J. Cent. South Univ. (2017) 24: 1986-1991

DOI: https://doi.org/10.1007/s11771-017-3607-5

Deformation characteristics and constitutive model of seafloor massive sulfides

HU Jian-hua(胡建华)^{1, 2, 3}, LIU Shao-jun(刘少军)^{1, 2, 3}, ZHANG Rui-qiang(张瑞强)^{1, 2, 3}, HU Qiong(胡琼)^{1, 2, 3}

1. School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China;

2. State Key Laboratory of Deep Sea Mineral Resources Development and Utilization Technology,Changsha 410012, China;

3. Shenzhen Research Institute of Central South University, Shenzhen 518000, China

Central South University Press and Springer-Verlag GmbH Germany 2017

Abstract: Deformation characteristics and constitutive model of seafloor massive sulfide (SMS) were selected as the research object. Uniaxial/triaxial compression test were carried out on the mineral samples, and the deformation characteristics of specimens under various conditions were studied. According to characteristics of the mineral, a new three stages constitutive equation was proposed. The conclusions are as follows: The axial strain, peak strain and maximum strength of seafloor massive sulfide increase with the confining pressure. The elastic modulus of the metal sulfide samples is decreased sharply with the increase of confining pressure. According to characteristics of seafloor massive sulfide, the constitutive equation is divided into three parts, the comparison between theoretical curves and experimental data shows that both of them are in good agreement, which also proves the correctness of the constitutive equation for uniaxial compression.

Key words: seafloor massive sulfide; constitutive model; failure characteristics; damage; uniaxial strain; triaxial strain

1 Introduction

Seafloor massive sulfide is following manganese nodules and cobalt-rich crusts found by humans and is a kind of deep-sea metal minerals. Seafloor massive sulfide is mainly composed of crystalline minerals, rich in Cu, Zn, Mn, Pb and some rare metals, hosted mainly in the depth of 1500 to 3000 m of the seabed [1, 2]. Because of the extremely precious metal sulfide samples, the existing samples are not much, and the study on rock mechanism of the seafloor massive sulfide is almost blank. Most researches are mainly focused on the maximum fracturing strength and other specific parameters of the mineral [3].

Seafloor massive sulfide is a very special rock, Some creative work has been carried out in the research of the constitutive model and deformation characteristics of normal rock. PAWEl et al [4] had proposed a viscoplastic constitutive model which described both yield stress and viscosity undergoing variation during the deformation process; OMID and MAHDI [5] had proposed an elasto-plastic constitutive model for intact rocks which includes the pre-peak elastic and the post-peak strain-softening behavior, as well as dilation; LI et al [6] had constructed a statistical damage constitutive model under triaxial compression condition based on unified strength theory, and the proposed constitutive equation is verified by triaxial experiment of soft rock. LI et al [7] had proposed a new nonlinear viscoelasto-plastic constitutive model of the silty mudstone under triaxial compression condition, which can describe the accelerative creep of the rock well. KATHLEEN [8] had employed a alternative constitutive model to examine localization condition for high porosity sandstone. MENG et al [9, 10] had studied mechanism of rock deformation and failure and monitoring analysis in water-rich soft rock roadway. KHALEDI et al [11] had conducted sensitivity analysis and parameter identification of a time-dependent constitutive model for rock salt. ZHANG et al [12] had given a statistical damage constitutive model for rocks based on the continuum damage mechanics theory and the statistical strength theory, and the Mohr-Coulomb criterion is introduced into this model, and the proposed damage constitutive model is verified by comparison with theory and experimental data. SEISUKE and KATSUNORI [13] had proposed a compliance-type constitutive equation based on experimental data and used to model force-penetration curves instead of the conventional linear springs. HUANG et al [14] had studied the multi-joint constitutive model of layered rock mass considering characteristics of multi-group structural plane, and the comparison between field tests and numerical simulation results proves the proposed model. ASADOLLAHI and TONON [15]had improved the Barton’s original model which can predict negative compressive dilatancy at small shear displacements. CHEMENDA [16] had conducted three-dimensional numerical modeling based on hydrostatic tests of porous rocks in a triaxial cell.

All of these achievements are based on ordinary rocks, usually, their researches are focused on a constitutive model and verified it by experimental data, and all the results and methods have a certain reference value for studying seafloor massive sulfide. However, different rocks have unique properties, even with the same method for seafloor massive sulfide, the results and conclusions will certainly be different. As the seafloor massive sulfide is located in the deep-sea, it is not easy to get, so the domestic and foreign theoretical and experimental study on rock mechanism on seafloor massive sulfide is almost blank. Mining scholars had mainly focused on the researches of the mining prospects and methods, mechanical fragmentation, simulation and so on [17-20], instead of on the view of rock mechanics. In fact, the study of rock mechanism is another aspect comparing with mechanical crushing. This is the reason that constitutive relation of seafloor massive sulfide is selected for further studying on the mining seafloor massive sulfide.

