Temporal homogenization formulation on general linear viscoelastic materials subjected to locally periodic loading

In this paper, the temporal homogenization formulation is proposed for the purpose of the accelerated prediction of time-dependent mechanical behavior of the general linear viscoelastic materials subjected to the locally cyclic loading. General linear viscoelastic materials can be described by higher-order differential equation form of the constitutive relation. This study is the first attempt to apply the two-scale time homogenization to the higher-order differential equation form of the linear viscoelastic constitutive relation. In conclusion, the original boundary value problems are divided into macro- and micro-chronological portions. For the verification of the proposed approach, three studies are conducted: (i) simple one-dimensional bar subjected to the locally cyclic loading, (ii) simple three-dimensional beam subjected to the locally cyclic loading, (iii) three-dimensional particulate composite material subjected to the locally cyclic loading. The verification results show that the proposed temporal homogenization results are well matched to the reference solutions, as the frequency of the imposed locally periodic loading is much higher than the material intrinsic relaxation time constants.