We perform molecular dynamics simulation on several relevant biological fibrils associated with neurodegenerative diseases such as Aβ40, Aβ42, and α-synuclein systems to obtain a molecular understanding and interpretation of nanomechanical characterization experiments. The computational method is versatile and addresses a new subarea within the mechanical characterization of heterogeneous soft materials. We investigate both the elastic and thermodynamic properties of the biological fibrils in order to substantiate experimental nanomechanical characterization techniques that are quickly developing and reaching dynamic imaging with video rate capabilities. The computational method qualitatively reproduces results of experiments with biological fibrils, validating its use in extrapolation to macroscopic material properties. Our computational techniques can be used for the co-design of new experiments aiming to unveil nanomechanical properties of biological fibrils from a point of view of molecular understanding. Our approach allows a comparison of diverse elastic properties based on different deformations , i.e., tensile (YL), shear (S), and indentation (YT) deformation. From our analysis, we find a significant elastic anisotropy between axial and transverse directions (i.e., YT > YL) for all systems. Interestingly, our results indicate a higher mechanostability of Aβ42 fibrils compared to Aβ40, suggesting a significant correlation between mechanical stability and aggregation propensity (rate) in amyloid systems. That is, the higher the mechanical stability the faster the fibril formation. Finally, we find that α-synuclein fibrils are thermally less stable than β-amyloid fibrils. We anticipate that our molecular-level analysis of the mechanical response under different deformation conditions for the range of fibrils considered here will provide significant insights for the experimental observations.