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VOLUMETRIC STRAIN OF A RECTANGULAR BODY SUBJECTED TO THREE MUTUALLY PERPENDICULAR FORCES

We were discussing “Determination of volumetric strain for a rectangular bar subjected with an axial load in the direction of length of the rectangular bar, “Total elongation of the bar due to its own weight” and “Thermal stress and strain” in our previous posts. 

Today we will see here the determination of volumetric strain for a rectangular bar subjected with three forces mutually perpendicular with each other with the help of this post. 

Before going ahead we must recall the basic concept of volumetric strain which is explained here briefly.

When an object will be subjected with a system of forces, object will undergo through some changes in its dimensions and hence, volume of that object will also be changed.

Volumetric strain will be defined as the ratio of change in volume of the object to its original volume. Volumetric strain is also termed as bulk strain.

Ԑv= Change in volume /original volume
Ԑv= dV/V

Let us consider one rectangular bar as displayed in following figure. x, y and z are the dimensions of the rectangular bar which is subjected with three forces and hence three direct tensile stresses mutually perpendicular with each other.
We can also say that we have following initial dimensions of the rectangular bar
x = Length of the rectangular bar
y = Width of the rectangular bar
z = Thickness or depth of the rectangular bar
Volume of the rectangular bar, V = xyz

Δx=Change in length of the rectangular bar
Δy=Change in width of the rectangular bar
Δz= Change in thickness or depth of the rectangular bar
ΔV= Change in volume of the rectangular bar

Let us determine the final dimensions of the rectangular bar

Final length of the rectangular bar = x+ Δx
Final width of the rectangular bar = y + Δy
Final thickness or depth of the rectangular bar = z + Δz
Final volume of the rectangular bar = (x+ Δx). (y + Δy). (t + Δt)
Let us ignore the product of small quantities and we will have
Final volume of the rectangular bar = xyz + y. z. Δx + x. z. Δy + x. y. Δz

Let us determine the change in volume of the rectangular bar

Change in volume of the rectangular bar = Final volume – initial volume
ΔV= (xyz + y. z. Δx + x. z. Δy + x. y. Δz) - xyz
ΔV= y. z. Δx + x. z. Δy + x. y. Δz

Volumetric strain also known as bulk strain will be determined as following
Ԑv= Change in volume /original volume
Ԑv= dV/V
Ԑv= (y. z. Δx + x. z. Δy + x. y. Δz)/ (xyz)
Ԑv= (Δx/x) + (Δy/y) + (Δz/z)
Ԑv= Ԑx + Ԑy + Ԑz

Let us consider the stresses in various directions and young’s modulus of elasticity
σx= Tensile stress in x-x direction
σy= Tensile stress in y-y direction
σz= Tensile stress in z-z direction
E= Young’s modulus of elasticity

Let us brief here first lateral strain before going ahead; lateral strain will be the strain at perpendicular or right angle to the direction of applied force.

Therefore we can define here lateral strain such as lateral strain will be basically defined as the ratio of change in breadth of the body to the original breadth of the body.

Tensile stress acting in x-x direction i.e. σx will produce a tensile strain in x-x direction and lateral strain in y-y direction and z-z direction.

Let us brief here first “Poisson ratio” before going ahead
Poisson ratio = Lateral strain /Linear strain
We can also say that, Lateral strain = Poisson ratio (ν) x Linear strain

As we have already seen that, lateral strain will be opposite in sign to linear strain and therefore above equation will be written as following 
Lateral strain = - Poisson ratio (ν) x Linear strain
Therefore tensile stress acting in x-x direction i.e. σx will produce a tensile strain (σx /E) in x-x direction and lateral strain  (ν. σx /E) in y-y direction and z-z direction.
Similarly, tensile stress acting in y-y direction i.e. σy will produce a tensile strain (σy /E) in y-y direction and lateral strain (ν. σy /E) in x-x direction and z-z direction.
Similarly, tensile stress acting in z-z direction i.e. σz will produce a tensile strain (σz /E) in z-z direction and lateral strain (ν. σz /E) in x-x direction and y-y direction. 

Strain produced in x-x direction

σx will produce a tensile strain (σx /E) in x-x direction
σy will produce a lateral strain i.e. compressive strain here (ν. σy /E) in x-x direction
σz will produce a lateral strain i.e. compressive strain here (ν. σz /E) in x-x direction

Total strain produced in x-x direction

Ԑx = (σx /E) - (ν. σy /E) - (ν. σz /E)
Ԑx = (σx /E) - ν [(σy + σz)/E]

Total strain produced in y-y direction

Ԑy = (σy /E) - ν [(σx + σz)/E]

Total strain produced in z-z direction

Ԑz = (σz /E) - ν [(σx + σy)/E]
Let us add total strain produced in each direction to find out the volumetric strain for a rectangular bar subjected with three forces mutually perpendicular with each other
Volumetric strain, Ԑv= Ԑx + Ԑy + Ԑz
Volumetric strain, Ԑv= [(σx + σy+ σz)/E-2 νx + σy+ σz)/E]

Volumetric strain, Ԑv= (σx + σy+ σz) (1-2 ν)/E

Do you have any suggestions? Please write in comment box

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

We will see another important topic i.e. Stress analysis of bars of varying sections in the category of strength of material.

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2 comments:

  1. When a stress acting on rectangular bar the breadth and height decreases then here we calculated the increases as d+d why

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