Rheologica Acta

, Volume 48, Issue 3, pp 245–284 | Cite as

The damping function in rheology



The damping function has been a concept introduced in rheology since more than 30 years ago, and although a similar concept was already earlier implemented in studying rubber materials, its implementation in the modeling of polymer melts was an essential step forward in the classification and understanding of nonlinear viscoelasticity phenomena. It is the objective of this contribution to give an overview on the theoretical background and physical interpretation of the concept of the damping function for different types of deformation, as well as a review on the experimental results including the experimental artefacts to be considered. Besides homopolymers, a summary is given on different investigations of other types of systems, where the concept of the damping function has also been applied, for example, rubbers, rubber-like materials, block copolymers, polymer composites, liquid crystals, polymer blends, suspensions, emulsions, micellar systems, and in food rheology.


Damping function Viscoelasticity Viscoelastic fluid Wagner model Rheology 


  1. Aoki Y, Hatano A, Tanaka T, Watanabe H (2001) Nonlinear stress relaxation of ABS polymers in the molten state. Macromolecules 34:3100–3107Google Scholar
  2. Archer LA (1999) Separability criteria for entangled polymer liquids. J Rheol 43(6):1555–1571Google Scholar
  3. Archer LA, Juliani (2004) Linear and nonlinear viscoelasticity of entangled multiarm (pom-pom) polymer liquids. Macromolecules 37:1076–1088MathSciNetGoogle Scholar
  4. Archer LA, Mhetar VR (1998) Differential constitutive equation for entangled polymers with partial strand extension. Rheol Acta 37:170–181Google Scholar
  5. Archer LA, Varshney SK (1998) Synthesis and relaxation dynamics of multiarm polybutadiene melts. Macromolecules 31:6348–6355Google Scholar
  6. Archer LA, Chen YL, Larson RG (1995) Delayed slip after step strains in highly entangled polystyrene solutions. J Rheol 39(3):519–525ADSGoogle Scholar
  7. Archer LA, Sanchez-Reyes J, Juliani (2002) Relaxation dynamics of polymer liquids in nonlinear step shear. Macromolecules 35:10216–10224Google Scholar
  8. Astarita G, Marrucci G (1974) Principles of non-Newtonian fluid mechanics. McGraw Hill, Great BritainGoogle Scholar
  9. Barrera MA, Vega JF, Aguilar M, Martínez-Salazar J (2006) Melt flow index on high molecular weight polyethylene: a comparative study of experiments and simulation. J Mater Process Technol 174:171–177Google Scholar
  10. Barroso VC, Maia JM (2002) Evaluation by means of stress relaxation (after a step strain) experiments of the viscoelastic behaviour of polymer melts in uniaxial extension. Rheol Acta 41:257–264Google Scholar
  11. Barroso VC, Ribeiro SP, Maia JM (2003) Stress relaxation after a step strain in uniaxial extension of polyisobutylene and polyethylene. Rheol Acta 42:345–354Google Scholar
  12. Bastian H (2001) Non-linear viscoelasticity of linear and long-chain-branched polymer melts in shear and extensional flows, Ph.D. Thesis, Stuttgart University, Germany. http://elib.uni-stuttgart.de/opus/volltexte/2001/894
  13. Bengoechea C, Puppo MC, Romero A, Cordobés F, Guerrero A (2008) Linear and non-linear viscoelasticity of emulsions containing carob protein as emulsifier. J Food Eng 87:124–135Google Scholar
  14. Bernstein B, Kearsley EA, Zapas LJ (1963) A study of stress relaxation with finite strain. Trans Soc Rheol 7:391–410Google Scholar
  15. Bick DK, McLeish TCB (1996) Topological contributions to nonlinear elasticity in branched polymers. Phys Rev Lett 76(14):2587–2590PubMedADSGoogle Scholar
  16. Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids, vol 1. Fluid mechanics. Wiley and Sons, USAGoogle Scholar
  17. Bishko G, McLeish TCB, Harlen OG, Larson RG (1997) Theoretical molecular rheology of branched polymers in simple and complex flows: the pom-pom model. Phys Rev Lett 79(12):2352–2355ADSGoogle Scholar
  18. Booij HC, Palmen JHM (1982) Some aspects of linear and nonlinear viscoelastic behaviour of polymer melts in shear. Rheol Acta 21:376–387Google Scholar
  19. Bruker I (1986) Measurements of the first normal-stress difference in a new Rheo-dilatometer for molten polymers: triple-step-shear-strain tests for all K-BKZ constitutive equations. Rheol Acta 25:501–506Google Scholar
  20. Byars JA, Jong L (2009) Flow properties of natural rubber composites filled with defatted soy flour. J Appl Polym Sci 111:2049–2055Google Scholar
  21. Campanella OH, Peleg M (1987) Analysis of the transient flow of mayonnaise in coaxial viscometer. J Rheol 31(6):439–452ADSGoogle Scholar
  22. Caram Y, Bautista F, Puig JE, Manero O (2006) On the rheological modeling of associative polymers. Rheol Acta 46:45–57Google Scholar
  23. Callaghan PT, Cates ME, Rofe CJ, Smeulders JBAF (1996) A study of the “spurt effect” in wormlike micelles using nuclear magnetic resonance microscopy. J Phys II France 6:375–393Google Scholar
  24. Carriere CJ, Thomas AJ, Inglett GE (2002) Prediction of the nonlinear transient and oscillatory rheological behavior of flour suspensions using a strain-separable integral constitutive equation. Carbohydr Polym 47:219–231Google Scholar
  25. Chang H, Lodge AS (1972) Comparison of rubberlike-liquid theory with stress-growth data for elongation of a low-density branched polyethylene melt. Rheol Acta 11:127–219Google Scholar
  26. Chang WV, Bloch R, Tschoegl NW (1976) On the theory of the viscoelastic behaviour of soft polymers in moderately large deformations. Rheol Acta 15:367–378MATHGoogle Scholar
