The strength–dilatancy characteristics embraced in hypoplasticity

Abstract

The strength-dilatancy characteristics of frictional materials embraced in the hypoplastic model proposed by Gudehus and Bauer are investigated and compared with the revised model suggested by Huang. In the latter the deviatoric stress in the model by Gudehus and Bauer is replaced by a transformed stress according to the stress transformation technique proposed by Matsuoka. The flow rule, the failure state surface equation and the strength-dilatancy relationship embraced in both models are derived analytically. The performance of the two hypoplastic models in reproducing the relationship between the peak strength and the corresponding dilation rate under triaxial compression, plane compression and plane shearing are then extensively investigated and compared with experimental results and with the predictions made by particular classical stress-dilatancy theories. Numerical investigations show that the performance in reproducing the strength-dilatancy relationship is quite satisfactory under triaxial compression stress state in both models and the predictions made by the transformed stress based model are closer to the results obtained from classical stress-dilatancy theories for plane compression and plane shearing problems.

Keywords

strength / dilatancy / hypoplasticity / frictional materials

Introduction

In engineering praxis, the soil material is often considered in an oversimplified manner either in a linear elastic state or in an ideal plastic state, i.e., deformation and strength are assumed to be independent of each other. Various evidences from both laboratory experiments and field investigations, however, indicate that the mechanical behavior of frictional materials like soil is inelastic, i.e., incrementally nonlinear, and strongly influenced by the pressure level, density and loading path. Thus for realistic simulation of the complex mechanical behavior of soils using a continuum description, more sophisticated constitutive models are needed. For the evaluation of a particular constitutive model, the calibratability and performance in reproducing the general properties of soils play an important role [1], for example the stress response envelope and the proportional loading properties [2]. In this paper particular attention will be focused on the strength-dilatancy characteristics of two particular hypoplastic constitutive models, i.e., the model proposed by Gudehus [1] and Bauer [2,3] (abbreviated as GB_HYPO hereafter) and the transformed stress based version by Huang [4] (abbreviated as TS_HYPO hereafter). In the latter the transformed stress suggested by Matsuoka [5] is embedded into the model proposed by Gudehus and Bauer.

The relationship between the strength and the dilatancy of granular materials like sand can be analyzed based on the so-called stress-dilatancy equation, which can be established from different points of view. For instance Taylor [6] discussed the shearing characteristics of sands and decomposed the shear strength into the frictional part and the interlocking part, where the latter is related to the dilatancy properties of the material. He assumed that at the peak state the amount of energy used during vertical expansion is the same as the power produced by the part of the shear stress contributed by interlocking. This relation can be expressed with the variables shown in Fig. 1 as follows:

(1)T_{\mathrm{11}}{\dot{\delta }}_{\mathrm{v}}=(T_{\mathrm{13}}-\mu T_{\mathrm{11}}){\dot{\delta }}_{\mathrm{h}}.

Equation (1) can be rewritten to

(2)\tan {\phi }_{\mathrm{m}}=\tan {\phi }_{\mathrm{c}}+\tan \psi ,

in which the mobilized friction angle {\phi }_{\mathrm{m}} and the dilation angle ψ are defined as \tan {\phi }_{\mathrm{m}}=T_{\mathrm{13}}/T_{\mathrm{11}} and tan\psi ={\dot{\delta }}_{\mathrm{v}}/{\dot{\delta }}_{\mathrm{h}}, respectively. The critical friction angle is related to the friction coefficient μ, i.e., \mu =\tan {\varphi }_{\mathrm{c}}.

Taylor’s concept was later employed by several authors for the stress-dilatancy characteristics of normally consolidated clay under triaxial compression [7]. The most frequently cited relation is the one conceived for Granta gravel and also used in developing the potential function in Cam-Clay model [7], i.e.,

(3){\displaystyle \frac{q}{p}}=M-{\displaystyle \frac{{\dot{ϵ}}_{\mathrm{v}}}{{\dot{ϵ}}_{\mathrm{d}}}}.

Herein the volumetric strain rate {\dot{ϵ}}_{\mathrm{v}} and the deviatoric strain rate {\dot{ϵ}}_{\mathrm{d}} are calculated by the axial strain rate D₁₁ and the radial strain rate D₃₃ via {\dot{ϵ}}_{\mathrm{v}}=D_{\mathrm{11}}+\mathrm{2}{D}_{\mathrm{33}} and {\dot{ϵ}}_{\mathrm{d}}=\mathrm{2}(D_{\mathrm{11}}-D_{\mathrm{33}})/\mathrm{3} while the mean stress p and the deviatoric stress q are calculated by the axial stress T₁₁ and the radial stress T₃₃ via p = (T₁₁ + 2T₃₃)/3 and q = (T₁₁- T₃₃). Similar to the coefficient μ in Eq. (1), M in Eq. (3) is also related to the critical friction angle ϕ_c, i.e., M=\mathrm{6}\sin {\phi }_c/(\mathrm{3}-\sin {\phi }_c) [7].

