# A hypoplastic constitutive model for debris materials

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## Abstract

Debris flow is a very common and destructive natural hazard in mountainous regions. Pore water pressure is the major triggering factor in the initiation of debris flow. Excessive pore water pressure is also observed during the runout and deposition of debris flow. Debris materials are normally treated as solid particle–viscous fluid mixture in the constitutive modeling. A suitable constitutive model which can capture the solid-like and fluid-like behavior of solid–fluid mixture should have the capability to describe the developing of pore water pressure (or effective stresses) in the initiation stage and determine the residual effective stresses exactly. In this paper, a constitutive model of debris materials is developed based on a framework where a static portion for the frictional behavior and a dynamic portion for the viscous behavior are combined. The frictional behavior is described by a hypoplastic model with critical state for granular materials. The model performance is demonstrated by simulating undrained simple shear tests of saturated sand, which are particularly relevant for the initiation of debris flows. The partial and full liquefaction of saturated granular material under undrained condition is reproduced by the hypoplastic model. The viscous behavior is described by the tensor form of a modified Bagnold’s theory for solid–fluid suspension, in which the drag force of the interstitial fluid and the particle collisions are considered. The complete model by combining the static and dynamic parts is used to simulate two annular shear tests. The predicted residual strength in the quasi-static stage combined with the stresses in the flowing stage agrees well with the experimental data. The non-quadratic dependence between the stresses and the shear rate in the slow shear stage for the relatively dense specimens is captured.

## Keywords

Constitutive modeling Debris flows Granular-fluid flows Hypoplastic model## 1 Introduction

Debris flow is a very common natural hazard in the mountainous areas of many countries. It represents the gravity-driven flow of a mixture of various sizes of sediment, water and air, down a steep slope, often initiated by heavy rainfall and landslides [17]. The highest velocity of debris flows can be more than \(30\,\hbox {m/s}\); however, typical velocities are less than \(10\,\hbox {m/s}\) [24]. The fast debris flows may cause significant erosion, while increasing the sediment charge and destructive potential. Such mass flows cause serious casualties and property losses in many countries around the world. The initiation mechanisms of debris flow and the predicted possible velocity are essential information for the design of protective measures. Numerical analysis plays an important role to obtain this information, where a competent constitutive model for debris materials is required. The main factors influencing the initiation of debris flow are, among others, the topography, material parameters, water and the initial stress state in the affected slope [22]. Earth slopes with inclinations ranging from \(26^\circ\) to \(45^\circ\) have been generally identified as most prone to debris flow initiation [40]. The common solid volume fraction of debris materials, defined as the ratio between the solid volume and the total volume of a representative volume element, varies between about 30 and 65 %. The water from heavy rainfall or snow melting makes the unconsolidated superficial deposit on a steep hillside saturated, thereby leading to a reduced shear strength due to the decreasing of matric suction, and further triggering a landslide. Such an upland landslide may develop into a hillside debris flow when the water in the sliding mass cannot be discharged quickly and therefore gives rise to excessive pore water pressure. In this case, based on the principles of soil mechanics, the effective stresses between solid particles will decrease to cause the reduction or complete loss of shear strength. Upon initiation of debris flow, debris material shows fluid-like behavior. As concluded by Iverson [19], debris flow can be mobilized by three processes: (*i*) widespread Coulomb failure along a rupture surface within a saturated soil or sediment mass, (*ii*) partial or complete liquefaction of a sliding mass due to high pore-fluid pressure and (*iii*) conversion of landslide translational energy to internal vibrational energy. In these processes, the development of high pore water pressure is likely the most significant triggering factor. In addition, experimental observation [18] shows that an almost constant excess pore water pressure persists during the runout and depositing of debris flows. Thus, a suitable constitutive model which can capture the solid-like behavior before failure and the fluid-like behavior after failure should has the capability to describe the developing of pore water pressure (or effective stresses) in the initiation stage and determine the residual effective stresses exactly. Some important material parameters such as solid volume fraction (or void ratio in soil mechanics) and the internal friction coefficient need to be taken into account. Actually, debris materials are normally simplified as solid spherical particle–viscous fluid mixture and treated as a fluid continuum with microstructural effect in the constitutive modeling [10, 11]. In most conventional models, constitutive equations for the static and dynamic regimes are formulated and applied separately, such as the models for the solid-like behaviors of granular materials [8, 27, 41, 43] and that for the fluid-like behaviors [1, 6, 21]. Although some models for granular-fluid flows have taken the stress state of the quasi-static stage into account, the employed theories for the static regime, such as Mohr–Coulomb criterion [34] and extended von Mises yield criterion [32], still fail to determine the changing of pore water pressure from the deformation directly. Hypoplasticity was proposed as an alternative to plasticity for the description of solid-like behavior of granular materials [41, 43]. The distinctive features of hypoplasticity are its simple formulation and capacity to capture some salient features of granular materials, such as non-linearity, dilatancy and yielding [42]. It may be the suitable choice for the description of solid-like behavior of debris materials.

