2020-04-22-101444.ipynb

Kernel: Python 3 (system-wide)

Constitutive equations

Compressible

$$(\text{Strain energy density})~W(\boldsymbol{E})=C_1(I_1-3-2\ln(J))+D_1(J-1)^2$$

Incompressible

$$(\text{Strain energy density})~W(\boldsymbol{E})=C_1(I_1-3)$$

E= 1 kPa How is this $E$ related to the parameters n the Neo-Hookean model

Spheroid

Young's modulus Spheroid= $2.5~\rm kPa$ )(Rafael suggestion $0.1$–$1.0~\rm kPa$. Yang found from literature $0.8$–$5.0~\rm kPa$ )
Poisson's ratio for Spheroid= $0.499$
Desnity= $1240 \rm~mg/mL$ (changed from the value of 1150 $\rm gm/mL$ suggested by Rafael)

Agarose

Young's modulus Agarose= $300~\rm kPa$
Poisson's ratio for Agarose= $0.49$
Density= $1640 \rm~mg/mL$

Cell Media

Density= $980 \rm~mg/mL$

import numpy as np
x=np.arange(0,1,0.01)
print(x)

[0.   0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1  0.11 0.12 0.13
 0.14 0.15 0.16 0.17 0.18 0.19 0.2  0.21 0.22 0.23 0.24 0.25 0.26 0.27
 0.28 0.29 0.3  0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4  0.41
 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5  0.51 0.52 0.53 0.54 0.55
 0.56 0.57 0.58 0.59 0.6  0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69
 0.7  0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8  0.81 0.82 0.83
 0.84 0.85 0.86 0.87 0.88 0.89 0.9  0.91 0.92 0.93 0.94 0.95 0.96 0.97
 0.98 0.99]

Radius of spheroid = 0.1 mm $d_0=15.92~\rm mm$
$r_0=172.165~\rm mm$
$r_1=178.69~\rm mm$
$V_{\rm aga}=298.925 ~\rm{mm}^3$, $0.298925 ~\rm{mL}$

$\text{Body force}: \rho(r) \omega^2(r+r_0)$
$\text{Pressure}:\rho_{\rm media}\omega^2(r_0r+\frac{r^2}{2})$

d0=15.92
r0= 172.165
r1=178.69
h=(r1-r0)
V=pi*pow((d0/2),2)*h
print(V/1000)

1.2988425994450112

from numpy import *

from units import unit

Finite element mesh

Numerical results for $\omega=419~\rm{Hz}$

Deformed configuration

$\sigma_{11}$

$\sigma_{22}$

$\sigma_{33}$

$\sigma_{12}$

Pressure