Rather different from the approach by Taylor, Rowe [8] established his stress-dilatancy theory by considering the static equilibrium of a packing of uniform rigid rods. Based on the assumption of least rate of internal work, Rowe proposed the following stress-dilatancy equation:

(4){\displaystyle \frac{T_{\mathrm{11}}}{T_{\mathrm{33}}}}=-{\tan }^{\mathrm{2}}\left({\displaystyle \frac{\mathrm{\pi }}{\mathrm{4}}}+{\displaystyle \frac{{\phi }_{\mathrm{u}}}{\mathrm{2}}}\right){\displaystyle \frac{D_{\mathrm{33}}}{D_{\mathrm{11}}}}.

in which T₁₁, T₃₃ and D₁₁, D₃₃ are the stress components and the strain rate components under biaxial condition (Fig. 2), ϕ_μ denotes the friction angle between the surfaces of the rods. By using the definitions of the mobilized Mohr-Coulomb friction angle {\phi }_m and the dilation angle ψ in plane compression problems as shown in Fig. 2, Eq. (4) can also be expressed as follows:

(5)\sin {\phi }_{\mathrm{m}}={\displaystyle \frac{\sin {\phi }_{\mathrm{u}}+\sin \psi }{\mathrm{1}+\sin {\phi }_{\mathrm{u}}\sin \psi }}.

The parameter ϕ_μ in Eq. (5) is often replaced by the critical friction angle ϕ_c since ϕ_μ equal to the mobilized friction angle when the dilation angle approaches zero.

Statistical researches based on compiled experimental results were also carried out. Bolton [9] investigated the strength-dilatancy characteristics of 17 sands in axisymmetric stress states and plane strain condition at different densities and confining pressures and he suggested that the stress-dilatancy equation by Rowe can be well approximated by the following empirical relation between the mobilized Mohr-Coulomb friction angle ϕ_m, the critical friction angle ϕ_c and the dilation angle ψ:

(6){\phi }_{\mathrm{m}}={\phi }_{\mathrm{c}}+\mathrm{0.8}\psi .

Equation (6) is initially proposed for plane compression problems and it indicates that the increase of the mobilized friction angle should be slower than the increase of the dilation angle. Bolton’s relation (6) was recently checked by Simoni and Houlsby [10] based on 87 direct shear tests on sand-gravel mixtures and it was found that Eq. (6) also reflects the properties in direct shear problem if the mobilized friction angle, the critical friction angle and the dilation angle are defined according to plane shear problem as that in Taylor’s relation.

The basic stress-dilatancy Eqs. (2), (3) and (5) were revisited and extended by some authors [11-13], leading to several more sophisticated relations taking the non-coaxial deformation [11,12] and the density and stress level dependencies of mechanical behavior [11] into consideration. However, refinement of an existing theory often results in an addition of parameters and in the present study only the basic Eqs. (2), (3), (5) and (6) were used to investigate the reliability of the relationships between the peak strength and the relevant dilation rate embraced in the two aforementioned hypoplastic models.

Throughout the paper, the sign convention in soil mechanics is adopted, i.e., compressive stress and strain are positive. Stress tensor and strain rate tensor are denoted by BoldItalic and BoldItalic, whose deviatoric parts are signified by BoldItalic* and BoldItalic* respectively. All the stresses involved in this paper are effective ones and a conventional superscript comma is omitted. Tensorial operations frequently used in this paper are tr(.) and ||.||, the former denotes the trace of a second-order tensor while the latter its Euclidean norm, e.g., tr(BoldItalic) = D_ijδ_ij, ||D|| = (D_ijD_ij)^1/2, where δ_ij is Kronecker delta and the summation over repeated indices is employed. Other symbols will be explained in the case of their first use.

The transformed stress based hypoplastic model

The constitutive equation proposed by Gudehus [1] and Bauer [2,3] reads:

(7)\stackrel{\circ }{T}=f_{\mathrm{s}}\left[{\widehat{a}}^{\mathrm{2}}D+\left(\widehat{T}:D\right)\widehat{T}-\widehat{a}{f}_{\mathrm{d}}\left(\widehat{T}+{\widehat{T}}^{*}\right)\Vert D\Vert \right],

in which \stackrel{\circ }{T} denotes the objective (Jaumann) stress rate and \widehat{T} the normalized Cauchy stress tensor, i.e., \widehat{T}=T/\mathrm{tr}T. The stiffness factor f_s is used to reflect the pressure- and density-dependence of the mechanical behavior and the density factor f_d is introduced to embed the concept of critical state into hypoplasticity. The function \widehat{a} in Eq. (7) is related to the critical friction angle {\phi }_{\mathrm{c}} and the relevant stress criterion [3]. For example, the use of the SMP criterion by Matsuoka et al. [5,14] as the stress condition for critical states in hypoplasticity gives the representation of \widehat{a} as follows [3]:

(8)\widehat{a}={\displaystyle \frac{\sin {\phi }_{\mathrm{c}}}{\mathrm{3}-\sin {\phi }_{\mathrm{c}}}}\left[\sqrt{{\displaystyle \frac{\mathrm{8}/\mathrm{3}-\mathrm{3}{\Vert {\widehat{T}}^{*}\Vert }^{\mathrm{2}}+\sqrt{\mathrm{3}/\mathrm{2}}\cos (\mathrm{3}\theta ){\Vert {\widehat{T}}^{*}\Vert }^{\mathrm{3}}}{\mathrm{1}+\sqrt{\mathrm{3}/\mathrm{2}}\cos (\mathrm{3}\theta )\Vert {\widehat{T}}^{*}\Vert }}}-\Vert {\widehat{T}}^{*}\Vert \right].

