Generalized Stress–Strain Curves for IBII Tests on Isotropic and Orthotropic Materials
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Abstract
This article presents a particular use of the Virtual Fields Method to exploit the results of ImageBased Inertial Impact (IBII) tests. This test consists on an edgeon impact of a freestanding thin flat rectangular coupon. The specimen response is recorded using an ultrahigh speed camera filming the deformation of a grid pattern printed at its surface. From these images, displacement fields are derived, from which strain and acceleration can be obtained. The Virtual Fields Method makes use of the acceleration fields to derive stress information. Until now, a very simple ‘stressgauge’ approach was used that could only provide relevant stressstrain information if the test was predominantly uniaxial. The alternative was to use the full inverse approach with the Virtual Fields Method but this would not allow the same degree of data understanding as the ‘stressgauge’ approach. This article proposes an extension to this ‘stressgauge’ approach for fully multiaxial tests. The equations are first derived and then validated using simulated and experimental IBII test data on isotropic and orthotropic materials.
Keywords
High strain rate High speed imaging Grid method Virtual fields method CompositesIntroduction
Many engineering materials exhibit timedependent properties. In the case of fast transient loadings like crash, blast or impact, the behaviour of a given structure can only be predicted if an accurate representation of the strain rate dependence of its constituent materials is available. While for certain classes of problems, high rate behaviour can be deduced from low rate tests [1], generally, this is not available and high strain rate tests have to be conducted. One of the main difficulties in high rate testing concerns the measurement of the applied load. For dropweight and fast hydraulic machines, the propagation of transient waves in the load cell causes ringing which prevents accurate measurement of the load sustained by the specimen. In Hopkinson bar systems, the transient waves propagating in the specimen violate the necessary quasistatic equilibrium required to perform the data reduction [2].
The advent of modern ultrahigh speed imaging systems based on single sensors [3, 4, 5, 6] in the late 2000s has democratized the recording of videos in the MHz range, making it possible to record images encoding the deformation associated with elastic mechanical waves arising from impact load. Using surface patterns, like a random speckle or a regular crosshatch grid, at the surface of the test piece, it is then possible to extract space and time resolved displacements using image correlation [7] or phaseshifting [8] image processing techniques. One of the key features that this enables is the derivation of time resolved acceleration maps. Combined with the dynamic mechanical equilibrium equations of a continuous solid, for instance expressed by the Principle of Virtual Work, acceleration maps provide direct stress information thus avoiding the need for external load measurement. The load information becomes embedded into the images and both stress and strain can be simultaneously extracted from them [9].
The first publications to exploit this new paradigm appeared in the late 2000s [10, 11] and it has flourished in the later part of the 2010s [12, 13, 14, 15, 16, 17]. A modern implementation of this idea uses purely inertial test configurations, inspired by the spalling tests in [13]. The principle was first published in 2014 [9] on an elastically isotropic material. It was then refined and extended to several classes of materials under the name ‘ImageBased Inertial Impact’ (IBII) test [18, 19, 20, 21]. This test consists of an edgeon impact of a thin rectangular test specimen with all other edges free from any constraints. This sends a compression wave into the specimen which reflects off the free edge as a tensile wave. The pulse length can be designed either to load and unload the specimen in compression [9, 18] or to develop enough tension after wave reflection so that tensile fracture occurs [19, 20, 21].
In this new class of high strain rate tests, two main approaches have been used to identify the constitutive parameters. The first one is based on a full implementation of the Virtual Fields Method (VFM), with parametrized constitutive models [12, 18, 22]. The second one uses a reconstruction of the average longitudinal stress across a vertical section of the test specimen to build up stressstrain curves, employing rigidbodylike virtual fields in the VFM. This has been referred to in the past as the ‘stressgauge’ approach. In the case of isotropic materials, this requires the a priori knowledge of Poisson’s ratio or the assumption of uniaxial stress [9]. Poisson’s ratio can be obtained using the full VFM analysis [19], but it would be convenient to enable both stiffness components to be identified with a stressgauge approach, which is the object of the present paper. For orthotropic materials tested ‘onaxis’ (i.e. with orthotropy axes aligned with the specimen axes), this is also possible. However, when a unidirectional composite is tested in the \(90^{\circ }\) orientation, Poisson’s effect can be neglected because of the high stiffness at \(0^{\circ }\) [19, 21]. The advantage of this second approach is that it leads naturally to spatially resolved properties, and provides an easy way to plot 2D curves from which the departure from possible linearity / elasticity is easily appraised.
In classical uniaxial tests, it is possible to plot stress versus strain to directly obtain Young’s modulus from the slope of the response in the linear range. However, this is generally not possible for multiaxial stress states when the stress field is unknown a priori, and inverse identification has to be used. Thanks to the possibility of reconstructing stress from acceleration in transient dynamics, the present article proposes a generalization of this stressstrain curve concept to the case of 2D multiaxial stress states for both isotropic and orthotropic IBII tests. The equations are first derived for isotropy. They are then extended to orthotropic properties by introducing the concept of an ‘angled stressgauge’ for the first time. This is validated on FE and experimental data for isotropic and orthotropic offaxis IBII tests. The experiments consider a tungsten carbide cermet (isotropy, [20]) and a \(45^{\circ }\) carbon/epoxy unidirectional composite. It should be noted that the quantities derived here can be applied to alternative inertial tests like the ImageBased Ultrasonic Shaking (IBUS) test. This test relies on an ultrasonic vibration to excite a thin rectangular test piece on its first resonance frequency to introduce high rate deformation [23].
Generalized Stress–Strain Curves in Isotropic Elasticity
The objective of this section is to present the derivation of equivalent stressstrain quantities that allow for the construction of generalized stressstrain curves for the case of isotropic elasticity.
Theoretical Derivation
Validation
Validation on Simulated Data
Identification results from FE simulated isotropic data
Stiffness in GPa  \(Q_{11}\)  \(Q_{12}\) 

