Detailed Description

Rigid body material.

A material specifies the properties of a rigid body: the density of the body, the coefficient of restitution and the coefficients of static and dynamic friction.
The pe module provides several predefined materials that can be directly used:

iron
copper
granite
oak
fir

In order to create a new custom material use the createMaterial() function:

// Creating a new material using the following material properties:
// - name/identifier: myMaterial
// - density: 2.54
// - coefficient of restitution: 0.8
// - coefficient of static friction: 0.1
// - coefficient of dynamic friction: 0.05
// - Poisson's ratio: 0.2
// - Young's modulus: 80.0
// - Contact stiffness: 100
// - dampingN: 10
// - dampingT: 11
MaterialID myMaterial = createMaterial( "myMaterial", 2.54, 0.8, 0.1, 0.05, 0.2, 80, 100, 10, 11 );

The following functions can be used to acquire a specific MaterialID or to get a specific property of a material:

// Searching a material
MaterialID myMaterial = Material::find( "myMaterial" );
 
// Getting the density, coefficient of restitution, coefficient of static and
// dynamic friction, Poisson's ratio and Young's modulus of the material
real_t density = Material::getDensity( myMaterial );
real_t cor     = Material::getRestitution( myMaterial );
real_t csf     = Material::getStaticFriction( myMaterial );
real_t cdf     = Material::getDynamicFriction( myMaterial );
real_t poisson = Material::getPoissonRatio( myMaterial );
real_t young   = Material::getYoungModulus( myMaterial ):

#include <Materials.h>

Inheritance diagram for walberla::pe::Material:

Public Member Functions
Constructor
	Material (const std::string &name, real_t density, real_t cor, real_t csf, real_t cdf, real_t poisson, real_t young, real_t stiffness, real_t dampingN, real_t dampingT)
	The constructor of the Material class. More...

Static Public Member Functions
Set functions
static void	setRestitution (MaterialID material1, MaterialID material2, real_t cor)
	Setting the coefficient of restitution between material material1 and material2. More...

static void	setStaticFriction (MaterialID material1, MaterialID material2, real_t csf)
	Setting the coefficient of static friction between material material1 and material2. More...

static void	setDynamicFriction (MaterialID material1, MaterialID material2, real_t cdf)
	Setting the coefficient of dynamic friction between material material1 and material2. More...

Private Types
using	SizeType = Materials::size_type
	Size type of the Material class. More...

Static Private Member Functions
Setup functions
static bool	activateMaterials ()
	Automatic registration of the default materials. More...

Get functions
const std::string &	getName () const
	Returns the name of the material. More...

real_t	getDensity () const
	Returns the density of the material. More...

real_t	getRestitution () const
	Returns the coefficient of restitution of the material. More...

real_t	getStaticFriction () const
	Returns the coefficient of static friction of the material. More...

real_t	getDynamicFriction () const
	Returns the coefficient of dynamic friction of the material. More...

real_t	getPoissonRatio () const
	Returns the Poisson's ratio of the material. More...

real_t	getYoungModulus () const
	Returns the Young's modulus of the material. More...

real_t	getStiffness () const
	Returns the stiffness in normal direction of the material's contact region. More...

real_t	getDampingN () const
	Returns the damping coefficient in normal direction of the material's contact region. More...

real_t	getDampingT () const
	Returns the damping coefficient in tangential direction of the material's contact region. More...

static MaterialID	find (const std::string &name)
	Searching for a registered material. More...

static std::vector< MaterialID >	findPrefix (const std::string &prefix)
	Searching for registered materials with a prefix. More...

static const std::string &	getName (MaterialID material)
	Returns the name of the given material. More...

static real_t	getDensity (MaterialID material)
	Returns the density of the given material. More...

static real_t	getRestitution (MaterialID material)
	Returns the coefficient of restitution of the given material. More...

static real_t	getRestitution (MaterialID material1, MaterialID material2)
	Returns the composite coefficient of restitution for a collision between two rigid bodies. More...

static real_t	getStaticFriction (MaterialID material)
	Returns the coefficient of static friction of the given material. More...

static real_t	getStaticFriction (MaterialID material1, MaterialID material2)
	Returns the coefficient of static friction for a collision between two rigid bodies. More...

static real_t	getDynamicFriction (MaterialID material)
	Returns the coefficient of dynamic friction of the given material. More...

