funcy  1.6.1
Biomechanics

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

Collaboration diagram for Biomechanics: ## 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.

## ◆ 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
 cCells scaling of the neo-Hookean model for the description of the adipocytes as cell foam. k1 stress-like parameter of the model for the interlobular septa k2 dimensionless parameter of the model for the interlobular septa kappa fiber dispersion parameter $$(0\le\kappa\le\frac{1}{3})$$. M structural tensor describing the fiber direction of the interlobular septa, i.e. $$M=v\otimes v$$ for a fiber direction $$v$$ d0 scaling of the penalty function for inflation d1 scaling of the penalty function for compression F initial 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
 d0 scaling of the penalty function for inflation d1 scaling of the penalty function for compression M structural tensor describing the fiber direction of the interlobular septa, i.e. $$M=v\otimes v$$ for a fiber direction $$v$$ F initial 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
 c first material parameter for the isotropic part b second material parameter for the isotropic part A first material parameter for the anisotropic part a second material parameter for the anisotropic part d0 material parameter for the penalty for inflation d1 material parameter for the penalty for compression M structural (rank-one) tensor describing the initial orientation of muscle fibers for $$F=I$$, where $$I$$ is the unit matrix. F deformation 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
 d0 material parameter for the penalty for inflation d1 material parameter for the penalty for compression M structural (rank-one) tensor describing the initial orientation of muscle fibers for $$F=I$$, where $$I$$ is the unit matrix. F deformation 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
 c0 scaling of the shifted first principal invariant c1 scaling of the product of shifted first and second principal invariant d0 scaling of the penalty function for inflation d1 scaling of the penalty function for compression F initial 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
 d0 scaling of the penalty function for inflation d1 scaling of the penalty function for compression F initial 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
 cCells scaling of the neo-Hookean model for the description of the adipocytes as cell foam. k1 stress-like parameter of the model for the interlobular septa k2 dimensionless parameter of the model for the interlobular septa kappa fiber dispersion parameter $$(0\le\kappa\le\frac{1}{3})$$. M structural tensor describing the fiber direction of the interlobular septa, i.e. $$M=v\otimes v$$ for a fiber direction $$v$$ F initial deformation gradient
Template Parameters
 offset number 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
 M structural tensor describing the fiber direction of the interlobular septa, i.e. $$M=v\otimes v$$ for a fiber direction $$v$$ F initial 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
 c first material parameter for the isotropic part b second material parameter for the isotropic part d first material parameter for the anisotropic part e second material parameter for the anisotropic part M structural (rank-one) tensor describing the initial orientation of muscle fibers for $$F=I$$, where $$I$$ is the unit matrix. F deformation gradient
Template Parameters
 offset number 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
 M structural (rank-one) tensor describing the initial orientation of muscle fibers for $$F=I$$, where $$I$$ is the unit matrix. F deformation 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
 c0 scaling of the shifted first principal invariant c1 scaling of the product of shifted first and second principal invariant F initial 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
 F initial deformation gradient