LAOS: The strain softening/strain hardening paradox

Numerous materials, from biopolymers to filled rubbers, exhibit strain softening at high strain amplitudes during a strain sweep in oscillatory rheology: The modulus decreases with increasing deformation. On the other hand, if the nonlinear elastic response is analyzed within a single oscillation cycle (described by a Lissajous curve), these systems are often reported to exhibit strain hardening. We compare strain sweeps and single cycle LAOS (large amplitude oscillatory shear) analyses of stress vs strain on three very different materials. We conclude that the reported strain hardening is due to the use of a tangent modulus in the LAOS analysis, and that the overall rheology remains strain softening. To show that this conclusion is robust, we demonstrate a rescaling of the modulus that collapses the data from all the oscillatory measurements onto a single master curve that clearly exhibits the correct strain softening behavior.