Strain and Strain components

In materials science and solid-state physics, a crystal is a solid material whose atoms or molecules are arranged in a highly ordered, repeating pattern extending in all three spatial dimensions. When a crystal is subjected to an external force, it can deform or change shape, and this deformation is characterized by strain components. Therefore, the change in shape or size due to application of external force is known as strain. Thus strain = change in shape or size / initial shape or size.

The three types of strain components in a crystal are:

1. Longitudinal strain component: This component describes the change in length of the crystal along a particular direction. It is denoted by εxx, εyy, and εzz.

2. Shear strain component: This component describes the change in shape of the crystal due to deformation. It is denoted by εxy, εxz, and εyz.

3. Volume strain component: This component describes the change in volume of the crystal due to deformation. It is denoted by εv.

Each of these strain components plays an important role in determining the mechanical behavior of a crystal. Understanding the strain components of a crystal is essential in many fields, including materials science, solid-state physics, and engineering.

Let us consider a crystal as a homogeneous continuous elastic medium rather as a periodic array of atoms. Let us consider 3 orthogonal unit vectors x, y, z be safely embedded in an unstrained crystal. Let us assume as straining crystal by a form to new non orthogonal vector and it can be written in terms of old vectors as

where coefficient E𝛼𝛽 define deformation, they are dimensionless and have value ≪ 1 for a small strain. The original axes (x, y, z) are unit length but new axes will not necessary be of unit length.

After deformation, the position of atom will be

Now deformation of atom under the action of deforming force is

where product of two E's and square of E's are neglected. The six dimensionless coefficient E𝛼𝛽 = E𝛽𝛼 completely define the drain.

Dilation : Dilation has many applications in mathematics, science, and engineering. For example, it can be used to study the properties of geometric shapes, to model the behaviour of materials under stress, and to design machines and structures that can withstand deformation. The fractional increase of volume associated with a deformation is called dilation. The dilation is negative for hydrostatic pressure. This process can occur as a result of thermal expansion, mechanical deformation, or other external factors that affect the interatomic distances and angles within the lattice. Dilation in crystals can have a significant impact on their physical and mechanical properties. For example, thermal expansion can cause changes in the lattice parameter and affect the crystal's thermal conductivity, while mechanical deformation can cause changes in the lattice symmetry and affect the crystal's mechanical strength. Dilation is also an important factor in crystal growth and can be controlled to produce crystals of specific sizes and shapes. For example, in the growth of semiconductor crystals, the lattice parameter must be carefully controlled to produce materials with specific electronic and optical properties

The volume of unit cube after deformation is

Overall, dilation plays a crucial role in the behaviour and properties of crystals, and understanding its effects is essential in the study and application of crystallography.

This note is a part of the Physics Repository.