# American Institute of Mathematical Sciences

March  2013, 18(2): 283-293. doi: 10.3934/dcdsb.2013.18.283

## Modelling of wheat-flour dough mixing as an open-loop hysteretic process

 1 CSIRO Mathematics, Informatics and Statistics, North Road, ANU Campus, Acton ACT, GPO Box 664, Canberra, ACT 2601, Australia 2 Institute of Information Theory and Automation of the ASCR, Pod vodárenskou věží 4, 182 08 Prague, Czech Republic

Received  October 2011 Revised  July 2012 Published  November 2012

Motivated by the fact that various experimental results yield strong confirmatory support for the hypothesis that the mixing of a wheat-flour dough is essentially a rate-independent process'', this paper examines how the mixing can be modelled using the rigorous mathematical framework developed to model an incremental time evolving deformation of an elasto-plastic material. Initially, for the time evolution of a rate-independent elastic process, the concept is introduced of an "energetic solution'' [24] as the characterization for the rate-independent deformations occurring. The framework in which it is defined is formulated in terms of a polyconvex stored energy density and a multiplicative decomposition of large deformations into elastic and nonelastic (plastic or viscous) components. The mixing of a dough to peak dough development is then modelled as a sequence of incremental elasto-nonelastic deformations. For such incremental processes, the existence of Sobolev solutions is guaranteed. Finally, the limit passage to vanishing time increment leads to the existence of an energetic solution to our problem.
Citation: Robert S. Anderssen, Martin Kružík. Modelling of wheat-flour dough mixing as an open-loop hysteretic process. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 283-293. doi: 10.3934/dcdsb.2013.18.283
##### References:
 [1] R. S. Anderssen and P. W. Gras, The hysteretic behaviour of wheat-flour dough during mixing, in "Wheat Gluten" (eds. P. R Schewry and A. S. Tatham), Royal Society of Chemistry Special Publications, (2000), 391-395. [2] R. S. Anderssen, I. G. Gotz and K. H. Hoffmann, The global behavior of elastoplastic and viscoelastic materials with hysteresis-type state equations, SIAM J. Appl. Math., 58 (1998), 703-723. [3] R. S. Anderssen, P. W. Gras and F. MacRitchie, Linking mathematics to data from the testing of wheat-flour dough, Chem. in Aust., 64 (1997), 3-5. [4] R. S. Anderssen, P. W. Gras and F. MacRitchie, The rate-independence of the mixing of wheat-flour dough to peak dough development, J. Cereal Sci., 27 (1998), 167-177. [5] B. Appelbe, D. Flynn, H. McNamara,P. O'Kane, A. Pimenov, A. Pokrovskii, D. Rachinskii and A. Zhezherun, Rate-independent hysteresis in terrestrial hydrologya vegetated soil model with preisach hysteresis, IEEE Control Syst. Mag., 29 (2009), 44-69. doi: 10.1109/MCS.2008.930923. [6] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., 63 (1977), 337-403. [7] Z. P. Bažant and M. Jirásek, Nonlocal integral formulation of plasticity and damage: A survey of progress, J. Engrg. Mech., 128 (2002), 1119-1149. doi: 10.1061/(ASCE)0733-9399(2002)128:11(1119). [8] C. Carstensen, K. Hackl and A. Mielke, Nonconvex potentials and microstructures in finite-strain plasticity, Proc. Roy. Soc. Lond. A, 458 (2002), 299-317. doi: 10.1098/rspa.2001.0864. [9] M. N. Charalambides, L. Wanigasooriya and J. G. Williams, Biaxial deformation of dough using the bubble inflation technique. II. Numerical modelling, Rheol. Acta, 41 (2002), 541-548. [10] P. G. Ciarlet and J. Ne\vcas, Injectivity and self-contact in nonlinear elasticity, Arch. Ration. Mech. Anal., 19 (1987), 171-188. [11] G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. reine angew. Math., 595 (2006), 55-91. [12] P. W. Gras, H. C. Carpenter and R. S. Anderssen, Modelling the developmental rheology of wheat-flour dough using extension tests, J. Cereal Sci., 31 (2000), 1-13. doi: 10.1006/jcrs.1999.0293. [13] M. E. Gurtin, On the plasticity of single crystals: Free energy, microforces, plastic-strain gradients, J. Mech. Phys. Solids, 48 (2000), 989-1036. doi: 10.1016/S0022-5096(99)00059-9. [14] R. H. Kilborn and K. H. Tipples, Factors affecting mechanical dough development 1. Effect of mixing intensity and work input, Cereal Chem., 49 (1972), 34-47. [15] J. Kratochvíl, M. Kružík and R. Sedláček, Energetic approach to strain gradient plasticity, Zeit. Angew. Math. Mech., 90 (2010), 122-135. [16] M. Kružík and J. Zimmer, A model of shape memory alloys accounting for plasticity, IMA J. Appl. Math., 76 (2011), 193-216. [17] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99. [18] A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci., 19 (2009). [19] A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances., Cont. Mech. Thermodyn., 15 (2002), 351-382. [20] A. Mielke, Evolution of rate-independent systems, in "Evolutionary equations, II, Handb. Differ. Equ.,'' Elsevier/North-Holland, Amsterdam, (2005), 461-559. [21] A. Mielke and T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys, Multiscale Model. Simul., 1 (2003), 571-597. doi: 10.1137/S1540345903422860. [22] A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity, ESAIM Math. Mod. Num. Anal., 43 (2009), 399-428. doi: 10.1051/m2an/2009009. [23] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, in "Models of Continuum Mechanics in Analysis and Engineering'' (eds. H.-D.Alder, R. Balean and R. Farwig), Shaker Verlag, Aachen, (1999), 117-129. [24] A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177. [25] T. S. K. Ng, G. H. McKinley and M. Padmanabhan, Linear to non-linear rheology of wheat flour dough, Applied Rheology, 16 (2006), 265-274. [26] N. PhanThien, M. SafariArdi and A. MoralesPatino, Oscillatory and simple shear flows of a flour-water dough: A constitutive model, Rheol. Acta, 36 (1997), 38-48. [27] I. Tsagrakis and E. C. Aifantis, Recent developments in gradient plasticity, J. Engrg. Mater. Tech., 124 (2002).

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##### References:
 [1] R. S. Anderssen and P. W. Gras, The hysteretic behaviour of wheat-flour dough during mixing, in "Wheat Gluten" (eds. P. R Schewry and A. S. Tatham), Royal Society of Chemistry Special Publications, (2000), 391-395. [2] R. S. Anderssen, I. G. Gotz and K. H. Hoffmann, The global behavior of elastoplastic and viscoelastic materials with hysteresis-type state equations, SIAM J. Appl. Math., 58 (1998), 703-723. [3] R. S. Anderssen, P. W. Gras and F. MacRitchie, Linking mathematics to data from the testing of wheat-flour dough, Chem. in Aust., 64 (1997), 3-5. [4] R. S. Anderssen, P. W. Gras and F. MacRitchie, The rate-independence of the mixing of wheat-flour dough to peak dough development, J. Cereal Sci., 27 (1998), 167-177. [5] B. Appelbe, D. Flynn, H. McNamara,P. O'Kane, A. Pimenov, A. Pokrovskii, D. Rachinskii and A. Zhezherun, Rate-independent hysteresis in terrestrial hydrologya vegetated soil model with preisach hysteresis, IEEE Control Syst. Mag., 29 (2009), 44-69. doi: 10.1109/MCS.2008.930923. [6] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., 63 (1977), 337-403. [7] Z. P. Bažant and M. Jirásek, Nonlocal integral formulation of plasticity and damage: A survey of progress, J. Engrg. Mech., 128 (2002), 1119-1149. doi: 10.1061/(ASCE)0733-9399(2002)128:11(1119). [8] C. Carstensen, K. Hackl and A. Mielke, Nonconvex potentials and microstructures in finite-strain plasticity, Proc. Roy. Soc. Lond. A, 458 (2002), 299-317. doi: 10.1098/rspa.2001.0864. [9] M. N. Charalambides, L. Wanigasooriya and J. G. Williams, Biaxial deformation of dough using the bubble inflation technique. II. Numerical modelling, Rheol. Acta, 41 (2002), 541-548. [10] P. G. Ciarlet and J. Ne\vcas, Injectivity and self-contact in nonlinear elasticity, Arch. Ration. Mech. Anal., 19 (1987), 171-188. [11] G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. reine angew. Math., 595 (2006), 55-91. [12] P. W. Gras, H. C. Carpenter and R. S. Anderssen, Modelling the developmental rheology of wheat-flour dough using extension tests, J. Cereal Sci., 31 (2000), 1-13. doi: 10.1006/jcrs.1999.0293. [13] M. E. Gurtin, On the plasticity of single crystals: Free energy, microforces, plastic-strain gradients, J. Mech. Phys. Solids, 48 (2000), 989-1036. doi: 10.1016/S0022-5096(99)00059-9. [14] R. H. Kilborn and K. H. Tipples, Factors affecting mechanical dough development 1. Effect of mixing intensity and work input, Cereal Chem., 49 (1972), 34-47. [15] J. Kratochvíl, M. Kružík and R. Sedláček, Energetic approach to strain gradient plasticity, Zeit. Angew. Math. Mech., 90 (2010), 122-135. [16] M. Kružík and J. Zimmer, A model of shape memory alloys accounting for plasticity, IMA J. Appl. Math., 76 (2011), 193-216. [17] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99. [18] A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci., 19 (2009). [19] A. Mielke, Energetic formulation of multiplicative elasto-plasticity using dissipation distances., Cont. Mech. Thermodyn., 15 (2002), 351-382. [20] A. Mielke, Evolution of rate-independent systems, in "Evolutionary equations, II, Handb. Differ. Equ.,'' Elsevier/North-Holland, Amsterdam, (2005), 461-559. [21] A. Mielke and T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys, Multiscale Model. Simul., 1 (2003), 571-597. doi: 10.1137/S1540345903422860. [22] A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity, ESAIM Math. Mod. Num. Anal., 43 (2009), 399-428. doi: 10.1051/m2an/2009009. [23] A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, in "Models of Continuum Mechanics in Analysis and Engineering'' (eds. H.-D.Alder, R. Balean and R. Farwig), Shaker Verlag, Aachen, (1999), 117-129. [24] A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal., 162 (2002), 137-177. [25] T. S. K. Ng, G. H. McKinley and M. Padmanabhan, Linear to non-linear rheology of wheat flour dough, Applied Rheology, 16 (2006), 265-274. [26] N. PhanThien, M. SafariArdi and A. MoralesPatino, Oscillatory and simple shear flows of a flour-water dough: A constitutive model, Rheol. Acta, 36 (1997), 38-48. [27] I. Tsagrakis and E. C. Aifantis, Recent developments in gradient plasticity, J. Engrg. Mater. Tech., 124 (2002).
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