  27. Chen CY, Wu SM, Chen ZR, Huang TJ, Hua CC (2003) Nonlinear stress relaxation of an entangled linear polystyrene in single step-strain flow: a quantitative theoretical investigation. J Polym Sci B Polym Phys 41:1281–1293Google Scholar
  28. Chodankar CD, Schieber JD, Venerus DC (2003a) Pom-pom theory evaluation in double-step strain flows. J Rheol 47(2):413–427ADSGoogle Scholar
  29. Chodankar CD, Schieber JD, Venerus DC (2003b) Evaluation of rheological constitutive equations for branched polymers in step shear strain flows. Rheol Acta 42:123–131Google Scholar
  30. Clemeur N, Debbaut B (2007) A pragmatic approach for deriving constitutive equations endowed with pom-pom attributes. Rheol Acta 46:1187–1196Google Scholar
  31. de Gennes PG (1971) Reptation of a polymer chain in the presence of fixed obstacles. J Chem Phys 55(2):572–579ADSGoogle Scholar
  32. Dealy JM, Wissbrun KF (1999) Melt rheology and its role in plastics processing. Theory and applications. Kluver Academic Publishers, NetherlandsGoogle Scholar
  33. Demarmels A, Meissner J (1986) Multiaxial elongations of polyisobutylene and the predictions of several network theories. Colloid Polym Sci 264:829–846Google Scholar
  34. Demarquette NR, Dealy JM (1992) Nonlinear viscoelasticity of concentrated polystyrene solutions: sliding plate rheometer studies. J Rheol 36(6):1007–1032ADSGoogle Scholar
  35. Doi M (1980) Stress relaxation of polymeric liquids after double-step strain. J Polym Sci B Polym Phys 18:1891–1905Google Scholar
  36. Doi M, Edwards SF (1978a) Dynamics of concentrated polymer systems. Part 1.—Brownian motion in the equilibrium state. Trans Faraday Soc 20:1789–1801Google Scholar
  37. Doi M, Edwards SF (1978b) Dynamics of concentrated polymer systems. Part 2.—Molecular motion under flow. Trans Faraday Soc 20:1802–1817Google Scholar
  38. Doi M, Edwards SF (1978c) Dynamics of concentrated polymer systems. Part 3.—The constitutive equations. Trans Faraday Soc 20:1818–1832Google Scholar
  39. Doi M, Edwards SF (1979) Dynamics of concentrated polymer systems. Part 4.- Rheological properties. Trans Faraday Soc 20:38–54Google Scholar
  40. Doi M, Edwards SF (1986) Theory of polymer dynamics. Oxford University PressGoogle Scholar
  41. Doi M, Takimoto J (2003) Molecular modelling of entanglement. Phil Trans R Soc Lond A 361:641–652ADSMathSciNetGoogle Scholar
  42. Ehrecke P, Wagner MH (1995) Untersuchungen zur Irreversibilität von Netzwerkentschlaufungen beim fließen von Polymerschmelzen. Macromol Chem Phys 196:2989–3004Google Scholar
  43. Einaga Y, Osaki K, Kurata M, Kimura S, Tamura M (1971) Stress relaxation of polymer solutions under large strain. Polymer J 2(4):550–552Google Scholar
  44. Einaga Y, Osaki K, Kurata M, Kimura S, Yamada N, Tamura M (1973) Stress relaxation of polymer solutions under large strain. Polymer J 5(1):91–96Google Scholar
  45. Erchiqui F (2005) Thermodynamic approach of inflation process of K-BKZ polymer sheet with respect to thermoforming. Polym Eng Sci 45(10):1319–1335Google Scholar
  46. Erchiqui F (2006) A new hybrid approach using the explicit dynamic finite element method and thermodynamic law for the analysis of the thermoforming and blow molding processes for polymer materials. Polym Eng Sci 46(11):1554–1564Google Scholar
  47. Fan Y, Liao H (2008) Experimental studies on the relaxation behavior of commercial polymer melts. J Appl Polym Sci 110:1520–1530Google Scholar
  48. Fan B, Kazmer DO, Bushko WC, Theriault RP, Poslinski AJ (2004) Birefringence prediction of optical media. Polym Eng Sci 44:814–824Google Scholar
  49. Feigl K, Öttinger HC, Meissner J (1993) A failure of a class of K-BKZ equations based on principal stretches. Rheol Acta 32:438–446Google Scholar
  50. Ferri JD (1980) Viscoelastic properties of polymers. John Wiley and Sons, USAGoogle Scholar
  51. Ferri D, Greco F (2006) Nonlinear stress relaxation of molten polymers: experimental verification of a new theoretical approach. Macromolecules 39:5931–5938Google Scholar
  52. Fukuda M, Osaki K, Kurata M (1975) Nonlinear viscoelasticity of polystyrene solutions. I. Strain-dependent relaxation modulus. J Polym Sci Polym Phys Ed 13:1563–1576Google Scholar
  53. Furuichi K, Nonomura Ch, Masubuchi Y, Ianniruberto G, Greco F, Marrucci G (2007) Primitive chain network simulations of damping functions for shear, uniaxial, biaxial and planar deformations. Nihon Reoroji Kakkaishi 35(2):73–77Google Scholar
  54. Gallegos C, Berjano M (1992) Linear viscoelastic behavior of commercial and model mayonnaise. J Rheol 36(3):465–478ADSGoogle Scholar
  55. Gevgilili H, Kalyon DM (2001) Step strain flow: wall slip effects and other error sources. J Rheol 45(2):467–475ADSGoogle Scholar
  56. Gianotti G, Cicuta A, Romanini D (1980) Long chain branching in low-density polyethylene: 1. Molecular structure. Polymer 21:1087–1091Google Scholar
  57. Gotsis AD, Zeevenhoven BLF, Tsenoglou C (2004) Effect of long branches on the rheology of polypropylene. J Rheol 48(4):895–914ADSGoogle Scholar
  58. Gottlieb M, Gaylord RJ (1987) Experimental tests of entanglement models of rubber elasticity. 3. Biaxial deformations. Macromolecules 20:130–138Google Scholar
  59. Greco F (2004) Entangled polymeric liquids: nonstandard statistical thermodynamics of a subchain between entanglement points and a new calculation of the strain measure tensor. Macromolecules 37:10079–10088MathSciNetGoogle Scholar
  60. Green MS, Tobolsky AV (1946) A new approach to the theory of relaxing polymeric media. J Chem Phys 14(2):80–92ADSGoogle Scholar
