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Shear-lag model for discontinuous fiber-reinforced composites with a membrane-type imperfect interface

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Abstract

A shear-lag model is developed for discontinuous fiber-reinforced composites with a membrane-type imperfect interface, across which the displacement vector is continuous but the traction vector suffers a jump that is governed by the generalized Young–Laplace equation. Closed-form expressions are obtained for the stress fields in both the fiber-reinforced region and the pure matrix regions and for the shear stress on the interface from both the fiber and matrix sides. To illustrate the newly developed analytical model, a numerical analysis is provided by directly using the general formulas derived. The numerical results reveal that the fiber aspect ratio and the interface parameter can both have significant effects on the stress distributions in the composite.

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Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 51739003, 11672099, 51909173 and 11772117). The authors also would like to thank Professor George Weng and two anonymous reviewers for their encouragement and helpful comments on earlier versions of the paper.

Author information

Authors and Affiliations

  1. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, 210098, China

    J.-Y. Wang, C.-S. Gu & H. Gu

  2. College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing, 210098, China

    J.-Y. Wang, C.-S. Gu & H. Gu

  3. National Engineering Research Center of Water Resources Efficient Utilization and Engineering Safety, Hohai University, Nanjing, 210098, China

    J.-Y. Wang, C.-S. Gu & H. Gu

  4. School of Civil Engineering, Chongqing University, Chongqing, 400044, China

    S.-T. Gu

  5. Department of Mechanical Engineering, Southern Methodist University, Dallas, TX, 75275-0337, USA

    X.-L. Gao

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  1. J.-Y. Wang
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  2. C.-S. Gu
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  3. S.-T. Gu
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  4. X.-L. Gao
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  5. H. Gu
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Appendix: Derivation of Eq. (15)

Appendix: Derivation of Eq. (15)

From Eqs. (11), (12) and (13a) and (14), it follows that

$$\begin{aligned}&{\mathrm {\sigma }}_\mathrm{s} ={\mathbb {L}}_\mathrm{s} :\varvec{\upvarepsilon }_\mathrm{s} ={\mathbb {L}}_\mathrm{s} :{\mathbb {T}}:{\mathbb {M}}^{(2)}:\varvec{\sigma }^{(2)}={\mathbb {L}}_\mathrm{s} :{\mathbb {M}}^{(2)}:\varvec{\sigma }^{(2)} \nonumber \\&\quad = (2k_\mathrm{s} {\mathbb {J}}_\mathrm{s} +2\mu _\mathrm{s} {\mathbb {K}}_\mathrm{s} ):\left( \frac{1}{3k^{(2)}}{\mathbb {J}}+\frac{1}{2\mu ^{(2)}}{\mathbb {K}}\right) :\varvec{\sigma }^{(2)} \nonumber \\&\quad = \frac{2k_\mathrm{s} }{3k^{(2)}}{\mathbb {J}}_\mathrm{s} :{\mathbb {J}}:\varvec{\sigma }^{(2)}+\frac{k_\mathrm{s} }{\mu ^{(2)}}{\mathbb {J}}_\mathrm{s} :{\mathbb {K}}:\varvec{\sigma }^{(2)}+\frac{2\mu _\mathrm{s} }{3k^{(2)}}{\mathbb {K}}_\mathrm{s} :{\mathbb {J}}:\varvec{\sigma }^{(2)}+\frac{\mu _\mathrm{s} }{\mu ^{(2)}}{\mathbb {K}}_\mathrm{s} :{\mathbb {K}}:\varvec{\sigma }^{(2)}\quad \text{ on } \Gamma _{{ c}}, \end{aligned}$$
(A1)

