Cubic Step Function Softening

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The Softening Law

A cubic step softening law has the following values:

      [math]\displaystyle{ f(\delta,s) = 1 + 2\left( {\delta\over \delta_{max} }\right)^3 - 3\left( {\delta\over \delta_{max} }\right)^2 }[/math]

where [math]\displaystyle{ \delta_{max} }[/math] is found from

      [math]\displaystyle{ sG_c = \int_0^{\delta_{max}} f(\delta,s) = {\delta_{max}\over 2} \quad{\rm or}\quad \delta_{max} = 2sG_c }[/math]

where s is the softening scaling term and Gc is toughness of the law (and the law's only property). The critical cracking strain, which depends on mesh size and crack orientation, is calculated above and is not a law property to be provided.

The area (or energy dissipation term) is

      [math]\displaystyle{ A(\delta,s) = {\delta\over 2}\left(1 +\left( {\delta\over \delta_{max} }\right)^2\left(1 - {\delta\over \delta_{max}} \right)\right) }[/math]

The stability condition is:

      [math]\displaystyle{ \max\bigl(-f'(\delta,s)\bigr) = {1.5\over \delta_{max}} = {3\over 4sG_c} }[/math]

Compared to a linear softening law, a cubic step function softening has zero derivative at both δ=0 and at δ=δmax. This change might smooth the damage process. The law, however, needs numerical methods to evolve damage (which are not needed by linear softening law) and will need slightly smaller elements (because of 50% higher maximum slope).

Softening Law Properties

Only one property is needed to define a cubic step function softening law:

Property Description Units Default
Gc The toughness associated with the this softening law energy release units none