  61. Goublomme A, Crochet MJ (1993) Numerical prediction of extrudate swell of a high-density polyethylene – Further results. J Non-Newton Fluid Mech 47:281–287Google Scholar
  62. Goublomme A, Draily B, Crochet MJ (1992) Numerical prediction of extrudate swell of a high-density polyethylene. J Non-Newton Fluid Mech 44:171–195Google Scholar
  63. Guerrero A, Partal P, Gallegos C (2000) Linear and non-linear viscoelasticity of low-in-cholesterol mayonnaise. Food Sci Tech Int 6(2):165–172Google Scholar
  64. Guth E, Wack PE, Anthony RL (1946) Significance of the equation of state for rubber. J Appl Phys 17:347–351ADSGoogle Scholar
  65. Halley PJ, Mackay ME (1994) The effect of metals on the processing of LLDPE through a slit die. J Rheol 38(1):41–51ADSGoogle Scholar
  66. Han CD, Kim SS (1994) Transient rheological behavior of a thermotropic liquid–crystalline polymer. III. Step strain experiment and shear stress relaxation modulus. J Rheol 38(1):31–40ADSMathSciNetGoogle Scholar
  67. Harry-O’kuru RE, Carriere CJ (2002) Synthesis, rheological characterization, and constitutive modeling of polyhydroxy triglycerides derived from milkweed oil. J Agric Food Chem 50:3214–3221PubMedGoogle Scholar
  68. Hepperle J, Münstedt H (2006) Rheological properties of branched polystyrene: nonlinear shear and extensional behaviour. Rheol Acta 45:717–727Google Scholar
  69. Holmqvist P, Castelletto V, Hamley IW, Hermsdorf N, Almdal K (2001) Stress relaxation experiments on a lamellar polystyrene-polyisoprene diblock copolymer melt. Polymer 42:7203–7208Google Scholar
  70. Huang SX, Lu CJ (2006) Stress relaxation characteristics and extrudate swell of the IUPAC-LDPE melt. J Non-Newton Fluid Mech 136:147–156Google Scholar
  71. Inoue T, Uematsu T, Yamashita Y, Osaki K (2002) Significance of the longest Rouse relaxation time in the stress relaxation process at large deformation of entangled polymer solutions. Maromolecules 35:4718–4724Google Scholar
  72. Isaki T, Takahashi M, Urakawa O (2003) Biaxial damping function of entangled monodisperse polystyrene melts: comparison with the Mead7–Larson–Doi model. J Rheol 47(5):1201–1210ADSGoogle Scholar
  73. Islam MT, Archer LA (2001) Nonlinear rheology of highly entangled polymer solutions in start-up and steady shear flow. J Poly Sci B Polym Phys 39:2275–2289Google Scholar
  74. Islam MT, Sanchez-Reyes J, Archer LA (2001) Nonlinear rheology of highly entangled polymer liquids: step shear damping function. J Rheol 45(1):61–82ADSGoogle Scholar
  75. Isono Y, Ferry JD (1985) Stress relaxation and differential dynamic modulus of polyisobutylene in large shearing deformations. J Rheol 29(3):273–280ADSGoogle Scholar
  76. Isono Y, Nishitake T (1995) Stress relaxation and change in entanglement structure of polyisobutylene in large shearing deformations. Polymer 36(8):1635–1638Google Scholar
  77. Isono Y, Itoh K, Komiyatani T, Fujimoto T (1991a) Differential dynamic modulus of polyisobutylene with high molecular weight 1. Single-step large shearing deformations. Macromolecules 24:4429–4432Google Scholar
  78. Isono Y, Shizuru K, Fujimoto T (1991b) Differential dynamic modulus of polyisobutylene with high molecular weight 2. Double-step large shearing deformations. Macromolecules 24:4433–4436Google Scholar
  79. Isono Y, Ohashi N, Kase T (1995) Chain contraction and change in entanglement structure of well-entangled polymer in large shearing deformations. Macromolecules 28:5154–5155Google Scholar
  80. Isono Y, Kamohara T, Takano A, Kase T (1997) Nonlinear viscoelastic properties and change in entanglement structure of linear polymer. 1. Single-step large shearing deformations. Rheol Acta 36:245–251Google Scholar
  81. Iza M, Bousmina M (2000) Nonlinear rheology of immiscible polymer blends: step strain experiments. J Rheol 44(6):1363–1384ADSGoogle Scholar
  82. Iza M, Bousmina M (2005) Damping function for narrow and large molecular weight polymers: comparison with the force-balanced network model. Rheol Acta 44:372–378Google Scholar
  83. Juliani, Archer LA (2001) Linear and nonlinear rheology of bidisperse polymer blends. J Rheol 45(3):691–708ADSGoogle Scholar
  84. Kajiwara T, Tomiyama H, Sueyoshi Y, Yamamura M, Adachi K (2001) Numerical simulation of extrudate swell problem and evaluation of applicability of viscoelastic constitutive models 1. A study of axisymmetric extrudate swell from a straight die. Nihon Reoroji Gakkaishi 29(1):47–52Google Scholar
  85. Kalyon DM, Yu DW, Moy FH (1988) Rheology and processing of linear low density polyethylene resins as affected by alpha-olefin comonomers. Polym Eng Sci 28(23):1542–1550Google Scholar
  86. Kasehagen LJ, Macosko CW (1998) Nonlinear shear and extensional rheology of long-chain randomly branched polybutadiene. J Rheol 42(6):1303–1327Google Scholar
  87. Kawamura T, Urayama K, Kohjiya S (2001) Multiaxial deformations of end-linked poly(dimethylsiloxane) networks. 1. Phenomenological approach to strain energy density function. Macromolecules 34:8252–8260Google Scholar
  88. Kaye A (1962) College of Astronautics. Cranford, U.K., Note No. 134Google Scholar
  89. Khan SA, Larson RG (1987) Comparison of simple constitutive equations for polymer melts in shear and biaxial and uniaxial extensions. J Rheol 31(3):207–234ADSGoogle Scholar
  90. Khan MMK, Tanner RI (1990) Rheology of an LDPE melt in reversing multi-step shear and elongational flows. Rheol Acta 29:281–297Google Scholar
  91. Khan SA, Prud’homme RK, Larson RG (1987) Comparison of the rheology of polymer melts in shear, biaxial and uniaxial extensions. Rheol Acta 26:144–151Google Scholar