Based on above reasons, this work had discussed the deformation characteristics and constitutive model of seafloor massive sulfide, and all the conclusions are significant for further understanding of seafloor massive sulfides and studying efficient exploitation of this kind of mineral.

2 Experimental results of SMS under different conditions

The specimens were tested under the triaxial rock mechanics testing system of MTS815.2 in the modern analysis and testing center of CSU. The samples were applied from China Ocean Sample Repository, and the specimen was processed into cylinder of 50 cm in diameter and 100 cm in height, also the specimen must meet the requirements of accuracy of rock mechanics test, and before testing the specimens are immersed into water for 24 h. The experiment is divided into three stages: 1) Preloading: The 0.1 kN load was forced to the specimen to make the specimens fully contact to the platform. 2) Confining pressure: The confining pressure was adjusted to a predetermined value to keep σ₂=σ_3, and for uniaxial test, σ₂ and σ₃ are set to 0. 3) Loading σ₁: Maintaining constantly confining pressure, and increasing σ₁ step by step to make the specimen damaged，and all the data and information are recorded during the test, Table 1 lists the experimental data of SMS.

Table 1 Experimental data of SMS

2.1 Results and analysis of uniaxial compressive test

Figure 1 shows the stress-strain curve of a typical ore specimen under uniaxial compression. In the stage from zero to 50% peak strain, the relationship between stress and strain is approximately linear, and the main performance of this stage is elastic. In the stage from 50% peak strain to 130% peak strain, the strain rapidly develops in the direction of the strain, and the stress decreases sharply along with the cracking after peak strain. In the stage from 130% peak strain to final, the main performance of this stage is elastic, but the value of elastic of this stage is much smaller than the first stage.

Fig. 1 Uniaxial stress-strain curve of specimen

2.2 Results and analysis of triaxial compressive test

Figure 2 shows the stress-strain curve of a typical ore specimen under triaxial compression condition. The stage from zero to 10% peak strain, the relationship between stress and strain is approximately linear, and the main performance of this stage is elastic, which is similar to the specimen under uniaxial compression condition. The stage from 10% peak strain to final, the relationship between stress and strain is no longer linear. Before the peak strain, the stress increases slowly with the increasing of the strain, and after the peak strain, the stress decreases with the increasing of the strain until the final.

Fig. 2 Triaxial stress-strain curve of a specimen

2.3 Comparative analysis of rock failure under uniaxial/triaxial compression condition

1) Axial strain and peak strain

Under uniaxial compressions, the axial strain of the ore is relatively small. With the increase of confining pressure, the axial strain increases gradually with the increasing of confining pressure under the triaxial compression condition. When confining pressure is 5 MPa, the peak strain is about 0.02, and when the confining pressure is 0 MPa, the peak strain is about 0.002, so the difference strain is more than one order of magnitude.

2) Peak strength and residual strength

Under uniaxial compression condition, the peak stress of the ore is usually small, and the uniaxial compressive strength is about 10 MPa, also the triaxial compressive strength is increased with the confining pressure. When the confining pressure is 6 MPa, the triaxial compressive strength reaches 36 MPa, and after testing, the residual strength of the specimen often exists, and the stress-strain curve is expressed as the residual elastic strength.

3) Damage characteristics

The failure modes of seafloor massive sulfide are transverse failure, shear failure and global failure. After statistical analysis, the failure characteristics of the specimen are closely related to the confining pressure.

4) Elastic modulus

When the confining is 0 MPa, the elastic modulus is about 11 GPa, and the elastic modulus is nearly 2 GPa when the confining is 6 MPa. Thus, the elastic modulus decreases in some way with the increase of confining pressure.

3 Three stages damage constitutive model

3.1 Uniaxial damage constitutive model based on Hoek-Brown criterion

Because seafloor massive sulfide is a heterogeneous mineral, which contains a lot of distributed randomly cracks and joints, so the strength characteristic is consistent with the statistical law, and the probability function of strength can be expressed with Weibull distribution [21].

              (1)

where F is the distributing variable of Weibull distribution for seafloor massive sulfide; m and F₀ are the scale and morphology parameters for Weibull distribution, respectively.

N_d is the number of damaging units; N is the total number of units. From a statistical point of view, the damage variable of this mineral can be defined by N_d and N.

                                 (2)

In any calculating interval [F, F+dF], P(x) is the breaking probability, and the number of damaged units is NP(x)dx. When the load reaches a certain level, the number of damaged units can be expressed as follows.

      (3)

Substituting formula (3) into (2) yields,

                        (4)

Formula (4) is the statistical damage evolution equation for this mineral, and the value of D is 0-1. All parts of ore are broken when D is 1, and all parts of ore are kept intact when D is 0.

