A good understanding of the physical properties of stress and strain is a prerequisite to utilize the many methods and results of analysis in design. This article provides the definitions and important relationships of stress and strain.
It is a distributed force on an external or an internal surface of a body. Consider an arbitrarily shaped body in the figure below. and are applied concentrated forces and applied surface force distributions, and and are support reactions force and surface force distributions. To find out the state of stress at point Q in the body, it is necessary to expose a surface containing the point Q. To do this, make a planar slice or break through the body intersecting at point Q. This slice has an arbitrary orientation, but it usually at a convinient plane where the state of stress can be easily found out.
As per the figure below, the first slice is orineted along the surface normal 'x'. This orientation gives us the 'yz' plane. In the figure below, the external forces on the remaining body and the internal stress distribution is shown acorss the exposed internal surface containing Q.
In this general case, the distribution will not be uniform along the surface and niether normal nor tangential to the surface at point Q. The forces at Q will have normal and tangential components. These components will be tensile or compressive and shear in nature.
Let us assume that the area surrounding Q is as shown in the figure below. The equivalent concentrated force across this area is . which is neither normal nor tangential to the surface and subscript x is used to denote that it is normal to the area. The components of in x, y and z directions are , and . The first subscript denoted the direction normal to the surface and the second gives the actual direction.
The average distributed force per unit area (average stress) in the x direction is, .
Stress is actually a point function, so we can obtain the exact stress in the x direction at point Q by allowing to approach to zero. Thus, or .
Stresses are brought into play from the tangential forces and as well, and because these forces are tangential, the stresses are shear stress. So, and .
By definition, represents a normal stress acting in the same direction as the corresponding surface normal, the double subscripts are useless and the standard practice is to denote as .
To define the complete state of stress at point Q, it is necessary to examine other surfaces by making different planar slices. The three surfaces are taken as mutually perpendicular surfaces, as shown in the figure below.
The state of the stress can now be written in a matrix form, .
Usually, the adjacent shear stresses are taken to be equal, i.e. , and , and the above stress matrix will now become, .
In many applications and problems, the stresses in one direction are zero. This kind of a situation is called plane stress. Usually, we can assume . This will make the last row and last column of the above stress matrix will become zero and can be eliminated, and can be written as,
The figure below, will depict the stress element when seen in 3D and down the z axis.
There are two strains that exist with the stresses, i.e. normal and shear strains. These are denoted by and , respectively. Normal strain is defined as the change in length of the stressed element in a given direction. Shear strain is defined as the measure of the distortion of the stressed element.
Consider the figure given below, in which only one normal stress is applied. It is visible that the element increases in length in x direction and decreases in length in y and z directions. The rate of increase in length is defined as the normal strain, and , and are the normal strains in the x, y and z directions respectively.
The new length in any direction will be the sum of original length and the rate if increase times the original length. This will give the following relations:
As per Hooke's law, for a linear, homogeneous and isotropic material the normal strain is directly proportional to the normal stress, and is represented mathematically as,
We hope you are able to understand the above given formulas and explanation. In the next article we will discuss about Stress Transformations.