  92. Kolkka RW, Malkus DS, Rose TR (1991) Finite rise time step strain modelling of nearly monodisperse polymer melts and solutions. Rheol Acta 30:430–446Google Scholar
  93. Kontou E (1994) Nonlinear viscoelasticity of a vulcanized elastomer. J Appl Polym Sci 54:1873–1877Google Scholar
  94. Kotsilkova R (2002) Rheology-structure relationship of polymer/layered silicate hybrids. Mech Time-Dependent Mater 6:283–300Google Scholar
  95. Kuhn R, Krömer H, Roßmanith G (1974) Struktur und Eigenschaften verschieden hergestellter Hochdruckpolyäthylene. Ang Makrom Chem 40/41:361–389Google Scholar
  96. Kurose T, Takahashi T, Sugimoto M, Taniguchi T, Koyama K (2005) Uniaxial elongational viscosity of PC/ A small amount of PTFE blend. J Soc Rheol Japan 33(4):173–182Google Scholar
  97. Kwon Y, Cho KS (2001) Time-strain nonseparability in viscoelastic constitutive equations. J Rheol 45(6):1441–1452ADSGoogle Scholar
  98. Larson RG (1984) A constitutive equation for polymer melts based on partially extending strand convection. J Rheology 28(5):545–571MATHADSGoogle Scholar
  99. Larson RG (1985) Nonlinear shear relaxation modulus for a linear low-density polyethylene. J Rheol 29(6):823–831ADSMathSciNetGoogle Scholar
  100. Larson RG (1988) Constitutive equations for polymer melts and solutions. Buttherworths, USAGoogle Scholar
  101. Larson RG (1999) The structure and rheology of complex fluids. Oxford University Press, USAGoogle Scholar
  102. Larson RG, Monroe K (1984) The BKZ as an alternative to the Wagner model for shifting shear and elongational flow data of an LDPE melt. Rheol Acta 23:10–13Google Scholar
  103. Larson RG, Monroe K (1987) Correction. Rheol Acta 26:208–209Google Scholar
  104. Larson RG, Valesano VA (1986) Are polymer melts visco-anelastic? J Rheol 30(6):1093–1108ADSGoogle Scholar
  105. Laun HM (1978) Description of the non-linear shear behaviour of a low density polyethylene melt by means of an experimentally determined strain dependent memory function. Rheol Acta 17:1–15Google Scholar
  106. Laun HM, Wagner MH, Janeschitz-Kriegl H (1979) Model analysis of nonlinear viscoelastic behaviour by use of a single integral constitutive equation: stresses and birefringence of a polystyrene melt in intermittent shear flows. Rheol Acta 18:615–622Google Scholar
  107. Le Meins JF, Moldenaers P, Mewis J (2002) Suspensions in polymer melts. 1. Effect of particle size on the shear flow behavior. Ind Eng Chem Res 41:6297–6304Google Scholar
  108. Leblans PJR (1987) Nonlinear viscoelasticity of polymer melts in different types of flow. Rheol Acta 26:135–143Google Scholar
  109. Leblans PJR, Sampers J, Booij HC (1985) Rheological properties of some polyolefine melts in transient uniaxial elongational flow, described with a special type of constitutive equation. J Non-Newton Fluid Mech 19:185–207MATHGoogle Scholar
  110. Lee JH, Orfanou K, Driva P, Iatrou H, Hadjichristidis N, Lohse DJ (2008) Linear and nonlinear rheology of dendritic star polymers: experiment. Macromolecules 41:9165–9178Google Scholar
  111. Lefebvre J (2006) An outline of the non-linear viscoelastic behaviour of wheat flour dough in shear. Rheol Acta 45:525–538Google Scholar
  112. Li TQ, Wolcott MP (2006) Rheology of wood plastics melt, part 3: nonlinear nature of the flow. Polym Eng Sci 46(1):114–121Google Scholar
  113. Lin YH, Das AK (2007) Monte Carlo simulations of stress relaxation of entanglement-free Fraenkel chains. II. Nonlinear polymer viscoelasticity. J Chem Phys 126:074903Google Scholar
  114. Lodge AS (1964) Elastic liquids. An introductory vector treatment of finite-strain polymer rheology. Academic Press, Great BritainGoogle Scholar
  115. Lodge AS (1968) Constitutive equation from molecular network theories for polymer solutions. Rheol Acta 7(4):379–392MATHGoogle Scholar
  116. Luo XL, Tanner RI (1988) Finite element simulation of long and short circular die extrusion experiments using integral models. Int J Numer Methods Eng 25(1):9–22MATHGoogle Scholar
  117. Mackley MR, Marshall RTJ, Smeulders JBAF, Zhao FD (1994) The rheological characterization of polymeric and colloidal fluids. Chem Eng Sci 49(16):2551–2565Google Scholar
  118. Macosko WCh (1994) Rheology. Principles, measurements, and applications. Wiley-VCH, USAGoogle Scholar
  119. Malkin AY, Isayev AI (2006) Rheology. Concepts, methods and applications. ChemTec Publisching, TorontoGoogle Scholar
  120. Marini L, Georgakis Ch (1984) Low-density polyethylene vessel reactors. Part I: steady state and dynamic modelling. AIChE 30(3):401–408Google Scholar
  121. Marrucci G, Grizzuti N (1983) The free energy function of the Doi-Edwards theory: analysis of the instabilities in stress relaxation. J Rheol 27(5):433–450ADSGoogle Scholar
  122. Marrucci G, Ianniruberto G (2003) Flow-induced orientation and stretching of entangled polymers. Phil Trans R Soc Lond A 361:677–688ADSGoogle Scholar
  123. Marrucci G, Greco F, Ianniruberto G (2000a) Possible role of force balance on entanglements. Macromol Symp 158:57–64Google Scholar
  124. Marrucci G, Greco F, Ianniruberto G (2000b) Simple strain measure for entangled polymers. J Rheol 44(4):845–854ADSGoogle Scholar
  125. Matsumiya Y, Watanabe H (2004) Nonlinear relaxation behaviour of diblock copolymer micellar dispersions: effects of corona-matrix and corona-corona entanglements. Macromolecules 37:9861–9871Google Scholar
  126. McLeish TCB, Larson RG (1998) Molecular constitutive equations for a class of branched polymers: the pom-pom polymer. J Rheol 42(1):81–110Google Scholar
