Stress: Definition, Formula, Unit and Types

Stress: A rubber band stretches when an external force is applied, but the rubber band restores its initial configuration when the force is removed. Likewise, a balloon expands when the air is forced into it, but the balloon returns to its original shape as the air rushes out of it. In this case, the property of the balloon or, in general, any deformed object by virtue of which it regains its original configuration after removing the deforming force is called Elasticity.

But what causes the objects to regain their original shape or size after the external forces are removed? This happens because of Stress. Stress in Physics is the internal force developed per unit area within the deformed object. This internal force will be equal in magnitude and opposite to the applied deforming force.

Stress Definition in Physics

Stress is defined as the restoring force acting per unit area of a body. Thus, stress is a quantity that describes the magnitude of forces that cause deformation on a unit area.

When forces pull on an object and cause elongation, like the stretching of an elastic band, we call it tensile stress. When forces cause compression of an object, we call it compressive stress. Bulk stress is observed in a body that is being compressed equally from all sides, for example, a submarine submerged underwater. In other situations, the acting forces may be neither tensile nor compressive and still produce a noticeable deformation. When deforming forces act tangentially to the object’s surface, we call them ‘shear’ forces and the stress they cause is called shear stress.

Formula to Calculate Stress

Stress, in physical sciences and engineering, is the force per unit area within a material that arises from externally applied forces. It gives an accurate description and prediction of elastic, plastic, and fluid behaviour. Since stress \(\left( \sigma \right)\) is expressed as a quotient of a force divided by an area, its formula can be given as:

\(\sigma = \frac{F}{A}\)

\(F\) is the restoring force, and \(A\) is the cross-sectional area of the given surface.

Units of Stress

Stress is defined as “The restoring force per unit area of the material.”

Thus, the units of stress will depend upon the units of force and area. The \({\rm{SI}}\) unit of force is newton \(\left( {\bf{N}} \right),\) and the \({\rm{SI}}\) unit of area is meter square \(\left( {{m^2}} \right).\) Thus, the SI unit of stress becomes:

\({\rm{Stress}} = \frac{{{\rm{Force}}}}{{{\rm{Area}}}} = \frac{{{\rm{newton}}}}{{{m^2}}} = {\rm{N\;}}{{\rm{m}}^{ – 2}} = {\rm{pascal\;}}\left( {{\rm{Pa}}} \right)\)

Thus, the \({\rm{SI}}\) unit of stress is the pascal. When one newton of force presses on a unit surface area of one meter squared, the resulting stress is one pascal, or, in \({\rm{SI}}\) base units, One pascal is one kilogram per meter per second squared, i.e., \(1{\rm{\;Pa}} = 1\,{\rm{kg}}{{\rm{s}}^{ – 1}}{{\rm{m}}^{ – 2}}.\) It is a tensor quantity.

In the British system of units, the unit of stress is ‘psi,’ which stands for ‘pound per square inch \(\left( {{\rm{lb}}/{\rm{i}}{{\rm{n}}^2}} \right).\) Another unit that is often used for Bulk stress is the atm (atmosphere). Conversion factors are:

\(1{\rm{\;psi}} = 6895{\rm{\;Pa}}\)

\(1{\rm{\;Pa}} = 1.450 \times {10^{ – 4{\rm{\;}}}}{\rm{\;psi}}\)

\(1{\rm{\;atm}} = 1.013 \times {10^2}{\rm{\;Pa}} = 14.7{\rm{\;psi}}\)

In mechanical engineering, we measure the stress of various buildings and bridges, and for that, we require larger units to formulate their stress. Some of these units are:

1. Kilo-Pascal: \(1{\rm{\;kPa}} = {10^3}{\rm{\;Pa}}\)

2. Mega-Pascal: \({\rm{\;}}1{\rm{\;MPa}} = {10^6}{\rm{\;Pa}}\)

3. Giga-Pascal: \({\rm{\;}}1{\rm{\;GPa}} = {10^9}{\rm{\;Pa}}\)

Types of Stress

Fig: Types of stress

Stress is classified into three types:

1. Normal Stress
2. Bulk Stress
3. Shearing/Tangential Stress

1. Normal Stress

Normal stress arises from forces that are perpendicular to a cross-sectional area of the material. The stress generated due to these forces leads to the change in the length or volume of the body. In hindsight, this stress occurs when an axial force loads a member. The value of the normal force for any prismatic section is simply the force divided by the cross-sectional area.  Normal stress will occur when a member is placed in tension or compression. Collar ties and columns are examples of pure normal forces. Longitudinal stress is a type of Normal stress.

Fig: Normal Stress

(i) Longitudinal Stress: Longitudinal stress is defined as restoring force per unit area when the force is applied to the cross-sectional area of the cylindrical body. Consider a cylinder which we have to deform. If we apply the force perpendicular to the cross-sectional area, a restoring force develops in the cylinder in the opposite direction. This restoring force per unit area is known as longitudinal stress. 
Longitudinal stress = Restoring force/cross-sectional area

Fig: Longitudinal Stress

To observe the increase in length, we can tie a heavy object to the cylinder with the help of threads. Let the cylinder’s initial length is \(L.\) After it gets stretched, its length increases by \(\Delta L\) due to the stress. As there is a change in the length, this type of stress is known as longitudinal stress. For example, in the below figure, if we attach a box to the cylinder, a force acts along the cross-sectional area of the cylinder, causing it to stretch, and as a result, there is a change in the length of the cylinder.

