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Elasticity Concepts and Moduli

Unit 5 discusses stress, strain, and elasticity. Stress is defined as the restoring force per unit area inside a body. Strain is the fractional deformation from stress. Longitudinal strain is the change in length per unit length. Hooke's law states that strain is directly proportional to stress below the elastic limit. Young's modulus is the ratio of stress to longitudinal strain and determines the elasticity of a material. Rigidity modulus is the ratio of shearing stress to strain. Bulk modulus is the ratio of volume stress to volume strain. Elastic materials return to their original shape after deformation, while plastic materials retain a permanent deformation.

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95 views2 pages

Elasticity Concepts and Moduli

Unit 5 discusses stress, strain, and elasticity. Stress is defined as the restoring force per unit area inside a body. Strain is the fractional deformation from stress. Longitudinal strain is the change in length per unit length. Hooke's law states that strain is directly proportional to stress below the elastic limit. Young's modulus is the ratio of stress to longitudinal strain and determines the elasticity of a material. Rigidity modulus is the ratio of shearing stress to strain. Bulk modulus is the ratio of volume stress to volume strain. Elastic materials return to their original shape after deformation, while plastic materials retain a permanent deformation.

Uploaded by

Rejo george
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Unit 5: Elasticity

Stress:
The restoring force set up per unit area inside the body is called stress.
𝐹
𝑆𝑡𝑟𝑒𝑠𝑠 = . The SI unit of stress is N/m2
𝐴

Strain:
Strain is the fractional deformation resulting from stress. It is measured by the ratio of the change in
dimension of the body to the original dimension in which the change took place.
Longitudinal strain:
It is the change in length per unit length.
𝑙
𝐿𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙𝑠𝑡𝑟𝑎𝑖𝑛 = 𝐿

Shearing strain (ɵ): The angular deformation due to shearing stress


is defined as the shearing strain.
Volume strain or bulk strain:
𝑑𝑣
It is defined as the change in volume to the original volume. 𝑉𝑜𝑙𝑢𝑚𝑒𝑠𝑡𝑟𝑎𝑖𝑛 =
𝑉

Hooks law:
Hooks law states that the strain produced in a body is directly proportional to the stress provided, the
stress should not be too large.
𝑆𝑡𝑟𝑒𝑠𝑠
= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑠𝑡𝑟𝑎𝑖𝑛
The constant is called modulus of elasticity and its SI unit is N/m2, 1 N/m2 = 1 pascal
Young’s modulus:
Young’s modulus is the ratio of the longitudinal stress to the longitudinal strain.
𝐹𝑜𝑟𝑐𝑒
𝑆𝑡𝑟𝑒𝑠𝑠 =
𝐴𝑟𝑒𝑎
𝐹
𝑆𝑡𝑟𝑒𝑠𝑠 =
𝐴
𝑙
𝑆𝑡𝑟𝑎𝑖𝑛 =
𝐿
(𝐹 ⁄𝐴) 𝐹𝐿
𝑌𝑜𝑢𝑛𝑔’𝑠𝑚𝑜𝑑𝑢𝑙𝑢𝑠𝑌 = =
(𝑙 ⁄𝐿) 𝐴𝑙
𝐼𝑓𝑡ℎ𝑒𝑏𝑜𝑑𝑦ℎ𝑎𝑠𝑐𝑖𝑟𝑐𝑢𝑙𝑎𝑟𝑐𝑟𝑜𝑠𝑠𝑠𝑒𝑐𝑡𝑖𝑜𝑛, 𝐴 = 𝜋𝑟 2
𝐹𝐿
𝐻𝑒𝑛𝑐𝑒𝑌 = 2
𝜋𝑟 𝑙
Determination of young’s modulus of a wire by using Searle’s method:
One end of the given wire AB along with the reference wire CD is to be hung on
the ceiling tightly and the other end with the weight holder with a dead weight. A
scale with millimeter precision is to be fixed on the reference wire and a vernier
scale on the other wire. The initial reading is measured as C0. The proceeding
readings to be taken while both loading and unloading weights by the steps of 1
kg Ci. The mean value of C0-Ci gives the extension for each load M. estimate the
average value of M/l and apply in the equation of the young’s modulus
𝑀𝑔𝐿
𝑌=
𝜋𝑟 2 𝑙
𝐿𝑔 𝑀
𝑌 = 2( )
𝜋𝑟 𝐿
Rigidity modulus:
Rigidity modulus is the ratio of shearing stress to the shearing strain.
𝐹 ⁄𝐴 𝐹
ɲ= =
ɵ 𝐴ɵ
Bulk modulus:
Bulk modulus is the ratio of the volume stress (pressure) to the volume strain
(𝐹 ⁄𝐴)
𝐾=
(𝑣 ⁄𝑉 )
𝑃 𝑃𝑉
𝐾= =
𝑣 ⁄𝑉 𝑣
𝑣
Compressibility: The reciprocal of is known as compressibility. 𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑦 = 𝑃𝑉

Elastic behavior: An elastic body regains its original size and shape by the removal of deforming force
applied. E.g. rubber.
Plastic behavior of solids: A plastic body cannot regain its original shape and size by removing the
deforming force. E.g. putty, paraffin wax etc.
Elastic limit: It is the magnitude of applied force which produces maximum amount of recoverable
deformation.
Breaking stress:
The maximum stress after which the wire begins to flow like a viscous liquid and break is called breaking
stress.
Brittle substance: the material breaks soon after the elastic limit is called brittle substances. E.g. glass.
Ductile substance: the object which undergoes large deformation after elastic limit before breaking is
called ductile substances. E.g. copper, iron etc
Elastic fatigue: The failure of an elastic material to withstand large fluctuations of stress is called the
elastic fatigue.

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