The stress‐strain rate relationships of four silicate melt compositions (high‐silica rhyolite, andesite, tholeiitic basalt, and nephelinite) have been studied using the fiber elongation method. Measurements were conducted in a stress range of 10–400 MPa and a strain rate range of 10⁻⁶ to 10⁻³ s⁻¹. The stress‐strain rate relationships for all the melts exhibit Newtonian behavior at low strain rates, but non‐Newtonian (nonlinear stress‐strain rate) behavior at higher strain rates, with strain rate increasing faster than the applied stress. The decrease in calculated shear viscosity with increasing strain rate precedes brittle failure of the fiber as the applied stress approaches the tensile strength of the melt. The decrease in viscosity observed at the high strain rates of the present study ranges from 0.25 to 2.54 log₁₀ Pa s. The shear relaxation times τ of these melts have been estimated from the low strain rate, Newtonian, shear viscosity, using the Maxwell relationship τ = η_s/G_∞. Non‐Newtonian shear viscosity is observed at strain rates ( ε˙=time−1) equivalent to time scales that lie 3 log₁₀ units of time above the calculated relaxation time. Brittle failure of the fibers occurs 2 log₁₀ units of time above the relaxation time. This study illustrates that the occurrence of non‐Newtonian viscous flow in geological melts can be predicted to within a log₁₀ unit of strain rate. High‐silica rhyolite melts involved in ash flow eruptions are expected to undergo a non‐Newtonian phase of deformation immediately prior to brittle failure.