  127. Menezes EV (1980) Some relations and tests on a constitutive equation with a factorized memory function. J Non-Newton Fluid Mech 7:45–62Google Scholar
  128. Mhetar V, Archer LA (1999) Nonlinear viscoelasticity of entangled polymeric liquids. J Non-Newton Fluid Mech 81:71–81MATHGoogle Scholar
  129. Milner ST, McLeish TCB, Likhtman AE (2001) Microscopy theory of convective constraint release. J Rheol 45(2):539–563ADSGoogle Scholar
  130. Minegishi A, Nishioka A, Takahashi T, Masubuchi Y, Takimoto J, Koyama K (2001) Uniaxial elongational viscosity of PS/a small amount of UHMW-PS blends. Rheol Acta 40:329–338Google Scholar
  131. Mongruel A, Cartault M (2006) Nonlinear rheology of styrene-butadiene rubber filled with carbon-black or silica particles. J Rheol 50(2):115–135ADSGoogle Scholar
  132. Morrison FA, Larson RG (1992) A study of shear-stress relaxation anomalies in binary of monodisperse polystyrenes. J Polym Sci B Polym Phys 30:943–950Google Scholar
  133. Muliawan EB, Hatzikiriakos SG (2008) The effect of refrigerated storage on the rheological properties of three commercial mozzarella cheeses. Int J Food Eng 4(4):1–19Google Scholar
  134. Ng TSK, McKinley GH (2008) Power law gels at finite strains: the nonlinear rheology of gluten gels. J Rheol 52(2):417–449ADSGoogle Scholar
  135. Ng TSK, McKinley GH, Padmanabhan M (2006) Linear to non-linear rheology of wheat flour-water dough. Appl Rheol 16(5):265–274Google Scholar
  136. Nielsen JK, Rasmussen HK, Hassager O, McKinley GH (2006a) Elongational viscosity of monodisperse and bidisperse polystyrene melts. J Rheol 50:453–476ADSGoogle Scholar
  137. Nielsen JK, Rasmussen HK, Denberg M, Almdal K, Hassager O (2006b) Nonlinear branch-point dynamics of multiarm polystyrene. Macromolecules 39:8844–8853Google Scholar
  138. Nishioka A, Takahashi T, Masubuchi Y, Takimoto J, Koyama K (2000) Description of uniaxial, biaxial, and planar elongational viscosities of polystyrene melt by the K-BKZ model. J Non-Newton Fluid Mech 89:287–301Google Scholar
  139. Nishioka A, Takahashi T, Masubuchi Y, Takimoto J, Koyama K (2002) Rheological characterization of ionic bounding in ethylene-ionomer melts with low neutralization degree. J Rheol 46(6):1325–1339ADSGoogle Scholar
  140. Nithi-Uthai N, Manas-Zloczower I (2002) Numerical studies of the effect of constitutive model parameters as reflecting polymer molecular structure on extrudate swell. Appl Rheol 12:252–259Google Scholar
  141. Noll W (1958) A mathematical theory of the mechanical behavior of continuous media. Arch Ration Mech Anal 2:197–226MATHGoogle Scholar
  142. Odian G (1981) Principles of polymerization. Wiley, USAGoogle Scholar
  143. Okamoto K, Takahashi M, Yamane H, Kashihara H, Watanabe H, Masuda T (1999) Shape recovery of a dispersed droplet phase and stress relaxation stress after application of step shear strains in a polystyrene/polycarbonate blend melt. J Rheol 43(4):951–965Google Scholar
  144. Osaki K (1993) On the damping function of shear relaxation modulus for entangled polymers. Rheol Acta 32:429–437Google Scholar
  145. Osaki K (1999) Constitutive equation and damping function for entangled polymers. Korea-Australia Rheol J 11(4):287–291Google Scholar
  146. Osaki K, Kurata M (1980) Experimental appraisal of the Doi-Edwards theory for polymer rheology based on the data for polystyrene solutions. Macromolecules 13:671–676Google Scholar
  147. Osaki K, Ohta S, Fukuda M, Kurata M (1976) Nonlinear viscoelasticity of polystyrene solutions. III. Stress development at the start of steady shear flow and an experimental check of some constitutive models. J Polym Sci, Polym Phys Ed 14:1701–1715Google Scholar
  148. Osaki K, Kim BS, Kurata M (1979) Rheology of copolymer solutions. IV. Nonlinear viscoelasticity of solutions of an SBS block copolymer. Polym J 11(1):33–42Google Scholar
  149. Osaki K, Kimura S, Kurata M (1981) Relaxation of shear and normal stresses in double-step shear deformations for a polystyrene solution. A test of the Doi–Edwards theory for polymer rheology. J Rheol 25(5):549–562Google Scholar
  150. Osaki K, Nishizawa K, Kurata M (1982) Material time constant characterizing the nonlinear viscoelasticity of entangled polymeric systems. Macromolecules 15:1068–1071Google Scholar
  151. Osaki K, Takatori E, Kurata M (1987) Nonlinear viscoelasticity of semidilute polystyrene solutions. Effect of molecular weight distribution. Macromolecules 20:1681–1687Google Scholar
  152. Osaki K, Takatori E, Kurata M, Watanabe H, Yoshida H, Kotaka T (1990) Viscoelastic properties of solutions of star-branched polystyrene. Macromolecules 23:4392–4396Google Scholar
  153. Osaki K, Takatori E, Watanabe H, Kotaka T (1993) Viscoelastic properties of semidilute poly(methyl methacrylate) solutions. Rheol Acta 32:132–139Google Scholar
  154. Osaki K, Watanabe H, Inoue T (1996) Damping function of the shear relaxation modulus and the chain retraction process of entangled polymers. Macromolecules 29:3611–3614Google Scholar
  155. Otsuki Y, Kajiwara T, Funatsu K (1997) Numerical simulations of annular extrudate swell of polymer melts. Polym Eng Sci 37(7):1171–1181Google Scholar
  156. Otsuki Y, Kajiwara T, Funatsu K (1999) Numerical simulations of annular extrudate swell using various types of viscoelastic models. Polym Eng Sci 39(10):1969–1981Google Scholar
  157. Papanastasiou AC, Scriven LE, Macosko CW (1983) An integral constitutive equation for mixed flows: viscoelastic characterization. J Rheol 27(4):387–410ADSGoogle Scholar
