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(rēŏl`əjē), branch of physics dealing with the deformation and flow of matter. It is particularly concerned with the properties of matter that determine its behavior when a mechanical forceforce,
commonly, a "push" or "pull," more properly defined in physics as a quantity that changes the motion, size, or shape of a body. Force is a vector quantity, having both magnitude and direction.
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 is exerted on it. Rheology is distinguished from fluid dynamics (see fluid mechanicsfluid mechanics,
branch of mechanics dealing with the properties and behavior of fluids, i.e., liquids and gases. Because of their ability to flow, liquids and gases have many properties in common not shared by solids.
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) in that it is concerned with all three of the traditional states of matterstates of matter,
forms of matter differing in several properties because of differences in the motions and forces of the molecules (or atoms, ions, or elementary particles) of which they are composed.
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 rather than only with liquids and gases. Unlike polymer physics it is concerned with macroscopic properties and behavior and not with molecular structure. The results of rheology provide a mathematical description of the viscoelastic behavior of matter (see elasticityelasticity,
the ability of a body to resist a distorting influence or stress and to return to its original size and shape when the stress is removed. All solids are elastic for small enough deformations or strains, but if the stress exceeds a certain amount known as the elastic
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; viscosityviscosity,
resistance of a fluid to flow. This resistance acts against the motion of any solid object through the fluid and also against motion of the fluid itself past stationary obstacles.
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). Applications of rheology are important in many areas of industry, involving metals, plastics, and many other materials.


In the broadest sense of the term, that part of mechanics which deals with the relation between force and deformation in material bodies. The nature of this relation depends on the material of which the body is constituted. It is customary to represent the deformation behavior of metals and other solids by a model called the linear or hookean elastic solid (displaying the property known as elasticity) and that of fluids by the model of the linear viscous or newtonian fluid (displaying the property known as viscosity). These classical models are, however, inadequate to depict certain nonlinear and time-dependent deformation behavior that is sometimes observed. It is these nonclassical behaviors which are the chief interest of rheologists and hence referred to as rheological behavior. See Viscosity

Rheological behavior is particularly readily observed in materials containing polymer molecules which typically contain thousands of atoms per molecule, although such properties are also exhibited in some experiments on metals, glasses, and gases. Thus rheology is of interest not only to mathematicians and physicists, who consider it to be a part of continuum mechanics, but also to chemists and engineers who have to deal with these materials. It is of special importance in the plastics, rubber, film, and coatings industries. See Fluid mechanics

Models and properties

Consider a block of material of height h deformed in the manner indicated in Fig. 1; the bottom surface is fixed and the top moves a distance w parallel to itself. A measure of the deformation is the shear strain γ given by Eq. (1). (1) 

Simple shearenlarge picture
Simple shear

To achieve such a deformation if the block is a linear elastic material, it is necessary to apply uniformly distributed tangential forces on the top and bottom of the block as shown in Fig. 1b. The intensity of these forces, that is, the magnitude of the net force per unit area, is called the shear stress S. For a linear elastic material, γ is much less than unity and is related

to S by Eq. (2), where the proportionality constant G is a property of the material known as the shear modulus.

If the material in the block is a newtonian fluid and a similar set of forces is imposed, the result is a simple shearing flow, a deformation as pictured in Fig. 1b with the top surface moving with a velocity dw/dt. This type of motion is characterized by a rate of shear = (dw/dt)/h, which is proportional to the shear stress S as given by Eq. (3), where &eegr; is a property of the material called the viscosity.


Linear viscoelasticity

If the imposed forces are small enough, time-dependent deformation behavior can often be described by the model of linear viscoelasticity. The material properties in this model are most easily specified in terms of simple experiments.

In a creep experiment a stress is suddenly applied and then held constant; the deformation is then followed as a function of time. This stress history is indicated in the solid line of Fig. 2a for the case of an applied constant shear stress S0. If such an experiment is performed on a linear elastic solid, the resultant deformation is indicated by the full line in Fig. 2b and for the linear viscous fluid in Fig. 2c. In the case of elasticity, the result is an instantly achieved constant strain; in the case of the fluid, an instantly achieved constant rate of strain. In the case of viscoelastic materials, there are some which eventually attain a constant equilibrium strain (Fig. 2d) and hence are called viscoelastic solids. Others eventually achieve constant rate of strain (Fig. 2e) and are called viscoelastic fluids. If the material is linear viscoelastic, the deformation γ(S0, t) is a function of the time t since the stress was applied and also a linear function of S0; that is, Eq. (4)

is satisfied, where J(t) is independent of S0. The function J(t) is a property of the material known as the shear creep compliance.

