TransformationFunction

These models are part of the PlanarDamageModel system. They map the stress normal to a slip plane $\sigma_{\bot}^{\left(i\right)}$ and an internal damage variable, defined with a SlipPlaneDamage object, to a suitable damage metric ranging from 0 (no damage) to 1 (no stiffness in the direction).

Superclass description

class TransformationFunction : public neml::NEMLObject 

Transformation functions: map the damage variable + ancillary info into the range [0,1]

Subclassed by neml::SigmoidTransformation, neml::SwitchTransformation

Public Functions

TransformationFunction(ParameterSet &params)

virtual double map(double damage, double normal_stress) = 0: Map from damage and the normal stress to [0,1].

virtual double d_map_d_damage(double damage, double normal_stress) = 0: Derivative of the map with respect to the damage.

virtual double d_map_d_normal(double damage, double normal_stress) = 0: Derivative of the map with respect to the normal stress.

Individual models

SigmoidTransformation

Overview

This transformation function is independent of the normal stress. It is a sigmoid function that maps the damage variable to the interval $[0,1]$ :

$T\left(d^{\left(i\right)},\sigma_{\bot}^{\left(i\right)}\right)=\begin{cases} \frac{1}{1+\left(\frac{d^{\left(i\right)}}{c-d^{\left(i\right)}}\right)^{-\beta}} & d^{\left(i\right)}<c\\ 1 & d^{\left(i\right)}\ge c \end{cases}$

The parameter $c^{(i)}$ is some critical value of the damage variable $d^{(i)}$ on plane $i$ and the exponent $\beta^{(i)}$ is a smoothness parameter where $\beta = 1$ represents a linear transition from 0 to 1 and higher values of $\beta$ represent a more abrupt onset of the effects of damage.

Parameters

Parameter	Object type	Description	Default
`c`	`double`	Critical damage value	No
`beta`	:c:type`double`	Abruptness of damage onset	No

Class description

class SigmoidTransformation : public neml::TransformationFunction 

Sigmoid function. x=0 -> y=0, x=c -> y=1, beta controls smoothing.

Public Functions

SigmoidTransformation(ParameterSet &params)

virtual double map(double damage, double normal_stress): Map from damage and the normal stress to [0,1].

virtual double d_map_d_damage(double damage, double normal_stress): Derivative of the map with respect to damage.

virtual double d_map_d_normal(double damage, double normal_stress): Derivative of the map with respect to the normal stress.

Public Static Functions

static std::string type(): String type for the object system.

static std::unique_ptr<NEMLObject> initialize(ParameterSet &params): Initialize from a parameter set.

static ParameterSet parameters(): Default parameters.

SwitchTransformation

Overview

This is a “meta-transformation” that takes another TransformationFunction as a parameter and modifies it to return a different result. Specifically, this function returns 0 (no damage) if the loading on the plane is compressive and returns the base TransformationFunction value if the stress on the plane is tensile. Mathematically:

$T\left(d^{\left(i\right)},\sigma_{\bot}^{\left(i\right)}\right)=\begin{cases} \tilde{T}\left(d^{\left(i\right)},\sigma_{\bot}^{\left(i\right)}\right) & \sigma_{\bot}^{\left(i\right)}\ge0\\ 0 & \sigma_{\bot}^{\left(i\right)}<0 \end{cases}$

where $\tilde{T}\left(d^{\left(i\right)},\sigma_{\bot}^{\left(i\right)}\right)$ is the base TransformationFunction.

Parameters

Parameter	Object type	Description	Default
`base`	`neml::TransformationFunction`	Base function	No

Class Description

class SwitchTransformation : public neml::TransformationFunction 

Normal stress switch: don’t damage compression.

Public Functions

SwitchTransformation(ParameterSet &params)

virtual double map(double damage, double normal_stress): Map from damage and the normal stress to [0,1]

virtual double d_map_d_damage(double damage, double normal_stress): Derivative of the map with respect to damage.

virtual double d_map_d_normal(double damage, double normal_stress): Derivative of the map with respect to the normal stress.

Public Static Functions

static std::string type(): String type for the object system.

static std::unique_ptr<NEMLObject> initialize(ParameterSet &params): Initialize from a parameter set.

static ParameterSet parameters(): Default parameters.