In this lecture, we'll cover:

• Extending the stress–strain relationships from two-dimensional to three-dimensional elements
• Incorporating the third normal stress component and additional shear components
• Formulating the full set of 3D elastic constitutive equations
• Expressing stress–strain relationships in matrix (vector–matrix) notation
• Introducing the material stiffness matrix $[\mathbf{C}]$ as the inverse of $[\mathbf{D}]$

In this lecture, we extend our existing understanding of stress–strain relationships from two-dimensional elements to fully three-dimensional elements. We begin by revisiting the 2D case and then systematically introduce the third dimension. By recognising that normal strain in any direction is influenced not only by the corresponding normal stress but also by stresses in the other two directions through Poisson’s ratio, we expand the constitutive equations to include the z-direction. We then complete the formulation by adding the remaining shear strain–stress relationships, giving us a full set of six equations that describe the elastic behaviour of a 3D element.

Once the full 3D stress–strain relationships are established, we reformulate them in compact matrix notation. We express the strain vector as the product of a material matrix $[\mathbf{D}]$ and the stress vector, clarifying the use of vector (curly bracket) and matrix (square bracket) notation. We then introduce the inverse relationship, defining the matrix $[\mathbf{C}]$ as the inverse of $[\mathbf{D}]$, allowing us to write stress as $[\mathbf{C}]$ times strain. We interpret this $[\mathbf{C}]$ matrix as the three-dimensional analogue of Young’s modulus in uniaxial loading, emphasising that it will play a central role in later developments, particularly in the derivation of stiffness matrices within the finite element method.

Next up:

In the next lecture, we simplify the full 3D formulation by introducing the plane stress condition — a common assumption for thin structures loaded in their own plane.