- Research article
- Open Access
Experimental characterization and constitutive modeling of the non-linear stress–strain behavior of unidirectional carbon–epoxy under high strain rate loading
- Hannes Koerber^{1},
- Peter Kuhn^{1},
- Marina Ploeckl^{1},
- Fermin Otero^{2, 3},
- Paul-William Gerbaud^{4},
- Raimund Rolfes^{5} and
- Pedro P. Camanho^{2, 6}Email authorView ORCID ID profile
https://doi.org/10.1186/s40323-018-0111-x
© The Author(s) 2018
Received: 21 February 2018
Accepted: 18 June 2018
Published: 28 June 2018
Abstract
The mechanical response of IM7-8552 carbon epoxy was investigated for transverse tension and transverse tension/in-plane shear loadings at static and dynamic strain rates using transverse tension and off-axis tension specimens. The dynamic tests were carried out on a split-Hopkinson tension bar at axial strain rates from 113 to 300 \(\hbox {s}^{-1}\). With the already available off-axis and transverse compression test data for IM7-8552, a comprehensive data set is available now, which can be used for validation and calibration of numerical models. The measured axial stress–strain response was simulated using a fully 3D transversely isotropic elastic–viscoplastic constitutive model. The constitutive model represents a viscoplastic extension of the transversely-isotropic plasticity model developed by the authors (Vogler et al. in Mech Mater 59:50–64, 2013). An invariant based failure criterion is added to the model to be able to predict the strength for a given orientation and strain rate accurately. The strain rate dependency of the elastic and ultimate strength properties is introduced in the model through scaling functions. A good correlation between the measured and numerically predicted stress–strain response and failure of the specimens was achieved for all specimen types and both strain rate regimes.
Keywords
- Composites
- Carbon–epoxy
- Strain rate effects
- Viscoplasticity
- Constitutive modeling
Introduction
Over the past years, the number of applications in which fiber reinforced polymer matrix composites (FRPMCs) are used in primary aeronautical (e.g. Airbus A350 and Boeing 787) and automotive (e.g. BMW i-project) structures has significantly increased. In both areas, composite structures may be subjected to high speed loading events and the simulation of the dynamic material response is therefore relevant for several loading scenarios, such as bird, tire and hail impact and crash.
Strain rate effects and non-linear stress–strain behavior should be captured by advanced composite material models to accurately predict the initiation and evolution of damage. This requires not only appropriate material models but also new experimental techniques for model identification and validation.
In the experimental part of this work the mechanical response of a UniDirectional (UD) carbon–epoxy composite is investigated under transverse tension and combined transverse tension/in-plane shear loading at quasi-static and dynamic strain rates. The presented test data complements static and dynamic experimental results from earlier and recent studies for the same material system, namely [2], where off-axis compression and transverse compression tests were performed [3, 4], where the longitudinal compressive response was studied and [5], where the effect of dynamic loading was investigated for the intralaminar fracture toughness associated with fiber compressive failure.
In the numerical part of this work the available tension and compression tests are simulated using a new fully 3D transversely isotropic elastic–viscoplastic constitutive model, which is able to predict the experimentally observed nonlinearities under off-axis loading prior to the onset of cracking [1]. The strain-rate dependency of the plastic yields, the elastic response and the ultimate strength is accurately taken into account.
It is understood that the mechanical behavior of a multidirectional laminate may differ from the mechanical response of a UD laminate or ply. In particular crack initiation and propagation in a multidirectional laminate strongly depends on the fiber orientation, thickness and stacking sequence of the individual plies in the laminate. The numerical model presented in this paper is used and has been validated with experimental data from unidirectional laminates. Further work is required to enhance and validate the herein presented model to also predict the mechanical response of arbitrary multidirectional laminates under static and high strain rate loading.
