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**Shear Modulus** is defined as the ratio of shear stress to the corresponding shear strain within a material’s proportional limit. Also known as modulus of rigidity and rigidity modulus, the shear modulus is denoted by “G” and can be experimentally determined from the slope of shear stress (τ) vs shear strain (γ) curve. The more the value of the shear modulus or modulus of rigidity, the more rigid the material is. The shear modulus values of engineering materials are determined by torsional tests.

## Shear Modulus Units

The SI unit of modulus of rigidity or shear modulus is the pascal (Pa). However, it is popularly expressed in gigapascals (GPa). The unit of shear modulus in the English system is thousand pounds per square inch (KSI). Its dimensional form is M^{1}L^{−1}T^{−2}.

## Shear Modulus Formula

On the application of a shear force on a body, it deforms laterally. So, the shear modulus is basically a measure of the material’s ability to resist transverse deformation. Large shear forces can create a flow in the material. Mathematically, the formula for shear modulus can be expressed as follows:

Shear Modulus, G=Shear Stress (τ)/Shear Strain (γ)

Now Shear stress=Shear Force (F)/ Area (A) and Shear Strain=Change in Length (Δ) in the lateral direction/Original length(L). Hence

### Formula for Shear Modulus, G=FL/AΔ

The relationship of Shear Modulus (G) with Young’s modulus (E) and Poisson’s ratio (μ) is given below:

**E=2G(1+μ)** or G = E/(2(1+**μ**)

## Characteristics of Shear Modulus or Modulus of Rigidity

- Modulus of rigidity is a material property and remains constant for a material at a specific temperature.
- Shear modulus is independent of the geometry of the material.
- With an increase in temperature, the modulus of rigidity decreases.
- A high value of modulus of rigidity means it will maintain its shape and a large force will be required to deform it, and a low value of shear modulus signifies the material to be soft or flexible.
- Fluids (Liquids and Gases) have the minimum value of rigidity modulus (0) and they started flowing with a little application of shear force.
- Diamond has the maximum value of shear modulus (478 GPa).
- Shear Strength is the maximum value of shear stress that a material can withstand without fracture or failure.

## Shear modulus of Steel and other materials

The shear modulus of carbon steel is 77 GPa and the same for stainless steel is 77.2 GPa. The modulus of rigidity of water is considered as zero. The modulus of rigidity values at room temperature for other common materials are tabulated below:

Material | Shear Modulus or Modulus of Rigidity in GPa | Material | Shear Modulus or Modulus of Rigidity in GPa |

Aluminum Alloys | 25-27 | Tin | 18 |

Aluminum | 24-28 | Titanium, Grade 2 | 41 |

Beryllium Copper | 48 | Titanium, Grade 5 | 41 |

Brass | 40 | Titanium, 10% Vanadium | 42 |

Bronze | 44.8 | Tungsten | 161 |

Cadmium | 19 | Wood, Douglas Fir | 13 |

Nickel | 76 | Zinc | 43 |

Cast Iron | 41 | Z-nickel | 76 |

Chromium | 115 | Steel, Cast | 78 |

Concrete | 21 | Steel, Cold-rolled | 75 |

Copper | 45 | Plywood | 0.62 |

Glass | 26.2 | Polycarbonate | 2.3 |

Glass, 96% silica | 19 | Polyethylene | 0.12 |

Inconel | 79 | Rubber | 0.0006 |

Iron, Ductile | 63 – 66 | Structural Steel | 79.3 |

Iron, Malleable | 64 | Monel metal | 66 |

Kevlar | 19 | Nickel Silver | 48 |

Lead | 13.1 | Nickel Steel | 76 |

Magnesium | 16.5 | Nylon | 4.1 |

Molybdenum | 118 | Phosphor Bronze | 41 |

Diamond | 478 | Wood | 4 |

Chalk | 3.2 | Granite | 24 |

**Table 1: Rigidity Modulus values for Common Materials**