Many different failure criteria, being part of soil constitutive models, can be found in the literature. The basic one to which all others are usually compared is the Mohr–Coulomb condition, due to its simplicity. It proved its usefulness in classic triaxial compression, where a cylindrical soil sample is subjected to an axisymmetric state of stress (σ_{1}, σ_{2} = σ_{3}, Fig. 1a). However, the problem of soil strength is more complex when true triaxial stress conditions are considered, and the principal stresses σ_{1}, σ_{2}, σ_{3} have different values (Fig. 1b).
This paper considers the most frequently used isotropic soil failure criteria (yield surfaces): Mohr–Coulomb, Drucker–Prager, Matsuoka–Nakai and Lade–Duncan. The detailed descriptions of the selected criteria are in Drucker and Prager (1952), Lade and Duncan (1975), Matsuoka and Nakai (1974) and Matsuoka and Nakai (1985).
The research on soil failure continues, and except listed above, there are also other criteria proposed in the literature. Georgiadis et al. (2004), Houlsby (1986) or Liu et al. (2012) suggest a yield surface which is a combination of the criteria mentioned above. Lagioia and Panteghini (2014) present a reformulation of the original Matsuoka–Nakai criterion to overcome the limitations which make its use in a stress point algorithm problematic. A novel soil strength criterion, where the cube root of principal stresses is constant, is proposed by Shao et al. (2017), and it shows that the Lade–Duncan criterion is not just an empirical one, as previously thought, but has a physical background.
Mohr–Coulomb failure condition is built on simplifying assumption that soil behaviour is governed by the difference between maximum and minimum principal stress (σ_{1}  σ_{3}) and does not depend on the intermediate principal stress (σ_{2}). It is clear that such simplification may be valid in some special conditions only. The influence of the intermediate principal stress σ_{2} on soil shear strength is discussed by Bishop (1971), Kulhawy and Mayne (1990) or Ochiai and Lade (1983). Barreto and O’Sullivan (2012) examined the effect of interparticle friction (
The contribution of the intermediate principal stress σ_{2} to the plane strain soil strength is particularly studied for practical reasons: to analyse longitudinal foundations, slopes, retaining walls and long excavations. Besides, many experimental techniques, like fullfield displacement measurements by digital image correlation (DIC), are usually performed on rectangular plane strain models. Also ground flow problems, static liquefaction and instability are studied in plane strain conditions (Wanatowski and Chu, 2007; Wanatowski et al. 2010).
Experimental investigations of dense granular soils have shown that the plane strain shear strength is higher than that in the axisymmetric conditions (Alshibli et al., 2003). In the case of loose soils, there is no such difference; see Cornforth (1964), Lee (1970), Rowe (1969) and Schanz and Vermeer (1996).
Soil strength can be defined depending on strain conditions (plane or threedimensional), but also on the range of strains (peak or critical strength). In this paper, the peak soil strength is considered a measure of soil failure state.
Peak strength is the maximum shear stress (maximum
Unlike critical soil strength, peak strength depends on the initial density of soil. Most of the research on soil peak strength uses Mohr–Coulomb condition and so the dependence of peak friction angle on soil density. Been and Jefferies (1985, 1986) have shown the relationship between the peak friction angle and the soil state parameter, defined as the difference in void ratio between the initial and steadystate, at the same mean effective stress.
Bolton (1986) studied the relationship between the mobilized friction angle, critical state friction angle and soil relative density in plane strain conditions and proposed the equation describing this relationship. Chakraborty and Salgado (2010) confirmed Bolton's theory for low confining pressures (triaxial and plane strain tests). The effect of confining pressure on peak friction angle in the process of grain crushing is shown in Yamamuro and Lade (1996): as confining pressure increases, the peak friction angle decreases. Sadrekarimi and Olson (2011) or Sarkar et al. (2019) show, in turn, that there exists no clear relationship between the peak friction angle and the effective stress.
The direct dependence of the friction angle on the initial soil porosity was shown already in Lee (1970), where analysed the data obtained in drained tests by Bishop (1961) and Cornforth (1964). The tests were conducted at confining pressure of 275 kPa, both in axisymmetric and plane strain conditions.
Fig. 3 shows their results, completed by the data obtained in this study for Skarpa sand at similar confining pressures between 278 kPa and 295 kPa (Tables 2 and 4). The same tendency: a decrease in the internal friction angle with growing sample porosity is observed in true triaxial tests on Skarpa sand under plane strain conditions, but no quantitative agreement is found, because they are two different soils.
The main purpose of this study is to establish, both experimentally and semitheoretically, the parameters characterizing different soil failure criteria, presented in Section 2, and their relationship to the internal friction angle in a given range of initial soil densities in plane strain state.
The experimental way of finding the parameters involves determining the set of principal stress values σ_{1}, σ_{2} and σ_{3}, corresponding to the soil peak strength (Fig. 2). The same parameters are calculated semitheoretically using the approach proposed by Vikash and Prashant (2010). The associated flow rule and plane strain condition are used to express the parameters as functions of the plane strain friction angle. The basic difference between the experimental and semitheoretical approach concerns the intermediate stress σ_{2}. In experiments, its value comes from the direct measurements; in calculations, it is determined on the base of the accepted assumptions and is different for each failure criterion. Validation of the Vikash and Prashant approach on the base of stress measurements is another purpose of the study.
The soil failure criteria are usually formulated using stress invariants, independent of the choice of the coordinate system. In standard triaxial conditions, two invariants of the stress tensor are enough to describe the state of stress and any load path in the stress space. In the true triaxial state, three different principal stresses exist; therefore, one more invariant is needed. Often, combinations of the basic invariants of the stress tensor are used to formulate failure criteria.
