plate-hole.md

Stress concentration around a hole

A plane strip (plane stress) of width 5 m, length 8 m with a centred hole of radius 0.5m is subjected to uniform tension of $\sigma_0$ = 100 Pa applied at the ends of the strip. Due to symmetry, a quarter of this plate is modelled.

Linear elastic solution for stress concentration

The elastic stress concentration around a hole in an infinite plate under uniaxial tension was first described by Ernst Gustav Kirsch [1]. The Kirch equations are

$\sigma_{rr}=\frac{\sigma_0}{2}(1-(\frac{a}{r})^2)+\frac{\sigma_0}{2}(1-4(\frac{a}{r})^2+3(\frac{a}{r})^4)cos2\theta$

$\sigma_{\theta\theta}=\frac{\sigma_0}{2}(1+(\frac{a}{r})^2)-\frac{\sigma_0}{2}(1+3(\frac{a}{r})^4)cos2\theta$

$\tau_{r\theta}=-\frac{\sigma_0}{2}(1+2(\frac{a}{r})^2-3(\frac{a}{r})^4)sin2\theta$

where a is hole radius, r and $\theta$ are radial coordinates and $\sigma_0$ is remote stress. The stress concentration factor, $K_t$ which is defined as the ratio of the maximum stress, $\sigma_{max}$ at hole to the remote stress, $\sigma_0$ for the infinite width plate is 3.

Howland's semi-analytical solution

For a finite width plate (see Fig. 2), the elastic stress state depends on the plate width (W) and the hole radius (R). The elastic stress state of a finite width plate with a circular hole is found by empirical relationships.

We use the relationship proposed by Howland [2] to compute the elastic stress at the circular hole as shown in Fig. 1. The stress ratios for a range of geometries (hole radius, R and plate finite width, 2W) are calculated in the table below.

Table 1: Howland's semi-analytical solution

R/W	Point A $\sigma_{xx}/\sigma_0$	Point B $\sigma_{yy}/\sigma_0$
0	3.00	-1.00
0.1	3.03	-1.03
0.2	3.14	-1.11
0.3	3.36	-1.26
0.4	3.74	-1.44
0.5	4.32	-1.58

MPM configuration

Analysis

Description	value
Type	Explicit USF
Velocity update	true
Total analysis time	30 s
dt	1.0E-4
Gravity	false

Mesh

Cell dimensions	value
x-length	0.0625 $m$
y-length	0.0625 $m$

Particles

Particle spacings	value
x-spacing	0.015625 $m$
y-spacing	0.015625 $m$
# material points /cell	16

Material

Description	value
Material	Linear Elastic
Young's modulus ($E$)	1.0E+6 $N/m^2$
Poisson ratio ($\nu$)	0.0
Density ($kg/m^3$)	2000.0

Results

MPM Explicit USF approach with velocity update is performed.

Stress concentration factors	Howland's	MPM USF
Point A ($\sigma_{xx}/\sigma_0$)	3.14	3.13
Point B ($\sigma_{xx}/\sigma_0$)	0.00	-0.029
Point B ($\sigma_{yy}/\sigma_0$)	-1.11	-1.08

The stresses obtained from the Explicit Update Stress First simulation is shown below.

Stress-XX

Stress-YY

References

[1] Kirsch, 1989, Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre. Zeitschrift des Vereines deutscher Ingenieure, 42, 797-807.

[2] Howland, R.C.J., 1930, On the Stresses in the Neighborhood of a Circular Hole in a Strip under Tension, Trans. Roy. Soc London, Series A, vol.229, 49-59.