Читать книгу The Rheology Handbook - Thomas Mezger - Страница 207

In order to analyze Maxwellian behavior during a load-and-removal cycle, the following differential equation is used (with the index “v” for the viscous portion and “e” for the elastic one):

Assumption 1: The total deformation is the sum of the individual deformations applied to the two model components.

γ = γv + γe

This applies also to the shear rates: γ ̇ = γ ̇ v + γ ̇ e

since γ ̇ = dγ/dt (as explained in Chapter 4.2.1)

Assumption 2: The same shear stress is acting on each one of the two components.

τ = τv = τe

The viscosity law applies to the viscous element: η = τv / γ ̇ vor γ ̇ v = τv /η

The elasticity law applies to the elastic element:G = τe /γeorγe = τe /G

and γ ̇ e = τ ̇ e /G respectively, with the change of the shear stress over time as τ ̇ = dτ/dt [Pa/s], this is the time derivative of τ.

The sum of the shear rates results in the differential equation according to Maxwell .:

Equation 5.1

γ ̇ = γ ̇ v + γ ̇ e = τv /η + τ ̇ e /G = τ/η + τ ̇ /G

The solution and use of this differential equation are described in Chapters 7.3.2c and 7.3.3.2 (relaxation tests) as well as in Chapter 8.4.2.1 (oscillatory tests/frequency sweeps).

End of the Cleverly section