Softening Laws

Introduction

Softening laws are used in softening materials to determine crack tractions as a function of effective crack opening displacements (i.e., as a function of the current state of damage). Each law defines a normalized function that starts at 1 for no damage and reaches 0 at the point of failure (or decohesion). Although it is common for softening materials to use laws that monotonically decay from 1 to 0, it is not a thermodynamics requirement (and some available laws rise to a peak before decaying to 0). The unnormalized crack traction is this softening law times its associated strength (entered in a damage initiation iaws). The area under the unnormalized law gives the fracture toughness associated with that law.

Each softening law also determines the evolution of one damage parameter. Each softening material needs two or more softening laws to model all possible types of damage and evolve their associated damage parameters.

Normalized Softening Law

The traction during softening is given by σ*f(δ) where f(δ) is the normalized softening law and σ is an initiation stress provided by the softening material (in its damage initiation iaws). The toughness associated with the law, G_c, is determined by the area under the law times a scaling factor to account for mesh size and the initiation stress σ :

      [math]\displaystyle{ {G}_c = {V_p \sigma\over A_c} \int_0^{\delta_{max}}f(\delta)d\delta }[/math]

where V_p is particle volume, A_c is area of the initiated crack within the particle domain, and δ_max is the critical cracking strain for failure (or for δ when the traction drops to zero). To fully define a softening law, we define a scaling factor

      [math]\displaystyle{ s = {A_c \over V_p \sigma} }[/math]

The softening material calculates s whenever softening law response is needed. All the softening law needs to evaluate is the normalized function value, f(δ,s), and the area under that law up to δ minus area under a linear return to the origin or:

      [math]\displaystyle{ \Omega(\delta,s) = \int_0^{\delta}f(\delta,s)d\delta - {1\over2}\delta f(\delta,s) }[/math]

Each softening law requires G_c and possibly other parameters to define the law's functional form. The laws do not need provide δ_max because it can be calculated from the law shape, s, G_c, and possible other parameters be solving

      [math]\displaystyle{ s{G}_c = \Omega(\delta_{max},s) = \int_0^{\delta_{max}}f(\delta,s)d\delta }[/math]

for δ_max. The reason for this approach as that δ_max is not a material property, but rather depends on mesh size. In contrast, G_c is treated as a material property for damage (although it may have some mesh dependencies as well).

Law Stability

For damage propagation to always make physical sense, damage cracking strain must increase whenever the associated applied strain is increasing and is causing damage. This stability condition requires:

      [math]\displaystyle{ \max\bigl(-f'(\delta,s)\bigr) \lt {1\over \varepsilon_0} }[/math]

which translates into maximum particle size of

      [math]\displaystyle{ \Delta x \le \eta \frac{G_c}{\sigma_0\varepsilon_0} = \eta\left(\frac{K_c}{\sigma_0}\right)^2 \qquad{\rm where} \qquad \eta = \frac{1}{sG_c\max\bigl(-f'(\delta,s)\bigr)} }[/math]

is a softening-law-dependent stability factor, [math]\displaystyle{ \sigma_0 }[/math] and [math]\displaystyle{ \varepsilon_0 }[/math] are damage initiation properties, and [math]\displaystyle{ K_c }[/math] is the corresponding stress-intensity factor toughness. It is always possible to get stable results by high enough resolution (or small enough particle size). The need for high resolution is most common for brittle materials (low [math]\displaystyle{ G_c }[/math] with high [math]\displaystyle{ \sigma_0\varepsilon_0 }[/math]). The most stable softening law is Linear Softening for which [math]\displaystyle{ \eta=2 }[/math].

Softening Laws

You create a softening law using one of "Softening" properties allowed for the selected softening materials. You pick the law by name or ID. The following table lists the available softening laws.

Name	ID	Description	Stability [math]\displaystyle{ \eta }[/math]
Linear	1	Linear softening law	2
Exponential	2	Exponential softening law	1
CubicStep	3	Cubic function softening law	effectively from 0.79 to 4/3
DoubleExponential	4	Two exponential softening law	effectively 1 to 1.49

Click each one for more details and information on their properties. After picking the law, all its properties are set using property commands within the softening material definition by prefacing the law property with appropriate codes. For example, to set mode I and mode II toughnesses in an isotropic softening material, the properties are entered as I-Gc and II-Gc where I- or II- preface the Gc softening law property.

If needed, it is relatively easy to implement custom softening laws. In general, the specific choice of softening law is less important than implied in literature on damage mechanics. Perhaps some problems will arise where the law's details matter. It is possible, for example, that relative mode I and mode II laws will influence mixed-mode failure simulations.