MECHANICS OF INDENTATION - Nanoindentation with Biological Applications - page 138

6.1.1. Linearly viscoelastic materials

The time-dependent mechanical behavior of polymeric and many
biological materials is often represented with a mechanical model formed
of spring elements that are linear elastic (Hookean):

(5-20) σ = G ε

and linearly viscous (Newtonian) “dashpots”:

d d t

(5-21) σ = η

Combinations of these simple relationships are made in arranging springs
and dashpots in series and parallel configurations ( Fig. 5-10 ) and then
solving for the stress-strain-time ( σ - ε - t ) relationship, a differential

equation in time.

Figure 5-10. Examples of linear viscoelastic models formed from combinations of springs
and dashpots with their independent constitutive responses as in Eqs. 5-20 and 5-21 .

The order of the differential equation depends on the number of
individual dashpots in the full system: the standard linear solid model
has a first order differential equation constitutive response while the
Burgers model is second order. From this constitutive relationship
the homogeneous (single-axis, uniform loading) creep and relaxation