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Elasticity

Solids resist both change in volume as well as change in shape when subjected to an external force. When external force attempts to deform a solid pressures develop inside the solid which are called elastic stresses. Within elastic limit deformations are proportional to stresses which cause them (Hooke’s law). The ratio of stress to relative deformation is a constant, and is called modulus of elasticity, for a given solid.

1. Young’s Modulus: When an external force F is applied normally on area A of slid, it produces a longitudinal extension ΔL in length L of the solid along the direction of force. Ratio of longitudinal stress to longitudinal strain is called Young’s modulus:


2. Bulk Modulus: If a solid is subjected to tensile or compressive stresses uniformly in all directions, then the ratio of stress to relative volume change is called Bulk modulus K of the solid:

where, Δ P is normal stress or pressure increment.

3. Modulus of Rigidity: Under uniform normal stress the shape of a solid remains same and only its volume changes. If forces are applied tangential to the surface of a solid, (and not perpendicular to its surface) then they cause a change in shape of the solid. Such a deformation is called shear strain.

Under the action of shear force F, a rectangular block of solid becomes oblique. Within first approximation, we assume that the height of the block does not change and hence its volume remains constant. The shearing strain is measured by angle θ = Δ x/y. The shear modulus, or modulus of rigidity is defined as


4. Poisson’s Ratio: When a wire or a rod is stretched longitudinally, it suffers a contraction in transverse (lateral) dimensions. The relative decrease in transverse dimension is proportional to longitudinal stress, or longitudinal extension. The ratio of relative transverse contraction to relative longitudinal extension is called Poisson’s ratio of the given material:

We will show later that, – 1 < σ < ½, for all solids. Thus, in principle, a solid material could exist with negative values of σ. A rod of such a material should become wider when it is stretched. No solid material having such properties are known to exist; hence, Poisson’s ratio in practice lies only between 0 and ½.

The above definitions are applicable to homogeneous, isotropic solids undergoing deformations. Deformation of anisotropic solids, such as single crystals is described by many more elastic constants; we shall not discuss anisotropic solids. However, we later discuss two cases where deformation is not uniform. First is the torsion of its cylindrical rod. This is an example of pure shear which is not uniform; different cross-sections of the rod are turned through different shear angle relative to one of its ends which is kept fixed. Second is the bending of a thin beam. When a beam is bent each volume element of the beam is either extended or compressed, deformation being different at different regions.