Herein θ denotes the Lode angle [2,3], i.e.,

(9)\theta ={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}\arccos \left[-{\displaystyle \frac{\sqrt{\mathrm{6}}\mathrm{tr}\left({\widehat{T}}^{*}\cdot {\widehat{T}}^{*}\cdot {\widehat{T}}^{*}\right)}{{\Vert {\widehat{T}}^{*}\Vert }^{\mathrm{3}}}}\right].

Huang et al. [4,15] investigated the strength characteristics embraced in Eq. (7) and found that the predicted peak friction angle is higher in triaxial compression than that in triaxial extension, which is not consistent with the experimental findings by Lade [16,17]. Furthermore, the failure surface predicted by Eq. (7) is somewhat concave for an initially dense material [15]. To eliminate the aforementioned limitations, Huang [4] replaced the normalized Cauchy stress tensor and its deviator in Eq. (7) with the following transformed quantities using the stress transformation technique by Matsuoka et al. [5]:

(10)\tilde{T}=s{\widehat{T}}^{*}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}I,{\tilde{T}}^{*}=s{\widehat{T}}^{*},

where BoldItalic is the unit tensor and the scale factor s reads [15]:

(11)s=\sqrt{{\displaystyle \frac{\mathrm{8}}{\mathrm{3}}}}\cdot {\left(\sqrt{{\displaystyle \frac{\mathrm{8}/\mathrm{3}-\mathrm{3}{\Vert {\widehat{T}}^{*}\Vert }^{\mathrm{2}}+\sqrt{\mathrm{3}/\mathrm{2}}\cos (\mathrm{3}\theta ){\Vert {\widehat{T}}^{*}\Vert }^{\mathrm{3}}}{\mathrm{1}+\sqrt{\mathrm{3}/\mathrm{2}}\cos (\mathrm{3}\theta )\Vert {\widehat{T}}^{*}\Vert }}}-\Vert {\widehat{T}}^{*}\Vert \right)}^{-\mathrm{1}}.

Therefore, the revised constitutive equation reads:

(12)\stackrel{\circ }{T}=f_{\mathrm{s}}\left[a^{\mathrm{2}}D+\left(\tilde{T}:D\right)\tilde{T}-a{f}_{\mathrm{d}}\left(\tilde{T}+{\tilde{T}}^{*}\right)\Vert D\Vert \right],

in which the stiffness factor f_s and the density factor f_d are the same as those in Eq. (1) while the stress-dependent function \widehat{a} is replaced by a constant a, i.e.,

(13)a=\sqrt{{\displaystyle \frac{\mathrm{8}}{\mathrm{3}}}}{\displaystyle \frac{\sin {\phi }_c}{\mathrm{3}-\sin {\phi }_c}}.

As pointed out by Huang, the hypoplastic model defines the critical state explicitly and let the peak state a model prediction, which depends on the effective pressure and the relative density. In the following part, the strength-dilatancy characteristics embraced in TS_HYPO will be investigated further. To this end, the peak state condition is used, i.e.,

(14)a^{\mathrm{2}}D+\left(\tilde{T}:D\right)\tilde{T}-a{f}_{\mathrm{d}}\left(\tilde{T}+{\tilde{T}}^{*}\right)\Vert D\Vert =0.

The inner products of the transformed stress tensor \tilde{T} and both sides of Eq. (14) yield the following equation for the “strain energy”:

(15)\tilde{T}:D=a{f}_{\mathrm{d}}{\displaystyle \frac{{\Vert \tilde{T}\Vert }^{\mathrm{2}}+{\Vert {\tilde{T}}^{*}\Vert }^{\mathrm{2}}}{a^{\mathrm{2}}+{\Vert \tilde{T}\Vert }^{\mathrm{2}}}}\Vert D\Vert .

Substituting Eq. (15) into Eq. (14) gives the flow rule for zero objective stress rate, i.e.,

(16){\displaystyle \frac{D}{\Vert D\Vert }}={\displaystyle \frac{f_{\mathrm{d}}}{a}}\left({\displaystyle \frac{a^{\mathrm{2}}-{\Vert {\tilde{T}}^{*}\Vert }^{\mathrm{2}}}{a^{\mathrm{2}}+{\Vert \tilde{T}\Vert }^{\mathrm{2}}}}{\displaystyle \frac{I}{\mathrm{3}}}+{\displaystyle \frac{\mathrm{2}{a}^{\mathrm{2}}+\mathrm{1}/\mathrm{3}}{a^{\mathrm{2}}+{\Vert \tilde{T}\Vert }^{\mathrm{2}}}}{\tilde{T}}^{*}\right).

Introducing the volumetric strain rate {\dot{ϵ}}_{\mathrm{v}}=\mathrm{tr}D and the deviatoric strain rate {\dot{ϵ}}_{\mathrm{d}}=\sqrt{\mathrm{2}{D}^{*}:D^{*}/\mathrm{3}}, as is done in elastoplasticity, the following strength-dilatancy equation can be derived from Eq. (16):

(17)d={\displaystyle \frac{{\dot{ϵ}}_{\mathrm{v}}}{{\dot{ϵ}}_{\mathrm{d}}}}={\displaystyle \frac{\sqrt{\mathrm{3}/\mathrm{2}}}{\mathrm{2}{a}^{\mathrm{2}}+\mathrm{1}/\mathrm{3}}}{\displaystyle \frac{a^{\mathrm{2}}-{\Vert {\tilde{T}}^{*}\Vert }^{\mathrm{2}}}{\Vert {\tilde{T}}^{*}\Vert }}.

