  The stress-gradient theory has a third order tensor as kinematic degree of
freedom, which is work-conjugate to the stress gradient. This tensor was called
micro-displacements just for dimensional reasons. Consequently, this theory
requires a constitutive relation between stress gradient and
micro-displacements, in addition to the conventional stress-strain relation.
The formulation of such a constitutive relation and identification of the
parameters therein is difficult without an interpretation of the
micro-displacement tensor. The present contribution presents an homogenization
concept from a Cauchy continuum at the micro-scale towards a stress-gradient
continuum at the macro-scale. Conventional static boundary conditions at the
volume element are interpreted as a Taylor series whose next term involves the
stress gradient. A generalized Hill-Mandel lemma shows that the
micro-displacements can be identified with the deviatoric part of the first
moment of the microscopic strain field. Kinematic and periodic boundary
conditions are provided as alternative to the static ones. The homogenization
approach is used to compute the stress-gradient properties of an elastic porous
material. The predicted negative size effect under uni-axial loading is
compared with respective experimental results for foams and direct numerical
simulations from literature.
