In this lecture, we'll cover the following:

• Revisiting the problem of shear locking in the Reissner–Mindlin element.
• The Kirchhoff constraint and its implication for transverse shear strains as thickness tends to zero.
• Expressing transverse shear strain in terms of nodal degrees of freedom and showing it forms a polynomial.
• Why directly enforcing the Kirchhoff constraint is too restrictive and leads to shear locking.
• Developing a strategy using modified sampling (integration) points to eliminate problematic terms.

In this lecture, we work through the root cause of shear locking in the Reissner–Mindlin formulation by examining what happens when we attempt to enforce the Kirchhoff constraint. We show that the transverse shear strain can be expressed as a polynomial in natural coordinates, with coefficients tied to the nodal degrees of freedom. By analysing these coefficients, we see that forcing the shear strains to vanish (as required by the Kirchhoff constraint for thin elements) imposes overly restrictive conditions, ultimately driving both rotations and displacements to zero. This explains why shear locking manifests as an artificially stiff response.

We then develop a strategy to overcome this issue without introducing new problems. Rather than enforcing the constraint directly, we instead modify the numerical integration scheme by carefully selecting sampling locations. This allows certain higher-order terms in the shear strain polynomial to vanish naturally, avoiding the restrictive constraints that cause locking. As a result, we can retain a $2 \times 2$ integration scheme (eliminating zero-energy modes) while also preventing shear locking. This leads to the idea of a substitute transverse shear strain–displacement matrix, which we will derive and implement in the next section.

Next up

The next section takes this plan forward, focusing on deriving and implementing the corrections needed to eliminate zero-energy displacements.