hydrostatic-column.md

Hydrostatic column

Hydrostatic analytical solution

The hydrostatic validation involves applying gravity loading to a column of material restrained in both lateral components and along the bottom plane. The Newtonian fluid model will yield hydrostatic pressures for all normal stress directions ($\sigma_{xx}$, $\sigma_{yy}$, $\sigma_{zz}$). The linear elastic material will yield geo-static stresses in the vertical direction and the horizontal stresses depend on the Poisson's ratio.

Hydrostatic column with a width w of 0.2 m and height h of 0.1 m. Restrained in both the lateral directions and along the bottom plane

MPM configuration

Mesh

|Cell dimensions | Case 1 | Case 2 | Case 3 | |-------------------|--------- -|------------|-------------| |x-length | 0.01 $m$ | 0.005 $m$ | 0.0025 $m$ | |y-length | 0.01 $m$ | 0.005 $m$ | 0.0025 $m$ |

Particles

Particle spacings	Case 1	Case 2	Case 3
x-spacing	0.002 $m$	0.001 $m$	0.0005 $m$
y-spacing	0.002 $m$	0.001 $m$	0.0005 $m$
Particles per cell	25	25	25

Analysis

Description	Case 1	Case 2	Case 3
Total analysis time	0.2 s	0.2 s	0.2 s
dt	0.00001 s	0.000005 s	0.0000025 s
Gravity	-9.81 $m/s^2$	-9.81 $m/s^2$	-9.81 $m/s^2$

Material

Description	value
Material	Newtonian
Density ($\rho$)	1800.0 $kg/m^3$
Bulk modulus ($K$)	1000000 $N/m^2$
Viscosity ($\mu$)	0.0

Description	value
Material	Linear Elastic
Density ($\rho$)	1800.0 $kg/m^3$
Young's modulus ($E$)	1000000 $N/m^2$
Poisson's ratio ($\nu$)	0.0

Hydrostatic analysis

Analysis are carried out using MPM Explicit USF and USL algorithms using velocity update. Note that the results converge with more refined mesh with smaller errors.

Results (Newtonian Fluid)

USF results at t = 0.2s

Parameter	Analytical	Case 1	Case 2	Case 3
$\sigma_{yy} (N/m^2)$	-1765.800	-1680.244	-1724.583	-1746.791
$\sigma_{xx} (N/m^2)$	-1765.800	-1680.244	-1724.583	-1746.791

USL results at t = 0.2s

Parameter	Analytical	Case 1	Case 2	Case 3
$\sigma_{yy} (N/m^2)$	-1765.800	-1680.182	-1724.550	-1746.774
$\sigma_{xx} (N/m^2)$	-1765.800	-1680.182	-1724.550	-1746.774

Results (Linear Elastic)

USF results at t = 0.2s

Parameter	Analytical	Case 1	Case 2	Case 3
$\sigma_{yy} (N/m^2)$	-1765.800	-1680.244	-1724.583	-1746.791
$\sigma_{xx} (N/m^2)$	0.000	0.000	0.000	0.000

USL results at t = 0.2s

Parameter	Analytical	Case 1	Case 2	Case 3
$\sigma_{yy} (N/m^2)$	-1765.800	-1680.182	-1724.550	-1746.774
$\sigma_{xx} (N/m^2)$	0.000	0.000	0.000	0.000

Plot of vertical stresses with time showing convergence of stress osciallations

Vertical stresses component results at t = 0.2s for USF linear elastic model