Debonding Resistance Evaluation in Virtual Testing of Sandwich Specimens

Abstract

Three different experimental methods used in fracture testing sandwich panels are studied in the context of providing an assessment of face sheet-to-core interface strength. For this reason, strain energy release rate (ERR), complex stress intensity factors (SIFs) and mode mixity phase angle are computed. The analytical models exploit both the framework of linear elastic fracture mechanics (LEFM) in a combination of analytical considerations and numerical results and one-dimensional (1D) beam theories, whereas the finite element predictions are conducted using the capabilities of the ABAQUS package and a standalone subroutine developed in MATLAB environment for post-processing the results of two-dimensional (2D) finite element analysis. The results presented in this research allow drawing conclusions on the accuracy of fracture analysis predictions for each of the three different specimens by comparing 2D numerical calculations with semi-analytical results and 1D analytical solutions.

Appendices

Appendix 1

Following the original notations in Østergaard and Sørensen (2007); Andrews and Massabò (2007); Barbieri et al. (2018), the geometrical and material dimensionless parameters of the sandwich beam cross-section are defined as follows:

$$\begin{array}{*{20}c} {\eta = \frac{{h_{f} }}{{h_{c} }}} \\ {\alpha = \frac{{\Sigma - 1}}{{\Sigma + 1}},{\text{with}}\;\Sigma = \frac{{\overline{E}_{f} }}{{\overline{E}_{c} }}} \\ {\beta = \frac{{G_{f} \left( {\overline{\kappa }_{c} - 1} \right) - G_{c} \left( {\overline{\kappa }_{f} - 1} \right)}}{{G_{f} \left( {\overline{\kappa }_{c} + 1} \right) + G_{c} \left( {\overline{\kappa }_{f} + 1} \right)}}} \\ {\varepsilon = \frac{1}{2\pi }\ln \left( {\frac{1 - \beta }{{1 + \beta }}} \right)} \\ \end{array}$$

(2.15)

where $\overline{E}_{i} = E_{i}$ for plane stress and $\overline{E}_{i} = \frac{{E_{i} }}{{1 - \nu_{i}^{2} }}$ for plane strain with $E_{i}$ and $\nu_{i}^{{}}$ the Young's modulus and the Poisson's ratio of the layer i = {f, c} and the shear modulus, $G_{i} = \frac{{E_{i} }}{{2(1 + \nu_{i} }})$; α and β stand for the Dundur's parameters with $\overline{\kappa }_{i} = \frac{{3 - \nu_{i} }}{{1 + \nu_{i} }}$ for plane stress and $\overline{\kappa }_{i} = 3 - 4\nu_{i}$ for plane strain; $\varepsilon$ is the oscillatory index.

The dimensionless distance of the neutral axis of the substrate at crack tip cross-section of unit width is defined by

$$\tilde{e}_{s} = \frac{{e_{s} }}{{h_{f} }} = \frac{{\overline{E}_{f} h_{f} }}{{\overline{E}_{c} h_{c} + \overline{E}_{f} h_{f} }}\frac{{h_{f} + h_{c} }}{{2h_{f} }} = \frac{{\Sigma \left( {\eta + 1} \right)}}{{2\left( {\Sigma \eta + 1} \right)}},$$

(2.16)

whereas the dimensionless bending stiffnesses of the substrate and fully bonded part (the base) of unit width cross-section are given by

$$\begin{aligned} \widetilde{D}_{s} & = \frac{{D_{s} }}{{\overline{E}_{f} h_{f}^{3} }} = \frac{1}{12} + \left( {\frac{\eta + 1}{{2\eta }} - \tilde{e}_{s} } \right)^{2} + \frac{1}{{\Sigma \eta }}\left( {\frac{1}{{12\eta^{2} }} + \tilde{e}_{s}^{2} } \right) \\ \widetilde{D}_{b} & = \frac{{D_{b} }}{{\overline{E}_{f} h_{f}^{3} }} = \left( {\frac{1}{6} + \frac{1}{{2\eta^{2} }}\left( {\eta + 1} \right)^{2} } \right) + \frac{1}{{12\Sigma \eta^{3} }} \\ \end{aligned}$$

(2.17)

The coefficients in (2.1) are calculated in the forms:

$$\begin{array}{*{20}c} {C_{1} = \frac{{\Sigma \eta }}{{1 + 2\Sigma \eta }}} \\ {C_{2} = \frac{1}{{\widetilde{D}_{b} }}\left( {\frac{1}{2} + \frac{1}{2\eta }} \right)} \\ {C_{3} = \frac{1}{{12\widetilde{D}_{b} }}} \\ \end{array}$$

(2.18)

Appendix 2

The positive dimensionless functions $f_{M} \left( {\Sigma ,\eta } \right)$, $f_{P} \left( {\Sigma ,\eta } \right)$ and the phase angle $\gamma_{M} \left( {\Sigma ,\eta } \right)$ define the energy release rates for arbitrary combinations of bending moments and axial forces (Fig. 2b). In terms of the dimensionless parameters they take the form (Barbieri et al. 2018):

$$\begin{array}{*{20}c} {f_{M} \left( {\Sigma ,\eta } \right) = \left( {\frac{1}{2}\left( {12 + \frac{1}{{\widetilde{D}_{s} }}} \right)} \right)^{\frac{1}{2}} } \\ {f_{P} \left( {\Sigma ,\eta } \right) = \left( {\frac{1}{2}\left( {1 + \frac{\Sigma \eta }{{1 + \Sigma \eta }} + \frac{1}{{\widetilde{D}_{s} }}\left( {\frac{1}{2} + \frac{1}{2\eta } + \tilde{e}_{s} } \right)^{2} } \right)} \right)^{\frac{1}{2}} } \\ {\gamma_{M} \left( {\Sigma ,\eta } \right) = \sin^{ - 1} \left( {\frac{1}{{2\widetilde{D}_{s} f_{P} f_{M} }}\left( {\frac{1}{2} + \frac{1}{2\eta } + \tilde{e}_{s} } \right)} \right)} \\ \end{array}$$

(2.19)