Moreover, the equation for the peak state surface in the principal stress space can also be derived from Eq. (16) using the condition

(D/\Vert D\Vert ):(D/\Vert D\Vert )=\mathrm{1}

, viz.,

(18){\left({\displaystyle \frac{\mathrm{2}{a}^{\mathrm{2}}+\mathrm{1}/\mathrm{3}}{a^{\mathrm{2}}+{\Vert \tilde{T}\Vert }^{\mathrm{2}}}}\right)}^{\mathrm{2}}{\Vert {\tilde{T}}^{*}\Vert }^{\mathrm{2}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\left({\displaystyle \frac{a^{\mathrm{2}}-{\Vert {\tilde{T}}^{*}\Vert }^{\mathrm{2}}}{a^{\mathrm{2}}+{\Vert \tilde{T}\Vert }^{\mathrm{2}}}}\right)}^{\mathrm{2}}={\left({\displaystyle \frac{a}{f_{\mathrm{d}}}}\right)}^{\mathrm{2}}.

Considering the identity {\Vert \tilde{T}\Vert }^{\mathrm{2}}={\Vert {\tilde{T}}^{*}\Vert }^{\mathrm{2}}+\mathrm{1}/\mathrm{3}, one can obtain the value of \Vert {\tilde{T}}^{*}\Vert for a given density factor f_d by solving Eq. (18). Then, the dilatancy ratio in Eq. (17) can be evaluated and the peak friction angle can be solved from the following Equation [3]:

(19)\Vert {\tilde{T}}_{\mathrm{p}}^{*}\Vert =\sqrt{{\displaystyle \frac{\mathrm{8}}{\mathrm{3}}}}{\displaystyle \frac{\sin {\phi }_{\mathrm{p}}}{\mathrm{3}-\sin {\phi }_{\mathrm{p}}}}.

Herein the subscript p signifies that the quantities are related to the peak state.

Figure 3 shows the relationship between the peak friction angle, the dilatancy ratio and the density factor at the peak state under triaxial compression. The parameters used are given as follows [18]:

{\phi }_{\mathrm{c}}={\mathrm{35.5}}^{\circ };h_{\mathrm{s}}=\mathrm{95}MPa;n=\mathrm{0.45};e_{i\mathrm{o}}=\mathrm{0.81};e_{\mathrm{c}\mathrm{o}}=\mathrm{0.62};e_{\mathrm{d}\mathrm{o}}=\mathrm{0.37};\alpha =\mathrm{0.20};\beta =\mathrm{1.05.}

It is evident that a denser material (with a lower f_d) has a higher peak friction angle and a higher dilatancy ratio. However, when the density factor f_d approaches 1, the dilation of the material tends to vanish and the peak friction angle is identical to the critical friction angle. These salient features agree well with the stress-dilatancy theories [6-8] and experimental observations [9,10].

It is important to point out that the scale factor s for the stress transformation is equal to 1 under triaxial compression and isotropic compression, i.e., the transformed stress tensors coincide with the pre-transformed stress tensors and \widehat{a} is identical to a. Therefore, the responses under triaxial compression and isotropic compression conditions predicted by GB_HYPO and TS_HYPO are exactly the same and the results shown in Fig. 3 are also applicable to GB_HYPO.

The strength-dilatancy characteristics in two hypoplastic models

In this part, the relationships between the peak strength and the dilation rate embraced in TS_HYPO will be explored. The components of the stress tensor and the strain rate tensor under different conditions are shown in Fig. 4.

Triaxial compression condition

To represent the relationship between the peak strength and the corresponding dilation rate in triaxial compression, we choose the peak stress ratio (T₁₁/T₃₃)_p to denote the strength and choose the ratio of volumetric strain rate to axial strain rate {\left({\dot{ϵ}}_v/{\dot{ϵ}}_{\mathrm{1}}\right)}_{\mathrm{p}} to denote the dilation rate. These two quantities can be expressed with the norm of the peak deviatoric stress and the dilatancy ratio respectively, i.e.,

(20){\left({\displaystyle \frac{T_{\mathrm{11}}}{T_{\mathrm{33}}}}\right)}_{\mathrm{p}}={\displaystyle \frac{\mathrm{1}+\mathrm{4}\sqrt{\mathrm{3}/\mathrm{8}}\Vert {\tilde{T}}_{\mathrm{p}}^{*}\Vert }{\mathrm{1}-\mathrm{2}\sqrt{\mathrm{3}/\mathrm{8}}\Vert {\tilde{T}}_{\mathrm{p}}^{*}\Vert }}.

and

(21){\left({\displaystyle \frac{{\dot{ϵ}}_{\mathrm{v}}}{{\dot{ϵ}}_{\mathrm{d}}}}\right)}_{\mathrm{p}}={\displaystyle \frac{\mathrm{3}{\left({\dot{ϵ}}_{\mathrm{v}}/{\dot{ϵ}}_{\mathrm{1}}\right)}_{\mathrm{p}}}{\mathrm{3}-{\left({\dot{ϵ}}_{\mathrm{v}}/{\dot{ϵ}}_{\mathrm{1}}\right)}_{\mathrm{p}}}}.

