Metal forming changes shape by driving the material beyond elastic response into sustained plastic deformation.

During plastic flow, the stress needed to continue deformation generally changes with accumulated strain. A simple idealization for strain hardening is

$$\sigma_f=K\varepsilon^n,$$

where $\sigma_f$ is flow stress, $K$ is a strength coefficient and $n$ is a strain-hardening exponent.

The plastic work per unit volume accumulated from strain $0$ to $\varepsilon$ is approximately

$$w=\int_0^{\varepsilon}\sigma_f,d\varepsilon.$$

For the power-law model,

$$w=\frac{K}{n+1}\varepsilon^{n+1}.$$

Forming force therefore depends on more than final part area. It is influenced by material flow stress, strain history, friction, contact geometry and constraint imposed by the tooling.

A useful distinction is between elastic recovery and plastic shape change. After the external load is removed, the elastic part of deformation recovers while the plastic part remains. This recovery is responsible for effects such as springback in bent sheet.

Flow stress provides the common mechanical language behind rolling, forging, extrusion, drawing and sheet-metal forming.