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Analytical modeling of the mixed-mode behavior in functionally graded coating/substrate systems

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Abstract

This work aims at studying the interfacial behavior of functionally graded coatings (FGCs) on different substrates, here modeled as asymmetric double cantilever beams, in line with the experimental tests. An enhanced beam theory (EBT) is proposed to treat the mixed-mode phenomena in such specimens, whose interface is considered as an assembly of two components of the coating/substrate system bonded together partially by an elastic interface. This last one is modeled as a continuous distribution of elastic–brittle springs acting along the tangential and/or normal direction depending on the interfacial mixed-mode condition. Starting with the Timoshenko beam theory, we determine the differential equations of the problem directly expressed in terms of the unknown interfacial stresses, both in the normal and tangential directions. Different distribution laws are implemented to define the functional graduation of the material in the thickness direction of the specimens, whose variation is demonstrated numerically to affect both the local and global response in terms of interfacial stresses, internal actions, energy quantities and load–displacement curves. The good accuracy of the proposed method is verified against predictions by a classical single beam theory (SBT), with interesting results that could serve as reference solutions for more expensive experimental investigations on the topic.

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Authors and Affiliations

  1. Department of Innovation Engineering, School of Engineering, University of Salento, 73100, Lecce, Italy

    Rossana Dimitri, Marco Trullo, Martina Rinaldi, Caterina Fai & Francesco Tornabene

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  1. Rossana Dimitri
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  2. Marco Trullo
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Appendix

Appendix

We here provide the extended expressions of constants \(d_{j} (j=7,....18)\), \(f_{j} (j=7,9,13)\), \(g_{j} (j=7,...,13)\), \(h_{7} \), as introduced in the closed form solution of the problem.

