Characteristic parameters of soil failure criteria for plane strain conditions – experimental and semi-theoretical study

Peak strength is the maximum shear stress (maximum q value, Eq. (7)) that the soil can transfer. It is influenced by inter-particle friction, grain arrangement, grain crushing and soil dilation (Mitchell and Soga, 2005). Fig. 2 shows the exemplary results of soil shearing in plane strain conditions (details of the experiment are included in Tables 2 and 3). Fig. 2a defines the peak strength q_max on q − ɛ₁ (axial strain) graph, and Fig. 2b shows the corresponding set of principal stress values. The major principal stress σ₁ is always the largest, whereas the lateral stress σ₂ is the intermediate one. In the paper, the soil mechanics sign convention is used (compression positive). The invariant q is defined under full stress conditions, including the meaning of σ₂; see formula (7).

Figure 2

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 steady-state, 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.

Figure 3

The main purpose of this study is to establish, both experimentally and semi-theoretically, 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 σ₁, σ₂ and σ₃, corresponding to the soil peak strength (Fig. 2). The same parameters are calculated semi-theoretically 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 semi-theoretical approach concerns the intermediate stress σ₂. 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 mean stress p and deviator stress q are most frequently employed for axisymmetric stress conditions. In the case of the true triaxial state, they are usually completed by Lode angle θ. In case of dry granular material invariants, p, q and θ can be expressed using the basic invariants of total or deviatoric stress tensor, defined by Eqs. (1)–(5):

• a)

  The basic invariants of the total stress tensor σ: (1) I1=σii=σ1+σ2+σ3, {I_1} = {\sigma _{{\rm{ii}}}} = {\sigma _1} + {\sigma _2} + {\sigma _3}, (2) I2=12(σiiσjj−σijσij)=σ1σ2+σ1σ3+σ2σ3, {I_2} = {1 \over 2}\left( {{\sigma _{{\rm{ii}}}}{\sigma _{{\rm{jj}}}} - {\sigma _{{\rm{ij}}}}{\sigma _{{\rm{ij}}}}} \right) = {\sigma _1}{\sigma _2} + {\sigma _1}{\sigma _3} + {\sigma _2}{\sigma _3}, (3) I3=detσ=σ1σ2σ3, {I_3} = \det {\boldsymbol {\sigma }} = {\sigma _1}{\sigma _2}{\sigma _3},

• b)

  The basic invariants of the deviatoric stress tensor s=σ−13tr(σ)1 {\boldsymbol {s}} = {\boldsymbol {\sigma }} - {1 \over 3}{\rm{tr}}\left( {\boldsymbol {\sigma }} \right){\bf{1}} : (4) J2=12s:s=12sijsij=16((σ1−σ2)2+(σ1−σ3)2+(σ2−σ3)2) {J_2} = {1 \over 2}{\boldsymbol {s:s}} = {1 \over 2}{s_{ij}}{s_{ij}} = {1 \over 6}\left( {{{\left( {{\sigma _1} - {\sigma _2}} \right)}^2} + {{\left( {{\sigma _1} - {\sigma _3}} \right)}^2} + {{\left( {{\sigma _2} - {\sigma _3}} \right)}^2}} \right) (5) J3=dets=13sijsjkski {J_3} = {\rm{det}}{\boldsymbol {s}} = {1 \over 3}{s_{ij}}{s_{jk}}{s_{ki}}

The mean stress p, deviator stress q and Lode angle θ are defined by Eqs. (6), (7) and (8).

