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Continuum mechanics

Rheology (pronounced /riˈɒlədʒi/) is the study of the flow of matter: mainly liquids but also soft solids or solids under conditions in which they flow rather than deform elastically[1]. It applies to substances which have a complex structure, including muds, sludges, suspensions, polymers, many foods, bodily fluids, and other biological materials. The flow of these substances cannot be characterized by a single value of viscosity (at a fixed temperature)[2] - instead the viscosity changes due to other factors. For example, ketchup can have its viscosity reduced by shaking, but water cannot. Since Isaac Newton originated the concept of viscosity, the study of variable viscosity liquids is also often called Non-Newtonian fluid mechanics.[1] The term rheology was coined by Eugene C. Bingham, a professor at Lafayette College, in 1920, from a suggestion by a colleague, Markus Reiner.[3] The term was inspired by the quotation mistakenly attributed to Heraclitus, (actually coming from the writings of Simplicius) panta rei, "everything flows". The experimental characterisation of a material's rheological behavior is known as rheometry, although the term rheology is frequently used synonymously with rheometry, particularly by experimentalists. Theoretical aspects of rheology are the relation of the flow/deformation behavior of material and its internal structure (e.g., the orientation and elongation of polymer molecules), and the flow/deformation behavior of materials that cannot be described by classical fluid mechanics or elasticity.



In practice, rheology is principally concerned with extending the "classical" disciplines of elasticity and (Newtonian) fluid mechanics to materials whose mechanical behavior cannot be described with the classical theories. It is also concerned with establishing predictions for mechanical behavior (on the continuum mechanical scale) based on the micro- or nanostructure of the material, e.g. the molecular size and architecture of polymers in solution or the particle size distribution in a solid suspension. Materials flow when subjected to a stress, that is a force per area. There are different sorts of stress[4] and materials can respond in various ways, so much of theoretical rheology is concerned with forces and stresses.[1]

Continuum mechanics Solid mechanics or strength of materials Elasticity
Plasticity Rheology
Fluid mechanics Non-Newtonian fluids
Newtonian fluids

Rheology unites the seemingly unrelated fields of plasticity and non-Newtonian fluids by recognizing that both these types of materials are unable to support a shear stress in static equilibrium. In this sense, a plastic solid is a fluid. Granular rheology refers to the continuum mechanical description of granular materials.

One of the tasks of rheology is to empirically establish the relationships between deformations and stresses, respectively their derivatives by adequate measurements. These experimental techniques are known as rheometry and are concerned with the determination with well-defined rheological material functions. Such relationships are then amenable to mathematical treatment by the established methods of continuum mechanics.

The characterisation of flow or deformation originating from a simple shear stress field is called shear rheometry (or shear rheology). The study of extensional flows is called extensional rheology. Shear flows are much easier to study and thus much more experimental data are available for shear flows than for extensional flows.


A rheologist is an interdisciplinary scientist who studies the flow of complex liquids or the deformation of soft solids. It is not taken as a primary degree subject, and there is no general qualification. He or she will usually have a primary qualification in one of several fields: mathematics, the physical sciences[5], engineering[6], medicine, or certain technologies, notably materials or food. A small amount of rheology may be given during the first degree, but the professional will extend this knowledge during postgraduate research or by attending short courses and by joining one of the professional associations (see below).


Rheology has applications in engineering, geophysics, physiology and pharmaceutics. In engineering, it affects the production and use of polymeric materials, but plasticity theory has been similarly important for the design of metal forming processes. Many industrially important substances such as concrete, paint and chocolate have complex flow characteristics. Geophysics includes the flow of lava, but in addition measures the flow of solid Earth materials over long time scales: those that display viscous behavior, e.g. granite [7], are known as rheids. In physiology, many bodily fluids have complex compositions and thus flow characteristics. In particular there is a specialist study of blood flow called hemorheology. The term biorheology is used for the wider field of study of the flow properties of biological fluids. Food rheology is important in the manufacture and processing of food products.[8]

Elasticity, viscosity, solid- and liquid-like behavior, plasticity

One generally associates liquids with viscous behavior (a thick oil is a viscous liquid) and solids with elastic behavior (an elastic string is an elastic solid). A more general point of view is to consider the material behavior at short times (relative to the duration of the experiment/application of interest) and at long times.

