A theoretical model for foam rheology that includes viscous forces is developed by considering the deformation of two‐dimensional, spatially periodic cells in simple shearing and planar extensional flow. The undeformed hexagonal cells are separated by thin liquid films. Plateau border curvature and liquid drainage between films is neglected. Interfacial tension and viscous tractions due to stretching lamellar liquid determine the individual film tensions. The network motion is described by a system of nonlinear ordinary differential equations for which numerical solutions are obtained. Coalescense and disproportionation of Plateau borders results in the relative separation of cells and provides a mechanism for yielding and flow. This process is assumed to occur when a film’s length reduces to its thickness. The time and position dependence of the cell‐scale dynamics are computed explicitly. The effective continuum stress of the foam is described by instantaneous and time‐averaged quantities. The capillary number, a dimensionless deformation rate, represents the relative importance of viscous and surface tension effects. The small‐capillary‐number or quasistatic response determines a yield stress. The dependence of the shear and normal stress material functions upon deformation rate, foam structure and physical properties is determined. A plausible mechanism for shear‐induced material failure, which would determine a shear strength, is revealed for large capillary numbers. The mechanism involves large cell distortion and film thinning, which provide favorable conditions for film rupture.