In viscoelastic materials, whose behavior is intermediate between those of liquids and solids, the total stress tensor comprises both viscous and elastic ("static") components.
In an arbitrary coordinate system, the viscous stress ε and the strain rate E at a specific point and time can be represented by 3 × 3 matrices of real numbers.
Let dF be the infinitesimal force due to viscous stress that is applied across that surface element to the material on the side opposite to dA.
In a fluid, elastic stress can be attributed to the increase or decrease in the mean spacing of the particles, that affects their collision or interaction rate and hence the transfer of momentum across the fluid; it is therefore related to the microscopic thermal random component of the particles' motion, and manifests itself as an isotropic hydrostatic pressure stress.
However in almost all practical situations these terms can be ignored, since they become negligible at the size scales where the viscous stress is generated and affects the motion of the medium.
In particular, the local strain rate E(p, t) is the only property of the velocity flow that directly affects the viscous stress ε(p, t) at a given point.
On the other hand, the relation between E and ε can be quite complicated, and depends strongly on the composition, physical state, and microscopic structure of the material.
where Ev and Es are the scalar isotropic and the zero-trace parts of the strain rate tensor E, and μv and μs are two real numbers.
The zero-trace part Es of E is a symmetric 3 × 3 tensor that describes the rate at which the medium is being deformed by shearing, ignoring any changes in its volume.
The part Ev of E acts as a scalar multiplier (like εv), the average expansion rate of the medium around the point in question.
It is numerically equal to 1/3 of the divergence of the velocity which in turn is the relative rate of change of volume of the fluid due to the flow.
Therefore, the scalar part εv of ε is a stress that may be observed when the material is being compressed or expanded at the same rate in all directions.