statmechcrack.mechanical

A module for the crack model treated mechanically.

This module consist of the class CrackMechanical which contains methods for computing quantities when treating the crack model mechanically rather than statistical mechanically.

class CrackMechanical(N=8, M=8, kappa=100, alpha=1, varepsilon=100)

Bases: BasicUtility

The crack model class treated mechanically.

N

The number of broken bonds.

Type:

int

M

The number of intact bonds.

Type:

int

L

The total number of bonds.

Type:

int

kappa

The nondimensional bending stiffness.

Type:

float

alpha

The nondimensional Morse parameter.

Type:

float

varepsilon

The nondimensional bond energy.

Type:

float

H_Pi(s)

The nondimensional Hessian of the total potential energy of the system,

\[\mathbf{H}^\Pi(\mathbf{s}) = \mathbf{H}^\Pi_0 +\mathbf{H}^\Pi_1(\boldsymbol{\lambda}).\]

where \(\lambda_j\equiv s_{N+j}\) and \(j=1,\ldots,M\).

Parameters:

s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Hessian.

Return type:

numpy.ndarray

H_Pi_0()

The nondimensional Hessian of the total potential energy of the system due to bending,

\[\begin{split}\left[\mathbf{H}^\Pi_0\right]_{mn} = \kappa\left(\phantom{\delta^{|m-n|}_1}\right. & 6\delta^m_n - 5\delta^m_1\delta^n_1 - \delta^m_2\delta^n_2 - 5\delta^m_{L+1}\delta^n_{L+1} - \delta^m_L\delta^n_L - 4\delta^{|m-n|}_1 \\ & + 2\delta^m_1\delta^n_2 + 2\delta^m_2\delta^n_1 + 2\delta^m_{L+1}\delta^n_L + 2\delta^m_L\delta^n_{L+1} + \delta^{|m-n|}_2 \left.\phantom{\delta^{|m-n|}_1}\right), \\ & \quad m,n=1,\ldots,L+1.\end{split}\]

Returns:

The nondimensional Hessian.

Return type:

numpy.ndarray

H_Pi_00()

The nondimensional Hessian of the total potential energy for the isolated bending system,

\[\begin{split}\left[\mathbf{H}^\Pi_b\right]_{mn} = \kappa\left(\phantom{\delta^{|m-n|}_1}\right. & 6\delta^m_n - 5\delta^m_1\delta^n_1 - \delta^m_2\delta^n_2 - 4\delta^{|m-n|}_1 \\ & + 2\delta^m_1\delta^n_2 + 2\delta^m_2\delta^n_1 + \delta^{|m-n|}_2 \left.\phantom{\delta^{|m-n|}_1}\right), \quad m,n=1,\ldots,N+1.\end{split}\]

Returns:

The nondimensional Hessian.

Return type:

numpy.ndarray

H_Pi_1(lambda_)

The nondimensional Hessian of the total potential energy of the system due to stretching intact bonds,

\[\left[\mathbf{H}^\Pi_1(\boldsymbol{\lambda})\right]_{mn} = \left[\mathbf{H}^U_1(\boldsymbol{\lambda})\right]_{mn}.\]

Parameters:

lambda (array_like) – The intact bond stretches.

Returns:

The nondimensional Hessian.

Return type:

numpy.ndarray

H_Pi_TS(s)

The nondimensional Hessian of the total potential energy of the system in its transition state.

Parameters:

s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Hessian.

Return type:

numpy.ndarray

H_U(s)

The nondimensional Hessian of the potential energy of the system,

\[\mathbf{H}^U(\mathbf{s}) = \mathbf{H}^U_0 +\mathbf{H}^U_1(\boldsymbol{\lambda}).\]

where \(\lambda_j\equiv s_{N+j}\) and \(j=1,\ldots,M\).

Parameters:

s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Hessian.