Creep and recovery; solid lines indicate creep; broken lines indicate recoveryenlarge picture
Creep and recovery; solid lines indicate creep; broken lines indicate recovery

Nonlinear viscoelasticity

If stresses become too high, linear viscoelasticity is no longer an adequate model for materials which exhibit time-dependent behavior. In a creep experiment, for example, the ratio of the strain to stress, γ(t, S0)/S0, is no longer independent of S0; this ratio generally decreases with increasing S0. Two examples of nonlinear viscoelasticity are shear thinning and thixotropy.

For polymer melts, solutions, and suspensions, generally speaking, the viscosity decreases as the shear rate increases. This type of behavior, called shear thinning, is of considerable industrial significance. For example, paints are formulated to be shear-thinning. A high viscosity at low flow rates keeps the paint from dripping from the brush or roller and prevents sagging of the paint film newly applied to a vertical wall. The lower viscosity at the high deformation rates while brushing or rolling means that less energy is required, and hence the painter's arm does not become overly tired.

Thixotropy is a property of suspensions (for example, bentonite clay in water) which, after remaining at rest for a long time, act as solids; for example, they cannot be poured. However, if it is stirred, such a suspension can be poured quite freely. If the suspension is then allowed to rest, the viscosity increases with time and finally sets again. This whole process is reversible; it can be repeated again and again. See Non-newtonian fluid



the science of the deformation and flow of matter. It studies processes associated with the irreversible permanent deformation and flow of various viscous and plastic materials, such as non-Newtonian fluids and disperse systems. It also deals with such phenomena as stress relaxation and elastic aftereffect.

The term “rheology” was introduced by the American scientist E. Bingham, who made valuable rheological investigations of fluids and disperse systems. The term was officially adopted at the Third Plasticity Symposium, which took place in 1929 in the USA. Some of the basic concepts of the science, however, were established long before this date. Rheology is closely connected with hydromechanics and the theories of elasticity, plasticity, and creep, and it makes extensive use of the methods of viscosimetry. The science of rheology is based on the laws of I. Newton concerning the resistance to motion of a viscous fluid, the Navier-Stokes equations of the motion of an incompressible viscous fluid, and the work of such scientists as J. Maxwell and W. Thomson. An important contribution to rheology has been made by the Russian scientists D. I. Mendeleev, N. P. Petrov, and F. N. Shvedov and by such Soviet scientists as P. A. Rebinder, M. P. Volarovich, and G. V. Vinogradov.

Rheological problems are encountered in engineering and technology. For example, the development of the technology of various industrial processes has to deal with such problems. Another example is design work and structural calculations involving a wide range of materials, such as metals (especially at high temperatures), composite materials, polymer systems, petroleum products, clays and other soils, rocks, construction materials (for example, concretes and silicates), and food products.

Rheology has several branches. Theoretical rheology (phenomenological rheology or macrorheology) may be considered a part of continuum mechanics and occupies an intermediate position between hydromechanics and the theories of elasticity, plasticity, and creep. It establishes relationships between the strains caused by mechanical stresses acting on a material and the changes of the strains over time. On the basis of the usual assumptions of continuum mechanics regarding the homogeneity and continuity of materials, theoretical rheology solves various boundary value problems of the deformation and flow of solids, liquids, and other materials. Attention is directed primarily to the complex rheological behavior of materials—for example, when viscous and elastic properties or viscous and plastic properties are evidenced simultaneously. A general rheological equation of the state of materials has not yet been established. Equations exist only for individual special cases. In order to describe the rheological behavior of materials, there are used mechanical models, for which differential equations are formulated that involve various combinations of elastic and viscous characteristics. Rheological models are used to study the mechanical properties of polymers, internal friction in solids, and other properties of real materials.

Experimental rheology (rheometry) determines various rheological properties of materials through the use of special instruments and testing machines.

Microrheology studies deformation and flow in microscopic volumes, for example, in volumes comparable in size to particles of the disperse phase in disperse systems or to atoms and molecules.

Biorheology studies the flow of various biological fluids (for example, blood, synovial fluids, and pleural fluids) and the deformation of various tissues (in muscles, bones, and blood vessels) in humans and animals.


Reologiia. Moscow, 1962. (Translated from English.)
Reiner, M. Reologiia. Moscow, 1965. (Translated from English.)
Volarovich, M. P., and N. I. Malinin. “Issledovaniia v oblasti fenomenologicheskoi relogii.” Inzhenerno-fizicheskii zhurnal, 1969, vol. 16, no. 2.
Uspekhi reologiipolimerov. Edited by G. V. Vinogradov. Moscow, 1970.
Rheology, vols. 1–5. New York, 1956–69.
Flow Properties of Blood and Other Biological Systems. Oxford, 1960.



The study of the deformation and flow of matter, especially non-Newtonian flow of liquids and plastic flow of solids.


The science dealing with flow of materials, including studies of deformation of hardened concrete, the handling and placing of freshly mixed concrete, and the behavior of slurries, pastes, and the like.
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