### 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,

** ****φ _{1} = σ_{1 }/ E –
μ * (σ_{2}/E)**

Where,

φ_{1 }- Total Strain created
along the x-direction

σ_{1 } - Stress crested along the x-direction due to
the force F1

E - Young’s modulus or modulus of Elasticity

μ - Poisson’s Ratio

σ_{2 } - 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,

** ****φ _{2} = σ_{2 }/ E –
μ * (σ_{1}/E)**

Where,

Φ_{2 }- Total Strain created
along the x-direction

σ_{2 } - Stress crested along the y-direction due to
the force F2

E – Young’s modulus or modulus of Elasticity

μ - Poisson’s Ratio

σ_{1 } - 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 = φ _{1 }+ φ**

_{2}

_{ }Where,

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

φ_{1 }- Total Strain in the x-direction

φ_{2 }- 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,

** ****φ _{1} = σ_{1 }/ E – μ (σ_{2}/E) **

**– μ (**

**σ3**

**/E)**

Where,

φ_{1 }- Total Strain created along the x-direction

σ_{1 } - Stress crested along the x-direction due to the force F1

E - Young’s modulus or modulus of Elasticity

μ - Poisson’s Ratio

σ_{2 } - 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,

** ****φ _{2} = σ_{2 }/ E – μ (σ_{1}/E)**

**– μ (**

**σ3**

**/E)**

Where,

φ_{2} _{ }- Total Strain created along the x-direction

σ_{2 } - Stress crested along the y-direction due to the force F2

E – Young’s modulus or modulus of Elasticity

μ - Poisson’s Ratio

σ_{1 } - 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 (σ_{1}/E)**

**– μ (**

**σ2**

**/E)**

Where,

φ3 _{ }- Total Strain created along the z-direction

σ_{2 } - Stress crested along the y-direction due to the force F2

E – Young’s modulus or modulus of Elasticity

μ - Poisson’s Ratio

σ_{1 } - 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

φ_{1 }- Total Strain in the x-direction

φ_{2 }- Total Strain in the y-direction

φ3_{ }- Total Strain in the z-direction

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