Elastic Constants and Their Relationship -Strength Of Material

Different types of stresses and their corresponding strains within elastic limit are related which are referred to as elastic constants.

The three types of elastic constants are:

  • Modulus of elasticity or Young’s modulus (E),
  • Bulk modulus (K) and
  • Modulus of rigidity or shear modulus (M, C or G).

Specifying how stress and strain are to be measured, including directions, allows for many types of elastic moduli to be defined. The three primary ones are:

  • Young’s modulus (E) describes tensile elasticity, or the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of tensile stress to tensile strain. It is often referred to simply as the elastic modulus.
  • The shear modulus or modulus of rigidity (G ) describes an object’s tendency to shear (the deformation of shape at constant volume) when acted upon by opposing forces; it is defined as shear stress over shear strain. The shear modulus is part of the derivation of viscosity.
  • The bulk modulus (K) describes volumetric elasticity, or the tendency of an object to deform in all directions when uniformly loaded in all directions; it is defined as volumetric stress over volumetric strain, and is the inverse of compressibility. The bulk modulus is an extension of Young’s modulus to three dimensions.

Within elastic limit the stress is directly proportional to strain. This is the statement of Hooke’s law and is true for direct (tensile or compressive) stress and strain as well as for shearing (including torsional shearing) stress and strain. The ratio of direct stress to direct strain is defined as modulus of elasticity (E) and the ratio of shearing stress and shearing strain is defined as modulus of rigidity (G). Both the moduli are called elastic constants. For isotropic material E and G are related with Poisson’s ratio

Elastic Constants and Their Relationship
Elastic Constants and Their Relationship

Poisson’s ratio which is the ratio of transverse to longitudinal strains (only magnitude) in tensile test specimen is yet another elastic constant. If stress acts in three directions at a point it is called volumetric stress and produces volumetric strain. The ratio of volumetric stress to volumetric strain according to Hooke’s law is a constant, called bulk modulus and denoted by K. It is important to remember that out of four elastic constants, for an isotropic material only two are independent and other two are dependent. Thus K can also be expressed as function of any two constants.

It may be understood that elastic constants E and G are not determined from tension or torsion test because the machines for these tests undergo adjustment of clearance and also some deformation, which is reflected in diagram ordinarily. The constants are determined from such devices, which show large deformation for comparatively smaller load.

For example, E is determined by measuring deflection of a beam under a central load and G is determined by measuring deflection of a close-coiled helical spring under

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