# A Statistic Damage Model of Rocks considering the Effect of Loading Rate

Authors: Kai Cui; Bin Hu; Jing Li
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

This paper develops a new statistic damage model for rock to mainly study the effect of a loading rate on its mechanical behaviours. The proposed model adopts a new loading rate-dependent damage density function and is capable of describing the macroscopic damage accumulation process for rock samples subjected to external high-speed dynamic loadings. The proposed model can also account for the residual strength of rocks by introducing a modified equivalent strain principle, which considers the contribution of the friction force to the strength of rocks. The friction force is generated by the movements of the nearby microcracks. The predicted stress-strain curves by the proposed model agree with the measured data of salty rock under the conditions of various confining pressures and loading rates. It can be found that both the peak strength and the corresponding axial strain are increased at high-speed loading conditions. At the same time, a transition from ductile failure to brittle failure can be observed in rock samples.


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## Body

## 1. Introduction

The mechanical behaviours such as strength and deformation of rock are affected by loading rates [1–4]. The microcracks distributed inside the rock samples will be propagated and result in macroscopic damage [5–9]. Mathematical description of the dynamic loading-induced damage accumulation process plays a crucial role in modelling the rock behaviours.In terms of strength, [10] proposed dynamic strength criteria for rock-like materials based on theoretical derivation to describe the uniaxial strength characteristics from quasistatic to dynamic uniformly [11] explored the effect of loading rate on the dynamic yield strength of sandstone through experiments. In addition, [12,13] carried out a series of dynamic experiments on the Bukit Timah granite. Based on the experimental results, they modified the Mohr-Coulomb strength criterion of rock to account for the effect of loading rates, which can accurately describe the dynamic response of rocks. Similarly, [14] proposed another strength criterion for the uniaxial dynamic strength of rocks based on the work of [15] and established a statistical damage model which can reasonably simulate the dynamic deformation process of rock.At the deformation aspect, [16–19] used the SHPB apparatus to conduct empirical research on the uniaxial dynamic behaviour of salt rock under different confining pressures and loading rates and obtained its steady and dynamic stress-strain curves. The experimental results showed that the salt peak strength of the rock corresponding axial strain would increase significantly with the increase of the loading rate. To simulate the dynamic deformation process of rock, [20] established a dynamic damage model of rock based on the viscoplastic theory by assuming that rock damage was only related to dynamic strain energy. However, [21] suggested that the stress level and the loading rate can also affect the damage law of rock.The statistic damage models can account for the propagation process of microcracks and characterise the strength and deformation behaviours of rocks, which have been widely applied in geotechnical engineering. [22]developed a damage model to study the effect of freeze-thaw cycles and confining pressures on the mechanical properties of rocks. [23] applied an equivalent continuum damage model for modelling the impact of weakness planes in rock masses on the stability of tunnels. The temperature may also affect the mechanical properties, which can be described using the statistic damage theory [24–26].Although the above statistic damage models are widely applied in practice, some of them cannot describe the residual strength of rocks. To overcome such limitation, [27] introduced a rock damage scale factor following the assumption that the damaged components of rocks can still resist external force-induced deformation. At the same time, [28] modified the Lemaitre strain equivalence principle to include the friction force between the nearby microcracks.This paper aimed to develop a dynamic statistic damage model to study the effect of loading rates on rocks’ strength and deformation behaviours. First, the evolution law of the damage factor was assumed to be affected by the loading rates to describe the macroscopic damage accumulation process when the rock samples were subjected to external dynamic loading. Second, the proposed model adopted the modified Lemaitre strain equivalence assumption to consider the residual strength of the rock. The proposed model was used to predict the experimental results of salt rock under different confining pressures and loading rate conditions.