(6) p=13(σ1+σ2+σ3)=3I1 p = {1 \over 3}\left( {{\sigma _1} + {\sigma _2} + {\sigma _3}} \right) = 3{I_1}

(7) q=12(σ1−σ2)2+(σ1−σ3)2+(σ2−σ3)2=3J2 q = {1 \over {\sqrt 2 }}\sqrt {{{\left( {{\sigma _1} - {\sigma _2}} \right)}^2} + {{\left( {{\sigma _1} - {\sigma _3}} \right)}^2} + {{\left( {{\sigma _2} - {\sigma _3}} \right)}^2}} = \sqrt {3{J_2}}

(8) θ=arccos(2σ1−σ2−σ3)2(σ1−σ2)2+(σ1−σ3)2+(σ2−σ3)2=13arccos(−33J32J23/2), \theta = \arccos {{\left( {2{\sigma _1} - {\sigma _2} - {\sigma _3}} \right)} \over {\sqrt 2 \sqrt {{{\left( {{\sigma _1} - {\sigma _2}} \right)}^2} + {{\left( {{\sigma _1} - {\sigma _3}} \right)}^2} + {{\left( {{\sigma _2} - {\sigma _3}} \right)}^2}} }} = {1 \over 3}\arccos \left( { - {{3\sqrt 3 {J_3}} \over {2{J_2}^{3/2}}}} \right),

In classical triaxial compression θ = 0°, in triaxial extension θ =60°, generally 0° < θ < 60°.

Lode angle is in some studies replaced by its alternative – the Bishop's parameter b is defined by Eq. (9) and related to θ by Eq. (10). (9) b=σ2−σ3σ1−σ3 b = {{{\sigma _2} - {\sigma _3}} \over {{\sigma _1} - {\sigma _3}}} (10) b=12(1+3tg(θ−30°)) b = {1 \over 2}\left( {1 + \sqrt 3 {\rm{tg}}\left( {\theta - 30^\circ } \right)} \right) An analysis of experimental studies shows that values of b under plane strain conditions lie within the range of 0.18–0.35 (assuming that 10° < θ < 20°) (Tatsuoka et al. 1986; Wanatowski and Chu 2007).

Mohr–Coulomb (M-C) failure surface F_M-C for non-cohesive soil can be described by equation (11): (11) FM-C=12(σ1−σ3)−12(σ1+σ3)sinφ=0, {F_{{\rm{M - C}}}} = {1 \over 2}\left( {{\sigma _1} - {\sigma _3}} \right) - {1 \over 2}\left( {{\sigma _1} + {\sigma _3}} \right)\sin \varphi = 0, where σ₁ and σ₃ are the maximum and the minimum principal stress and ϕ_M-C is the internal friction angle. On the deviatoric plane, the M-C criterion is represented by a hexagonal contour (Fig. 4) and ϕ_M-C is constant. F_M-C does not depend on the intermediate stress σ₂.

Drucker–Prager (D-P) failure criterion was established as a generalization of the Mohr–Coulomb condition; see Drucker and Prager (1952), and this criterion for non-cohesive soil is described by equation (12): (12) FD−P=J2−κD−PI1=0, {F_{{\rm{D}} - {\rm{P}}}} = \sqrt {{J_2}} - {\kappa _{{\rm{D}} - {\rm{P}}}}{I_1} = 0, where I₁ and J₂ are given by Eqs. (1) and (4) and κ_D-P is the specific soil parameter. On the deviatoric plane, the D-P is represented by a circle (Fig. 4).

Lade–Duncan (L-D) criterion – true triaxial experiments (especially in plane strain conditions) prove that shear strength for intermediate values of Lode angle (0° < θ < 60°) is higher than that resulting from the M-C criterion in the classical triaxial test; see Eekelen (1980), Lade and Duncan (1973) and Sławińska (2018). An empirical criterion which takes into account these observations is called Lade–Duncan (L-D) criterion (Eq. (13)): (13) FL−D=I13I3−κL−D=0, {F_{{\rm{L}} - {\rm{D}}}} = {{I_1^3} \over {{I_3}}} - {\kappa _{{\rm{L}} - {\rm{D}}}} = 0, where κ_L-D is the specific soil parameter and I₁, I₃ are defined by Eqs. (1) and (3).