Note that Eqs. (20) and (21) are valid under triaxial compression condition.

Given a value of the strain rate ratio {\left({\dot{ϵ}}_{\mathrm{v}}/{\dot{ϵ}}_{\mathrm{1}}\right)}_{\mathrm{p}}, the dilatancy ratio {\left({\dot{ϵ}}_{\mathrm{v}}/{\dot{ϵ}}_{\mathrm{d}}\right)}_{\mathrm{p}} can be obtained via Eq. (21) and the norm of the transformed deviatoric stress tensor \Vert {\tilde{T}}_{\mathrm{p}}^{*}\Vert can be solved from Eq. (17); substituting the obtained \Vert {\tilde{T}}_{\mathrm{p}}^{*}\Vert into Eq. (20) gives the peak stress ratio (T₁₁/T₃₃)_p.

Figure 5(a) shows the relationships between (T₁₁/T₃₃)_p and {\left({\dot{ϵ}}_{\mathrm{v}}/{\dot{ϵ}}_{\mathrm{1}}\right)}_{\mathrm{p}} for three different critical friction angles. For comparison, the predictions made by Eq. (3) are also plotted in Fig. 5(a). It can be seen that the strength-dilatancy relation predicted by TS_HYPO and that by Eq. (3) are very close to each other, especially when the dilatancy ratio is relatively low.

Oda [19] conducted triaxial compression experiments on two kinds of sand and compiled the relationship between the peak stress ratio (T₁₁/T₃₃)_p and the corresponding strain rate ratio {\left({\dot{ϵ}}_{\mathrm{v}}/{\dot{ϵ}}_{\mathrm{1}}\right)}_{\mathrm{p}} as shown in Fig. 5(b). Although Oda has not give any information about the critical friction angle of the sands, we can approximately estimate it according to the experimental data, i.e., {\phi }_{\mathrm{c}}\approx {\mathrm{29}}^{\circ }. With the calibrated critical friction angle, both the strength-dilatancy relation predicted by TS_HYPO and that predicted by Eq. (3) are plotted together with the experimental results in Fig. 5(b). Satisfactory agreement among the prediction by TS_HYPO and the experimental results are observed, indicating that the transformed stress based model is capable of capturing the strength-dilatancy characteristics of granular materials under triaxial compression condition.

As pointed out previously, the constitutive equation of GB_HYPO is the same as that of TS_HYPO under triaxial compression. Therefore, the results and conclusion obtained here are also applicable to GB_HYPO. Namely, both GB_HYPO and TS_HYPO are capable of capturing the strength-dilatancy characteristics of frictional materials under triaxial compression condition satisfactorily.

Plane compression condition

It is useful to expand Eq. (12) for the discussion of the strength-dilatancy characteristics at the peak state in plane compression. Hence, the componential equations are given as follows:

(22)\stackrel{\circ }{T_{\mathrm{22}}}=f_{\mathrm{s}}\left[a^{\mathrm{2}}{D}_{\mathrm{22}}+\left(\tilde{T}:D\right){\tilde{T}}_{\mathrm{22}}-a{f}_{\mathrm{d}}\left({\tilde{T}}_{\mathrm{22}}+{\tilde{T}}_{\mathrm{22}}^{*}\right)\sqrt{D_{\mathrm{11}}^{\mathrm{2}}+D_{\mathrm{33}}^{\mathrm{2}}}\right]

And

(23)$$\begin{gathered} \{a^{\mathrm{2}}{D}_{\mathrm{11}}+\left(\tilde{T}:D\right){\tilde{T}}_{\mathrm{11}}-a{f}_{\mathrm{d}}\left({\tilde{T}}_{\mathrm{11}}+{\tilde{T}}_{\mathrm{11}}^{*}\right)\sqrt{D_{\mathrm{11}}^{\mathrm{2}}+D_{\mathrm{33}}^{\mathrm{2}}}=\mathrm{0} \\ {a}^{\mathrm{2}}{D}_{\mathrm{33}}+\left(\tilde{T}:D\right){\tilde{T}}_{\mathrm{33}}-a{f}_{\mathrm{d}}\left({\tilde{T}}_{\mathrm{33}}+{\tilde{T}}_{\mathrm{33}}^{*}\right)\sqrt{D_{\mathrm{11}}^{\mathrm{2}}+D_{\mathrm{33}}^{\mathrm{2}}}=\mathrm{0}. \end{gathered}$$

Simple operations on the two expressions in Eq. (23) yield the following equation:

(24)a\left(\mathrm{1}-{\displaystyle \frac{{\tilde{T}}_{\mathrm{11}}}{{\tilde{T}}_{\mathrm{33}}}}{\displaystyle \frac{D_{\mathrm{33}}}{D_{\mathrm{11}}}}\right)+{\displaystyle \frac{f_{\mathrm{d}}}{\mathrm{3}}}\left(\mathrm{1}-{\displaystyle \frac{{\tilde{T}}_{\mathrm{11}}}{{\tilde{T}}_{\mathrm{33}}}}\right)\sqrt{\mathrm{1}+{\left({\displaystyle \frac{D_{\mathrm{33}}}{D_{\mathrm{11}}}}\right)}^{\mathrm{2}}}=\mathrm{0.}