Experimental work
Material and specimens
Quasi-static and dynamic off-axis tension and transverse tension tests were carried out, using the UD prepreg system IM7-8552. While being a high performance prepreg material system often used for primary composite structures, this toughened-epoxy composite exhibits considerable non-linear stress–strain behavior, which is used here to verify the proposed constitutive material model. In accordance with the specified heating cycle, a UD panel with a nominal thickness of 1.5 mm was manufactured by curing a 12-ply prepreg layup in a hot press. From the manufactured panel, off-axis tension specimens with fiber orientation angles of \(\theta =\) 15\(^\circ \), 30\(^\circ \) and 45\(^\circ \) and transverse tension specimens (\(\theta =\) 90\(^\circ \)) were cut on a water-cooled diamond saw.
The nominal dimensions of the 15\(^\circ \) off-axis specimens were 72 \(\times \) 8 \(\times \) 1.5 mm\(^3\) and 62 \(\times \) 8 \(\times \) 1.5 mm\(^3\) for all other specimens (Fig. 1a).
To attach the specimens to the split-Hopkinson tension bar system, slotted steel adapters with an M12 \(\times \) 1.25 outside thread were manufactured. The rectangular composites specimens were then glued into the adapters using the structural adhesive 3M Scotch-Weld DP490 (Fig. 1b).
The 12-ply UD laminate with a thickness of 1.5 mm was chosen to guarantee that the ultimate load of the strongest specimen (high strain rate test of the 15\(^\circ \) off-axis tension specimen) can be transmitted through the bond in the adapters and that failure occurs in the free gauge section of the specimen instead of the bond.
Quasi-static and dynamic test setup
Rings of 2 mm thick silicon rubber, wrapped around the impact flange at the end of the loading bar, were used for pulse shaping. The resulting ramp-shaped pulse was ideally suited for the pre-dominantly linear stress–strain behavior expected for the dynamic test, due to the relatively low plastic strains measured during the quasi-static tests.
From a previous study [2] is was observed that, depending on the off-axis angle, the strain rate acting in the fracture plane of the off-axis specimens can be significantly higher than the strain rate applied in axial (loading) direction. For an accurate comparison of the strain rate effect, the strain rates acting in the fracture plane should be similar for all tested specimen types. In this work, the strain rate in the loading direction was therefore increased with increasing off-axis angle in order to capture the corresponding shift from an in-plane shear to a transverse tensile material response. To reach a shear strain rate of 350 s\(^{-1}\) in the fracture plane of the respective off-axis specimens, the axial strain rate for the 15\(^\circ \), 30\(^\circ \) and 45\(^\circ \) off-axis specimens was adjusted to approximately 110, 180 and 300 s\(^{-1}\), respectively. For the transverse tension specimens, the axial strain rate is the governing value. Due to the striker-bar velocity of the used split-Hopkinson tension bar system, together with the chosen specimen geometry, the average attainable strain rate for the 90\(^\circ \) specimens was 271 s\(^{-1}\) and was therefore slightly lower.
Setup parameters of high speed camera
Specimen type | Frames per second \(\varvec{(\mathrm{s}^{-1})}\) | Resolution \(\varvec{(\mathrm{pixel}^2)}\) |
---|---|---|
15\(^\circ \) | 186.000 | 392 \(\times \) 96 |
30\(^\circ \) | 186.000 | 256 \(\times \) 112 |
45\(^\circ \) | 300.000 | 192 \(\times \) 80 |
90\(^\circ \) | 300.000 | 192 \(\times \) 80 |
Data reduction
Experimental results
The axial stress–strain curves of the off-axis tension and transverse tension specimens for both strain rate regimes are shown together with the predictions of the numerical model in “Results” section, Figs. 12, 13, 14 and 15.^{1}
All off-axis tension loaded specimens show non-linear stress–strain behavior at quasi-static loading and a tendency to more linear stress–strain curves under dynamic loading. For the transverse tensile specimens, the stress–strain behavior is linear at both strain rate regimes (see Fig. 15).