Reference  54.95  16.48 
\(1+2\)  54.92  16.34 
\(1+2\) difference (%)  \(\)0.04  \(\)0.83 
\(1+3\)  54.84  16.11 
\(1+2\) difference (%)  \(\)0.18  \(\)2.26 
\(2+3\)  54.62  16.07 
\(2+3\) difference (%)  \(\)0.57  \(\)2.48 
Validation on Experimental Data
The equations above are now validated using experimental data from [20]. IBII tests were performed on Tungsten Carbide cermets with Cobalt and Nickel binders. The specimen considered here is 2WCF13Co as reported in Table 5 in [20], which has 13% Cobalt binder. All experimental details are provided in this reference, only the main features are briefly recalled here. The specimens were \(60 \times 30\times 4\,\hbox {mm}^3\). They were impacted at a nominal speed of \(50\,\hbox {m.s}^{1}\). 15CDV6 high strength steel impactors and waveguides were used. Contrary to the simulation before, impact occurs over the full specimen width. The reason why this was possible here and not on the simulation is that experimentally, the boundary conditions are imperfect and some heterogeneity was present along vertical slices, which was enough to provide meaningful values for the two terms in Eq. 4. Grids were transferred onto the specimens using the technique detailed in [26]. A grid pitch of 0.7 mm was used. The grids were imaged at 5 Mfps with a Shimadzu HPVX camera providing \(400 \times 250\,\hbox {pixel}^2\). The field of view was \(56 \times 35\,\hbox {mm}^2\). From the grid images, displacements were derived using the incremental version of the grid method as detailed in [8]. Acceleration and strain fields were then obtained by smoothing / differentiation, with details in [20].
In fact, it is possible to identify these two stiffness components for all transverse sections. The results are illustrated in Fig. 7 for equations \(1+2\). The greyed out areas at both ends correspond to half the Gaussian smoothing kernel where the data are corrupted. In most of the central part of the specimen, the response is very stable with only small local variations of the identified stiffness, most probably caused by the measurement noise as this material is very stiff and does not deform much. However, between 38 and 50 mm, the modulus ramps up significantly. To investigate this, both terms of the first line of Eq. 7 were plotted for all time steps, as well as their difference. The results are shown in Fig. 8. The big red blob at the top of the difference map corresponds to the presence of a crack which clearly violates the continuum assumption underlying the principle of virtual work on which this is based. It is also clear that down to about 40 mm from the free edge, the equation is well verified until the initiation of the crack. However, in the 15 mm closer to the impact edge, significant discrepancies exist, confirming the plot in Fig. 7. This is probably caused by uneven contact between the wave guide and the specimen, which fades away a certain distance away from the impact. A similar effect was also reported in [27]. However, this would need to be confirmed by using backtoback cameras to image both sides of the specimens in the future. Finally, the results from equations \(1+3\) and \(2+3\) show exactly the same trends, just with more oscillations on \(Q_{12}\).
Identification results the isotropic experimental data, specimen 2WCF13Co (Tungsten Carbide cermet) from [20], averaged between 3 and 38 mm from the free edge
Generalized Stress–Strain Curves in Orthotropic Elasticity
Section Parallel to the Fibres
Section Transverse to the Fibres
Validation on Simulated Data
A finite element model was developed using ABAQUS Explicit. The reason for switching from LSDYNA to ABAQUS was that some inconsistencies were found in the LSDYNA results when attempting to validate the generalized stressstrain relationships. This is currently unresolved. In all other aspects, the model was similar to the isotropic one. The inplane dimensions were \(70 \times 44\,\hbox {mm}^2\), with a 1 mm thickness. The elements were CPS4R (4node, reduced integration, plane stress). The mesh consisted of \(280 \times 176\; {\text{elements}}^2\). The time step was floating at 0.8 \(t_{crit}\) and beta damping of \(1.10^{8}\) s was employed. The data were output every \(0.5 \,\mu \hbox {s}\) mimicking the experimental frame rate (see next section). A fibre angle of \(45^{\circ }\) was selected, with inplane stiffness components close to the material in the experiment described in the next section: \(Q_{11} = 124.7\,\hbox {GPa}\), \(Q_{22} = 8.35\,\hbox {GPa}\), \(Q_{12} = 2.51\,\hbox {GPa}\) and \(Q_{66} = 3.70\,\hbox {GPa}\). The loading was applied through a triangular compressive pressure pulse rising to 300 MPa in \(9\,\mu \hbox {s}\) and decreasing back to zero in the same amount of time. The pressure was uniformly distributed over the specimen edge, contrary to the isotropic simulation. The offaxis configuration was found to be sufficient to ensure enough stress and strain heterogeneity so that none of the generalized stressstrain equations degenerated into zero equals zero.
Contrary to the isotropic example, acceleration averages have to be calculated over an irregularly shaped area, while strain averages need to be obtained along angled slices. It was therefore necessary to perform data interpolation. This will also be the case for the experimental data which are obtained on a regular grid of points. The \(\hbox {Matlab}^{\circledR }\) ‘scatteredinterpolant’ command was used (linear scattered interpolant) to calculate strain values over angled sections at \(45^{\circ }\). The number of data points along each angled slice was kept the same as the number of points in a vertical slice, so as not to ‘overinterpolate’. The effect of interpolation parameters on the accuracy of the results is beyond the scope of the present paper but will have to be studied in the future. Only the slices of constant length were kept to ensure that enough data were averaged. Therefore, only a number of sections in the central part of the specimen were considered. They cover a length of 26 mm of the 70 mm of the specimen, for both sets of slices. To calculate the acceleration averages, a first test was performed to check whether the centroid of a given element was inside the considered area (\(S_A\) or \(S_B\) according to Fig. 9). The average was then calculated only from the elements inside the area. This resulted in a ‘staircase’ pattern which was deemed to be a reasonable approximation for the present validation. Split elements could be considered prorata in the future. The detailed interpolation procedure can be found in the \(\hbox {Matlab}^{\circledR }\) routines provided in the data repository linked to this article.
Validation on Experimental Data
Conclusion and Future Work