Soil failure state is graphically represented, in 3D stress space, by the surface which separates the allowable stress states from the states of uncontrolled plastic flow and is called yield surface. It is accepted in this study that the yield surface corresponds to the stress states at which a soil reaches its maximum (peak) strength. The failure criteria considered in this paper describe the shape of the yield surface and differ in crosssection on the deviatoric plane, perpendicular to the hydrostatic axis σ_{1} = σ_{2} = σ_{3} (Fig. 4).
The mean stress
The basic invariants of the total stress tensor σ:
The basic invariants of the deviatoric stress tensor
The mean stress
In classical triaxial compression
Lode angle is in some studies replaced by its alternative – the Bishop's parameter
Soil parameters
Transforming equations (11)–(14) gives the expressions (15)–(18) which allow to determine the parameters of failure criteria using the measured values of σ_{1}^{max}, σ_{2} and σ_{3}:
The associated flow rule comes from the rigorous formulation of the plasticity theory, while the nonassociated one is only semitheoretically postulated. The latter was introduced for soils to reduce the mismatch between measured and theoretically predicted volumetric strains in element tests. However, the solution of this particular problem introduces another problem, namely the noncoaxiality of the stress and strain tensors. It is difficult to measure experimentally and limits the application of the upperbound theorem, based on the assumption of the associated flow rule and frequently used in limit analysis solutions of soil mechanics boundary value problems (e.g. Deusdado et al., 2016, di Santolo et al., 2012).
As a result, both the associated and nonassociated flow rules are still used to model soil behaviour, depending on the nature of the problems studied.
Liu (2013) shows various results of triaxial compression tests on sands and their simulations using both the associated and nonassociated flow rules. Similar simulations for more complex stress paths in the
Plastic flow rule can describe soil deformation at failure state. Its general formulation, as a nonassociated flow rule, is given by Eq. (19):
In the Appendix, the set (20) is presented in the expanded form, specific for each failure condition. The set can be completed by Eq. (21), valid in plane strain conditions, to express
To solve (20), first the intermediate principal stress σ_{2} is determined as a function of σ_{1} and σ_{3}, separately for DP, LD and MN yield conditions:
It can be seen from Eqs. (22)–(24) that each condition gives the different expression for the intermediate stress σ_{2}. Finally, the following expressions for
The series of tests in true triaxial apparatus in plane strain conditions were performed to verify both approaches of determining failure criteria parameters. Limiting the strains to plane (twodimensional) case induces some partly controlled threedimensional stress state, where σ_{2} comes from the soil reaction and cannot be applied a priori, but has to be measured. The nonzero σ_{2} in plane strain condition has always been a problem in interpreting the results of standard 2D soil mechanics tests.
The tests in this study were carried out on Skarpa sand samples at different confining pressures and initial void ratios. All the tests were performed in dry conditions. Basic properties of Skarpa sand are collected in Table 1.
Parameters of Skarpa sand.
Specific density [kg/m^{3}]  2650 
Mean particle size [mm]  
Uniformity coefficient [ − ]  
Minimum void ratio [ − ]  
Maximum void ratio [ − ] 
The plane strain tests, with deformation fixed in
Fig. 7a shows the sample ready for the test, with the side platens fixed. After installing the sample in the testing chamber (measurement cell, Fig. 6), its doors are closed and it is filled with water, then the side platens are gently pressed against the specimen until the difference between horizontal stresses σ_{2}  σ_{3} is about 2–3 kPa, to secure the proper contact. To reduce friction between the membrane and the side plates, the plates are lubricated with a special lubricant.
Specimen base pedestal is connected to the pressure/volume controller (back pressure controller, Fig. 6), which is used to apply and measure the pore water pressure and volume changes. Cell pressure is controlled by a pneumatichydraulic system (a cell pressure controller), where required value can be set. In addition, TTA is also equipped with a cell pressure transducer (see Fig. 6) located inside the chamber.
Vertical actuators shown in Fig. 6 are used to apply the major σ_{1} principal stress. The horizontal stress σ_{3} is applied through the water pressure in the chamber, and σ_{2} is recorded by a gauge located on vertical actuators.
In order to measure the specimen displacements in
Soil samples are prepared in a membranelined split mould by air pluviation. This method involves preparing a soil sample using a funnel with a nozzle of approximately 5 mm. The weighed sand is placed in the funnel at the selected distance from the centre of the mould. The height of the funnel and the mass of sand are determined by ‘trial and error’ to obtain the appropriate relative density (Li et al., 2018).
Eleven tests were carried out according to the same procedure. The tests consisted of two phases:
Phase 1 (isotropic compression): the sample is loaded isotropically by increasing water pressure in the testing chamber (σ_{3}^{c} in Table 2). This is not carried out under plane strain conditions. The side plates, pressed against the sample, move along with it, while a constant set value of lateral stress (about 2–3 kPa) is maintained.
Phase 2 (shear): the sample is vertically loaded with a constant vertical displacement rate of 15 mm/hour at constant chamber pressure (σ_{1} = σ_{3}^{c} = const) in plane strain conditions.
The samples’ porosities were between 0.316 and 0.36 (the relative densities
Table 2 contains the initial conditions of all tested samples: the initial void ratio at the start of Phase 1 (
The basic results of the eleven tests listed in Table 1 are gathered in Figs 8–11. Each of the figures shows the full set of curves for all the tests: deviator stress (Fig. 8), maximum principal stress (Fig. 9), principal stress in the direction of fixed strain (Fig. 10) and volumetric strain (Fig. 11), as functions of the axial strain.
The sets are highly varied because they correspond to the samples of different densities (medium to very dense), tested at different confining pressures (Table 2). The axial strain at which the peak strength occurs is determined for each test on the base of Fig. 8, and then, the corresponding values of σ_{1}^{max}, σ_{2} and σ_{3} are established, like it is demonstrated in Fig. 2.
Initial test conditions.



