J. Cent. South Univ. Technol. (2009) 16: 0488-0494

DOI: 10.1007/s11771-009-0082-7

Viscoelasto-plastic rheological experiment under circular increment step load and unload and nonlinear creep model of soft rocks

ZHAO Yan-lin(赵延林)^{1, 2}, CAO Ping(曹 平)¹, WANG Wei-jun(王卫军)²,

WAN Wen(万 文)², LIU Ye-ke(刘业科)¹

(1. School of Resources and Safety Engineering, Central South University, Changsha 410083, China;

2. Hunan Provincial Key Laboratory of Safe Mining Techniques of Coal Mines,

Hunan University of Science and Technology, Xiangtan 411201, China)

Abstract: The viscoelastic-plastic creep experiments on soft ore-rock in Jinchuan Mine Ⅲ were performed under circular increment step load and unload. The experimental data were analyzed according to instantaneous elastic strain, visco-elastic strain, instantaneous plastic strain and visco-plastic strain. The result shows that instantaneous deformation modulus tends to increase with the increase of creep stress; soft rocks enhance the ability to resist instantaneous elastic deformation and instantaneous plastic deformation during the multi-level of load and unload in the cyclic process. In respect of specimen JC1099, the ratio of visco-elastic strain to visco-plastic strain varies from 3.15 to 6.58, and the ratio has decreasing tendency with stress increase as a whole; creep deformation tends to be a steady state at low stress level; soft rocks creep usually embodies accelerated creep properties at high stress level. With the damaging variable and the hardening function introduced, a nonlinear creep model of soft rocks is established, in which the decay creep is described by the nonlinear hardening function H of viscidity coefficient. The model can describe the accelerated creep of soft rocks since the nonlinear damaging evolvement variable D of deformation parameter of rocks is introduced. Three stages of soft rocks creep can be described with the uniform creep equation in the nonlinear creep model. With this nonlinear creep model applied to the creep experiments of the ore-rock of Jinchuan Mine Ⅲ, the nonlinear creep model’s curves are in good agreement with experimental data.

Key words: rheological experiment; nonlinear creep; damaging; hardening; soft rocks

1 Introduction

Rocks rheology is the process that the mineral configuration of rocks adjusts and the stress and strain of rocks change with time. Compared with other rocks, the rheological properties of soft rocks highlight relatively. The researches in rheology and time-dependent behavior of soft rocks, especially creep characteristics, are important base in explaining and analyzing the phenomena of geotechnical tectonic movement, as well as predicting the long-term stability for soft rocks engineering [1-4]. Laboratory rheological experiments can open out rheology properties of rocks at different stress levels by means of circular increment step load and unload, and can help to set up appropriate rehological model and supply related rehological parameters to carry through rheological numerical analysis on geotechnical engineering [5-6]. Many researchers have worked on laboratory and in-situ rheological experiments and built many nonlinear viscoelasto-plastic rheological models [7-12]. TSAI et al [7] presented systematic experimental data regarding time-dependent deformation of a typical weak sandstone, researched viscoelasto-plastic behaviors of the weak sandstone and established the associated flow rules to model the time-dependent deformational behavior of weak sandstone. SUN [8] studied nonlinear properties of viscidity coefficient η in Bingham model in which η was considered as a function of time. ZHOU  et al [9] proposed a unified approach for modeling of elastic-plastic and visco-plastic behaviour coupled with induced damage within the unique constitutive model in quasi-brittle rocks. CHEN et al [10] brought forward two nonlinear elements: creep body and cranny-plastic body, and obtained a new combinatorial rheological mechanics model that can describe the accelerated creep stage of soft rocks well. CHALLAMEL et al [11] presented a time-dependent nonlinear softening model applied to quasi-brittle materials such as rock or concrete based on strong thermodynamical arguments. XU et al [l2] built a nonlinear viscoelastic-plastic rheological model of rocks, namely Hehai model based on the triaxial creep tests on greenschist specimens, which was made up of a nonlinear visco-plastic body and five-element linear visco-elastic model in series. Besides, a nonlinear rheological numerical program of Hehai model was

