(a) Experiment of Johnson and Beatty (1993) showing the Mullins effect for cycles with increasing amplitude; (b) Scheme of the prototypical model; (c) Energy density function; (d) Stress-strain diagram for a breakable chain.
Regions of broken, active, and unloaded chains in the probability space . In (a) the current strain coincides with the maximum value attained in the loading history . In (b) we represent the case . Observe that for the loading paths with no new chain breaks and the behavior is elastic (ideal Mullins effect).
Special distribution of broken chains, with a probability function concentrated on the straight line marked with bold line. In the figure we show the segments corresponding to active, unloaded, and broken chains. In (a) the present strain coincides with the maximum value attained in the loading history. In (b) we show the case .
Mullins effect for a system with a probability (2.11) represented in (a), with , , and . We represent in (b) the elastic energy, in (c) the fracture energy, and in (d) the stress on the primary loading curve. In (f) we show the Mullins effect under the loading history represented in (e). Here, we take , , and a fraction of elastic chains with an energy density .
Sketch of the experimental tests to deduce the probability function. In (a) we show the experiment needed under the simplifying assumption (2.8). In (b) we show the experiment for the general case to evaluate the average value of on a discrete grid shown in (c). In the figure .
Hysteresis cycles for a system with a neo-Hookean type energy (4.1) with and . The breakable fraction of the material is assigned by a probability (4.5) with and a Gaussian probability with , , and is chosen by the normalizing condition (observe that by the incompressibility condition it follows that ). Here, .
Mullins effect for azimuthal shear with under the hypothesis of a neo-Hookean energy (4.1) with , , and . The breakable material fraction is assigned by a probability (4.5) with and if and , otherwise. In (a) we show the probability function and in (b) the loading program. In (d) and (e) we show the rotation angle and the deformation on the primary loading curve as a function of for different values of the assigned moment. In (c) we show the moment-rotation relation. In (f) we represent the damage effect by plotting in a gray scale the fraction of broken chains for different values of the maximum moment.
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