The damping ratio of rocks is essential for evaluating rock mass stability under dynamic loads. This study investigates energy evolution and damping characteristics through single cyclic loading–unloading uniaxial compression tests and acoustic emission (AE) monitoring on sandstone and granite, considering stress history. A strong linear relationship between cumulative AE energy and cumulative damage energy validates the theory of viscoelastic-plastic energy conversion in rocks. Additionally, the study discovered the linear energy damping (LED) law, characterized by a consistent linear relationship between damping energy and input energy, and a constant linear energy damping coefficient (LEDC) defined by linear fitting parameter A. Based on the LED law, two damping ratio calculation formulas (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4466_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="256" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda_{LED,i} = (A_{2} u_{i} + B_{2} )/\pi (A_{1} u_{i} + B_{1} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mrow> <mi>L</mi> <mi>E</mi> <mi>D</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mn>2</mn> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>π</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mn>1</mn> </msub> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4466_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="133" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda_{I - LED} = A_{2} /\pi A_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mrow> <mi>I</mi> <mo>-</mo> <mi>L</mi> <mi>E</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo stretchy="false">/</mo> <mi>π</mi> <msub> <mi>A</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>) were proposed. The results showed that the LED law enables the calculation of damping strain energy density at peak strength. The LEDC serves as an effective parameter for characterizing the damping capacity of various rock types <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4466_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda_{LED,i}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mrow> <mi>L</mi> <mi>E</mi> <mi>D</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> represents the evolution curve of the damping ratio, while <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="603_2025_4466_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda_{I - LED}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mrow> <mi>I</mi> <mo>-</mo> <mi>L</mi> <mi>E</mi> <mi>D</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> provides accurate estimates of the damping ratio at peak strength compared to traditional methods and reflects reliable stability. Stress history significantly affects the linear energy storage law fitting parameter B and the |B/A|. The research findings offer a novel method for calculating the energy relationship and damping ratio under ultimate stress states, enhancing the reliability of geotechnical engineering designs, and reducing disaster risks.

A Novel Rock Damping Ratio and Damping Coefficient Based on Linear Energy Damping Law

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发表日期：

2025年3月3日

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Haowei Yang,
Bing Sun,
Jie Cui,
Sheng Zeng,
Yi Shan

摘要

The damping ratio of rocks is essential for evaluating rock mass stability under dynamic loads. This study investigates energy evolution and damping characteristics through single cyclic loading–unloading uniaxial compression tests and acoustic emission (AE) monitoring on sandstone and granite, considering stress history. A strong linear relationship between cumulative AE energy and cumulative damage energy validates the theory of viscoelastic-plastic energy conversion in rocks. Additionally, the study discovered the linear energy damping (LED) law, characterized by a consistent linear relationship between damping energy and input energy, and a constant linear energy damping coefficient (LEDC) defined by linear fitting parameter A. Based on the LED law, two damping ratio calculation formulas ( $\lambda_{LED,i} = (A_{2} u_{i} + B_{2} )/\pi (A_{1} u_{i} + B_{1} )$ $λ_{L E D, i} = (A_{2} u_{i} + B_{2}) / π (A_{1} u_{i} + B_{1})$ , $\lambda_{I - LED} = A_{2} /\pi A_{1}$ $λ_{I - L E D} = A_{2} / π A_{1}$ ) were proposed. The results showed that the LED law enables the calculation of damping strain energy density at peak strength. The LEDC serves as an effective parameter for characterizing the damping capacity of various rock types $\lambda_{LED,i}$ $λ_{L E D, i}$ represents the evolution curve of the damping ratio, while $\lambda_{I - LED}$ $λ_{I - L E D}$ provides accurate estimates of the damping ratio at peak strength compared to traditional methods and reflects reliable stability. Stress history significantly affects the linear energy storage law fitting parameter B and the |B/A|. The research findings offer a novel method for calculating the energy relationship and damping ratio under ultimate stress states, enhancing the reliability of geotechnical engineering designs, and reducing disaster risks.