  158. Partal P, Guerrero A, Berjano M, Gallegos C (1999) Transient flow of o/w sucrose palmitate emulsions. J Food Eng 41:33–41Google Scholar
  159. Petrie CJS (1979) Measures of deformation and convected derivatives. J Non-Newton Fluid Mech 5:147–176MATHGoogle Scholar
  160. Phan-Thien N (2002) Understanding viscoelasticity. Basics of rheology. Springer, GermanyMATHGoogle Scholar
  161. Pol HV, Joshi YM, Tapadia PS, Lele AK, Mashelkar RA (2007) A geometrical solution to the sharkskin instability. Ind Eng Chem Res 46:3048–3056Google Scholar
  162. Raible T, Stephenson SE, Meissner J, Wagner MH (1982) Constant force elongational flow of a low-density polyethylene melt-experiment and theory. J Non-Newton Fluid Mech 11:239–256Google Scholar
  163. Rasmussen HK, Nielsen JK, Bach A, Hassager O (2005) Viscosity overshoot in the start-up of uniaxial elongation of low density polyethylene melts. J Rheol 49(2):369–381ADSGoogle Scholar
  164. Ravindranath S, Wang SQ (2007) What are the origins of stress relaxation behaviors in step shear of entangled polymer solutions? Macromolecules 40:8031–8039Google Scholar
  165. Ren J, Krishnamoorit R (2003) Nonlinear viscoelastic properties of layered-silicate-based intercalated nanocomposites. Macromolecules 36:4443–4451Google Scholar
  166. Riscardo MA, Moros JE, Franco JM, Gallegos C (2005) Rheological characterization of salad-dressing-type emulsions stabilized by egg yolk/sucrose distearate blends. Eur Food Res Technol 220:380–388Google Scholar
  167. Rivlin RS, Sawyers KN (1971) Nonlinear continuum mechanics of viscoelastic fluids. Ann Rev Fluid Mech 3:117–146ADSGoogle Scholar
  168. Rolón-Garrido VH, Wagner MH (2007) The MSF model: relation of nonlinear parameters to molecular structure of long-chain branched polymer melts. Rheol Acta 46:583–593Google Scholar
  169. Roovers J (1984) Melt rheology of H-shaped polystyrenes. Macromolecules 17:1196–1200Google Scholar
  170. Rubio P, Wagner MH (2000) LDPE melt rheology and the pom-pom model. J Non-Newton Fluid Mech 92:245–259MATHGoogle Scholar
  171. Samurkas T, Larson RG, Dealy JM (1989) Strong extensional and shearing flows of a branched polythylenes. J Rheol 33:559–578ADSGoogle Scholar
  172. Sanchez-Reyes J, Archer LA (2002) Step shear dynamics of entangled polymer liquids. Macromolecules 35:5194–5202Google Scholar
  173. Shikata T, Hirata H, Takatori E, Osaki K (1988) Nonlinear viscoelastic behaviour of aqueous detergent solutions. J Non-Newton Fluid Mech 28:171–182Google Scholar
  174. Shiraishi Y, Narazaki N, Kikutani T (2001) The application of an integral type constitutive equation to numerical flow analyses of viscoelastic fluid in unsteady flow. Polym Eng Sci 41(10):1695–1704Google Scholar
  175. Sodeifian G, Haghtalab A (2004) Discrete relaxation spectrum and K-BKZ constitutive equation for PVC, NBR and their blends. Appl Rheol 14:180–189Google Scholar
  176. Sofou S, Muliawan EB, Hatzikiriakos SG, Mitsoulis E (2008) Rheological characterization and constitutive modeling of bread dough. Rheol Acta 47:369–381Google Scholar
  177. Soskey PR, Winter HH (1984) Large step shear strain experiments with parallel-disk rotational rheometers. J Rheol 28(5):625–645ADSGoogle Scholar
  178. Soskey PR, Winter HH (1985) Equibiaxial extension of two polymer melts: polystyrene and low density polyethylene. J Rheol 29(5):493–517ADSGoogle Scholar
  179. Stadler FJ, Auhl D, Münstedt H (2008) Influence of the molecular structure of polyolefins on the damping function in shear. Macromolecules 41:3720–3726Google Scholar
  180. Stephenson SE (1980) Biaxial extensional flow of polymer melts and its realization in a newly developed rheometer. PhD Thesis ETH No. 6664Google Scholar
  181. Sugimoto M, Suzuki Y, Hyun K, Ahn KH, Ushioda T, Nishioka A, Taniguchi T, Koyama K (2006) Melt rheology of long-chain-branched polypropylenes. Rheol Acta 46(1):33–44Google Scholar
  182. Sui C, McKenna GB (2007) Nonlinear viscoelastic properties of branched polyethylene in reversing flows. J Rheol 51(3):341–365ADSGoogle Scholar
  183. Sui C, McKenna GB, Puskas JE (2007) Nonlinear viscoelastic response of dendritic (arborescent) polyisobutylenes in single- and reversing double-step shearing flows. J Rheol 51(6):1143–1169ADSGoogle Scholar
  184. Sun J, Phan-Thien N, Tanner RI (1996) Extrudate swell through an orifice die. Rheol Acta 35:1–12Google Scholar
  185. Takahashi M, Isaki T, Takigawa T, Masuda T (1993) Measurement of biaxial and uniaxial extensional flow behaviour of polymer melts at constant strain rates. J Rheol 37(5):827–846ADSGoogle Scholar
  186. Takahashi T, Toda H, Minagawa K, Takimoto J, Iwakura K, Koyama K (1995) Nonlinear stress properties of poly(syterene-block-butadiene-block-styrene) melt under elongational and shear deformation. J Appl Polym Sci 56:411–417Google Scholar
  187. Takahashi T, Takimoto J, Koyama K (1998) Elongational viscosities of random and block copolymer melts. J Appl Polym Sci 69:1765–1774Google Scholar
  188. Takahashi T, Takimoto J, Koyama K (1999) Uniaxial elongational viscosity of various molten polymer composites. Polym Compos 20(3):357–366Google Scholar
  189. Tanner RI (1988) From A to (BK)Z in constitutive relations. J Rheol 32(7):673–702ADSGoogle Scholar
  190. Tanner RI (2006) On the congruence of some network and pom-pom models. Korean-Australia Rheol J 18(1):9–14Google Scholar
  191. Tobolsky AV, Andrews RD (1945) Systems manifesting superposed elastic and viscous behavior. J Chem Phys 13:3–27ADSGoogle Scholar