009_17_MC_5  0.585  0.376  391  0.563  0.465  0.36 
033_17_MC_14  0.559  0.482  293  0.548  0.527  0.354 
012_18_MC_21  0.541  0.555  292  0.532  0.592  0.347 
013_18_MC_22  0.519  0.645  146  0.514  0.665  0.339 
010_18_MC_19  0.517  0.653  195  0.508  0.690  0.337 
001_18_MC_15  0.521  0.637  191  0.499  0.727  0.333 
010_15_MC_1  0.496  0.739  278  0.490  0.763  0.329 
009_18_MC_18  0.488  0.771  292  0.480  0.804  0.324 
008_18_MC_17  0.489  0.767  295  0.476  0.820  0.322 
028_17_MC_12  0.467  0.857  52  0.462  0.878  0.316 
031_17_MC_13  0.469  0.849  99  0.462  0.878  0.316 
Fig. 11 shows that dilative behaviour is observed in all the tested samples. The volumetric strain reaches the maximum (compression is positive), which is typical for dense samples. The maximum value of the deviator stress
Characteristics of peak strength state for the tested samples.
009_17_MC_5  1402  653  391  815  909  0.26  14.46 
033_17_MC_14  1072  479  293  615  705  0.24  13.21 
012_18_MC_21  1184  459  292  645  821  0.19  10.14 
013_18_MC_22  678  262  146  362  485  0.22  11.97 
010_18_MC_19  902  355  195  484  642  0.23  12.46 
001_18_MC_15  870  332  191  464  621  0.21  11.35 
010_15_MC_1  1291  529  278  699  914  0.25  13.76 
009_18_MC_18  1483  528  292  768  1092  0.20  10.78 
008_18_MC_17  1396  506  295  732  1012  0.19  10.40 
028_17_MC_12  287  109  52  149  212  0.24  13.44 
031_17_MC_13  508  191  99  266  372  0.22  12.38 
Figs 8–11 represent the whole course of the experimental tests, showing pre and postpeak behaviour. Only the prepeak part of the tests, defined on the basis of Figs 8–11, is analysed in the paper. Before the onset of localization, the deviator peak strength (not the critical strength) is used. The literature shows that localization is observed after the deviator peak strength has been reached, e.g. Leśniewska et al (2012) and Desrues and Viggiani (2004).
Fig. 12 shows the relationship between principal stress components, corresponding to the peak soil strength
Fig. 13 collects the values of Lode angle calculated using Eq. (8) and the data from Table 3, and it suggests that there is no statistically significant difference in Lode angle due to varying principal stress σ_{2} or confining pressure σ_{3}. The Lode angle is considered constant at
The constant and relatively low value of Lode angle obtained for all the tests from Table 3 confirms that in plane strain conditions, the influence of the intermediate stress σ_{2} on the soil peak behaviour is limited and does not depend on the confining pressure. The question remains, how to estimate the magnitude of this influence.
Values of parameters
Characteristic parameters of Drucker–Prager, Matsuoka–Nakai and Lade–Duncan soil failure criteria, obtained from direct stress measurements (A) and the associated flow rule assuming plane strain conditions (B).