$$\begin{aligned} d_{7}= & {} g_{7} {T}\nonumber \\ d_{8}= & {} {N}-\frac{t_{1} }{t_{2} }d_{9} \nonumber \\ d_{9}= & {} \frac{1}{f_{9} }\left( {{M}_{1} +{N}\frac{1}{A_{11}^{\left( 1 \right) } }\left( {-B_{11}^{\left( 1 \right) } +\frac{F^{\left( 1 \right) }}{F^{\left( 2 \right) }}\frac{A_{11}^{\left( 2 \right) } }{A_{11}^{\left( 1 \right) } }\frac{1}{f_{13} }} \right) } \right) \nonumber \\ d_{10}= & {} g_{10} {T} \nonumber \\ d_{11}= & {} g_{11} {T} \nonumber \\ d_{12}= & {} d_{9} \frac{F^{\left( 1 \right) }}{A_{11}^{\left( 1 \right) } }\left( {\frac{A_{11}^{\left( 2 \right) } }{F^{\left( 2 \right) }}g_{13} -\left( {\frac{t_{1} }{t_{2} }B_{11}^{\left( 1 \right) } +\frac{B_{11}^{\left( 2 \right) } }{F^{\left( 2 \right) }}} \right) } \right) -\frac{{N}}{A_{11}^{\left( 1 \right) } }\left( {-B_{11}^{\left( 1 \right) } +\frac{F^{\left( 1 \right) }}{F^{\left( 2 \right) }}\frac{A_{11}^{\left( 2 \right) } }{A_{11}^{\left( 1 \right) } }\frac{1}{f_{13} }} \right) \nonumber \\ d_{13}= & {} d_{9} g_{13} +\frac{B_{11}^{\left( 1 \right) } }{f_{13} F^{\left( 1 \right) }}\left( {\frac{B_{11}^{\left( 1 \right) } }{A_{11}^{\left( 1 \right) } }-\frac{H_{1} }{2}} \right) \widehat{N}\nonumber \\ d_{14}= & {} d_{15} +\frac{\bar{{b}}}{F^{\left( 1 \right) }}\left( {B_{11}^{\left( 1 \right) } d_{12} +\frac{1}{2}\bar{{b}}B_{11}^{\left( 1 \right) } d_{10} -D_{11}^{\left( 1 \right) } d_{8} } \right) +\sum \limits _{i=1}^2 {\frac{t_{i} \bar{{b}}^{2}d_{7} }{2F^{\left( i \right) }}\left( {D_{11}^{\left( i \right) } +(-1)^{i}B_{11}^{\left( i \right) } \frac{H_{i} }{2}} \right) } \end{aligned}$$
(A1)
$$\begin{aligned}{} & {} +\frac{\bar{{b}}}{F^{\left( 2 \right) }}\left( {-B_{11}^{\left( 2 \right) } d_{13} -\frac{1}{2}\bar{{b}}B_{11}^{\left( 2 \right) } d_{11} +D_{11}^{\left( 2 \right) } d_{9} } \right) \nonumber \\ d_{15}= & {} -\frac{\bar{{b}}D_{11}^{\left( 2 \right) } }{F^{\left( 2 \right) }}\left( {d_{9} +\frac{t_{2} \bar{{b}}d_{7} }{2}} \right) +\frac{\bar{{b}}B_{11}^{\left( 2 \right) } }{F^{\left( 2 \right) }}\left( {d_{13} +\frac{\bar{{b}}}{2}\left( {d_{11} -\frac{H_{2} }{2}t_{2} d_{7} } \right) } \right) \nonumber \\ d_{16}= & {} d_{17} +\frac{d_{10} }{A_{44}^{\left( 1 \right) } }-\frac{d_{11} }{A_{44}^{\left( 2 \right) } } \nonumber \\ d_{17}= & {} \frac{2}{H}\left( {d_{15} -d_{14} -\frac{d_{7} }{k_{x} }-\frac{H_{1} }{2}\left( {\frac{d_{10} }{A_{44}^{\left( 1 \right) } }-\frac{d_{11} }{A_{44}^{\left( 2 \right) } }} \right) } \right) \nonumber \\ d_{18}= & {} \bar{{b}}\left( {d_{16} -\frac{d_{10} }{A_{44}^{\left( 1 \right) } }} \right) +\frac{\bar{{b}}^{3}}{6F^{\left( 1 \right) }}\left( {A_{11}^{\left( 1 \right) } \left( {d_{10} -\frac{H_{1} }{2}t_{1} d_{7} } \right) +B_{11}^{\left( 1 \right) } t_{1} d_{7} } \right) +\frac{B_{11}^{\left( 1 \right) } \bar{{b}}^{2}}{2F^{\left( 1 \right) }}\left( {d_{12} -d_{8} } \right) \end{aligned}$$
(A2)