static real_t	getDynamicFriction (MaterialID material1, MaterialID material2)
	Returns the coefficient of dynamic friction for a collision between two rigid bodies. More...

static real_t	getPoissonRatio (MaterialID material)
	Returns the Poisson's ratio of the given material. More...

static real_t	getYoungModulus (MaterialID material)
	Returns the Young's modulus of the given material. More...

static real_t	getYoungModulus (MaterialID material1, MaterialID material2)
	Returns the (effective) Young's modulus for a collision between two rigid bodies. More...

static real_t	getStiffness (MaterialID material)
	Returns the stiffness in normal direction of the material's contact region. More...

static real_t	getStiffness (MaterialID material1, MaterialID material2)
	Returns the stiffness in normal direction of the contact between two materials. More...

static real_t	getDampingN (MaterialID material)
	Returns the damping coefficient in normal direction of the material's contact region. More...

static real_t	getDampingN (MaterialID material1, MaterialID material2)
	Returns the damping in normal direction of the contact between two materials. More...

static real_t	getDampingT (MaterialID material)
	Returns the damping coefficient in tangential direction of the material's contact region. More...

static real_t	getDampingT (MaterialID material1, MaterialID material2)
	Returns the damping in tangential direction of the contact between two materials. More...

static std::string	toString (const MaterialID &v)

Member variables
std::string	name_
	The name of the material. More...

real_t	density_
	The density of the material. More...

real_t	restitution_
	The coefficient of restitution (COR) of a self-similar collision \( [0..1] \). More...

real_t	static_
	The coefficient of static friction (CSF) \( [0..\infty) \). More...

real_t	dynamic_
	The coefficient of dynamic friction (CDF) \( [0..\infty) \). More...

real_t	poisson_
	The Poisson's ratio for the material \( [-1..0.5] \). More...

real_t	young_
	The Young's modulus for the material \( (0..\infty) \). More...

real_t	stiffness_
	The stiffness of the contact region \( (0..\infty) \). More...

real_t	dampingN_
	The damping at the contact region in normal direction \( [0..\infty) \). More...

real_t	dampingT_
	The damping at the contact region in tangential direction \( [0..\infty) \). More...

static Materials	materials_
	Vector for the registered materials. More...

static MatN	corTable_
	Table for the coefficients of restitution. More...

static MatN	csfTable_
	Table for the coefficients of static friction. More...

static MatN	cdfTable_
	Table for the coefficients of dynamic friction. More...

static bool	materialsActivated_
	Helper variable for the automatic registration process. More...

static unsigned int	anonymousMaterials_ = 0
	Counter for the amount of anonymous materials. More...

Member Typedef Documentation

◆ SizeType

using walberla::pe::Material::SizeType = Materials::size_type

private

Size type of the Material class.

Constructor & Destructor Documentation

◆ Material()

walberla::pe::Material::Material	(	const std::string &	name,
		real_t	density,
		real_t	cor,
		real_t	csf,
		real_t	cdf,
		real_t	poisson,
		real_t	young,
		real_t	stiffness,
		real_t	dampingN,
		real_t	dampingT
	)

inlineexplicit

The constructor of the Material class.

Parameters

name	The name of the material.
density	The density of the material \( (0..\infty) \).
cor	The coefficient of restitution (COR) of the material \( [0..1] \).
csf	The coefficient of static friction (CSF) of the material \( [0..\infty) \).
cdf	The coefficient of dynamic friction (CDF) of the material \( [0..\infty) \).
poisson	The Poisson's ratio of the material \( [-1..0.5] \).
young	The Young's modulus of the material \( (0..\infty) \).
stiffness	The stiffness in normal direction of the material's contact region.
dampingN	The damping coefficient in normal direction of the material's contact region.
dampingT	The damping coefficient in tangential direction of the material's contact region.

Member Function Documentation

◆ activateMaterials()

bool walberla::pe::Material::activateMaterials ( )

staticprivate

Automatic registration of the default materials.

Returns: true after the materials have been registered.

◆ find()

MaterialID walberla::pe::Material::find ( const std::string & name )

static

Searching for a registered material.

Parameters

name	The name of the material.

Returns: The MaterialID of the material if the material is found, invalid_material otherwise.