Quasi-static and high rate off-axis tension and transverse tension properties
Fiber angle \(\varvec{\theta }\) (\(^\circ \)) | Ultimate strength (MPa) | Shear angle d\(\varvec{\theta }\) (\(^\circ \)) | Axial strain rate \(\varvec{{\dot{\varepsilon }}_{xx}\; (\mathrm {s^{-1}})}\) | Shear strain rate \(\varvec{{\dot{\gamma }}_{12}\; (\mathrm {s^{-1}})}\) | ||
---|---|---|---|---|---|---|
Quasi-static | 15 | Mean | 364 | 1.56 | 2.1\(\cdot 10^{-4}\) | 7.8\(\cdot 10^{-4}\) |
STDV | 27.7 | 0.32 | 2.8\(\cdot 10^{-5}\) | 1.8\(\cdot 10^{-4}\) | ||
CV (%) | 7.6 | 20.5 | 13.3 | 15.3 | ||
30 | Mean | 175 | 0.82 | 2.9\(\cdot 10^{-4}\) | 6.0\(\cdot 10^{-4}\) | |
STDV | 12.5 | 0.09 | 7.7\(\cdot 10^{-6}\) | 4.2\(\cdot 10^{-5}\) | ||
CV (%) | 7.2 | 11.0 | 2.7 | 7.0 | ||
45 | Mean | 114 | 0.35 | 2.6\(\cdot 10^{-4}\) | 3.8\(\cdot 10^{-4}\) | |
STDV | 9.5 | 0.03 | 2.9\(\cdot 10^{-5}\) | 3.3\(\cdot 10^{-5}\) | ||
CV (%) | 8.3 | 8.6 | 11.3 | 8.6 | ||
90 | Mean | 62 | – | 2.8\(\cdot 10^{-4}\) | – | |
STDV | 14.0 | – | 1.7\(\cdot 10^{-5}\) | – | ||
CV (%) | 22.6 | – | 5.9 | – | ||
High rate | 15 | Mean | 503 | 1.45 | 113 | 358 |
STDV | 10.7 | 0.20 | 19 | 51 | ||
CV (%) | 2.1 | 13.8 | 16.8 | 14.2 | ||
30 | Mean | 223 | 0.71 | 177 | 337 | |
STDV | 3.3 | 0.05 | 10 | 10 | ||
CV (%) | 1.4 | 7.0 | 5.6 | 3.0 | ||
45 | Mean | 148 | 0.34 | 300 | 410 | |
STDV | 16.7 | 0.04 | 44 | 64 | ||
CV (%) | 11.3 | 11.8 | 14.7 | 15.6 | ||
90 | Mean | 79 | – | 271 | – | |
STDV | 11.9 | – | 13 | – | ||
CV (%) | 15.1 | – | 4.8 | – |
For the transverse tension specimens, pronounced scatter in the measured strength was found for both strain rate regimes. Nevertheless, the calculated mean value for quasi-static loading coincides well with the transverse strength listed in the HexPly IM7-8552 material data sheet [10].
Constitutive model
The strain rate dependency of the elastic and ultimate strength properties is introduced in the present model by fitting the experimental data using suitable scaling functions. Then, the material property for a given strain rate is the quasi-static one multiplied by the scaling function. A simple function i.e. \(f({\dot{\varvec{\varepsilon } }})= 1 + \sqrt{K{\dot{\varvec{\varepsilon } }}}\), where the rate dependency of the properties depends on an appropriate selection of the parameter K was proposed by Wiegand [19].
Failure criterion
When \(r = \text {max} [r_{\text {matrix}},r_{\text {fiber}}]\) reaches a value of zero (i.e. \(r_{\text {matrix}}(\varvec{\sigma } ,\varvec{A}) = 0\) or \(r_{\text {fiber}}(\varvec{\sigma } ,\varvec{A}) = 0\)), it means that one of the failure conditions is fulfilled and then, the material fails completely. These two failure conditions allow the model to predict the ultimate strength in the composite and also to differentiate which component-dominated failure mechanism happens first.