$$\begin{aligned} f_{7}= & {} \sum \limits _{i=1}^2 {\frac{t_{i} }{F^{\left( i \right) }}\left( {B_{11}^{\left( i \right) } +(-1)^{i}\frac{H_{i} }{2}A_{11}^{\left( i \right) } } \right) \,\,\,\,} \nonumber \\ f_{9}= & {} \frac{F^{\left( 1 \right) }}{F^{\left( 2 \right) }}\frac{A_{11}^{\left( 2 \right) } }{A_{11}^{\left( 1 \right) } }g_{13} -\frac{F^{\left( 1 \right) }}{A_{11}^{\left( 1 \right) } }\left( {\frac{t_{1} }{t_{2} }\frac{B_{11}^{\left( 1 \right) } }{F^{\left( 1 \right) }}+\frac{B_{11}^{\left( 2 \right) } }{F^{\left( 2 \right) }}} \right) \nonumber \\ f_{13}= & {} \frac{1}{F^{\left( 2 \right) }}\left( {\left( {B_{11}^{\left( 2 \right) } -B_{11}^{\left( 1 \right) } \frac{A_{11}^{\left( 2 \right) } }{A_{11}^{\left( 1 \right) } }} \right) +\frac{A_{11}^{\left( 2 \right) } }{2}\left( {\frac{H_{1} }{F^{\left( 1 \right) }}+H_{2} } \right) } \right) \nonumber \\ g_{7}= & {} \frac{\frac{1}{2F^{\left( 1 \right) }F^{\left( 2 \right) }}\left( {\frac{A_{11}^{\left( 1 \right) } A_{11}^{\left( 2 \right) } }{2}H+B_{11}^{\left( 2 \right) } A_{11}^{\left( 1 \right) } -B_{11}^{\left( 1 \right) } A_{11}^{\left( 2 \right) } } \right) }{h_{7} \left( {\frac{A_{11}^{\left( 2 \right) } }{F^{\left( 2 \right) }}+\frac{t_{1} }{t_{2} }\frac{A_{11}^{\left( 1 \right) } }{F^{\left( 1 \right) }}} \right) +f_{7} \frac{1}{2}\left( {\frac{t_{1} }{t_{2} }\frac{1}{F^{\left( 1 \right) }}\left( {\frac{H_{1} }{2}A_{11}^{\left( 1 \right) } -B_{11}^{\left( 1 \right) } } \right) -\frac{1}{F^{\left( 2 \right) }}\left( {\frac{H_{2} }{2}A_{11}^{\left( 2 \right) } +B_{11}^{\left( 2 \right) } } \right) } \right) } \nonumber \\ g_{8}= & {} -\frac{t_{1}^{2} }{t_{2} f_{9} } \nonumber \\ g_{9}= & {} \frac{t_{1} }{f_{9} } \nonumber \\ g_{10}= & {} 1-\frac{t_{1} }{t_{2} }g_{11} \nonumber \\ g_{11}= & {} \frac{\left( {f_{7} g_{7} +\frac{A_{11}^{\left( 1 \right) } }{F^{\left( 1 \right) }}} \right) }{\left( {\frac{A_{11}^{\left( 2 \right) } }{F^{\left( 2 \right) }}+\frac{t_{1} }{t_{2} }\frac{A_{11}^{\left( 1 \right) } }{F^{\left( 1 \right) }}} \right) } \nonumber \\ g_{12}= & {} \frac{t_{1} }{f_{9} }\frac{F^{\left( 1 \right) }}{A_{11}^{\left( 1 \right) } }\left( {\frac{A_{11}^{\left( 2 \right) } }{F^{\left( 2 \right) }}g_{13} -\left( {\frac{t_{1} }{t_{2} }\frac{B_{11}^{\left( 1 \right) } }{F^{\left( 1 \right) }}+\frac{B_{11}^{\left( 2 \right) } }{F^{\left( 2 \right) }}} \right) } \right) \nonumber \\ g_{13}= & {} \frac{1}{f_{13} }\left( {\frac{-t_{1} }{t_{2} }\frac{1}{F^{\left( 1 \right) }}\left( {\frac{H_{1} }{2}B_{11}^{\left( 1 \right) } -D_{11}^{\left( 1 \right) } } \right) +\frac{1}{F^{\left( 2 \right) }}\left( {\frac{H_{2} }{2}B_{11}^{\left( 2 \right) } +D_{11}^{\left( 1 \right) } } \right) -\frac{1}{A_{11}^{\left( 1 \right) } }\left( {B_{11}^{\left( 1 \right) } -\frac{H_{1} }{2}A_{11}^{\left( 1 \right) } } \right) \left( {\frac{B_{11}^{\left( 2 \right) } }{F^{\left( 2 \right) }}+\frac{t_{1} }{t_{2} }\frac{B_{11}^{\left( 1 \right) } }{F^{\left( 1 \right) }}} \right) } \right) \nonumber \\ h_{7}= & {} \sum \limits _{i=1}^2 {\frac{t_{i} }{2F^{\left( i \right) }}\left( {\frac{H_{i}^{2} A_{11}^{\left( i \right) } }{4}+(-1)^{i}H_{i} B_{11}^{\left( i \right) } +D_{11}^{\left( i \right) } } \right) \,\,\,\,} \end{aligned}$$
(A3)

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Dimitri, R., Trullo, M., Rinaldi, M. et al. Analytical modeling of the mixed-mode behavior in functionally graded coating/substrate systems. Continuum Mech. Thermodyn. (2024). https://doi.org/10.1007/s00161-024-01301-8

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