The function searches for a registered material with the given name. If the material is found, the corresponding MaterialID is returned. Otherwise, invalid_material is returned.

◆ findPrefix()

std::vector< MaterialID > walberla::pe::Material::findPrefix ( const std::string & prefix )

static

Searching for registered materials with a prefix.

Parameters

prefix The prefix common to the names of the materials.

Returns: A std::vector object containing the MaterialIDs of all materials found.

The function collects all registered materials with names beginning with the given string. Their IDs are assembled in an std::vector object. If no Materials are found, the container is empty.

◆ getDampingN() [1/3]

real_t walberla::pe::Material::getDampingN ( ) const

inline

Returns the damping coefficient in normal direction of the material's contact region.

Returns: The damping coefficient in normal direction of the material's contact region.

◆ getDampingN() [2/3]

real_t walberla::pe::Material::getDampingN ( MaterialID material )

inlinestatic

Returns the damping coefficient in normal direction of the material's contact region.

Parameters

material The material to be queried.

Returns: The damping in normal direction of the contact region of the given material.

◆ getDampingN() [3/3]

real_t walberla::pe::Material::getDampingN	(	MaterialID	material1,
		MaterialID	material2
	)

inlinestatic

Returns the damping in normal direction of the contact between two materials.

Parameters

material1	The material of the first colliding rigid body.
material2	The material of the second colliding rigid body.

Returns: The damping in normal direction of the contact between two materials.

Rigid body theory assumes that deformation during contact is localized to the contact region. Therefore the contact region is often modelled simplified as a spring-damper. When two bodies are in contact the spring-dampers are serially connected and thus the contact damping can be expressed as the series connection of two viscous dampers: \( c_*^{-1} = c_1^{-1} + c_2^{-1}\).

◆ getDampingT() [1/3]

real_t walberla::pe::Material::getDampingT ( ) const

inline

Returns the damping coefficient in tangential direction of the material's contact region.

Returns: The damping coefficient in tangential direction of the material's contact region.

◆ getDampingT() [2/3]

real_t walberla::pe::Material::getDampingT ( MaterialID material )

inlinestatic

Returns the damping coefficient in tangential direction of the material's contact region.

Parameters

material The material to be queried.

Returns: The damping in tangential direction of the contact region of the given material.

◆ getDampingT() [3/3]

real_t walberla::pe::Material::getDampingT	(	MaterialID	material1,
		MaterialID	material2
	)

inlinestatic

Returns the damping in tangential direction of the contact between two materials.

Parameters

material1	The material of the first colliding rigid body.
material2	The material of the second colliding rigid body.

Returns: The damping in tangential direction of the contact between two materials.

Rigid body theory assumes that deformation during contact is localized to the contact region. Therefore the contact region is often modelled simplified as a spring-damper. When two bodies are in contact the spring-dampers are serially connected and thus the contact damping can be expressed as the series connection of two viscous dampers: \( c_*^{-1} = c_1^{-1} + c_2^{-1}\).

◆ getDensity() [1/2]

real_t walberla::pe::Material::getDensity ( ) const

inline

Returns the density of the material.

Returns: The density of the material.

◆ getDensity() [2/2]

real_t walberla::pe::Material::getDensity ( MaterialID material )

inlinestatic

Returns the density of the given material.

Parameters

material The material to be queried.

Returns: The density of the given material.

◆ getDynamicFriction() [1/3]

real_t walberla::pe::Material::getDynamicFriction ( ) const

inline

Returns the coefficient of dynamic friction of the material.

Returns: The coefficient of dynamic friction of the material.

◆ getDynamicFriction() [2/3]

real_t walberla::pe::Material::getDynamicFriction ( MaterialID material )

inlinestatic

Returns the coefficient of dynamic friction of the given material.

Parameters

material The material to be queried.

Returns: The coefficient of dynamic friction of the given material.

◆ getDynamicFriction() [3/3]

real_t walberla::pe::Material::getDynamicFriction	(	MaterialID	material1,
		MaterialID	material2
	)

inlinestatic

Returns the coefficient of dynamic friction for a collision between two rigid bodies.

Parameters

material1	The material of the first colliding rigid body.
material2	The material of the second colliding rigid body.

Returns: The resulting coefficient of dynamic friction of the collision.