Matsuoka–Nakai (M-N) criterion is proposed on the basis of spatially mobilized plane concept (Matsuoka and Nakai, 1974). This criterion, based on theoretical considerations, is described by the relation (14): (14) FM−N=I1I2I3−κM−N=0, {F_{{\rm{M}} - {\rm{N}}}} = {{{I_1}{I_2}} \over {{I_3}}} - {\kappa _{{\rm{M}} - {\rm{N}}}} = 0, where κ_M-N is the specific soil parameter and I₁, I₂, I₃ are defined by Eqs. (1)–(3).

Transforming equations (11)–(14) gives the expressions (15)–(18) which allow to determine the parameters of failure criteria using the measured values of σ₁^max, σ₂ and σ₃: (15) κM−Cexp=arcsin(σ1−σ3σ1+σ3)=φps \kappa _{{\rm{M}} - {\rm{C}}}^{\exp } = \arcsin \left( {{{{\sigma _1} - {\sigma _3}} \over {{\sigma _1} + {\sigma _3}}}} \right) = {\varphi _{ps}} (16) κD−Pexp=J2I1 \kappa _{{\rm{D}} - {\rm{P}}}^{\exp } = {{\sqrt {{J_2}} } \over {{I_1}}} (17) κM−Nexp=I1I2I3 \kappa _{{\rm{M}} - {\rm{N}}}^{\exp } = {{{I_1}{I_2}} \over {{I_3}}} (18) κL−Dexp=I13I3 \kappa _{{\rm{L}} - {\rm{D}}}^{\exp } = {{I_1^3} \over {{I_3}}}

The associated flow rule comes from the rigorous formulation of the plasticity theory, while the non-associated one is only semi-theoretically 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 non-coaxiality of the stress and strain tensors. It is difficult to measure experimentally and limits the application of the upper-bound 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 non-associated 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 non-associated flow rules. Similar simulations for more complex stress paths in the p - q - θ space can be found in Chan (1988) and Ling and Liu (2003). Their research demonstrates that the associated flow rule predicts well the results of triaxial compression tests for medium and medium-dense soils in the pre-peak stage and under drained conditions. The non-associated flow rule, on the other hand, gives better results for the unstable response of saturated loose sand and under undrained condition. According to these findings, it is justified to use the associated flow rule to analyse the behaviour of medium and medium-dense sand samples up to the peak strength, if volume changes are still significant and no distinct localization occurs.

Plastic flow rule can describe soil deformation at failure state. Its general formulation, as a non-associated flow rule, is given by Eq. (19): (19) ε˙ij=λ∂g∂σij {\dot \varepsilon _{ij}} = \lambda {{\partial g} \over {\partial {\sigma _{ij}}}} where ε˙ij {\dot \varepsilon _{ij}} means the strain rate tensor, λ is the scalar coefficient and g is the plastic potential. The special case of the non-associated is the associated flow rule, where the plastic potential g coincides with the yield surface F (g = F). In such a case, the plastic potentials for D-P, M-N and L-D criteria are given by Eqs. (12)–(14), and it is possible to determine parameters κ_D-P, κ_L-D and κ_M-N theoretically by solving the set of equations (20). The set comes from the assumption of soil failure state (F = 0, where F means F_D-P, F_L-D or F_M-N) and the plane strain condition dɛ₂ = 0, which substituted to the flow rule (19) gives ∂F∂σ2=0 {{\partial F} \over {\partial {\sigma _2}}} = 0 . Such way of calculating parameters κ_D-P, κ_L-D and κ_M-N was proposed by Vikash and Prashant (2010). (20) { F=0dε2=∂F∂σ2=0 \left\{ {\matrix{ {F = 0} \hfill \cr {d{\varepsilon _2} = {{\partial F} \over {\partial {\sigma _2}}} = 0} \hfill \cr } } \right.

