Functions for the description of isotropic elastic behavior. More...

Functions
constexpr double	E (const double K, const double G)

constexpr double	nu (const double K, const double G)

constexpr double	shearModulus (const double E, const double nu)

constexpr double	lameParameter (const double E, const double nu)

Matrix6d	complianceTensor (const double E, const double nu)

Matrix6d	stiffnessTensor (const double E, const double nu)

Matrix6d	stiffnessTensorKG (const double K, const double G)

Detailed Description

Functions for the description of isotropic elastic behavior.

Function Documentation

◆ E()

constexpr double Marmot::ContinuumMechanics::Elasticity::Isotropic::E	(	const double	K,
		const double	G
	)

constexpr

Computes the isotropic young's modulus E from the compression modulus K and shear modulus G

\[ \displaystyle E = \frac{9\,K\,G}{3\,K + G)} \]

◆ nu()

constexpr double Marmot::ContinuumMechanics::Elasticity::Isotropic::nu	(	const double	K,
		const double	G
	)

constexpr

Computes the isotropic poisson's ratio \( \nu \) from the compression modulus K and shear modulus G

\[ \displaystyle \nu = \frac{3\,K - 2\,G}{6\,K + 2\,G)} \]

◆ shearModulus()

constexpr double Marmot::ContinuumMechanics::Elasticity::Isotropic::shearModulus	(	const double	E,
		const double	nu
	)

constexpr

Computes the isotropic shear modulus G from the young's modulus E and poisson's ratio \( \nu \).

\[ \displaystyle G = \frac{E}{2\,(1 + \nu)} \]

◆ lameParameter()

constexpr double Marmot::ContinuumMechanics::Elasticity::Isotropic::lameParameter	(	const double	E,
		const double	nu
	)

constexpr

Computes the isotropic lame parameter \( \lambda \) from the young's modulus E and poisson's ratio \( \nu \)

\[ \displaystyle \lambda = \frac{E\,\nu}{(1 + \nu)(1 - 2\,\nu)} \]

◆ complianceTensor()

Matrix6d Marmot::ContinuumMechanics::Elasticity::Isotropic::complianceTensor	(	const double	E,
		const double	nu
	)

Computes the isotropic compliance tensor:

\[ \displaystyle \mathbb{ C }^{-1} = \begin{bmatrix} \frac{1}{E} & \frac{-\nu}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \\ \frac{-\nu}{E} & \frac{1}{E} & \frac{-\nu}{E} & 0 & 0 & 0 \\ \frac{-\nu}{E} & \frac{-\nu}{E} & \frac{1}{E} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{G} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{G} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{G} \end{bmatrix} \]

from the young's modulus E and the poisson's ratio \( \nu \) with

\[ \displaystyle G = \frac{E}{2\,(1 + \nu)} \]

◆ stiffnessTensor()

Matrix6d Marmot::ContinuumMechanics::Elasticity::Isotropic::stiffnessTensor	(	const double	E,
		const double	nu
	)

Computes the isotropic stiffness tensor:

\[ \displaystyle \mathbb{ C } = \frac{E\,(1-\nu)}{(1+\nu)(1-2\,\nu)} \begin{bmatrix} 1 & \frac{\nu}{1-\nu} & \frac{\nu}{1-\nu} & 0 & 0 & 0 \\ \frac{\nu}{1-\nu} & 1 & \frac{\nu}{1-\nu} & 0 & 0 & 0 \\ \frac{\nu}{1-\nu} & \frac{\nu}{1-\nu} & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1-2\,\nu}{2\,(1-\nu)} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1-2\,\nu}{2\,(1-\nu)} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1-2\,\nu}{2\,(1-\nu)} \end{bmatrix} \]

from the young's modulus E and the poisson's ratio \( \nu \).

◆ stiffnessTensorKG()

Matrix6d Marmot::ContinuumMechanics::Elasticity::Isotropic::stiffnessTensorKG	(	const double	K,
		const double	G
	)

Computes the isotropic stiffness tensor \mathbb{ C } from the bulk modulus K and the shear modulus G.