Compared to existing processing methods for the IBII test, these new relationships provide a step forward. They are more extensive than the simple ‘stressgauge’ version presented initially in [9]. They do not need to assume Poisson’s ratio as was previously the case with the standard stressgauge. The new equations allow the different stiffness components to be retrieved directly from various combinations of averages of the kinematic fields. For the first time, they have been successfully applied to an offaxis IBII test on a unidirectional composite.

They are analytical and so, no complex inverse identification is required. They can be run as a quick diagnostic just after a test to evaluate the quality of the data.

They provide spatiallyresolved stiffness information without the need for parametrization. This is ideal to study the development of damage for instance.

These relationships can help understand the consistency of the data, as illustrated in Fig. 8.

They consider linearelastic behaviour only, so would generally only apply to brittle or quasibrittle materials. However, they can still be useful to study the transition between elasticity and say, plasticity or damage, as the earlier part of the response will generally be linear elastic. The stress equations like Eq. 2, 3 and 4 can also be used in the nonlinear Virtual Fields Method [18, 22].

They are limited to IBIIlike configurations, in which only one specimen edge is loaded. This enables the use of rigidbodylike virtual fields without introducing another unknown stress distribution at the other end of the considered slice. They will also apply to the IBUS test configuration [23]. Some work is underway on this. They may also be extended to ‘anvillike’ tests with loads at both ends, when considering the wave dominated regime. This will be investigated in the future.
Finally, it is clear that these are still early days in the design and exploitation of imagebased transient dynamic tests and that many opportunities are opening up with the availability of quality ultrahigh speed cameras. This work is a step towards this longterm goal which will provide the next generation of high strain rate tests beyond the current split Hopkinson bar methodologies.
Data Provision
Data supporting this study are openly available from the University of Southampton repository at https://doi.org/10.5258/SOTON/D0915.
Notes
Acknowledgements
The authors would like to thank Mr Jared Van Blitterswyk for his contribution to edge data reconstruction and for preparing the composite sample used for the offaxis tests.
Funding
The authors acknowledge support from EPSRC through Grant EP/L026910/1 (Established Career Fellowship) as well as from the Air Force Research Laboratory, under agreement number FA95501510293. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory or the U.S. Government. The authors would like to thank Dr David Garner (EOARD) for supporting this work.
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