<p>Beyond the utilization of linear elastic foundations like the Winkler, Pasternak, and Hetenyi among others, this study deployed a nonlinear elastic foundation to model the deflection of plates amid longitudinal and transverse initial pre-stresses. A promising analytical method has been utilized to construct various exact solutions for the model, which hugely contribute to the experimental and numerical studies, in addition to the analyses of overall linearized dispersion relation and the model’s stability. The numerical examination of the model has it that an increase in both the initial pre-stresses and the coefficient of the cubic nonlinearity coefficient <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4458_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">${M}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> opposes the deflection of waves in the plate. In contrast, an increase in the coefficient of the quadratic nonlinearity <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4458_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">${M}_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> increases the vibrational displacement in the medium. In addition, the resulting approximate dispersion relation has it that an increase in both the attenuation parameter <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4458_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\eta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation> and transverse initial pre-stress <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4458_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${N}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> smoothly increases the dispersion of flexural waves, while an increase in the longitudinal initial pre-stress <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4458_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">${N}_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> opposes the dispersion of waves in the medium. Moreover, future work can be directed toward incorporating various forms of highly nonlinearly terms into the governing plate equation, in addition to an in-depth search for an optimal analytical procedure to perfectly tackle the nonlinearity terms.</p>

Dispersion of flexural waves on an initially pre-stressed thin plate resting on nonlinear elastic foundations

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发表日期：

2025年8月8日

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Saad Althobaiti,
Ali M. Mubaraki,
Rahmatullah Ibrahim Nuruddeen

摘要

Beyond the utilization of linear elastic foundations like the Winkler, Pasternak, and Hetenyi among others, this study deployed a nonlinear elastic foundation to model the deflection of plates amid longitudinal and transverse initial pre-stresses. A promising analytical method has been utilized to construct various exact solutions for the model, which hugely contribute to the experimental and numerical studies, in addition to the analyses of overall linearized dispersion relation and the model’s stability. The numerical examination of the model has it that an increase in both the initial pre-stresses and the coefficient of the cubic nonlinearity coefficient ${M}_{2}$ $M_{2}$ opposes the deflection of waves in the plate. In contrast, an increase in the coefficient of the quadratic nonlinearity ${M}_{1}$ $M_{1}$ increases the vibrational displacement in the medium. In addition, the resulting approximate dispersion relation has it that an increase in both the attenuation parameter $\eta$ $η$ and transverse initial pre-stress ${N}_{2}$ $N_{2}$ smoothly increases the dispersion of flexural waves, while an increase in the longitudinal initial pre-stress ${N}_{1}$ $N_{1}$ opposes the dispersion of waves in the medium. Moreover, future work can be directed toward incorporating various forms of highly nonlinearly terms into the governing plate equation, in addition to an in-depth search for an optimal analytical procedure to perfectly tackle the nonlinearity terms.