Fluid and solid character are relevant at long times

We consider the application of a constant stress (a so-called creep experiment):

  • if the material, after some deformation, eventually resists further deformation, it is considered a solid
  • if, by contrast, the material flows indefinitely, it is considered a fluid
By contrast, elastic and viscous (or intermediate, viscoelastic) behavior is relevant at short times (transient behavior)

We again consider the application of a constant stress:

  • if the material deformation strain increases linearly with increasing applied stress , then the material is purely elastic
  • if the material deformation rate increases linearly with increasing applied stress, then the material is purely viscous
  • if neither the deformation strain, nor its derivative with time (rate) follows the applied stress, then the material is viscoelastic
Plasticity is equivalent to the existence of a yield stress

A material that behaves as a solid under low applied stresses may start to flow above a certain level of stress, called the yield stress of the material. The term plastic solid is often used when this plasticity threshold is rather high, while yield stress fluid is used when the threshold stress is rather low. There is no fundamental difference, however, between both concepts.

Dimensionless numbers in rheology

Deborah number

When the rheological behavior of a material includes a transition from elastic to viscous as the time scale increases (or, more generally, a transition from a more resistant to a less resistant behavior), one may define the relevant time scale as a relaxation time of the material. Correspondingly, the ratio of the relaxation time of a material to the timescale of a deformation is called Deborah number. Small Deborah numbers correspond to situations where the material has time to relax (and behaves in a viscous manner), while high Deborah numbers correspond to situations where the material behaves rather elastically.[9]

Note that the Deborah number is relevant for materials that flow on long time scales (like a Maxwell fluid) but not for the reverse kind of materials (like the Voigt or Kelvin model) that are viscous on short time scales but solid on the long term.

Reynolds number

In fluid mechanics, the Reynolds number is a measure of the ratio of inertial forces (vsρ) to viscous forces (μ/L) and consequently it quantifies the relative importance of these two types of effect for given flow conditions. Under low Reynolds numbers viscous effects dominate and the flow is laminar, whereas at high Reynolds numbers inertia predominates and the flow may be turbulent. However, since rheology is concerned with fluids which do not have a fixed viscosity, but one which can vary with flow and time, calculation of the Reynolds number can be complicated.

It is one of the most important dimensionless numbers in fluid dynamics and is used, usually along with other dimensionless numbers, to provide a criterion for determining dynamic similitude. When two geometrically similar flow patterns, in perhaps different fluids with possibly different flow rates, have the same values for the relevant dimensionless numbers, they are said to be dynamically similar.

Typically it is given as follows:

 \mathit{Re} = {\rho v_{s}^2/L \over \mu v_{s}/L^2} = {\rho v_{s} L\over \mu} = {v_{s} L\over \nu}


  • vs - mean fluid velocity, [m s-1]
  • L - characteristic length, [m]
  • μ - (absolute) dynamic fluid viscosity, [N s m-2] or [Pa s]
  • ν - kinematic fluid viscosity: ν = μ / ρ, [m² s-1]
  • ρ - fluid density, [kg m-3].


Rheometers are instruments used to characterize the rheological properties of materials, typically fluids and melts. These instruments impose a specific stress field or deformation to the fluid, and monitor the resultant deformation or stress. Instruments can be run in steady flow or oscillatory flow, in both shear and extension.

Notes and references

  1. ^ a b c W. R. Schowalter (1978) Mechanics of Non-Newtonian Fluids Pergamon ISBN 0-08021778-8
  2. ^ While the viscosity of liquids normally varies with temperature, it is variations with other factors which are studied in rheology
  3. ^ J. F. Steffe (1996) Rheological Methods in Food Process Engineering 2nd ed ISBN 0-9632036-1-4 page 1
  4. ^ for example, a shear stress or extensional stress
  5. ^ mainly chemistry, physics, biology
  6. ^ mainly mechanical, chemical or civil engineering
  7. ^ Kumagai, Naoichi; Sadao Sasajima, Hidebumi Ito (15 February 1978). "Long-term Creep of Rocks: Results with Large Specimens Obtained in about 20 Years and Those with Small Specimens in about 3 Years". Journal of the Society of Materials Science (Japan) (Japan Energy Society) 27 (293): 157–161. Retrieved 2008-06-16. 
  8. ^ B.M. McKenna, and J.G. Lyng. "Texture in food > Introduction to food rheology and its measurement". Retrieved 2009-09-18. 
  9. ^ M. Reiner (1964) Physics Today volume 17 no 1 page 62 The Deborah Number

See also

Further reading

External links

Journals covering rheology
Organizations concerned with the study of rheology
Rheology Conferences


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