<p>Rock is a fundamental material in mining engineering, and its creep behavior plays a critical role in determining the long-term stability of roadways. Consequently, investigating the creep constitutive models of rocks with varying brittleness holds significant practical importance. To investigate the evolution law of energy in uniaxial creep of rocks and establish a constitutive model, this study systematically examined the evolution of creep energy and energy distribution in four kinds of rocks—coal, mudstone, white sandstone, and red sandstone—through uniaxial creep-unloading tests. We constructed the fractional derivative damage constitutive models by introducing fractional derivative elements based on energy dissipation damage variables. The findings reveal the following: (1) The elastic strain energy density (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9796_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mi mathvariant="normal">e</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$u^{\mathrm{e}}$</EquationSource> </InlineEquation>) of rocks exhibits a linear decreasing trend with prolonged creep time, indicating a linear attenuation characteristic. (2) A method for calculating rock creep energy was proposed, leveraging the linear attenuation characteristics of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9796_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mi mathvariant="normal">e</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$u^{\mathrm{e}}$</EquationSource> </InlineEquation>. (3) The dissipated strain energy density (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9796_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mi mathvariant="normal">d</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$u^{\mathrm{d}}$</EquationSource> </InlineEquation>) and input strain energy density (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9796_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>u</mi> </math></EquationSource> <EquationSource Format="TEX">$u$</EquationSource> </InlineEquation>) of the four rocks with varying brittleness levels increase over time, and this growth can be partitioned into three stages: decay growth, steady growth, and accelerated growth. (4) As creep time increases, the proportion of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9796_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mi mathvariant="normal">d</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$u^{\mathrm{d}}$</EquationSource> </InlineEquation>/<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9796_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>u</mi> </math></EquationSource> <EquationSource Format="TEX">$u$</EquationSource> </InlineEquation> gradually rises, reaching its maximum at the end of accelerated creep. Rocks with higher brittleness exhibit a greater proportion of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9796_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mi mathvariant="normal">d</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$u^{\mathrm{d}}$</EquationSource> </InlineEquation> at these critical points. (5) A fractional derivative damage constitutive model was successfully developed, with the parameter <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9796_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> <EquationSource Format="TEX">$\alpha $</EquationSource> </InlineEquation> of the fractional derivative element reflecting the degree of rock brittleness.</p>

Energy evolution and fractional-order damage constitutive model of rock creep

期刊论文
发表日期：

2025年6月26日

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Zhixi Liu,
Dandan Ye

摘要

Rock is a fundamental material in mining engineering, and its creep behavior plays a critical role in determining the long-term stability of roadways. Consequently, investigating the creep constitutive models of rocks with varying brittleness holds significant practical importance. To investigate the evolution law of energy in uniaxial creep of rocks and establish a constitutive model, this study systematically examined the evolution of creep energy and energy distribution in four kinds of rocks—coal, mudstone, white sandstone, and red sandstone—through uniaxial creep-unloading tests. We constructed the fractional derivative damage constitutive models by introducing fractional derivative elements based on energy dissipation damage variables. The findings reveal the following: (1) The elastic strain energy density ( $u^{e}$ $u^{\mathrm{e}}$ ) of rocks exhibits a linear decreasing trend with prolonged creep time, indicating a linear attenuation characteristic. (2) A method for calculating rock creep energy was proposed, leveraging the linear attenuation characteristics of $u^{e}$ $u^{\mathrm{e}}$ . (3) The dissipated strain energy density ( $u^{d}$ $u^{\mathrm{d}}$ ) and input strain energy density ( $u$ $u$ ) of the four rocks with varying brittleness levels increase over time, and this growth can be partitioned into three stages: decay growth, steady growth, and accelerated growth. (4) As creep time increases, the proportion of $u^{d}$ $u^{\mathrm{d}}$ / $u$ $u$ gradually rises, reaching its maximum at the end of accelerated creep. Rocks with higher brittleness exhibit a greater proportion of $u^{d}$ $u^{\mathrm{d}}$ at these critical points. (5) A fractional derivative damage constitutive model was successfully developed, with the parameter $α$ $\alpha $ of the fractional derivative element reflecting the degree of rock brittleness.