Figure 1.
Stress-strain curves for an oscillatory shear strain .
(a, b) Experiment: passive transient F-actin/HMM gels ( mg/ml,
) sheared at strain amplitudes of
and
, corresponding to a weakly/strongly non-linear response, respectively. The upward bending of the ellipses signals stiffening, their concave regions near maximum strain imply softening. The softening and the ensuing “shakedown” of the stress-strain curves towards a limit cycle are indicative of inelastic fluidization. (c, d) Corresponding theory curves from the inelastic glassy wormlike chain (i Gwlc) model [25] (parameters
,
,
,
Hz; single-polymer displacement and force were converted to network strain and stress as described in Methods). The absolute stress and strain scales in theory and experiment are compatible on the present (mean-field) level of modeling, but the theory somewhat overestimates the stiffening during the initial large-amplitude loading cycle, and, as a consequence, also the peak force and the strength of the shakedown.
Figure 2.
Nonlinear inelastic response of F-actin/HMM networks.
(a) Schematic of the oscillatory driving protocol (the strain amplitude is increased in steps after every 30 cycles, driving frequency
Hz). (b) Measured reduced nonlinear modulus
(peak stress over peak strain) as a function of the cycle number
. The shaded background indicates the monotonic increase of the strain amplitude
(indicated in percent). Note that the modulus responds nonmonotonically to both transient and stationary loading, hinting at antagonistic mechanisms with multiple time scales. Inset: Theory curve from the i Gwlc model [25] reproducing the key features, transient and stationary stiffening and softening with the parameters from Fig. 1 (see also Methods and Fig. E in Supporting Information S1).
Figure 3.
Fluidization and slow mechanical recovery of an F-actin/HMM gel after a transient strain pulse (inset).
Stiffness is quantified by the normalized (“n”) real part of the linear shear modulus, measured by small sinusoidal oscillations at fixed oscillation frequency
, before and after the stretch. The softening immediately after the stretch is found to be sensitive to the maximum strain
(circles) and
(squares) of the pulse, albeit less pronounced as for cells, where the same pattern is observed at 3–4 times smaller strain amplitudes [4]. Error bars are SE, ensemble size is
; lines represent theoretical (exponential) fits by the i Gwlc model [25]; see Methods and Supporting Information S1 for further explanations.
Figure 4.
Constitutive diagram for the iGWLC model.
The central panel gives a qualitative graphical summary of the mechanical response predicted by the model as a function of the amplitude and characteristic rate of an imposed deformation pulse. At low amplitudes, in the linear regime, it exhibits power-law rheology (upper panel, log-log scale). At low rates, in the quasistatic regime, it exhibits stiffening at low amplitudes, where entropic stiffening of the polymer backbone dominates, and softening at high amplitudes, where the stiffening is eventually overruled by the exponential bond softening (left panel, linear scale). This mechanism underlies the initially ascending and later descending steps in the nonlinear modulus in Fig. 2. At high rates and high amplitudes, a steep initial stiffening with subsequent fluidization and slow recovery governs the response (central panel). The schematic stress-strain curves for oscillatory driving exemplify the salient features of the nonlinear response in the various parameter regions.