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# Modulus of Elasticity

## Modulus of Elasticity Definition

Last updated date: 28th Mar 2023
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Modulus of Elasticity also referred to as Elastic Modulus or just Modulus is the quantification of the ratio of a material's elasticity. Modulus of elasticity measures the resistance of a material to non-permanent or elastic deformation when a ratio of stress is applied to its body. When under stress, materials will primarily expose their elastic properties. The stress induces them to deform, but the material will resume to its earlier state after the stress is eliminated. After undergoing the elastic region and through their production point, materials enter a plastic region, where they reveal everlasting deformation even after the tensile stress is removed.

### Graphical Representation of Modulus of Elasticity

Graphically, a Modulus is described as being the slope of the straight-line part of stress, denoted by (σ), and strain, denoted by (ε), curve. Focusing on the elastic region, if the slope is between two stress-strain points, the modulus will be the change in stress divided by the change in strain. Thus, Modulus =σ2−σ1 / ε2−ε1.

In this, the stress (σ) is force divided by the specimen's cross-sectional region and strain (ε) is the alteration in the length of the material divided by the material's original measure length. Seeing that both stress and strain are normalized quantifications, modulus exhibits a consistent material property that can be differentiated between specimens of different sizes. A huge steel specimen will have a similar modulus as a small steel specimen, though the large specimen would need a greater maximum force to deform the material.

Note: Brittle materials such as plastics, aluminum, copper and composites will reveal a steeper slope and higher modulus value than ductile materials such as iron, rubber, steel, etc.

Refer below for the Strain-Stress curve.

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### Modulus of Plastic or Elastomers

In contrast to brittle materials like metals and plastics, elastomeric materials do not display a yield point and continue to deform the material body elastically until they break. In the case of synthetic polymers having elastic properties like rubber, the modulus is simply expressed as a measure of the force at a given elongation. For example, in the graph below, the modulus is reported as stress at different levels for various materials.

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### Types of Modulus of Elasticity

The modulus of elasticity of a material is the quantification of its stiffness and for most materials remains consistent over a range of stress. Following are the different types of modulus of elasticity:

### Young's Modulus

The ratio proportion of the longitudinal strain to the longitudinal stress is known as Young's modulus.

### Bulk Modulus

The ratio of the stress applied to the body on the body's fractional decrease in volume is called the bulk modulus. Thus, when a body is subjected to three mutually perpendicular stresses of the same intensity, the ratio of direct stress to the corresponding volumetric strain is what we call the Bulk Modulus. Bulk Modulus is generally denoted with the letter K.

### Shear Modulus

Also known as Rigidity Modulus, the ratio of the tangential force applied per unit area to the angular deformation in radians is called the shear modulus. Shear Modulus is generally denoted with the letter C.

### How to Calculate Different Types of Modulus of Elasticity

There are different types of modulus of elasticity and specific ways of calculating types of modulus of elasticity which we will be discussing below.

### Calculating Different Types of Modulus of Elasticity

Calculating the modulus of elasticity is generally required by users recording modulus. Hence, they should be well acquainted that there are various ways to measure the slope of the initial linear portion of a stress/strain curve. Remember that, when comparing outcomes of modulus for a given material between different laboratories, it is crucial to know which type of modulus calculation has been selected.

Thus, slopes are measured on the initial linear portion of the curve by employing least-squares fit on test data. The steepest slope is concluded as the modulus.

Let’s see below how to calculate different types of modulus of elasticity:

### Young’s Modulus

Young’s Modulus, usually denoted by (Y) = Longitudinal Stress ÷ Longitudinal Strain Nm-² or pascals.

### Bulk Modulus

Bulk Modulus of material is easily calculated in the following manner.

Bulk Modulus (K) = $\frac {(F/A)} {(V/V)}$ = $\frac {FV} {VA}$

Thus, we get

K= $\frac {PV} {v Nm^2}$

Where, $\frac {F} {A}$ = volume stress or bulk stress

$\frac {v} {V}$ = volume strain or bulk strain

Thus, P = $\frac {F} {A}$

### Shear Modulus

We determine the Shear Modulus in the following way.

Shear Modulus (n) = tangential stress ÷ Shearing strain

= $\frac {F} {A Nm^2}$

### Chord Modulus

To determine the chord modulus, we have to choose a beginning strain point and an end strain point. A line segment is needed to be drawn between the two points and the slope of that line is reported as the modulus.

### E-Modulus

Elastic modulus is identified using a standard linear regression strategy. The part of the curve to be used for the computation is chosen automatically and does not include the initial and final parts of the elastic deformation at the position where the stress-strain curve is non-linear.

### Hysteresis Modulus

Modulus is identified easily by a hysteresis loop produced by a portion of loading and reloading.

### Secant Modulus

Using the zero stress/strain point as the beginning value and a user-selected strain point as the end value, we can determine this type of modulus. A segment is constructed between the two points, and the slope of that line is reported as the modulus.

### Segment Modulus

We need to choose a start strain point and an end strain point. Using the least-squares fit on all points between the start and the endpoints, a line segment is drawn. The slope of the best fit line is thus recorded as the modulus.