The IQC is tested with the parameters given in Table 3. The transverse shear strength is assumed identical to the transverse tensile strength for the same reasons as evoked by Camanho et al. in [24].
On Fig. 6 can be seen the experimental failure surfaces for both quasi-static and dynamic compression and tensile tests (blue and pink crosses)t. Also shown in that figure are the failure surfaces predicted by the IQC, both quasi-static and dynamic loadings. They were plotted assuming values of the scale function of \(f_u(\text {quasi-static}) = 1.0\) and \(f_u(\text {dynamic}) = 1.4\) accordingly to the experimental results (see “Calibration of the scaling functions” section).
Red and yellow crosses indicate the simulated failure points using one specimen-sized single element and confirm the proper functioning of the criterion in the finite elements code by matching the IQC surfaces. The tiny variation between the plotted IQC surfaces and the simulated points is due to the use of the scaling functions (for the 15\(^\circ \) tensile test as an example, the strain rates led to \(f_u(4 \cdot 10^{-4}\, \mathrm {s^{-1}}) = 1.0146\) for the quasi-static tests, \(f_u(122\, \mathrm {s^{-1}}) = 1.3427\), for the dynamic tests) which modifies the strength parameters in each case (different loading speeds) while the surfaces are computed for one value of \(f_u\) each.
The IQC uses the same formulation as the yield criterion for the matrix failure which leads to similar shaped yield and failure surface. In “Results” section are shown the failure points simulated on stress–strain diagrams with the further described meshes.
Material data and calibration of the viscoplastic parameters
Hereafter, the material data preparation using the test data and the calibration of the viscoplastic model parameters are briefly discussed.
Nonlinear behavior and hardening curves
The transverse compression hardening curve was directly obtained from the transverse compression test [2]. The transverse shear hardening curve is assumed to be similar to the in-plane shear behavior [21], because the transverse shear behavior is not very sensitive in the off-axis and transverse compression and tension tests. For a more detailed examination, the transverse shear behavior could be calibrated using 3 point bending tests with a relatively low span length.
Calibration of viscoplasticity parameter m and \(\upeta \)
The calibration of the two viscoplastic parameters m and \(\upeta \) of the Perzyna type overstress model introduced in Eq. (18) is briefly discussed. For the current material IM7-8552, just two strain rate regimes are tested. Consequently, the parameter m is set to \(m=1\) and an approximately linear dependency of the viscoplastic yield stress on the logarithmic strain rate can be modelled using the parameter \(\upeta \). Although such a linear dependency from the logarithmic strain rate for carbon–epoxy is reported by [28], it cannot be assumed for arbitrary matrix materials. Thermoplastics for instance, or thermoplastic toughened resins exhibit a nonlinear dependency on the logarithmic strain rate [21]. To account for this nonlinear dependency, the parameter m of the viscoplastic model can be used. Therefore, test data for at least 3 strain rate regimes are required for a calibration of the parameter m.
To calibrate the remaining viscoplastic parameter \(\upeta \), the axial stress–strain curves from the 45\(^\circ \) off-axis compression tests are used. These tests were performed at two different strain rates, at quasi-static rate (\(0.0004\, \hbox {s}^{-1}\)) and at an axial strain rate of approximate \(280\, \hbox {s}^{-1}\). The calibration is done with a simple approximation of \(\upeta \) using a single element test and finally checked using a fine mesh, whereby the parameter \(\upeta \) = 4.0E−4 \(\hbox {Ns\,mm}^{2}\) gives the best approximation. It is assumed that the viscosity is independent from the hydrostatic pressure and, hence, the strain rate dependency on the yielding behavior is similar both in tension and in compression.
