Most engineering structure are designed to undergo relatively small deformations, involving only the straight line portion of the corresponding stress strain diagram. The relation is known as hook's law. The coefficient "E"  is called the modulus of elasticity of the material involved or young's modulus. Since the strain is dimensionless quantity, the modulus "E" is expressed in the same units as the stress namely in pascals or one of its multiples.

Actually at the grass root level if you see that Hook's law is basically the small displacement of atoms and ions of the matter from their actual position due to application of force.

The largest value of stress for which hook's law can be used for a given material is known as proportional limit of that material. In case of ductile materials, the proportional limit almost coincides with the yield point. For outer material proportional limit cannot be determined easily.

MODULUS OF ELASTICITY IS CAN ALSO BE DEFINED AS ability to resist a deformation within the  linear range, in the same.

By bending, twisting, pulling or compressing a material or solid, elasticity can be observed in the material under elastic limit. This is due to hook's law.Consider F= kX where F is the applied force and X is the elongation. k depends on shape,size and type of elastic material. Hook's law is valid for elastic region up to point till force applied is proportional to elongation.

Hook's law can also be written as F= -kX, where F is not the applied force rather it is restoring force which comes in picture when body is coming to its original shape after deformation in elastic bodies. 

Elastic behavior:-
  • If the strain caused in the test specimen the application of a given load disappear when the load is removed,the material is said to have elastically.
  • The largest value of the stress for which the material behaves elastically is called the ELASTIC LIMIT of the material.
  • Ductile material has well defined yield point, elastic limit, proportional limit.
  • In other words, the material behaves elastically and linearly as long as the stress is kept below the yield point.
Plastic behavior

After the material is stressed or loaded beyond elastic region it can never return to its original shape by itself this is commonly known as plastic deformation. In very simple word we can say that the strain does not return to zero after the load has been removed which indicates that permanent set of plastic deformation of material has taken place. For most of the materials plastic deformation depends not only upon the maximum value reached by the stress, but also upon the time elapsed before the load is removed. Each and every material has different value beyond which plastic deformation occurs.
  • The stress dependent part of plastic deformation is referred to as SLIP. The time dependent part which is also influenced by the temperature is called CREEP.

  • When a specimen is subjected to axial loading and the maximum stress in the specimen does not exceed the elastic limit of material, the specimen return back to the original condition after removing the load.
  • This loading may repeat hundred times or thousand times provided the stress remain in the elastic range.
  • If the loading repeated is of very high order (say million times) the failure may occur at stress much lower than  the static breaking strength this phenomenon is called fatigue.
  • Fatigue failure is brittle in nature. Even ductile material fractures in brittle manner.
  • Fatigue is considered as deigns of all structural member which are subjected to fluctuating or repeated load.
  • Some loading are fluctuating in nature. For e.g. the passage of traffic over bridge is will cause stress levels that will fluctuate about the stress level due to bridge weight.

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