009_17_MC_5  34.3°  0.21  11.7  40.9  1181.5  740.4  896.5  0.179  11.7  39.6 
033_17_MC_14  34.8°  0.22  11.9  41.7  904.8  560.4  682.5  0.181  11.8  40.0 
012_18_MC_21  37.2°  0.25  12.5  45.7  1007.5  588.0  738  0.190  12.3  42.5 
013_18_MC_22  40.2°  0.26  13.2  49.4  583.8  314.6  412  0.202  13.1  46.3 
010_18_MC_19  40.1°  0.26  13.1  49.0  776.3  419.3  548.5  0.201  13.1  46.2 
001_18_MC_15  39.8°  0.26  13.1  49.0  747.8  407.6  530.5  0.200  13.0  45.7 
010_15_MC_1  40.2°  0.25  13.1  48.6  1111.5  599.1  784.5  0.202  13.1  46.3 
009_18_MC_18  42.1°  0.27  13.8  53.4  1287.1  658.1  887.5  0.209  13.7  49.1 
008_18_MC_17  40.6°  0.27  13.4  50.9  1203.9  641.7  745.5  0.203  13.2  46.9 
028_17_MC_12  43.9°  0.27  14.3  55.3  251.0  122.1  169.5  0.215  14.3  52.0 
031_17_MC_13  42.4°  0.27  13.8  52.9  441.3  224.3  303.5  0.209  13.7  46.5 
The data included in Table 4 are presented in Figs 14–17. First, the relationship between the intermediate principal stress σ_{2} for Drucker–Prager, Matsuoka–Nakai and Lade–Duncan criteria, obtained from Eqs. (22)–(24) and measured in the experiments, is analysed, and then Vikash and Prashant solution, given by Eqs. (25)–(27), is verified experimentally.
Fig. 14a presents σ_{2}^{DP}, σ_{2}^{MN} and σ_{2}^{LD} as functions of the experimental σ_{2}, taken from Table 3. A perfect fit between the calculated and measured values would mean that they lie on the dashed line σ_{2}^{calc} = σ_{2}^{exp}, shown in the figure.
A linear relationship between the calculated and measured intermediate stress σ_{2} for all three selected criteria is found, but only Matsuoka–Nakai is close to the perfect fit. The ratio R = σ_{2}^{calc} / σ_{2}^{exp} plotted in Fig. 14b shows three constant trends: 2.15 for Drucker–Prager, 1.52 for Lade–Duncan and 1.17 for Matsuoka–Nakai condition, with perfect fit equal to 1. It means that the Vikash and Prashant (2010) approach, highly overestimates the influence of the intermediate stress in plane strain conditions for Drucker–Prager, gives about 50% overestimation for Lade–Duncan and is close to measured values for Matsuoka–Nakai criterion.
Fig. 15 shows the dependence of the intermediate stress σ_{2}, measured and calculated by Eqs. (25)–(27), on the initial relative density of Skarpa sand. There is no clear tendency visible because the tests presented in the paper were performed at different confining pressures. If the ratio of σ_{2} to the confining pressure σ_{3}^{c} is examined instead (Fig. 16), the linear trends appear both in the case of measured and calculated values and again the Matsuoka–Nakai criterion is closest to reality.
Parameters
It is possible in Vikash and Prashant approach to replace σ_{1} and σ_{3} by the internal friction angle
Fig. 17(a) shows the relation between the MC friction angle
There is no such a good fit in case of Drucker–Prager and Lade–Duncan conditions. As Matsuoka–Nakai criterion is most commonly used to estimate soil strength in complex stress states, this finding can help to determine parameters necessary for numerical analysis of plane strain problems in a relatively simple way.
All three failure conditions give linear increase of
The linear fits for the failure criteria parameter
The linear fits
Drucker–Prager 