## 2. Dynamic Stress-Strain Curve of Rock

To study the effect of loading rates on the mechanical behaviours of salty rocks, [16] conducted a series of triaxial tests using the SPHB apparatus under the conditions of various confining pressures and loading rates. The measured stress-strain curves are presented in Figure 1; it can be found that the tangent modulus of rocks samples is gradually decreased during the early loading process, which is mainly induced by the damage of microstructure and the accumulation of plastic deformation. Besides, the stress-strain curves exhibit the feature of strain-softening after the peak strength state as the shear strength decreases considerably against the axial strain. At a given confining pressure, increasing the loading rate will lead to a higher peak shear strength, together with a larger corresponding axial strain, indicating that the strength of salt rock is enhanced at a high-speed loading condition.Figure 1
Dynamic stress-strain curves of salt rock [16] under different confining pressures and loading rates. (a) Confining pressure σ3=5MPa. (b) Confining pressure σ3=15MPa. (c) Confining pressure σ3=25MPa.
(a)(b)(c)Figure2 shows the dynamic yield strength of the salty rock samples which is not affected by the loading rate ranging from 0.1–1.0/s, and the relationship between the yield strength and the confining pressure under the quasi-static loading conditions can be described using the conventional Mohr-Coulomb strength criteria. However, further increasing the loading rate will significantly increase the dynamic yield strength of rock samples; the most apparent increment happens in the case of σ3=5 MPa, where the dynamic yield strength is approximately increased from 20 MPa to 62 MPa for the loading rates of 1.0 and 103/s, respectively. To address such a strength enhancement phenomenon, [14] suggested that the strength of rocks consists of two different components: one is the quasi-static strength that only depends on the confining pressure, whereas the other is the inertia force-induced strength. Increasing the loading rate will increase the inertia force-induced strength, which will play a primary role in forming the strength of rocks.Figure 2
Dynamic yield strength of salt rock [16] corresponding to the loading rate at different confining pressures.


## 3. Statistical Damage Rock Model

### 3.1. Damage Factor

When rock is subjected to external forces, microcracks will gradually initiate and propagate to form random-distributed microcracks. The macroscopic damage accumulation process will decrease the tangent elastic modulus of rock samples and lead to a final failure.Within the statistic damage theory, rock is assumed to consist of many microscope elements, and some of them will be damaged due to external forces. AssumingN is the number of all microscopic elements within rock and Nd is the number of damaged elements, the damage factor D is usually defined as the ratio between Nd and N as(1)D=NdN,0≤D≤1.The Weibull distribution is used herein to describe the strength of microscopic elements, namely,(2)PF=mF0FF0m−1exp−FF0m,where F is the element strength parameter corresponding to the failure criterion of rocks, which may be a function of the stress level or axial strain. m and F0 are the shape and scale parameters, respectively.We assume that the incremental number of damaged microscopic elements subjected to external forces is calculated as(3)dNd=NPFdF.Substituting equations (1) and (2) into (3), the damage factor D can be integrated as(4)D=∫0FPFdF=1−exp−FF0m.The damage factorD can quantify the macroscopic damage accumulation process. When D equals 1.0, rock samples will be completely damaged and lose the capability of resisting further deformation.Figure3 shows the effect of parameters m and F0 on the evolution of D concerning F. On the one hand, it can be found that the evolution curve will conduct counterclockwise rotation by increasing the value of m. On the other hand, increasing the values of F0 will significantly accelerate the macroscopic damage accumulation process.Figure 3
Effect of parametersm and F0 on the evolution of D concerning F. (a) Parameter m. (b) Parameter F0.
(a)(b)


### 3.2. Stress-Strain Relation

This study assumes that only a part of the rock sample will be damaged when subjected to external forces. The intact component will perform elastic response obeying the generalized Hook’s law:(5)σi'=Eεi'+μσj'+σk',i,j,k=1,2,3...,where σ′ and ε′ are the undamaged stress and strain, respectively. E is the elastic modulus, and μ is the Poisson’s ratio.According to the equivalent strain principle, it can be considered that the rock is composed of damaged and undamaged materials under external load, and the damaged part of the material does not have any bearing capacity. The equivalent stressσ is defined as(6)σi=σi′1−D,i=1,2,3.Substituting equation (6) into (5), the stress-strain relation of rock considering the macroscopic damage accumulation is given by(7)σi=Eεi1−D+μσj+σk.We note that equation (7) cannot predict the residual strength of rock samples at different confining pressures because D will turn to be zero after the rock samples are entirely damaged. To overcome this limitation, [28] suggested that the friction force between the nearby microcracks can contribute to the residual strength of rocks. After that, the stress-strain relation is modified to account for the residual strength σresidual as(8)σi=Eεi1−D+Dσresidual+μσj+σk,where σresidual is the residual strength of the rock. When σresidual=0, (8) will degenerate to the original Lemaitre strain equivalence assumption.To consider the effect of the loading rate on the mechanical behavior of rocks, the evolution law of the damage ratioD in (9) is modified as(9)D=1−exp−ε1F01+αlnε˙1εrefm,where α is a non-negative material constant, and εref is the reference loading rate that is chosen as ε·ref=10−5.


