Shear Modulus: Definition, Formula, Factors and Sample Questions

Shear modulus or modulus of rigidity in materials is a measure of the elastic shear stiffness of a material. It is also defined as the ratio of shear stress to the shear strain. It is denoted by G, or less commonly by S or μ. A large shear modulus value denotes that a solid is highly rigid. Shear Modulus is used to define how a material resists transverse deformations. The formula of Shera Modulus is, \(G = \frac{\tau_{xy}}{\gamma_{xy}}\) \(=\frac{\frac{(F)}{(A)}}{\Delta x \over l} = \frac{Fl}{A \Delta x}\). 

The dimensional formula of shear modulus is defined as M¹L^-1T^-2.  Typically, the shear modulus G explains a material's response to shear stress (such as, cutting materials with dull scissors). In case a material is extremely resistant to attempted shearing, thus it will further transmit the shear energy quickly.

Key Terms: Shear Modulus, Transverse Internal Forces, Stress, Rigidity of Modulus, Shear Modulus formula, Elastic constant

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What is Shear Modulus?

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The modulus of rigidity, commonly known as shear modulus, is a constant number that characterizes the elastic properties of a solid when transverse internal forces, such as stress, are applied.

Shear Modulus

The shear modulus, for example, is the twisting of metal about its own axis in torsion. Any of the small volumes of the material is distorted in these sorts of materials in such a way that both faces slide parallel to each other for a short distance and the other two faces shift from square to diamond shape.

The shear modulus assesses an object's ability to sustain flexible deformations and is a good predictor of behaviour that only lasts a short time before the thing returns to its original configuration. The excessive shear force causes permanent material deformation, rotation, and fracture.

Also Read: Elastic moduli

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Modulus of Rigidity Formula

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Shear modulus, also known as modulus of stiffness, is symbolized by the letter ‘G' in physics and materials science, but it can alternatively be written as ‘S' or ‘μ'. The ratio of shear stress to shear strain is the definition of shear modulus, which is a measure of a material's elastic shear stiffness.

Where:

• \(\tau_{xy}\) = F/A = Shear stress
• The force acting on the item is denoted by F.
• The area on which the force acts is denoted by the letter A.
• γ_xy = shear strain. In engineering Δx/l = tanθ, 
• The transverse displacement is denoted by x'.
• The starting length of the area is denoted by the letter ‘l.'

Although the shear modulus is measured in pascals (pa), it is more often expressed in gigapascals (GPa), or thousand per square inch (KSI). M¹L⁻¹T⁻², where force is replaced by mass times acceleration, is the dimensional formula for the shear modulus.

Also Read: Yield Strength

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Explanation of Shear Modulus

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Shear modulus is one of many terms used to describe the stiffness of materials, all of which are derived from Hooke's law in general. 

• The material's strain (change in length relative to its original length) in response to uniaxial tension in the direction of the stress is described by Young's modulus ‘E’ (like pulling on the ends of a wire, with the wire getting longer).
• The Poisson's ratio, designated by the letter ‘v,' is the ratio that describes the reaction to this uniaxial stress in orthogonal directions (the wire getting thinner compared to the previous thickness).
• The material's bulk modulus 'k' specifies its response to hydrostatic (uniform) pressure (like the pressure at the bottom of the swimming pool).
• The shear modulus G of a material represents the reaction of the shear stress; these moduli are not independent, and they are coupled by equations 2G(1 + v) = E = 3K(1 – 2v)  for isotropic materials.

When a solid-state experiences the same force as one of its surfaces, the shear modulus is concerned with the conversion. The oppositional face, on the other hand, is confronted with conflicting forces (such as friction).

It will be disabled as a parallelepiped in the event of a rectangular prism object. When tested in different directions, anisotropic materials such as wood, paper, and essentially all single crystals reveal a varied reaction of substances to stress or difficulty. In this situation, rather than a single scalar value of the material, a full-strength accent of the elastic constant may be required.

Also Read: Tensile Stress

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Factors Affecting Shear Modulus

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There are several factors that affect Shear Modulus, including:

Isotropic and Anisotropic Materials

When it comes to sheer, some materials are isotropic, which means that the deformation in response to a force is the same regardless of orientation. Other materials are anisotropic, meaning they react to stress or strain differently depending on their orientation.

