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A unified evolution equation for the Cauchy stress tensor of an isotropic elasto-visco-plastic material. II: Normal stress difference in a viscometric flow, and an unsteady flow with a moving boundary. (English) Zbl 1170.74308
Summary: This paper continues Part I [ibid. 19, No. 7, 423–440 (2008; Zbl 1170.74308)], in which a unified evolution equation for the Cauchy stress tensor, which takes elastic, viscous, and plastic features of the material simultaneously into account, was proposed. Hypoplasticity in particular was incorporated to account for the plastic characteristics. In the present paper, the stress model is applied to study normal stress differences in the context of viscometric flow, and the unsteady flow characteristics of an elasto-visco-plastic fluid between two infinite parallel plates driven by a sudden motion of the plate, to estimate the performance and limitations of the proposed method. Numerical calculations show that, in the context of viscometric flow, different degrees of plasticity and the associated first and second normal stress differences can be addressed appropriately by the stress model. For the unsteady flow situation the results show that the complex behavior of the fluid, in particular after the start of the driving motion, can be described to some extent by the model. In addition, different relaxation and retardation spectra with plastic characteristics can be simulated by varying the model parameters. These findings suggest the applicability of the proposed stress model, for example, in the fields of granular/debris and polymeric flows.

##### MSC:
 74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity) 76A10 Viscoelastic fluids 76M20 Finite difference methods applied to problems in fluid mechanics
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##### References:
 [1] Fang, C., Wang, Y., Hutter, K.: A unified evolution equation for the Cauchy stress tensor of an isotropic elasto-visco-plastic material. I. On thermodynamically consistent evolution (2007) (in press) · Zbl 1170.74309 [2] Kolymbas D. (1991). An outline of hypoplasticity. Arch. Appl. Mech. 61: 143–151 · Zbl 0734.73023 [3] Kolymbas D. (2000). Introduction to Hypoplasticity. Balkema, Rotterdam [4] Wu W. (2006). On high-order hypoplastic models for granular materials. J. Eng. Math. 56: 23–34 · Zbl 1103.74019 · doi:10.1007/s10665-006-9040-7 [5] Wu W. and Kolymbas D. (2000). Hypoplasticity then and now. In: Kolymbas, D. (eds) Constitutive Modelling of Granular Materials., pp 57–105. Springer, Berlin [6] Wang Y. (2006). Time-dependent Poiseuille flows of visco-elasto-plastic fluids. Acta Mech. 186: 187–201 · Zbl 1106.76011 · doi:10.1007/s00707-006-0376-x [7] Barnes H.A., Hutton J.F. and Walters K. (1989). An Introduction to Rheology. Elsevier, Amsterdam · Zbl 0729.76001 [8] Tanner R.I. (1992). Engineering Rheology. Oxford University Press, Oxford · Zbl 1012.76002 [9] Bird R.B. and Wiest J.M. (1995). Constitutive equations for polymeric liquids. Ann. Rev. Fluid Mech. 27: 169–193 · doi:10.1146/annurev.fl.27.010195.001125 [10] Hayat T., Siddiqui A.M. and Asghar S. (2001). Some simple flows of an Oldroyd-B fluids. Int. J. Eng. Sci. 39: 135–147 · doi:10.1016/S0020-7225(00)00026-4 [11] Kolkka R.W., Malkus D.S., Hansen M.G., Ierly G.R. and Worthing R.A. (1988). Spurt phenomenon of the Johnson-Segelman fluid and related models. J. Non-Newton. Fluid Mech. 29: 303–335 · doi:10.1016/0377-0257(88)85059-6 [12] O’Reilly O.M. and Srinivasa A.R. (2001). On the nature of constraint forces in dynamics. Proc. R. Soc. A 457: 1307–1313 · Zbl 1009.70012 · doi:10.1098/rspa.2000.0717 [13] Rajagopal K.R. (2003). On implicit constitutive equations. Appl. Math. 48: 279–319 · Zbl 1099.74009 · doi:10.1023/A:1026062615145 [14] Rajagopal K.R. and Srinivasa A.R. (2005). On the nature of constraints for continua undergoing dissipative processes. Proc. R. Soc. A 461: 2785–2795 · Zbl 1186.74008 · doi:10.1098/rspa.2004.1385 [15] Denn M.M. (2004). Fifty years of non-Newtonian fluid dynamics. AIChE. J. 50: 2335–2345 · doi:10.1002/aic.10357 [16] Tanner R.I. and Walters K. (1998). Rheology: An Historical Perspective. Elsevier, Amsterdam · Zbl 0946.76002 [17] Bird R.B., Armstrong R.C. and Hassager O. (1987). Dynamics of Polemetric Liquids, vol. I. Fluid Dynamics, 2nd edn. Wiley, New York [18] Barnes H.A. and Walters K. (1985). The yeidl stress myth? Rheol. Acta 24: 323–326 [19] Schurz J. (1990). The yield stress–an empirical reality. Rheol. Acta 29: 170–171 · doi:10.1007/BF01332384 [20] Barnes H.A. (1999). The yield stress–a review or ’ $$\pi\alpha\nu\tau\alpha\,\rho\varepsilon\iota$$ ’–everything flows? J. Non-Newtonian Fluid Mech. 81: 133–178 · Zbl 0949.76002 · doi:10.1016/S0377-0257(98)00094-9 [21] Evans I.D. (1992). Letter to the editor: on the nature of the yield stress. J. Rheol. 36: 1313–1316 · doi:10.1122/1.550262 [22] White F.M. (1991). Viscous Fluid Flow, 2nd edn. McGraw-Hill, New York [23] Anderson D.A., Tannehill J.C. and Pletcher R.H. (1984). Computational Fluid Mechanics and Heat Transfer. McGraw-Hill, New York · Zbl 0569.76001
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