  We perform comprehensive density-functional theory calculations on strained
two-dimensional phosphorus (P), arsenic (As) and antimony (Sb) in the
monolayer, bilayer, and bulk $\alpha$-phase, from which we compute the key
mechanical and electronic properties of these materials. Specifically, we
compute their electronic band structures, band gaps, and charge-carrier
effective masses, and identify the qualitative electronic and structural
transitions that may occur. Moreover, we compute the elastic properties such as
the Young's modulus $Y$; shear modulus $G$; bulk modulus $\mathcal{B}$; and
Poisson ratio $\nu$ and present their isotropic averages of as well as their
dependence on the in-plane orientation, for which the relevant expressions are
derived. We predict strain-induced Dirac states in the monolayers of As and Sb
and the bilayers of P, As, and Sb, as well as the possible existence of Weyl
states in the bulk phases of P and As. These phases are predicted to support
charge velocities up to $10^6$~$\textrm{ms}^{-1}$ and, in some highly
anisotropic cases, permit one-dimensional ballistic conductivity in the
puckered direction. We also predict numerous band gap transitions for moderate
in-plane stresses. Our results contribute to the mounting evidence for the
utility of these materials, made possible by their broad range in tuneable
properties, and facilitate the directed exploration of their potential
application in next-generation electronics.
