




• Note instantaneous reversal of stress vs. strain




• Note instantaneous reversal of stress vs. strain (dictated by “Stokesian reversibility”).

Model (continued)

Model (continued)

cf. F. Tatsuoka et al. 2005, P. Jop et al. 2005

Model (continued)

Model (continued)

* Huang, N. et al. PRL, 94, 028301, 2005; Tsai & Gollub, PRE, 70, 031303, 2004.

Simplified Modelfor

Stokesian Suspensions

Assumed model

n

Assumed model

*Cowin, S. C., Mech. Materials, 4,137, 1985.

n

Inferences fromStokesian dynamics

Postulatedevolution ofanisotropy

Reversal of steady shearing*

Reversal of steady shearing*

*cf. Coleman and Dill, JMPS, 1971: index of refraction tensor in simple materials.

Comparison to Experimentswith Reversal of Steady Shear

Experimental details*

* Kolli, G. et al. J. Rheol., 46, 321, 2002; split-ring shear cell

relativetorque

relativethrust

Nonlinear least-squares fit with two relaxation modes

• Inferred ratio of normal stresses N2! / N1! ≈ 10 for data on right is out of range of other experiments, possibly due to split-ring geometry of Kolli et al.

• One relaxation strain is ≈ 1, reflecting large-scale rearrangement, but one is ≈ 0.0001, accounting for incomplete reversal of stress and representing another Stokesian symmetry breaking, e.g. by short-range interparticle forces, of a type employed in “Stokesian dynamics” simulations.

Remarks on data fitting*

* http://www.levenspiel.com/octave/elephant.htm

“… spoilsports like to quote the statementattributed to … Gauss…

‘Give me four parameters, and I will draw an elephant for you; with five I shall have him raise and lower his trunk and his tail.’

Wei (1) challenged this assertion andproceeded to draw the best elephant hecould with various numbers of parameters.Here … his results with 5, 10, 20 and 30adjustable parameters.

(1) J. Wei, Chemtech,1975, May 128-IBC …”

Conclusions

** cf. Phan-Thien, N, J. Rheol., 1995, for a dense suspension model.

*JDG, JNNFM,14,141,1984.

**

Conclusions for suspension and granular media

** cf. Phan-Thien, N, J. Rheol., 1995, for a dense suspension model.

*JDG, JNNFM,14,141,1984.

**

Possible consequences for more general elastoplasticity

(J. Rice, 2004)

Possible consequences (cont’d)

dissipative materials, ilyushin's postulate and...

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