Relationship of Bulk Modulus and Young's Modulus

    Welcome to “The Civil Engineer”.  About myself, I am a Civil Engineer with 9+ Years of Experience and I write articles about Civil Engineering on this website regularly. In this article, I explained what is Bulk Modulus and the relationship between Young's Modulus and Bulk Modulus. 

Bulk Modulus 

    When an object is subject to mutually perpendicular-like and equal direct stress, the ratio of the direct stress to the volumetric strain is a constant within a limit, such constant is known as the bulk modulus. The Build modulus is represented by "K" 

K = Direct Stress / Volumetric Strain = σ / (dV/V) - eq 1 

Relationship Between Bulk Modulus and Young's Modulus 

     Before we proceed further, read the relationship between stress and strain in Three Dimensional systems for better understanding. 
 

    Let us take a cube and it is subjected to three mutually perpendicular tensile forces of equal intensity. 
 

    Let  
 

    L - length of the Cube 
 

    dl - Change in the length of the cube 
 

    E- young's modulus 
 

    K - Bulk modulus 
 

    σ - Tensile stress on the faces of the object 
 

    μ - Poisson's Ratio 
 

    Now, the Volume is given by V= L3 

    From the relationship between stress and strain in Three Dimensional systems, the strain in one direction is given by, 

    φ = σ/E - μ (σ/E) - μ (σ/E) 

     φ or dL /L = σ/E (1-2μ) - eq 2 

Here φ is the Change in the length of the object in one Face, then the total change in volume is given by, 

By differentiating the volume, we get the change in the volume, hence mathematically, by the power rule, 

     V= L3  

Becomes dV = 3L2 . dL  

i.e, dV / V = 3L2 . dL / L3 = 3dL / L – eq3 

Substitute eq3 in eq1, we get 

K = σ / (3dL / L) - eq4 

Substitute eq2 in eq4, we get 

K = σ / 3(σ/E (1-2μ)) 

K = E / 3 (1-2μ)  

E = 3K (1-2μ) - eq4 

eq4 gives the relationship between the Bulk modulus and Young’s modulus. By using equation 4, if we know Young’s Modulus and Poisson’s Ratio, we can find Bulk Modulus Likewise if we know any two values, we can calculate the third one. This equation is very essential for determining the strength of materials. 

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