To calculate shear rigidity, also known as the shear modulus or modulus of rigidity (often denoted by G or μ), you use the relationship between shear stress and shear strain. Shear rigidity is defined as the ratio of shear stress to shear strain in an elastic material.
Understanding the Formula
The fundamental formula for shear rigidity (G) is:
G = Shear Stress / Shear Strain
Using the definitions provided in the references:
- Shear Stress (τxy) is the force acting parallel to an area divided by that area. According to reference 1, τxy = FA. [Note: Based on standard physics definitions and the context of Force (F) and Area (A) as defined in references 2 and 3, this formula is understood as τxy = F/A].
- Shear Strain (γxy) is the measure of the deformation that occurs perpendicular to the applied force. According to reference 4, γxy = Δxl. [Note: Based on standard physics definitions and the context of Transverse displacement (Δx) and Initial length (l) as defined in references 5 and 6, this formula is understood as γxy = Δx/l].
Therefore, substituting these definitions into the shear rigidity formula:
G = (τxy) / (γxy)
G = (F / A) / (Δx / l)
This can be rewritten as:
*G = (F / A) (l / Δx)**
Breaking Down the Components
Based on the provided references (and standard physics definitions), here's what each term in the calculation represents:
- τxy (Shear Stress): A measure of the stress component acting parallel to a cross-section. It is calculated using the applied force and the area it acts upon.
- Reference 1: τxy=FA is the shear stress.
- F (Force): The external force applied parallel to the surface of the object.
- Reference 2: F represents the force acting on the object.
- A (Area): The surface area of the object on which the force is acting parallel.
- Reference 3: A is the area on which the force is acting.
- γxy (Shear Strain): A measure of the deformation caused by shear stress, represented by the ratio of transverse displacement to the initial length.
- Reference 4: γxy=Δxl is the shear strain.
- Δx (Transverse Displacement): The distance a point on the object moves perpendicular to the direction of the initial length due to the applied shear force.
- Reference 5: Δx is the transverse displacement.
- l (Initial Length): The original dimension of the object perpendicular to the direction of the applied force and the transverse displacement.
- Reference 6: l is the initial length.
Summary of Terms
| Term | Description | Reference | Standard Formula |
|---|---|---|---|
| τxy | Shear Stress | Ref 1: τxy=FA (understood as F/A) | F/A |
| F | Force acting on the object | Ref 2: F represents the force acting on the object | F |
| A | Area on which the force is acting | Ref 3: A is the area on which the force is acting | A |
| γxy | Shear Strain | Ref 4: γxy=Δxl (understood as Δx/l) | Δx/l |
| Δx | Transverse displacement | Ref 5: Δx is the transverse displacement | Δx |
| l | Initial length | Ref 6: l is the initial length | l |
| G | Shear Rigidity (Shear Modulus) | (Derived from definitions) | τxy / γxy |
In essence, you calculate shear rigidity by determining the shear stress and shear strain experienced by a material under a shearing force and then dividing the stress value by the strain value.
