Growing biological media develop residual stresses to make compatible elastic and inelastic growth-induced deformations, which in turn remodel the tissue properties modifying the actual elastic moduli and transforming an initially isotropic and homogeneous material into a spatially inhomogeneous and anisotropic one. This process is crucial in solid tumor growth mechanobiology, the residual stresses directly influencing tumor aggressiveness, nutrients walkway, necrosis and angiogenesis. With this in mind, we here analyze the problem of a hyperelastic sphere undergoing finite heterogeneous growth, in
cases of different boundary conditions and spherical symmetry. By following an analytical approach, we obtain the explicit expression of the tangent elasticity tensor at any point of the material body as a function of
the prescribed growth, by involving a small-on-large procedure and exploiting exact solutions for layered media. The results allowed to gain several new insights into how growth-guided mechanical stresses
and remodeling processes can influence the solid tumor development. In particular, we highlight that— under hypotheses consistent with mechanical and physiological conditions—auxetic (negative Poisson
ratio) transformations of the elastic response of selected growing mass districts could occur and contribute to explain some not yet completely understood phenomena associated to solid tumors. The general approach proposed in the present work could be also helpfully employed to conceive composite materials where ad hoc pre-stress distributions can be designed to obtain auxetic or other selected mechanical properties
