#
# Class for cracking
#
import numpy as np
import pybamm
from .base_mechanics import BaseMechanics
[docs]
class CrackPropagation(BaseMechanics):
"""
Cracking behaviour in electrode particles. See :footcite:t:`Ai2019` for mechanical
model (thickness change) and :footcite:t:`Deshpande2012` for cracking model.
Parameters
----------
param : parameter class
The parameters to use for this submodel
domain : str
The domain of the model either 'Negative' or 'Positive'
x_average : bool
Whether to use x-averaged variables (SPM, SPMe, etc) or full variables (DFN)
options: dict
A dictionary of options to be passed to the model.
See :class:`pybamm.BaseBatteryModel`
phase : str, optional
Phase of the particle (default is "primary")
"""
def __init__(self, param, domain, x_average, options, phase="primary"):
super().__init__(param, domain, options, phase)
self.x_average = x_average
pybamm.citations.register("Ai2019")
pybamm.citations.register("Deshpande2012")
[docs]
def get_fundamental_variables(self):
domain, Domain = self.domain_Domain
phase_name = self.phase_param.phase_name
if self.x_average:
if self.size_distribution:
l_cr_av_dist = pybamm.Variable(
f"X-averaged {domain} {phase_name}particle crack length distribution [m]",
domains={
"primary": f"{domain} particle size",
"secondary": "current collector",
},
scale=self.phase_param.l_cr_0,
)
l_cr_dist = pybamm.SecondaryBroadcast(
l_cr_av_dist, f"{domain} electrode"
)
l_cr_av = pybamm.size_average(l_cr_av_dist)
else:
l_cr_av = pybamm.Variable(
f"X-averaged {domain} {phase_name}particle crack length [m]",
domain="current collector",
scale=self.phase_param.l_cr_0,
)
l_cr = pybamm.PrimaryBroadcast(l_cr_av, f"{domain} electrode")
else:
if self.size_distribution:
l_cr_dist = pybamm.Variable(
f"{Domain} {phase_name}particle crack length distribution [m]",
domains={
"primary": f"{domain} particle size",
"secondary": f"{domain} electrode",
"tertiary": "current collector",
},
scale=self.phase_param.l_cr_0,
)
l_cr = pybamm.size_average(l_cr_dist)
else:
l_cr = pybamm.Variable(
f"{Domain} {phase_name}particle crack length [m]",
domain=f"{domain} electrode",
auxiliary_domains={"secondary": "current collector"},
scale=self.phase_param.l_cr_0,
)
variables = self._get_standard_variables(l_cr)
if self.size_distribution:
variables.update(self._get_standard_size_distribution_variables(l_cr_dist))
return variables
[docs]
def get_coupled_variables(self, variables):
domain, Domain = self.domain_Domain
phase_name = self.phase_param.phase_name
variables.update(self._get_standard_surface_variables(variables))
variables.update(self._get_mechanical_results(variables))
if self.size_distribution:
variables.update(self._get_mechanical_size_distribution_results(variables))
T = variables[f"{Domain} electrode temperature [K]"]
k_cr = self.phase_param.k_cr(T)
m_cr = self.phase_param.m_cr
b_cr = self.phase_param.b_cr
if self.size_distribution:
stress_t_surf = variables[
f"{Domain} {phase_name}particle surface tangential stress distribution [Pa]"
]
else:
stress_t_surf = variables[
f"{Domain} {phase_name}particle surface tangential stress [Pa]"
]
if self.size_distribution:
l_cr = variables[
f"{Domain} {phase_name}particle crack length distribution [m]"
]
else:
l_cr = variables[f"{Domain} {phase_name}particle crack length [m]"]
# # compressive stress will not lead to crack propagation
dK_SIF = stress_t_surf * b_cr * pybamm.sqrt(np.pi * l_cr) * (stress_t_surf >= 0)
dl_cr = k_cr * (dK_SIF**m_cr) / 3600 # divide by 3600 to replace t0_cr
variables.update(
{
f"{Domain} {phase_name}particle cracking rate [m.s-1]": dl_cr,
f"X-averaged {domain} {phase_name}particle cracking rate [m.s-1]": pybamm.x_average(
dl_cr
),
}
)
return variables
[docs]
def set_rhs(self, variables):
domain, Domain = self.domain_Domain
phase_name = self.phase_param.phase_name
if self.x_average is True:
if self.size_distribution:
l_cr = variables[
f"X-averaged {domain} {phase_name}particle crack length distribution [m]"
]
else:
l_cr = variables[
f"X-averaged {domain} {phase_name}particle crack length [m]"
]
dl_cr = variables[
f"X-averaged {domain} {phase_name}particle cracking rate [m.s-1]"
]
else:
if self.size_distribution:
l_cr = variables[
f"{Domain} {phase_name}particle crack length distribution [m]"
]
else:
l_cr = variables[f"{Domain} {phase_name}particle crack length [m]"]
dl_cr = variables[f"{Domain} {phase_name}particle cracking rate [m.s-1]"]
self.rhs = {l_cr: dl_cr}
[docs]
def set_initial_conditions(self, variables):
domain, Domain = self.domain_Domain
phase_name = self.phase_param.phase_name
l_cr_0 = self.phase_param.l_cr_0
if self.x_average is True:
if self.size_distribution:
l_cr = variables[
f"X-averaged {domain} {phase_name}particle crack length distribution [m]"
]
l_cr_0 = pybamm.PrimaryBroadcast(l_cr_0, f"{domain} particle size")
else:
l_cr = variables[
f"X-averaged {domain} {phase_name}particle crack length [m]"
]
else:
if self.size_distribution:
l_cr = variables[
f"{Domain} {phase_name}particle crack length distribution [m]"
]
l_cr_0 = pybamm.PrimaryBroadcast(l_cr_0, f"{domain} electrode")
l_cr_0 = pybamm.PrimaryBroadcast(l_cr_0, f"{domain} particle size")
else:
l_cr = variables[f"{Domain} {phase_name}particle crack length [m]"]
l_cr_0 = pybamm.PrimaryBroadcast(l_cr_0, f"{domain} electrode")
self.initial_conditions = {l_cr: l_cr_0}
[docs]
def add_events_from(self, variables):
domain, Domain = self.domain_Domain
phase_name = self.phase_param.phase_name
if self.x_average is True:
l_cr = variables[
f"X-averaged {domain} {phase_name}particle crack length [m]"
]
else:
l_cr = variables[f"{Domain} {phase_name}particle crack length [m]"]
self.events.append(
pybamm.Event(
f"{domain} {phase_name} particle crack length larger than particle radius",
1 - pybamm.max(l_cr) / self.phase_param.R_typ,
)
)