Lade–Duncan 


Matsuoka–Nakai 

The linear fits
Drucker–Prager 


Lade–Duncan 


Matsuoka–Nakai 

Values of Pearson's correlation coefficient
To estimate more quantitatively the difference between the two approaches of determining the failure criteria parameters, two statistical measures are employed:
Similarly,
Table 7 shows the summary of the calculations carried out for each of the criteria.
Average relative difference of parameters
Drucker–Prager 


Lade–Duncan 


Matsuoka–Nakai 


The smallest average differences are for Matsuoka–Nakai criterion
A series of shear tests on Skarpa sand was carried out in true triaxial apparatus in plane strain conditions. Experiments covered a wide range of initial soil densities and confining pressures, with the initial relative density index
The semitheoretical approach proposed by Vikash and Prashant (2010) is compared with the results of the calculations based on the measurements of the full set of principal stresses. Both approaches differ in the way of obtaining the value of intermediate stress σ_{2}, which can be measured independently or calculated on the base of σ_{1} and σ_{3} measurement, assuming associative flow rule.
The most important conclusion for soil testing in plane strain conditions is that using Matsuoka–Nakai failure condition, the associated flow rule takes properly into account the effect of nonzero intermediate stress σ_{2}, without a need to measure it.
Linear fits to describe the relationship between
The constant and relatively low value of Lode angle obtained for all the TTA tests confirms that in plane strain conditions, the influence of the intermediate stress σ_{2} on the soil peak behaviour is limited and does not depend significantly on the confining pressure. The trend for Lode angle is constant (
Average relative difference of parameters κ and intermediate principal stress σ2, determined by the two approaches: full set of principal stresses and Vikash and Prashant proposal, for Drucker–Prager, Lade–Duncan and Matsuoka–Nakai failure criteria.
Drucker–Prager 


