Source code for pulse.compressibility

r"""This module defines compressibility models for the material models.
We define two compressibility models: `Incompressible` and `Compressible`.

An incompressible material is a material that does not change its volume under
deformation. The volume change is described by the Jacobian :math:`J = \det(F)`,
where :math:`F` is the deformation gradient.

The `Incompressible` model is defined by the strain energy density function
:math:`\Psi = p (J - 1)`, where :math:`p` is a function representing the
Lagrange multiplier. The `Compressible` model is defined by the strain energy
density function :math:`\Psi = \kappa (J \ln(J) - J + 1)`, where :math:`\kappa`
is a material parameter representing the bulk modulus. Higher values of
:math:`\kappa` correspond to more incompressible material.
"""

import abc
import logging
from dataclasses import dataclass, field

import dolfinx
import numpy as np
import ufl

from . import exceptions
from .units import Variable

logger = logging.getLogger(__name__)




class Compressibility(abc.ABC):
    """Base class for compressibility models."""



    @abc.abstractmethod
    def strain_energy(self, C: ufl.core.expr.Expr) -> ufl.core.expr.Expr:
        """Strain energy density function"""
        pass




    def register(self, *args, **kwargs) -> None:
        pass




    def S(self, C: ufl.core.expr.Expr) -> ufl.core.expr.Expr:
        """Cauchy stress tensor for the compressibility model."""
        return 2.0 * ufl.diff(self.strain_energy(C), C)




    def P(self, F: ufl.core.expr.Expr) -> ufl.core.expr.Expr:
        """First Piola-Kirchhoff stress tensor for the compressibility model."""
        C = F.T * F
        return ufl.diff(self.strain_energy(C), F)




    @abc.abstractmethod
    def is_compressible(self) -> bool:
        """Returns True if the material model is compressible."""
        pass






@dataclass(slots=True)
class Incompressible(Compressibility):
    r"""Incompressible material model

    Strain energy density function is given by

    .. math::
        \Psi = p (J - 1)

    """

    p: dolfinx.fem.Function = field(default=None, init=False)

    def __post_init__(self):
        logger.debug("Created Incompressible compressibility model")

    def __str__(self) -> str:
        return "p (J - 1)"



    def register(self, p: dolfinx.fem.Function) -> None:
        self.p = p




    def strain_energy(self, C: ufl.core.expr.Expr) -> ufl.core.expr.Expr:
        J = ufl.sqrt(ufl.det(C))
        if self.p is None:
            raise exceptions.MissingModelAttribute(attr="p", model=type(self).__name__)
        return self.p * (J - 1.0)




    def is_compressible(self) -> bool:
        return False






@dataclass(slots=True)
class Compressible(Compressibility):
    r"""Compressible material model

    Strain energy density function is given by

    .. math::
        \Psi = \kappa (J \ln(J) - J + 1)

    """

    kappa: Variable = field(default_factory=lambda: Variable(1e6, "Pa"))

    def __post_init__(self):
        if not isinstance(self.kappa, Variable):
            unit = "kPa"
            logger.warning("Setting mu to %s %s", self.kappa, unit)
            self.kappa = Variable(self.kappa, unit)

        # Check that value are positive
        if not exceptions.check_value_greater_than(
            self.kappa.value,
            0.0,
            inclusive=True,
        ):
            raise exceptions.InvalidRangeError(
                name="kappa",
                expected_range=(0.0, np.inf),
            )
        logger.debug(f"Created Compressible compressibility model: {str(self)}")
        logger.debug(f"Material parameters: kappa = {self.kappa}")

    def __str__(self) -> str:
        return "\u03ba (J ln(J) - J + 1)"



    def strain_energy(self, C: ufl.core.expr.Expr) -> ufl.core.expr.Expr:
        J = ufl.sqrt(ufl.det(C))
        kappa = self.kappa.to_base_units()
        return kappa * (J * ufl.ln(J) - J + 1)




    def is_compressible(self) -> bool:
        return True






@dataclass(slots=True)
class Compressible2(Compressible):
    r"""Compressible material model

    Strain energy density function is given by

    .. math::
        \Psi = \kappa / 4 (J^2 - 1 - 2 \ln(J))

    """

    kappa: Variable = field(default_factory=lambda: Variable(1e6, "Pa"))

    def __str__(self) -> str:
        return "\u03ba / 4 (J ** 2 - 1 - 2 ln(J))"



    def strain_energy(self, C: ufl.core.expr.Expr) -> ufl.core.expr.Expr:
        J = ufl.sqrt(ufl.det(C))
        kappa = self.kappa.to_base_units()
        return 0.25 * kappa * (J**2 - 1 - 2 * ufl.ln(J))






@dataclass(slots=True)
class Compressible3(Compressible):
    r"""Compressible material model used in Usyk et al. 2002

    Strain energy density function is given by

    .. math::
        \Psi = \kappa / 2 (J - 1) * \ln(J)

    """

    kappa: Variable = field(default_factory=lambda: Variable(5e4, "Pa"))

    def __str__(self) -> str:
        return "\u03ba / 2 * (J - 1) * ln(J)"



    def strain_energy(self, C: ufl.core.expr.Expr) -> ufl.core.expr.Expr:
        J = ufl.sqrt(ufl.det(C))
        kappa = self.kappa.to_base_units()
        return 0.5 * kappa * (J - 1) * ufl.ln(J)