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.

Ԑ_v= Change in volume /original volume

Ԑ_v= 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

Ԑ_v= Change in volume /original volume

Ԑ_v= dV/V

Ԑ_v= (y. z. Δx + x. z. Δy + x. y. Δz)/ (xyz)

Ԑ_v= (Δx/x) + (Δy/y) + (Δz/z)

Ԑ_v= Ԑ_x + Ԑ_y + Ԑ_z

Let us consider the stresses in various directions and young’s modulus of elasticity

σ_x= Tensile stress in x-x direction

σ_y= Tensile stress in y-y direction

σ_z= 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. σ_x 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. σ_x will produce a tensile strain (σ_x /E) in x-x direction and lateral strain  (ν. σ_x /E) in y-y direction and z-z direction.

Similarly, tensile stress acting in y-y direction i.e. σ_y will produce a tensile strain (σ_y /E) in y-y direction and lateral strain (ν. σ_y /E) in x-x direction and z-z direction.

Similarly, tensile stress acting in z-z direction i.e. σ_z will produce a tensile strain (σ_z /E) in z-z direction and lateral strain (ν. σ_z /E) in x-x direction and y-y direction. 

Strain produced in x-x direction

σ_x will produce a tensile strain (σ_x /E) in x-x direction

σ_y will produce a lateral strain i.e. compressive strain here (ν. σ_y /E) in x-x direction

σ_z will produce a lateral strain i.e. compressive strain here (ν. σ_z /E) in x-x direction

Total strain produced in x-x direction

Ԑ_x = (σ_x /E) - (ν. σ_y /E) - (ν. σ_z /E)

Ԑ_x = (σ_x /E) - ν [(σ_y + σ_z)/E]

Total strain produced in y-y direction

Ԑ_y = (σ_y /E) - ν [(σ_x + σ_z)/E]

Total strain produced in z-z direction

Ԑ_z = (σ_z /E) - ν [(σ_x + σ_y)/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, Ԑ_v= Ԑ_x + Ԑ_y + Ԑ_z

Volumetric strain, Ԑ_v= [(σ_x + σ_y+ σ_z)/E-2 ν (σ_x + σ_y+ σ_z)/E]

Volumetric strain, Ԑ_v= (σ_x + σ_y+ σ_z) (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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