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uniaxial-stress.md

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uniaxial-stress.md

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Uniaxial stress

In an uniaxial stress state, only the axial stress is non-zero while all other stress components are zero.

Under an axial loading condition, the strains for an isotropic linear elastic material are:

$$ \epsilon_{axial} = \frac{1}{E}(\sigma_{axial} - 2 \nu \sigma_{lateral})$$

$$ \epsilon_{lateral} = \frac{1}{E}\left[(1 - \nu)(\sigma_{lateral} - \nu \sigma_{axial} \right]$$

For an uniaxial stress ($\sigma_{lateral} = 0 $), the above equation becomes:

$$ \sigma_{axial} = E \epsilon_{axial} $$

$$ \epsilon_{lateral} = - \nu \epsilon_{axial} $$

Analytical solution

Fo an axial loading condition, the axial strain and stress at a given time $t$ are:

$$ \epsilon_{yy} = u \times \frac{(t - t_0)}{l} $$

$$ \sigma_{yy} = E \epsilon_{yy} = E \times u \times \frac{(t - t_0)}{l} $$

and the lateral strains are:

$$ \epsilon_{xx} = \epsilon_{zz} = -\nu \times \epsilon_{yy} $$

where,

$u$ is the applied velocity at both ends, $l$ is the length of the block, and $E$ is the young's modulus of the system.

uniaxial stress

MPM configuration

Mesh

Cell dimensions value
x-length 1.0 $m$
y-length 1.0 $m$
z-length 1.0 $m$

Particles

Particle spacings value
x-spacing 0.5 $m$
y-spacing 0.5 $m$
z-spacing 0.5 $m$

Analysis

Description value
Total analysis time 0.1 s
Gravity false

Material

Description value
Material Linear Elastic
Young's modulus ($E$) 1000 $N/m^2$
Poisson ratio ($\nu$) 0.2

Analysis

Analysis are carried out using MPM Explicit USF and USL algorithms.

Cases

Description Case I Case II Case III
Density $\rho$($kg/m^3$) 1.0 1.0 0.01
dt (s) 0.01 0.001 0.001
nsteps 10 100 1000

Results

USF Results at 0.1s

Parameter Analytical Case I Case II Case III
$\epsilon_{yy}$ -0.001 -0.001 -0.001 -0.001
$\epsilon_{xx}$ or $\epsilon_{zz}$ 0.0002 0.000231 0.00023 0.000201
$\sigma_{yy} (N/m^2)$ -1.00 -0.982638 -0.983478 -0.999406
$\sigma_{xx}$ or $\sigma_{zz} (N/m^2)$ 0.00 0.043406 0.041306 0.001485

USL Results at 0.1s

Parameter Analytical Case I Case II Case III
$\epsilon_{yy}$ -0.001 -0.001 -0.001 -0.001
$\epsilon_{xx}$ or $\epsilon_{zz}$ 0.0002 0.000201 0.000224 0.0002
$\sigma_{yy} (N/m^2)$ -1.00 -0.999476 -0.986596 -1.0
$\sigma_{xx}$ or $\sigma_{zz} (N/m^2)$ 0.00 0.0131 0.033509 1.595325e-07