18.17
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 withi…
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.
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Q1: What happens to a cubic element when an axial load is applied to a slender bar?
When an axial load is applied, the cubic element undergoes deformation in both axial and transverse directions. Depending on its orientation, the cube transforms into either a rectangular parallelepiped or a rhombus, inducing shearing strain. This combined deformation results in both normal and shearing strains within the element.
Q2: Why do maximum shearing stresses occur at 45 degrees to the load axis?
At a 45-degree angle to the load axis, elements experience equal normal and shearing stresses simultaneously. This orientation produces the maximum shearing strain, as the element's geometry is optimally positioned to develop both stress components. The diagonal plane intersection creates a prismatic element that exhibits this peak shearing effect.
Q3: How is the relationship between maximum shearing strain and axial strain derived?
The relationship is determined by applying the mathematical formula for the tangent of the difference between two angles to the prismatic element's deformation. The element's internal angles and sides change proportionally to the load-induced strains, allowing engineers to express maximum shearing strain as a function of axial strain.
Q4: What role does Hooke's law play in relating elastic constants?
Hooke's law establishes correlations among the elastic constants: Poisson's ratio, modulus of elasticity, and modulus of rigidity. These relationships allow engineers to determine one constant from the other two, providing a comprehensive understanding of how materials respond to combined normal and shearing stresses under axial loading.
Q5: How does a diagonal plane intersecting a cubic element affect its geometry?
When a diagonal plane intersects the cubic element, it forms a prismatic element. This prismatic element modifies its internal angles and sides in proportion to the strains generated by the axial load. These geometric changes directly reflect the combined effects of normal and shearing deformations occurring within the material.
Q6: What is the relationship between transverse and axial deformation in an axially loaded bar?
Transverse deformation is related to axial deformation through Poisson's ratio, one of the elastic constants. When an axial load causes axial strain, it simultaneously produces transverse strain in the perpendicular directions. This coupled behavior is fundamental to understanding the complete deformation pattern and stress distribution in the material.
Q7: Why is understanding elastic constant relationships important for material analysis?
Understanding elastic constant relationships enables engineers to predict material behavior under complex loading conditions. Since one constant can be determined from the other two using Hooke's law, engineers need only measure or know two constants to fully characterize a material's elastic response to both normal and shearing stresses.