Dimensional Formula of Modulus of Rigidity

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Shear Modulus

The modulus of rigidity or Shear modulus which is denoted by G, or sometimes S or μ, in materials science, it is defined as the ratio of shear stress to the shear strain.

Pascal (Pa) is the SI unit of shear modulus, and usually it is expressed in gigapascals (GPa) or in thousand pounds per square inch (ksi). Shear modulus dimensional form is M1L−1T−2, replacing force by mass times acceleration.

Explanation

For measuring the stiffness of materials the shear modulus is one of several quantities. In the generalized Hooke's law All of them arise:

  • The material's strain response to uniaxial stress in the direction of this stress (like putting a weight on top of a column or pulling on the ends of a wire, with the wire getting longer and the column losing height), described by the Young's modulus E. 

  • The response in the directions orthogonal to this uniaxial stress (the wire getting thinner and the column thicker), is described by the Poisson's ratio ν.  

  • Hydrostatic pressure (uniform) of the material's response (like the pressure at the bottom of the ocean or a deep swimming pool) is described by the bulk modulus K,

  • The material's response to shear stress (like cutting it with dull scissors) of the shear modulus is described by G. For isotropic materials they are connected via the equations 2G(1+υ) = E = 3K(1−2υ), and these moduli are not independent. 


The deformation of a solid is concerned with the shear modulus when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force (such as friction). In the case of a rectangular prism, it will deform into a parallelepiped. Wood, paper and also essentially all single crystals like Anisotropic materials exhibit differing material response to stress or strain when tested in different directions. One may need to In this case, use the full tensor-expression of the elastic constants, rather than a single scalar value.

A fluids one possible definition of would be a material with zero shear modulus.

Dimensional Formula and Derivation

modulus of rigidity dimensional formula of is given by,

[M1 L-1 T-2]

Where,

  • M = Mass

  • L = Length

  • T = Time

Derivation

Rigidity Modulus (μ) = shear stress × [shear strain]-1 . . . . (1)

Since, strain = ΔL/L = Dimensionless Quantity . . . . (2)

And, Stress = Force × [Area]-1 . . . . . (3)

The dimensional formula of,

Area = [M0 L2 T0] . . . . (4)

Force = [M1 L1 T-2] . . . . . (5)

While substituting the equation (4) and the equation (5) in the equation (3) we get,

Stress = [M1 L1 T-2] × [M0 L2 T0]-1

∴ The dimensions of stress = [M1 L-1 T-2] . . . . (6)

While substituting the equation (2) and the equation (6) in the equation (1) we get,

Rigidity Modulus (μ) = shear stress × [shear strain]-1

Or, μ = [M1 L-1 T-2] × [M0 L0 T0]-1 = [M1 L-1 T-2].

The Universal Gravitational Constant is dimensionally represented as [M1 L-1 T-2].

Shear Modulus of Metals

With increasing temperature the shear modulus of metal is usually observed to decrease. The shear modulus also appears to increase at high pressures, with the applied pressure. The vacancy formation energy, melting temperature, and the shear modulus correlations have been observed in many metals.

attempt to predict the shear modulus of metals (and possibly that of alloys) is existing in Several models. In plastic flow computations the shear modulus models that have been used include:

  1. Mechanical Threshold Stress (MTS) plastic flow stress model is being used in the MTS shear modulus model developed by and used in conjunction. 

  2. The Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model is used in the  Steinberg-Cochran-Guinan (SCG) shear modulus model developed by and used in conjunction.

  3. Lindemann theory  is used in the Nadal and LeSpac (NP) shear modulus model that determines the temperature dependence and the SCG model for pressure dependence of the shear modulus.

Modulus of Rigidity

Material property with a value equal to the shear stress divided by the shear strain is defined for The rigidity modulus, also known as shear modulus.

Shear modulus general formula is written as shown below:

    τ = γ*G

where the shear stress is τ in a given member, which has the unit of force divided by area (N/m 2 or lbf/ft2); γ which does not have a unit  is the shear strain (strain is the change of length divided by the original length); and the shear modulus or the modulus of rigidity is G, which has the unit of force divided by the area. Shear modulus is another form of the generalized Hook's Law similar to the modulus of elasticity.

By performing a tensile stress test the modulus of rigidity can be determined, where stress vs strain is plotted. The modulus of rigidity is equal to the slope of the line. Since shear force over area is equal to the shear stress, and strain is equal to the change in length divided by initial length, we get the equation.

FAQ (Frequently Asked Questions)

Q1. Explain how Modulus of Rigidity is Calculated?

Ans: Shear modulus or modulus of rigidity is the rate of change of unit shear stress with respect to unit shear strain for the pure shield condition within the proportional limit. 

The modulus of rigidity formula is G=E/(2(1+v)), and modulus of rigidity is denoted by G, elastic modulus is denoted by E and poisson’s ratio is v in the formula.

Q2. Explain what is Modulus of Rigidity.

Ans: Shear modulus in the material science or we say the modulus of rigidity is denoted by G or sometimes by S it is defined as the ratio of the shear stress , where the force is the shear stress which acts in an area is = shear strain. 

Q3. Explain the Axial Rigidity.

Ans: The axial rigidity can be obtained as the EA of the uncracked session+ reinforcement in the cracked session, in the uncracked region as the reinforcement does not actually contribute to axial stiffness.

Q4. Explain Rigidity.

Ans: It is defined as the property exhibited by the solid to change its shape. That is why we apply external force to the solid material there won't be any changes in the shape. This shows how closely particles are packed and the attraction between them.