### 3.3. Model Parameter Calibration

The proposed statistic damage model for rocks includes six material parameters. The values of the elastic modulusE, Poisson’s ratio μ, and the residual strength of σresidual can be determined through the measured stress-strain curve of rock samples at a specific confining pressure. Parameter α should be calibrated by conducting another test with a different loading rate.To calibrate the parametersm and F0, the measured peak strength state of the rock samples should be used. For a conventional triaxial test (σ2=σ3), the stress-strain relation represented in (8) is simplified as(10)σ1=Eε11−D+Dσresidual+2μσ3.Once the peak strength state is reached, the derivative ofσ1 with the corresponding ε1 should be zero, namely,(11)∂σ1∂ε1|ε1=εpeak,σ1=σpeak=0,where σpeak and εpeak are the peak strength and the related axial strain.Substituting (10) into (9) and calculating the derivatives, we obtain(12)ε1F01+αlnε˙1εrefm−1=EF0mEεpeak−mσresidual.The stress-strain relation at the peak strength state can be expressed as(13)σpeak=1−D⋅Eεpeak+D⋅σresidual+2vσ3.Solving equations (12) and (13) will lead to the exact expression of parameters m and F0 as(14)m=−Eεpeak1+αlnε˙1/εrefEεpeak−σresiduallnσpeak−2vσ3−σresidual/Eεpeak−σresidual,F0=mEεpeak−mσresidualEεpeak1+αlnε˙1εrefm−11/m.


### 3.4. Parameter Experiment

To investigate the performance of the proposed statistical damage model in modelling the dynamic mechanical behaviours of rocks, a series of case studies were conducted here by letting the confining pressure be 0.5 MPa, and the values of the corresponding model parameters are given in Table1.Table 1
Model parameters for the proposed model.
Material parametersValueElastic modulusE (MPa)800.0Poisson’s ratioμ0.5Residual strengthσresidual (MPa)1.5Peak strengthσpeak (MPa)10.0Axial strain at the peak strength stateεpeak0.025Rate-related material constantα1.0First, we studied the effect of the residual strength on the stress-strain curves of rocks. Unlike the previous statistical damage models, the proposed model introduced a modified equivalent strain principle to account for the contribution of the friction force between the propagated nearby microcracks and the strength of rocks. In the cases ofσresidual were chosen to be 1.0, 2.0, 3.0, and 4.0 MPa, the predicted stress-strain curves shown in Figure 4(a) indicated that the strain-softening feature was weakened during the post-failure process, and the rock samples turn to be more ductile, which was confirmed by the evolution of the damage factor against the axial strain, as shown in Figure 4(b).Figure 4
Effect of residual strengthσresidual on stress-strain curve and damage curve of rock. (a) Stress-strain curve. (b) Damage curve.
(a)(b)Second, this paper explored the influence of loading rate on rock dynamic stress-strain curve and damage evolution law, and the calculation results are shown in Figure5. From Figure 5(a), it can be seen that the loading rate does not affect the residual strength of rock. However, the peak strength of rock σpeak will gradually increase with a gradual increase of ε·1. On the contrary, the peak strength of rock σpeak will gradually decrease. The calculation results are consistent with the experimental results of rock under dynamic loading.Figure 5
Effect of loading rates on stress-strain curve and damage curve of rockε·1. (a) Stress-strain curve. (b) Damage curve.
(a)(b)According to Figure5(b), the damage factor D will gradually decrease with the increase of loading rate in a small strain range (ε1≤0.025), and the tangent modulus of rock will also increase. However, with the further increase of axial strain, the growth rate is accelerated with the increase of loading rate, leading to brittle failures of rock more efficiently, and the stress-strain curve will have a noticeable “stress drop” phenomenon.The experimental results show that the rock statistical damage model established in this paper can better describe the effect of loading rate on the dynamic, progressive failure, and the residual strength of rock.


