## 1 Introduction  
The term critical state is defined as a limit state in which a soil element continues to deform at constant void ratio (or volumetric strain), mean effective stress, and deviatoric stress [1; 2; 3; 4]. [PERSON] [5] defined this state symbolically as  
\[\left\{ \begin{array}{l}\mathrm{d}\epsilon_{q}\
eq 0\\ \mathrm{d}e=\mathrm{d}e_{v}=\mathrm{d}p=\mathrm{d}q=\mathrm{d}\eta=0\end{array} \right. \tag{1}\]  
where \(\epsilon_{q}\), \(e\), \(e_{v}\), \(p\), \(q\), and \(\eta\) (=\(q/p\)) are the deviatoric strain, void ratio, volumetric strain, mean effective stress, deviatoric stress, and stress ratio of the soil element, respectively. The concept of critical state is well accepted and understood in soil mechanics [6; 7; 8; 9; 10; 11] and rock mechanics [12; 13]. This concept is also considered to have applications to new types of composite geo-materials in civil engineering, such as soil-waste mixtures [14; 15]. It is not unusual to use standard drained triaxial compression tests (in which \(\mathrm{d}q/\mathrm{d}p=3\) and \(\epsilon_{q}=\epsilon_{a}-\epsilon_{v}/3\), where \(\epsilon_{a}\) is the axial strain) to obtain the critical state locus in the confinement-shear-volume (\(p\)-\(q\)-\(e\)) space [4; 14; 16]. The locus is normally characterized using a linear failure envelope in the \(q\)-\(p\) plane and a critical state line in the \(e\)-\(p\) plane. The slope of the failure envelope, \(q_{\mathrm{cs}}/p_{\mathrm{cs}}\) (where the subscript cs indicates 'critical state'), is called the critical state stress (or friction) ratio \(\eta_{\mathrm{cs}}\), which is often denoted by \(M\). The critical state line depicts the relationship between the void ratio at the critical state \(e_{\mathrm{cs}}\) and the corresponding mean effective stress \(p_{\mathrm{cs}}\). For dense ('dilative') cohesionless granular geo-materials, the critical state is typically attained at large strains upon the end of the strain-softening process. As a consequence, it is not uncommon that, within the limits of the triaxial equipment, the critical state is not reached at the end of the tests (e.g., [14; 17; 18; 19; 20]). In this situation, it becomes necessary to estimate the values of \(e_{\text{cs}}\), \(p_{\text{cs}}\), and \(M\) from measured data that are free of interference from strain localization into shear bands. This interference could be minimized by utilizing the testing devices and procedures recommended by [21; 22]. Three methods are available for estimating the critical state void ratio \(e_{\text{cs}}\) or volumetric strain \(\varepsilon_{\text{pcs}}\) (which is given by \(\ln[(1+e_{0})/(1+e_{\text{cs}})]\), where \(e_{0}\) is the initial ('pre-shearing') void ratio). The first method extrapolates the \(\varepsilon_{\text{v}}\)-\(\varepsilon_{q}\) (or \(\varepsilon_{\text{v}}\)-\(\varepsilon_{a}\)) data to a specified'most probable' value of \(\varepsilon_{q}\) (or \(\varepsilon_{a}\)) postulated to be at the onset of the critical state, say, 35% [14; 18; 23], to an \(\varepsilon_{q}\) (or \(\varepsilon_{a}\)) value approaching infinity [24; 25; 26] or to an \(\varepsilon_{q}\) value estimated from extrapolation of the \(\eta\)-\(\varepsilon_{q}\) data to the critical state stress ratio \(\eta_{\text{cs}}\) (\(M\)) [27]. Different curve-fitting functions, data sets, and data set sizes have been used in this extrapolation, including hyperbolic [25] and exponential [26] functions for fitting the entire data, and sine functions for fitting a subset of the data from the minimum volumetric strain to the end of the test [27]. It seems that the selection of a curve-fitting function and a data set was based on personal observation and preference. The second method extrapolates the void ratio-stress ratio (\(e\)-\(\eta\)) trend to an estimated critical state stress ratio \(\eta_{\text{cs}}\)[28]. The \(e\)-\(\eta\) data set from the maximum ('peak') stress ratio \(\eta_{\text{max}}\) to the end of the test is used for extrapolation.