PSI - Issue 40

I.A. Bannikova et al. / Procedia Structural Integrity 40 (2022) 32–39 I. A. Bannikova at al. / Structural Integrity Procedia 00 (2022) 000 – 000

33

2

Keywords: gaped momentum states; viscosity anomalies in solids and liquids; EEW; spall strength; VISAR; self-similar.

1. Introduction The GMS (Gapped Momentum States) emerge in the Maxwell – Frenkel approach to liquid viscoelasticity as the gaps in the dispersion relations (DR) leading to a continuous change from the energy to the momentum space (Baggioli et al. (2020)). Frenkel’s idea (Frenkel (1947)) is that shear modes exist in liquids at the time determined by the Frenkel frequency F  : 1 F F      , where G F    is the relaxation time in the Maxwell equation d dt σ G σ η d dt ε    1 , where  is the shear strain, and  is the shear stress. The hydrodynamic viscous flow is equivalent to the substitution of G by the operator   1 1 M G A   in the momentum conservation law, which has the form of the Navier – Stokes equation for velocity v . (1) Representation of the solution Eq. (1) as = 0 ( ( − )) yields the dispersion equation 2 2 2 0 F G i c k       , leading to the complex frequency = − (2 ) ± √ 2 2 − 1 (4 2 ⁄ ) ⁄ . From this it follows that the gap in the k-space emerges in the liquid transverse spectrum at g k k  , where   1 / 2 g F k = c τ . The k-gap characterizes the propagating shear modes and can be related to the finite propagation length of shear waves in a liquid: F c τ determines the propagation length of shear waves, or liquid elasticity length el F d c   . This definition of the k-gap means that the liquid can be considered as an ensemble of dynamical regions of characteristic size F c τ , where the solid-like response manifests itself as the shear waves. The gap occurs in k-space but not in the frequency space. This is the consequence of a difference between the local nature of the relaxation event (particle jump introduced by Frenkel (1947) and the extended character of the wave. These statements are confirmed in Bazaron et al. (1990) by the measurements of relaxation spectra in a simple shear flow of a liquid accompanying the superposition of shear oscillations in the liquid at a frequency of 10 5 Hz, at which the existence of shear elasticity was established. The explanation for this long relaxation time anomaly is related to a coordinated displacement and reorientation of groups of molecules, which requires longer time. GMS as a coordinated displacement of groups of molecules (similarly to coordinated relative slip of blocks or grains in solids) can be realized during the nucleation of defects (microshears) in the space between these groups of molecules. The structural image of the k-gap, reflecting the rigidity transition at low packing fractions can be introduced as the localized shear collective mode corresponding to the coordinated movement of groups of molecules in the elastic field of shear stresses. Statistical thermodynamics and kinetics of a condensed matter with microshears was developed by Naimark (2004) and the nonequilibrium free energy was represented as a generalized form of the Ginzburg-Landau equation for the microshear-induced strain p , t v ρ +   t v ρτ = x v η   2 2 F   2 2 .

   

6     p

   

2     p



 2

1

1

1

δ

δ

4

,

(2)

F = A

Bp + C

χ D p+ σ

p



2 1

4

6 1

δ

δ

l

*

с

where  is the structural-scaling parameter, reflecting the current sensitivity of the matter to the defects growth and representing the ratio of the mean size of defects to the spacing between them. The bifurcation points , с    play the role similar to the role of characteristic temperatures in the Ginzburg-Landau phase transition theory. The gradient term in Eq. (2) describes the non-local interaction in the ensemble of mesodefects: , , , A B C D and  are the phenomenological parameters. The kinetic equation for parameter p is

  

   

  

  

l p x x        χ

  

  

  

dp

δ

δ

,

(3)

1

1

= Γ A  

3 p Bp +C

5 p Dσ

 



 

p

dt

δ

δ

*

c

l