## 3.1. Damage Factor

When rock is subjected to external forces, microcracks will gradually initiate and propagate to form random-distributed microcracks. The macroscopic damage accumulation process will decrease the tangent elastic modulus of rock samples and lead to a final failure.Within the statistic damage theory, rock is assumed to consist of many microscope elements, and some of them will be damaged due to external forces. AssumingN is the number of all microscopic elements within rock and Nd is the number of damaged elements, the damage factor D is usually defined as the ratio between Nd and N as(1)D=NdN,0≤D≤1.The Weibull distribution is used herein to describe the strength of microscopic elements, namely,(2)PF=mF0FF0m−1exp−FF0m,where F is the element strength parameter corresponding to the failure criterion of rocks, which may be a function of the stress level or axial strain. m and F0 are the shape and scale parameters, respectively.We assume that the incremental number of damaged microscopic elements subjected to external forces is calculated as(3)dNd=NPFdF.Substituting equations (1) and (2) into (3), the damage factor D can be integrated as(4)D=∫0FPFdF=1−exp−FF0m.The damage factorD can quantify the macroscopic damage accumulation process. When D equals 1.0, rock samples will be completely damaged and lose the capability of resisting further deformation.Figure3 shows the effect of parameters m and F0 on the evolution of D concerning F. On the one hand, it can be found that the evolution curve will conduct counterclockwise rotation by increasing the value of m. On the other hand, increasing the values of F0 will significantly accelerate the macroscopic damage accumulation process.Figure 3
Effect of parametersm and F0 on the evolution of D concerning F. (a) Parameter m. (b) Parameter F0.
(a)(b)


## 3.2. Stress-Strain Relation

This study assumes that only a part of the rock sample will be damaged when subjected to external forces. The intact component will perform elastic response obeying the generalized Hook’s law:(5)σi'=Eεi'+μσj'+σk',i,j,k=1,2,3...,where σ′ and ε′ are the undamaged stress and strain, respectively. E is the elastic modulus, and μ is the Poisson’s ratio.According to the equivalent strain principle, it can be considered that the rock is composed of damaged and undamaged materials under external load, and the damaged part of the material does not have any bearing capacity. The equivalent stressσ is defined as(6)σi=σi′1−D,i=1,2,3.Substituting equation (6) into (5), the stress-strain relation of rock considering the macroscopic damage accumulation is given by(7)σi=Eεi1−D+μσj+σk.We note that equation (7) cannot predict the residual strength of rock samples at different confining pressures because D will turn to be zero after the rock samples are entirely damaged. To overcome this limitation, [28] suggested that the friction force between the nearby microcracks can contribute to the residual strength of rocks. After that, the stress-strain relation is modified to account for the residual strength σresidual as(8)σi=Eεi1−D+Dσresidual+μσj+σk,where σresidual is the residual strength of the rock. When σresidual=0, (8) will degenerate to the original Lemaitre strain equivalence assumption.To consider the effect of the loading rate on the mechanical behavior of rocks, the evolution law of the damage ratioD in (9) is modified as(9)D=1−exp−ε1F01+αlnε˙1εrefm,where α is a non-negative material constant, and εref is the reference loading rate that is chosen as ε·ref=10−5.


## 3.3. Model Parameter Calibration

The proposed statistic damage model for rocks includes six material parameters. The values of the elastic modulusE, Poisson’s ratio μ, and the residual strength of σresidual can be determined through the measured stress-strain curve of rock samples at a specific confining pressure. Parameter α should be calibrated by conducting another test with a different loading rate.To calibrate the parametersm and F0, the measured peak strength state of the rock samples should be used. For a conventional triaxial test (σ2=σ3), the stress-strain relation represented in (8) is simplified as(10)σ1=Eε11−D+Dσresidual+2μσ3.Once the peak strength state is reached, the derivative ofσ1 with the corresponding ε1 should be zero, namely,(11)∂σ1∂ε1|ε1=εpeak,σ1=σpeak=0,where σpeak and εpeak are the peak strength and the related axial strain.Substituting (10) into (9) and calculating the derivatives, we obtain(12)ε1F01+αlnε˙1εrefm−1=EF0mEεpeak−mσresidual.The stress-strain relation at the peak strength state can be expressed as(13)σpeak=1−D⋅Eεpeak+D⋅σresidual+2vσ3.Solving equations (12) and (13) will lead to the exact expression of parameters m and F0 as(14)m=−Eεpeak1+αlnε˙1/εrefEεpeak−σresiduallnσpeak−2vσ3−σresidual/Eεpeak−σresidual,F0=mEεpeak−mσresidualEεpeak1+αlnε˙1εrefm−11/m.