## 2 The framework of constitutive modeling for debris materials

As stated in the preceding section, debris materials show solid-like behavior before failure and fluid-like behavior after failure. This particular phenomenon cannot be modeled only within the framework of statics or dynamics. An applicable model may need to combine a static and a dynamic portion and make the transition from solid-like to fluid-like behavior turns out as an outcome [42].

*P*and

*T*are the normal and shear stresses for the solid phase; \(P_0\) and \(T_0\) are the normal and shear stress caused by prolonged contact between particles; \(T_v\), \(T_i\) and \(P_i\) are slightly modified Bagnold’s constitutive relations for a gravity-free dispersion of solid spheres sheared in Newtonian liquids. The stresses \(P_0\) and \(T_0\) are the static portion of the framework which satisfy a generalized Mohr–Coulomb type yield criterion [9, 31]. Thus,

*U*is the shear velocity as shown in Fig. 1 and \({{\mathrm {d}}}U/{{\mathrm {d}}}y\) denotes the shear rate changing along the depth direction;

*C*is the mean solid volume fraction and \(C_c\) is the maximum solid volume fraction to assure a full shearing to occur;

*n*is a fitting parameter and \(\mu\) is the dynamic viscosity of the interstitial fluid; \(\lambda\) is a dimensionless parameter termed linear concentration. For perfectly spherical particles, \(\lambda\) is defined as

*s*is the mean free distance between two particles; \(C_\infty\) is the asymptotic limit of the maximum measured solid volume fraction as the container dimensions approach infinity, which is also related to the size of the particles [16]. The shear stress for the ‘grain-inertia’ regime, \(T_i\), is formulated as

*d*denote the material density and mean diameter of the grains, respectively; the tangent of the angle \(\alpha _i\) corresponds to the ratio between the shear and normal stress in the ‘grain-inertia’ regime. Therefore, the expression of the normal stress in the ‘grain-inertia’ regime is

*g*is the gravity acceleration;

*h*is the depth along the

*y*axis which is normal to the flow bed. Then we get the stress ratio

In the above analysis, the simple formula for the initial value of \(P_0\), (13), is only applicable for free surface dry granular flows. As pointed out in the preceding section, debris materials are saturated solid–fluid mixtures which will be partially or fully liquefied in the initiation of debris flows. The normal stress \(P_0\) is the effective stress and obtained by subtracting the excess pore water pressure from the total normal stress in this case. A proper theory is required to capture the partial or complete liquefaction, and further determine the residual strength \(P_0\) and \(T_0\). As introduced before, hypoplasticity may be the suitable choice for the description of solid-like behavior of debris materials. In the following section, we study the capability of a specific hypoplastic model for capturing the main properties of debris materials in the quasi-static stage.

## 3 The applicability of hypoplastic models for debris materials

Hypoplastic constitutive equations are based on nonlinear tensorial functions with the major advantages of simple formulation and few parameters. Two hypoplastic models, the one developed by Wu et al. [41] and the one by Gudehus [13], are compared in the selection of the static portion for the framework (1). In the more recent models by Gudehus [13], mainly the stiffness is modified by the two factors, \(f_b\) and \(f_e\), which take into account the influence of stress state and density, respectively. In modeling debris flow, however, the strength is very important and the stiffness is not important. Moreover, his model makes use of the exponential functions for the dependence of critical void ratio and minimum void ratio on pressure. For each function the parameters reduce from 3 to 2. However, there are only few data in the literature for the exponential functions. Therefore, in this paper, we will embark on the model proposed by Wu et al. [41] which is the first hypoplastic model with critical state to verify that, by employing an appropriate hypoplastic model as the static portion, the combined model based on the framework (1) can fulfill an entire and quantitative description of stress state for debris materials from quasi-static stage to fast flow stage.