This is a linear equation about the transformed stress ratio and the solution reads:

(25){\displaystyle \frac{{\tilde{T}}_{\mathrm{11}}}{{\tilde{T}}_{\mathrm{33}}}}={\displaystyle \frac{\mathrm{3}a+f_{\mathrm{d}}\sqrt{\mathrm{1}+{(D_{\mathrm{33}}/D_{\mathrm{11}})}^{\mathrm{2}}}}{\mathrm{3}a(D_{\mathrm{33}}/D_{\mathrm{11}})+f_{\mathrm{d}}\sqrt{\mathrm{1}+{(D_{\mathrm{33}}/D_{\mathrm{11}})}^{\mathrm{2}}}}}.

Equation (25) establishes the strength-dilatancy relationship in plane compression. Given the density factor f_d and the Lode angle θ, the relationship between the transformed stress ratio {\tilde{T}}_{\mathrm{11}}/{\tilde{T}}_{\mathrm{33}} and the strain rate ratio D_{\mathrm{33}}/D_{\mathrm{11}} can be obtained, which can be transformed to the relationship between the peak friction angle ϕ_p and the dilation angle ψ.

Figure 6 shows the relationships between the peak friction angle and the corresponding dilation angle for different densities and Lode angles predicted by Eq. (25). The predictions by Rowe’s stress-dilatancy theory [8] and Bolton’s empirical relation [9] are also plotted in Fig. 6 for comparison. It is rather striking that the strength-dilatancy relationship embraced in TS_HYPO is either close to Rowe’s theoretical prediction or close to Bolton’s empirical prediction. In particular, when the Lode angle lies within the range [20°, 40°] (which is the common range of Lode angle at peak state in plane compression according to series of numerical simulations), the strength-dilatancy relationship in TS_HYPO is very close to Bolton’s predictions. Since Bolton’s empirical relation was verified by many documented experimental data of cohesionless materials, it can be concluded that in plane compression the strength-dilatancy characteristics of granular materials can be captured effectively by TS_HYPO.

The strength-dilatancy relationship in GB_HYPO can be obtained simply by replacing the transformed stress components in Eq. (19) with the pre-transformed ones and the constant a with the stress-dependent \widehat{a}, i.e.,

(26){\displaystyle \frac{{\widehat{T}}_{\mathrm{11}}}{{\widehat{T}}_{\mathrm{33}}}}={\displaystyle \frac{\mathrm{3}\widehat{a}+f_{\mathrm{d}}\sqrt{\mathrm{1}+{(D_{\mathrm{33}}/D_{\mathrm{11}})}^{\mathrm{2}}}}{\mathrm{3}\widehat{a}(D_{\mathrm{33}}/D_{\mathrm{11}})+f_{\mathrm{d}}\sqrt{\mathrm{1}+{(D_{\mathrm{33}}/D_{\mathrm{11}})}^{\mathrm{2}}}}}.

While Eq. (25) gives a reasonable strength-dilatancy relationship in TS_HYPO, Eq. (26) is not equivalently effective. This is illustrated in Fig. 7: The deviation of the relationship between the peak friction angle and the dilation angle predicted by GB_HYPO from the predictions made by Rowe’s theory and Bolton’s equation is much more evident than that shown in Fig. 6. In most of the selected cases, the peak friction angle predicted by GB_HYPO is lower than the one predicted by Rowe’s and Bolton’s theories, and the discrepancy tends to be increasingly evident when the dilation angle increases. Comparison of Figs. 6 and 7 indicates that the performance in reproducing the strength-dilatancy relationship of granular materials is much closer to the classical stress-dilatancy theories in TS_HYPO by using the stress transformation technique.

Plane shearing condition

To explore the strength-dilatancy relationship in simple shearing, Eq. (12) is expanded as follows:

(27)$$\begin{gathered} \{\stackrel{\circ }{T_{\mathrm{22}}}=f_{\mathrm{s}}\left[\left(\tilde{T}:D\right){\tilde{T}}_{\mathrm{22}}-a{f}_{\mathrm{d}}\left({\tilde{T}}_{\mathrm{22}}+{\tilde{T}}_{\mathrm{22}}^{*}\right)\sqrt{D_{\mathrm{11}}^{\mathrm{2}}+\mathrm{2}{D}_{\mathrm{13}}^{\mathrm{2}}}\right] \\ \stackrel{\circ }{T_{\mathrm{33}}}=f_{\mathrm{s}}\left[\left(\tilde{T}:D\right){\tilde{T}}_{\mathrm{33}}-a{f}_{\mathrm{d}}\left({\tilde{T}}_{\mathrm{33}}+{\tilde{T}}_{\mathrm{22}}^{*}\right)\sqrt{D_{\mathrm{11}}^{\mathrm{2}}+\mathrm{2}{D}_{\mathrm{13}}^{\mathrm{2}}}\right]. \end{gathered}$$

