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.

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

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

Functions

Detailed Description

Function Documentation

◆ compressible_adipose_tissue_sommer_holzapfel() [1/2]

◆ compressible_adipose_tissue_sommer_holzapfel() [2/2]

◆ compressible_muscle_tissue_martins() [1/2]

◆ compressible_muscle_tissue_martins() [2/2]

◆ compressible_skin_hendriks() [1/2]

◆ compressible_skin_hendriks() [2/2]

◆ incompressible_adipose_tissue_sommer_holzapfel() [1/2]

◆ incompressible_adipose_tissue_sommer_holzapfel() [2/2]

◆ incompressible_muscle_tissue_martins() [1/2]

◆ incompressible_muscle_tissue_martins() [2/2]

◆ incompressible_skin_hendriks() [1/2]

◆ incompressible_skin_hendriks() [2/2]