Softening Laws
Introduction
Softening laws are used in softening materials to determine crack tractions as a function of effective crack opening displacements. Each law defines a normalized function that decays from 1 to 0 as damage develops up to the point of failure. The area under the law (times an associated stress) gives the fracture toughness associated with that law. These laws also control the evolution of one damage parameter. Various softening materials will need two or more softening laws to model all possible types of damage.
Normalized Softening Law
The traction during softening is given by σf(δ) where f(δ) is the normalized softening law and σ is an initiation stress provided by the softening material. The toughness associated with the law, Gc, is determined by the area under the law times a scaling factor to account for mesh size and the initiation stress σ [1]:
[math]\displaystyle{ {G}_c = {V_p \sigma\over A_c} \int_0^{\delta_{max}}f(\delta)d\delta }[/math]
where Vp is particle volume, Ac is area of the initiated crack within the particle domain, and δmax is the critical cracking strain for failure (or for δ when the traction drops to zero). To define a scaling 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 evaulate 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{ A(\delta,s) = \int_0^{\delta}f(\delta,s)d\delta - {1\over2}\delta f(\delta,s) }[/math]
Each softening law requires Gc and possibly other parameters to define the law's functional form. The laws do not need to be provided δmax because it can be calculated from the law shape, s, and Gc from total area under the law. Furthermore, δmax depends on mesh size while Gc is a material property for damage.
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 translate into maximum particle size of
[math]\displaystyle{ \Delta x \le \eta \frac{G_c}{\sigma_0\varepsilon_0} }[/math]
where [math]\displaystyle{ \eta }[/math] is a softening-law-dependent stability factor. 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 |
|---|---|---|
| Linear | 1 | Linear softening law. |
| Exponential | 2 | Exponential softening law. |
| CubicStep | 3 | Cubic step function softening law. |
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.
References
- ↑ J. A. Nairn, C. Hammerquist, and Y. E. Aimene (2017), Numerical Implementation of Anisotropic Damage Mechanics, Int. J. for Numerical Methods in Engineering, 112(12), 1846-1868.