# Investigation on the Linear Energy Storage and Dissipation Laws of Rock Materials Under Uniaxial Compression

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## Abstract

To investigate the energy evolution characteristics of rock materials under uniaxial compression, the single-cyclic loading–unloading uniaxial compression tests of four rock materials (Qingshan granite, Yellow sandstone, Longdong limestone and Black sandstone) were conducted under five unloading stress levels. The stress–strain curves and failure characteristics of rock specimens under the single-cyclic loading–unloading uniaxial compression tests basically corresponded with those of under uniaxial compression, which indicates that single-cyclic loading–unloading has minimal effects on the variations in the loading–deformation response of rocks. The input energy density, elastic energy density and dissipated energy density of four rocks under five unloading stress levels were calculated using the graphical integration method, and variation characteristics of those three energy density parameters with different unloading stress levels were explored. The results show that all three energy density parameters above increased nonlinearly with increasing unloading stress level as quadratic polynomial functions. Meanwhile, both the elastic and dissipated energy density increased linearly when the input energy density increased, and the linear energy storage and dissipation laws for rock materials were observed. Furthermore, a linear relationship between the dissipated and elastic energy density was also proposed. Using the linear energy storage or dissipation law, the elastic and dissipated energy density at any stress levels can be calculated, and the internal elastic (or dissipated) energy density at peak compressive strength (the peak elastic and dissipated energy density for short) can be obtained. The ratio of the elastic energy density to dissipated energy density with increasing input energy density was investigated using a new method, and the results show that this ratio tends to be constant at the peak compressive strength of rock specimens.

## Keywords

Rock materials Input energy density Elastic energy density Linear energy storage law Linear energy dissipation law Peak elastic energy density Single-cyclic loading–unloading uniaxial compression## List of Symbols

*D*Diameter of specimen

*E*_{u}Linear modulus of ideal unloading curve at peak strength

*E*_{s}Unloading tangential modulus

*E*_{l}Loading curve’s Young’s modulus

*H*Height of specimen

*i*Actual unloading stress level

*k*Setting unloading stress level

- \( u_{{}}^{{\rm e}} \)
Elastic energy density of rock at peak compressive strength (the peak elastic energy density for short)

- \( u_{{}}^{{\rm d}} \)
Dissipated energy density of rock at peak compressive strength (the peak dissipated energy density for short)

- \( u_{{}}^{{\rm o}} \)
Input energy density of rock at peak compressive strength (the peak input energy density for short)

- \( u_{i}^{{\rm d}} \)
Dissipated energy density at actual unloading stress level

*i*- \( u_{i}^{{\rm e}} \)
Elastic energy density at actual unloading stress level

*i*- \( u_{\text{d}}^{{\text {o}}} \)
Input energy density at any loading time before the peak compressive strength

- \( u_{i}^{{\rm o}} \)
Input energy density at actual unloading stress level

*i*- \( u_{i}^{{\rm d}} /u_{i}^{{\rm o}} \)
Ratio of the dissipated energy density to the input energy density

- \( u_{i}^{\text{e}} /u_{i}^{\text{o}} \)
Ratio of the elastic energy density to the input energy density

- \( v \)
Longitudinal wave speed of the specimen

*V*Volume of specimen

- \( W_{\text{et}}^{\text{d}} \)
Ratio of the elastic energy density to the dissipated energy density when the input energy density is \( u_{\text{d}}^{\text{o}} \)

- \( W_{\text{et}}^{i} \)
Ratio of the elastic energy density to the dissipated energy density at actual unloading stress level

*i*and is equal to \( u_{i}^{{\rm e}} \) / \( u_{i}^{{\rm d}} \)- \( W_{\text{et}}^{{\rm p}} \)
Ratio of the peak elastic energy density to the peak dissipated energy density and is equal to \( u_{{}}^{{\rm e}} \) / \( u_{{}}^{{\rm d}} \)

## Greek Symbols

- \( \rho \)
Density of specimen

- \( \sigma_{{\rm c}}^{{\rm e}} \)
Uniaxial compressive strength of specimen

- \( \sigma_{{\rm c}}^{k} \)
Peak compressive strength of specimen under single-cyclic loading–unloading uniaxial compression when setting unloading stress level is

*k*

## Abbreviations

- DED
Dissipated energy density

- EED
Elastic energy density

- IED
Input energy density

- SCLUC
Single-cyclic loading–unloading uniaxial compression

- UC
Uniaxial compression

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 41877272 and 41472269).

### Compliance with ethical standards

### Conflicts of interest

The authors declare no conflict of interest.

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