  The Bialas-Bzdak model of elastic proton-proton scattering is generalized to
the case when the real part of the parton-parton level forward scattering
amplitude is non-vanishing. Such a generalization enables the model to describe
well the dip region of the differential cross-section of elastic scattering at
the ISR energies, and improves significantly the ability of the model to
describe also the recent TOTEM data at sqrt{s} = 7 TeV LHC energy. Within this
framework, both the increase of the total cross-section, as well as the
decrease of the location of the dip with increasing colliding energies, is
related to the increase of the quark-diquark distance and to the increase of
the "fragility" of the protons with increasing energies. In addition, we
present and test the validity of two new phenomenological relations: one of
them relates the total p+p cross-section to an effective, model-independent
proton radius, while the other relates the position of the dip in the
differential elastic cross-section to the measured value of the total
cross-section.
