funcy  1.6.0
Functions
Biomechanics

Models for the description of different biologial soft tissues. More...

Functions

template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::incompressible_adipose_tissue_sommer_holzapfel (double cCells, double k1, double k2, double kappa, const Mat &A, const Mat &F)
 Model for adipose tissue of [Sommer2013]. More...
 
template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::incompressible_adipose_tissue_sommer_holzapfel (const Mat &A, const Mat &F)
 Model for adipose tissue of [Sommer2013]. More...
 
template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::compressible_adipose_tissue_sommer_holzapfel (double cCells, double k1, double k2, double kappa, double d0, double d1, const Mat &M, const Mat &F)
 Compressible version of the model for adipose tissue of [Sommer2013]. More...
 
template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::compressible_adipose_tissue_sommer_holzapfel (double d0, double d1, const Mat &M, const Mat &F)
 Compressible version of the model for adipose tissue of [Sommer2013]. Material parameters are taken from the same publication, Table 2, i.e. \(c_\mathrm{Cells}=0.15 (\,\mathrm{kPa})\), \(k_1=0.8 (\,\mathrm{kPa})\), \(k_2=47.3\) and \(\kappa=0.09\). More...
 
template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::incompressible_muscle_tissue_martins (double c, double b, double d, double e, const Mat &A, const Mat &F)
 Incompressible version of the model for muscle tissue of [Martins1998]. More...
 
template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::incompressible_muscle_tissue_martins (const Mat &A, const Mat &F)
 Incompressible version of the model for muscle tissue of [Martins1998]. More...
 
template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::compressible_muscle_tissue_martins (double c, double b, double A, double a, double d0, double d1, const Mat &M, const Mat &F)
 Compressible version of the model for muscle tissue of [Martins1998]. More...
 
template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::compressible_muscle_tissue_martins (double d0, double d1, const Mat &M, const Mat &F)
 Compressible version of the model for muscle tissue of [Martins1998]. More...
 
template<linalg::Matrix Mat, int n = linalg::dim< Mat >()>
auto funcy::incompressible_skin_hendriks (double c0, double c1, const Mat &F)
 Model for skin tissue of [Hendriks2005]. More...
 
template<linalg::Matrix Mat, int n = linalg::dim< Mat >()>
auto funcy::incompressible_skin_hendriks (const Mat &F)
 Model for skin tissue of [Hendriks2005]. More...
 
template<class InflationPenalty , class CompressionPenalty , linalg::Matrix Mat, int n = linalg::dim< Mat >()>
auto funcy::compressible_skin_hendriks (double c0, double c1, double d0, double d1, const Mat &F)
 Compressible version of the model for skin tissue of [Hendriks2005]. More...
 
template<class InflationPenalty , class CompressionPenalty , linalg::Matrix M, int n = linalg::dim< M >()>
auto funcy::compressible_skin_hendriks (double d0, double d1, const M &F)
 Compressible version of the model for skin tissue of [Hendriks2005]. More...
 

Detailed Description

Models for the description of different biologial soft tissues.

Function Documentation

§ compressible_adipose_tissue_sommer_holzapfel() [1/2]

template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::compressible_adipose_tissue_sommer_holzapfel ( double  cCells,
double  k1,
double  k2,
double  kappa,
double  d0,
double  d1,
const Mat &  M,
const Mat &  F 
)

Compressible version of the model for adipose tissue of [Sommer2013].

Implementation of the stored energy function \( W(F)= c_\mathrm{Cells}(\iota_1-3) + \frac{k_1}{k_2}\exp(k_2(\kappa\iota_1+(1-3\kappa)*\iota_4)^2-1) + d_0\Gamma_\mathrm{Inflation}(\det(F)) + d_1\Gamma_\mathrm{Compression} \), where \( \iota_1,\iota_4 \) are the first and first mixed invariant of the strain tensor \(F^T F\).

