Experimental Mechanics (2016) 56:3–23DOI 10.1007/s11340-015-0078-1

Exploration of Saint-Venant’s Principle in Inertial HighStrain Rate Testing of Materials

H. Zhu1,2 · F. Pierron2

Received: 20 October 2014 / Accepted: 16 July 2015 / Published online: 18 August 2015© Society for Experimental Mechanics 2015

Abstract Current high strain rate testing procedures ofmaterials are limited by poor instrumentation which leadsto the requirement for stringent assumptions to enable dataprocessing and constitutive model identification. This is thecase for instance for the well known Split Hopkinson Pres-sure Bar (SHPB) apparatus which relies on strain gaugemeasurements away from the deforming sample. This paperis a step forward in the exploration of novel tests based ontime and space resolved kinematic measurements obtainedthrough ultra-high speed imaging. The underpinning idea isto use acceleration fields obtained from temporal differen-tiation of the full-field deformation maps measured throughtechniques like Digital Image Correlation (DIC) or the gridmethod. This information is then used for inverse identi-fication with the Virtual fields Method. The feasibility ofthis new methodology has been verified in the recent paston a few examples. The present paper is a new contributiontowards the advancement of this idea. Here, inertial impacttests are considered. They consist of firing a small steel

Electronic supplementary material The online version of thisarticle (doi:10.1007/s11340-015-0078-1) contains supplementarymaterial, which is available to authorized users.

� H. Zhuhaibin.zhu@soton.ac.uk

F. Pierronf.pierron@soton.ac.uk

1 LASMIS Universite de Technologie de Troyes,12 Rue Marie Curie, CS 42060, 10004, Troyes Cedex, France

2 Faculty of Engineering and the Environment,University of Southampton, Southampton SO17 1BJ, UK

ball impactor at rectangular free standing quasi-isotropiccomposite specimens. One of the main contributions of thework is to investigate the issue of through-thickness het-erogeneity of the kinematic fields through both numericalsimulations (3D finite element model) and actual tests. Theresults show that the parasitic effects arising from non-uniform through-the-thickness loading can successfully bemitigated by the use of longer specimens, making use ofSaint-Venant’s principle in dynamics.

Keywords Virtual fields method · High strain rates ·Inertial effects · Full-field measurements · Grid method

Introduction

Fibre-reinforced composite materials are currently used inmany applications where they are potentially submitted tohigh strain rate events (runway debris impact on aircraftpanels, crash of automotive components etc.). The mechan-ical behaviour of these materials at high rates is knownto be significantly different from that under static loading.Therefore, it is essential to investigate and to characterisethe strain rate dependent behaviour of these materials witha view to structural design and numerical simulation. Overthe years, different strategies for characterising the mechan-ical behaviour of materials at high strain rates have beendeveloped in the scientific community. A review of the mainexperimental techniques for high rate tests is available in[1]. Among these techniques the most popular is the so-called the Split Hopkinson Pressure Bar (SHPB) or Kolskybar. The original idea was proposed a century ago by Hop-kinson [2], while the current split bars system was designedby Kolsky [3]. In the recent decades the SHPB techniquehas been applied for high strain rate testing of a number of

4 Exp Mech (2016) 56:3–23

materials. The review paper [1] has many citations on theuse of the SHPB, for instance. However, the SHPB proce-dure suffers a number of shortcomings. First, it is based onthe one-dimensional wave theory; therefore, it strictly relieson the assumption of uniaxial and homogeneous stress state.Then, another stringent assumption is the fact that the stan-dard SHPB analysis based on the strain gauges readings onthe input and output bars requires quasi-static loading con-ditions, i.e., no inertial effects. As a consequence, specimensusually have to be very short to minimize the time overwhich stress waves travel back and forth within the speci-men and fade away. This is even worse for materials withlow sound speeds like soft materials and biological tissues.Although some authors proposed improvements to addresssome of these issues [4, 5], the process of reconstructing thewave and load histories still requires some very constrainingassumptions.

One of the reasons for the limitations of the SHPB pro-cedure is the rather poor instrumentation it relies on. Only afew strain gauges are used to analyse the complex behaviourof a material submitted to a high rate impact. Unlike conven-tional strain measurement techniques such as extensometersor strain gauges, full-field measurement techniques such asdigital image correlation (DIC) [6] and the grid method [7]provide much richer experimental information. It is there-fore natural to expect that this new wealth of informationwill lead to an in-depth revisit of the high strain rate test-ing procedures. In the past few years, DIC has been usedto acquire full-field information in the SHPB tests [8–10].However, full-field deformation measurements were mainlyused to provide average strains over a certain area like anon-contact strain gauge, which did not take full advan-tage of the strain imaging capability to measure nominallyheterogeneous strain states.

