Differentiate between divergent and gradient terminologies used in mechanics and derive the differential form of the governing stress equation for a loaded body under equilibrium.
Understand the Problem
The question is asking for a distinction between the terms 'divergent' and 'gradient' as they apply to mechanics, and also requests the derivation of the differential form of the governing stress equation for a loaded body that is in equilibrium. This involves explaining these concepts and applying mechanics principles to derive a relevant equation.
Answer
Gradient is a vector field, divergence is a scalar field. The differential form of the governing stress equation is ∇·σ + f = 0.
The gradient is a vector indicating the direction of the greatest rate of increase of a scalar field. Divergence is a scalar field representing the rate of change of density at a point in a vector field. The differential form of the governing stress equation in equilibrium is ∇·σ + f = 0.
Answer for screen readers
The gradient is a vector indicating the direction of the greatest rate of increase of a scalar field. Divergence is a scalar field representing the rate of change of density at a point in a vector field. The differential form of the governing stress equation in equilibrium is ∇·σ + f = 0.
More Information
The divergence of the stress tensor abla·σ represents the internal forces acting within the body. In mechanical equilibrium, they balance the external body forces f, leading to the equilibrium equation.
Tips
A common mistake is confusing vector and scalar fields when calculating gradient and divergence. Ensure proper usage to avoid errors.
Sources
- What is the difference between the divergence and gradient? - Quora - quora.com
- What is the difference between divergence and gradient - Reddit - reddit.com
- 1 Differential Equations for Solid Mechanics - pkel015.connect.amazon.auckland.ac.nz
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