In this paper, the first-order ordinary differential constitutive equations of endochronic theory were combined with the principle of virtual work for simulating the response of creep (moment is kept constant for a period of time) or relaxation (curvature is kept constant for a period of time) of thin-walled tubes subjected to pure bending with different curvature-rates at the preloading stage. A group of Fourier series was used to describe the circumferential displacements of the tube. Thus, a system of nonlinear algebraic equations was determined. This system of equations can be solved by numerical method. Experimental data tested by Pan and Fan [1] were compared with the theoretical simulations in this study. It is shown that the theoretical formulations effectively simulate the experimental data.

A material function of endochronic theory is proposed for investigating the plastic behaviors of material. Depending on the material parameters properly chosen, the present model can be classified into four categories, and is appropriate for describing various materials behaving cyclic strain hardening inherently with respect to the deformation history. Experimental verification of the theory was demonstrated using the experimental results of Shiao [1] and Lamba and Sidebottom [2]. The theory is in good agreement with experimental results obtained by Shiao [1] through comparing the stress-strain hysteresis loops of SAE 4340 steel under axisymmetrically cyclic loading condition with various amplitudes. In addition, the present model is shown to be capable of describing the behavior of erasure of memory of materials, as experimentally observed by Lamba and Sidebottom [2].