developed and the model was applied to Jinping first-degree water-power plant geotechnical engineering. The nonlinear rheological models of rocks which are improved from the linear rheological model can reflect accelerated creep. However, we usually have to use different creep equations to describe different creep stages. In this work, the soft ore-rocks of Jinchuan Mine Ⅲ were chosen as study objects. By means of viscoelastic-plastic rheological experiments of soft rocks under circular increment step load and unload which provide a complete partition of the elastoplastic response of rocks, the viscoelastic-plastic deformation characteristics of soft ore-rocks in different creep stages were acquired. A uniform nonlinear creep equation was established after the creep experiment curves were analyzed and the damage variable and the hardening function were introduced to improve Burgers model.

2 Experimental

2.1 Equipment, methods and data processing of experiment

Soft rock specimens were obtained from drilling A2 nickel ore-rock in Jinchuan Mine Ⅲ area. Soft rock samples were cut into d 50 mm×100 mm cylinder standard specimens. The apparatus of rheological experiment is RYL600 computer controlling rock shear rheological machine. The machine uses advanced numeral exchange servo fast controlling system made in Panasonic, Japan. Fig.1 shows RYL600 computer controlling rock shear rheological machine. In the creep experiment the load way of circular increment step load was adopted. Planning-putting maximal load was divided into 5 steps, according to uniaxial compressive strength σ_c (σ_c=44.5 MPa) obtained by uniaxial compressive experiments on soft rock specimens and then applied load from low to high on the same specimen. The duration of experiments would depend on the displacement rates under step loads. When the rate of axial deformation was less than 0.001 mm/h, the creep under this load level was considered to be basically stable, the load applied on the specimen was unloaded fully, then the delayed visco-elastic recovery deformation

Fig.1 Creep testing device under step load and unload

was observed. When delayed recovery deformation could not be seen in 24 h, the next load level could be executed afterwards. The loading rate in the creep experiment was 0.5 MPa/s. The above-mentioned operation was repeated till the specimen failed. Fig.2 shows the experimental loading process under circular increment step load and unload.

Fig.2 Circular increment step load and unload

As the example, the soft rock specimen JC1099 of A2 lean nickel ore obtained from directional drill ZK06 which located at -1 250 m level of Jinchuan Mine Ⅲ area was used to study visco-plastic deformation characteristics of this kind of rocks. Its geological joints grew up. During the process of load and unload in cyclic uniaxial compression creep experiments on the specimens, instantaneous strain, creep strain, delayed elastic recovery strain and residual strain of the rock specimens under various stress levels are observed and identified. Total strain ε⁽ⁱ⁾ is made up of instantaneous elastic strain instantaneous plastic strain  visco-elastic strainand visco-plastic strain  The total strain of rocks can be expressed as follows:

Instantaneous strain  at the ith step stress level is composed of two parts: instantaneous elastic strain  and instantaneous plastic strain cumulation  Instantaneous strain  is obtained form Eqn.(2) considering the history of loading.

Creep strain  at the ith step stress level is composed of two parts:  visco-elastic strain  and visco-elastic plastic strain cumulation  Creep strain  is obtained form Eqn.(3) considering the history of loading.

where  represent instantaneous plastic strain increment and visco-plastic strain increment caused by current step stress increment Δσ_i (Δσ_i=σ_i-σ_i-1) respectively, the value of which is equal to the reading of residual strain. Visco-elastic strain  at the ith step stress level is equal to the delayed elastic recovery strain  at the unloading moment on the assumption that visco-elastic strain curve has the same route with the unloaded delayed elastic recovery curve which can be obtained by unloading experiment. Experimental data are listed in Table 1.