It is worth mentioning that the hypoplastic model with critical state is just one of the choices for describing the initiation of debris flows. Recently some improved models have been available, e.g. [12, 23, 35], which are developed from some widely used versions of hypoplastic model [28, 37] and aim to improve the dependence of stiffness on pressure and density. However, the capability of these models for capturing the phenomenon of liquefaction and the stability in the cases of large deformation or low confining pressure still need to be verified. A more concise hypoplastic model with the former mentioned capability and stability can be employed to determine the stress state in the quasi-static stage of debris materials.

Taking the four material constants \(c_1\sim c_4\) as unknowns, a system of four linear equations can be obtained by substituting the corresponded stress and strain rate of the two stress states into the model (16). Therefore, the material constants are determined as functions of the well-established parameters in soil mechanics, the initial tangent modulus \(E_i\), the initial Poisson ratio \(\upsilon _i\), the friction angle \(\phi _0\) and the dilatancy angle \(\psi\). It should be pointed out that these parameters are related to a specific confining pressure, all the sets of material constants used in this paper are obtained with a confining pressure \(\mathbf{T }_h(3,3)=100\,\hbox {kPa}\). In addition, the deviatoric loading in the initial hydrostatic state is considered to be zero, i. e. the initial Poisson ratio \(\upsilon _i=0\).

*a*is a material parameter related to the stress level and

*e*is the void ratio; \(e_{{\mathrm{min}}}\) and \(e_{{\mathrm{crt}}}\) are the minimum and the critical void ratio, respectively. The effect of void ratio and stress level on the behavior of granular materials is taken into account in the model (21) by using the following expressions,

*e*is equal to the critical value \(e_{{\mathrm{crt}}}\) from (22) and (23). It means, for same material, same constants \(c_1\sim c_4\) will be obtained for the original and extended models in the case of \(e=e_{{\mathrm{ecrt}}}\). Thus, the material constants emerging in the model (21) can be determined by the same way as done for (16). The dilatancy angle \(\psi\) is equal to zero since there is no volume deformation in this case [44]. About the material parameters \(p_i (i=1, \ldots , 3)\) and \(q_i (i=1, \ldots , 3)\), some theoretical and experimental analyses are presented in [41]. \(p_1\) is the critical void ratio when the confining pressure approaches infinity, since \(p_3\) is negative. The value of \(p_1\) should be close to the minimum void ratio under a high confining pressure. For the case of zero confining pressure, the critical void ratio is equal to \(p_1+p_2\) which may close to the maximum void ration measured with very low confining pressure. \(q_1\) is assumed to be always equal to 1 and \(q_3\) is a negative value. For the case of \({\text{tr}}\mathbf{T }_h \rightarrow \infty\), the difference between dense and loose packing tends to disappear since the parameter \(a\rightarrow 1\). Based on the numerical parametric study [41], \(q_2\) is suggested to lie in the range (−0.3, 0.0). \(p_3\) and \(q_3\) for quartz sand are assumed to be −0.0001 kPa. In the case of very low confining pressure, such as the state of liquefaction, relatively higher values of \(q_2\), \(p_3\) and \(q_3\) may be needed to keep the sensitivity of \(I_e\) to the stress level.

Parameters for \(e_{{\mathrm{crt}}}\) and *a* in the simulation of the experiments in [46]

\(p_1\) | \(p_2\) | \(p_3\) | \(q_1\) | \(q_2\) | \(q_3\) |
---|---|---|---|---|---|

[–] | [–] | [\({\hbox {kPa}}^{-1}\)] | [–] | [–] | [\({\hbox {kPa}}^{-1}\)] |

0.53 | 0.45 | −0.0018 | 1.0 | −0.4 | −0.0001 |

*a*. Both changes increase the density function \(I_e\) and then limit the developing of hardening. Conversely, when the softening occurs, \(I_e\) will decrease to restrict softening and liquefaction. Due to the regulatory function of \(I_e\), the model (21) can describe the shear softening and the residual strength of very loose granular materials. It can be used as the static portion of the new model for debris materials. As shown in Fig. 5, the normal stresses \(\sigma _{ii} (i=1, 2, 3)\) of the very loose specimen with \(e=0.876\) tend to be isotropic when the shear strain is large enough, no matter what is the initial stress state. The isotropic normal stress in the large deformation stage corresponds to the former mentioned thermodynamic pressure \(P_0\).

