A homogenization theory for time-dependent deformation of composites with periodic internal structures

A homogenization theory for time-dependent deformation such as creep and viscoplasticity of composites with periodic internal structures is developed. The deviation of microscopic displacement rate from macroscopic one in the macroscopically uniform case, i.e., the first order perturbation of displacement rate in the macroscopically nonuniform case is decomposed into elastic and viscous parts. Thus, the constitutive relation between macroscopic stress and strain rates and the evolution equation of microscopic stress are derived using Y-periodic functions introduced into the elastic and viscous parts, and two unit cell problems to determine the Y-periodic functions are formulated. The theory is described first in the macroscopically uniform case in a rate form, and then it is extended to the macroscopically nonuniform case in an incremental form using the asymptotic expansion of field variables. As an application of the theory, transverse creep of metal matrix composites reinforced unidirectionally with continuous fibers is analyzed numerically to discuss the effect of fiber arrays on the anisotropy in such creep.