# Shear Modulus

## What is Shear Modulus?

Shear modulus is also known as the modulus of rigidity, it is a constant number that describes the elastic properties of a solid, under the use of transverse internal forces such as arise. For example, in torsion, twisting of the metal about its own axis is known as the shear modulus. In these types of material, any of the small volumes of the material is distorted, in a way that both faces of the material slide parallel to each other for a small distance and the other two faces change from the square to the diamond shape.

Rigidity Modulus Definition - It is the measure of the rigidity of the body, represented as the ratio of shear stress to the shear strain.

The shear modulus measures the strength of an object to withstand flexible deformities and is a valid indicator of behaviour that extends only to a small deformity, after which the object is able to return to its original configuration. Excessive shear force leads to the permanent deformation and rotation or breakage of the material.

### Mathematically

In physics materials science, shear modulus or modulus of rigidity is denoted by the letter ‘G’, or sometimes it is also denoted as  ‘S’ or ‘μ’. Shear modulus is the measure of elastic shear stiffness of a material and the shear modulus definition - is the ratio of shear stress to the shear strain:

G = $\frac{\tau_{xy}}{\gamma_{xy}}$ = $\frac{F/A}{\frac{\Delta x}{l}}$ = $\frac{Fl}{A\Delta x}$

where,

$\tau$$_{xy}$ = F/A = shear stress.

‘F’ is the force that acts.

‘A’ is the area of the material, on which force will be acting.

$\gamma$$_{xy}$ = shear strain. In engineering $\frac{\Delta x}{l}$ = tanθ,

elsewhere it will be ‘θ’

$\Delta$x’ is the transverse displacement.

‘l’ is the initial length of the area.

The SI unit of the shear modulus is the pascal (pa), but it is usually expressed in the gigapascal (GPa) or some of the time it is also expressed in thousand per square inch (KSI). And the dimensional formula of the shear modulus is given as  M1L−1T−2, replacing force by mass times acceleration.

### Explanation

Shear modulus is one among the various quantities used for measuring the stiffness of the materials, all of them arising keeping Hooke’s law in general.

• Young's modulus denoted by ‘E’, describes the strain of the material (change in length to the original length) response to uniaxial stress in the direction of this stress (like pulling on the ends of a wire, with the wire getting longer).

• The Poisson's ratio denoted by ‘ν’, is the ratio that describes the response in the orthogonal directions to this uniaxial stress (the wire getting thinner compared to the previous thickness).

•  The bulk modulus ‘k’ of the material describes the response to hydrostatic (uniform) pressure (like the pressure at the bottom of the swimming pool).

• The shear modulus G  of a material describes the response of the  shear stress, these moduli are not independent, and for isotropic materials, they are connected through the equations 2G(1 + v) = E = 3K(1 - 2v)

The shear modulus is concerned with the conversion of a solid-state when it encounters the same force as one of its surfaces. While its opposing face encounters opposing forces (such as friction). In the case of a rectangular prism object, it will be disabled as a parallelepiped. Anisotropic materials such as wood, paper and basically all single crystals show a different reaction of substances to stress or difficulty when tested in different directions. In this case, one may need to use a full-strength accent of the elastic constant, rather than a single scalar value of the material.

## Shear Modulus Value of Different Materials

 Material Typical Values for Shear Modulus (GPa)(at Room Temperature) Diamond 478.0 Steel 79.3 Iron 52.5 Copper 44.7 Titanium 41.4 Glass 26.2 Aluminium 25.5 Polyethylene 0.117 Rubber 0.0006 Granite 24 Shale 1.6 Limestone 24 Chalk 3.2 Sandstone 0.4 Wood 4

From the above-given table, we get to know the value of the shear modulus of the different material for example shear modulus of steel/modulus of rigidity of steel is 79.3 GPa.

### Example Problem

1. Consider, a block of unknown material kept on a table (The square face of the material is placed on the table), is under a shearing force. Some of the data are given below to calculate the shear modulus of the material, dimensions = 60 mm x 60 mm x 20 mm, Shearing Force = 0.245 N, displacement = 5 mm.

Solution:

Substituting the values in the formula we get-

Shear stress = $\frac{F}{A}$ = $\frac{0.245}{60 \times 20 \times 10^{-6}}$ = $\frac{2450}{50}$ N/m2

Shear strain = $\frac{\Delta x}{l}$ = $\frac{5}{60}$ = $\frac{1}{12}$

Thus,

Shear modulus, G = $\frac{\text{shear stress}}{\text{shear strain}}$ = $\frac{2450 \times 12}{12}$

= 2450 N/m2.

### Do you know?

What is the relation between Elastic Constants?

The elastic moduli of a material, like Young’s Modulus, Bulk Modulus, Shear Modulus are specific forms of Hooke’s law, which states that, for an elastic material, the strain experienced by the corresponding stress applied is proportional to that stress. Thus, we can write the relation between elastic constants by the following equation: 2G(1 + v) = E = 3K(1 - 2v)

Where,

• G is the Shear Modulus

• E is the Young’s Modulus

• K is the Bulk Modulus elasticity

• υ is Poisson’s Ratio

Q1. How Does Rigidity Modulus Be Related to Other Elastic Moduli?

Ans: The rigidity modulus of the material is related to other elastic moduli as 2G(1 + v) = E = 3K(1 - 2v), where G is the Shear Modulus, E is the Young’s Modulus, K is the Bulk Modulus elasticity, and υ is Poisson’s Ratio.

Q2. What Would Happen to the Shear Modulus, If the Shear Force Applied Increases?

Ans: When the applied shear force is increased, the value of the shear modulus will also increase, they are dependent.

Q3. What Does it Mean if the Shear Modulus of One Material is X Pascals and Material Two is 30x Pascals?

Ans: If the shear modulus of elasticity of material one is x pascals and Material two is 30x pascals, it means that material two is 30 times more rigid compared to material one.

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