where

$$\begin{aligned}&{\mathbb {J}}_\mathrm{s} :{\mathbb {J}}:\varvec{\sigma }^{\left( 2\right) }=\left( \frac{1}{2}{\mathbf {T}}\otimes {\mathbf {T}}\right) :\left( \frac{1}{3}{\mathbf {I}}\otimes {\mathbf {I}}\right) :\varvec{\sigma }^{\left( 2\right) }=\frac{1}{6}{\mathbf {T}}\left( {\mathbf {T}}:{\mathbf {I}}\right) \left( {\mathbf {I}}:\varvec{\sigma }^{\left( 2\right) }\right) =\frac{1}{3}{\mathbf {T}}\hbox {tr}^{\left( 2\right) } \nonumber \\&\quad =\frac{1}{3}\left( \sigma _{rr}^{\left( 2\right) } +\sigma _{\theta \theta }^{\left( 2\right) } +\sigma _{zz}^{\left( 2\right) } \right) {\mathbf {T}}\quad \text{ on } \Gamma _{{ c}}, \end{aligned}$$
(A2)
$$\begin{aligned}&{\mathbb {J}}_\mathrm{s} :{\mathbb {K}}:\varvec{\sigma }^{\left( 2\right) }={\mathbb {J}}_\mathrm{s} :\left( {\mathbb {I}}-{\mathbb {J}}\right) :\varvec{\sigma }^{\left( 2\right) }=\left( \frac{1}{2}{\mathbf {T}}\otimes {\mathbf {T}}\right) :\left( {\mathbb {I}}-\frac{1}{3}{\mathbf {I}}\otimes {\mathbf {I}}\right) :\varvec{\sigma }^{\left( 2\right) } \nonumber \\&\quad =\frac{1}{2}\left( {\mathbf {T}}\otimes {\mathbf {T}}\right) :{\mathbb {I}}:\varvec{\sigma }^{\left( 2\right) }-\frac{1}{6}{\mathbf {T}}\left( {\mathbf {T}}:{\mathbf {I}}\right) \left( {\mathbf {I}}:\varvec{\sigma }^{\left( 2\right) }\right) =\frac{1}{2}{\mathbf {T}}\left( {\mathbf {T}}:\varvec{\sigma }^{\left( 2\right) }\right) -\frac{1}{3}{\mathbf {T}}\hbox {tr}^{\left( 2\right) } \nonumber \\&\quad ={\mathbf {T}}\left( \frac{1}{2}{\mathbf {T}}:\varvec{\sigma }^{\left( 2\right) }-\frac{1}{3}\hbox {tr}\varvec{\sigma }^{\left( 2\right) }\right) ={\mathbf {T}} \left[ \frac{1}{2}\left( \sigma _{\theta \theta }^{\left( 2\right) } +\sigma _{zz}^{\left( 2\right) } \right) -\frac{1}{3}\left( \sigma _{rr}^{\left( 2\right) } +\sigma _{\theta \theta }^{\left( 2\right) } +\sigma _{zz}^{\left( 2\right) } \right) \right] \nonumber \\&\quad =\left( -\frac{1}{3}\sigma _{rr}^{\left( 2\right) } +\frac{1}{6}\sigma _{\theta \theta }^{\left( 2\right) } +\frac{1}{6}\sigma _{zz}^{\left( 2\right) } \right) {\mathbf {T}}\,\, \text{ on } \Gamma _{{ c}} , \end{aligned}$$
(A3)
