Validation: NASA 5.3.1

The Panelyze BEM implementation is validated against Example 5.3.1 from NASA CR-125596 (also referenced in subsequent BEM documentation).

Problem Description

A circular hole of radius \(R=5.0\) is located at the center of a large orthotropic plate. The plate is subjected to a remote uniaxial stress \(\sigma_{\infty} = 100\) in the x-direction.

Material Properties (Orthotropic)

The material properties used for verification are:

  • \(E_1 = 10,000\)

  • \(E_2 = 5,000\)

  • \(\nu_{12} = 0.3\)

  • \(G_{12} = 3,000\)

Theoretical Solution

For an infinite plate, the theoretical stress concentration factor (SCF) at the pole of the hole (\(\theta = 90^\circ\) or \(270^\circ\)) is given by:

\[K = 1 + \sqrt{2 \left( \sqrt{\frac{E_1}{E_2}} - \nu_{12} \right) + \frac{E_1}{G_{12}}}\]

For the given properties:

\[K = 1 + \sqrt{2 \left( \sqrt{\frac{10000}{5000}} - 0.3 \right) + \frac{10000}{3000}} \approx 3.3583\]

BEM Results

The validation script nasa_531.py discretizes the hole into 400 linear elements and uses a 200x200 panel (width = \(40R\)) to approximate an infinite plate.

At a distance \(r = 1.01R\) from the center, the calculated stresses are:

Comparison of Stresses at \(r = 1.01R\)

Angle (\(\theta\))

Theoretical \(\sigma_{xx}\)

BEM \(\sigma_{xx}\)

Error

\(90^\circ\)

335.83

329.14

2.0%

\(45^\circ\)

84.14

\(0^\circ\)

0.0

-6.70

Discussion

The 2% error is attributable to two primary factors:

  1. Domain Size: The BEM models a finite panel (\(W=40R\)), whereas the theoretical solution is for an infinite sheet.

  2. Evaluation Point: The stress is evaluated at \(r=1.01R\) to ensure numerical stability when using interior point kernels.

Despite these approximations, the BEM solver correctly captures the high-gradient stress concentration and the directionality of the orthotropic response.