Foam rheology: Relation between extensional and shear deformations in high gas fraction foams

A two-dimensional hexagonal foam cell model is used to derive analytic expressions for the bulk stress tensor and foam microstructure for any small homogeneous deformation. We show that calculations done for deformations where the principal axes of stress and strain coincide, such as in extension, are sufficient to provide all information about shear deformation. The stresses and foam structure for any given strain and initial cell orientation in shear bears a unique relation to a different strain and orientation in extension. Such a mapping is obtained using the assumption that the principal axes of strain and stress corotate with each other. This in turn implies that high gas fraction foams follow the Lodge-Meissner relation, i.e. the ratio of the normal-stress difference to the shear stress equals the shear strain. The spatially periodic structure of foam along with the fact that the cell centers move affinely with the bulk, makes the above assumption a justifiable one.