Elastic constants of materials

ELASTIC CONSTANTS OF MATERIALS

There are three Elastic Constants of materials, viz., Young's Modulus, Modulus of Rigidity and Bulk Modulus of Elasticity. These moduli describe the relationships between stress and strain and are explained in the following sections:

 

1. HOOKE'S LAW

Hooke's Law was formulated by Robert Hooke in 1678. Hooke's Law states:

“When an elastic material is stresses within elastic limit, the stress induced in the material is proportional to the strain. In other words, the ratio of axial stress to the corresponding axial strain is a constant within the limit of proportionality.” That is, Stress oc Strain, i.e., foc e.

Therefore, f = E e, where E is a constant. Hooke's Law is applicable to tension, compression and shear. Hooke's Law forms the basis for evaluating the three elastic constants.

 

2. YOUNG'S MODULUS or MODULUS OF ELASTICITY (E)

The constant of proportionality concept was introduced by Thomas Young in 1807.

According to Hooke's Law, stress f = Ee, where E is a proportionality constant. E is known as Modulus of Elasticity or Young's Modulus.

Young's Modulus of a material is defined as the ratio of the axial stress to the corresponding axial or linear strain, within the elastic limit.

Young's Modulus = E = Axial Stress / Axial Strain = f / e

Unit for Young's Modulus is N/mm² or Pa (Pascal). The above relation is valid only for homogeneous materials of uniform cross-section, when loaded axially within the limit of proportionality.

 

3. YOUNG'S MODULUS AND ELONGATION OF A BAR UNDER AXIAL LOAD

Let a bar of uniform cross-section A and length 1 is subjected to an axial tensile load P. Let dl be the elongation or deformation of the bar. E is the Young's Modulus of the material of the bar.

 

4. MODULUS OF RIGIDITY or SHEAR MODULUS (G)

Hookes's Law is assumed to apply to shear also. Modulus of Rigidity or Shear Modulus of a material is the ratio of shear stress to the corresponding shear strain. This ratio is a constant up to the elastic limit of the material. It is denoted by G.

 

5. BULK MODULUS OF ELASTICITY (K)

See Fig. 8. When a body is subjected to uniform direct stresses of equal intensity (either tensile or compressive) in all the three mutually perpendicular directions, then the ratio of the direct stress (f) to the corresponding volumetric strain (ev) is constant up to the elastic limit. This elastic constant is called Bulk Modulus of the material. It is denoted by K.

Bulk Modulus = K = Direct Stress / Volumetric Strain

Volumetric Strain (ev)

Due to external loads, the longitudinal and lateral strains occur in an elastic body. Therefore, the volume of the elastic body changes. The change in volume dv (increase or decrease) of the elastic body on unit original volume V is called the Volumetric Strain. It is denoted by ev.

Volumetric Strain = ev = Change in Volume / Original Volume = dv / V

Therefore, Bulk Modulus = K = Direct Stress / Volumetric Strain = f / ev

= (P /A) / (dv/V)  = PV/ A(dv).

 

6. POISSON'S RATIO (µ or 1/m )

When a body is subjected to tensile load (axial load), it is subjected to axial strain. There is elongation in the direction of the tensile load. Simultaneously, there is a reduction in the transverse dimensions (lateral dimensions). The ratio of the transverse strain (lateral strain) to the corresponding axial strain (longitudinal strain) is constant within the proportionality limit. This ratio is called Poisson's Ratio.

Poisson's Ratio is named after Poisson, a French Mathematician. It is denoted by µ or 1/m.

Poisson's Ratio = µ = 1/m = Lateral or Transverse Strain / Axial or Longitudinal Strain

 

7. WORKING STRESS or SAFE STRESS or PERMISSIBLE STRESS

While designing the size of structural members, Working Stress much less than the Proportional Limit is used. Working Stress is actually the stress developed in the material when it is loaded. Allowable Stress or Permissible Stress is the maximum stress which a material can safely withstand.

For all practical purposes, working stress is equal to allowable stress. It should not exceed the Proportional Limit. Only then, the stress-strain relationship of Hooke's Law will be satisfied. Working Stress is also known as Safe Stress or Permissible Stress. The working stress shall always be less than the Ultimate Stress.

Working Load = Working Stress / Area of Cross-section

 

8. FACTOR OF SAFETY

Designs are based on the principle that the stresses to which structural members are subjected are less than the proportional limit. On this basis, the Working Stresses are prescribed. In structural steel, the working stress is kept much below the yield strength to avoid excessive permanent deformations. The ratio of yield strength to permissible working stress is called the Factor of Safety.

For homogeneous and uniform materials like steel, aluminium, etc., a factor of safety of 4 is used. For other materials like timber, which are highly non-uniform due to the presence of knots, higher factor of safety is used.

In concrete, there is no defined yield strength. But, the ultimate strength is clear. Hence, for concrete, the factor of safety is fixed with reference to the ultimate strength.

Factor of Safety = Ultimate Stress / Working Stress

Factor of safety is always greater than 1. Allowable Stresses are pre-determined for different materials under different loading conditions. The determination of an appropriate value for the factor of safety is a complex matter. It requires considerable engineering judgment. Factor of safety depends on the following considerations:

• Quality of material and its degree of reliability

• Type of loading whether static or dynamic, dead or live load, concentrated or distributed load, etc., including nature and conditions of loading

• Function of the member under design

• Possible manufacturing or fabrication errors and workmanship

• Possible type of maintenance

The recommended factor of safety for different civil structures as per IS codes is given in the following table:

For steel structures, factor of safety is fixed with reference to the yield strength and hence it is generally higher. Since the factor of safety is fixed with reference to the ultimate strength for concrete structures, it is generally lower.

 

9. RELATION BETWEEN ELASTIC CONSTANTS

Each elastic material has four elastic constants. These constants are inter-related with one another. If the values of any two constants are known for a material, the other two constants may be easily determined using the following equations: