  Recently, a widely applicable system of hyperbolic partial differential
equations has been derived that enables the deterministic computation of a full
heterogeneous stress field from a measured deformation field, for example, from
a strain field obtained via digital image correlation. This information enables
the determination of material properties, making this approach an alternative
to finite element model updating or the virtual fields method. This article
focuses on developing a finite volume discretization of this system of
equations to address instabilities that arise from violations of the
Courant-Friedrichs-Lewy condition. The developed discretization enables the
system of equations to be applied to irregular geometries and finite
deformation. We determine how, in general, one may translate knowledge of the
traction at the boundary into boundary conditions, so that the numerical method
can be applied to a variety of loading conditions. We analyse the solution
structure in the case of deformation discontinuities, and the results are
applied to account for discontinuities at the interfaces between finite volumes
(this has relevance to other applications such as composite materials).
Interestingly, the discontinuities cause reflection and transmission of the
principal stresses.
  The finite volume discretization is validated using data output from
commercial finite element software. Furthermore, the discretization is applied
to an experimental uniaxial tension test with plastic deformation and necking.
The strain field is obtained using digital image correlation, and the stress
field is computed using the developed finite volume discretization. Together,
these give the stress-strain behavior for each material element.
