In this lecture, we'll cover:

• A brief review of uniaxial stress and strain and the fundamental elastic relationships
• The extension from uniaxial to two-dimensional (biaxial) normal stress
• The definition and interpretation of stress components on an infinitesimal 2D element
• The role of Young’s modulus and Poisson’s ratio in relating 2D stress and strain
• Shear stress and shear strain in two dimensions, including the shear modulus

In this lecture, we begin by revisiting the basic concepts of uniaxial stress and strain using the familiar example of a bar in axial tension. We restate the fundamental relationships: stress as force over area, strain as change in length over original length, and Hooke’s law linking them through Young’s modulus. This provides the foundation for extending the discussion to two-dimensional stress states.

We then move to a 2D infinitesimal element subjected to biaxial normal stresses acting along orthogonal axes. We carefully define normal stresses as those acting perpendicular to a cut face and introduce the notation for stress components. We establish the strain–stress relationships in two dimensions, showing how strain in one direction depends not only on the corresponding normal stress but also on the perpendicular stress through Poisson’s ratio. This highlights the coupling between orthogonal directions that does not appear in the uniaxial case.

Finally, we introduce shear stress and shear strain for a 2D element, defining shear stresses as acting parallel to the cut face and explaining the associated deformation in terms of angular distortion. We present the relationship between shear stress and shear strain via the shear modulus, and show how the shear modulus is related to Young’s modulus and Poisson’s ratio. Together, these relationships form the core set of equations needed to describe stress–strain behaviour in two dimensions.

Next up:

In the next lecture, we establish the relationship between strain and displacement, completing the set of equations needed to describe how deformation relates to the forces acting on an element.