MECHANICS OF INDENTATION - Nanoindentation with Biological Applications - page 140

viscoelastic approach for three-dimensional stress and strain analysis,
these basic functional forms are assumed to hold between individual
tensor components of the overall stress and strain tensors. In practice,
indentation problems are most commonly written in the single scalar
form relating axial load to axial displacement, and in this context
the conversion from viscoelastic stress-strain space to axial load-
displacement space is trivial.
Approaches to the analytical viscoelastic indentation problem utilize
the correspondence between elastic and viscoelastic elasticity
problems 25,26 ; the only difference between an elastic and a viscoelastic
problem is in the choice of a time-dependent constitutive law for the
material. 26 Elastic-viscoelastic correspondence analysis was examined
within the context of the Hertzian contact problem by Lee and Radok 27
and is summarized in Johnson's text. 4 For analysis of indentation
loading conditions in which the contact area is non-decreasing, the
problem is fairly straightforward; for decreasing contact areas an adapted
analysis 28 is required. Since the two most common time-dependent
experiments (creep or relaxation at fixed load or displacement, or
loading at various load- or displacement-rates) are for non-decreasing
contact areas, the straight-forward analysis of Lee and Radok 27 is the
sole analysis considered here in the context of elastic-viscoelastic
correspondence.
The elastic solutions for indentation—written in terms of an axial
load and axial displacement—are rewritten as viscoelastic expressions by
replacing the elastic constants by viscoelastic operators, and integrating
over the applied load- or displacement-perturbation. For the viscoelastic
indentation analysis, an elastic indentation solution ( Eqs. 5-5, 7, 8 ) is
rewritten in terms of the shear modulus, where G = E /[2(1+)] for the
general case (requiring fixed Poisson's ratio) and G = E /3 for the
incompressible case (and the plane strain modulus thus equates to 4 G ).
From this point we will emphasize the incompressible case; data can
always be back-corrected in the instance where the Poisson's ratio is
known (and known to be less than 0.5). 29
In cases where step-loading can be assumed, there is a
straightforward replacement of P /2 G with the product of the creep load,
P 0 , and the creep function, J ( t ):