A note on the differences between Drucker-Prager and Mohr-Coulomb shear strength criteria

In soil and rock mechanics, the Mohr-Coulomb (MC) shear strength criterion, along with its parameters, namely friction angle and cohesion, is treated as a kind of standard and reference concept for other shear strength criteria. This is due to the fact, that it fits well the experimental data, where asymmetric strength response in triaxial compression (TXC) and triaxial extension (TXE) tests is observed. Moreover, the MC criterion parameters have clear engineering interpretation and they are typically obtained in most geotechnical laboratories. On the other hand the MC concept discards the influence of the intermediate principal stress on the shear strength of the material, whereas this influence is visible when true triaxial testing is performed [1, 9]. Additionally, the MC failure surface, defined in the principal stress space, contains sharp edges, which introduces some difficulties in the implementation of the criterion for numerical analysis purposes. These shortcomings are often opposed to the straightforward implementation of the smooth failure functions, with Drucker-Prager (DP) criterion being one of such examples [2, 7].

Of continuous interest to researchers is the question regarding how the different shear strength critera used in engineering and computational practice compare to the reference MC predictions. For this purpose, a concept of equivalent friction angle is usually used [3, 5, 6, 8]. This angle is defined as the friction angle of the MC surface that would pass through the particular stress point given by the shear strength criterion under consideration. Its variations with the changing stress state and the parameters of the criterion being compared are then analysed.

In this paper, another approach is used. Instead of defining and analysing the equivalent friction angle, the shear strengths predicted by the MC and DP criteria are compared directly. In case of these particular criteria, an analytical formula can be derived for this purpose. To the knowledge of the author, such a formula has not been published explicitly yet. From this result, it is deduced that the DP criterion generates shear strength between 0.6 and 3 times the MC strength, for the same friction angle and cohesion parameters. The appropriate conditions for obtaining equal strengths for both criteria are also analysed. Additionally, some new DP failure surfaces that minimize differences with MC criterion are proposed.

Let us assume that the principal stresses in the isotropic material are given by σ₁ ≥ σ₂ ≥ σ₃ and let us assume the positive sign for compressive stresses. The invariants of the stress state used in the following text can be written as expressed in Eqs (1)-(3):

where I₁ is the first invariant of the stress tensor and J₂, J₃ are the second and the third invariants of the stress deviator. The intermediate principal stress can always be represented as a linear combination of two other stresses:

where:

is the so called principal stress ratio. Let us now introduce the maximum shear plane stresses

Please note, that the same values are often denoted in the literature as s, t or σ_m, τ_m instead of p, q. Also, they should not be confused with the invariants I1/3,3J2′,${{I}_{1}}/3,\,\sqrt{3{{J}_{{{2}'}}},}$respectively.

The principal stresses can be now written as follows:

where:

is an equivalent measure of principal stress ratio. Introducing equations (8)-(10) into the definitions of invariants the following relations are obtained:

Finally the Lode angle θ ϵ 〈-π/6, π/6〉 is introduced by Eq. (15):

It is straightforward to show that Lode angle is related to the parameter a via Eq. (16):

and it can be viewed as yet another measure of the principal stress ratio. For all three measures namely b,σ,θ, the left limit of their values stands for the σ₁ = σ₂ case, i.e. triaxial extension (TXE), whereas the right limit signifies σ₂ = σ₃, i.e. triaxial compression (TXC).

In the following sections p, q and θ will be used as the representation of the stress states in the material.

The MC strength criterion can be written in the following form:

where ϕ is the friction angle and c is the material cohesion. It is quite obvious that the failure shear stress q^MC does not depend on θ, which means also that the intermediate stress σ₂ does not influence material shear strength. On the other hand the DP condition is usually expressed via the following relation:

where α and k are parameters of this criterion. Inserting Eqs. (12), (13) and (16) into Eq. (18) the expression for the failure shear stress in the DP condition is derived as following:

Comparison of the free coefficients and the coefficients standing by p in these two criteria leads us to the expressions for α and k in relation to ϕ and c:

Clearly, the derived values of α and k depend on the Lode angle, so they are not constant for constants ϕ and c, but rely also on the intermediate principal stress σ₂.