## 4 A new constitutive model for debris materials

What need to be mentioned is that the static portion, \(\mathbf{T }_h\), is rate independent. It is varying due to the accumulation of the shear strain rather than the changing of the shear rate. By merging with the dynamic portion, the total effective stress (32) becomes rate dependent. As shown in Fig. 5, the normal stresses reach a residual constant when the shear strain is approximately 0.4. This process is finished with very small shear velocity in the so-called quasi-static stage. Thus, in the simulation, the shear rate must be kept at a small value before the failure of the granular-fluid mixture to make sure that the static portion is the dominant part in the total effective stress \(\mathbf{T }\). The static portion should be much greater than the dynamic portion at the point *A* in Fig. 6. One approach to meet this requirement in numerical calculations is using small shear strain acceleration and increasing the time steps for the stage before failure.

## 5 Performance of the proposed model

In this section, the new model, (32), will be used to predict the stress–strain relations of granular-fluid flows with different materials and experimental apparatus in some element tests. The experimental data of two annular shear tests as undrained simple shear tests are employed to verify the applicability of the new model. In our former work [15], these two experiments are also simulated by a constitutive model which cannot capture the shear softening of granular-fluid materials in the quasi-static stage. The former simulation results can be used as a control group to highlight the function of the hypoplastic portion in the new model.

### 5.1 Dry granular materials

Parameters for the static portion in the simulation of dry granular flows

\(c_1\) | \(c_2\) | \(c_3\) | \(c_4\) | \(e_{{\mathrm{min}}}\) | \(p_1\) | \(p_2\) | \(p_3\) | \(q_1\) | \(q_2\) | \(q_3\) |
---|---|---|---|---|---|---|---|---|---|---|

[–] | [–] | [–] | [–] | [–] | [–] | [–] | [\({\hbox {kPa}}^{-1}\)] | [–] | [–] | [\({\hbox {kPa}}^{-1}\)] |

\(-50\) | \(-746.55\) | \(-746.55\) | 1855.13 | 0.563 | 0.65 | 0.55 | −0.11 | 1.0 | −0.24 | −0.013 |

Parameters for the dynamic portion in the simulation of dry granular flows

| \(C_\infty\) | \(C_c\) | \(\rho _s\) | \(\mu\) | \({{\mathrm {tan}}}\alpha _i\) |
---|---|---|---|---|---|

\({[}\hbox {mm}]\) | [–] | [–] | \([\hbox {kg/m}^3]\) | \([\hbox {Pa}\cdot \hbox {s}]\) | [–] |

1.0 | 0.64 | 0.62 | 1095 | \(1.83\times 10^{-5}\) | \(0.40\sim 0.51\) |

Stress state in quasi-static stage calculated by hypoplastic model

Solid volume fraction, | 0.461 | 0.483 | 0.504 | 0.524 |

Initial void ratio, | 1.17 | 1.07 | 0.98 | 0.91 |

\(P_0 [{\mathrm{Pa}}]\) | 0 | 0 | 0 | 81 |

\(T_0 [{\mathrm{Pa}}]\) | 0 | 0 | 0 | 36 |

### 5.2 Granular-water mixture

Parameters for the static portion in the simulation of granular-fluid flows

\(c_1\) | \(c_2\) | \(c_3\) | \(c_4\) | \(e_{{\mathrm{min}}}\) | \(p_1\) | \(p_2\) | \(p_3\) | \(q_1\) | \(q_2\) | \(q_3\) |
---|---|---|---|---|---|---|---|---|---|---|

[–] | [–] | [–] | [–] | [–] | [–] | [–] | [\({\hbox {kPa}}^{-1}\)] | [–] | [–] | [\({\hbox {kPa}}^{-1}\)] |

\(-50\) | \(-511.31\) | \(-511.31\) | 680.53 | 0.64 | 0.65 | 0.55 | −0.11 | 1.0 | −0.12 | −0.013 |

