# Assessing the State of Structural Foundations in Permafrost Regions by Means of Acoustic Testing
## 2 Methodology. Problem Statement  
### Soil Model Selection  
According to the norms and regulations governing the construction of buildings and structures on the territory of the Russian Federation in permafrost regions [25; 26; 27], it is mandatory to monitor foundation deformations during the entire life cycle of a structure. The following criterion is used to monitor foundation condition [27]:  
\[S_{f}\leq S_{u} \tag{1}\]  
where \(S_{f}\) is the deformation of the plastic-frozen foundation under the load caused by the structure, \(S_{u}\) is the maximum permissible deformation of the structural foundation for the estimated life cycle, which is determined according to [27]. To calculate foundation settlement, various mathematical models are used. Let us consider two key models: the linear elastic-perfectly plastic Mohr-Coulomb (MC) model [28; 29] and the Hardening Soil (HS) model [30; 31; 32]. The major disadvantage of the MC model is the invariance of the deformation modulus over the entire stress range. The relationship between stresses and strains in the MC model is described by [PERSON]'s law [28; 29]:  
\[\{\sigma\}=[D]\{\epsilon\} \tag{2}\]  
\[[D]=\frac{E_{\text{G}}}{(1-2\mu)(1+\mu)}\left[ \begin{array}{cccccc}A&\mu&\mu &0&0&0\\ 0&A&\mu&0&0&0\\ 0&0&A&0&0&0\\ 0&0&0&B&0&0\\ 0&0&0&\mu&B&0\\ 0&0&0&\mu&\mu&B\end{array} \right] \tag{3}\]  
\[A=1-\mu \tag{4}\]  
\[B=(1-2\mu)\backslash 2 \tag{5}\]where \(\{\sigma\}\) is the stress tensor, \([D]\) is the secant elasticity matrix, \(\{\epsilon\}\) is the strain tensor, \(E_{G}\) is the deformation modulus, and \(\mu\) is Poisson's ratio. Unlike the MC model, the HS model takes into account the change in the deformation modulus depending on the stress-strain state. To do this, it factors in three moduli: the modulus of deformation that takes into account stiffness under deviator stress \(E_{50}\), the modulus of unloading and reloading \(E_{ur}\), and the modulus that takes into account stiffness under compression \(E_{ood}\). Each of them depends on the value of confining pressure [30; 31; 32]:  
\[E_{i}=E_{i}^{ref}\left(\frac{c\cos\varphi-\ \sigma_{3}\sin\varphi}{c\cos \varphi+p^{ref}\sin\varphi}\right)^{m} \tag{6}\]  
where \(E_{i}=E_{50}\); \(E_{ur}\); \(E_{ood}\) is the modulus value for a given elementary volume of soil, \(E_{i}^{ref}=E_{50}^{ref}\); \(E_{ur}^{ref}\); \(E_{ood}^{ref}\) is the reference value of the modulus, \(p^{ref}\) is the reference pressure, \(c\) is adhesion, and \(\varphi\) is the angle of internal friction. The key factor determining whether it is possible to use one model or another to solve problems of monitoring the state of permafrost soils is the possibility of finding the parameters of the model in the course of field measurements on site. The authors of [32; 33; 34; 29] analysed the methods used to find soil model parameters. Table 1 summarizes their results. As Table 1 shows, in order to find the parameters of the HS model, it is necessary to conduct laboratory tests. Direct field methods have not yet been developed. As for the MC model, Expression (2) and Table 1 show that its parameters can be determined on site. The authors of [28; 36] note that the HS model produces better results than the MC model, in comparison with the results of on-site settlement measurements. However, the MC model still produces sufficient results and is widely used in Russia for preliminary calculations [36]. The MC theory serves as a foundation for the deformation theory of plasticity (DTP), in which the deformation modulus \(E_{G}\) depends on the maximum principal stress \(\sigma_{1}\)[37]:  
\[E_{G}\equiv E_{G}(\sigma_{1}) \tag{7}\]  
Therefore, in order to find deformations using DTP, it is necessary to find (7) in the course of tests over the entire range of the stress acting on the soil. To find the components of the stress-strain state of a permafrost foundation, let us consider a standard model of a permafrost foundation (Figure 1).