Let us consider now that α and k have been established for some fixed Lode angle, say θ₀, and inserted back to the DP criterion definition, namely Eq. (19). This will lead us, after some not very tedious algebraic transformations, to the following relation:

where:

The DP shear strength can be then represented by the MC strength multiplied by a parameter A dependent on the friction angle ϕ and the following two Lode angles: θ₀, for which the DP parameters have been established, and θ, representing the current stress state.

Clearly if A = 1, then the DP and MC criteria are equivalent. This will occur in two cases: firstly if θ=θ₀ - which is quite obvious, and secondly if the following condition holds:

One can note that Eq. (24) can be considered as a measure of equivalent friction angle (Griffiths 1990) for the DP condition when θ₀ and θ are given.

Further analysis of A shows that A>1, i.e. q^DP>q^MC, when one of the following occurs:

or

On the other hand if:

or

then A<1 and q^DP<q^MC are obtained. Additionally, considering the physical constraints on the friction angle, i.e. ϕϵ〈0^°,90^°〉, it is observed that the inequality sinϕ>3tan (− (θ +θ0) / 2)$\sin \phi \gt \sqrt 3 \tan \left( { - \left( {\theta + {\theta _0}} \right)/2} \right)$is always true if only θ+θ₀<0. These variability considerations are presented compactly in Figure 1.

Figure 1

In computational practice some specific values of θ₀ are usually selected for fitting α and k parameters of the DP criterion with ϕ and c values. By setting θ₀ as -30^°, 0 and 30^°, the tensile, shear and compressive meridians, respectively, of the MC criterion are met. Furthermore, when the minimum of α coefficient is considered, then θ0=−arctan(sinϕ/3)${{\theta }_{0}}=-\arctan \left( \sin \phi /\sqrt{3} \right)$is obtained and the DP yield surface inscribes the MC envelope. However, any other θ₀ can reasonably be used (see Section 5 for examples). table 1 shows the expressions for A along with its minimum, maximum and average values for these typical θ₀ choices. Additionally, in Figure 2 the distribution of deviation of A from unity is presented. One can observe, that for θ₀= 30^° the Drucker-Prager shear strength is always greater (or equal) to the Mohr-Coulomb strength and it is exactly opposite for θ0=−arctan(sinϕ/3).${{\theta }_{0}}=-\arctan \left( \sin \phi /\sqrt{3} \right).$For all other cases both possibilities are present, depending on θ and ϕ values. One can also verify that the overall maximum for A takes the value 3.0 and is achieved for (θ0,θ,ϕ)=(30∘,−30∘,90∘),$\left( {{\theta }_{0}},\theta ,\phi \right)=\left( {{30}^{\circ }},-{{30}^{\circ }},{{90}^{\circ }} \right),$whereas the overall minimum is equal to 0.6 and is achieved for (θ0,θ,ϕ)=(−30∘,30∘,90∘).$\left( {{\theta }_{0}},\theta ,\phi \right)=\left( -{{30}^{\circ }},{{30}^{\circ }},{{90}^{\circ }} \right).$Thus, in general, the following inequality is valid:

Figure 2

which means that the DP shear strength cannot be lower then 0.6 and greater than 3.0 times its MC counterpart.

An average value of the coefficient A in terms of θ₀ can be expressed as follows:

where Ω=[θ]×[ϕ]$\Omega =\left[ \theta \right]\times \left[ \phi \right]$is the chosen integration area. It would be of interest to look for such values of θ₀ for which A¯=1,$\bar{A}=1,$i.e. the difference between MC and DP criteria vanishes, in this average sense. Another possibly interesting θ₀ point would be located at the minimum of A¯.$\bar{A}.$

Unfortunately the explicit formula for the integral defined by Eq. (30) seems to be impossible to be given, thus the integration has to be done numerically. Indeed, when A is integrated over the whole range of variability of θ and ϕ, i.e. for Ω=[-30∘,30∘]×[0∘,90∘],$\Omega =\left[ \text{-}{{30}^{\circ }},{{30}^{\circ }} \right]\times \left[ {{0}^{\circ }},{{90}^{\circ }} \right],$then it reaches unity at θ0=4.22∘${{\theta }_{0}}={{4.22}^{\circ }}$and it attains its minimum at θ₀=-19.35^° (see Figure 3). These θ₀ values can be treated as possible choices for establishing parameters of the DP criterion, especially when the best overall agreement with the MC shear strength is expected. See Table 1 and Figure 2 for more details about these θ₀ values.