Calibration of the scaling functions
Elastic properties
Results
Meshes and boundary conditions
Tension results
The stress–strain curves become more linear under dynamic loading (see Figs. 12, 13, 14, 15). The present model provides strain–stress curves which match the experimental ones accurately. The viscous effects noticed in both, elastic and plastic range are correctly predicted for the four orientations, validating the model for tensile simulations.
Compression results
Figures 16, 17, 18, 19, 20 and 21 show the measured and predicted axial stress–strain curves for 15\(^\circ \), 30\(^\circ \), 45\(^\circ \), 60\(^\circ \), 75\(^\circ \) off-axis compression and 90\(^\circ \) transverse compression under quasi-static and dynamic loading. Generally, a good prediction of the nonlinear behavior has been achieved for both, quasi-static and dynamic loading case. For compression loading as for the tensile loading, the viscous effects are well taken into account by the presented model and the scaling functions, allowing a correct prediction of the stress–strain behavior.
Summary and conclusion
The viscoplastic behavior of the IM7-8552 carbon–epoxy was investigated using off-axis and transverse tension and compression tests under quasi-static (\(0.0004\, \hbox {s}^{-1}\)) and dynamic (113–\(300\, \hbox {s}^{-1}\)) loading. The available off-axis and transverse tension and compression test data was simulated using a fully 3D transversely isotropic elastic–viscoplastic constitutive model, able to predict nonlinearities under off-axis loading conditions prior to the onset of cracking. A representative in-plane shear curve for pure shear loading was deduced from the available off-axis tension and compression test data and the influence of triaxiality on the in-plane shear characterization was discussed. The calibration of the parameter \(\upeta \) of the presented viscoplastic model was shown, assuming an approximated linear dependency of the yield stress on the logarithmic strain rate, since experimental data was available from only two strain rate regimes. The strain rate dependency of the elastic moduli and the ultimate strength properties was introduced in the proposed model by fitting the available experimental data with suitable scaling functions. For the prediction of failure, an Invariant based Quadratic failure Criterion (IQC) based on the work of Vogler et al. [11, 24] was implemented. The model provides good predictions of the stress–strain state, regardless of the size of the finite elements, meaning that using a single element provides excellent quality results in a very short time. It therefore makes it a very efficient model for stress–strain predictions at the ply scale. The correlation between experimental data and the stress–strain response predicted by the constitutive model was well achieved for all specimen types and for both strain rate regimes. A shift from nonlinear to predominantly linear stress–strain response was observed for the dynamic off-axis tensile tests. However, the importance of assuming a nonlinear hardening curve for transverse tension beyond the point of failure in order to predict combined in-plane shear—transverse tensile stress states was discussed. The failure observed in the experiments presented here is caused by matrix failure. An good prediction of the stress level at failure was achieved for both tension and compression simulations.
It is noted that the experimental results of the off-axis and transverse tension tests were first shown in [9]. Due to a later improved calibration of the split-Hopkinson tension bar, the stresses had to be corrected and were about 5% lower than initially presented in [9]. For completeness, the corrected data is used in this work for Table 2, Figs. 4, 12, 13, 14, 15, 16, 17, 18, 19, 20 and 21.
Declarations
Author's contributions
HK, PK and MP defined, executed and analysed the experimental tests. FO and P-WG defined and implemented the numerical model. RR and PPC supervised the development of the numerical model. All authors contributed to the text presented in the paper. All authors read and approved the final manuscript.
Acknowledgements
This paper is dedicated to the memory of Dr. Matthias Vogler, an exceptional young scientist that sadly left us too soon. P. P. Camanho and F. Otero would like to acknowledge the funding of Project NORTE-01-0145-FEDER-000022—SciTech—Science and Technology for Competitive and Sustainable Industries, co-financed by Programa Operacional Regional do Norte (NORTE2020), through Fundo Europeu de Desenvolvimento Regional (FEDER).
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Not applicable.
Consent for publication
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Ethics approval and consent to participate
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Funding
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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