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 κ_D-P, κ_L-D and κ_M-N as functions of the internal friction angle ϕ_ps. (21) σ1σ3=1+sinφps1−sinφps {{{\sigma _1}} \over {{\sigma _3}}} = {{1 + \sin {\varphi _{{\rm{ps}}}}} \over {1 - \sin {\varphi _{{\rm{ps}}}}}}

To solve (20), first the intermediate principal stress σ₂ is determined as a function of σ₁ and σ₃, separately for D-P, L-D and M-N yield conditions: (22) σ2D−P=σ32+σ12σ1+σ3 \sigma _2^{{\rm{D}} - {\rm{P}}} = {{\sigma _3^2 + \sigma _1^2} \over {{\sigma _1} + {\sigma _3}}} (23) σ2L−D=σ1+σ32 \sigma _2^{{\rm{L}} - {\rm{D}}} = {{{\sigma _1} + {\sigma _3}} \over 2} (24) σ2M−N=σ1σ3 \sigma _2^{{\rm{M}} - {\rm{N}}} = \sqrt {{\sigma _1}{\sigma _3}}

It can be seen from Eqs. (22)–(24) that each condition gives the different expression for the intermediate stress σ₂. Finally, the following expressions for κ_D-P, κ_L-D and κ_M-N are obtained from the set (20), taking into account Eqs. (21)–(24): (25) κD−P=σ1−σ323σ32+σ1σ3+σ12=sinφps33+sin2φps {\kappa _{{\rm{D}} - {\rm{P}}}} = {{{\sigma _1} - {\sigma _3}} \over {2\sqrt 3 \sqrt {\sigma _3^2 + {\sigma _1}{\sigma _3} + \sigma _1^2} }} = {{\sin {\varphi _{{\rm{ps}}}}} \over {\sqrt 3 \sqrt {3 + {{\sin }^2}{\varphi _{{\rm{ps}}}}} }} (26) κL−D=27(σ32+2σ1σ3+σ12)4σ1σ3=274−sin2φps {\kappa _{{\rm{L}} - {\rm{D}}}} = {{27\left( {\sigma _3^2 + 2{\sigma _1}{\sigma _3} + \sigma _1^2} \right)} \over {4{\sigma _1}{\sigma _3}}} = {{27} \over {4 - {{\sin }^2}{\varphi _{ps}}}} (27) κM−N=σ32+3σ1σ3+σ12+22σ12σ32+2σ13σ3+σ1σ33σ1σ3=5−sin2φpscos2φps+4cosφps {\kappa _{{\rm{M}} - {\rm{N}}}} = {{\sigma _3^2 + 3{\sigma _1}{\sigma _3} + \sigma _1^2 + 2\sqrt {2\sigma _1^2\sigma _3^2 + 2\sigma _1^3{\sigma _3} + {\sigma _1}\sigma _3^3} } \over {{\sigma _1}{\sigma _3}}} = {{5 - {sin^2}{\varphi _{ps}}} \over {{cos^2}{\varphi _{ps}}}} + {4 \over {{cos}{\varphi _{ps}}}}

The plane strain tests, with deformation fixed in x₂ principal stress direction (ɛ₂ = 0, Fig. 5), were carried out in true triaxial apparatus. The Electro-Mechanical True Triaxial Apparatus (EMTTA) is manufactured by the British company GDS Instruments Ltd. and is shown in Fig. 6. Plane strain condition can be imposed on the rectangular samples of 150 × 75 × 75 mm dimensions in this type of true triaxial apparatus by fixing the position of the side platens to prevent their movement (Fig. 7). Depending on the way the stresses are exerted on the test specimen, EMTTA is mixed type, with flexible membrane (horizontal stress) and rigid end platens (axial stress).

Figure 5

Figure 6

Figure 7

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 σ₂ - σ₃ 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 pneumatic-hydraulic 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 σ₁ principal stress. The horizontal stress σ₃ is applied through the water pressure in the chamber, and σ₂ is recorded by a gauge located on vertical actuators.

In order to measure the specimen displacements in x₃ direction, two proximity transducers are installed on the inner side of both cell doors, exactly at their centre (Fig. 7b). When the sample deforms, the changes in a distance between the square aluminium plate (Fig. 7a), adhered to the sample and the transducers, are recorded, giving the local value of the displacement. Based on this measurement, ɛ₃ is calculated. Displacement in x₁ direction is measured by LVDT transducers attached to each actuator; the vertical strain ɛ₁ is calculated on its base.