Return type:

numpy.ndarray

H_U_0()

The nondimensional Hessian of the potential energy of the system due to bending,

\[\left[\mathbf{H}^U_0\right]_{mn} = \left[\mathbf{H}^\pi_0\right]_{m+1,n+1}, \quad m,n=1,\ldots,L.\]

Returns:

The nondimensional Hessian.

Return type:

numpy.ndarray

H_U_00()

The nondimensional Hessian of the potential energy for the isolated bending system,

\[\left[\mathbf{H}^U_00\right]_{mn} = \kappa\left(6\delta^m_n - \delta^m_1\delta^n_1 - 4\delta^{|m-n|}_1 + \delta^{|m-n|}_2\right), \quad m,n=1,\ldots,N.\]

Returns:

The nondimensional Hessian.

Return type:

numpy.ndarray

H_U_1(lambda_)

The nondimensional Hessian of the potential energy of the system due to stretching intact bonds,

\[\left[\mathbf{H}^U_1(\boldsymbol{\lambda})\right]_{mn} = 2\alpha^2\varepsilon e^{-\alpha(\lambda_m-1)}\left( 1 - 2e^{-\alpha(\lambda_m-1)} \right)\delta^m_n.\]

Parameters:

lambda (array_like) – The intact bond stretches.

Returns:

The nondimensional Hessian.

Return type:

numpy.ndarray

H_U_TS(s)

The nondimensional Hessian of the potential energy of the system in its transition state.

Parameters:

s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Hessian.

Return type:

numpy.ndarray

beta_Pi(p, v, s)

The nondimensional total potential energy of the system,

\[\beta \Pi(p,v,\mathbf{s}) = \beta U(v,\mathbf{s}) - pv.\]

Parameters:

• p (array_like) – The nondimensional end force.

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional total potential energy.

Return type:

numpy.ndarray

beta_Pi_0(p, v, s)

The nondimensional total potential energy of the reference system,

\[\beta \Pi_{0}(p,v,\mathbf{s}) = \beta U_{0}(v,\mathbf{s}) - pv.\]

Parameters:

• p (array_like) – The nondimensional end force.

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

beta_Pi_00(p, v, s)

The nondimensional total potential energy for the isolated bending system,

\[\beta \Pi_b(p,v,\mathbf{s}) = \beta U_{00}(v,\mathbf{s}) - pv.\]

Parameters:

• p (array_like) – The nondimensional end force.

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

beta_U(v, s)

The nondimensional potential energy of the system,

\[\beta U(v,\mathbf{s}) = \beta U_0(v,\mathbf{s}) + \beta U_1(\boldsymbol{\lambda}),\]

where \(\lambda_j\equiv s_{N+j}\) and \(j=1,\ldots,M\).

Parameters:

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional potential energy.

Return type:

numpy.ndarray

beta_U_0(v, s)

The nondimensional potential energy of the system due to bending, i.e. of the reference system,

\[\beta U_0(v, \mathbf{s}) = \sum_{i=1}^{L-1} \frac{\kappa}{2} \left( s_{i-1} - 2s_i + s_{i+1} \right)^2,\]

where \(s_0\equiv v\).

Parameters:

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional potential energy.

Return type:

numpy.ndarray

beta_U_00(v, s)

The nondimensional potential energy of the isolated bending system,

\[\beta U_{00}(v, \mathbf{s}) = \sum_{i=1}^{N+1} \frac{\kappa}{2} \left( s_{i-1} - 2s_i + s_{i+1} \right)^2,\]

where \(s_0\equiv v\), and \(s_{N+1}\equiv\lambda_1\), and \(s_{N+2}\equiv\lambda_2\).

Parameters:

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional potential energy.

Return type:

numpy.ndarray

beta_U_01(lambda_)

The nondimensional potential energy due to bending within the intact region,

\[\begin{split}\beta U_{01}(\boldsymbol{\lambda}) &= \beta U_0(v, \mathbf{s}) - \beta U_{00}(v, \mathbf{s}) \\ &= \sum_{i=N+2}^{L-1} \frac{\kappa}{2} \left( s_{i-1} - 2s_i + s_{i+1} \right)^2 \\ &= \sum_{j=2}^{M-1} \frac{\kappa}{2} \left( \lambda_{j-1} - 2\lambda_j + \lambda_{j+1} \right)^2,\end{split}\]

where \(s_0\equiv v\), and \(s_{N+1}\equiv\lambda_1\), and \(s_{N+2}\equiv\lambda_2\).