Parameters
cCellsscaling of the neo-Hookean model for the description of the adipocytes as cell foam.
k1stress-like parameter of the model for the interlobular septa
k2dimensionless parameter of the model for the interlobular septa
kappafiber dispersion parameter \((0\le\kappa\le\frac{1}{3})\).
Mstructural tensor describing the fiber direction of the interlobular septa, i.e. \(M=v\otimes v\) for a fiber direction \(v\)
d0scaling of the penalty function for inflation
d1scaling of the penalty function for compression
Finitial deformation gradient

§ compressible_adipose_tissue_sommer_holzapfel() [2/2]

template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::compressible_adipose_tissue_sommer_holzapfel ( double  d0,
double  d1,
const Mat &  M,
const Mat &  F 
)

Compressible version of the model for adipose tissue of [Sommer2013]. Material parameters are taken from the same publication, Table 2, i.e. \(c_\mathrm{Cells}=0.15 (\,\mathrm{kPa})\), \(k_1=0.8 (\,\mathrm{kPa})\), \(k_2=47.3\) and \(\kappa=0.09\).

Implementation of the stored energy function \( W(F)= c_\mathrm{Cells}(\iota_1-3) + \frac{k_1}{k_2}\exp(k_2(\kappa\iota_1+(1-3\kappa)*\iota_4)^2-1) + d_0\Gamma_\mathrm{Inflation}(\det(F)) + d_1\Gamma_\mathrm{Compression} \), where \( \iota_1,\iota_4 \) are the first and first mixed invariant of the strain tensor \(F^T F\).

Parameters
d0scaling of the penalty function for inflation
d1scaling of the penalty function for compression
Mstructural tensor describing the fiber direction of the interlobular septa, i.e. \(M=v\otimes v\) for a fiber direction \(v\)
Finitial deformation gradient

§ compressible_muscle_tissue_martins() [1/2]

template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::compressible_muscle_tissue_martins ( double  c,
double  b,
double  A,
double  a,
double  d0,
double  d1,
const Mat &  M,
const Mat &  F 
)

Compressible version of the model for muscle tissue of [Martins1998].

Implementation of the stored energy function \( W(F)=c(\exp(b(\bar\iota_1-3))-1) + A(\exp(a(\bar\iota_6-1)^2)-1) + d_0\Gamma_\mathrm{Inflation}(\det(F)) + d_1\Gamma_\mathrm{Compression} \), where \(\bar\iota_1,\bar\iota_6=\bar\iota_4\) are the first modified principal and the third modified mixed invariant of the strain tensor \(F^T F\).

Parameters
cfirst material parameter for the isotropic part
bsecond material parameter for the isotropic part
Afirst material parameter for the anisotropic part
asecond material parameter for the anisotropic part
d0material parameter for the penalty for inflation
d1material parameter for the penalty for compression
Mstructural (rank-one) tensor describing the initial orientation of muscle fibers for \(F=I\), where \(I\) is the unit matrix.
Fdeformation gradient

§ compressible_muscle_tissue_martins() [2/2]

template<class Inflation , class Compression , linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::compressible_muscle_tissue_martins ( double  d0,
double  d1,
const Mat &  M,
const Mat &  F 
)

Compressible version of the model for muscle tissue of [Martins1998].

Implementation of the stored energy function \( W(F)=c(\exp(b(\bar\iota_1-3))-1) + A(\exp(a(\bar\iota_6-1)^2)-1) + d_0\Gamma_\mathrm{Inflation}(\det(F)) + d_1\Gamma_\mathrm{Compression}(\det(F))\), where \(\bar\iota_1,\bar\iota_6=\bar\iota_4\) are the first modified principal and the third modified mixed invariant of the strain tensor \(F^T F\).

Material parameters taken from the above mentioned publication, i.e. \(a=0.387 (\,\mathrm{kPa})\), \( b = 23.46 \), \( A = 0.584 (\,\mathrm{kPa}) \) and \( a = 12.43\).

