Wave Propagation and Dynamic Correction Factors for Composite Structures

Abstract

Shear and normal correction factors are used within first-order Equivalent Single Layer theories in order to improve the description of the transverse strains and the accuracy of the solutions of static and dynamic problems in layered structures. Dynamic correction factors may be derived by imposing that certain wave propagation frequencies, e.g., the first cut-off frequencies of the thickness-modes as originally proposed by Mindlin, match those of three-dimensional elasticity. The exact frequencies can be derived only through complex computational procedures and, as a consequence, correction factors obtained for homogeneous plates are typically used also for layered plates, at the expense of accuracy. In this chapter, we consider cross-ply laminated plates with arbitrary layups and elastic constants and show how using a homogenized zig-zag structural theory allows the accurate closed-form derivation of the first thickness-modes and the dynamic correction factors of first-order equivalent single layer theories. The correction factors accurately reproduce those obtained by matching the exact elasticity solutions in highly inhomogeneous bilayer media.

Appendices

Appendix 1

The coefficients in the expressions of the macro-scale displacement field of the homogenized model assuming perfect bonding of the layers are [4]:

$$ R_{S22}^{k} (x_{3} ) = \sum\limits_{i = 1}^{k - 1} {\left[ \begin{aligned} \hfill \\ \hfill \\ \end{aligned} \right.} \varLambda_{22}^{{\left( {1;i} \right)}} \left( {x_{3} - x_{3}^{i} } \right)\left. \begin{aligned} \hfill \\ \hfill \\ \end{aligned} \right] $$

$$ R_{N22}^{k} (x_{3} ) = \sum\limits_{i = 1}^{k - 1} {\left[ {\varLambda_{22}^{{\left( {1;i} \right)}} x_{3}^{1} + \sum\limits_{l = 2}^{i} {\varLambda_{22}^{{\left( {l;i} \right)}} \left( {x_{3}^{l} - x_{3}^{l - 1} } \right)\left( {1 + \sum\limits_{j = 1}^{l - 1} {\varLambda_{33}^{\left( j \right)} } } \right)} } \right.} \left. \begin{aligned} \hfill \\ \hfill \\ \end{aligned} \right]\left( {x_{3} - x_{3}^{i} } \right) $$

$$ \varLambda_{22}^{{\left( {1;i} \right)}} = {}^{(1)}C_{44} \left( {\frac{1}{{{}^{(i + 1)}C_{44} }} - \frac{1}{{{}^{(i)}C_{44} }}} \right) $$

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$$ \varLambda_{33}^{\left( i \right)} = - {}^{(1)}C_{33} \left( {\frac{{{}^{(i + 1)}C_{33} - {}^{(i)}C_{33} }}{{{}^{(i + 1)}C_{33} {}^{(i)}C_{33} }}} \right) $$

The coefficients in the equations of the homogenized model which describe the thickness-shear and -stretch modes assuming perfect bonding of the layers are given below [3, 30]:

$$ (R_{0}^{{}} \,,R_{1}^{{}} ,R_{2}^{{}} ,R_{1}^{N} ,R_{2}^{N} ) = \int\limits_{h} {\rho (1,x_{3} ,x_{3}^{2} ,x_{3} ,x_{3}^{2} )dx_{3} + (0,R^{0S} ,2R^{1S} + R^{S2} ,R^{0N} ,2R^{1N} + R^{N2} } ) $$

$$ A_{44}^{{}} \, = k_{44} C_{44}^{P}, $$

$$ A_{33}^{{}} \, = k_{3} C_{33}^{P}, $$

$$ C_{44}^{P} = \sum\limits_{k = 1}^{n} {{}^{(k)}C_{44}^{{}} \int\limits_{{x_{3}^{k - 1} }}^{{x_{3}^{k} }} {\left( {1 + \sum\limits_{i = 1}^{k - 1} {\varLambda_{22}^{{\left( {1;i} \right)}} } } \right)^{2} } dx_{3} }, $$

$$ C_{33}^{P} = \sum\limits_{k = 1}^{n} {{}^{(k)}C_{33}^{{}} \int\limits_{{x_{3}^{k - 1} }}^{{x_{3}^{k} }} {\left( {1 + \sum\limits_{i = 1}^{k - 1} {\varLambda_{33}^{\left( j \right)} } } \right)^{2} } dx_{3} } , $$

