MSR/ressources/Viscoelasticity.md

Viscous materials, like water, resist both shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed. The study of such materials is covered by Rheology.

A viscoelastic substance dissipates energy when a load is applied, then removed.

In most of the analyses, extensional/compressional strains of the viscoelastic layer are not taken into account since the damping comes mostly from the shear strain.

The properties of a viscoelastic layer change with the frequency of excitation .

Models

(see Rheology#Linear viscosity for the definition of f and g, and for the sources)

Perfectly elastic solid

Rheologic equation $$\varepsilon(t)=J\tau(t)

Where J is the elastic compliance. G=\frac{1}{J} is called the elastic modulus

the creep function f(t) is then :

f(t)=J

Newtonian viscous fluid

Rheologic equation (see Rheology#Dynamic viscosity)

\frac{d\varepsilon(t)}{dt}=\frac{\tau(t)}{\eta}

initial conditions : t<0 ; \tau(t)=0 ; \varepsilon(t)=0, so

\varepsilon(t)=\frac{1}{\eta}\int_{0}^{t}\tau(t')dt'

under a constant stress \tau_0 :

\varepsilon(t)=\frac{\tau_0}{\eta}t

the creep function is obtained with a constant unit stress, so with \tau_0=1

f(t)=\frac{t}{\eta}

Associations

Parallel

• stress to the whole is the sum of the stresses on each branch
• strain to the whole is equal to the strains in each branch

Series

• stress to the whole is equal to the stresses on each branch
• strain to the whole is the sum of the strains in each branch