Soil samples are prepared in a membrane-lined 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:

• a)

  Phase 1 (isotropic compression): the sample is loaded isotropically by increasing water pressure in the testing chamber (σ₃^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.

• b)

  Phase 2 (shear): the sample is vertically loaded with a constant vertical displacement rate of 15 mm/hour at constant chamber pressure (σ₁ = σ₃^c = const) in plane strain conditions.

The samples’ porosities were between 0.316 and 0.36 (the relative densities IDc I_D^c after isotropic compression between 0.465 and 0.878), and thus medium-dense, dense and very dense samples were tested. The applied confining pressures were in a range of 50–400 kPa.

Table 2 contains the initial conditions of all tested samples: the initial void ratio at the start of Phase 1 (e), the void ratio and porosity after isotropic compression, at the beginning of Phase 2 (e^c, n^c) and the applied confining pressure (σ₃^c). Also, the relative density I_D, defined by Eq. (28), is given. (28) ID=emax−eemax−emin {I_D} = {{{e_{{max}}} - e} \over {{e_{{max}}} - {e_{{min}}}}} where e_max and e_min are maximum and minimum void ratios determined in the Proctor test. The porosity n^c is calculated on the base of a void ratio, using Eq. (29): (29) n=ee+1 n = {e \over {e + 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.

Figure 8

Figure 9

Figure 10

Figure 11

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 σ₁^max, σ₂ and σ₃ are established, like it is demonstrated in Fig. 2.

Table 2

Test	e	ID	σc3[kPa]	ec	IDc {\boldsymbol {I}}_{\boldsymbol {D}}^{\boldsymbol {c}}	nc
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 q was observed in test ‘009_17_MC_5’ (Fig. 8). The corresponding axial stress equals σ₁^max = 1402 kPa (Fig. 9). At this range of stress, no grain crushing is possible in case of silica sand, so it is accepted that no grain crushing occurred in the tests; see Yang et al. (2010). Table 3 presents the measured values of σ₁^max, σ₂, σ₃, defining the peak strength of each sample, and p, q, θ and b, calculated on their base.

Figs 8–11 represent the whole course of the experimental tests, showing pre- and post-peak behaviour. Only the pre-peak 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 q: σ₁^max (σ₃) and σ₂(σ₃), which can be estimated with sufficient accuracy by straight lines (r = 0.94 and r = 0.9, where r is Pearson's coefficient). Fig. 12 also confirms that in all the tests, the measured σ₂ is always the intermediate principal stress.

Figure 12

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 σ₂ or confining pressure σ₃. The Lode angle is considered constant at θ = 12.6° (calculated as an arithmetic average), but its scatter around this value is considerable, which is probably related to the unavoidable problem of forming soil samples of repeatable structure. The value of the Lode angle may also depend in a more complex way on the initial density of the soil sample or confining pressure. This problem is open.

Figure 13

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 σ₂ 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 κM−Cexp(φps) \kappa _{{\rm{M}} - {\rm{C}}}^{\exp }\left( {{\varphi _{ps}}} \right) , κD−Pexp \kappa _{{\rm{D}} - {\rm{P}}}^{\exp } , κM−Nexp \kappa _{{\rm{M}} - {\rm{N}}}^{\exp } and κL−Dexp \kappa _{{\rm{L}} - {\rm{D}}}^{\exp } , obtained from Eqs. (15)–(18) by substituting σ₁^max, σ₂ and σ₃, which correspond to the peak strength of the tested samples, are collected in Table 4, column ‘A’. Values of parameters κ_D-P, κ_L-D and κ_M-N, calculated from Eqs. (25)–(27) by substituting σ₁^max and σ₃, are given in Table 4, column ‘B’.

