Elasticity (redirected from Demand elasticity)

elasticity, 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 limit, a permanent deformation is produced. Both the resistance to stress and the elastic limit depend on the composition of the solid. Some different kinds of stresses are tension, compression, torsion, and shearing (see strength of materials). For each kind of stress and the corresponding strain there is a modulus, i.e., the ratio of the stress to the strain; the ratio of tensile stress to strain for a given material is called its Young's modulus.

Hooke's law [for Robert Hooke] states that, within the elastic limit, strain is proportional to stress.

---

elasticity

Ability of a deformed material body to return to its original shape and size when the forces causing deformation are removed. Most solids show some elastic behaviour, but there is usually a limit—the material's “elastic limit”—to the force from which recovery is possible. Stresses beyond its elastic limit cause the material to yield, or flow, and the result is permanent deformation or breakage. The limit depends on the material's internal structure; for example, steel, though strong, has a low elastic limit and can be extended only about 1% of its length, whereas rubber can be elastically extended up to about 1,000%. Robert Hooke, one of the first to study elasticity, developed a mathematical relation between tension and extension.

---

Elasticity

The property whereby a solid material changes its shape and size under the action of opposing forces, but recovers its original configuration when the forces are removed. The theory of elasticity deals with the relations between the forces acting on a body and the resulting changes in configuration, and is important in many branches of science and technology, for instance, in the design of structures, in the theory of vibration and sound, and in the study of the forces between atoms in crystal lattices.

The forces acting on a body are expressed as stresses and measured as force per unit area. Thus if a bar ABCD of square cross section (illus. a) is fixed at one end and subjected to a force F uniformly distributed over the other end DC, the stress is F/(DC)². This stress causes the bar to become longer and thinner and to assume the shape A′B′C′D′. The strain is measured by the ratio (change in length)/(original length), that is, by (B′C′ - BC)/(BC). According to Hooke's law, stress is proportional to strain, and the ratio of stress to strain is therefore a constant, in this case the Young's modulus, denoted by E, so that E = F(BC)/(DC)² (B′C′ - BC). See Stress and strain, Young's modulus

Poisson's ratio σ is the ratio of lateral strain to longitudinal strain so that σ = BC(DC - D′C′)/DC(B′C′ - BC). The bar of illustration a is in a state of tension, and the stress is tensile; if the force F were reversed in direction, the stress would be compressive. Stresses of this type are called direct or normal stresses; a second type of stress, known as tangential or shear stress, is shown in illus. b. In this case, the configuration ABCD becomes ABC′D′, with the shear forces F acting in the directions AB and CD. The shear strain is measured by the angle Θ, and if the body is originally a cube, the shear stress is F/(DC). The ratio of stress to strain, F/(DC)² Θ, is the shear or rigidity modulus G, which measures the resistance of the material to change in shape without change in volume.

Stresses on a bar

A further elastic constant, the bulk modulus k, measures the resistance to change in volume without changes in shape, and is shown in illus. c. The original configuration is represented by the circle AB, and under a hydrostatic (uniform) pressure P, the circle AB becomes the circle A′B′. The bulk modulus is then k = Pv/Δv, where Δv/v is the volumetric strain. The reciprocal of the bulk modulus is the compressibility.

The elastic constants may be determined directly in the way suggested by their definitions; for instance, Young's modulus can be determined by measuring the relative extension of a rod or wire subjected to a known tensile stress. Less direct methods are, however, usually more convenient and accurate. Prominent among these are the dynamic methods involving frequency of vibration and velocity of sound propagation. The elastic constants can be expressed in terms of frequency of (or velocity in) regularly shaped specimens, together with the dimensions and density, and by measuring these quantities, the elastic constants can be found. The elastic constants can also be determined from the flexure and torsion of bars.

In practice, stress is only proportional to strain, and the strain is only completely recoverable within certain limits called the elastic limits of the material. Above the elastic limits, the material is subject to time-dependent effects, and as the stress is further increased, the ultimate strength of the material is approached.