<p>A three-dimensional continuum model is illustrated to analyze the mechanics of fiber-reinforced composites (FRC) subjected to a combination of lateral pressure and bilateral extension. This model incorporates the Neo-Hookean strain energy function for the matrix material and considers the kinematic contribution of bidirectional reinforcing fibers. The strain energy of the bidirectional fibers is characterized by accounting for the stretching, bending, and twisting responses being computed through first- and second-order gradient deformation. To derive the equilibrium equations, differential geometry is employed to define the FRC surface configurations, while the variational principle is used to establish the Euler equation and boundary conditions. Numerical results demonstrate the model’s validation in analyzing both out-of-plane and in-plane deformations of the matrix material, as well as the bending, twisting, and stretching of the bidirectional fiber network. The novelty of this research lies in its theoretical framework for understanding the mechanics of FRC subjected to simultaneous lateral pressure and bilateral stretching, particularly addressing the effects of interaction between lateral pressure and bilateral extension on the FRC deformation via the characterization of both matrix material and fiber meshwork deformation. The findings reveal that increased lateral pressure leads to greater out-of-plane deformation and strain (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4264_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon _{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ε</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>), while bilateral stretching reduces out-of-plane deformation and transverse strain (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4264_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon _{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ε</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>), while longitudinal strain (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4264_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon _{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ε</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>) distribution remains unchanged. Additionally, the top and bottom boundaries of the FRC exhibit the most pronounced curvatures due to the shrinking effects at these boundaries. Bias extension test results further showcase significant shear strain in the central domain, with the resultant forces from lateral pressure tension and bilateral stretch affecting the shear angle’s enlargement or reduction. The theoretical analysis of fiber unit extension and flexure provides a coherent explanation for the overall deformation of the fiber meshwork, supporting the hypothesis that the fibers’ microstructural deformations govern the macroscopic deformation of the FRC meshwork. The continuum model demonstrates its practical validity by providing reasonable explanations for the shaping process of woven fabrics, Sikken-type stiffeners, fiber-reinforced thermoplastics, and bamboo polylactic acid (PLA) composites.</p>

The mechanics of bidirectionally reinforced elastomeric sheet subjected to the combination of lateral pressure and bilateral stretch

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发表日期：

2025年2月25日

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Wenhao Yao,
Yaoning Sun,
Chun I. L. Kim

摘要

A three-dimensional continuum model is illustrated to analyze the mechanics of fiber-reinforced composites (FRC) subjected to a combination of lateral pressure and bilateral extension. This model incorporates the Neo-Hookean strain energy function for the matrix material and considers the kinematic contribution of bidirectional reinforcing fibers. The strain energy of the bidirectional fibers is characterized by accounting for the stretching, bending, and twisting responses being computed through first- and second-order gradient deformation. To derive the equilibrium equations, differential geometry is employed to define the FRC surface configurations, while the variational principle is used to establish the Euler equation and boundary conditions. Numerical results demonstrate the model’s validation in analyzing both out-of-plane and in-plane deformations of the matrix material, as well as the bending, twisting, and stretching of the bidirectional fiber network. The novelty of this research lies in its theoretical framework for understanding the mechanics of FRC subjected to simultaneous lateral pressure and bilateral stretching, particularly addressing the effects of interaction between lateral pressure and bilateral extension on the FRC deformation via the characterization of both matrix material and fiber meshwork deformation. The findings reveal that increased lateral pressure leads to greater out-of-plane deformation and strain ( $\varepsilon _{1}$ $ε_{1}$ ), while bilateral stretching reduces out-of-plane deformation and transverse strain ( $\varepsilon _{1}$ $ε_{1}$ ), while longitudinal strain ( $\varepsilon _{2}$ $ε_{2}$ ) distribution remains unchanged. Additionally, the top and bottom boundaries of the FRC exhibit the most pronounced curvatures due to the shrinking effects at these boundaries. Bias extension test results further showcase significant shear strain in the central domain, with the resultant forces from lateral pressure tension and bilateral stretch affecting the shear angle’s enlargement or reduction. The theoretical analysis of fiber unit extension and flexure provides a coherent explanation for the overall deformation of the fiber meshwork, supporting the hypothesis that the fibers’ microstructural deformations govern the macroscopic deformation of the FRC meshwork. The continuum model demonstrates its practical validity by providing reasonable explanations for the shaping process of woven fabrics, Sikken-type stiffeners, fiber-reinforced thermoplastics, and bamboo polylactic acid (PLA) composites.