Table 4

Test	A. Direct stress measurements Eqs. (15)–(18)				B. Flow rule and plane strain condition Eqs. (25)–(27)
	ϕps	κD−Pexp \kappa _{{\rm{D}} - {\rm{P}}}^{\exp }	κM−Nexp \kappa _{{\rm{M}} - {\rm{N}}}^{\exp }	κL−Dexp \kappa _{{\rm{L}} - {\rm{D}}}^{\exp }	σ2D-P \sigma _2^{{\rm{D - P}}}	σ2M-N \sigma _2^{{\rm{M - N}}}	σ2L-D \sigma _2^{{\rm{L - D}}}	κD-Pflow rule \kappa _{D{\rm{ - }}P}^{f{\rm{low}}\,{\rm{rule}}}	κM-Nflow rule \kappa _{M{\rm{ - }}N}^{f{\rm{low}}\,{\rm{rule}}}	κL-Dflow rule \kappa _{L{\rm{ - }}D}^{f{\rm{low}}\,{\rm{rule}}}
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

Fig. 14a presents σ₂^D-P, σ₂^M-N and σ₂^L-D as functions of the experimental σ₂, taken from Table 3. A perfect fit between the calculated and measured values would mean that they lie on the dashed line σ₂^calc = σ₂^exp, shown in the figure.

A linear relationship between the calculated and measured intermediate stress σ₂ for all three selected criteria is found, but only Matsuoka–Nakai is close to the perfect fit. The ratio R = σ₂^calc / σ₂^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 σ₂, 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 σ₂ to the confining pressure σ₃^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 ϕ_ps, κ_D-, κ_L-D and κ_M-N, obtained by Eqs. (15)–(18), requiring knowledge of the full set of principal stresses σ₁, σ₂ and σ₃, and by Vikash and Prashant approach, which requires measurement of two principal stresses only (σ₁ and σ₁), are shown in Fig. 17, as functions of the relative density I_D, for all the true triaxial tests (Tables 2–4).

It is possible in Vikash and Prashant approach to replace σ₁ and σ₃ by the internal friction angle ϕ_ps (Eqs. (25)–(27)). The internal friction angle for plane strain condition can be obtained directly from the relation (21) and play the role of the independent variable that specifies κ_D-P, κ_L-D and κ_M-N.

Fig. 17(a) shows the relation between the M-C friction angle ϕ_ps and the relative density IDc I_D^c of Skarpa sand, which is linear within the given density range, thus allowing for the equivalent presentation of κ_D-P, κ_L-D and κ_M-N as functions of ϕ_ps or IDc I_D^c independent variables. IDc I_D^c was selected for Fig. 17(b)–(d). The figure confirms the previous finding that the Vikash and Prashant approach gives a very good result for the Matsuoka–Nakai failure criterion – the κ_M-N values obtained in this case by the two approaches almost coincide in Fig. 17c.

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 κ_M-N, κ_D-P and κ_M-N with soil relative density, regardless of the way they are obtained, as expected.

The linear fits for the failure criteria parameter κ^exp, obtained from the direct stress measurement of σ₁^max, σ₂ and σ₃ (Table 3) and expressed as a function of the relative soil density (I_D) or the plane strain friction angle ϕ_ps, are given in Tables 5 and 6, respectively.

Table 5

	Linear fit	Pearson's coefficient r
Drucker–Prager	κD−Pexp=0.16IDc+0.13 \kappa _{{\rm{D}} - {\rm{P}}}^{\exp } = 0.16I_D^c + 0.13	rD-P = 0.94
Lade–Duncan	κL−Dexp=26.33IDc+31.64 \kappa _{{\rm{L}} - {\rm{D}}}^{\exp } = 26.33I_D^c + 31.64	rL-D = 0.96
Matsuoka–Nakai	κM−Nexp=9.20IDc+5.48 \kappa _{{\rm{M}} - {\rm{N}}}^{\exp } = 9.20I_D^c + 5.48	rM-N = 0.96