Figure 3

However, one can also investigate other integration ranges for A, matching specific material properties and loading scenarios. For example in the case of natural sandy and gravelly soils the friction angle varies usually between 30^° and 45^°. If the compressive loading of such soil is assumed, i.e. θ∈〈0∘,30∘〉,$\theta \in \left\langle {{0}^{\circ }},{{30}^{\circ }} \right\rangle ,$then the condition A¯=1,$\theta \in \left\langle {{0}^{\circ }},{{30}^{\circ }} \right\rangle ,$will be obtained at θ₀=-15.5^°, which is quite different from the previous result (see Figure 3).

This way the general method for obtaining the DP parameters that best fit the material behaviour and loading data is obtained.

Elasto-plastic material models are often accompanied by plane strain conditions, which allow for dimension reduction from 3D to 2D. In this case, the in-plane principal stresses ^σ₁, ^σ₃ are used to compute the principal out-of-plane^σ₂ stress via the following relation:

where ν is the Poisson ratio of the material. When the plastic flow occurs, it is very common to assume ν=0.5. Following the notation from Section 2 this choice corresponds to assuming the Lode angle θ=0^° (or b=0.5 or a=0). In this specific case, the DP criterion become fully equivalent to the MC criterion, independently on the friction angle of the material, if only θ₀=0^° is taken for fitting α and k parameters (see upper right graph in Figure 2). However, the validity of assuming the apparent Poisson ratio equal to 0.5 is arguable [10, 11]. It should be emphasized that any other choice for ν makes the MC and DP equivalence disappear in plane strain conditions and the general 3D Drucker-Prager criterion has to be considered.

In this paper the formula relating material shear strength predictions generated by the MC and the DP criteria is derived and analysed. This relation is of the form q^DP=Aq^MC, where A depends on the friction angle ϕ, Lode angle θ₀ for which the DP coefficients have been derived and Lode angle θ describing the current stress state. It should be also noted that A does not depend on the cohesion of the material. The variability considerations of this relation are summarized as follows:

The interval of variability of A value, i.e. Aϵ〈0.6,3〉 can be considered as surprisingly wide. However the extreme values are obtained for friction angle ϕ=90^°, which is rather seldom observed, and for the maximum possible discrepancy between θ₀ and θ values (equal to 60^°). Unfortunately, even for more realistic friction angles, i.e. for ϕ≤45^°, the interval Aϵ〈0.68, 1.89〉 is obtained, which is obviously narrower, but still significant. It seems that the key for achieving best overall agreement between the MC and DP predictions is the proper choice of θ₀ value. Indeed, quite good agreement is obtained for θ₀=0^°, where the maximum difference between θ₀ and θ is equal to 30^° (see Figure 2). For this θ₀ choice and for ϕϵ〈0^°, 90^°〉 and θϵ〈-30^°, 30^°〉 the interval Aϵ〈0.67, 1.22〉 is obtained. Furthermore, if we constrain the possible friction angles to ϕ≤45^° then this interval reduces to Aϵ〈0.72, 1.09〉. Even better agreement, in the average sense, is possible to be achieved by means of the procedure described in Section 5, if only the variability ranges of the friction and Lode angles can be reasonably estimated for the problem under consideration. The value θ₀=4.22^° is recommended for the most general case of ϕ and θ variability. For the reduced range of the friction angle, the value θ₀=7.92^° can be used instead. It is generally discouraged to use in the computations the limiting values of Lode angle (θ₀=-30^°, θ₀=30^°) for fitting DP criterion parameters, unless it is really known that the considered loading conditions will be close to the triaxial extension or compression case.

The approach used in the paper may become a useful strategy for comparing also other shear strength criteria to the MC predictions. However, the linear relation of the explicit form given by Eqs. (22) and (23) might be simply impossible to be obtained in other cases and some more sophisticated functions should be investigated.