### Tangent Modulus

Choosing a tangent point on the stress/strain curve, we can calculate the tangent type of modulus. The slope of the tangent line is thus recorded as the modulus.

### Does Young's Modulus Change with Change in Radius of the Wire?

No, Young's modulus does not change with a change in radius. Young's modulus is a material property, and for a wire its value remains constant. It doesn't depend on the dimension of material or the load(force) applied.

## FAQs on Modulus of Elasticity

1. What is the difference between Modulus of Elasticity and Modulus of Rigidity?

The main differences between Modulus of Elasticity and Modulus of Rigidity are:

• Modulus of Elasticity describes the deformation of a material when a force acts at a right angle to the surface of the object whereas Modulus of Rigidity describes a material's deformation when a force is applied in a parallel direction to the surface of the object.

• Change in Shape - When an object undergoes deformation force due to Modulus of Elasticity, it either gets lengthened or shortened. On the other hand, When an object undergoes some amount of force due to modulus of rigidity, it gets displaced with respect to another surface.

• Relative Size - For most materials, the modulus of elasticity is more than the modulus of rigidity. But, the exception of this rule is the 'auxetic materials'. These materials have a negative value of Poisson’s ratios. Poisson's ratio is the change in the width of the material per unit width. The value of the Poisson's ratio is equal to the negative of the ratio of transverse strain to axial strain i.e, ( -transverse strain/axial strain).

2. Explain the relation between

(i)  Modulus of Elasticity and Modulus of Rigidity

(ii) Modulus of Elasticity and bulk modulus

(i) The relation between Modulus of Elasticity and Modulus of Rigidity is given by:

E = 2G (1+μ)

Here, v refers to the number called the 'Poisson ratio' of the particular material.

Also, The fractional increase in length is defined as the axial strain and the fractional reduction in length is defined as the traverse strain on the object. The S.I unit of the relation between Young's modulus of Elasticity and Modulus of Rigidity is N/m2 or pascal(Pa).

(ii) The relation between Young’s modulus (modulus of elasticity) and bulk modulus is given by: E=3K(1−2μ).

Here, G = modulus of rigidity

.S.I unit between Young’s modulus and bulk modulus is N/m2 or pascal(Pa).

Also, the relation between Young’s modulus and bulk modulus k and modulus of rigidity is represented by :

E = $\frac {9KG} {G+3K}$

where, E = Young’s modulus or modulus of Elasticity.

3. Explain the following terms related to Young's Modulus:

(i) Stress

(ii) Strain

Terms related to Young’s modulus or modulus of elasticity:

(i) Stress: Stress is the force experienced by the unit area of a material. Stress can be characterized on the basis of the direction of the deforming forces acting on the body. Magnitude of stress = F.A = Force.area

(ii) Strain: A force that stretches or affects the material and makes it reach a damaging degree. Strain = $\frac {dL} {L}$

, where dL = total elongation and L = original length.

4. What are the real-life examples of the Modulus of Elasticity?

Following are some practical and real-life examples of Young's modulus:

• While walking up or down through the stairway, it is assumed that the young's modulus of the boards are good enough that they resist breaking, when we put our full body weight on them.

• Springs are used in many mechanical and electronic devices in this modern world. Spring stores elastic energy which handles the load applied to it.

• Aircraft and racing bicycles use young's modulus of elasticity. In these transport applications, stiffness is required at minimum weight.

•  When we apply a force on the rubber, then, there are major changes occurring in it. One of them is the 'stretching of rubber'. Due to Young's modulus of elasticity, rubber doesn't break but changes its length and handles the force.

• The beams used in the bridge are an example of Young's modulus. They play an important role in reducing the stress and the bridge can withstand the huge load of the moving traffic.

5. Solve the following numerical on Young's Modulus of Elasticity :

A 45-kilogram traffic light is suspended from two steel cables of equal length and radii 0.50 cm. If each cable makes a 15° angle with the horizontal, what is the fractional increase in their length due to the weight of the light?

Given & Known Data:

• m = 45 kg is the mass of the traffic light.

• r = 0.50 cm = 0.0050 m is the radius of the steel cable.

• 0= 15° is the angle made by the cable with the horizontal.

• E = 20 × 1010 N m2 is Young's modulus of steel.

The traffic light is in static equilibrium condition. So in the vertical direction, we will be using the translational equilibrium condition.

sum Fy =0

T x sin + T x sinθ - W = 0

T x sinθ + T x sinθ = W

T x sinθ + T x sinθ = mg

T - $\frac {(mg)} {(2\sin\theta)}$

= 45 kg x 9.81 m s2 2 x sin15

= approx 852.82 N

Therefore, the tension in the cable is 852.82 N.

Now, we'll calculate the fractional increase in the length of the cable :

Delta x L = $\frac {(TL)} {(AE)}$

(Delta x L)/L = $\frac {T} {(AE)}$

= T ( pi r2 )E

= 5.4 x 105

Hence, Due to the weight of traffic lights, the fractional increase in length of the cable is 5.4 x 105