# A Fundamental Investigation of the Tensile Failure of Rock Using the Three-Dimensional Lattice Spring Model

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## Abstract

A fundamental study on the tensile failure of rock is conducted using the three-dimensional lattice spring model. The model covers three aspects: (1) the relationship between the mesoscopic tensile/shear failure and the corresponding macroscopic tensile failure; (2) the effects of the size, shape, and location of the initial defect on the macroscopic tensile failure; and (3) the effects of the porosity, heterogeneity, crystal structure, mesoscopic constitutive model, and model scale on its macroscopic tensile responses. Through investigation, this study reveals that the mesoscopic strength heterogeneity affects the macroscopic pre-peak response of rock, and the initial defect could control its macroscopic post-peak response. The post-peak characteristics of the mesoscopic constitutive model influence both the macroscopic pre-peak and post-peak responses, which are scale independent and scale dependent, respectively. Based on these investigations, a parameter-selection method for the mesoscopic constitutive model is established to fully utilize the macroscopic tensile experimental data.

## Keywords

Lattice spring model Tensile failure Post-peak responses Constitutive model Scale effect## List of Symbols

## Roman Alphabet

- \(D\left( \cdot \right)\)
The damage amount of the spring bond

- \({D_{\text{n}}}\left( \cdot \right)\)
The damage function of the normal spring

- \({D_{\text{s}}}\left( \cdot \right)\)
The damage function of the shear spring

- \(f\left( \cdot \right)\)
A mathematic function for damage description

- \({{\mathbf{F}}_{{\text{n}},ij}}\)
The normal interaction forces (vectors)

- \({{\mathbf{F}}_{{\text{s,}}ij}}\)
The shear interaction forces (vectors)

- \({k_{\text{n}}}\)
The shiftiness of the normal spring

- \({k_{\text{s}}}\)
The shiftiness of the shear spring

- \(\bar {l}\)
The mean length of the spring bond in the lattice model

- \(l_{i}^{{}}\)
The length of the

*i*-th spring bond in the lattice model- \(m\)
A shape coefficient of the Weibull distribution

- \({m_p}\)
The particle mass

- \({\mathbf{n}}\)
The initial unit vector of the connected particles

- \({{\mathbf{n}}_s}\)
The direction unit vector of the tangential deformation of the spring bond

- \({{\mathbf{u}}_i}\)
The displacement of particle\(i\)

- \({{\mathbf{u}}_j}\)
The displacement of particle\(j\)

- \({{\mathbf{u}}_{{\text{n}},ij}}\)
The relative normal deformation vector between particles

- \({{\mathbf{u}}_{{\text{s}},ij}}\)
The relative tangential deformation vector between particles

- \({\mathbf{\dot {u}}}\)
The particle velocity

- \({u_{\text{n}}}\)
The normal deformations of the spring

- \(u_{{\text{n}}}^{*}\)
The maximum tensile deformation of the spring

- \({u_{\text{s}}}\)
The tangential deformations of the spring

- \(\left| {{u_{\text{s}}}} \right|\)
The absolute value of the shear deformation

- \(u_{{\text{s}}}^{*}\)
The maximum shear deformation of the spring

- \({u^ * }\)
The ultimate deformation

- \(u\)
The current deformation

- \(\upsilon\)
The Poisson’s ratio

- \(V\)
The represented macroscopic volume of the computational model

- \(x\)
The non-dimensional spring deformation

- \({x_i}\)
The initial coordinate of particle \(i\)

- \({y_i}\)
The initial coordinate of particle \(i\)

- \({z_i}\)
The initial coordinate of particle \(i\)

- \({x_j}\)
The initial coordinate of particle \(j\)

- \({y_j}\)
The initial coordinate of particle \(j\)

- \({z_j}\)
The initial coordinate of particle \(j\)

- \(\Delta t\)
The time step

## Greek Symbols

- \(\alpha\)
A non-dimensional parameter for the mesoscopic constitutive model

- \(\beta\)
A parameter for the mesoscopic constitutive model

- \(\zeta\)
A random number that satisfies a certain distribution

- \({\xi _i}\)
The random number assigned to particle

*i*- \({\xi _j}\)
The random number assigned to particle

*j*- \({\xi ^{{\text{3D}}}}\)
The lattice coefficient

- \({\left[ \varepsilon \right]_{{\text{bond}}}}\)
The local strain of a spring bond

- \({\bar {\sigma }_t}\)
The macroscopic tensile strength

- \(\sigma _{t}^{{}}\)
The macroscopic tensile strength

- \(\lambda\)
A scale increase factor

## Notes

### Acknowledgements

This research is financially supported by the National Key Research and Development Program of China (2016YFC0401900), National Natural Science Foundation of China (U1765202, 1177020290) and Program of Introducing Talents of Discipline to Universities (B14012).

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