# Moduli of Elasticity

## Moduli of Elasticity

quantities that characterize the elastic properties of materials. In the case of small deformations when the Hooke law is valid (that is, when there is a linear relationship between the stresses and strains), moduli of elasticity are the proportionality constants for the relationships.

The modulus of elongation *E* (Young’s modulus) corresponds in the direction of tension to a unilateral normal stress σ generated by simple tension or compression. It is equal to the ratio of the normal stress σ to the relative elongation e resulting from the stress in the direction of its action (*E = σ/e*) and characterizes the ability of the material to resist extension.

The shear modulus *G* corresponds to a pure shear stressed state in which only tangential stresses τ act on two mutually perpendicular areas. The shear modulus is equal to the ratio of the tangential stress τ to the angle of shear λ, which is determined by the distortion of the right angle between the planes upon which the tangential stresses are acting. Thus, *G* = τ/λ. The shear modulus determines the ability of a material to resist a change in shape while maintaining its volume. The compression modulus *K*, which is the bulk modulus of elasticity, corresponds to a normal stress *or* that is uniform in all directions (generated, for example, by hydrostatic pressure). It is equal to the ratio of the normal stress γ to the relative bulk compression A resulting from it: *K* γ/Δ. The bulk modulus of elasticity characterizes the ability of a material to resist a change in its volume not accompanied by a change in shape.

The Poisson ratio *v* is another constant that characterizes the elastic properties of materials. It is equal to the ratio of the absolute value of the compressive transverse strain of a cross section ∊′ (for unilateral extension) to the relative elongation ∊, that is, *v* = ǀ*∊’*ǀ/*∊*.

*In* the case of a uniform isotropic body, the moduli of elasticity are the same in all directions. The four constants *E, G, K*, and v are related to one another by two formulas:

Thus, only two of these constants are independent quantities, and the elastic properties of an isotropic body are determined by two elastic constants.

In the case of an anisotropic material, the constants *E, G*, and *v* assume different values in different directions, and their values may vary widely. The number of moduli of elasticity of an anisotropic material depends on the structure of the material. An anisotropic body lacking any symmetry with respect to elastic properties has 21 moduli of elasticity. When symmetry is present in the material, the number of moduli of elasticity is reduced.

Moduli of elasticity are determined experimentally by mechanical testing of samples of materials. Moduli of elasticity are not rigorous constants for the same material, and they vary depending on the chemical composition of the material and its previous treatment (heat treatment, rolling, and forging). Values of moduli of elasticity also depend on the temperature of the material.

### REFERENCE

Fridman, la. B.*Mekhanicheskie svoistva metallov*, 2nd ed. Moscow, 1952.