Isotropic and Anisotropic Material

Shear is far more vulnerable to anisotropic materials along one axis than the other. Consider the behaviour of a block of wood when a force is applied parallel to the wood grain versus a force applied perpendicular to the grain. Consider how a diamond reacts to a force applied to it. The orientation of the force with regard to the crystal lattice determines how easily the crystal shears.

Effect of Temperature and Pressure

The response of a material to an applied force varies with temperature and pressure, as one might expect. Shear modulus in metals normally falls as temperature rises. With increasing pressure, rigidity diminishes.

The Mechanical Threshold Stress (MTS) plastic flow stress model, the Nadal and LePoac (NP) shear modulus model, and the Steinberg-Cochran-Guinan (SCG) shear modulus model are three models used to estimate the effects of temperature and pressure on shear modulus. In metals, there is a temperature and pressure range where the change in shear modulus is linear. Modeling behavior outside of this range is more difficult.

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Features of Shear Modulus

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When a shear force causes lateral deformation, the modulus of rigidity is the elastic coefficient. It allows us to determine how stiff a body is. Everything you need to know about stiffness modulus is summarised in the table below.

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Calculation of Shear Modulus

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Consider a shearing force acting on a block of unknown material on a table (the material's square face is on the table). Some of the information needed to calculate the material's shear modulus is provided below.

Dimensions of the block = 60 mm x 60 mm x 20 mm

Shearing Force = 0.245 N

Displacement = 5 mm

We get the following results by substituting the values in the formula:

Shear stress = F/A = 0.245/60×20×10^-6  = 2450/50 Nm^-2

Shear strain = Δx/l = 5/60 = 1/12

Thus,

Shear modulus, G = shear stress/shear strain = 2450 × 12/12

= 2450 N/m²

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Relation between Elastic Constants

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The elastic constants can be expressed as Gibbs free energy’s second derivative in respect to strain  Young's Modulus, Bulk Modulus, and Shear Modulus are special formulations of Hooke's law, which asserts that the strain experienced by the corresponding stress imposed is proportionate to that stress for an elastic material. As a result, the following equation can be used to express the relationship between elastic constants.

Where,

• Shear Modulus is denoted by the letter G
• Young's Modulus is denoted by the letter E
• The Bulk Modulus of Elasticity is denoted by the letter K
• υ is the Poisson Ratio

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Previous Year Questions

1. A copper wire of length 2.2m and a steel wire  … [KCET 2013]
2. A light rod of length 100 cm is suspended from the ceiling horizontally … [AP EAPCET]
3. A copper wire and a steel wire of the same diameter and length … [BHU UET]
4. A steel wire can withstand a load up to … [BITSAT 2008]
5. A load of 1 kg weight is attached to one end of a steel wire of area … [BITSAT 2008]
6. There is some change … [BITSAT 2008]
7. If the ratio of lengths, radii and Young's modulus of steel … [VITEEE 2009]
8. One end of a horizontal thick copper wire of length … [JEE Advanced 2013]
9. When a rod is heated but prevented from expanding the stress … [BHU UET]
10. Two wires of same length and same material … [KEAM]
11. The breaking stress of a wire of length ll and radius r is … 
12. Two wires are made of the same material and have the same volume … [NEET 2018]
13. When a block of mass … [NEET 2019]
14. Two wires A and B are stretched by the same load … [KCET 2018]
15. Four wires of the same material are stretched by the same load … 
16. Two wires are made of the same material and have the same volume … 
17. Elastic strain energy of 10 J is stored in a loaded copper rod … [TS EAMCET 2018]
18. If the ratio of diameters, lengths and Young's modulus of steel … [NEET 2003]

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Things to Remember

• Shear modulus, also called the modulus of rigidity, is a constant number that characterizes the elastic properties of a solid when transverse internal forces, such as stress, are applied.
• The shear modulus, for example, is the twisting of metal about its own axis in torsion.
• When it comes to sheer, some materials are isotropic, which means that the deformation in response to a force is the same regardless of orientation.
• The response of a material to an applied force varies with temperature and pressure, as one might expect. Shear modulus in metals normally falls as temperature rises. With increasing pressure, rigidity diminishes.
• The Mechanical Threshold Stress (MTS) plastic flow stress model, the Nadal and LePoac (NP) shear modulus model, and the Steinberg-Cochran-Guinan (SCG) shear modulus model are three models used to estimate the effects of temperature and pressure on shear modulus.
• When a shear force causes lateral deformation, the modulus of rigidity is the elastic coefficient.

Read More: Impulse Units Notes