In this lecture, we'll cover the following:

• Revisit the assumed transverse shear strain field and its simplified polynomial form that avoids shear locking.
• Transform transverse shear strains from Cartesian to natural coordinates using the Jacobian.
• Sample shear strains at midpoints of element edges to determine the unknown field coefficients.
• Derive a relationship between sampled shear strains and the polynomial coefficients via matrix transformations.

In this lecture, we begin the derivation of a substitute transverse shear strain matrix by first restating the assumed shear strain field in a simplified polynomial form that avoids shear locking. We express these strains in the natural coordinate system and use the Jacobian to relate them back to Cartesian coordinates. We then rewrite the shear strain field in a compact matrix form, where it is defined by a set of unknown coefficients that we aim to determine.

To find these coefficients, we introduce the concepts of natural directions along element edges, their associated angles, and the corresponding shear strain components. By sampling shear strains at the midpoints of each element edge, where higher-order locking terms vanish, we construct a system of equations that links these sampled values to the unknown coefficients. Combining these relationships, we derive a matrix expression that allows us to compute the coefficients directly from the sampled shear strains, establishing a key step towards building the substitute shear strain matrix.

Next up

In the next lecture, we will continue the derivation by expressing the interpolated shear strains in terms of nodal degrees of freedom and assembling the full substitute shear strain matrix.