Dimensional Formula of Stress

A solid body has tightly packed atoms that won’t move from one position to another like fluids. However, some solids exhibit elastic properties, which means that their dimensions can be changed by applying force. Once the deforming forces are withdrawn, the body will return to its shape. Elasticity plays a significant role in mechanical physics, especially in calculating the load handling capacity of a rope, the stretchability of a body, and so on. When we consider elasticity, the two most important factors that describe the phenomenon are stress and strain. Out of these, stress acts as the leading player in defining the elastic behaviour of any solid, which is why learning about the dimensional formula of potential difference is crucial.

What is stress, and how does it relate to elasticity?

When an external force is applied to the surface of an elastic object, it impacts the entire surface area. Therefore, stress is the force applied per unit area of a solid and rigid body. According to Newton’s law, force is considered a physical entity that can bring a moving body to rest and vice versa.

However, stress involves a force that causes a change in the dimensions of an object. For example, let’s consider a rubber band. When you stretch it, you are applying force to it. This force is not causing the rubber band to start moving from its initial resting position. Instead, its application on the surface area increases the rubber band’s length. This is what we know as stress. To grasp the concept of stress, one needs to understand its proper dimensional formula of potential difference for real-time applications.

σ = F/A

Dimensional expression and formula of stress

When we have to work on deriving the dimensional formula of the potential difference of any physical attribute, we have to start with the independent dimensions. These are length, mass, and time. Length is denoted by L, M denotes mass, and time is denoted by S.

These three independent dimensions are expressed both in SI and CGS units. Therefore,

· The SI unit of L will be m, and the CGS unit will be cm.

· For M, the SI unit is kg, and the CGS unit is g.

· For T, both the SI and CGS units are expressed as sec.

Since stress depends on force and surface area, we have to move step by step to derive its dimensional expression.

Force = mass x acceleration

Or, force = mass x velocity/time

Or, force = (mass x displacement)/time2

Writing force as f, mass as m, acceleration with a velocity as v, displacement as s, and time as t:

f = ma

Or, f = (mv)/t

Or, f = (ms)/t2

Now, putting the dimensional values of all the units expressed in the above equations:

f = ( [M1L1]) / T2

Or, f = [M1L1T-2]

Putting the dimensional formula of force in the stress equation:

σ = F/A

σ = [M1L1T-2] / [L2]

σ = [M1T-2] / [L1]

σ = [M1L-1T-2]

Putting the SI units of each dimension in the above expression:

σ = Kg1m-1sec-2

σ = kgm-1sec-2 or Pa

Types of stress and their applications

Stress can be divided into two primary forms: normal and shear. These two types depend on the point of application of the external force and its direction.

Normal stress

When the force is applied on a solid body at a point x perpendicular to the surface orientation, the stress is defined as normal. Here, the angle between the line of force and the surface is 90 degrees.

If we consider the normal force to be FN and the total surface area to be A, the normal stress can be written as:

σN = FN / A

Here, the SI unit of FN is N (Newtons), and that of A is (m2). Therefore, the SI unit of σ is Nm-2 or Pa.

Shear stress

On the contrary, when a force is applied along the object’s surface, that is, parallel to the surface, the force per unit surface area is termed shear stress. Here, the angle between the line of force and the surface is 0 degrees.

If we consider the force to be FS and the surface area to be A, shear stress is written as:

σS = FS / A

Or, T = FS / A

where σS = T = shear stress

Here, the SI unit of the force, FS, is given by N, while the surface area is expressed as m2. Therefore, the SI unit of shear stress is considered Nm-2or Pa.

Conclusion

Grasping how stress depends on the force applied to an object and its surface area is crucial to understanding the importance of the dimensional formula of potential difference and its derivation. All three parameters—area, force, and stress—are dependent dimensions, so they need to be expressed in terms of the independent dimensions for the derivation of the SI unit for stress. Apart from this, with the dimensional formula of potential difference, it becomes clear how a body will behave in the context of applied stress. For example, the stress amount will decrease with greater surface area, as they are inversely proportional. Similarly, stress will increase because of direct proportionality with more mass. Therefore, it is crucial to study how the dimensional formula of stress is derived and used for further analysis.