In this lecture, we'll cover:

• How to define the constitutive matrix that links stresses and strains,
• The assumptions of homogeneous and isotropic material behaviour,
• How to split the stress vector into in-plane stresses and transverse shear stresses,
• The in-plane and transverse shear constitutive matrices, $D'_p$ and $D'_s$,
• How these stress components are represented on an infinitesimal element,
• How bending and membrane actions combine to form the in-plane stress resultants.

We begin by defining the constitutive matrix as the matrix analogue of Young’s modulus: it is the quantity that relates stress to strain when we are working with vector and matrix quantities rather than a single scalar relationship. We assume a homogeneous, isotropic material, and we organise the local stress vector into three in-plane stresses and two transverse shear stresses. These are then linked to the corresponding strain vector through a block constitutive matrix, with a 3×3 in-plane matrix $D'_p$ and a $2\times 2$ transverse shear matrix $D'_s$, while the off-diagonal blocks are zero. The in-plane matrix is the familiar plane stress form involving $E$ and Poisson’s ratio, $\nu$, and the shear matrix is simply based on the shear modulus $G$.

We also spend time visualising what these stresses mean on an infinitesimal element. For the in-plane components, we see how bending stresses vary linearly through the depth, while membrane stresses remain constant; the combined in-plane stresses are the sum of both effects. For the shear component, we note that the transverse shear stress is taken as constant through the thickness in the Reissner–Mindlin idealisation, even though the actual distribution is parabolic. This gives us a clear physical picture of the stress fields

Next up

In the next lecture, we will integrate these stress components through the depth to obtain stress resultants.