Elastic and plastic strain

Internal strain within a metal is either elastic or plastic. In the case of elastic strain this is observed as a distortion of the crystal lattice, in the case of plastic strain this is observed by the presence of dislocations –the displacement of part of the crystal lattice. Such strain effects can result in unwanted cracking of the material, as is the case with residual plastic strain. In other cases deliberate introduction of plastic strain results in a strengthening of the material and other performance enhancing behaviors, for example in the manufacture of semi-conductors and solar cells.

As an illustration, if you hang a weight on a spring it extends in direct proportion to the load. That is the same effect that occurs in the elastic deformation part of the standard tensile test.

This is normally written as: applied stress = Young’s modulus * strain

That is: σ = Y * e

Commonly known as: Hooke’s Law.

Where stress is a force ( a vector property) divided by the cross sectional area. Strain is the displacement (also a vector quantity) per unit length and is also a vector quantity. A vector quantity requires three numbers to define its direction. It can be represented then by a number with one subscript, Ui, where i takes the numbers 1 to 3. Such a quantity can be referred to as a tensor of rank 1. The rank refers to the number of subscripts. The strains e can be formally written as a differential or gradient. So that, if u1 was the extension in the x direction, we would write the strain as δu1/δx. We also know that when we stretch something it gets thinner in the direction normal to the stretch direction. This is the Poisson effect and the ratio extension /contraction is the Poisson ratio.

This coupling of the different strains caused in a sample by the application of a stress classifies strain as a tensor property. In fact it is a second rank tensor because it consists of a 3 x 3 matrix and hence each element will need two suffixes to locate any one of its components in the matrix, i.e. eij. The physical significance of this will be apparent later.

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