Parameters
d0material parameter for the penalty for inflation
d1material parameter for the penalty for compression
Mstructural (rank-one) tensor describing the initial orientation of muscle fibers for \(F=I\), where \(I\) is the unit matrix.
Fdeformation gradient

§ compressible_skin_hendriks() [1/2]

template<class InflationPenalty , class CompressionPenalty , linalg::Matrix Mat, int n = linalg::dim< Mat >()>
auto funcy::compressible_skin_hendriks ( double  c0,
double  c1,
double  d0,
double  d1,
const Mat &  F 
)

Compressible version of the model for skin tissue of [Hendriks2005].

Implementation of the stored energy function \(W(F)=c_0(\iota_1-3) + c_1(\iota_1-3)(\iota_2-3) + d_0\Gamma_\mathrm{Inflation}(\det(F)) + d_1\Gamma_\mathrm{Compression}\), where \(\iota_1,\iota_2\) are the first and second principal invariants of the strain tensor \(F^T F\).

Parameters
c0scaling of the shifted first principal invariant
c1scaling of the product of shifted first and second principal invariant
d0scaling of the penalty function for inflation
d1scaling of the penalty function for compression
Finitial deformation gradient

§ compressible_skin_hendriks() [2/2]

template<class InflationPenalty , class CompressionPenalty , linalg::Matrix M, int n = linalg::dim< M >()>
auto funcy::compressible_skin_hendriks ( double  d0,
double  d1,
const M &  F 
)

Compressible version of the model for skin tissue of [Hendriks2005].

Implementation of the stored energy function \(W(F)=c_0(\iota_1-3) + c_1(\iota_1-3)(\iota_2-3) + d_0\Gamma_\mathrm{Inflation}(\det(F)) + d_1\Gamma_\mathrm{Compression}\), where \(\iota_1,\iota_2\) are the first and second principal invariants of the strain tensor \(F^T F\).

Material parameters are taken from [Xu2011], i.e \(c_0=9.4 (\,\mathrm{kPa})\) and \( c_1 = 82 (\,\mathrm{kPa}) \).

Parameters
d0scaling of the penalty function for inflation
d1scaling of the penalty function for compression
Finitial deformation gradient

§ incompressible_adipose_tissue_sommer_holzapfel() [1/2]

template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::incompressible_adipose_tissue_sommer_holzapfel ( double  cCells,
double  k1,
double  k2,
double  kappa,
const Mat &  A,
const Mat &  F 
)

Model for adipose tissue of [Sommer2013].

Implementation of the stored energy function \( W(F)= c_\mathrm{Cells}(\iota_1-3) + \frac{k_1}{k_2}\exp(k_2(\kappa\iota_1+(1-3\kappa)*\iota_4)^2-1) \), where \( \iota_1,\iota_4 \) are the first and first mixed invariant of the strain tensor \(F^T F\).

Parameters
cCellsscaling of the neo-Hookean model for the description of the adipocytes as cell foam.
k1stress-like parameter of the model for the interlobular septa
k2dimensionless parameter of the model for the interlobular septa
kappafiber dispersion parameter \((0\le\kappa\le\frac{1}{3})\).
Mstructural tensor describing the fiber direction of the interlobular septa, i.e. \(M=v\otimes v\) for a fiber direction \(v\)
Finitial deformation gradient
Template Parameters
offsetnumber of rows/columns of F, this is only required to adjust the offset of the energy functional such that \(W(F)=0\) for \(F=I\).

§ incompressible_adipose_tissue_sommer_holzapfel() [2/2]

template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::incompressible_adipose_tissue_sommer_holzapfel ( const Mat &  A,
const Mat &  F 
)

Model for adipose tissue of [Sommer2013].

Implementation of the stored energy function \( W(F)= c_\mathrm{Cells}(\iota_1-3) + \frac{k_1}{k_2}\exp(k_2(\kappa\iota_1+(1-3\kappa)*\iota_4)^2-1) \), where \( \iota_1,\iota_4 \) are the first and first mixed invariant of the strain tensor \(F^T F\).