$$\begin{aligned}&{\mathbb {K}}_\mathrm{s} :{\mathbb {J}}:\varvec{\sigma }^{\left( 2\right) }=\left( {\mathbb {T}}-{\mathbb {J}}_\mathrm{s} \right) :{\mathbb {J}}:\varvec{\sigma }^{\left( 2\right) }=\left( {\mathbb {T}}-\frac{1}{2}{\mathbf {T}}\otimes {\mathbf {T}}\right) :\left( \frac{1}{3}{\mathbf {I}}\otimes {\mathbf {I}}\right) :\varvec{\sigma }^{\left( 2\right) } \nonumber \\&\quad =\frac{1}{3}{\mathbb {T}}:\left( {\mathbf {I}}\otimes {\mathbf {I}}\right) :\varvec{\sigma }^{\left( 2\right) }-\frac{1}{6}{\mathbf {T}}\left( {\mathbf {T}}:{\mathbf {I}}\right) \left( {\mathbf {I}}:\varvec{\sigma }^{\left( 2\right) }\right) =\frac{1}{3}\left( {\mathbb {T}}:{\mathbf {I}}\right) \left( {\mathbf {I}}:\varvec{\sigma }^{\left( 2\right) }\right) -\frac{1}{3}{\mathbf {T}}\hbox {tr}^{\left( 2\right) } \nonumber \\&\quad =\frac{1}{3}{\mathbf {T}}\hbox {tr}^{\left( 2\right) }-\frac{1}{3}{\mathbf {T}}\hbox {tr}\varvec{\sigma }^{\left( 2\right) }=0\,\, \text{ on } \Gamma _{{ c}} , \end{aligned}$$
(A4)
$$\begin{aligned}&{\mathbb {K}}_\mathrm{s} :{\mathbb {K}}:\varvec{\sigma }^{\left( 2\right) }=\left( {\mathbb {T}}-{\mathbb {J}}_\mathrm{s} \right) :\left( {\mathbb {I}}-{\mathbb {J}}\right) :\varvec{\sigma }^{\left( 2\right) }= \left[ {\mathbb {T}}:{\mathbb {I}}-{\mathbb {J}}_\mathrm{s} :{\mathbb {I}}-\left( {\mathbb {T}}-{\mathbb {J}}_\mathrm{s} \right) :{\mathbb {J}}\right] :\varvec{\sigma }^{\left( 2\right) } \nonumber \\&\quad =\left[ {\mathbb {T}}:{\mathbb {I}}-{\mathbb {J}}_\mathrm{s} :\left( {\mathbb {K}}+{\mathbb {J}}\right) -{\mathbb {K}}_\mathrm{s} :{\mathbb {J}}\right] :\varvec{\sigma }^{\left( 2\right) }= \left[ {\mathbb {T}}:{\mathbb {I}}-{\mathbb {J}}_\mathrm{s} :{\mathbb {K}}-{\mathbb {J}}_\mathrm{s} :{\mathbb {J}}\right] :\varvec{\sigma }^{\left( 2\right) } \nonumber \\&\quad =\mathbf {T\varvec{\sigma } }^{\left( 2\right) }{\mathbf {T}}-\left( -\frac{1}{3}\sigma _{rr}^{\left( 2\right) } +\frac{1}{6}\sigma _{\theta \theta }^{\left( 2\right) } +\frac{1}{6}\sigma _{zz}^{\left( 2\right) } \right) {\mathbf {T}}-\frac{1}{3}\left( \sigma _{rr}^{\left( 2\right) } +\sigma _{\theta \theta }^{\left( 2\right) } +\sigma _{zz}^{\left( 2\right) } \right) {\mathbf {T}}\nonumber \\&\quad =\mathbf {T\varvec{\sigma } }^{\left( 2\right) }{\mathbf {T}}-\frac{1}{2}\left( \sigma _{\theta \theta }^{\left( 2\right) } +\sigma _{zz}^{\left( 2\right) } \right) {\mathbf {T}} \text{= }\sigma _{\theta \theta }^{\left( 2\right) } {\mathbf {e}}_{\theta } \otimes {\mathbf {e}}_{\theta } +\sigma _{zz}^{\left( 2\right) } {\mathbf {e}}_{z} \otimes {\mathbf {e}}_{z} -\frac{1}{2}\left( \sigma _{\theta \theta }^{\left( 2\right) } +\sigma _{zz}^{\left( 2\right) } \right) {\mathbf {T}}\,\, \text{ on } \Gamma _{{ c}} . \end{aligned}$$
(A5)