The longitudinal stress can further be classified based on whether the object under stress is elongated or compressed. Types of longitudinal stress are:

• a. Compressive Stress
• b. Tensile Stress

a. Compressive Stress: When there is a reduction or decrease in the object’s length due to the applied deforming force, the stress associated with it is called Compressive stress. It can also be described as the force that leads to the deformation of the material such that there is a reduction in the material’s volume.  For example: When a toy is squeezed in or when a wire is compressed by pushing it, compressive stress is generated.

Fig: Compressive Stress

b. Tensile Stress: When there is an increase in length or elongation of the object due to the application of an external force, the stress associated with it is called tensile stress. We can easily observe tensile stress in the case of a rubber band.

Fig: Tensile Stress

2. Bulk Stress or Volumetric Stress

Bulk stress causes a change in the volume of the object or medium and is caused by forces acting on the body from all directions, perpendicular to its surface. It occurs when forces are acting along a body across all the dimensions uniformly. Since these lead to a change in the volume, they are referred to as volumetric stress. The uniform pressure on all sides of the diver is Bulk stress (or volume stress)

\({\rm{Bulk Stress}}\,{\rm{ = Force/Area}}\)

\({\rm{Force}}\,{\rm{ = Pressure}} \times {\rm{Area}}\)

Thus, Bulk stress \(= \frac{{{\rm{Pressure}} \times {\rm{Area}}}}{{{\rm{Area}}}} = {\rm{Pressure}}\)

Fig: Bulk Stress

3. Shearing/Tangential Stress

When a force acts parallel to the surface of an object, it exerts shear stress. In general, it is the component of stress coplanar with a material cross-section. This stress results in a change in the shape of the body

• a. Shear stress in solids results from twisting a metal bar about a longitudinal axis like tightening a screw.
• b. Shear stress in fluids results from actions such as the flow of liquids and gases through pipes, the sliding of a metal surface over a liquid lubricant, and the passage of a plane through the air.
• c. Shear stresses, however small, applied to proper fluids produce continuous deformation or flow as layers of the fluid move over each other at different velocities like individual cards in a spread deck.

\({\rm{Shear}}\,{\rm{ stress = force/surface area}}\)

Fig: Shear Stress

Some Other Types of Stress

Let us look at other types of Stress:

1. Breaking Stress: The minimum value of stress required to break a wire is called breaking stress.
   Breaking stress is fixed for material, but breaking force varies with the area of cross-section of the wire.
   \({\rm{Safety\,factor = Breaking\,stress / Working\,stress}}\)
2. Thermal Stress: When the temperature of a rod fixed at both ends is changed, the produced stress is called thermal stress.
   \({\text{Thermal}}\,{\text{Stress}} = \frac{F}{A} = \gamma \alpha \Delta \theta \)
   Where \(\alpha = \) coefficient of linear expansion of the material of the rod.
   When the temperature of a gas enclosed in a vessel is changed, then the thermal stress produced is equal to the change in pressure \(\left( {\Delta p} \right)\) of the gas.
   \({\text{Thermal}}\,{\text{Stress}} = \Delta p = K\gamma \Delta \theta \)
   \(K =\) bulk modulus of elasticity and \(\gamma =\) coefficient of cubical expansion of the gas and Interatomic force constant,  \(K = Y{r_0}\) (\({r_0} = \) interatomic distance).
3. Hydraulic Stress: It is the restoring force per unit area when a fluid applies force on the body. It can be visualized in a case of a ball that is dipped in a lake. Due to water pressure from all directions, the force acts on the ball; the ball seems slightly contracted. Now, because of the force exerted by the water, a restoring force develops within the ball that is equal in magnitude but opposite in direction to the force applied by the water. This type of stress is known as hydraulic stress.

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Frequently Asked Questions on Stress

Let us look at some of the frequently asked questions about Stress:

Q1. What is deformation?
Ans: A change in shape due to the application of a force is known as a deformation. Even minimal forces are known to cause some deformation. Deformation is experienced by objects or physical media under the action of external forces— for instance, this may be squashing, ripping, squeezing, twisting, shearing, or pulling the objects apart.

Q.2. What is stress in Physics?
Ans: Stress is defined as the restoring force acting per unit area on a material.

Q.3. What is longitudinal stress?
Ans: Longitudinal stress is equal to the deforming force per unit area. Longitudinal strain is the ratio of change in length to the original length.

Q.4. What is bulk stress?
Ans: Bulk or volume stress is equal to the change in pressure. Bulk or volume strain is the ratio of change in volume to the original volume.

Q.5. Which stress leads to a decrease in the length of an object?
Ans: Compressive stress leads to a decrease in the length of an object.

Q.6. Why is stress generated in a material?
Ans: When a body is deformed, the intermolecular distance of separation changes from its equilibrium value. This causes intermolecular forces to be generated, restoring the body to its original state once the deforming external forces are removed. The sum total of intermolecular restoring forces per unit area is called stress.

Q.7. What is the SI unit of stress?
Ans: Stress is measured in terms of Pascal. One pascal is equal to one-newton force over one squared meter area.

Q.8. What is tangential stress?
Ans: Shear or tangential stress is equal to a tangential deforming force per unit area. Shear strain is the angle through which the dimension perpendicular to the faces under deforming forces rotates.

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