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Fracture mechanics analysis of an anti-plane crack in gradient elastic sandwich composite structures

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Abstract

The strain gradient elasticity theory is applied to the solution of a mode III crack in an elastic layer sandwiched by two elastic layers of infinite thickness. The model includes volumetric and surface strain gradient characteristic length parameters. Both the near-tip asymptotic stresses and the crack displacement are obtained. Due to stain gradient effects, the magnitudes of the stress ahead of the crack tip are significantly higher than those in the classical linear elastic fracture mechanics. When the gradient parameters reduce to sufficiently small, all results reduce to the conventional linear elastic fracture mechanics results. In addition to the single crack in the finite layer, the solution and the results for two collinear cracks are also established and given.

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References

  • Chan, Y.S., Fannjiang, A.C., Glaucio, H.: Integral equations with hypersingular kernels—theory and applications to fracture mechanics. Int. J. Eng. Sci. 41, 683–720 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Chan, Y.S., Paulino, G.H., Fannjiang, A.C.: Gradient elasticity theory for mode III fracture in functionally graded materials-part II: crack parallel to the material gradation. J. Appl. Mech. 75, 061015 (2008)

    Article  Google Scholar 

  • Exadaktylos, G.: Gradient elasticity with surface energy: mode-I crack problem. Int. J. Solids Struct. 35, 421–456 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Exadaktylos, G., Vardoulakis, I., Aifantis, E.: Cracks in gradient elastic bodies with surface energy. Int. J. Fract. 79, 107–119 (1996)

    Article  MATH  Google Scholar 

  • Fang, T.H., Li, W.L., Tao, N.R., Lu, K.: Revealing extraordinary intrinsic tensile plasticity in gradient nano-grained copper. Science 331, 1587–1590 (2011)

    Article  Google Scholar 

  • Fannjiang, A.C., Paulino, G.H., Chan, Y.S.: Strain gradient elasticity for antiplane shear cracks: a hypersingular integrodifferential equation approach. SIAM J. Appl. Math. 62, 1066–1091 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42, 475–487 (1994)

    Article  Google Scholar 

  • Giannakopoulos, A., Stamoulis, K.: Structural analysis of gradient elastic components. Int. J. Solids Struct. 44, 3440–3451 (2007)

    Article  MATH  Google Scholar 

  • Joseph, R.P., Wang, B.L., Samali, B.: Strain gradient fracture in an anti-plane cracked material layer. Int. J. Solids Struct. 146, 214–223 (2018)

    Article  Google Scholar 

  • Karimipour, I., Fotuhi, A.R.: Anti-plane analysis of an infinite plane with multiple cracks based on strain gradient theory. Acta Mech. 228, 1793–1817 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Lam, D.C., Yang, F., Chong, A., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)

    Article  MATH  Google Scholar 

  • Ma, Q., Clarke, D.R.: Size dependent hardness of silver single crystals. J. Mater. Res. 10, 853–863 (1995)

    Article  Google Scholar 

  • McElhaney, K.W., Vlassak, J.J., Nix, W.D.: Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments. J. Mater. Res. 5, 1300–1306 (1998)

    Article  Google Scholar 

  • McFarland, A.W., Colton, J.S.: Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15, 1060–1067 (2005)

    Article  Google Scholar 

  • Mousavi, S.M., Aifantis, E.: A note on dislocation-based mode III gradient elastic fracture mechanics. J. Mech. Behav. Mater. 24, 115–119 (2005)

    Google Scholar 

  • Paulino, G., Fannjiang, A., Chan, Y.-S.: Gradient elasticity theory for mode III fracture in functionally graded materials—part I: crack perpendicular to the material gradation. J. Appl. Mech. 70, 531–542 (2003)

    Article  MATH  Google Scholar 

  • Piccolroaz, A., Mishuris, G., Radi, E.: Mode III interfacial crack in the presence of couple-stress elastic materials. Eng. Fract. Mech. 80, 60–71 (2012)

    Article  MATH  Google Scholar 

  • Poole, W.J., Ashby, M.F., Fleck, N.A.: Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Mater. 34, 559–564 (1996)

    Article  Google Scholar 

  • Qiu, Y., Wu, H., Wang, J., Lou, J., Zhang, Z., Liu, A., Chai, G.: The enhanced piezoelectricity in compositionally graded ferroelectric thin films under electric field: a role of flexoelectric effect. J. Appl. Phys. 123, 084103 (2018)