Using Eqs. (A2)–(A5) in Eq. (A1) yields

$$\begin{aligned} \varvec{\sigma }_\mathrm{s}= & {} \left[ \left( \frac{2k_\mathrm{s} }{9k^{\left( 2\right) }}-\frac{k_\mathrm{s} }{3\mu ^{\left( 2\right) }}\right) \sigma _{rr}^{\left( 2\right) } +\left( \frac{2k_\mathrm{s} }{9k^{\left( 2\right) }}+\frac{k_\mathrm{s} }{6\mu ^{\left( 2\right) }}+\frac{\mu _\mathrm{s} }{2\mu ^{\left( 2\right) }}\right) \sigma _{\theta \theta }^{\left( 2\right) }\right. \nonumber \\&\quad \left. +\left( \frac{2k_\mathrm{s} }{9k^{\left( 2\right) }}+\frac{k_\mathrm{s} }{6\mu ^{\left( 2\right) }}-\frac{\mu _\mathrm{s} }{2\mu ^{\left( 2\right) }}\right) \sigma _{zz}^{\left( 2\right) } \right] {\mathbf {e}}_{\theta } \otimes {\mathbf {e}}_{\theta } \nonumber \\&\quad +\left[ \left( \frac{2k_\mathrm{s} }{9k^{\left( 2\right) }}-\frac{k_\mathrm{s} }{3\mu ^{\left( 2\right) }}\right) \sigma _{rr}^{\left( 2\right) } +\left( \frac{2k_\mathrm{s} }{9k^{\left( 2\right) }}+\frac{k_\mathrm{s} }{6\mu ^{\left( 2\right) }}-\frac{\mu _\mathrm{s} }{2\mu ^{\left( 2\right) }}\right) \sigma _{\theta \theta }^{\left( 2\right) } \right. \nonumber \\&\quad \left. +\left( \frac{2k_\mathrm{s} }{9k^{\left( 2\right) }}+\frac{k_\mathrm{s} }{6\mu ^{\left( 2\right) }}+\frac{\mu _\mathrm{s} }{2\mu ^{\left( 2\right) }}\right) \sigma _{zz}^{\left( 2\right) } \right] {\mathbf {e}}_{z} \otimes {\mathbf {e}}_{z} \,\, \text{ on } \Gamma _{{ c}} . \end{aligned}$$
(A6)

Based on the third shear-lag assumption listed in Eq. (9),

$$\begin{aligned} \sigma _{rr}^{(i)}<<\sigma _{zz}^{(i)} \text{, }\,\,\,\,\,\,\sigma _{\theta \theta }^{(i)}<<\sigma _{zz}^{(i)}\,\, \text{( }i=1,\,2). \end{aligned}$$
(A7)

Then, Eq. (A6) can be simplified as, upon using Eq. (A7),

$$\begin{aligned} \varvec{\sigma }_\mathrm{s}= & {} \left[ \left( \frac{2k_\mathrm{s} }{9k^{\left( 2\right) }}+\frac{k_\mathrm{s} }{6\mu ^{\left( 2\right) }}-\frac{\mu _\mathrm{s} }{2\mu ^{\left( 2\right) }}\right) \sigma _{zz}^{\left( 2\right) } \right] {\mathbf {e}}_{\theta } \otimes {\mathbf {e}}_{\theta } \nonumber \\&+\left[ \left( \frac{2k_\mathrm{s} }{9k^{\left( 2\right) }}+\frac{k_\mathrm{s} }{6\mu ^{\left( 2\right) }}+\frac{\mu _\mathrm{s} }{2\mu ^{\left( 2\right) }}\right) \sigma _{zz}^{\left( 2\right) } \right] {\mathbf {e}}_{z} \otimes {\mathbf {e}}_{z} \,\, \text{ on } \Gamma _{{ c}} . \end{aligned}$$
(A8)

From Eq. (A8), it follows that

$$\begin{aligned} \sigma _{\theta \theta }^\mathrm{s} =\left( \frac{2k_\mathrm{s} }{9k^{\left( 2\right) }}+\frac{k_\mathrm{s} }{6\mu ^{\left( 2\right) }}-\frac{\mu _\mathrm{s} }{2\mu ^{\left( 2\right) }}\right) \sigma _{zz}^{\left( 2\right) } ,\quad \tau _{\theta z}^\mathrm{s} =0,\quad \sigma _{zz}^\mathrm{s} =\left( \frac{2k_\mathrm{s} }{9k^{\left( 2\right) }}+\frac{k_\mathrm{s} }{6\mu ^{\left( 2\right) }}+\frac{\mu _\mathrm{s} }{2\mu ^{\left( 2\right) }}\right) \sigma _{zz}^{\left( 2\right) } . \end{aligned}$$
(A9)

These are the surface stress components listed in Eq. (15), thereby completing the derivation.

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Wang, JY., Gu, CS., Gu, ST. et al. Shear-lag model for discontinuous fiber-reinforced composites with a membrane-type imperfect interface. Acta Mech 231, 4717–4734 (2020). https://doi.org/10.1007/s00707-020-02768-7

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  • DOI: https://doi.org/10.1007/s00707-020-02768-7

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