  Soft glassy materials such as mayonnaise, wet clays, or dense microgels
display under external shear a solid-to-liquid transition. Such a shear-induced
transition is often associated with a non-monotonic stress response, in the
form of a stress maximum referred to as "stress overshoot". This ubiquitous
phenomenon is characterized by the coordinates of the maximum in terms of
stress $\sigma_\text{M}$ and strain $\gamma_\text{M}$ that both increase as
weak power laws of the applied shear rate. Here we rationalize such power-law
scalings using a continuum model that predicts two different regimes in the
limit of low and high applied shear rates. The corresponding exponents are
directly linked to the steady-state rheology and are both associated with the
nucleation and growth dynamics of a fluidized region. Our work offers a
consistent framework for predicting the transient response of soft glassy
materials upon start-up of shear from the local flow behavior to the global
rheological observables.
