## 5 Conclusions  
The exact asymptotic representation of solutions in the vicinity of envelopes of characteristics has been found for the double slip and rotation model under plane strain and axially symmetric conditions. This representation coincides with the behavior of specific solutions found in [11; 12; 13; 16; 17]. This representation also coincides with that for isotropic rigid plastic solids obeying an arbitrary smooth yield criterion [18], and for some anisotropic yield criteria [19]. It is seen from Equations (33), (34) and (38) that the stress and velocity fields are singular in the vicinity of the envelope of characteristics, in the sense that some derivatives of the stress and velocity components approach infinity near the envelope.  
The applied aspect of the asymptotic representation of solutions derived in the present paper is twofold. Firstly, as known from other models of rigid plasticity, numerical solutions using traditional finite elements do not converge if the exact solution is described by equations similar to (33), (34) and (38) [32; 33]. The asymptotic representation given in (33), (34) and (38) can be used in conjunction with the extended finite element method, outlined in [29], for developing an accurate numerical approach for solving corresponding boundary value problems.  
Secondly, the asymptotic representation given in (33), (34) and (38) can be used in conjunction with models for soil-structure interfaces [21; 22; 23]. In particular, an envelope of characteristics usually coincides with the interface between a structure and material, for example between a structure and soil. In this case, experiment confirms some mathematical features of the general theoretical solution. If an envelope of characteristics coincides with the interface then the regime of sliding occurs independently of the friction law chosen. This means that the friction stress is independent of the roughness of the contact surface. It is in general known that the surface roughness affects the material behavior near the interface [34; 35]. However, this effect is usually significant only if the surface roughness is small enough. For example, in the case of the interface between Yongdinghe sand and low carbon steel the friction stress is practically independent of the surface roughness if \(R_{n}>R_{cr}=0.1\) where \(R_{n}\) is the relative interface roughness and \(R_{cr}\) is its critical value [35]. In [35], the soil-structure interfaces are classified as smooth or rough depending on the value of \(R_{n}\). When \(R_{n}<R_{cr}\) the interface is smooth. If \(R_{n}>R_{cr}\) then the interface is rough. In the latter case, strain localization occurs near the interface. This feature of material behavior is in agreement with (38) as this equation predicts that the shear strain rate approaches infinity as \(x_{2}\to 0\). Moreover, it has been found in [35] that the residual friction angle is close to the internal friction angle of sand if \(R_{n}>R_{cr}\). This feature is in agreement with the second equation in (33). Therefore, it is reasonable to assume that the condition that an envelope of characteristics coincides with the soil-structure interface is adequate when \(R_{n}>R_{cr}\). In this case, it is natural to expect the similarity between the behavior of soil and interface. This similarity has been investigated in [(23)].  
Conceptualization, S.A.; writing, Y.W.; formal analysis, E.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Natural Science Foundation of Hebei Province of China with Grant No. E2019202224 and the RFBR (Project No.18-51-76001).  
The authors declare no conflict of interest.