  Recent Atomic Force Microscope (AFM) nanoindentation experiments measuring
mechanical response of the protein shells of viruses have provided a
quantitative description of their strength and elasticity. To better understand
and interpret these measurements, and to elucidate the underlying mechanisms,
this paper adopts a course-grained modeling approach within the framework of
three-dimensional nonlinear continuum elasticity. Homogeneous, isotropic,
elastic, thick shell models are proposed for two capsids: the spherical Cowpea
Chlorotic Mottle Virus (CCMV), and the ellipsocylindrical bacteriophage $\phi
29$. As analyzed by the finite element method, these models enable parametric
characterization of the effects of AFM tip geometry, capsid dimensions, and
capsid constitutive descriptions. The generally nonlinear force response of
capsids to indentation is shown to be insensitive to constitutive details, and
greatly influenced by geometry. Nonlinear stiffening and softening of the force
response is dependent on the AFM tip dimensions and shell thickness. Fits of
the models capture the roughly linear behavior observed in experimental
measurements and result in estimates of Young's moduli of $\approx$280--360 MPa
for CCMV and $\approx$4.5 GPa for $\phi 29$.