and

(28)$$\begin{gathered} \{a^{\mathrm{2}}{D}_{\mathrm{11}}+\left(\tilde{T}:D\right){\tilde{T}}_{\mathrm{11}}-a{f}_{\mathrm{d}}\left({\tilde{T}}_{\mathrm{11}}+{\tilde{T}}_{\mathrm{11}}^{*}\right)\sqrt{D_{\mathrm{11}}^{\mathrm{2}}+\mathrm{2}{D}_{\mathrm{13}}^{\mathrm{2}}}=\mathrm{0} \\ {a}^{\mathrm{2}}{D}_{\mathrm{13}}+\left(\tilde{T}:D\right){\tilde{T}}_{\mathrm{13}}-a{f}_{\mathrm{d}}\left({\tilde{T}}_{\mathrm{13}}+{\tilde{T}}_{\mathrm{11}}^{*}\right)\sqrt{D_{\mathrm{11}}^{\mathrm{2}}+\mathrm{2}{D}_{\mathrm{13}}^{\mathrm{2}}}=\mathrm{0}. \end{gathered}$$

Mathematical operations on both expressions in Eq. (28) yield the strength-dilatancy equation, i.e.,

(29)a\left({\displaystyle \frac{{\tilde{T}}_{\mathrm{13}}}{{\tilde{T}}_{\mathrm{11}}}}{\displaystyle \frac{D_{\mathrm{11}}}{D_{\mathrm{13}}}}-\mathrm{1}\right)+{\displaystyle \frac{f_{\mathrm{d}}}{\mathrm{3}}}{\displaystyle \frac{{\tilde{T}}_{\mathrm{13}}}{{\tilde{T}}_{\mathrm{11}}}}\sqrt{\mathrm{2}+{\left({\displaystyle \frac{D_{\mathrm{11}}}{D_{\mathrm{13}}}}\right)}^{\mathrm{2}}}=\mathrm{0},

the solution of which reads:

(30){\displaystyle \frac{{\tilde{T}}_{\mathrm{13}}}{{\tilde{T}}_{\mathrm{11}}}}={\displaystyle \frac{\mathrm{3}a}{\mathrm{3}a(D_{\mathrm{11}}/D_{\mathrm{13}})+f_{\mathrm{d}}\sqrt{\mathrm{2}+{(D_{\mathrm{11}}/D_{\mathrm{13}})}^{\mathrm{2}}}}}.

For a given density factor f_d and Lode angle θ, the strength-dilatancy relationship can be derived numerically as is done for plane compression problems. However, it deserves to pay a closer attention to the stress state in simple shearing problems implied in TS_HYPO. Generally, the specimen is first vertically compressed without horizontal expansion before shearing. This means the specimen is in K₀ consolidation state at the beginning of shearing. Bearing this initial condition in mind and examining Eq. (27) reveal an important feature of TS_HYPO in simulating simple shearing problems, namely, the objective stress rates of T₂₂ and T₃₃ are equal. Therefore, the stress component T₂₂ is identical to the component T₃₃ throughout the whole process of shearing, i.e.,

(31)T_{\mathrm{22}}\equiv T_{\mathrm{33}};{\tilde{T}}_{\mathrm{22}}\equiv {\tilde{T}}_{\mathrm{33}};{\widehat{T}}_{\mathrm{22}}\equiv {\widehat{T}}_{\mathrm{33}}.

Moreover, numerical simulations of simple shearing using TS_HYPO show that the horizontal stress exceeds the vertical stress at peak states, i.e., . This condition confines the range of Lode angle within [0°, 30°].

Figure 8 shows the relationship between the peak friction angle and the dilation angle for different densities and different Lode angles. In the numerical calculations, the peak friction angle is evaluated according to Coulomb criterion (ϕ_p = arctan(T₁₃/T₃₃)_p). The predictions made by Taylor’s stress-dilatancy theory [6] and Bolton’s empirical relation [9] are also plotted in Fig. 8. It can be seen that the strength-dilatancy relationship embraced in TS_HYPO are either close to Taylor’s theoretical predictions or close to Bolton’s empirical predictions. In particular when the Lode angle lies within the range [20°, 30°] (which is the common range of Lode angle at the peak state in simple shearing), the strength-dilatancy relationship in TS_HYPO is quite close to Bolton’s predictions. Since Bolton’s empirical relationship is also verified by direct shear experiments [10], it’s reasonable to argue that the strength-dilatancy characteristics of granular materials in plane shearing can also be captured by TS_HYPO satisfactorily.

Similar to Eq. (30), the strength-dilatancy relationship for plane shearing problems embraced in GB_HYPO can be obtained by replacing {\tilde{T}}_{ij} and a in Eq. (30) with {\widehat{T}}_{ij} and \widehat{a}, respectively, i.e.,

(32){\displaystyle \frac{{\tilde{T}}_{\mathrm{13}}}{{\tilde{T}}_{\mathrm{11}}}}={\displaystyle \frac{\mathrm{3}a}{\mathrm{3}a(D_{\mathrm{11}}/D_{\mathrm{13}})+f_{\mathrm{d}}\sqrt{\mathrm{2}+{(D_{\mathrm{11}}/D_{\mathrm{13}})}^{\mathrm{2}}}}}.

The relationship predicted by Eq. (26) is plotted in Fig. 9. Once again, the peak friction angle at a given dilation angle predicted by GB_HYPO is lower than the one given by Taylor’s and Bolton’s theories, and the deviation is considerably reduced in the transformed stress based constitutive model by Huang [4].

