## 2 Theoretical Analyses
### Fill State in Backfilled Openings  
In the precedent theoretical (idealized) cases, the soil is initially at rest, and would remain at-rest if the retaining wall is immobile. The motion (displacement) of the wall then controls the stress state in the soil. The situation is different in backfilled openings (trenches, silos, mine slopes, and behind fill retaining walls), in which the fill is placed into a preexisting rigid confining structure. Numerical analyses performed by [PERSON] et al. [24] showed that fills near the center of vertical openings can reach an active state because of some local yielding, even if the walls displacement upon filling is negligible. Simulation results reported by [PERSON] et al. [28; 29] revealed that the stresses along the VCL of similar openings can be close to an active or at-rest state, depending on the relationships (independent or related) between \(\varphi^{\prime}\) and \(\
u\). Near the opening walls, the numerical results of [PERSON] et al. [28] showed that the ratios of the minor over major principal effective stresses approach Rankine's active earth pressure coefficient \(K_{\text{a}}\), even though the confining walls remain unmoved. Thus, the stress state in such cases cannot be solely linked to the movement of the confining wall(s). However, the mechanism behind this behavior has not been explicitly addressed. In the following, the state of fills placed in a vertical opening is analyzed based on an elasto-plastic model with varying values of \(\
u\) and \(\varphi^{\prime}\). It will be seen that the backfill can approach active or at-rest state along the VCL of a backfilled opening, depending on the relations between the respective values of \(\varphi^{\prime}\) and \(\
u\) (even for fixed confining walls). When obeying the Mohr-Coulomb elastoplastic criterion, the fill exhibits a linear elastic behavior under relatively small stresses before reaching a yield state. Upon placement, the major \(\sigma^{\prime}{}_{1}\) and minor \(\sigma^{\prime}{}_{3}\) principal stresses along the VCL of the opening, for an elastic state, are \(\sigma^{\prime}{}_{\rm v}\) and \((K_{0})_{\rm v}\sigma^{\prime}{}_{\rm v}\), respectively. As shown in Figure 1a, two situations can arise for a given friction angle \(\varphi^{\prime}\), considering the limit stress state (\(\sigma^{\prime}{}_{\rm v}\), \(K_{\rm a}\sigma^{\prime}{}_{\rm v}\)), with the Mohr circle becoming tangential to the Coulomb yield envelope (\(\tau=\sigma\) tang\(\varphi^{\prime}\), where \(\tau\) is the maximum shear stress for a given normal stress \(\sigma\)). When \((K_{0})_{\rm v}\sigma^{\prime}{}_{\rm v}\leq K_{\rm a}\sigma^{\prime}{}_{\rm v}\), the corresponding Mohr circle (dotted line in Figure 1) theoretically exceeds the yield envelope. Since this is not allowed, yielding with plastic strain occurs and the horizontal stress is controlled by the circle tangent to the yield envelope. Such stress state corresponds to an active state in the fill, and can be associated with Rankine's active coefficient \(K_{\rm a}\). On the other hand, if \((K_{0})_{\rm v}\sigma^{\prime}{}_{\rm v}>K_{\rm a}\sigma^{\prime}{}_{\rm v}\), the Mohr circle (solid line) remains below the Coulomb yield envelope. The backfill along the VCL can then be expected to be near the at-rest (elastic) state (with \(K=(K_{0})_{\rm v}\)). In Figure 1, \(\varphi^{\prime}{}_{\rm c}\) and \(\
u_{\rm c}\) denote the critical values of the fill's internal friction angle \(\varphi^{\prime}\) and Poisson's ratio \(\
u\) (defined below). Similarly, Figure 1b illustrates that, for a given Poisson's ratio \(\
u\), the resulting Mohr circle based on the initial elastic stress state (\(\sigma^{\prime}{}_{\rm v}\), \((K_{0})_{\rm v}\sigma^{\prime}{}_{\rm v}\)) may or may not theoretically exceed the Coulomb yield envelope, depending on the value of \(\varphi^{\prime}\). When \(\varphi^{\prime}\) is small enough, \((K_{0})_{\rm v}\sigma^{\prime}{}_{\rm v}\leq K_{\rm a}\sigma^{\prime}{}_{\rm v}\), and the stress state will be active (with \(K=K_{\rm a}\)).