Lade–Duncan 


Matsuoka–Nakai 


The linear fits κexp(ϕps)and the corresponding statistics Pearson's correlation coefficients.
Drucker–Prager 


Lade–Duncan 


Matsuoka–Nakai 

Characteristics of peak strength state for the tested samples.
009_17_MC_5  1402  653  391  815  909  0.26  14.46 
033_17_MC_14  1072  479  293  615  705  0.24  13.21 
012_18_MC_21  1184  459  292  645  821  0.19  10.14 
013_18_MC_22  678  262  146  362  485  0.22  11.97 
010_18_MC_19  902  355  195  484  642  0.23  12.46 
001_18_MC_15  870  332  191  464  621  0.21  11.35 
010_15_MC_1  1291  529  278  699  914  0.25  13.76 
009_18_MC_18  1483  528  292  768  1092  0.20  10.78 
008_18_MC_17  1396  506  295  732  1012  0.19  10.40 
028_17_MC_12  287  109  52  149  212  0.24  13.44 
031_17_MC_13  508  191  99  266  372  0.22  12.38 
Initial test conditions.



009_17_MC_5  0.585  0.376  391  0.563  0.465  0.36 
033_17_MC_14  0.559  0.482  293  0.548  0.527  0.354 
012_18_MC_21  0.541  0.555  292  0.532  0.592  0.347 
013_18_MC_22  0.519  0.645  146  0.514  0.665  0.339 
010_18_MC_19  0.517  0.653  195  0.508  0.690  0.337 
001_18_MC_15  0.521  0.637  191  0.499  0.727  0.333 
010_15_MC_1  0.496  0.739  278  0.490  0.763  0.329 
009_18_MC_18  0.488  0.771  292  0.480  0.804  0.324 
008_18_MC_17  0.489  0.767  295  0.476  0.820  0.322 
028_17_MC_12  0.467  0.857  52  0.462  0.878  0.316 
031_17_MC_13  0.469  0.849  99  0.462  0.878  0.316 
Characteristic parameters of Drucker–Prager, Matsuoka–Nakai and Lade–Duncan soil failure criteria, obtained from direct stress measurements (A) and the associated flow rule assuming plane strain conditions (B).











009_17_MC_5  34.3°  0.21  11.7  40.9  1181.5  740.4  896.5  0.179  11.7  39.6 
033_17_MC_14  34.8°  0.22  11.9  41.7  904.8  560.4  682.5  0.181  11.8  40.0 
012_18_MC_21  37.2°  0.25  12.5  45.7  1007.5  588.0  738  0.190  12.3  42.5 
013_18_MC_22  40.2°  0.26  13.2  49.4  583.8  314.6  412  0.202  13.1  46.3 
010_18_MC_19  40.1°  0.26  13.1  49.0  776.3  419.3  548.5  0.201  13.1  46.2 
001_18_MC_15  39.8°  0.26  13.1  49.0  747.8  407.6  530.5  0.200  13.0  45.7 
010_15_MC_1  40.2°  0.25  13.1  48.6  1111.5  599.1  784.5  0.202  13.1  46.3 
009_18_MC_18  42.1°  0.27  13.8  53.4  1287.1  658.1  887.5  0.209  13.7  49.1 
008_18_MC_17  40.6°  0.27  13.4  50.9  1203.9  641.7  745.5  0.203  13.2  46.9 
028_17_MC_12  43.9°  0.27  14.3  55.3  251.0  122.1  169.5  0.215  14.3  52.0 
031_17_MC_13  42.4°  0.27  13.8  52.9  441.3  224.3  303.5  0.209  13.7  46.5 
The linear fits κexp(IDc) {\kappa ^{\exp }}\left( {I.D^c} \right) and the corresponding Pearson's correlation coefficients.
Drucker–Prager 


Lade–Duncan 


Matsuoka–Nakai 

Parameters of Skarpa sand.
Specific density [kg/m^{3}]  2650 
Mean particle size [mm]  
Uniformity coefficient [ − ]  
Minimum void ratio [ − ]  
Maximum void ratio [ − ] 
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