Papkovich-Neuber and Galerkin Vector Solutions for 3D Elasticity Formulations A Papkovich-Neuber (PN) vector solution represents the elastic deformation...
Papkovich-Neuber and Galerkin Vector Solutions for 3D Elasticity Formulations
A Papkovich-Neuber (PN) vector solution represents the elastic deformation of a material point or surface in a 3D elastic analysis. It is an extension of the 2D Green-Lagrange (GL) solution and captures both the normal and shear elastic response of the material.
The PN vector can be derived from the Galerkin (GL) equations, which are derived from the principle of minimum energy. The GL equations involve finding the equilibrium configuration of the material that minimizes the total energy of the system.
In other words, the PN vector represents the direction and magnitude of the velocity gradient at a given material point. This information is then used to calculate the stress and strain tensors at that point.
Example:
Consider a thin plate subjected to a uniaxial tensile force. The PN solution would provide the normal and shear stress vectors at each point on the plate, allowing for the determination of the material's deformation