<p>This study investigates the stress generated in a stationary transversely isotropic rock medium resting on the ocean floor and subjected to a moving load. The upper rock layer incorporates a parabolic surface irregularity, and a comparative analysis is performed for three scenarios: parabolic, rectangular, and no surface irregularity. To solve the resulting nonlinear second-order partial differential equation governing the displacement field, Fourier Integral Transform Method is employed due to its effectiveness in handling such problems. The analysis reveals that the incremental normal stresses vanish directly beneath the boundary, while the incremental shear stresses attain their maximum at the same point. Because the boundary surface is irregular, a perturbation method is employed to obtain a linearized approximation of the governing equations. Since the perturbation parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\epsilon$</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation> is assumed to be very small <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\left( { \in < < 1} \right)$</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mrow> <mo>∈</mo> <mo><</mo> <mo><</mo> <mn>1</mn> </mrow> </mfenced> </math></EquationSource> </InlineEquation>, terms involving higher powers of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\epsilon$</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation> are neglected, and only first-order terms are considered in the analysis. A detailed parametric study is conducted to evaluate the influence of various factors, including layer thickness, depth, type of surface irregularity, and heterogeneity parameters, on the shear and normal stresses. Exponential heterogeneity is assumed in the rock medium, and comparisons are made between parabolic and rectangular heterogeneity profiles using graphical illustrations. The shear stress is computed numerically, and the results are presented graphically to support the findings. Additional insights are provided regarding the significant roles of the depth factor and the frictional coefficient in stress behavior in both media.</p>

Dynamic stress response of a transversely isotropic rock layer with parabolic irregularity under moving load: a perturbation approach

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发表日期：

2026年7月21日

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Manoj Kumar Singh,
Juhi Baroi,
Soniya Chaudhary,
Amit Kumar Rahul,
Ram Kumar,
Kamlesh Kumar Pankaj

摘要

This study investigates the stress generated in a stationary transversely isotropic rock medium resting on the ocean floor and subjected to a moving load. The upper rock layer incorporates a parabolic surface irregularity, and a comparative analysis is performed for three scenarios: parabolic, rectangular, and no surface irregularity. To solve the resulting nonlinear second-order partial differential equation governing the displacement field, Fourier Integral Transform Method is employed due to its effectiveness in handling such problems. The analysis reveals that the incremental normal stresses vanish directly beneath the boundary, while the incremental shear stresses attain their maximum at the same point. Because the boundary surface is irregular, a perturbation method is employed to obtain a linearized approximation of the governing equations. Since the perturbation parameter $\epsilon$ $ϵ$ is assumed to be very small $\left( { \in < < 1} \right)$ $(\in < < 1)$ , terms involving higher powers of $\epsilon$ $ϵ$ are neglected, and only first-order terms are considered in the analysis. A detailed parametric study is conducted to evaluate the influence of various factors, including layer thickness, depth, type of surface irregularity, and heterogeneity parameters, on the shear and normal stresses. Exponential heterogeneity is assumed in the rock medium, and comparisons are made between parabolic and rectangular heterogeneity profiles using graphical illustrations. The shear stress is computed numerically, and the results are presented graphically to support the findings. Additional insights are provided regarding the significant roles of the depth factor and the frictional coefficient in stress behavior in both media.