It is supposed that the failure criterion of seafloor massive sulfide is in accordance with Hoek-Brown strength criterion, and then the H-B criterion can be expressed as follows:

                      (5)

where σ_c is uniaxial compressive strength of rock specimen; m and s are two non-dimensional coefficients of characterization of rock mass. The parameter σ_c is 10.243 MPa according to the experimental data, then

                   (6)

Usually, rock failure criterion can be used for [22], where f(σ*) is rock strength random variable, σ^* is effective stress, K is the constant related to rock, so the H-B criterion can be expressed as follows:

          (7)

Substituting formula (7) into (4) yields,

    (8)

According to Ref. [23], the constitutive relation of breaking seafloor massive sulfide can be expressed as follows:

                             (9)

where λ changes in the range from 0 to 1, because to and the effective stressesand in triaxial compression condition can be expressed by the nominal stress σ₁, σ₃, strain ε₁, elastic modulus E and Poisson ratio μ.

(10)

                            (11)

                       (12)

From formulae (11), (12) and (13), then

                            (13)

                        (14)

Substituting formulae (13) and (14) into formula (8), then the damage of seafloor massive sulfide in triaxial compression condition is expressed as follows:

           (15)

From formulae (11), (12), (13) and (14),

                           (16)

where σ₃=0 in uniaxial compression condition, from formulae (15) and (16), the damage constitutive model of seafloor massive sulfide in uniaxial compression condition is deduced based on the H-B criterion.

         (17)

3.2 Definition of three stages constitutive model

According to the deformation characteristics of rock, the stress-strain curve can be divided into three stages. The first stage is linear and the relationship of stress and strain is in accordance with Hooke’s law. The second stage is H-B criterion’s stage; the relationship of stress and strain is in accordance with H-B criterion. The third stage represents residual strength of rock, and this stage also is linear.

        (18)

The strain ranges are determined by the approximate fitting of the stress and strain data, where k₁ is the elastic modulus of rock failure, and k₂ is the residual elastic modulus after rock failure, when the strain ranges from 0.5ε_F to 1.3ε_F, the stress-strain curve is in accordance with the strength failure criterion. The parameters F₀ and m of constitutive equation is determined by above uniaxial/triaxial compression experiments.

4 Parameters determination and model verification

4.1 Parameters m and s determination

According to Eq. (6), m and s are the parameters of strength criterion of Hoek-Brown. The fitting curve of Hoek-Brown criterion is shown in Fig. 3 according to Eq. (6) and the data of Table 1. From the fitting result, the parameter m is 11.869, and the parameter s is 0.6917.

4.2 Parameter F₀ determination

Based on the results of rock testing, the physical parameters k₁, k₂, F₀ of damage constitutive model can be determined by uniaxial experimental data. The parameters of specimens 1, 2, 3 of uniaxial constitutive model were listed under uniaxial conditions in Table 2. From Table 2, residual strength (k₂) of the different test specimen after destruction is different from one to another and the residual strengths(k₂) of some specimens are relatively small for the thorough destruction, however, the residual strengths(k₂) of some specimens are relative large because of incomplete destruction.

Fig. 3 Fitting curve of Hoek-Brown criterion

Table 2 Damage mechanical parameters of SMS three stages model

4.3 Model verification and analysis

Figure 4 represents comparison of stress-strain of curve between model and three specimens. From Fig. 3, the three stages constitutive model can reflect the strength and deformation failure process. The theoretical curve is in good agreement with the experimental data. But there are some deviations in some stages, which is due to the anisotropy of the rock, sometimes, it is difficult to predict accurately the strength and deformation process of the rock. The slope of the first stage is close to the elastic modulus, while the second stage is related closely with H-B criterion, and the relationship of stress and strain is in accordance with H-B criterion, however, the third stage represents residual strength of the rock, which is similar to the first stage.

Fig. 4 Comparison of stress-strain of curve between model and three specimens

5 Conclusions

1) The residual strength can be characterized as a new elastic modulus, which is much smaller than that of the initial elastic modulus.

2) According to the characteristics of seafloor massive sulfide, the constitutive equation is divided into three stages: the elastic linear stage, the nonlinear stage and the residual strength stage after failure. The initial elastic modulus can be characterized as the initial elastic modulus, and the nonlinear stage can be characterized as the H-B failure criterion, and residual strength stage can be characterized as residual elastic modulus.

3) Compared with the experimental data, the curve of model can be found in good agreement with the experimental data. The model can be used to determine the strength of seafloor massive sulfide and select mining tools.

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(Edited by HE Yun-bin)

Cite this article as: HU Jian-hua, LIU Shao-jun, ZHANG Rui-qiang, HU Qiong. Deformation characteristics and constitutive model of seafloor massive sulfides [J]. Journal of Central South University, 2017, 24(9): 1986–1991. DOI:https://doi.org/10.1007/s11771-017-3607-5.

Foundation item: Project(2012AA091291) supported by the National High-tech Research and Development Program of China; Project(51074179) supported by the National Natural Science Foundation of China; Projects(JCYJ20130401160614378, JCYJ20140506150310437) supported by Shenzhen Science and Technology Innovation Basic Research Foundation, China

Received date: 2016-01-28; Accepted date: 2016-04-29

Corresponding author: LIU Shao-jun, Professor; Tel: +86-13923770811; E-mail: 42975254@qq.com