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 some scaling 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 scaling stress σ:
[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 (of for 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]
or maximum slope of the softening law is less then one over strain to initiate that damage process. In general, the maximum slope will be proportional to particle size. Thus, it is always possible to get stable softening laws by high enough resolution (or small enough particle size). The need for high resolution is most common for brittle materials.
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.
Softening Law Implementation
It is easy to add new softening laws, although not clear if that change would be much different then using the softening laws already available. As explained in Ref. [1], the addition of a softening only requires code to return:
[math]\displaystyle{ f(\delta,s), \quad f'(\delta,s), \quad A(\delta,s), \quad{\rm and}\quad\max\bigl(-f'(\delta,s)\bigr) }[/math]
One internal calculation is to evolve damage. That calculation can be done from the above softening law properties by solving the discrete differential equation[1]
[math]\displaystyle{ d\varepsilon = d\delta + \varepsilon_0 \bigl(f(\delta+d\delta,s) - f(\delta,s)\bigr) }[/math]
where [math]\displaystyle{ \delta }[/math] is a maximum cracking strain associated with strain increment [math]\displaystyle{ \varepsilon }[/math]. The task is to solve for [math]\displaystyle{ d\delta }[/math] or the increment in cracking strain, which represents the evolution of damage in the current time step. The generic softening law code solves this equation using Newton's method with bracketing. The initial brackets are:
[math]\displaystyle{ d\varepsilon \le d\delta \le d\varepsilon + \varepsilon_0f(\delta,s) }[/math]
The lower limit is found because the second term is less than or equal to zero for monotonic softening. The upper limit is if failure occurs such that [math]\displaystyle{ f(\delta+d\delta,s)=0 }[/math]. If [math]\displaystyle{ d\delta_n }[/math] is the Newton's method sequence approaching the solution, the next increment by Newton's method is derived to be:
[math]\displaystyle{ d\delta_{n+1} = d\delta_n - {{d\delta_n-d\varepsilon\over \varepsilon_0f(\delta,s)} + {f(\delta+d\delta_n,s)\over f(\delta,s)}-1\over {1\over \varepsilon_0f(\delta,s)} + {f'(\delta+d\delta_n,s)\over f(\delta,s)}} }[/math]
When the next increment is outside the current brackets, a custom bracketing calculation is used to make the solution always stable. Although generic code can solve this differential equation for any softening law with provided properties listed above, some softening laws may have more efficient solutions. For example Linear Softening can exactly solve the differential equation without needing Newton's method and Exponential Softening can evaluate the ratio terms slightly more efficiently then separate evaluations of the numerators and the denominators. Any new softening laws can override generic functions for Newton's method if they have a more efficient approach.