# 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.

## Fundamentals

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.

## Illustration of Poisson effect

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.

We set up a co-ordinate system based on some reference axes in the sample. A convenient one for the case illustrated above is included in the figure. It is a right handed system when reading the axes in the sequence x y z. If we start at a point on the positive x axis and rotate towards the positive y axis in a clockwise direction then positive z axis would point in the direction of movement of a right handed screw.

To keep it as clear as possible which reference axes we are using it is conventional often to refer to these axes as 1 2 and 3. This is to allow easy identification of the strain terms as depicted in the tensor 3x3 matrix. The directions 1 2 and 3 are synonymous with the x y and z directions.

In a sample strained in tension (uni-axial tension) as in the above figure, the z axis has been strained in the positive sense, i.e. it becomes longer. The strain term for this deformation is e33. The applied force is acting in the z direction. The first subscript refers to the axis that is being strained and the second to the direction it is being strained. Thus the first subscript 3 refers to the strain is occurring along the z axis and the second subscript to the fact that the strain is in the direction of the z axis. Likewise for the strains e11 and e 22. The strains e 11, e 22 and e 33 are known as tensile or normal strains. The force along z can also be resolved onto inclined planes causing these planes to shear resulting in shear strains. It will also cause the crystal to rotate (called a rigid body rotation). To illustrate this I include a more general deformation, as shown next.

Illustration of tensile and shear strains plus rotation in x z plane. x y z right handed axis system, u1 u2 u3 displacements in x y z directions. The figure shows only deformation, that is change of length and direction, in the x z plane for clarity. The tensile strains e11 and e 33 are shown plus the two strains e 13 and e 31. Note the differential form of these latter strains as shown in the figure. Let us consider for now that there is no rotation. Then e 13 and e 31 represent the shear components alone and are in fact equal. The first of the subscript in both of e 13 and e31 signifies the plane that is sheared and the second component the axis defining the direction of shear. Hence the first shear component e13 describes the shear of the plane normal to the x axis and in the z direction whilst the second, e31 defines the shear of the plane normal to the z axis in the x direction. Simple (mechanical) shear versus physics Now some of you may say, ‘surely shear of a plane distorts a crystal’ as shown in the left sketch below.

Simple shear. (Engineering) Shear (Physics) Well this is true and is how engineers express it. As shown in the right hand figure, the shear strain as used in engineering is twice that used by physicists. Engineering strain measures the total strain in the xz plane and is often given the symbol γ. On the other hand, e31, which remember = δu3/δx and in this case where there is no rotation, is simply the average of the strains on the z and x faces i.e. ε31 = ½(e31+e13). Note I have changed the symbol from e31 to ε31. The symbol e31 refers to the strains as measured and may contain a rotational component. The symbol ε31 refers specifically to the shear strain component included in the e31 term. You may need to note this when involved in discussion with engineers. For consistency the Greek symbol ε is used instead of e for the tensile strains as well. Tensors in Strain Tensors: Geometric entities introduced into mathematics and physics to extend the notion of scalars, (geometric) vectors, and matrices. Many physical quantities are naturally regarded not as vectors themselves, but as correspondences between one set of vectors and another.

## Techniques for measuring strain

There is a wide variety of strain measurement techniques, depending on the resolution and precision required. Larger scale methods include the use of strain gauges. For higher precision Raman Spectroscopy and recently emerging techniques such as EBSD are used. Traditional measurement of residual strain in metals is through x-ray diffraction. Unfortunately this has major shortcomings in today’s applications and material’s development environment:

• Probe size (width) is too large for nano-scale material structures.
• Probe size (depth) is typically too large for nano-scale material structures.

Result: Diffraction volume is too large and individual grains (crystal lattices) cannot be observed or measured accurately.

The advanced application of the alternative diffraction technique of Electron Backscatter Diffraction (EBSD) overcomes these limitations. Software applications such as CrossCourt3 have enabled practical measurement at the nanoscale quite possible.

## Electron backscatter diffraction

The technique of EBSD produces patterns from the metal’s crystallography that consist of a well-defined geometric arrangement of bright bands edged by sharp lines. These are known as Kikuchi bands and Kikuchi lines respectively. Elastic strain changes the width of these bands, and the angles between them plus a general rotation of the entire pattern. The total result is a distortion that involves dilation of the crystal lattice (or unit cell) as a result of tensile and compressive strains, shear strains and rotation. A complete description of internal elastic strain requires that all these crystallographic parameters be measured. This is a benefit of advanced or high accuracy EBSD.[1]