Physical Intuition: Moving Beyond Constant Strain

In our journey through two-dimensional (2D) finite element formulations, the Constant Strain Triangle (CST) served as the gateway to plane elasticity. While the CST—a three-node, linear-displacement triangular element—is conceptually simple and computationally cheap, it exhibits severe physical and mathematical limitations. Specifically, because the displacement field within a CST is linear, the resulting strain and stress fields are spatially constant. In any real-world structural component, stress is rarely uniform; it concentrates, peaks, and decays in response to geometric features (such as holes, fillets, notches, or crack tips) and localized boundary conditions.

To approximate these rapid changes—known as stress concentration gradients—using CST elements, an analyst must refine the mesh to an extreme degree. Even then, the constant-stress approximation results in a jagged, discontinuous representation of stress across element boundaries, which converges slowly and demands massive computational resources. Furthermore, CST elements suffer from shear locking (or parasitic shear) when subjected to bending, artificially stiffening the structure because a linear displacement field cannot represent the curved, parabolic shear strain profiles of flexural deformation.

To resolve these shortcomings, we introduce the Linear Strain Triangle (LST) element. The LST is a quadratic, higher-order triangular element. By adding three midside nodes along the edges of the triangle, we increase the total number of nodes from three to six. This structural upgrade allows us to interpolate displacements quadratically rather than linearly.

Mathematically, the quadratic displacement field produces a linear strain field (and consequently, a linear stress field) within the element. This allows a single LST element to represent linear variations in strain and stress, making it exceptionally proficient at capturing steep stress gradients and bending behaviors with a fraction of the mesh density required by CST elements.

Core Concepts

1. Higher-Order Elements and Pascal's Triangle

The choice of interpolation polynomial determines the kinematic capability of a finite element. For two-dimensional elements, we can visualize the polynomial terms using Pascal's Triangle:

$$\begin{array}{ccccccccc} & & & & 1 & & & & \\ & & & x & & y & & & \\ & & x^2 & & xy & & y^2 & & \\ & x^3 & & x^2y & & xy^2 & & y^3 & \\ x^4 & & x^3y & & x^2y^2 & & xy^3 & & y^4 \end{array}$$

The completeness of the quadratic polynomial is crucial. Because it contains all terms up to the second degree ($x^2, xy, y^2$), the element possesses geometric isotropy (or spatial invariance). This means the element shows no bias toward any coordinate direction, and its physical behavior is independent of the coordinate system's orientation.

If we were to drop any of the quadratic terms (e.g., using a 4-node or 5-node triangle), the element would lose coordinate invariance, violating basic physical principles.

2. The Role of Midside Nodes

Midside nodes are positioned at the midpoints of the element boundaries. They serve two vital functions:

3. Continuous vs. Discontinuous Stress Fields ($C^0$ vs. $C^{-1}$)

In displacement-based FEM, we enforce compatibility (continuity of displacements) across element boundaries. For both CST and LST elements, the displacement field is continuous across shared edges, which represents $C^0$ continuity.

However, because strain and stress are obtained by taking derivatives of the displacement field:

4. Area Coordinates (Natural Coordinates)

Integrating functions over arbitrary triangular shapes in Cartesian coordinates $(x,y)$ is mathematically tedious because the limits of integration are coupled. To simplify the mathematical formulations and integrations, we use Area Coordinates (also called natural or triangular coordinates), denoted by $L_1, L_2,$ and $L_3$.

Consider a triangle with vertices 1, 2, and 3. Any point $P$ inside the triangle divides it into three sub-triangles: $\Delta_1$ (opposite Node 1, formed by $P$, 2, and 3), $\Delta_2$ (opposite Node 2, formed by $P$, 3, and 1), and $\Delta_3$ (opposite Node 3, formed by $P$, 1, and 2).

We define the area coordinates as the ratio of these sub-areas to the total area $A$:

Area coordinates possess several unique properties:

This relationship implies that only two of the three area coordinates are independent. If we know $L_1$ and $L_2$, then $L_3 = 1 - L_1 - L_2$.

Similarly:

5. Comparison: CST vs. LST

To appreciate the advancement of the LST formulation, let us compare the two elements: