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A homotopy of the identity map to some other map.
Any alteration of shape or dimensions of a body caused by stresses, thermal expansion or contraction, chemical or metallurgical transformations, or shrinkage and expansions due to moisture change.


An act of deforming or changing the shape or an alteration in form that a structure undergoes when subjected to the action of a weight or load.



the change in the relative positions of the particles of a body associated with their displacement. It results from a change in interatomic distances and the regrouping of blocks of atoms. Deformation is usually accompanied by an alteration in the magnitudes of the interatomic forces, a measure of which is elastic stress.

The simplest forms of deformation of a body as a whole are extension-compression, shear, flexure, and torsion. In most cases the observed deformation is a number of types of deformation simultaneously. Ultimately, however, it is possible to reduce any deformation to two of the simplest forms, extension (or compression) and shear. The deformation of a body is completely determinable if the displacement vector for each of its points is known. The deformation of solids in connection with their structural peculiarities is studied in solid-state physics, and the displacements and stresses in solids that are being deformed are investigated by the theory of elasticity and plasticity. In liquids and gases, whose particles have high mobility, the study of deformation is replaced by the study of instantaneous velocity distribution.

Deformation of a solid can be manifested as a consequence of phase transitions associated with a change in volume and with thermal expansion, magnetization (the magnetostrictive effect), and the appearance of an electric charge (the piezoelectric effect), or as a result of the action of external forces. The deformation is called elastic if it disappears after the removal of the load that caused it and plastic if it does not disappear (or does not disappear completely) after the load is removed. Upon deformation all real solids have plastic properties to a greater or lesser extent. Under certain conditions the plastic properties of bodies may be disregarded, as is done in elasticity theory. A solid may with sufficient accuracy be considered elastic, that is, not exhibiting appreciable plastic deformation, as long as the load does not exceed a certain limit.

The nature of plastic deformation may vary depending on temperature, the duration of action of the load, and the rate of deformation. If the load applied to the body is constant, the deformation changes with time; this phenomenon is called creep. The creep rate increases with temperature. Relaxation and elastic aftereffect are special cases of creep. Relaxation is the process of spontaneous decrease in internal stress over time at constant deformation. The process of spontaneous increase in deformation at constant stress is called an aftereffect. One of the theories explaining the mechanism of plastic deformation is the theory of dislocations in crystals.

In the theory of elasticity and plasticity, bodies are regarded as continuous. Continuity, which is the ability to fill the entire volume occupied by the material of a body, without any empty space, is one of the basic properties ascribed to real bodies. The concept of continuity also applies to the elementary volumes into which a body can be divided mentally. A change in the distance between the centers of each of two contiguous infinitely small volumes in a body not subject to fracture must be small compared with the initial value of that distance.

The simplest elementary deformation is the relative elongation of a certain element: є = (l1- 1)ll, where l1 is the length of an element after deformation and l is the original length of the element. In practice, small deformations are more often encountered, so that є < < 1.

Deformation is measured either in the process of testing materials to ascertain their mechanical properties or in the study of structures by actual measurement on them or on models to evaluate the stresses. Elastic deformations are quite small, and high accuracy is necessary when measuring them. Strain gauges are most generally used to study deformation. Extensive use is also made of resistance strain gauges, the optical polarization method of studying stresses, and X-ray structural analysis. In assessing local plastic deformation, a grid is etched on the surface of an article or the surface is covered with an easily cracked lacquer.


Rabotnov, Iu. N. Soprotivlenie materialov. Moscow, 1950.
Kuznetsov, V. D. Fizika tverdogo tela, vols. 2-4, 2nd ed. Tomsk 1941-47.
Sedov, L. I. Vvedenie v mekhaniku sploshnoi sredy. Moscow, 1962.


Any change of form, shape, or dimensions produced in a body by a stress or force, without a breach of the continuity of its parts.
References in periodicals archive ?
In the case of a high viscosity ratio blend, the dispersed phase near the flow front slightly deforms during the filling stage, and breakup of the dispersed phase is not possible during the cooling stage.
In case of high viscosily ratio blends, the hard particles scarcely break up even though they deform in the barrel-to-sprue zone: they relax and return to their original shape instead of breaking up since the deformation is not high enough to cause the hard particles to break up.
In the most extensive computer calculations of their type to date, the scientists used the ACSI White supercomputer, which was built last year by IBM for LLNL, to create and then deform simulated cubes of as many as 1 billion atoms.
Rubber can deform under four basic modes of deformation: uniaxial, equi-biaxial, planar and volumetric.
The results revealed that the two artificial materials, like real skin, can deform to closely match an outside force.
Did the material stretch or deform before breaking?
The deformation inducing means, including panel and elastically stretchable members, has transverse ends secured in a vicinity of the deformation assisting means near the panel member, so as to convexly deform the panel member about the longitudinal axis toward the skin-contactable side as elastically stretchable member elastically contracts.
In addition to improving dampening over the current solid pads, PPI recommended that (i) pads absorb mis-match between ties and (ii) cells deform under the weight of concrete panels.
The input energy to deform a sample is therefore equal to the sum of the energies for hysteresis and resilience.
The force increases in direct proportion to the thermal energy of the chain, kT; it will be more difficult to deform a chain as temperature is increased.
Last year, a group of Australian researchers reported something different in the data coming from three strain-sensing instruments they had installed near Parkfield The instruments lie in bore-holes and are designed to sense forces within the crust that gently deform the originally circular holes.
Many geodesists get around that problem by putting antennas on top of buried concrete pillars, but Blewitt rejected this approach because concrete can deform and shrink over the years.