Parameters:

lambda (array_like) – The intact bond stretches.

Returns:

The nondimensional potential energy.

Return type:

numpy.ndarray

beta_U_1(lambda_)

The nondimensional potential energy of the system due to stretching intact bonds,

\[\beta U_1(\boldsymbol{\lambda}) = \sum_{j=1}^M \varepsilon\beta u(\lambda_j).\]

Parameters:

lambda (array_like) – The intact bond stretches.

Returns:

The nondimensional potential energy.

Return type:

numpy.ndarray

beta_u(lambda_)

The nondimensional potential energy of a single bond as a function of the bond stretch, given by the Morse potential [1],

\[\beta u(\lambda) = \varepsilon \left[ 1 - e^{-\alpha(\lambda - 1)} \right]^2.\]

Parameters:

lambda (array_like) – The bond stretch.

Returns:

The nondimensional potential energy.

Return type:

numpy.ndarray

beta_u_p(lambda_)

The first derivative of the potential energy of a single bond as a function of the bond stretch,

\[\beta u'(\lambda) = 2\alpha\varepsilon e^{-\alpha(\lambda - 1)} \left[ 1 - e^{-\alpha(\lambda - 1)} \right]^2.\]

Parameters:

lambda (array_like) – The bond stretch.

Returns:

The first derivative of the nondimensional potential energy.

Return type:

numpy.ndarray

beta_u_pp(lambda_)

The second derivative of the potential energy of a single bond as a function of the bond stretch,

\[\beta u''(\lambda) = 2\alpha^2\varepsilon e^{-\alpha(\lambda - 1)} \left[ 2e^{-\alpha(\lambda - 1)} - 1 \right]^2.\]

Parameters:

lambda (array_like) – The bond stretch.

Returns:

The second derivative of the nondimensional potential energy.

Return type:

numpy.ndarray

j_Pi(p, v, s)

The nondimensional Jacobian of the total potential energy of the system,

\[\mathbf{j}^\Pi(p,v,\mathbf{s}) = \mathbf{j}^\Pi_0(p,v,\mathbf{s}) +\mathbf{j}^\Pi_1(\boldsymbol{\lambda}).\]

where \(\lambda_j\equiv s_{N+j}\) and \(j=1,\ldots,M\).

Parameters:

• p (array_like) – The nondimensional end force.

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Jacobian.

Return type:

numpy.ndarray

j_Pi_0(p, v, s)

The nondimensional Jacobian of the total potential energy of the system due to bending,

\[\mathbf{j}^\Pi_0(p,v,\mathbf{s}) = \mathbf{H}^\Pi_0 \cdot \left( v, s_1, \cdots, s_L \right)^T - \left( p, 0, \ldots, 0 \right)^T\]

Parameters:

• p (array_like) – The nondimensional end force.

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Jacobian.

Return type:

numpy.ndarray

j_Pi_00(p, v, s)

The nondimensional Jacobian of the total potential energy of the system for the isolated bending system,

\[\mathbf{j}^\Pi_b(p,v,\mathbf{s}) = \mathbf{H}^\Pi_b \cdot \left( v, s_1, \cdots, s_N \right)^T - \kappa\left( p/\kappa, 0, \ldots, 0, -\lambda_1, 4\lambda_1-\lambda_2 \right)^T\]

Parameters:

• p (array_like) – The nondimensional end force.

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Jacobian.

Return type:

numpy.ndarray

j_Pi_1(lambda_)

The nondimensional Jacobian of the total potential energy of the system due to stretching intact bonds,

\[\mathbf{j}^\Pi_1(\boldsymbol{\lambda}) = \mathbf{j}^U_1(\boldsymbol{\lambda}).\]

Parameters:

lambda (array_like) – The intact bond stretches.

