Statistical Mechanics of Viscoelasticity

Abstract

The theory of viscoelasticity is developed from the point of view of statistical mechanics. The general transport relations (for linear processes) are treated with a modified form of the Chapman-Enskog method, the modification being such as to take into account the existence of large internal relaxation times. The stress is then found to depend on the past history of the strain or strain rate, with relaxation functions which are time-dependent correlation functions. Certain basic properties of the relaxation functions then follow almost immediately. In particular the shear and volume relaxation functions are shown to be positivedefinite functions of the time. The theory is linear, but no assumptions are made regarding the reference state, which may be an arbitrary state of strain. The thermodynamics of a stressed medium, which is necessary for application of the Chapman-Enskog method, is discussed in an appendix.

Mechanical Properties of Cellularly Responsive Hydrogels and Their Experimental Determination

“…Passive tracer microrheology has a rich historical background that can be traced back to the work of Jean Perrin and a team of students who performed impressively precise position evolution measurements on colloidal particles undergoing Brownian motion, demonstrating the accuracy of the Stokes–Einstein relationship (〈Δ r 2 ( t )〉 = tk B T / πaη , where 〈Δ r 2 ( t )〉 is the mean‐squared displacement of a spherical particle having a radius a in a fluid with viscosity η ) 93. More recently, the Brownian motion of a spherical particle in viscoelastic materials was shown to follow the generalized Stokes–Einstein relationship, or GSER (〈Δ r̃ 2 ( s )〉 = tk B T / πas $ \tilde\eta $ ( s ),94–96 where the mean‐squared displacement is now a complex function expressed here in the Laplace domain, s ); this result is the solution of Equation 1, where F is a stochastic Brownian force 96–98. The complex viscosity is conveniently transformed to the creep compliance in the time domain as J ( t ) = 〈Δ r 2 ( t )〉 πa/k B T 67, 99.…”

Stress tensor and elastic properties for hard and soft spheres

“…For the special case of equilibrium it follows that g (q 1 , 12 ) so the average stresses (12) and (13) become equal. They are proportional to the unit tensor with proportionality constant equal to the pressure of (5).…”

“…Formal linear response methods have been applied to this problem [13], and the result applied to soft spheres is…”

“…In the original derivation of [13] an additional contribution from the local equilibrium reference state was included, i.e. the contributions from the elastic constants.…”

“…In reference [13] an integration by parts is performed to get a representation of viscoelasticity theory…”