Table 6

	Linear fit	Pearson's coefficient r
Drucker–Prager	κD−Pexp=0.006353 φps+0.001497 \kappa _{{\rm{D}} - {\rm{P}}}^{\exp } = 0.006353\,{\varphi _{{\rm{ps}}}} + 0.001497	rD-P = 0.970
Lade–Duncan	κL−Dexp=1.5 φps−10.613 \kappa _{{\rm{L}} - {\rm{D}}}^{\exp } = 1.5\,{\varphi _{{\rm{ps}}}} - 10.613	rL-D = 0.993
Matsuoka–Nakai	κM−Nexp=0.2605 φps+2.767 \kappa _{{\rm{M}} - {\rm{N}}}^{\exp } = 0.2605\,{\varphi _{{\rm{ps}}}} + 2.767	rM-N = 0.997

Values of Pearson's correlation coefficient r in Tables 5 and 6 lay between 0.970 and 0.997, which proves very good linear fit in each case. The Statistica software was used to determine the fits by the least-squares method.

To estimate more quantitatively the difference between the two approaches of determining the failure criteria parameters, two statistical measures are employed:v_κ and v_σ2.

v_κ is a measure of the average relative difference between κ^{flow rule} = κ(σ₁, σ₃) = κ(ϕ_ps)and κ^exp (Table 4) and is given by the formula (30): (30) νκ=∑i=1n| κexp−κflow rule |κexp*100%n, {\nu _\kappa } = {{\sum\limits_{{\rm{i}} = 1}^{\rm{n}} {{{\left| {{\kappa ^{\exp }} - {\kappa ^{{\rm{flow}}\,{\rm{rule}}}}} \right|} \over {{\kappa ^{\exp }}}}*100\% } } \over {\rm{n}}}, where ‘n’ means number of tests (n = 11).

Similarly, v_σ2 is the average relative difference between the semi-theoretically and experimentally validated σ₂, and it is calculated according to the relationship (31): (31) νσ2=∑i=1n| σ2exp−σ2flow rule |σ2exp*100%n {\nu _{{\sigma _2}}} = {{\sum\limits_{{\rm{i}} = 1}^{\rm{n}} {{{\left| {\sigma _2^{\exp } - \sigma _2^{{\rm{flow}}\,{\rm{rule}}}} \right|} \over {\sigma _2^{\exp }}}*100\% } } \over {\rm{n}}}

Table 7 shows the summary of the calculations carried out for each of the criteria.

Table 7

	vκ	vσ2
Drucker–Prager	νκD−P=21.21 % \nu _\kappa ^{{\rm{D}} - {\rm{P}}} = 21.21\,\%	νσ2D−P=119.0 % \nu _{{\sigma _2}}^{{\rm{D}} - {\rm{P}}} = 119.0\,\%
Lade–Duncan	νκL−D=5.99 % \nu _\kappa ^{{\rm{L}} - {\rm{D}}} = 5.99\,\%	νσ2L−D=55.5 % \nu _{{\sigma _2}}^{{\rm{L}} - {\rm{D}}} = 55.5\,\%
Matsuoka–Nakai	νκM−N=0.66 % \nu _\kappa ^{{\rm{M}} - {\rm{N}}} = 0.66\,\%	νσ2M−N=19.4 % \nu _{{\sigma _2}}^{{\rm{M}} - {\rm{N}}} = 19.4\,\%

The smallest average differences are for Matsuoka–Nakai criterion (νκM−N=0.66%, νσ2M−N=19.4%) \left( {\nu _\kappa ^{{\rm{M}} - {\rm{N}}} = 0.66\% ,\,\,\nu _{{\sigma _2}}^{{\rm{M}} - {\rm{N}}} = 19.4\% } \right) and the biggest in the case of Drucker–Prager (νσ2D−P=21.21%, νσ2D−P=119.0%) \left( {\nu _{{\sigma _2}}^{{\rm{D}} - {\rm{P}}} = 21.21\% ,\,\,\nu _{{\sigma _2}}^{{\rm{D}} - {\rm{P}}} = 119.0\% } \right) . The relatively big average relative difference of the intermediate stress σ₂ has little influence on the corresponding difference of κ_M-N, being less than 1%.