Parameters for the dynamic portion in the simulation of granular-fluid flows

| \(C_\infty\) | \(C_c\) | \(\rho _s\) | \(\mu\) | \({{\mathrm {tan}}}\alpha _i\) |
---|---|---|---|---|---|

\({[}\hbox {mm}]\) | [–] | [–] | \([\hbox {kg/m}^3]\) | \([\hbox {Pa}\cdot \hbox {s}]\) | [–] |

1.85 | 0.61 | 0.52 | 2780 | \(1.0\times 10^{-3}\) | 0.59 |

Stress state in quasi-static stage calculated by hypoplastic model

Solid volume fraction, | 0.49 | 0.51 |

Initial void ratio, | 1.04 | 0.96 |

\(P_0 [{\mathrm{Pa}}]\) | 0 | 173 |

\(T_0 [{\mathrm{Pa}}]\) | 0 | 102 |

## 6 Conclusions

In the initiation of debris flows, the development of excess pore water pressure is considered as the most significant triggering factor. Debris materials are normally simplified as granular-fluid mixture for constitutive modeling. Therefore, a theory which can be used to describe the solid-like behavior of debris materials should have the ability to capture the changing of pore water pressure. Moreover, a constitutive relation for the debris materials in the flowing stage should be rate dependent, in which some important material parameters in a granular-fluid flow, such as solid volume fraction, fluid viscosity and particle density, are taken into account. In a former developed framework [15], a static portion for the friction component and a dynamic portion for the viscous component are combined. The dynamic portion is composed of a linear term for drag force of the fluid and a quadratic term for the collisional force. For a dry granular flow on an inclined plane, the linear term is negligible since the viscous effect of air is insignificant compare to the frictional and collisional effect of particles. In this case, the model predicts a steady uniform flow over a slope range, which is consistent with the experimental observation [3]. The constitutive relations based on this framework can describe the stress state throughout the shear process from yielding to high-speed shearing. Moreover, a smooth transition is obtained between the so-called macro-viscous and grain-inertia regimes. The applicability of hypoplastic model in describing debris flows before failure is studied by simulating the undrained simple shear test of saturated granular material. Such test condition is particularly relevant to the initiation mechanism of debris flow. Three types of stress–strain behavior in which the ‘liquefaction’ is regarded as the main factor of debris flow mobilization are reproduced by the hypoplastic model. It is shown that the hypoplastic model has the capability to describe the changes of pore water pressure and further capture the shear softening and hardening behavior of granular-fluid mixtures. Therefore, it is chosen as the static portion of the new model for debris flows. Then, this static part is combined with the tensor form of the modified Bagnold’s dynamic model to obtain a new complete model for the modeling of debris materials from static to dynamic state. The new model is employed to simulate two annular shear tests with dry and water-saturated granular materials. In the case of dry granular flow with constant volume, the hypoplastic portion predicts that only the densest one of the four specimens has residual strength. It implies a non-quadratic dependence between the stresses and the shear rate in the slow flowing stage which was observed in the experiments. Similar conclusion is also obtained in the case of water-saturated granular flow. Comparing to the dry granular flow, the linear term \(T_v\), which characterizes the effect of the interstitial fluid, is non-negligible in the granular-fluid flow. The element test results show that the new model is applicable to the modeling of granular materials with different interstitial fluid. The predicted stress–strain curves agree well with the experimental data.

Further verification is still needed for the new model. It is our intention to implement this model in some numerical codes for large deformation, such as SPH and computational fluid dynamics (CFD) codes, to simulate granular-fluid flows in an inclined channel or a rotating drum. As mentioned before, a hypoplastic model developed by Wang and Wu [39] has been implemented in SPH for large deformation analysis [29]. Therefore, SPH will be the preferred choice for further verification of the new model. As mentioned before, the models in the rate form may have the capability to account for the different behaviors for loading and unloading. It will be an interesting exploration to develop the rate form expression for the dynamic portion in which the loading and unloading process can be distinguished.

## Notes

### Acknowledgments

Open access funding provided by University of Natural Resources and Life Sciences Vienna (BOKU). The authors wish to thank the European Commission for the financial support to the following projects: Multiscale Modelling of Landslides and Debris Flows (MUMOLADE), Contract Agreement No. 289911 within the program Marie Curie ITN, 7th Framework Program.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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