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Theory Manual Version 3.6
 Subsubsection 5.8.1.4: Damage Measures Up Subsection 5.8.1: Theoretical Formulation Section 5.9: Hydraulic Permeability 

5.8.1.5 Cumulative Damage Distribution Functions

The final set of constitutive relations required to fully define an elastoplastic damage material are the two CDFs, and . As shown by [78], the only requirement imposed by the Clausius-Duhem inequality is that these be monotonically increasing functions.
figure ../Figures/FigReactivePlasticityDamage.png
Figure 5.4 Parametric study of the effect of the damage parameter for a Weibull distribution, with no intact damage taking place. (a) As increases, the onset of noticeable damage shifts to higher strains and becomes more rapid. (b) Plot of the damage variable . The response becomes more nonlinear as deviates from unity. Other plasticity and damage parameters are , MPa, MPa, , , , and .
Whereas these CDFs may be characterized directly from experimental data, here we illustrate the FEBio elastoplastic damage framework using a Weibull distribution of the form where (same units as ) is the value of at which the fraction of bonds have failed and the exponent (unitless) controls the slope of the response, such that approaches a step function with a jump at as . Therefore, each damage function has two free parameters and . Based on experimental evidence, we may let be given by the von Mises (effective) stress, while is taken to be the effective plastic strain (Section 5.8.1.4↑). Figure 5.4↑ shows the effect of the Weibull parameter on the stress-strain and damage-strain responses, with fixed. The damage response as a function of plastic strain is identically the prescribed CDF (Figure 5.4↑b). The shape of the CDF changes from logarithmic-like to exponential as increases, demonstrating the ability of this formulation to recover a broad variety of experimentally measured damage-strain behaviors [29].]


 Subsubsection 5.8.1.4: Damage Measures Up Subsection 5.8.1: Theoretical Formulation Section 5.9: Hydraulic Permeability