```python
%matplotlib notebook
import numpy as np
import matplotlib.pyplot as plt
import sympy as sym
#sym.init_printing(use_unicode=False, wrap_line=True)
```


```python
s = sym.Symbol('alpha')
k = sym.Symbol('self%k')
asym  = (1-s)**2/(1+(k-1)*(1-1-s)**2)
print('a:  ',asym)
dasym = sym.diff(asym, s)
print("a': ",dasym)
d2asym = sym.diff(dasym, s)
print('a": ',d2asym)

def a(s,k):
    return (1 - s)**2/(s**2*(k - 1) + 1)
def da(s,k):
    return -2*s*(1 - s)**2*(k - 1)/(s**2*(k - 1) + 1)**2 + (2*s - 2)/(s**2*(k - 1) + 1)
def d2a(s,k):
    return 8*s**2*(1 - s)**2*(k - 1)**2/(s**2*(k - 1) + 1)**3 - 4*s*(k - 1)*(2*s - 2)/(s**2*(k - 1) + 1)**2 - 2*(1 - s)**2*(k - 1)/(s**2*(k - 1) + 1)**2 + 2/(s**2*(k - 1) + 1)
def w(s):
    return 1-(1-s)**2
```

    a:   (1 - alpha)**2/(alpha**2*(self.k - 1) + 1)
    a':  -2*alpha*(1 - alpha)**2*(self.k - 1)/(alpha**2*(self.k - 1) + 1)**2 + (2*alpha - 2)/(alpha**2*(self.k - 1) + 1)
    a":  8*alpha**2*(1 - alpha)**2*(self.k - 1)**2/(alpha**2*(self.k - 1) + 1)**3 - 4*alpha*(2*alpha - 2)*(self.k - 1)/(alpha**2*(self.k - 1) + 1)**2 - 2*(1 - alpha)**2*(self.k - 1)/(alpha**2*(self.k - 1) + 1)**2 + 2/(alpha**2*(self.k - 1) + 1)
    cw:  Integral(Piecewise((I*(1 - alpha)*(alpha - 1)**3/(2*((alpha - 1)**2 - 1)**(3/2)) - I*(1 - alpha)*(alpha - 1)/(2*((alpha - 1)**2 - 1)**(3/2)) + 3*I*(alpha - 1)**2/(2*sqrt((alpha - 1)**2 - 1)) - I/sqrt((alpha - 1)**2 - 1), (alpha - 1)**2 > 1), ((1 - alpha)*(alpha - 1)/(2*sqrt(1 - (alpha - 1)**2)) + sqrt(1 - (alpha - 1)**2)/2 + 1/(2*sqrt(1 - (alpha - 1)**2)), True)), (alpha, 0, 1))



```python
alpha = np.linspace(0,1,101)[1:-1]
fig, ax = plt.subplots(figsize=(8,6))
for k in (0,0.25,0.5,0.75,1.):
    ax.plot(alpha,a(alpha,k),label=r'$s=${0}'.format(k))
ax.set_xlabel(r'$\alpha$')
ax.legend(loc=0)
ax.set_title(r'$a(\alpha)$')
ax.plot()
```


    <IPython.core.display.Javascript object>









    []




```python
alpha = np.linspace(0,1,101)[1:-1]
fig, ax = plt.subplots(figsize=(8,6))
for k in (0,0.25,0.5,0.75,1.):
    ax.plot(alpha,da(alpha,k),label=r'$s=${0}'.format(k))
ax.set_xlabel(r'$\alpha$')
ax.legend(loc=0)
ax.set_title(r"$a'(\alpha)$")
ax.plot()
```


    <IPython.core.display.Javascript object>









    []




```python
alpha = np.linspace(0,1,101)[1:-1]
fig, ax = plt.subplots(figsize=(8,6))
for k in (0,0.25,0.5,0.75,1.):
    ax.plot(alpha,d2a(alpha,k),label=r'$s=${0}'.format(k))
ax.set_xlabel(r'$\alpha$')
ax.legend(loc=0)
ax.set_title(r'$a"(\alpha)$')
ax.plot()
```


    <IPython.core.display.Javascript object>









    []




```python
alpha = np.linspace(0,1,101)[1:-1]
fig, ax = plt.subplots(figsize=(8,6))
ax.plot(alpha,w(alpha),label=r'$w$')
ax.set_xlabel(r'$\alpha$')
#ax.legend(loc=0)
ax.set_title(r"$w(\alpha)$")
ax.plot()
```


    <IPython.core.display.Javascript object>









    []




```python

```

98c4d35335a1b14f2683a9ab6e6bf04f3713056d
ipynb
Jupyter Notebook
m_DefMech/LinSoft model.ipynb
jeanmichelscherer/mef90
48b9b7d8bdaccb846a76833853f6ea81ce6fc9b1
BSD-2-Clause
2019-12-04T01:38:56.000Z
2022-02-13T17:35:06.000Z
m_DefMech/LinSoft model.ipynb
jeanmichelscherer/mef90
48b9b7d8bdaccb846a76833853f6ea81ce6fc9b1
BSD-2-Clause
2022-02-19T21:38:52.000Z
2022-02-19T21:38:52.000Z
m_DefMech/LinSoft model.ipynb
jeanmichelscherer/mef90
48b9b7d8bdaccb846a76833853f6ea81ce6fc9b1
BSD-2-Clause
2021-01-20T01:57:25.000Z
2022-02-17T18:11:38.000Z
Qwen/Qwen-72B
1. YES
2. YES