International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. D. ch. 1.3, pp. 80-81

Section 1.3.3.1. Hooke's law

A. Authiera* and A. Zarembowitchb

a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and bLaboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France
Correspondence e-mail:  aauthier@wanadoo.fr

1.3.3.1. Hooke's law

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Let us consider a metallic bar of length [l_{o}] loaded in pure tension (Fig. 1.3.3.1[link]). Under the action of the uniaxial stress [T = F/{\cal A}] (F applied force, [{\cal A}] area of the section of the bar), the bar elongates and its length becomes [l = l_{o} + \Delta l]. Fig. 1.3.3.2[link] relates the variations of Δl and of the applied stress T. The curve representing the traction is very schematic and does not correspond to any real case. The following result, however, is common to all concrete situations:

  • (i) If [0 \,\,\lt\,\, T \,\,\lt\,\, T_{o} ], the deformation curve is reversible, i.e. if one releases the applied stress the bar resumes its original form. To a first approximation, the curve is linear, so that one can write Hooke's law: [{\Delta l \over l} = {1 \over E} T, \eqno(1.3.3.1)]where E is the elastic stiffness, also called Young's modulus. The physical mechanism at the origin of elasticity is the deformation of the chemical bonds between atoms, ions or molecules in the solid, which act as so many small springs. The reaction of these springs to an applied stress is actually anharmonic and Hooke's law is only an approximation: a Taylor expansion up to the first term. A rigorous treatment of elasticity requires nonlinear phenomena to be taken into account. This is done in Section 1.3.6[link]. The stress below which the strain is recoverable when the stress is removed, [T_{o}], is called the elastic limit.

    [Figure 1.3.3.1]

    Figure 1.3.3.1 | top | pdf |

    Bar loaded in pure tension.
    [Figure 1.3.3.2]

    Figure 1.3.3.2 | top | pdf |

    Schematic stress–strain curve. T: stress; [T_{o}]: elastic limit; [\Delta l /l]: elongation; the asterisk symbolizes the rupture.

  • (ii) If [T \,\,\gt\,\, T_{o}], the deformation curve is no longer reversible. If one releases the applied stress, the bar assumes a permanent deformation. One says that it has undergone a plastic deformation. The region of the deformation is ultimately limited by rupture (symbolized by an asterisk on Fig. 1.3.3.2[link]). The plastic deformation is due to the formation and to the movement of lattice defects such as dislocations. The material in its initial state, before the application of a stress, is not free in general from defects and it possesses a complicated history of deformations. The proportionality constant between stresses and deformations in the elastic region depends on the interatomic force constants and is an intrinsic property, very little affected by the presence of defects. By contrast, the limit, [T_{o}], of the elastic region depends to a large extent on the defects in the material and on its history. It is an extrinsic property. For example, the introduction of carbon into iron modifies considerably the extent of the elastic region.

    The extents of the elastic and plastic regions vary appreciably from one material to another. Fragile materials, for instance, have a much reduced plastic region, with a clear break.








































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