◆ getName() [1/2]

const std::string & walberla::pe::Material::getName ( ) const

inline

Returns the name of the material.

Returns: The name of the material.

◆ getName() [2/2]

const std::string & walberla::pe::Material::getName ( MaterialID material )

inlinestatic

Returns the name of the given material.

Parameters

material The material to be queried.

Returns: The name of the given material.

◆ getPoissonRatio() [1/2]

real_t walberla::pe::Material::getPoissonRatio ( ) const

inline

Returns the Poisson's ratio of the material.

Returns: The Poisson's ratio of the material.

◆ getPoissonRatio() [2/2]

real_t walberla::pe::Material::getPoissonRatio ( MaterialID material )

inlinestatic

Returns the Poisson's ratio of the given material.

Parameters

material The material to be queried.

Returns: The Poisson's ratio of the given material.

◆ getRestitution() [1/3]

real_t walberla::pe::Material::getRestitution ( ) const

inline

Returns the coefficient of restitution of the material.

Returns: The coefficient of restitution of the material.

◆ getRestitution() [2/3]

real_t walberla::pe::Material::getRestitution ( MaterialID material )

inlinestatic

Returns the coefficient of restitution of the given material.

Parameters

material The material to be queried.

Returns: The coefficient of restitution of the given material.

◆ getRestitution() [3/3]

real_t walberla::pe::Material::getRestitution	(	MaterialID	material1,
		MaterialID	material2
	)

inlinestatic

Returns the composite coefficient of restitution for a collision between two rigid bodies.

Parameters

material1	The material of the first colliding rigid body.
material2	The material of the second colliding rigid body.

Returns: The resulting composite coefficient of restitution of the collision.

◆ getStaticFriction() [1/3]

real_t walberla::pe::Material::getStaticFriction ( ) const

inline

Returns the coefficient of static friction of the material.

Returns: The coefficient of static friction of the material.

◆ getStaticFriction() [2/3]

real_t walberla::pe::Material::getStaticFriction ( MaterialID material )

inlinestatic

Returns the coefficient of static friction of the given material.

Parameters

material The material to be queried.

Returns: The coefficient of static friction of the given material.

◆ getStaticFriction() [3/3]

real_t walberla::pe::Material::getStaticFriction	(	MaterialID	material1,
		MaterialID	material2
	)

inlinestatic

Returns the coefficient of static friction for a collision between two rigid bodies.

Parameters

material1	The material of the first colliding rigid body.
material2	The material of the second colliding rigid body.

Returns: The resulting coefficient of static friction of the collision.

◆ getStiffness() [1/3]

real_t walberla::pe::Material::getStiffness ( ) const

inline

Returns the stiffness in normal direction of the material's contact region.

Returns: The stiffness in normal direction of the material's contact region.

◆ getStiffness() [2/3]

real_t walberla::pe::Material::getStiffness ( MaterialID material )

inlinestatic

Returns the stiffness in normal direction of the material's contact region.

Parameters

material The material to be queried.

Returns: The stiffness in normal direction of the contact region of the given material.

◆ getStiffness() [3/3]

real_t walberla::pe::Material::getStiffness	(	MaterialID	material1,
		MaterialID	material2
	)

inlinestatic

Returns the stiffness in normal direction of the contact between two materials.

Parameters

material1	The material of the first colliding rigid body.
material2	The material of the second colliding rigid body.

Returns: The stiffness in normal direction of the contact between two materials.

Rigid body theory assumes that deformation during contact is localized to the contact region. Therefore the contact region is often modelled simplified as a spring-damper. When two bodies are in contact the spring-dampers are serially connected and thus the contact stiffness can be expressed as the series connection of two springs: \( k_*^{-1} = k_1^{-1} + k_2^{-1}\).

◆ getYoungModulus() [1/3]

real_t walberla::pe::Material::getYoungModulus ( ) const

inline

Returns the Young's modulus of the material.

Returns: The Young's modulus of the material.

◆ getYoungModulus() [2/3]

real_t walberla::pe::Material::getYoungModulus ( MaterialID material )

inlinestatic

Returns the Young's modulus of the given material.

Parameters

material The material to be queried.

Returns: The Young's modulus of the given material.

◆ getYoungModulus() [3/3]

real_t walberla::pe::Material::getYoungModulus	(	MaterialID	material1,
		MaterialID	material2
	)

inlinestatic

Returns the (effective) Young's modulus for a collision between two rigid bodies.