$$ R_{{}}^{rS} = \sum\limits_{k = 1}^{n} {{}^{(k)}\rho \int\limits_{{x_{3}^{k - 1} }}^{{x_{3}^{k} }} {\left( {x_{3} } \right)^{r} \sum\limits_{i = 1}^{k - 1} {\left[ {\varLambda_{22}^{{\left( {1;i} \right)}} \left( {x_{3} - x_{3}^{i} } \right)} \right]} dx_{3} } }, $$

$$ R_{{}}^{S2} = \sum\limits_{k = 1}^{n} {{}^{(k)}\rho \int\limits_{{x_{3}^{k - 1} }}^{{x_{3}^{k} }} {\left( {\sum\limits_{i = 1}^{k - 1} {\left[ {\varLambda_{22}^{{\left( {1;i} \right)}} \left( {x_{3} - x_{3}^{i} } \right)} \right]} } \right)^{2} dx_{3} } } , $$

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$$ R_{{}}^{rN} = \sum\limits_{k = 1}^{n} {{}^{(k)}\rho \int\limits_{{x_{3}^{k - 1} }}^{{x_{3}^{k} }} {\left( {x_{3} } \right)^{r} \sum\limits_{i = 1}^{k - 1} {\varLambda_{33}^{\left( i \right)} \left( {x_{3} - x_{3}^{i} } \right)} dx_{3} } }, $$

$$ R^{N2} = \sum\limits_{k = 1}^{n} {{}^{(k)}\rho \int\limits_{{x_{3}^{k - 1} }}^{{x_{3}^{k} }} {\left\{ {\sum\limits_{i = 1}^{k - 1} {\varLambda_{33}^{\left( i \right)} \left( {x_{3} - x_{3}^{i} } \right)} } \right\}^{2} dx_{3} } }. $$

Under the assumptions of the first-order equivalent single layer theories the coefficients above simplify as:

$$ (R_{0}^{{}} \,,R_{1}^{ESL} ,R_{2}^{ESL} ,R_{1}^{N,ESL} ,R_{2}^{N,ESL} ) = \int\limits_{h} {\rho (1,x_{3} ,x_{3}^{2} ,x_{3} ,x_{3}^{2} )dx_{3} }, $$

$$ A_{44}^{ESL} = k_{44}^{ESL} \int\limits_{h} {C_{44} dx_{3} }, $$

$$ A_{33}^{ESL} = k_{33}^{ESL} \int\limits_{h} {C_{33} dx_{3} } .$$

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Appendix 2

The elasticity frequency equation of the thickness-shear mode of propagation in an isotropic bilayer with f and c the thicknesses of the layers and $G_{f}$ and $G_{c}$ the corresponding shear moduli in the plane $x_{2} - x_{3}$, has been derived in [8, 16] as the limit for wavelengths approaching infinity of the Rayleigh–Lamb equation:

$$ \beta_{f} \sin (\beta_{f} f)\cos \beta_{c} c + \left( {{{G_{c} } \mathord{\left/ {\vphantom {{G_{c} } {G_{f} }}} \right. \kern-0pt} {G_{f} }}} \right)\beta_{c} \sin (\beta_{c} c)\cos (\beta_{f} f) = 0 $$

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with $\beta_{i}^{2} = \omega^{2} \rho_{i} /G_{i}$. The exact cut-off frequency of the first thickness-stretch mode for a bilayer with $M_{f}$ and $M_{c}$ the P-waves moduli are defined by the first root of the frequency equation:

$$ \left( {{{\alpha_{c} } \mathord{\left/ {\vphantom {{\alpha_{c} } {\alpha_{f} }}} \right. \kern-0pt} {\alpha_{f} }}} \right)\beta_{f}^{2} \sin (\alpha_{f} f)\cos (\alpha_{c} c) + \left( {{{G_{c} } \mathord{\left/ {\vphantom {{G_{c} } {G_{f} }}} \right. \kern-0pt} {G_{f} }}} \right)\beta_{c}^{2} \sin (\alpha_{c} c)\cos (\alpha_{f} f) = 0 $$

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with $\alpha_{i}^{2} = \omega^{2} \rho_{i} /M_{i}$. The P-waves modulus in an isotropic material is given by $M_{i} = C_{33i} = \frac{{E_{i} \left( {1 - \nu_{i} } \right)}}{{\left( {1 + \nu_{i} } \right)\left( {1 - 2\nu_{i} } \right)}}$.