Given the Airy Stress Function φ = ay2
What are the corresponding stress components and displacement components for the Airy stress function φ = ay2? How can we identify the rigid body terms in the displacement?
Stress Components:
σx = 2a
σy = 0
τxy = 0
Displacement Components:
The displacement components require complex calculations, but any constant or linear term in the stress function indicates rigid body motion. Since these terms are absent in the given Airy stress function, the rigid body terms in the displacement are zero.
Explanation:
In solid mechanics, the Airy stress function, φ, is a potential function that helps derive stress components. For φ = ay2, the stress components can be determined by taking derivatives. The stress components σx, σy, and τxy are calculated as:σx = ∂²φ/∂y²
σy = ∂²φ/∂x²
τxy = -∂²φ/∂x∂y
Therefore, for φ = ay2:
σx = 2a
σy = 0
τxy = 0 To find displacement components, compatibility equations and integration are needed. The presence of constant or linear terms in the stress function indicates rigid body motion. As the given stress function lacks these terms, there are no rigid body terms in the displacement.