Stiffness identification of boundary conditions by using thin-layer element for parameterization

2.1. Thin-layer element

The size of thin-layer element is $l_1$×$l_2$×$d$, according to the principle of virtual displacement derived the following equation:

$l_1$ and $l_2$ are the lengths of the thin-layer elements in the $x$ and $y$ directions in the local coordinate system, and $d$ is the $z$-direction size. Stiffness matrix [$K$] of thin-layer element is calculated by isoparametric transformation:

Fig. 1 shows the isoparametric transformation of thin-layer element. Where $\xi$, $\eta$, $\zeta$ are natural coordinate symbols, [$J$] is the Jacobi ratio matrix, and when the coordinates of the natural and local coordinates are consistent, which has the following form:

The following numerical expression for stiffness matrix [$K$] of thin-layer element can be obtained by the calculation of two-node Gaussian integration method:

where $w_{\xi }$, $w_{\eta }$, and $w_{\zeta }$ are the Gaussian integral weight function.

It is assumed that the thickness $d$ of the thin-layer element is much smaller than the feature sizes $l_1$ and $l_2$. In this case, the in-plane strain components (${\epsilon }_x$, ${\epsilon }_y$, ${\gamma }_{xy}$) and the stress components (${\sigma }_x$, ${\sigma }_y$, ${\tau }_{xy}$) of the element are ignored. Which leads to the strain component ${\epsilon }_x={\epsilon }_y={\gamma }_{xy}\approx$ 0. Therefore, only three strain components at the Gaussian point of the thin-layer element are not zero. The strain component can be simplified to $\epsilon ={\left[{\epsilon }_z{\gamma }_{yz}{\gamma }_{zx}\right]}^T$. According to the previous analysis, assuming that the normal and tangential contact properties of the interface are independent and the two tangential contact properties are consistent.

Fig. 1Isoparametric change of thin-layer element

In the finite element analysis, according to the basic theory of thin-layer element, the constitutive equation of the material will be transformed to the following form:

In this equation, the normal elastic constant and the tangent elastic constant are correlated. The selection of different thickness of thin layer element have a great influence on the calculation results. Desai et al. [17] recommended the width to thickness ratio for obtaining the exact result, the ratio is expressed as:

Calculating the contact stiffness is one of the key problems on thin-layer element for simulating the bolt connected structures. When the contact surface maintains a linear bonding state. The constitutive relation of the material can be integrated into the thin-layer element and the finite element dynamic analysis of the connection structure can be carried out. However, there are some shortcomings that need to be tested separately for the contact stiffness in different directions, and the requirements for the test device and the test method.

2.2. Parameter identification

In this paper, material parameter identification of thin-layer elements is transformed as an optimization problem. The parameters to be identified are the elastic modulus $E$ and the shear modulus $G$ of the thin-layer element. And the minimum value of the objective function is expressed:

where ${\epsilon }_f=z^m-z^a\left(p\right)$ is the discrepancy between the experimental data and the numerical results. $W$ is the weighted matrix, which reflects the relative weight of modal parameters. In general, the expression of $W$ is the identity matrix $\mathbf{I}$ or ${\left[\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(z^m\right)\right]}^{-2}$. Until the numerical modal data satisfy the requirement of precision, the accurate contact surface material parameters are obtained. The procedure of the above optimization problem is shown in Fig. 2.

Fig. 2The flow chat of elastic parameters identification

3.1. Equivalent modeling

Neglecting lateral shearing stress, the honeycomb core is equivalent to an orthotropic material of uniform thickness. As a result, the face sheets are modeled as the upper and the lower panels, the core is equivalent to a solid structure. The interlaminar shear effect caused by the weak layer has an important influence on the macroscopic mechanical properties of the sandwich material. Therefore, the influence of the adhesive layer on the dynamic characteristics of the honeycomb sandwich composite can’t be neglected. Fig. 4 shows the finite element model of honeycomb plate. Clamped boundary condition is modeled by isotropic thin-layer elements.

3.2. Parameter identification

The influence of different weighting matrices on the results of parameter identification is studied by establishing a thin-layer element with $R=$ 100. After the identification, the accuracy of the inherent frequency is greatly improved. The error of the first three order natural frequency is less than 2.63 %, the convergence error can be effectively reduced by $W_{\epsilon }={\left[\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(z^m\right)\right]}^{-2}$.

Based on the above results, parameter identification are carried out when the $R$ of thin layer elements is 10, 50, 100, 500, respectively. The convergence error is reduced with the increase of the width and thickness ratio. However, the more iterations are needed for convergence. Table 2 is the comparison of calculation natural frequency and parameter identification results of thin-layer element under different $R$.

In this case, the stiffness properties of the contact surface are determined by the elastic parameters and the thickness of the thin layer element.

Fig. 4Numerical model of the clamped honeycomb plate

Fig. 5Frequency convergence under different R: a) R= 10, b) R= 50, c) R= 100, d) R= 500

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b)

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d)