2.2 Experimental results analysis

As shown in Figs.3 and 4, instantaneous strain  is composed of instantaneous elastic strain  and instantaneous plastic strain cumulation Instantaneous elastic strain  and instantaneous plastic strain cumulation  increase with the increase of stress level. The experimental result shows instantaneous elastic strain increment, instantaneous plastic strain increment and instantaneous total strain increment under unit stress increment have a decreasing tendency with stress increase as a whole. Instantaneous strain increment reaches 0.544 92×10^-3 under unit stress increment at the first stress level. The experimental results show that deformation of soft rocks caused by the hole and cranny compression is obvious at the first low stress level. Fig.4 shows instantaneous elastic strain increment  instantaneous plastic strain increment  and instantaneous total strain increment  are respectively 9.356×10^-5, 7.010×10^-5 and     0.163 66×10^-3 at the second stress level and 7.424×10^-5, 2.230×10^-5, 9.654×10^-5 at the fifth stress level under unit stress increment. This shows the deformation characteristics of soft rocks that the ability to resist instantaneous deformation is enhanced under circular increment step load and unload.

The ratio of instantaneous stress increment ?σ_i to instantaneous strain increment  at each stress level is defined as instantaneous deformation modulus E_i, E_i=?σ_i/. Fig.5 shows instantaneous deformation modulus E_i tends to increase with the increase of stress. Instantaneous deformation modulus (E₅=10.343 GPa) of the fifth stress level E₅ is 5 times more than that of the first stress level E₁ (E₁=1.922 GPa). This reveals that soft rocks have tendency of instantaneous strain hardening obviously under step increment load.

Creep strain  is made up of visco-elastic strain  and visco-plastic strain  Figs.4 and 6 show that creep stain increment tends to increase under unit stress increment. Fig.7 shows that the ratio of visco- elastic strain  to visco-plastic strain  at various stress levels varies from 3.15 to 6.58, and  has decreasing tendency with stress increase as a whole. This means that the possessive proportion of visco-plastic strain in creep strain will increase with stress increase as a whole.

Figs.8 and 9 show creep rate decreases gradually with time and soft rocks present time-dependent strain hardening at decay creep stage. After entering steady creep stage, creep rate fluctuates limitedly, and steady creep rate increases with the increase of creep stress. Creep rate of steady creep stage is low and creep deformation tends to be steady at low stress level. At the first stress level, the decay creep lasts for 4.812 h, creep deformation tends to be stable, and the total strain is    4.536 59×10^-3. Thereinto, creep strain is 0.275 56×10^-3 and accounts for 6.074% of the total strain. Soft rocks creep usually embodies accelerated creep properties at high stress level. The creep rate increases quickly and damage intenerating of soft rocks speeds up at accelerated creep stage. At the fifth stress level, after the decay creep lasts for 17.021 h, the specimen JC1099 goes into a steady creep stage in which the creep rate keeps at about 1.241 1×10^-5 h^-1. After 208.832 h, the creep rate increases fast and the accelerated creep is evident. With the accelerated creep lasting for 9.253 h, the soft rock specimen collapsed.

3 Soft rock nonlinear creep model

The elements combinatorial rheological models have been applied the most widely to rheological models of rocks, because of having clear physical meaning  and reflecting diversified rheological characteristics of

Table 1 Testing results of viscoelastic-plastic strain under step load and unload

Fig.3 Relationship curves between instantaneous strain and step stress

Fig.4 Strain increment under unit stress increment

Fig.5 Curve of instantaneous deformation modulus

rheological medium. The elements and compoundings, based on existing linear model theories, can describe decay creep and steady creep and but not accelerated creep. Many scholars added nonlinear elements to improve combinatorial models and attempted to establish some nonlinear elements combinatorial rheological models for studying rock nonlinear rheology. However, rheological equations have to be divided into subsections to describe different creep stages. Consequently, we modified Burgers model by introducing damage variable and hardening function and tried to establish an uniform nonlinear creep equation.

Fig.6 Relationship curves between creep strain and step stress

Fig.7 Curve of ratio of visco-elasticity strain to visco-plasticity strain

Fig.8 Creep rate curves of step-loading

3.1 Creep damage intenerating variable and hardening function

Burgers model is usually used to describe rule of

Fig.9 Curve of creep strain under step loads

soft rocks creep, which is made up of Maxwell body and Kelvin body in series. Maxwell model is equal to linear intenerating Hoke body. Its creep equations are listed as follows:

where   is the equivalent deformation modulus and intenerates with time.

Kelvin body is equal to linear hardening Newton body. Its creep equations are listed as follows:

where   is equivalent viscidity coefficient and hardens with time. Burgers model can reflect intenerating and hardening properties of rock material, but it cannot describe creep nonlinear and accelerated creep stages. To establish nonlinear creep model, in this work, the nonlinear damage intenerating function and the hardening function were introduced. Soft rocks creep was made up of the nonlinear intenerating Maxwell body and the nonlinear hardening Kelvin body in series.

3.1.1 Damage intenerating function

Kachanov creep damage law under uniaxial stress state is calculated as [13]:

where  A and γ are material constants. Damage variable D varies from 0 (rock non-damage state) to 1 (rock damage destroy state). From Eqn., critical destructive time t_R of creep damage is obtained.

t_R=[A(γ+1)σ^γ]^-1                              

Damage evolvement Eqn. is obtained from Eqns. and .

D=1-(1-t/t_R)^1/(γ+1)                           

Creep destructive time t_R is in an inverse proportion to σ^γ in Eqn., which is coincident with experimental results. The equivalent deformation parameter  of nonlinear intenerating Maxwell body is supposed as follows:

E₀(1-D)=E₀(1-t/t_R)^1/(γ+1)                  

3.1.2 Hardening function

During the creep of soft rocks, creep ratio  varies with creep time t or creep strain ε_c, the reason for the changes of creep ratio lies in deformation modulus E nonlinear damage and viscidity coefficient η nonlinear hardening with creep time or creep deformation.

Hardening function H can describe the hardening of microcosmic configuration of soft rocks introduced. The hardening function H of viscidity coefficient η is assumed to be power function in which σ and t are viewed as independent variables. To study the nonlinear hardening of soft rocks creep, based on the research of Ref.[14], the equation below is adopted:

H=cσ^νt^1-α                                  

where  c, ν and α are material constants. When 0＜α＜1, the equivalent viscidity coefficient  of the nonlinear hardening Kelvin body is calculated as

3.2 Nonlinear creep equation

Soft rock creep model is made up of the nonlinear intenerating Maxwell body and the nonlinear hardening Kelvin body in series. Nonlinear creep uniform equation can be expressed as follows:

4 Soft rock nonlinear creep model validation

The creep data of A2 lean ore specimen JC1099 which located at -1 250 m level of Jinchuan Mine Ⅲ area were used to validate the assumed nonlinear creep model. Material parameters k, γ and α are assumed to be constants and independent of stress and load time. Using optimized program based on FORTRAN language to analyze experimental results, model parameters of non- linear creep equations are obtained and listed in Table 2.

Fig.10 shows the experimental and theoretical creep fitting curves of soft rock specimen JC1099. The points are creep experimental data points of soft rock specimen at various stress levels. The real lines represent nonlinear

Table 2 Creep parameters of nonlinear creep model of soft rocks at step stress levels

Fig.10 Experimental and theoretical creep curves of soft rock specimen

creep model theoretical curves. From Fig.10, it can be seen that the nonlinear creep model’s curves are in good agreement with experimental data.

In Table 2, initial deformation modulus E₀ increases with the increase of stress, which accords with results of creep experiments [15-16]. This shows that soft rocks have the ability to resist the increase of instantaneous deformation under circular increment step load and unload. Initial viscidity coefficient η₀ reduces with the increase of stress, indicating that visco fluid velocity of soft rocks accelerates with stress addition [17-18]. This phenomenon is also coincident with experimental results of Fig.8.

Putting the fitting data of Table 2 into Eqns. and , the damage evolutive equation D(σ, t) and the hardening function H(σ, t) of soft rocks creep are obtained. Fig.11 shows the creep damage curve under the fifth step stress, and Fig.12 shows the creep hardening curves at various stress levels.

Creep damage expansion is calm and approximately increases linearly with the increase of time at decay creep stage and steady creep stage. But when soft rocks enter accelerated creep stages, its creep damage expands fleetly. At the fifth stress level (σ=33.807 MPa), when creep time is 140, 175, 208 and 215 h, the corresponding creep damage values are 0.096, 0.146, 0.252 and 0.707, respectively. Compared Fig.11 with Fig.10, the duration

Fig.11 Creep damage curve at the fifth step stress level (σ=33.807 MPa)

Fig.12 Creep hardening curves at various stress levels

in which creep damage variable D speeds up is corresponding to accelerated creep stage. This is also the reason why the nonlinear creep equation can represent the accelerated creep stage of soft rocks after creep damage nonlinear intenerating function is introduced into nonlinear creep model.

Hardening function H is a monotony increasing function in which stress s and time t are viewed as independent variables. Fig.12 shows the creep hardening curves of soft rock specimen at various stress levels. Hardening function H represents the decay creep of soft rocks.

Decay creep is caused by the nonlinear increase of viscidity coefficient η with the increase of time. Accelerated creep roots in nonlinear damage of deformation modulus E. After the damage function and the hardening function are introduced into nonlinear creep model at the same time, they are both continuous and monotony functions. When the hardening mechanism is in the highest flight, creep presents decay creep stage. When the damage mechanism is in the highest flight, creep presents accelerated creep stage. When they are close, creep enters approximatively steady creep stage.

5 Conclusions

(1) Soft rocks have relatively strong rheological properties. Soft rocks rheological deformation includes instantaneous elastic deformation, instantaneous plastic deformation, visco-elastic deformation, and visco-plastic deformation. Instantaneous deformation modulus tends to increase with the increase of creep stress. Soft rocks enhance ability to resist instantaneous elastic deformation and instantaneous plastic deformation during the multi-level of load and unload in the cyclic process.

(2) The occurrence and duration of each creep stage depend on the load level applied. In respect of soft rocks, creep deformation is obvious at creep decay stage. The steady stage in which the creep ratio is approximately constant is sustained for longer time. Besides, the steady creep rate increases with the increase of creep stress. At high stress level the accelerated creep stage in which the creep rate increases and leads to specimen collapse usually happens in short time.

(3) In respect of soft rock specimen JC1099, the ratio of visco-elastic strain to visco-plastic strain varies from 3.15 to 6.58, and the ratio has a decreasing tendency with stress increase as a whole.

(4) A nonlinear creep model of soft rocks is established after the damaging variable and the hardening function are introduced. The model can describe all three stages of soft rocks creep with uniform creep equation. Creep damage expansion is calm at decay creep stage and steady creep stage. But when soft rock enters accelerated creep stage, its creep damage expands fleetly. With nonlinear damaging intenerating function introduced, the established nonlinear creep model can describe the accelerated creep stage.

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(Edited by YANG You-ping)

Foundation item: Project(2007CB209400) supported by the Major State Basic Research and Development Program of China; Project(50774093) supported by the National Natural Science Foundation of China; Project(200801) supported by Open Research Fund of Hunan Provincial Key of Safe Mining Techniques of Coal Mines

Received date: 2008-08-26; Accepted date: 2008-10-30

Corresponding author: ZHAO Yan-lin, PhD; Tel: +86-13974963257; E-mail: yanlin_8@tom.com