Test methodComparison of laminate stiffness as measured by three experimental methods
Introduction
Stiffness is a fundamental material property and is typically measured from quasi-static testing on unidirectional specimens. The material is tested in different directions to obtain the full set of orthotropic elastic constants. Properties of composite laminates with fibers running in different directions can, subsequently, be calculated by laminate theory. If no material damage is present, the stiffness of a composite material is determined solely by its intrinsic properties.
Various non-destructive test methods can identify laminate stiffness indirectly, i.e. not by directly measuring the force-deformation response. These techniques can be divided into wave propagation and vibration based categories. Waves propagating along plate-like waveguides are commonly referred as Lamb waves, after the discoverer of these phenomena. High frequency (hundreds of kilohertz), dispersive phase velocities of Lamb waves are measured along different composite plate directions. Vibration based methods employ periodic, low frequency structural movements, typically up to a few thousand hertz. When testing at similar temperature and humidity, the main difference between static, vibration and Lamb wave methods is due to the strain rate and maximum strain applied to the specimens. This potentially creates issues for characterizing materials with viscous properties or with non-linear stress-strain curves.
Through thickness bulk wave measurements [1] or vibrating beam tests [2], [3] are commonly used for isotropic materials, revealing stiffness information about one material direction. Lamb wave and natural vibration based methods can also be applied to flat plates, characterizing in-plane and out-of-plane properties, making these more suitable for anisotropic materials. Recent research about stiffness determination by Lamb wave methods can be found in [4], [5], [6], [7], [8], [9], [10]. Summarized reviews and research about vibration testing of plates and its use for the elastic constant determination are presented in [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. Although advanced modelling also employs damping or attenuation, e.g. [1], [13], [25], these energy dissipation mechanisms can typically be disregarded for thermoset laminates without introducing overly large errors to the identified stiffnesses.
The two indirect methods based on vibration or wave velocity measurements are quite different in terms of ease of practical application, complexity of equipment and the number of obtained constants. Very few comparisons exist about measurements conducted on the same material with several advanced methods — e.g. [26] where resonant plate, resonant beam and tensile test results are compared. Static, vibration or Lamb wave based methods are often separately employed to evaluate composite materials. Various different composite material systems are studied and, for this reason, individual results in the literature are typically incomparable. There have not been any comparative studies to characterize composite materials by these three independent measurement techniques. In order to have confidence in individual methods the results need to agree.
A significant data processing effort is required to obtain the elastic constants from raw measurement data when working with plate specimens. This is because the elastic constants are only indirectly measured and need to be back-calculated. Parameter identification inverse problems are typically solved by an optimization algorithm working iteratively with a numerical calculation model. In recent decades, researchers have started to use different versions of genetic algorithms (GA) to identify elastic constants or to detect damage, e.g. [7], [8], [10], [24], [27], [28], [29]. Although converging slowly, GA optimization does not require an initial guess and easily escapes local minima. Simplex optimization is, however, faster, but local minima can often be obtained.
This research summarizes an experimental technique based on natural frequency measurements. The obtained elastic constants are compared to results of static and Lamb wave measurements. Unidirectional and cross-plied plates are studied. Coupons in principal material directions are measured by tensile and flexural testing. The structure of the paper is as follows. First, the specifics of test specimens are discussed, followed by descriptions of experimental set-ups. The inversion process describes how the elastic moduli are obtained from measured data. This is explained in detail for vibration measurements, whereas a previously developed method from literature is employed for the Lamb wave data. In the final sections, the results from three independent methods are compared and conclusions drawn.
Section snippets
Test specimens
Non-destructive testing of glass fiber material is less frequently reported compared to composites made from advanced fibers e.g. carbon. In addition, the current study on virgin specimens is followed by an investigation of matrix cracking, which has a more pronounced effect on glass fiber laminate stiffness. Therefore, glass/epoxy and glass/vinylester material systems are used in this work. The details of materials and production are laid out in Table 1. The plate specimens are labeled EP and
Optimization algorithms
Laminate stiffness constants are inversely calculated from natural frequencies of rectangular plates or experimental traces of Lamb wave dispersion curves. Optimization algorithms can be employed to solve these inverse problems. A global, stochastic search method based on simple genetic algorithm (SGA) is used for both of these tasks. Simplex algorithm is used only for Lamb wave dispersion curves. A very brief description of these algorithms is given below.
Quasi-static measurements
The static stiffness measurement results for virgin specimens are presented in Table 2. Length to thickness ratios L/h are reported for flexural tests. Since testing was done at very low strains, less than 0.2 %, the stress vs. strain curves were linear and Young's moduli could easily be obtained.
Experimental data
The physical properties of composite plates were measured, as reported in Table 3. The first five natural frequencies of the plates were identified from impulse testing, as seen in Table 4.
Numerical model and error estimates
A finite
Comparison of results and discussion
Table 8, Table 9 compare experimentally measured stiffnesses obtained by different measurement methods (quasi-static, natural vibrations and Lamb waves). Table 8 shows good agreement of results for quasi-static and vibration based measurements of flexural stiffness. The results in Table 6 also showed that in-plane shear stiffnesses Gxy, measured from the natural vibrations of plates with the same material but different layup ([902/02/902] and [0/904/0]), result in similar values (4.5 GPa and
Conclusions
The main conclusions from current work are summarized as follows.
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Quasi-static, natural frequency and Lamb wave based measurement methods were employed to characterize elastic stiffnesses of virgin glass-fiber laminates. The agreement for elastic and shear moduli was very good.
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Stiffnesses of cross-ply laminates can be measured either by static or vibration testing. As demonstrated, ply moduli E1, E2 can then be back-calculated form these laminate stiffnesses using approaches from [35].
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Poisson's
Acknowledgements
The research was supported by the Estonian Research Council, with institutional research funding grant IUT1917.
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