# Relation between Poisson’s ratio, Modulus of Elasticity and Modulus of Rigidity

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Mechanical Engineering
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JoVE Central Mechanical Engineering
Relation between Poisson’s ratio, Modulus of Elasticity and Modulus of Rigidity

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A slender bar subjected to an axial load undergoes deformation in the axial and transverse directions. The deformation affects the cubic elements within it.

Depending on its orientation, the cube is transformed into a rectangular parallelepiped or a rhombus, resulting in shearing strain.

The axial loading on the element results in a combination of shearing and normal strains.

Applying an axial load triggers normal and shearing stresses on elements that are oriented at an angle of 45 degrees to the load axis.

The cubic element, when intersected with a diagonal plane, forms a prismatic element. The prismatic element modifies its internal angles and sides in a manner proportional to the strains generated by the load.

By applying the formula for the tangent of the difference of two angles, the relationship between the maximum shearing strain and the axial strain is determined.

From Hooke's law, the relation between the constants is obtained. One constant can be determined from the other two.

## Relation between Poisson’s ratio, Modulus of Elasticity and Modulus of Rigidity

Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis. Significantly, the shearing strain achieves its maximum level at this particular orientation.

When a diagonal plane intersects the cubic element, a prismatic element forms. This prismatic element alters its internal angles and sides in proportion to the load-induced strains. This deformation process illuminates the effects of stress on the structural integrity of the element. The relationship between maximum shearing strain and axial strain can be derived by applying the expression for the tangent of the difference between two angles.

This mathematical relationship offers insights into the interaction between these two strain types under axial load conditions. Hooke's law correlates among the constants, a pivotal principle in studying how materials behave under load. With this correlation, one constant can be determined from the other two, enriching our understanding of a material's response to stress and strain.