Material parameters are taken from the above mentioned publication, Table 2, i.e. \(c_\mathrm{Cells}=0.15 (\,\mathrm{kPa})\), \(k_1=0.8 (\,\mathrm{kPa})\), \(k_2=47.3\) and \(\kappa=0.09\).

Parameters
Mstructural tensor describing the fiber direction of the interlobular septa, i.e. \(M=v\otimes v\) for a fiber direction \(v\)
Finitial deformation gradient

§ incompressible_muscle_tissue_martins() [1/2]

template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::incompressible_muscle_tissue_martins ( double  c,
double  b,
double  d,
double  e,
const Mat &  A,
const Mat &  F 
)

Incompressible version of the model for muscle tissue of [Martins1998].

Implementation of the stored energy function \( W(F)=c(\exp(b(\bar\iota_1-3))-1) + A(\exp(a(\bar\iota_6-1)^2)-1) \), where \(\bar\iota_1,\bar\iota_6=\bar\iota_4\) are the first modified principal and the third modified mixed invariant of the strain tensor \(F^T F\).

Parameters
cfirst material parameter for the isotropic part
bsecond material parameter for the isotropic part
dfirst material parameter for the anisotropic part
esecond material parameter for the anisotropic part
Mstructural (rank-one) tensor describing the initial orientation of muscle fibers for \(F=I\), where \(I\) is the unit matrix.
Fdeformation gradient
Template Parameters
offsetnumber of rows/columns of F, this is only required to adjust the offset of the energy functional such that \(W(F)=0\) for \(F=I\).

§ incompressible_muscle_tissue_martins() [2/2]

template<linalg::Matrix Mat, int offset = linalg::dim< Mat >()>
auto funcy::incompressible_muscle_tissue_martins ( const Mat &  A,
const Mat &  F 
)

Incompressible version of the model for muscle tissue of [Martins1998].

Implementation of the stored energy function \( W(F)=c(\exp(b(\bar\iota_1-3))-1) + A(\exp(a(\bar\iota_6-1)^2)-1) \), where \(\bar\iota_1,\bar\iota_6=\bar\iota_4\) are the first modified principal and the third modified mixed invariant of the strain tensor \(F^T F\).

Material parameters taken from the same above mentioned publication, i.e. \(a=0.387 (\,\mathrm{kPa})\), \( b = 23.46 \), \( A = 0.584 (\,\mathrm{kPa}) \) and \( a = 12.43\).

Parameters
Mstructural (rank-one) tensor describing the initial orientation of muscle fibers for \(F=I\), where \(I\) is the unit matrix.
Fdeformation gradient

§ incompressible_skin_hendriks() [1/2]

template<linalg::Matrix Mat, int n = linalg::dim< Mat >()>
auto funcy::incompressible_skin_hendriks ( double  c0,
double  c1,
const Mat &  F 
)

Model for skin tissue of [Hendriks2005].

Implementation of the stored energy function \(W(F)=c_0(\iota_1-3) + c_1(\iota_1-3)(\iota_2-3)\), where \(\iota_1,\iota_2\) are the first and second principal invariants of the strain tensor \(F^T F\).

Parameters
c0scaling of the shifted first principal invariant
c1scaling of the product of shifted first and second principal invariant
Finitial deformation gradient

§ incompressible_skin_hendriks() [2/2]

template<linalg::Matrix Mat, int n = linalg::dim< Mat >()>
auto funcy::incompressible_skin_hendriks ( const Mat &  F)

Model for skin tissue of [Hendriks2005].

Implementation of the stored energy function \(W(F)=c_0(\iota_1-3) + c_1(\iota_1-3)(\iota_2-3)\), where \(\iota_1,\iota_2\) are the first and second principal invariants of the strain tensor \(F^T F\).

Material parameters are taken from [Xu2011], i.e \(c_0=9.4 (\,\mathrm{kPa})\) and \( c_1 = 82 (\,\mathrm{kPa}) \).

Parameters
Finitial deformation gradient