A Class implementing regularized nonlinear peridynamic model. More...

#include <rnpBond.h>

Inheritance diagram for material::pd::RNPBond:

Collaboration diagram for material::pd::RNPBond:

Public Member Functions
std::pair< util::Point3, double >	getBondEF (size_t i, size_t j)
	Returns energy and force state between node i and node j. More...

util::Point3	getBondForceDirection (const util::Point3 &dx, const util::Point3 &du) const
	Get direction of bond force. More...

double	getInfFn (const double &r) const
	Returns the value of influence function. More...

double	getS (const util::Point3 &dx, const util::Point3 &du)
	Returns the bond strain. More...

double	getS (size_t i, size_t j)
	Returns the bond strain. More...

double	getSc (const double &r)
	Returns critical bond strain. More...

double	getSc (size_t i, size_t j)
	Returns critical bond strain. More...

	RNPBond (inp::MaterialDeck deck, data::DataManager dataManager)
	Constructor. More...

Public Member Functions inherited from material::pd::BaseMaterial
	BaseMaterial (const size_t &dim, const double &horizon)
	Constructor. More...

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

double	getHorizon () const
	Returns horizon. More...

virtual util::Matrix33	getStrain (size_t i)
	Returns strain tensor. More...

virtual util::Matrix33	getStress (size_t i)
	Returns stress tensor. More...

bool	isStateActive () const
	Returns true if state-based potential is active. More...

std::string	name () const
	Returns name of the material. More...

virtual void	print (int nt=0, int lvl=0) const
	Prints the information about the instance of the object. More...

virtual std::string	printStr (int nt=0, int lvl=0) const
	Returns the string containing information about the instance of the object. More...

virtual void	update ()
	Let the material class in the quasi-static case know that there is a new loading step.

Private Member Functions
void	computeMaterialProperties (inp::MaterialDeck *deck, const double &M)
	Computes elastic and fracture properties from the rnp material parameters. More...

void	computeParameters (inp::MaterialDeck *deck, const double &M)
	Computes rnp material parameters from elastic constants. More...

Private Attributes
material::pd::BaseInfluenceFn *	d_baseInfluenceFn_p
	Base object for influence function.

double	d_contact_Kn = 0.
	Kn coefficient for normal contact force between broken bonds.

double	d_contact_Rc = 0.
	Contact radius for normal contact force between broken bonds.

data::DataManager *	d_dataManager_p
	Store pointer to datamanager.

const inp::MaterialDeck *	d_deck
	Pointer to the material deck.

double	d_factorSc
	Factor to multiply to critical strain to check if bond is fractured. More...

double	d_invFactor
	Inverse of factor = $\epsilon \|B_\epsilon(0)\|$ .

bool	d_irrevBondBreak
	Flag which indicates if the breaking of bond is irreversible.

double	d_rbar
	Inflection point of nonlinear function = $1/\sqrt{2\beta}$ .

Material parameters
double	d_C
	Parameter C.

double	d_beta
	Parameter $\beta$ .

Detailed Description

A Class implementing regularized nonlinear peridynamic model.

Provides method to compute energy and force using nonlinear bond-based model introduced and studied in Lipton 2016, Jha and Lipton 2018, and Lipton and Jha 2019.

The pairwise energy is
$E_{bond} = \int_D \frac{1}{|B_\epsilon(x)|} \int_{B_\epsilon(x)} |y-x| W_{bond}(S(y,x; u)) dy dx$
where $B_\epsilon(x)$ is a ball (circle in 2-d) centered at of radius $\epsilon$ , $|B_\epsilon(x)|$ is the volume (area in 2-d) of ball, $S(y,x;u) = \frac{u(y) - u(x)}{|y-x|} \cdot \frac{y-x}{|y-x|}$ is the linearized bond strain (assuming small deformation).
Bond energy density $W_{bond}$ is given by
$W_{bond} (S(y,x;u)) = J^\epsilon(|y-x|) \frac{1}{\epsilon |y-x|} \psi (|y-x| S(y,x;u)^2)$
where $\psi(r) : \textsf{R}^+ \to \textsf{R}$ is positive, smooth, concave function with following properties
$\lim_{r\to 0^+} \frac{\psi(r)}{r} = \psi'(0), \quad \lim_{r\to \infty} \psi(r) = \psi_\infty < \infty.$
$J^\epsilon(|y-x|)$ is the influence function.
Force at material point is given by
$f_{bond}(x) = \frac{4}{|B_\epsilon(x)|} \int_{B_\epsilon(x)} \frac{J^\epsilon(|y-x|)}{\epsilon} \psi'(|y-x| S(y,x;u)^2) S(y,x;u) \frac{y-x}{|y-x|} dy.$
In this class, we will assume
$\psi(r) = C ( 1-\exp[-\beta r]$
where $C, \beta$ are the peridynamic material parameter determined from the elastic and fracture properties of the material.

Constructor & Destructor Documentation

◆ RNPBond()

material::pd::RNPBond::RNPBond	(	inp::MaterialDeck *	deck,
		data::DataManager *	dataManager
	)

Constructor.

Parameters

deck	Pointer to the input deck
DataManager	Pointer to the data manager object

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Member Function Documentation

◆ computeMaterialProperties()

void material::pd::RNPBond::computeMaterialProperties	(	inp::MaterialDeck *	deck,
		const double &	M
	)

private

Computes elastic and fracture properties from the rnp material parameters.

This function does opposite of pd::Material::RNPBond::computeParameters. From peridynamic material properties, it uses the relation between lame parameters and peridynamic parameters to compute the lame parameters, and from lame parameters it computes the elastic constants.

Parameters

deck	Input material deck
M	Moment of influence function

Referenced by RNPBond().

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◆ computeParameters()

void material::pd::RNPBond::computeParameters	(	inp::MaterialDeck *	deck,
		const double &	M
	)

private

Computes rnp material parameters from elastic constants.

Either Young's modulus E or bulk modulus K, and either critical energy release rate Gc or critical stress intensity factor KIc are needed. Assuming Poisson's ratio $\nu = \frac{1}{4}$ we compute lame parameters $\lambda, \mu$ where $\mu = \lambda$ for bond-based.

With lame parameters, we use following formula, see Equation (5.7) & (5 .8) of Lipton 2016

if
$\lambda = \mu = \frac{\psi'(0)}{4} M_2, \qquad Gc = \frac{4 \psi_{\infty}}{\pi} M_2.$
if
$\lambda = \mu = \frac{\psi'(0)}{5} M_3, \qquad Gc = \frac{3 \psi_{\infty}}{2} M_3.$

Where $M_2, M_3$ are defined by

$M_2 =\int_0^1 r^2 J(r) dr, \qquad M_3 = \int_0^1 r^3 J(r) dr.$

For potential function $\psi(r) = c ( 1-\exp[-\beta r])$ , we have $\psi'(0) = c\beta, \psi_{\infty} = c$ . Thus, the values of $c, \beta$ are given by

if
$c = \frac{\pi G_c}{4} \frac{1}{M_2}, \qquad \beta = \frac{4 \lambda}{c} \frac{1}{M_2} .$
if
$c = \frac{2 G_c}{3} \frac{1}{M_3}, \qquad \beta = \frac{5 \lambda}{c} \frac{1}{M_3} .$

Parameters

deck Input material deck

M Moment of influence function