Returns:

The nondimensional Jacobian.

Return type:

numpy.ndarray

j_Pi_TS(p, v, s)

The nondimensional Jacobian of the total potential energy of the system in its transition state.

Parameters:

• p (array_like) – The nondimensional end force.

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Jacobian.

Return type:

numpy.ndarray

j_U(v, s)

The nondimensional Jacobian of the potential energy of the system,

\[\mathbf{j}^U(v,\mathbf{s}) = \mathbf{j}^U_0(v,\mathbf{s}) +\mathbf{j}^U_1(\boldsymbol{\lambda}).\]

where \(\lambda_j\equiv s_{N+j}\) and \(j=1,\ldots,M\).

Parameters:

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Jacobian.

Return type:

numpy.ndarray

j_U_0(v, s)

The nondimensional Jacobian of the potential energy of the system for the reference system,

\[\mathbf{j}^U_0(v,\mathbf{s}) = \mathbf{H}^U_0 \cdot \mathbf{s} - \kappa\left( 2v, -v, 0, \ldots, 0 \right)^T\]

Parameters:

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Jacobian.

Return type:

numpy.ndarray

j_U_00(v, s)

The nondimensional Jacobian of the potential energy of the system for the isolated bending system,

\[\mathbf{j}^U_00(v,\mathbf{s}) = \mathbf{H}^U_00 \cdot \mathbf{s} - \kappa\left( 2v, -v, 0, \ldots, 0, -\lambda_1, 4\lambda_1-\lambda_2 \right)^T.\]

Parameters:

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Jacobian.

Return type:

numpy.ndarray

j_U_1(lambda_)

The nondimensional Jacobian of the potential energy of the system due to stretching intact bonds,

\[\left[\mathbf{j}^U_1(\boldsymbol{\lambda})\right]_j = 2\alpha\varepsilon e^{-\alpha(\lambda_j-1)}\left( 1 - e^{-\alpha(\lambda_j-1)} \right).\]

Parameters:

lambda (array_like) – The intact bond stretches.

Returns:

The nondimensional Jacobian.

Return type:

numpy.ndarray

j_U_TS(v, s)

The nondimensional Jacobian of the potential energy of the system in its transition state.

Parameters:

• v (array_like) – The nondimensional end separation.

• s (array_like) – The nondimensional configuration.

Returns:

The nondimensional Jacobian.

Return type:

numpy.ndarray

minimize_beta_Pi(p, transition_state=False)

Function to minimize the total potential energy of the system.

Parameters:

• p (array_like) – The nondimensional end force.

• transition_state (bool, optional, default=False) – Whether or not to fix the crack tip bond stretch at the transition state stretch.

Returns:

• (numpy.ndarray) - The minimized nondimensional total potential energy.

• (numpy.ndarray) - The corresponding nondimensional end separation.

• (numpy.ndarray) - The corresponding nondimensional positions.

Return type:

tuple

minimize_beta_Pi_00(p, lambda_)

Function to minimize the total potential energy of the isolated bending system.

Parameters:

• p (array_like) – The nondimensional end force.

• lambda (array_like) – The intact bond stretches.

Returns:

• (numpy.ndarray) - The minimized nondimensional total potential energy.

• (numpy.ndarray) - The corresponding nondimensional end separation.

• (numpy.ndarray) - The corresponding nondimensional positions.

Return type:

tuple

minimize_beta_U(v, transition_state=False)

Function to minimize the potential energy of the system.

Parameters:

• v (array_like) – The nondimensional end separation.

• transition_state (bool, optional, default=False) – Whether or not to fix the crack tip bond stretch at the transition state stretch.

Returns:

• (numpy.ndarray) - The minimized nondimensional potential energy.

• (numpy.ndarray) - The corresponding nondimensional positions.

