The carbon partitioning and lengthening rate of bainitic ferrite (α_b) are excellent experimental parameters to estimate our level of understanding of the mechanism of bainitic transformation from a continuum perspective and our ability to capture it in analytical expressions. For Fe-C alloys and relatively simple steels the classical Zener-Hillert theory captures the bainitic transformation rather well but mispredicts the level of carbon in solution in the bainite and overestimates the lengthening rates for transformations at lower temperatures. To address this issue, this paper presents a new thermo-kinetic model based on the Zener-Hillert theory and the Gibbs energy balance concept to simulate the lengthening behavior of α_b in the Fe-C and low alloyed steels. The model incorporates the effect of the temperature dependent carbon diffusion within the migrating interface via a temperature dependent ferrite/austenite interfacial energy and a temperature dependent diffusion coefficient but does not impose local equilibrium across the interface. The good agreement between the model predictions and nine sets of published experiments indicates that both the carbon supersaturation in α_b and the slower lengthening rate are caused by carbon diffusion within the migrating interface. It is found that the degree of carbon supersaturation in α_b increases significantly with decreasing temperature. Consequently, the enhanced carbon solute drag effect, resulting from carbon diffusion within the interface, strongly retards the lengthening rates of α_b at lower temperatures. Transformation strain is shown to have a modest effect on the lengthening rates but to lower the degree of carbon supersaturation.

The stability and collapse mechanism of tunnel faces are simplified conservatively to two-dimensional plane strain models along the longitudinal middle line of tunnel. Using the upper bound finite element method with rigid translatory moving element (UBFEM-RTME), a series of stability factors N_cr and collapse mechanisms displayed with active discontinuities are deduced. The influences of dimensionless buried depth ratio H/D, internal friction angle φ and dilatancy angle ψ on the variations of N_cr and mesh-like collapse mechanisms that are identical to the form of slip lines are discussed. A fitting formula of N_cr for the influence factors H/D and φ is deduced, and the effects of numbers and locations of active discontinuities are also investigated. This study illustrates that the UBFEM-RTME with combination of mesh adaptive updating strategies and reasonable and sufficient mesh density can improve the accuracy of the obtained N_cr values and the refinement of mesh-like collapse mechanism. The results reveal the main characteristics of the ultimate collapse mechanisms of tunnel faces, and they can provide theoretical supports for the stability evaluations of tunnel faces and pre-reinforcement scheme of soil strata.