In this lecture, we'll cover:

• A recap of the role of the material $[\mathbf{C}]$ matrix in finite element analysis
• The different stress–strain models derived: uniaxial, full 3D, plane stress and plane strain
• How these models relate to element behaviour in the direct stiffness method
• The key differences between plane stress and plane strain conditions
• How this section prepares us for deriving the finite element equations

In this lecture, we wrap up our study of elastic material behaviour by consolidating the different stress–strain relationships we have derived. We revisit the idea that the $[\mathbf{C}]$ matrix represents the material behaviour, linking stress and strain for different modelling assumptions. We summarise the progression from a simple axially loaded bar, governed by Young’s modulus alone, to the full three-dimensional triaxial stress state, and then show how this general 3D formulation can be reduced to plane stress and plane strain conditions for two-dimensional elements. In each case, we identify the appropriate constant or matrix that characterises the material response.

We emphasise that these material models are essential for the finite element method, particularly within the direct stiffness approach, because they define how individual elements respond to loading. A key conceptual clarification is that plane stress and plane strain are distinct conditions arising from different structural assumptions and do not occur simultaneously. An element in plane stress experiences strain in the out-of-plane direction, whereas an element in plane strain experiences stress in that direction. With this foundation in elastic material behaviour established, we are now prepared to move on to formulating the finite element equations using the principle of minimum potential energy.

Next up:

In the next lecture, we begin Section 3, where we shift focus from material behaviour to deriving the finite element equations and the stiffness matrix for an axially loaded bar.