Return type:

tuple

Example

Plot the rescaled minimized nondimensional potential energy as a function of the nondimensional end separation for the mechnically-treated system for an increasing number of broken bonds \(N\) and compare to the thermodynamic limit:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackMechanical
>>> v = np.linspace(1, 11, 33)
>>> _ = plt.figure()
>>> for N in [5, 10, 25]:
...     model = CrackMechanical(N=N)
...     beta_U = model.minimize_beta_U(v)[0]
...     _ = plt.plot(v - 1, model.N**3/3/model.kappa*beta_U,
...                  label='$N=$'+str(N))
...     p_m = model.p_mechanical(v)
...     beta_Pi = model.minimize_beta_Pi(p_m)[0]
...     _ = plt.plot(v - 1,
...                  model.N**3/3/model.kappa*(beta_Pi+p_m*v)
...                  , 'k:')
>>> _ = plt.plot(v - 1, (v - 1)**2/2, 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v$')
>>> _ = plt.ylabel(r'minimized $(N^3/3\kappa)\beta U$')
>>> _ = plt.legend()
>>> plt.show()

minimize_beta_U_00(v, lambda_)

Function to minimize the potential energy of the isolated bending system.

Parameters:

• v (array_like) – The nondimensional end separation.

• lambda (array_like) – The intact bond stretches.

Returns:

• (numpy.ndarray) - The minimized nondimensional potential energy.

• (numpy.ndarray) - The corresponding nondimensional end separation.

• (numpy.ndarray) - The corresponding nondimensional positions.

Return type:

tuple

Example

Plot the rescaled minimized nondimensional potential energy as a function of the nondimensional end separation for the mechnically-treated isolated bending system for an increasing number of broken bonds \(N\) and compare to the thermodynamic limit:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackMechanical
>>> v = np.linspace(1, 11, 33)
>>> _ = plt.figure()
>>> for N in [5, 10, 25]:
...     model = CrackMechanical(N=N)
...     beta_U = model.minimize_beta_U_00(v, [1, 1])[0]
...     _ = plt.plot(v - 1, model.N**3/3/model.kappa*beta_U,
...                  label='$N=$'+str(N))
...     p_m = model.p_0_mechanical(v, [1, 1])
...     beta_Pi = model.minimize_beta_Pi_00(p_m, [1, 1])[0]
...     _ = plt.plot(v - 1,
...                  model.N**3/3/model.kappa*(beta_Pi+p_m*v)
...                  , 'k:')
>>> _ = plt.plot(v - 1, (v - 1)**2/2, 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v$')
>>> _ = plt.ylabel(r'minimized $(N^3/3\kappa)\beta U$')
>>> _ = plt.legend()
>>> plt.show()

p_0_mechanical(v, lambda_)

The nondimensional end force as a function of the nondimensional end separation for the mechnically-treated reference system, calculated by minimizing the potential energy.

Parameters:

• v (array_like) – The nondimensional end separation.

• lambda (array_like) – The intact bond stretches.

Returns:

The nondimensional end force.

Return type:

numpy.ndarray

Example

Plot the rescaled nondimensional end force as a function of the nondimensional end separation for the mechnically-treated reference system for an increasing number of broken bonds \(N\) and compare to the thermodynamic limit:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackMechanical
>>> v = np.linspace(1, 11, 33)
>>> _ = plt.figure()
>>> for N in [5, 10, 25]:
...     model = CrackMechanical(N=N)
...     p_m = model.p_0_mechanical(v, [1, 1])
...     r_p_m = model.N**3/3/model.kappa*p_m
...     _ = plt.plot(v - 1, r_p_m, label='$N=$'+str(N))
...     v_m = model.v_0_mechanical(p_m, [1, 1])
...     _ = plt.plot(v_m - 1, r_p_m, 'k:')
>>> _ = plt.plot(v - 1, v - 1, 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v$')
>>> _ = plt.ylabel(r'$(N^3/3\kappa)p_m$')
>>> _ = plt.legend()
>>> plt.show()

p_b_mechanical(v, lambda_)

The nondimensional end force as a function of the nondimensional end separation for the mechnically-treated reference system, calculated by minimizing the potential energy.