Parameters

material1	The material of the first colliding rigid body.
material2	The material of the second colliding rigid body.

Returns: The resulting (effective) Young's modulus of the collision.

This function returns the effective Young's modulus for a collision between two rigid bodies. The effective Young's modulus is calculated as

   \f[ \frac{1}{E_{eff}} = \frac{1 - \nu_1^2}{E_1} + \frac{1 - \nu_2^2}{E_2}, \f]

where \( E_1 \) and \( E_2 \) are the Young's modulus for the first and second material, respectively, and \( \nu_1 \) and \( \nu_2 \) are the Poisson's ratio for the materials.

◆ setDynamicFriction()

void walberla::pe::Material::setDynamicFriction	(	MaterialID	material1,
		MaterialID	material2,
		real_t	cdf
	)

inlinestatic

Setting the coefficient of dynamic friction between material material1 and material2.

Parameters

material1	The material of the first colliding rigid body.
material2	The material of the second colliding rigid body.
cdf	The coefficient of dynamic friction between material1 and material2.

Returns: void

◆ setRestitution()

void walberla::pe::Material::setRestitution	(	MaterialID	material1,
		MaterialID	material2,
		real_t	cor
	)

inlinestatic

Setting the coefficient of restitution between material material1 and material2.

Parameters

material1	The material of the first colliding rigid body.
material2	The material of the second colliding rigid body.
cor	The coefficient of restitution between material1 and material2.

Returns: void

◆ setStaticFriction()

void walberla::pe::Material::setStaticFriction	(	MaterialID	material1,
		MaterialID	material2,
		real_t	csf
	)

inlinestatic

Setting the coefficient of static friction between material material1 and material2.

Parameters

material1	The material of the first colliding rigid body.
material2	The material of the second colliding rigid body.
csf	The coefficient of static friction between material1 and material2.

Returns: void

◆ toString()

std::string walberla::pe::Material::toString ( const MaterialID & v )

static

Member Data Documentation

◆ anonymousMaterials_

unsigned int walberla::pe::Material::anonymousMaterials_ = 0

staticprivate

Counter for the amount of anonymous materials.

◆ cdfTable_

MatN walberla::pe::Material::cdfTable_

staticprivate

Table for the coefficients of dynamic friction.

◆ corTable_

MatN walberla::pe::Material::corTable_

staticprivate

Table for the coefficients of restitution.

◆ csfTable_

MatN walberla::pe::Material::csfTable_

staticprivate

Table for the coefficients of static friction.

◆ dampingN_

real_t walberla::pe::Material::dampingN_

private

The damping at the contact region in normal direction \( [0..\infty) \).

Rigid body theory assumes that the deformation during contact is localized to the contact region. This local compliance in normal direction can be modelled simplified as a spring-damper. The viscous damping coefficient corresponds to this parameter.

◆ dampingT_

real_t walberla::pe::Material::dampingT_

private

The damping at the contact region in tangential direction \( [0..\infty) \).

Friction counteracts the tangential relative velocity and thus can be modelled as a viscous damper with a limited damping force. The viscous damping coefficient corresponds to this parameter.

◆ density_

real_t walberla::pe::Material::density_

private

The density of the material.

◆ dynamic_

real_t walberla::pe::Material::dynamic_

private

The coefficient of dynamic friction (CDF) \( [0..\infty) \).

The CDF is a dimensionless, non-negative quantity representing the amount of dynamic friction between two touching rigid bodies. Dynamic friction occurs in case the relative tangential velocity between the two bodies is greater than 0. Then the force magnitudes of the normal and friction force are related by an inequality:

\[ |\vec{f_t}| = -\mu_d |\vec{f_n}| \frac{\vec{v_t}}{|\vec{v_t}|} \]

◆ materials_

Materials walberla::pe::Material::materials_

staticprivate

Vector for the registered materials.

◆ materialsActivated_

bool walberla::pe::Material::materialsActivated_

staticprivate

Helper variable for the automatic registration process.

◆ name_

std::string walberla::pe::Material::name_

private

The name of the material.

◆ poisson_

real_t walberla::pe::Material::poisson_

private

The Poisson's ratio for the material \( [-1..0.5] \).

