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Yield stress fluids (YSFs) display a dual nature highlighted by the existenceof a yield stress such that YSFs are solid below the yield stress, whereas theyflow like liquids above it. Under an applied shear rate $\dot\gamma$, thesolid-to-liquid transition is associated with a complex spatiotemporalscenario. Still, the general phenomenology reported in the literature boilsdown to a simple sequence that can be divided into a short-time responsecharacterized by the so-called "stress overshoot", followed by stressrelaxation towards a steady state. Such relaxation can be either long-lasting,which usually involves the growth of a shear band that can be only transient orthat may persist at steady-state, or abrupt, in which case the solid-to-liquidtransition resembles the failure of a brittle material, involving avalanches.Here we use a continuum model based on a spatially-resolved fluidity approachto rationalize the complete scenario associated with the shear-induced yieldingof YSFs. Our model provides a scaling for the coordinates of the stress maximumas a function of $\dot\gamma$, which shows excellent agreement withexperimental and numerical data extracted from the literature. Moreover, ourapproach shows that such a scaling is intimately linked to the growth dynamicsof a fluidized boundary layer in the vicinity of the moving boundary. Yet, suchscaling is independent of the fate of that layer, and of the long-term behaviorof the YSF. Finally, when including the presence of "long-range" correlations,we show that our model displays a ductile to brittle transition, i.e., thestress overshoot reduces into a sharp stress drop associated with avalanches,which impacts the scaling of the stress maximum with $\dot\gamma$. Our workoffers a unified picture of shear-induced yielding in YSFs, whose complexspatiotemporal dynamics are deeply connected to non-local effects.

Continuum modelling of shear start-up in soft glassy materials