Flow stress

In materials science the flow stress, typically denoted as Y_f (or ${\displaystyle \sigma _{\text{f}}}$), is defined as the instantaneous value of stress required to continue plastically deforming a material - to keep it flowing. It is most commonly, though not exclusively, used in reference to metals. On a stress-strain curve, the flow stress can be found anywhere within the plastic regime; more explicitly, a flow stress can be found for any value of strain between and including yield point (${\displaystyle \sigma _{\text{y}}}$) and excluding fracture (${\displaystyle \sigma _{\text{F}}}$): ${\displaystyle \sigma _{\text{y}}\leq Y_{\text{f}}<\sigma _{\text{F}}}$.

The flow stress changes as deformation proceeds and usually increases as strain accumulates due to work hardening; although the flow stress could decrease due to any recovery process. In continuum mechanics, the flow stress for a given material will vary with changes in temperature, ${\displaystyle T}$, strain, ${\displaystyle \varepsilon }$, and strain-rate, ${\displaystyle {\dot {\varepsilon }}}$, therefore it can be written as some function of those properties:[1]

${\displaystyle Y_{\text{f}}=f(\varepsilon ,{\dot {\varepsilon }},T)}$

The exact equation to represent flow stress depends on the particular material and plasticity model being used. Hollomon's equation is commonly used to represent the behavior seen in a stress-strain plot during work hardening:[2]

${\displaystyle Y_{\text{f}}=K\varepsilon _{\text{p}}^{\text{n}}}$

Where ${\displaystyle Y_{\text{f}}}$ is flow stress, ${\displaystyle K}$ is a strength coefficient, ${\displaystyle \varepsilon _{\text{p}}}$ is the plastic strain, and ${\displaystyle n}$ is the strain hardening exponent. Note that this is an empirical relation and does not model the relation at other temperatures or strain-rates (though the behavior may be similar).

Generally, raising the temperature of an alloy above 0.5 T_m results in the plastic deformation mechanisms being controlled by strain-rate sensitivity, whereas at room temperature metals are generally strain-dependent. Other models may also include the effects of strain gradients.[3] Independent of test conditions, the flow stress is also affected by: chemical composition, purity, crystal structure, phase constitution, microstructure, grain size, and prior strain.[4]

The flow stress is an important parameter in the fatigue failure of ductile materials. Fatigue failure is caused by crack propagation in materials under a varying load, typically a cyclically varying load. The rate of crack propagation is inversely proportional to the flow stress of the material.