Parameters:

• v (array_like) – The nondimensional end separation.

• lambda (array_like) – The intact bond stretches.

Returns:

The nondimensional end force.

Return type:

numpy.ndarray

p_mechanical(v)

The nondimensional end force as a function of the nondimensional end separation for the mechnically-treated system, calculated by minimizing the potential energy.

Parameters:

v (array_like) – The nondimensional end separation.

Returns:

The nondimensional end force.

Return type:

numpy.ndarray

Example

Plot the rescaled nondimensional end force as a function of the nondimensional end separation for the mechnically-treated system for an increasing number of broken bonds \(N\) and compare to the thermodynamic limit:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackMechanical
>>> v = np.linspace(1, 11, 33)
>>> _ = plt.figure()
>>> for N in [5, 10, 25]:
...     model = CrackMechanical(N=N)
...     p_m = model.p_mechanical(v)
...     r_p_m = model.N**3/3/model.kappa*p_m
...     _ = plt.plot(v - 1, r_p_m, label='$N=$'+str(N))
...     v_m = model.v_mechanical(p_m)
...     _ = plt.plot(v_m - 1, r_p_m, 'k:')
>>> _ = plt.plot(v - 1, v - 1, 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v$')
>>> _ = plt.ylabel(r'$(N^3/3\kappa)p_m$')
>>> _ = plt.legend()
>>> plt.show()

v_0_mechanical(p, lambda_)

The nondimensional end separation as a function of the nondimensional end force for the mechnically-treated reference system, calculated by minimizing the total potential energy.

Parameters:

• p (array_like) – The nondimensional end force.

• lambda (array_like) – The intact bond stretches.

Returns:

The nondimensional end separation.

Return type:

numpy.ndarray

v_b_mechanical(p, lambda_)

The nondimensional end separation as a function of the nondimensional end force for the mechnically-treated isolated bending system, calculated by minimizing the total potential energy.

Parameters:

• p (array_like) – The nondimensional end force.

• lambda (array_like) – The intact bond stretches.

Returns:

The nondimensional end separation.

Return type:

numpy.ndarray

v_mechanical(p)

The nondimensional end separation as a function of the nondimensional end force for the mechnically-treated system, calculated by minimizing the total potential energy.

Parameters:

p (array_like) – The nondimensional end force.

Returns:

The nondimensional end separation.

Return type:

numpy.ndarray

Example

Plot the nondimensional end separation as a function of the rescaled nondimensional end force for the mechnically-treated system for an increasing nondimensional bond energy \(\varepsilon\) and compare to the limit given by the mechanically-treated reference system:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from statmechcrack import CrackMechanical
>>> v = np.linspace(1, 11, 33)
>>> _ = plt.figure()
>>> for varepsilon in [50, 100, 500]:
...     model = CrackMechanical(varepsilon=varepsilon)
...     r_p = np.linspace(0, 10, 33)
...     p = 3*model.kappa/model.N**3*r_p
...     v_m = model.v_mechanical(p)
...     _ = plt.plot(v_m - 1, r_p,
...                  label=r'$\varepsilon=$'+str(varepsilon))
...     p_m = model.p_mechanical(v_m)
...     r_p_m = model.N**3/3/model.kappa*p_m
...     _ = plt.plot(v_m - 1, r_p_m, 'k:')
>>> _ = plt.plot(model.v_0_mechanical(p, [1, 1]) - 1,
...              r_p, 'k--', label='limit')
>>> _ = plt.xlabel(r'$\Delta v_m$')
>>> _ = plt.ylabel(r'$(N^3/3\kappa)p$')
>>> _ = plt.legend()
>>> plt.show()

---

References

[1]

Philip M Morse. Diatomic molecules according to the wave mechanics. II. Vibrational levels. Physical Review, 34(1):57, 1929. doi:10.1103/PhysRev.34.57.