When a material is compressed in one direction, it usually tends to expand in the other two directions perpendicular to the direction of compression. This effect is called Poisson effect. In this context, the Poisson's ratio is the ratio of the contraction or transverse strain (perpendicular to the applied load) to the extension or axial strain (in the direction of the applied load). For stable, isotropic, linear elastic materials this ratio cannot be less than -1.0 nor greater than 0.5 due to the requirement that Young's modulus has positive values.

◆ restitution_

real_t walberla::pe::Material::restitution_

private

The coefficient of restitution (COR) of a self-similar collision \( [0..1] \).

The COR represents the energy dissipated during a collision between self-similar bodies, that is bodies with similar materials. A value of 0 corresponds to completely inelastic collision where all energy is dissipated, a value of 1 corresponds to a completely elastic collision where no energy is lost. The COR is assumed to be rate-independent. The COR is often determined experimentally by measuring the pre- and post-impact relative velocities:

\[ C_R = \frac{V_{2,after}-V_{1,after}}{V_{2,before}-V_{1,before}} \]

During a collision, the COR values of the two colliding rigid bodies can be used by the collision response mechanism to determine the restitution factor of the contact point.

◆ static_

real_t walberla::pe::Material::static_

private

The coefficient of static friction (CSF) \( [0..\infty) \).

The CSF is a dimensionless, non-negative quantity representing the amount of static friction between two touching rigid bodies. Static friction occurs in case the relative tangential velocity between the two bodies is 0. Then the force magnitudes of the normal and friction force are related by an inequality:

\[ |\vec{f_t}| \leq \mu_s |\vec{f_n}| \]

The direction of the friction must oppose acceleration if sliding is imminent and is unrestricted otherwise.

◆ stiffness_

real_t walberla::pe::Material::stiffness_

private

The stiffness of the contact region \( (0..\infty) \).

Rigid body theory assumes that the deformation during contact is localized to the contact region. This local compliance can be modelled simplified as a spring-damper. The spring constant corresponds to this parameter.

◆ young_

real_t walberla::pe::Material::young_

private

The Young's modulus for the material \( (0..\infty) \).

The Young's modulus is a measure for the stiffness of an isotropic elastic material. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's law holds. The SI unit for Young's modulus is \( Pa \) or \( N/m^2 \).

The documentation for this class was generated from the following files:

/builds/administration/walberla-website/walberla/src/pe/Materials.h
/builds/administration/walberla-website/walberla/src/pe/Materials.cpp

Detailed Description

Public Member Functions

Static Public Member Functions

Private Types

Static Private Member Functions

Get functions

Member variables

Member Typedef Documentation

◆ SizeType

Constructor & Destructor Documentation

◆ Material()

Member Function Documentation

◆ activateMaterials()

◆ find()

◆ findPrefix()

◆ getDampingN() [1/3]

◆ getDampingN() [2/3]

◆ getDampingN() [3/3]

◆ getDampingT() [1/3]

◆ getDampingT() [2/3]

◆ getDampingT() [3/3]

◆ getDensity() [1/2]

◆ getDensity() [2/2]

◆ getDynamicFriction() [1/3]

◆ getDynamicFriction() [2/3]

◆ getDynamicFriction() [3/3]

◆ getName() [1/2]

◆ getName() [2/2]

◆ getPoissonRatio() [1/2]

◆ getPoissonRatio() [2/2]

◆ getRestitution() [1/3]

◆ getRestitution() [2/3]

◆ getRestitution() [3/3]

◆ getStaticFriction() [1/3]

◆ getStaticFriction() [2/3]

◆ getStaticFriction() [3/3]

◆ getStiffness() [1/3]

◆ getStiffness() [2/3]

◆ getStiffness() [3/3]

◆ getYoungModulus() [1/3]

◆ getYoungModulus() [2/3]

◆ getYoungModulus() [3/3]

◆ setDynamicFriction()

◆ setRestitution()

◆ setStaticFriction()

◆ toString()

Member Data Documentation

◆ anonymousMaterials_

◆ cdfTable_

◆ corTable_

◆ csfTable_

◆ dampingN_

◆ dampingT_

◆ density_

◆ dynamic_

◆ materials_

◆ materialsActivated_

◆ name_

◆ poisson_

◆ restitution_

◆ static_

◆ stiffness_

◆ young_