    Article  Google Scholar 

  • Shi, M., Wu, H., Li, L., Chai, G.: Calculation of stress intensity factors for functionally graded materials by using the weight functions derived by the virtual crack extension technique. Int. J. Mech. Mater. Des. 10, 65–77 (2014)

    Article  Google Scholar 

  • Stolken, J.S.: The Role of Oxygen in Nickel–Sapphire Interface Fracture. Ph.D. Dissertation, University of California, Santa Barbara (1997)

  • Thevamaran, R., Lawal, O., Yazdi, S., Jeon, S., Lee, J.-H., Thomas, E.: Dynamic creation and evolution of gradient nanostructure in single-crystal metallic microcubes. Science 354, 312–316 (2016)

    Article  Google Scholar 

  • Vardoulakis, I., Exadaktylos, G., Aifantis, E.: Gradient elasticity with surface energy: mode-III crack problem. Int. J. Solids Struct. 33, 4531–4559 (1996)

    Article  MATH  Google Scholar 

  • Wei, Y.: A new finite element method for strain gradient theories and applications to fracture analyses. Eur. J. Mech. A. Solids 25, 897–913 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Wu, H., Li, L., Chai, G., Song, F., Kitamura, T.: Three-dimensional thermal weight function method for the interface crack problems in bimaterial structures under a transient thermal loading. J. Therm. Stress. 39, 371–385 (2016a)

    Article  Google Scholar 

  • Wu, H., Ma, X., Zhang, Z., Zhu, J., Wang, J., Chai, G.: Dielectric tunability of vertically aligned ferroelectric-metal oxide nanocomposite films controlled by out-of-plane misfit strain. J. Appl. Phys. 119, 154102 (2016b)

    Article  Google Scholar 

  • Zeng, Z., Li, X., Xu, D., Lu, L., Gao, H., Zhu, T.: Gradient plasticity in gradient nano-grained metals. Extreme Mech. Lett. 8, 213–219 (2016)

    Article  Google Scholar 

  • Zhang, L., Huang, Y., Chen, Y.J., Hwang, K.C.: The mode III full-field solution in elastic materials with strain gradient effects. Int. J. Fract. 92, 325–348 (1998)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Project Nos. 11502101, 11672084, 11372086), Research Innovation Foundation of Jinling Institute of Technology, China (Project No. jit-b-201515), and Research Innovation Fund of Shenzhen City of China (Project No. JCYJ20170413104256729).

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Correspondence to Jine Li.

Appendix

Appendix

The following formulas (Chan et al. 2008; Fannjiang et al. 2002; Paulino et al. 2003) are used in deriving the hypersingular integral equations:

$$ \frac{1}{\pi }\int_{ - 1}^{1} {\frac{{U_{m} (r)\sqrt {1 - r^{2} } }}{ (r - x)}{\text{d}}r} = \left\{ {\begin{array}{*{20}l} { - T_{m + 1} (x),} \hfill & {m \ge 0 ,\left| x \right| < 1} \hfill \\ { - \, \left[ {x - \frac{\left| x \right|}{x}\sqrt {x^{2} - 1} } \right]^{m + 1} ,} \hfill & {m \ge 0 ,\left| x \right| > 1} \hfill \\ \end{array} } \right\}, $$
(43)
$$ \begin{aligned} & \frac{1}{\pi }\int_{ - 1}^{1} {\frac{{U_{m} (r)\sqrt {1 - r^{2} } }}{{ (r - x)^{3} }}{\text{d}}r} \\ & \quad = \left\{ {\begin{array}{*{20}l} {[(m^{2} + m)U_{m + 1} (x) - (m^{2} + 3m + 2)U_{m - 1} (x)]/[4(1 - x^{2} )],} \hfill & {m \ge 1 ,\left| x \right| < 1} \hfill \\ { - \frac{1}{2} (m + 1 ) { }\left[ {x - \frac{\left| x \right|}{x}\sqrt {x^{2} - 1} } \right]^{m - 1} \left[ {m\left( {1 - \frac{\left| x \right|}{{\sqrt {x^{2} - 1} }}} \right)^{2} + \frac{{\left[ {x - \frac{\left| x \right|}{x}\sqrt {x^{2} - 1} } \right]}}{{(x^{2} - 1)^{3/2} }}} \right],} \hfill & {m \ge 0 ,\left| x \right| > 1} \hfill \\ \end{array} } \right\}, \\ \end{aligned} , $$
(44)

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Li, J., Wang, B. Fracture mechanics analysis of an anti-plane crack in gradient elastic sandwich composite structures. Int J Mech Mater Des 15, 507–519 (2019). https://doi.org/10.1007/s10999-018-9425-6

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