Conclusions

In this study, the strength-dilatancy characteristics embraced in TS_HYPO were investigated on the element level. Particular attention was focused on the relationship between the peak strength and the corresponding dilation rate under triaxial compression, plane compression and simple shearing. The relationships predicted by two constitutive equations were compared with those predicted by classical stress-dilatancy theories. It is concluded that:

1) The strength-dilatancy characteristics embraced in TS_HYPO and GB_HYPO under triaxial compression are the same, both of which are rather comparable to the predictions made by the stress-dilatancy equation used in the Cam-Clay model and are in good agreement with the experimental results of Oda.

2) The strength-dilatancy characteristics in plane compression predicted by TS_HYPO are either close to Rowe’s theoretical results or close to Bolton’s empirical results. In particular, when the Lode angle at the peak state lies within the common range in plane compression problems, the relationship embraced in TS_HYPO are in good agreement with Bolton’s experiments-based empirical relation. On the other hand, the performance of GB_HYPO in capturing the classical stress-dilatancy theories by Rowe and Bolton is less satisfactory, i.e., the peak friction angle at a given dilation angle predicted by GB_HYPO is lower than the one given by both classical theories, and the discrepancy tends to be increasingly evident when the dilation angle increases.

3) The strength-dilatancy characteristics for plane shearing problems predicted by TS_HYPO are either close to Taylor’s theoretical predictions or close to Bolton’s empirical predictions. Furthermore, when the Lode angle at the peak state lies within the normal range in plane shearing, the predictions made by TS_HYPO agree well with Bolton’s empirical results again. Similar as in plane compression problems, the performance of GB_HYPO in capturing the strength-dilatancy characteristics is evidently enhanced by employing the stress transformation technique.

Based on the above conclusions, it is reasonable to argue that the deformation and strength of granular materials are well combined on the element level in the hypoplastic models, particularly in the transformed stress based model. However, it’s important to note that the strain localization effect within the granular bodies near the peak state was not taken into account in this paper and the results presented here are based on the presupposition of a homogeneous deformation field, which also serves as the footstone of most of the existing constitutive models.

References

[1] Gudehus G. A comprehensive constitutive equation for granular materials. Soil and Foundation, 1996, 36(1): 1–12

[2] Bauer E. Calibration of a comprehensive hypoplastic model for granular materials. Soil and Foundation, 1996, 36(1): 13–26

[3] Bauer E. Conditions for embedding Casagrande’s critical states into hypoplasticity. Mechanics of Cohesive-Frictional Materials, 2000, 5(2): 125–148

[4] Huang W X, Sloan S, Fityus S. Incorporating a predefined limit condition in a hypoplastic model by means of stress transformation. Mechanics of Materials, 2008, 40(10): 796–802

[5] Matsuoka H, Yao Y P, Sun D A. The cam-clay models modified by the SMP criterion. Soil and Foundation, 1999, 39(1): 81–95

[6] Taylor D W. Fundamentals of Soil Mechanics. New York: John & Wiley, 1948

[7] Schofield A N, Wroth C P. Critical State Soil Mechanics. London: McGraw-Hill, 1968

[8] Rowe P W. The stress dilatancy relation for static equilibrium of an assembly of particles in contact. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1962, 269(1339): 500–527

[9] Bolton M D. The strength and dilatancy of sands. Geotechnique, 1986, 36(1): 65–78

[10] Simoni A, Houlsby G T. The direct shear strength and dilatancy of sand-gravel mixtures. Geotechnical and Geological Engineering, 2006, 24(3): 523–549

[11] Vardoulakis I, Georgopoulos I O. The ‘stress-dilatancy’ hypothesis revisited: shear-banding related instabilities. Soil and Foundation, 2005, 45(2): 61–76

[12] Iai S. A new look at the stress dilatancy relation in Cam-Clay model. Soil and Foundation, 1994, 34(2): 1–12

[13] Cubrinovski M, Ishihara K. Modelling of sand behaviour based on state concept. Soil and Foundation, 1998, 38(3): 115–127

[14] Matsuoka H, Nakai T. Relationship among Tresca, Mises, Mohr-Coulomb and Matsuoka-Nakai failure criteria. Soil and Foundation, 1985, 25(4): 123–128

[15] Huang W X, Sun D A, Sloan S W. Analysis of the failure mode and softening behavior of sands in true triaxial tests. International Journal of Solids and Structures, 2007, 44(5): 1423–1437

[16] Lade P V. Assessment of test data for selection of 3-D failure criterion for sand. International Journal for Numerical and Analytical Methods in Geomechanics, 2006, 30(4): 307–333

[17] Lade P V. Instability, shear banding, and failure in granular materials. International Journal of Solids and Structures, 2002, 39(13-14): 3337–3357

[18] Fu Z Z, Bauer E. Hypoplastic constitutive modeling of the long term behaviour and wetting deformation of weathered granular materials. In: Bauer E, Semprich S, Zenz G eds. In: Proceeding of the 2nd International Conference on Long Term Behaviour of Dams. Graz, 2009, 473–478

[19] Oda M. Deformation mechanism of sand in triaxial compression tests. Soil and Foundation, 1972, 12(4): 45–63

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