A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the material is said to have become plastic. Further deformation of the material causes the stress state to remain on the yield surface, even though the shape and size of the surface may change as the plastic deformation evolves. This is because stress states that lie outside the yield surface are non-permissible in rate-independent plasticity, though not in some models of viscoplasticity.

The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space (${\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}}$), a two- or three-dimensional space spanned by stress invariants (${\displaystyle I_{1},J_{2},J_{3}}$) or a version of the three-dimensional Haigh–Westergaard stress space. Thus we may write the equation of the yield surface (that is, the yield function) in the forms:

The first principal invariant (${\displaystyle I_{1}}$) of the Cauchy stress (${\displaystyle {\boldsymbol {\sigma }}}$), and the second and third principal invariants (${\displaystyle J_{2},J_{3}}$) of the deviatoric part (${\displaystyle {\boldsymbol {s}}}$) of the Cauchy stress are defined as:

where (${\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}}$) are the principal values of ${\displaystyle {\boldsymbol {\sigma }}}$, (${\displaystyle s_{1},s_{2},s_{3}}$) are the principal values of ${\displaystyle {\boldsymbol {s}}}$, and

where ${\displaystyle {\boldsymbol {I}}}$ is the identity matrix.

A related set of quantities, (${\displaystyle p,q,r\,}$), are usually used to describe yield surfaces for cohesive frictional materials such as rocks, soils, and ceramics. These are defined as

where ${\displaystyle \sigma _{\mathrm {eq} }}$ is the equivalent stress. However, the possibility of negative values of ${\displaystyle J_{3}}$ and the resulting imaginary ${\displaystyle r}$ makes the use of these quantities problematic in practice.

Another related set of widely used invariants is (${\displaystyle \xi ,\rho ,\theta \,}$) which describe a cylindrical coordinate system (the Haigh–Westergaard coordinates). These are defined as: