Failure mechanism of rock under ultra-high strain rates

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State Key Laboratory of Hydraulic Engineering Simulation and Safety; School of Civil Engineering, Tianjin University, Tianjin 300350, P. R. China)

Abstract: We explored the failure mechanism of rock under ultra-high strain rates using 3D numerical modeling of the light gas gun test. Based on numerical results, it concluded that mesoscopic hydro-compressive failure rather than mesoscopic shear or tensile failure is the main mechanism of rock failure under the condition of shock compressive loading. The shock wave, indicated by the stress signals of two stress gauges in the rock specimen, can be well reproduced by numerical simulation with the quasi-static rather than the dynamic elastic parameters. The simulation results indicate that the compressive shock wave involves a compressive failure loading process similar to that shown in the conventional uniaxial compressive failure test rather than the ultrasonic test. A mesoscopic-rate-dependent failure model was developed to take the dynamic effect into account. Our results revealed that larger rock porosity could result in an decrease in dynamic strength and dynamic effect under shock compressive loading.

Keywords: dynamic failure; light-gas gun test; shock compressive loading; lattice spring model

1 Introduction

The light-gas gun, first developed for hypervelocity research in aeronautics, was later used in the field of material science. The light-gas gun test is mainly used in material science for: 1) obtaining the Hugoniot curve of the material[1-3], 2) measuring dynamic failure strength[4], 3) studying the high-pressure phase transition[5], and 4) investigating impact-induced chemical reactions[6]. The light-gas gun test has also been applied to rock and rock-like materials. Grote et al.[7]conducted a flat plate impact test on cement mortar and concrete. They reported that the average flow stress of cement mortar and concrete increased to 1.3 GPa and 1.7 GPa, respectively, compared with the unconfined quasi-static compressive strength of 46 MPa and 30 MPa. The dynamic effect of reinforced concrete under shock compressive loading was also explored using the light-gas gun test[8]. The bearing capacity of the reinforced concrete was found to be improved with the increase of impact velocity and reinforcement ratio[8]. Recently, similar phenomena were observed by Zhang et al.[1]with marble and gabbro.

The dynamic elasticity theory for solid materials was adopted to interpret experimental data of the light-gas gun test[1,7-8]. The constitutive model based on the dynamic elasticity theory can be used with the finite element method (FEM) for numerical simulation of the light-gas gun test. Liu et al.[9]simulated the dynamic fracture mode of tungsten alloy by LS-DYNA (an explicit FEM code) and captured the characteristics in their experimental results. Lopatnikov et al.[10]simulated the light-gas gun test of foamed aluminum plates using LS-DYNA and demonstrated that the dynamic deformation-time relationship and kinetic energy changes of the impact plates at different impact velocities predicted by the FEM were consistent with their theoretical model. Nevertheless, as pointed out by Riedel et al.[11], the intrinsic heterogeneity of rock-like material might result in challenges to the determination of the equation of state parameters and the need for a mesomechanical model. Duan et al.[12]incorporated a statistical isotropic elastic micro-crack model into the FEM to simulate the propagation of plane shock waves in soda-lime glass and reproduced their experimental data. Recently, besides FEM with the mesomechanical model[13-14], a number of discontinuum-based numerical methods[15-20]have also been applied in the study of the dynamic failure of rock. The molecular dynamics (MD)[15]and the lattice type model[18-20]were successfully applied to study the dynamic failure of rock. However, most of these studies only reproduced the failure mode of the rock using a mesoscopic constitutive model with tensile failure. The implementation and practicability of the model to simulate the light-gas gun test is still unclear.

In this work, we used a lattice type model to explore the light-gas gun test conducted on marble by Zhang et al.[1]. Our main purposes were to answer the following questions:

1) What kind of mesoscopic damage information can be obtained from the experimental data of the light-gas gun test?

2) What kind of elastic parameters, the dynamic elastic ones or the quasi-static elastic ones, control the shock loading propagation within the rock specimen？

3) How to describe the rate dependency of rock under shock compressive loading in the light-gas gun test using a mesoscopic constitutive model?

4) What is the role of the rock’s mesostructure in the dynamic strength and dynamic effects (strain/loading rate dependency) of rock under shock compressive loading?

In this work, the distinct lattice spring model (DLSM) was adopted as the numerical tool due to its advantages in modeling rock failure based on simple mechanical elements, e.g., Newton’s second law and springs with simple constitutive models. These characteristics of the DLSM make it a suitable numerical tool for failure mechanism study. In addition, the DLSM is computationally appropriate for full 3D simulation. The following sections of this paper include a brief introduction of the physical test and numerical model, a brief description of the development of rate-dependent constitutive models and a comparison of model response with published data of the light-gas gun test conducted by Zhang et al.[1], based on which the failure mechanism of rock under shock compressive loading is explored. Finally, a few conclusions are derived to answer the aforementioned questions.

2 Methods

2.1 The light-gas test

The light-gas test data reported by Zhang et al.[1], obtained using the experimental facilities at the Cavendish Laboratory, Cambridge, UK[21], are adopted in this work. Fig.1 shows the basic working principle of the experimental facilities as well as the components of the marble specimen and the copper flyer. Different from the traditional rock mechanics tests, two stress gauges were placed in the composite specimen made of marble and PMMA (see Fig.1(b)). The two stress gauges were used to obtain the stress history curve of the corresponding location inside the specimen during the test, that is, the waveform of the shock wave. Since there is a certain distance between the two stress gauges, the velocity of the shock wave in the rock specimen can be obtained based on these signals. After the light-gas gun test, both the composite rock specimen and the flyer became powdery and dissipated, as a result, no morphological data was recorded for the failure process, or the failure pattern of the rock specimen. Another important parameter of the light-gas gun test is the impact velocity of the copper flyer, which is the only controllable input of the light-gas gun test. A number of different impact velocities were adopted by Zhang et al.[1]to obtain the Hugoniot parameters of the marble. Basic material properties of the marble, PMMA and copper flyer were also provided in reference [1]. The longitudinal and shear wave velocities were obtained using an ultrasonic transducer, which can be further used to obtain the dynamic elastic parameters. The quasi-static elastic parameters of the marble were obtained by the standard uniaxial compression test, the parameters of which are listed in Table 1. The dynamic elastic parameters of PMMA and copper were close to their quasi-static counterparts, therefore, no quasi-static tests were conducted on the copper and PMMA.

2.2 Distinct Lattice Spring Model (DLSM)

The lattice spring model(LSM) was first developed by Hrennikoff in 1941[22]. Its basic principle is to represent the mechanical responses of a solid through a group of spring-like interactions. The LSM is regarded as the ancestor of both the FEM and the discrete element method (DEM). Due to its simplicity, the LSM has been widely used in the study of many fundamental mechanical phenomena of solids[23-25]. The distinct lattice spring model (DLSM) was developed by Zhao et al.[26]to overcome the Poisson’s limitation in the classical LSM. As shown in Fig.2(a), the basic principle of the DLSM is to represent the solid as a group of particles linked through spring bonds, which consist of a normal spring and a shear spring. Defining the normal unit vector:n=(nxnynz)Tis directed from particleito particlej, which are connected by a normal spring, and the normal deformation of the spring is defined as

(1)

whereuij=uj-uiis the relative displacement between particlesjandi. In the DLSM[26], the most commonly used constitutive model for the normal spring is

(2)

The multi-body shear spring is one of thedistinctive features of the DLSM. Its principle is to calculate the shear deformation through local strain rather than the displacement of the two particles. In the DLSM, the shear deformation is given as

(3)

where [ε]bondis the local strain of the spring bond, which can be obtained from an average operation over the local strain defined at the two particles using a least square method. The main feature of the multi-body shear spring is the ability to represent different Poisson’s ratios without violating the rotation invariance[26]. The failure criterion of the shear spring is given as

(4)

Eqs.(2) and (4) are the constitutive models adopted in the DLSM for brittle solids. Fig. 3(a) and (b) show the constitutive model for the normal spring and the constitutive model for the shear spring, respectively. Since a solid is represented as a spring network in the DLSM (as shown in Fig. 2(a)), the fracturing process is manifested by the gradual breakage of these springs (mesoscopic failure events). It has the following advantages in simulating the fracturing of brittle materials. First, these constitutive models are easily implemented, and the simulation results are easily explained, as fewer parameters are required compared to other, discontinuous models, e.g., the bonded DEM. Third, the straight forward parallelization of the DLSM[27-28]gives the model relatively high computational performance.

Fig.2 Basic principle of the DLSM and the computational model for the light-gas gun

Fig.3 Classical brittle constitutive models used in the DLSM[26] represented in a dimensionless

Another distinctive feature of the DLSM is that the relationship between the spring parameters(knandks) and the macroscopic material properties (Eandv) is derived from the hyper-elastic theory. No calibration is required for the mesoscopic elastic parameters, as they can be calculated using the following equations.

(5)

(6)

whereα3Dis a lattice coefficient that can be calculated according to

(7)

whereliis the initial length of theithspring andVmis the volume representing the computational model. More details of the DLSM and its recent development can be found in reference [25-27].

2.3 Computational model for the light-gas gun test

(8)

whereAgis the area of the stress gauge and sign(·) is the sign operation, which is given as

(9)

Eq. (8) gives the time history of the normal stress on the stress gauge.

The particle size used in this work is 1 mm and there are about 100 000 particles. The input parameters of the light-gas gun test are based on experimental tests. The output of the numerical test is controlled by the selection of the meso-mechanical constitutive model and the corresponding constitutive parameters. A key point of this work is to minimize the difference between the numerical prediction and the physical experimental result. Thus, the failure mechanism of the rock under shock compressive loading may be interpreted from the searching process of the meso-mechanical constitutive model and its parameters.

2.4 Tri-linear hydro-compressive model

Many researchers suggest that macroscopic compression and shear failure is essentially tensile failure at the mesoscopic level[28-30]. However, most of those studies are limited to tensile failure of the normal spring[18, 20, 30]. Fig.3(a) shows the simple brittle constitutive model of the normal spring considering only tensile failure. More comprehensive forms, e.g., considering the damage and plastic deformation, can be developed. Jiang and Zhao[31]recently developed a coupled damage plasticity model to describe the dynamic crack propagation of a gypsum-like 3D printing material. When considering the interaction between two particles, shear failure is a natural logical extension. However, there are very few studies on shear failure in classical LSMs due to the absence of shear interaction. In the DLSM, shear failure can be considered due to the introduction of the multi-body shear spring. The corresponding constitutive model is illustrated in Fig.3(b). Shear failure was widely adopted in the bonded DEM, which usually takes into account both the influence of the cohesion of the contact bond between two particles and the frictional angle. The shear constitutive model presented in Fig.3(b) can be viewed as a special case when the frictional angle is zero. Therefore, the prediction of shear fracturing of the DLSM using the constitutive model shown in Fig.3(b) is not conservative.

(10)

Fig.4 A tri-linear constitutive model considering the hydro-compressive failure of the normal spring

To introduce the effects of plasticity and damage, the same approach as that adopted by Jiang and Zhao[31]is used. First, rewrite Eq.(10) as follows.

F(u)=S(u)knu

(11)

whereS(u) is a state parameter of the normal spring, which is defined as follows.

(12)

The interpretation of Eq.(10) can be expressed as the following formula if it is based on damage mechanics.

F(u)=(1-D(u))knu

(13)

Then, the damage function can be expressed as follows.

D(u)=1-S(u)

(14)

In the calculation of the spring force, the maximum damage variable corresponding to the loading history is recorded asD*(u), then the damage constitutive equation considering the loading and unloading can be written as

F(u)=(1-D*(u))knu

(15)

The loading and unloading response of this constitutive model is as shown in Fig.4(a). If Eq.(10) is regarded as a plastic response, it can be expressed as

F(u)=(u-up)kn

(16)

Compare Eq.(14), we will have the plastic deformationupas

up=u-S(u)u

(17)

Similarly, to record the maximum plastic deformationup*experienced by the spring at the moment of loading, the pure plastic constitutive model considering loading and unloading can be written as

F(u)=(u-up*(u))kn

(18)

The corresponding pure plastic constitutive model is shown in Fig.4(b). Combined with Eq.(15) and (18), a coupled damage-plastic model can be constructed by introducing the damage-plastic coupling coefficient as follows.

F(u)=(1-λdp)(u-up*(u))kn+

λdp(1-D*(u))knu

(19)

Whenλdp=1, it represents the pure damage constitutive model, and whenλdp=0, it represents the pure plastic constitutive model. In this work, we consider the copper behavior as pure plastic material, while the rock has a pure damage response.

Under dynamic loading, the material has obvious dynamic rate effects. To capture this phenomenon, it is convenient to introduce a rate-dependent model. The approach developed by Zhao et al.[18]is adopted in this work to bring rate dependency into the tri-linear constitutive model. The idea is to change the compressive strength parameters of the trilinear constitutive model as a function of the spring deformation rate. It was found that the instantaneous value of the spring deformation rate may not be able to reproduce the correct macroscopic dynamic effect[18]. To solve this problem, Zhao et al.[18]proposed the concept of time non-localization, which uses the average value of the spring deformation rate from the start time to the current time. In this work, following a similar idea, the local average deformation rate is given as

(20)

The average deformation rate of the spring is only calculated when the spring is deformed beyond 0.99uc1. Based on this concept, the strength parameters of the corresponding dynamic tri-linear constitutive model are given by the following formula.

(21)

(22)

Fig.5 Dynamic tri-linear hydro-compressive constitutive

3 Numerical modeling and discussion

3.1 Elastic parameters selection

Fig.6 Shock waveform predicted by the DLSM with

Fig.7 The shock waveform predicted by the DLSM with different mesoscopic failure

3.2 Failure mechanism of the marble specimen in the light-gas gun test

Fig.8 Light-gas gun test predicted by the DLSM considering the mesoscopic tensile

Fig.9 The shock waveform predicted by the DLSM using the tri-linear constitutive

Fig.10 Influence of the failure constitutive parameters of the tri-linear hydro-compressive constitutive model on the shock waveform predicted by the

3.3 Dynamic effect of the mesoscopic hydro-compressive failure

Fig.11 Dynamic strength of the rock specimen in the light-gas gun test under different impact velocities predicted by the DLSM with the rate independent (RI) constitutive model and rate dependent

3.4 Influence of mesostructure

In this section, the influence of the mesostructure on the dynamic failure of rock under shock loading is explored. Fig.12 shows the corresponding computational models for the rock specimens with different mesostructures. These computational models are formed by randomly removing a given number of particles from the marble specimen. The porosity, which was used to characterizs different computational models, is defined asn=number of particles being excavated / total number of particles in the original model. As shown in Fig.12, four pore models with a porosity of 5%, 10%, 15%, and 20% are constructed. All the constitutive parameters are the same as those of the previous section. Fig.13 shows the numerical results of the predicted strength of those different porosity models under different impact velocities. Overall, larger porosity results in lower dynamic strength, which seems logically correct. However, there are some variations, for example, when porosity is 5%, a higher strength is obtained. The reason might be that the inter-particle velocities become more violent under shock compressive loading when a small number of particles were removed from the original model. It might trigger the dynamic effect of the constitutive model and consequently, result in a higher strength. Nevertheless, when the porosity continuously increases, the inter-particle velocity increase may be released in the lateral direction due to the existence of a large mesoscopic free surface. This may result in the increase of strength due to the dynamic effect becoming inconspicuous (as shown in Fig.13).

Fig.12 Computational models of rock specimens

Fig.13 Numerical prediction of the dynamic strength of rock specimens with different porosities under various

4 Conclusions

In this work,light-gas gun tests on marble were numerically investigated using the DLSM to study the failure mechanism of rock under compressive shock loading. Based on a detailed comparison of the numerical response with published experimental data, it was found that the elastic parameters controlling the compressive shock wave propagation in the rock specimen should be determined through quasi-static tests rather than ultrasonic tests. Moreover, mesoscopic tensile failure and shear failure between rock grains are not the mechanisms of rock failure under compressive shock loading. To reproduce the shock stress waveform observed in the light-gas gun test, a mesoscopic hydro-compressive constitutive model is needed. In other words, it is reasonable to conclude that the light-gas gun test may be able to provide information about the hydro-compressive failure of rock under dynamic loading and provide calibration data for the DLSM or other numerical methods to determine the constitutive equations under dynamic hydro-compression loading conditions. It is noted that a rate-dependent hydro-compressive constitutive model is also necessary to reproduce the experimental observation of the strength increase under different impact velocities. Finally, the influence of the mesostructure in terms of porosity was found to decrease both the dynamic strength and the dynamic effect. The findings in this work may provide a better understanding of rock failure under compressive shock loading, which might be useful for a more rational protective design of underground rock engineering structures under blasting loading[33-36].

Acknowledgements

This research is financially supported by the National Key R&D Program of China (Grant No. 2018YFC0406804) and the National Natural Science Foundation of China (Grant No. 11772221).