  192. Tsenoglou CJ, Voyiatzis E, Gotsis AD (2006) Simple constitutive modelling of nonlinear viscoelasticity under general extension. J Non-Newton Fluid Mech 138:33–43Google Scholar
  193. Urakawa O, Takahashi M, Masuda T, Ebrahimi NG (1995) Damping functions and chain relaxation in uniaxial and biaxial extensions: comparison with the Doi–Edwards theory. Macromolecules 28:7196–7201Google Scholar
  194. Urayama K, Kawamura T, Kohjiya S (2001) Multiaxial deformations of end-linked poly(dimethylsiloxane) networks. 2. Experimental tests of molecular entanglement models of rubber elasticity. Macromolecules 34:8261–8269Google Scholar
  195. Valencia C, Sánchez MC, Ciruelos A, Latorre A, Madiedo JM, Gallegos C (2003) Non-linear viscoelasticity modeling of tomato paste products. Food Res Int 36:911–919Google Scholar
  196. Vasquez PA, McKinley GH, Cook LP (2007) A network scission model for wormlike micellar solutions I. Model formulation and viscometric flow predictions. J Non-Newton Fluid Mech 144:122–139Google Scholar
  197. Vega DA, Milner ST (2007) Shear damping function measurements for branched polymers. J Polym Sci Part B Polym Phys 45:3117–3136Google Scholar
  198. Venerus DC (2005) A critical evaluation of step strain flows of entangled linear polymer liquids. J Rheol 49(1):277–295ADSGoogle Scholar
  199. Venerus DC, Kahvand H (1994) Doi-Edwards theory evaluation in double-step strain flows. J Polym Sci B Polym Phys 32:1531–1542Google Scholar
  200. Venerus DC, Nair R (2006) Stress relaxation dynamics of an entangled polystyrene solution following step strain flow. J Rheol 50(1):59–75ADSGoogle Scholar
  201. Venerus DC, Vrentas CM, Vrentas JS (1990) Step strain deformations for viscoelastic fluids: experiment. J Rheol 34(5):657–683ADSGoogle Scholar
  202. Venerus DC, Tariq SA, Bernstein B (1993) On the use of stress growth data to determine strain-dependent material functions for factorable K-BKZ equations. J Non-Newton Fluid Mech 49:299–315Google Scholar
  203. Venerus DC, Brown EF, Burghardt WR (1998) The nonlinear response of a polydisperse polymer solution to step strain deformations. Macromolecules 31:9206–9212Google Scholar
  204. Vinogradov GV, Malkin AYa (1980) Rheology of polymers. Mir Publishers URSSGoogle Scholar
  205. Vrentas CM, Graessley WW (1981) Relaxation of shear and normal stress components in step-strain experiments. J Non-Newton Fluid Mech 9:339–355Google Scholar
  206. Vrentas CM, Graessley WW (1982) Study of shear stress relaxation in well-characterized polymer liquids. J Rheol 26(4):359–371ADSGoogle Scholar
  207. Vrentas JS, Vrentas CM (1993) Strain-coupling effects in extensional flows. J Appl Polym Sci 49:733–740Google Scholar
  208. Vrentas JS, Venerus DC, Vrentas CM (1991a) Step-strain deformations for viscoelastic fluids: formulation of a strain-coupling constitutive equation. J Polym Sci B Polym Phys 29:537–545Google Scholar
  209. Vrentas JS, Vrentas CM, Venerus DC (1991b) Evaluation of the Wagner irreversible constitutive equation. Rheol Acta 30:175–179Google Scholar
  210. Wagner MH (1976a) Analysis of stress-growth data for simple extension of a low-density branched polyethylene melt. Rheol Acta 15:133–135Google Scholar
  211. Wagner MH (1976b) Analysis of time-dependent non-linear stress-growth data for shear and elongational flow of a low-density branched polyethylene melt. Rheol Acta 15:136–142Google Scholar
  212. Wagner MH (1978) A constitutive analysis of uniaxial elongational flow data of a low-density polyethylene melt. J Non-Newton Fluid Mech 4:39–55Google Scholar
  213. Wagner MH (1979) Zur Netzwerktheorie von Polymer-Schmelzen. Rheol Acta 18:33–50Google Scholar
  214. Wagner MH (1990) The nonlinear strain measure of polyisobutylene melt in general biaxial flow and its comparison to the Doi-Edwards model. Rheo Acta 29:594–603Google Scholar
  215. Wagner MH (1992) The slip-link model: a constitutive equation for general biaxial extension of polymer melts. Makromol Chem Macromol Symp 56:13–24Google Scholar
  216. Wagner MH (1994a) Analysis of small angle neutron scattering data on poly(dimethylsiloxane) network unfolding. Macromolecules 27:5223–5226Google Scholar
  217. Wagner MH (1994b) The origin of the C2 term in rubber elasticity. J Rheol 38(3):655–679ADSGoogle Scholar
  218. Wagner MH, Demarmels A (1990) A constitutive analysis of extensional flows of polyisobutylene. J Rheol 34(6):943–958ADSGoogle Scholar
  219. Wagner MH, Ehrecke P (1998) Dynamics of polymer melts in reversing shear flows. J Non-Newton Fluid Mech 76:183–197MATHGoogle Scholar
  220. Wagner MH, Meissner J (1980) Network disentanglement and time-dependent flow behaviour of polymer melts. Makromol Chem 181:1533–1550Google Scholar
  221. Wagner MH, Rolón-Garrido VH (2008) Verification of branch point withdrawal in elongational flow of pom-pom polystyrene melt. J Rheol 52(5):1049–1068Google Scholar
  222. Wagner MH, Schaeffer J (1992) Nonlinear measures for general biaxial extension of polymer melts. J Rheol 36(1):1–26ADSGoogle Scholar
  223. Wagner MH, Schaeffer J (1993) Rubbers and polymer melts: universal aspects of nonlinear stress-strain relations. J Rheol 37(4):643–661ADSGoogle Scholar
  224. Wagner MH, Schaeffer J (1994) Assessment of nonlinear strain measures for extensional and shearing flows of polymer melts. Rheol Acta 33:506–516Google Scholar
  225. Wagner MH, Stephenson SE (1979a) The spike-strain test for polymeric liquids and its relevance for irreversible destruction of network connectivity by deformation. Rheol Acta 18:463–468Google Scholar
