Direct Evaluation of Euler–Almansi Strains for the 8-Node Isoparametric Quadrilateral Finite Element

01

Q8 extension

This section extends the direct evaluation of Euler–Almansi strains presented in Section 15.3 to the Q8 element. The element has eight nodes, 16 element displacement DOFs and quadratic interpolation. It uses $3\times3$ Gauss quadrature.

The remaining Updated Lagrangian formulation, including $\htmlClass{fea-math-bold-italic}{\boldsymbol B}_0$, $\htmlClass{fea-math-bold-italic}{\boldsymbol B}_L$, the geometrical stiffness terms and the Newton–Raphson procedure, is unchanged from the formulation already presented in Section 15.3.

02

Direct Euler–Almansi strain

With $\htmlClass{fea-math-bold-italic}{\boldsymbol J}_0$ and $\htmlClass{fea-math-bold-italic}{\boldsymbol J}_t$ evaluated from the initial and current trial nodal coordinates, respectively, the total deformation gradient is:

The left Cauchy–Green tensor and the Euler–Almansi strain tensor are:

The engineering components are:

The Q8 element uses nine Gauss points, formed from the following coordinates in each natural-coordinate direction:

with the corresponding one-dimensional weights:

03

Cantilever example

The same cantilever example as in the other sections of Chapter 15 is analysed using five load steps. The meshes below are given as length × height; displacements are in millimetres.

The Q8 element converges much faster than Q4 for this example. The coarse Q8 mesh $20\times3$, containing only 60 elements, already gives results very close to the finest Q8 result. In contrast, the Q4 mesh $180\times24$ contains 4320 elements and still gives slightly smaller displacement values.

04

MATLAB program

A MATLAB program can be downloaded below. Run main.m.