## 3.4. Parameter Experiment

To investigate the performance of the proposed statistical damage model in modelling the dynamic mechanical behaviours of rocks, a series of case studies were conducted here by letting the confining pressure be 0.5 MPa, and the values of the corresponding model parameters are given in Table1.Table 1
Model parameters for the proposed model.
Material parametersValueElastic modulusE (MPa)800.0Poisson’s ratioμ0.5Residual strengthσresidual (MPa)1.5Peak strengthσpeak (MPa)10.0Axial strain at the peak strength stateεpeak0.025Rate-related material constantα1.0First, we studied the effect of the residual strength on the stress-strain curves of rocks. Unlike the previous statistical damage models, the proposed model introduced a modified equivalent strain principle to account for the contribution of the friction force between the propagated nearby microcracks and the strength of rocks. In the cases ofσresidual were chosen to be 1.0, 2.0, 3.0, and 4.0 MPa, the predicted stress-strain curves shown in Figure 4(a) indicated that the strain-softening feature was weakened during the post-failure process, and the rock samples turn to be more ductile, which was confirmed by the evolution of the damage factor against the axial strain, as shown in Figure 4(b).Figure 4
Effect of residual strengthσresidual on stress-strain curve and damage curve of rock. (a) Stress-strain curve. (b) Damage curve.
(a)(b)Second, this paper explored the influence of loading rate on rock dynamic stress-strain curve and damage evolution law, and the calculation results are shown in Figure5. From Figure 5(a), it can be seen that the loading rate does not affect the residual strength of rock. However, the peak strength of rock σpeak will gradually increase with a gradual increase of ε·1. On the contrary, the peak strength of rock σpeak will gradually decrease. The calculation results are consistent with the experimental results of rock under dynamic loading.Figure 5
Effect of loading rates on stress-strain curve and damage curve of rockε·1. (a) Stress-strain curve. (b) Damage curve.
(a)(b)According to Figure5(b), the damage factor D will gradually decrease with the increase of loading rate in a small strain range (ε1≤0.025), and the tangent modulus of rock will also increase. However, with the further increase of axial strain, the growth rate is accelerated with the increase of loading rate, leading to brittle failures of rock more efficiently, and the stress-strain curve will have a noticeable “stress drop” phenomenon.The experimental results show that the rock statistical damage model established in this paper can better describe the effect of loading rate on the dynamic, progressive failure, and the residual strength of rock.


## 4. Model Verification

To further verify the performance of the proposed model, especially in describing the effect of loading rate on the dynamic yield strength of rocks, the experimental results of salty rocks are adopted here again. Table2 gives the values of the model parameters with different confining pressures and loading rates.Table 2
Model parameters of different kinds of rock.
σ3/MPaε˙/S−1E/GPAμmF054267.120.251.1820.0535197.210.251.1550.057154767.130.251.1190.0556317.240.251.2090.063254336.080.250.9320.0465136.950.251.0940.052The measured and predicted stress-strain curves by the proposed model are shown in Figures6–8, together with the predictions by [14] as a comparison. Although the two statistic damage models predict similar results and can fit the measured dynamic yield strength quite well, the stress-strain curve predicted by [14] has an unreasonable shear strength before the external force is applied. Such limitation is overcome by the proposed model, which can better account for the mechanical response of rock samples during the early loading stage. It is noted that the proposed model can be conveniently extended based on further experimental investigation to consider the effect of loading rate on the residual strength of rocks.Figure 6
Comparison between experimental and model results of dynamic stress-strain curves of salt rock [16] under confining pressure σ3=5MPa. (a) ε˙1=426/s. (b) ε˙1=519/s.
(a)(b)Figure 7
Comparison between experimental and model results of dynamic stress-strain curves of salt rock [16] under confining pressure σ3=15MPa. (a) ε˙1=476/s. (b) ε˙1=631/s.
(a)(b)Figure 8
Comparison between experimental and model results of dynamic stress-strain curves of salt rock [16] under confining pressure σ3=25MPa. (a) ε˙1=433/s. (b) ε˙1=513/s.
(a)(b)


## 5. Conclusion

This paper develops a new statistic damage model for rock to study its dynamic mechanical behaviours. The evolution law of the damage factor is assumed to depend on the loading ratio. Besides, the proposed model adopts a modified equivalent strain principle to improve its performance in modelling the residual strength of rocks. The main conclusions are summarized as follows:(1)
The microscopic damage accumulation process is affected by the loading rate, resulting in a higher peak strength of rocks. The evolution law of the damage factor is in terms of the axial strain and loading rate.(2)
The modified equivalent strain principle enables the proposed model to predict a reasonable residual strength of rocks mainly formed by the friction force generated by the movement of the nearby microcracks.(3)
The proposed model can well describe the stress-strain curves of salty rocks, providing a promising tool to study the dynamic behaviours of rocks.



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*Source: 9124208-2022-02-09.xml*