Materials and Structures

, Volume 46, Issue 1–2, pp 289–311 | Cite as

Extension of the Reiner–Riwlin equation to determine modified Bingham parameters measured in coaxial cylinders rheometers

  • Dimitri FeysEmail author
  • Jon E. Wallevik
  • Ammar Yahia
  • Kamal H. Khayat
  • Olafur H. Wallevik
Original Article


The determination of the exact rheological properties, in fundamental units, of cementitious materials has become gradually a necessary step in the domain of concrete science. Several types of rheometers and their corresponding transformation equations are described in the literature. In this paper, the Reiner–Riwlin transformation equation, valid for coaxial cylinders rheometers, is developed for the modified Bingham model, which is an extension of the Bingham model with a second order term in the shear rate. The established transformation is shown to be compatible with the Reiner–Riwlin equation for the Bingham and Herschel–Bulkley models. Its validation is further proven by means of numerical simulations applied on experimental data. The yield stress values for the three rheological models (applied on the same experimental data) are compared with the yield stress calculated by means of slump flow values. Results showed that the modified Bingham model results in the most stable yield stress values, which are independent of the non-linear behaviour.


Rheology Modified Bingham Yield stress Shear-thickening Coaxial cylinders rheometer 

List of symbols


Area (m2)


Second order term of second order curve in T-N graph: modified Bingham model (Nm s2)


Second order term: modified Bingham model (Pa s2)


Intercept of curve in T-N graph with T-axis (Nm)


Gravitational acceleration (m/s2)


Inclination of straight line in T-N graph: Bingham model (Nm s)


First order term of second order curve in T-N graph: modified Bingham model (Nm s)


Height of the inner cylinder submerged in concrete in coaxial cylinders rheometer (m)


Unit dyadic


Consistency factor: Herschel–Bulkley model (Pa sn)


Rotational velocity measured in coaxial cylinders rheometer (rps)


Consistency index (power); Herschel–Bulkley model (−)


Unit normal vector


Pressure (Pa)


Spread radius of slump flow (m)


Inner cylinder radius of coaxial cylinders rheometer (m)


Outer cylinder radius of coaxial cylinders rheometer (m)


Theoretical plug radius (m)


Min(R p, R o) = Boundary between sheared and unsheared material (m)


Radial coordinate (m)


Vector location (m)


Deviator stress tensor (Pa)


Torque measured in coaxial cylinders rheometer (Nm)


Extra stress tensor (Pa)


Time (s)


Velocity vector (m/s)


Volume of slump flow sample (m3)

\( \dot{\gamma } \)

Shear rate (s−1)


Regularization parameter for yield stress in numerical simulations (s−1)

\( {\dot{\varvec{\varepsilon }}} \)

Rate of deformation tensor (s−1)


Apparent viscosity (Pa s)


Linear term: modified Bingham model (Pa s)


Plastic viscosity: Bingham model (Pa s)


Density (kg/m3)

\( {\varvec{\sigma}} \)

(total) Stress tensor (Pa)


Shear stress (Pa)


Yield stress (Pa)


Angular velocity at outer radius (rad/s)


Angular velocity at radius r (rad/s)


Second invariant of deviator stress tensor (Pa2)



The authors would like to thank the Belgian Building Research Institute (BBRI) and the Katholic University Leuven (KULeuven) for the use of their ConTec Viscometer 5.


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Copyright information

© RILEM 2012

Authors and Affiliations

  • Dimitri Feys
    • 1
    Email author
  • Jon E. Wallevik
    • 2
  • Ammar Yahia
    • 1
  • Kamal H. Khayat
    • 1
    • 3
  • Olafur H. Wallevik
    • 1
    • 2
  1. 1.Concrete Research Division, Department of Civil EngineeringUniversité de SherbrookeSherbrookeCanada
  2. 2.ICI Rheocenter, Innovation Center IcelandReykjavik UniversityReykjavikIceland
  3. 3.Department of Civil, Architectural and Environmental EngineeringMissouri University of Science and TechnologyRollaUSA

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