  We investigate the rheology of strain-hardening spherical capsules, from the
dilute to the concentrated regime under a confined shear flow using
three-dimensional numerical simulations. We consider the effect of capillary
number, volume fraction and membrane inextensibility on the particle
deformation and on the effective suspension viscosity and normal stress
differences of the suspension. The suspension displays a shear-thinning
behaviour which is a characteristic of soft particles such as emulsion
droplets, vesicles, strain-softening capsules, and red blood cells. We find
that the membrane inextensibility plays a significant role on the rheology and
can almost suppress the shear-thinning. For concentrated suspensions a
non-monotonic dependence of the normal stress differences on the membrane
inextensibility is observed, reflecting a similar behaviour in the particle
shape. The effective suspension viscosity, instead, grows and eventually
saturates, for very large inextensibilities, approaching the solid particle
limit. In essence, our results reveal that strain-hardening capsules share
rheological features with both soft and solid particles depending on the ratio
of the area dilatation to shear elastic modulus. Furthermore, the suspension
viscosity exhibits a universal behaviour for the parameter space defined by the
capillary number and the membrane inextensibility, when introducing the
particle geometrical changes at the steady-state in the definition of the
volume fraction.