  226. Wagner MH, Stephenson SE (1979b) The irreversibility assumption of network disentanglement in flowing polymer melts and its effects on elastic recoil predictions. J Rheol 23(4):489–504ADSGoogle Scholar
  227. Wagner MH, Raible T, Meissner J (1979) Tensile stress overshoot in uniaxial extension of a LDPE melt. Rheol Acta 18:427–428Google Scholar
  228. Wagner MH, Ehrecke P, Hachmann P, Meissner J (1998) A Constitutive analysis of uniaxial, equibiaxial and planar extension of a commercial linear high-density polyethylene melt. J Rheol 42(3):621–638Google Scholar
  229. Wagner MH, Bastian H, Hachmann P, Meissner J, Kurzbeck S, Münstedt H, Langouche F (2000) The strain-hardening behaviour of linear and long-chain branched polyolefin melts in extensional flows. Rheol Acta 39:97–109Google Scholar
  230. Wagner MH, Rubio P, Bastian H (2001) The molecular stress function model for polydisperse polymer melts with dissipative convective constraint release. J Rheol 45(6):1387–1412ADSGoogle Scholar
  231. Wagner MH, Yamaguchi M, Takahashi M (2003) Quantitative assessment of strain hardening of low-density polyethylene melts by the molecular stress function model. J Rheol 47(3):779–793ADSGoogle Scholar
  232. Wagner MH, Hepperle J, Münstedt H (2004) Relating rheology and molecular structure of model branched polystyrene melts by molecular stress function theory. J Rheol 48(3):489–503ADSGoogle Scholar
  233. Wagner MH, Rolón-Garrido VH, Nielsen JK, Rasmussen HK, Hassager O (2008) A constitutive analysis of transient and steady-state elongational viscosities of bidisperse polystyrene blends. J Rheol 52(1):67–86ADSGoogle Scholar
  234. Wang CF, Kokini JL (1995) Simulation of the nonlinear rheological properties of gluten dough using the Wagner constitutive model. J Rheol 39(6):1465–1482ADSGoogle Scholar
  235. Watanabe H, Matsumiya Y (2005) Rheology of diblock copolymer micellar dispersions having soft cores. Macromolecules 38:3808–3819Google Scholar
  236. Watanabe H, Sato T, Osaki K, Yao ML (1996) Relaxation of spherical micellar systems of styrene-isoprene diblock copolymers. 2. Nonlinear stress relaxation behavior. Macromolecules 29:3890–3897Google Scholar
  237. Watanabe H, Yao ML, Sato T, Osaki K (1997) Non-newtonian flow behaviour of diblock copolymer micelles: shear-thinning in a nonentangling matrix. Macromolecules 30:5905–5912Google Scholar
  238. Watanabe H, Sato T, Osaki K, Aoki Y, Li L, Kakiuchi M, Yao ML (1998a) Rheological images of poly(vinyl chloride) gels. 4. Nonlinear behavior in a critical gel state. Macromolecules 31:4198–4204Google Scholar
  239. Watanabe H, Osaki K, Matsumoto M, Bossev DP, McNamee CE, Nakahara M, Yao ML (1998b) Nonlinear rheology of threadlike micelles of tetraethylammonium perfluorooctyl sulfonate. Rheol Acta 37:470–485Google Scholar
  240. Watanabe H, Yao ML, Osaki K, Shikata T, Niwa H, Morishima Y (1999) Nonlinear rheology of concentrated spherical silica suspensions: 3. Concentration dependence. Rheol Acta 38:2–13Google Scholar
  241. Watanabe H, Matsumiya Y, Ishida S, Takigawa T, Yamamoto T, Vlassopoulos D, Roovers J (2005) Nonlinear rheology of multiarm star chains. Macromolecules 38:7404–7415Google Scholar
  242. Waton G, Michels B, Steyer A, Schosseler F (2004) Shear-induced demixing and shear-banding instabilities in dilute triblock copolymer solutions. Macromolecules 37:2313–2321Google Scholar
  243. Wekumbura C, Stastna J, Zanzotto L (2005) Stress growth coefficient in polymer modified asphalt. Mater Struct 38:755–760Google Scholar
  244. Winter HH (1978) On the network models of molten polymers: loss of junctions due to stretching of material planes. Rheol Acta 17:589–594Google Scholar
  245. Yamaguchi M, Takahashi M (2001) Rheological properties of low-density polyethylenes produced by tubular and vessel processes. Polymer 42:8663–8670Google Scholar
  246. Yamamoto T, Ohta Y, Takigawa T, Masuda T (2002) Stress relaxation of multi-arm star polystyrenes in the molten state. Nihon Reoroji Gakkaishi 30(3):129–132Google Scholar
  247. Yin G, Solomon MJ (2008) Soft glassy rheology model applied to stress relaxation of a thermoreversible colloidal gel. J Rheol 52(3):785–800ADSGoogle Scholar
  248. Zapas LJ, Phillips JC (1971) Simple shearing flows in polyisobutylene solutions. J Res Natl Bur Stand 75A:33–40Google Scholar
  249. Zdilar AM, Tanner RI (1992) The recoil of rigid PVC. Rheol Acta 31:44–54Google Scholar
  250. Zdilar AM, Tanner RI (1993) Erratum. Rheol Acta 32:114Google Scholar
  251. Zdilar AM, Tanner RI (1994) Role of structure in rigid PVC recoil. J Rheol 38(4):909–920ADSGoogle Scholar
  252. Zeng XS, Takahashi M, Yamane H, Takigawa T, Masuda T (1999a) Dynamic viscoelasticity of ionomers based on ethylene-co-methacrylic acid copolymer in the melt state. J Soc Rheol Japan 27(1):53–57Google Scholar
  253. Zeng XS, Takahashi M, Yamane H, Masuda T (1999b) Stress relaxation under large step strain for ionomers based on ethylene-co-methacrylic acid copolymer in the melt state. J Soc Rheol Japan 27(1):59–62Google Scholar
  254. Zheng Q, Wang W, Yu Q, Yu J, He L, Tan H (2006) Nonlinear viscoelastic behaviour of styrene-[ethylene-ethylene-propylene)]-styrene block copolymer. J Polym Sci B Polym Phys 44:1309–1319Google Scholar
  255. Zhou L, Vasquez PA, Cook LP, McKinley GH (2008) Modeling the inhomogeneous response and formation of shear bands in steady and transient flows of entangled liquids. J Rheol 52(2):591–623ADSGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Víctor H. Rolón-Garrido
    • 1
  • Manfred H. Wagner
    • 1
  1. 1.Polymer Engineering/Polymer PhysicsTU BerlinBerlinGermany

Personalised recommendations