Relationship between Stress and Strain

      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 on a regular basis. In this article, I explained the Relationship between Stress and Strain,

Relationship in Two Dimensional system

    The relationship between stress and strain is obtained from Poisson's ratio but when we apply the ratio in a two-dimensional system, it becomes like this. Let me explain it with an example. Before reading further, read Poisson's ratio to better understand the concept.

    Let us take a square object (ABCD) as the corners of the object, Let the Tensile force(F1) acts on the sides AB and CD i.e x- direction, and let there be another Tensile force (F2) act on the sides AD and BC i.e y-direction. 

    Now from the above set up Force (F1) creates Tensile stress (σ1)  and Tensile strain (φ1), Force (F2) creates Tensile stress (σ2)  and Tensile strain (φ2),

    The Force F1 also creates lateral strain along the y-direction even though it acts in the x-direction in addition to Tensile stress (σ1)  and Tensile strain (φ1), Mathematically Expressing,

The strain Created along the x-direction is given by,

            φ₁ = σ₁ / E – μ * (σ₂/E)

            Where,

            φ₁ - Total Strain created along the x-direction

            σ₁  - Stress crested along the x-direction due to the force F1

            E - Young’s modulus or modulus of Elasticity

            μ - Poisson’s Ratio

            σ₂  - Stress crested along the y-direction due to the force F2

In the same manner, the total strain created in the Y-direction is given by,

            φ₂ = σ₂ / E – μ * (σ₁/E)

            Where,

            Φ₂ - Total Strain created along the x-direction

            σ₂  - Stress crested along the y-direction due to the force F2

            E – Young’s modulus or modulus of Elasticity

            μ - Poisson’s Ratio

            σ₁  - Stress crested along the x-direction due to the force F1

Hence the Total Stress and Strain Relationship in a two-dimensional system is given by,

            Φ = φ₁ + φ₂

                Where,

            Φ – Total Strain in the object due to the two-dimensional stress and strain

            φ₁  - Total Strain in the x-direction

            φ₂  - Total Strain in the y-direction.

Relationship in a Three Dimensional System

  Let us take an object, Let the Tensile force(F1) acts in the x direction, let there be another Tensile force (F2) act in the y direction and let there be another Tensile force (F3) act in the z-direction

    Now from the above set up Force (F1) creates Tensile stress (σ1)  and Tensile strain (φ1), Force (F2) creates Tensile stress (σ2)  and Tensile strain (φ2) and Force (F3) creates Tensile stress (σ3)  and Tensile strain (φ3)

    The Force F1 also creates lateral strain along the y & z-direction even though it acts in the x-direction in addition to Tensile stress (σ1)  and Tensile strain (φ1), Mathematically Expressing,

The strain created along the x-direction is given by,

            φ₁ = σ₁ / E – μ (σ₂/E) – μ (σ3/E)

            Where,

            φ₁ - Total Strain created along the x-direction

            σ₁  - Stress crested along the x-direction due to the force F1

            E - Young’s modulus or modulus of Elasticity

            μ - Poisson’s Ratio

            σ₂  - Stress created along the y-direction due to the force F2

             σ3 - Stress created along the z-direction due to the force F3

In the same manner, the total Strain Created in the y-direction is given by,

            φ₂ = σ₂ / E – μ (σ₁/E)– μ (σ3/E)

            Where,

            φ₂  - Total Strain created along the x-direction

            σ₂  - Stress crested along the y-direction due to the force F2

            E – Young’s modulus or modulus of Elasticity

            μ - Poisson’s Ratio

            σ₁  - Stress crested along the x-direction due to the force F1

           σ3 - Stress created along the z-direction due to the force F3

In the same manner, the total Strain Created in the z-direction is given by,

            φ3 = σ3 / E – μ (σ₁/E)– μ (σ2/E)

            Where,

            φ3  - Total Strain created along the z-direction

            σ₂  - Stress crested along the y-direction due to the force F2

            E – Young’s modulus or modulus of Elasticity

            μ - Poisson’s Ratio

            σ₁  - Stress crested along the x-direction due to the force F1

           σ2 - Stress created along the z-direction due to the force F2

Hence the Total Stress and Strain Relationship in a three-dimensional system is given by,

            Φ = φ1 + φ2 + φ3

                Where,

            Φ – Total Strain in the object due to the three-dimensional stress and strain

            φ₁  - Total Strain in the x-direction

            φ₂  - Total Strain in the y-direction

            φ3 - Total Strain in the z-direction