In this lecture, we'll cover the following:

• Introducing the constitutive matrix as the link between stress and strain.
• Assumptions of homogeneous, isotropic material behaviour and no in-plane axial strain.
• Extending Young’s modulus from a scalar to a matrix form for 3D stress–strain relationships.
• Separating bending and transverse shear components of stress and strain.
• Formulating constitutive matrices for plane stress and transverse shear behaviour.
• Expressing stress–strain relations in terms of generalised strains and variation through thickness.

In this lecture, we establish the formal relationship between stress and strain by introducing the constitutive matrix, which generalises Young’s modulus to multi-dimensional problems. We begin by clarifying key assumptions: the material is homogeneous and isotropic, and membrane (in-plane axial) behaviour is neglected. These assumptions simplify the formulation and allow us to focus on how material properties govern the stress–strain relationship.

We then expand from the familiar scalar form of Hooke’s law to a vector–matrix formulation, where a vector of stresses is related to a vector of strains via the constitutive matrix. We separate this relationship into bending and transverse shear components, defining distinct constitutive matrices for each. The bending matrix corresponds to the standard plane stress formulation, while the shear matrix is expressed using the shear modulus. Finally, we reinterpret these relationships in terms of generalised strains, highlighting the linear variation of bending stresses through the thickness and the constant nature of transverse shear, setting up a foundation for later derivations involving element stiffness and stress resultants.

Next up

In the next lecture, we will see how these stress–strain relationships are integrated through the plate thickness to produce stress resultants — the moments and shear forces used directly in design.