Using a more complex test configuration has conse-quences on how material parameters can be extracted. Awayfrom the restrictive assumptions of the classical SHPBanalysis, there is no more direct link between the strain mea-surements and the material parameters to be identified. Asa consequence, it is necessary to resort to some so-called‘inverse resolution’. A number of strategies (e.g. the FiniteElement Model Updating (FEMU) method [11], the Vir-tual Fields Method (VFM) [12, 13], etc.) based on full-fieldmeasurements have been proposed in the past. It is beyondthe scope of the present paper to recall the details of thesemethods, information can be found in the following reviewpaper [14]. FEMU has been applied in the past to iden-tify the constitutive parameters of materials from full-fieldmeasurements at high strain rates [15]. However, in thatwork, the cost function used to identify the model param-eters only included the measured and simulated impactforces, also meaning that the assumptions necessary toobtain the force from the strain gauge readings on the input

bar still had to hold. In particular, wave dispersion and iner-tial effects had to be neglected. These assumptions are notalways true at high strain rates, especially for specimenswith complex shapes or made from soft materials [16, 17].A more recent study [18] combined impact force and strainat the centre of the specimen to construct the cost functionbut again, inertial effects were neglected and force measure-ments relied on specific hypotheses. Finally, strain mapshave been used in FEMU on a SHPB test [19] but again, theforce needed to be measured with a Hopkinson bar. It shouldalso be noted that all these studies were looking at ratherlarge plastic strains where inertial effects were not a prob-lem anymore. To the best knowledge of the present authors,such a FEMU methodolgy has not been applied to materialsin the elastic range.

An alternative to the popular FEMU technique is the Vir-tual Fields Method (VFM) [13]. In a nutshell, the VFMuses the full-field measurements to derive stresses using anassumed constitutive model and identifies the parameters ofthis model by checking stress equilibrium using the global(or weak) form of stress equilibrium known as the principleof virtual work. The main benefit from this is that itera-tive solving of finite element models is avoided, resulting inmuch faster computations. This is even more critical at highstrain rates where a single finite element model may alreadytake several to several tens of minutes to run and this willhave to be done many times to evaluate the cost functionuntil convergence, leading to several hours of computationsuntil convergence (2.5 to 3 h in [19] for one parameter only,private communication from the corresponding author, J.Kajberg).

The present paper makes use of the VFM to extractthe constitutive parameters from the full-field kinematicmeasurements. Here, only linear elastic behaviour will beconsidered as a first step towards more complex constitutivemodels. Linear elasticity is also one of the most challengingsituations as the strains are small and inertial effects signif-icant when not predominant. In quasi-static situations, it isnecessary to measure a load information in order to iden-tify stiffness components. Otherwise, only stiffness ratioscan be obtained. In dynamics however, the acceleration fielddue to inertial effects can be used as an alternative loadcell provided that the density of the material is known. Thisimportant underpinning concept has already been demon-strated in the recent past [20–22]. It has great potential tochange in depth the way high strain rate testing is performedin the future.

The current work complements the three already pub-lished contributions cited above. First, it relies on purelyinertial loading and as a consequence, presents more chal-lenges in virtual fields definition than [22]. The specialoptimized virtual field approach already reported in [21] isalso used here but in the present case, the test configuration

Exp Mech (2016) 56:3–23 5

is designed to lead to a more heterogeneous state of stress.Indeed, while the line impact employed in [21] led to amainly unidirectional stress state, the ball impact configu-ration in the present work generates a more complex stressand strain state and is therefore one step further towardstest complexity. Finally, one of the important novelties ofthe present work compared to [21] is the systematic numer-ical and experimental study of the effects of non-uniformityof the kinematic fields through the thickness and the sug-gestions for ways to mitigate these effects. Full 3D finiteelement models have been developed and specimens withdifferent lengths have been tested in order to investigate theapplicability of Saint-Venant’s principle in such dynamicinertial tests.

In the present paper, three quasi-isotropic compositespecimens with different lengths were impacted by a smallball-bullet. The microsecond-level events were recorded bya digital ultra-high speed (UHS) camera (Shimadzu HPV-X). The grid method was used to perform full-field defor-mation measurements at high strain rates. The experimentalprocedure and measurement performance are introduced inSection “Experimental Procedure”. The VFM with iner-tial effects is then recalled in Section “The Virtual FieldsMethod”. 3D simulations of the ball impact tests includingthe misalignment of the contact point and the effect of spec-imen length are provided in Section “Finite Element Simu-lation”. Experimental results and related identification usingdifferent methodologies are detailed in Section “Experimen-tal Results”.

Experimental Procedure

Tests and Materials

The specimens tested in this paper were laminated to thefollowing [0/45/ – 45/90] s stacking sequence from CYTECMTM58FRB carbon/epoxy prepreg. The thickness of thespecimens is around 3.7 mm. The nominal quasi-static stiff-ness parameters are: E11 = 124 GPa, E22 = 7.5 GPa,ν12 = 0.31, G12 = 4.0 GPa [23]. Due to the nature ofthe quasi-isotropic layup, the in-plane elastic stiffness com-ponents of the specimens only depend on two parameters.Based on lamination theory, the in-plane Young’s modulusof this quasi-isotropic laminate is 47.1 GPa and Poisson’sratio is 0.31. Additionally, the specimens are expected toexhibit low strain rate dependence because the behaviour ofthis quasi-isotropic lay-up is highly dominated by the fibreswhich do not show any significant strain rate dependence.Therefore, the quasi-static Young’s modulus and Poisson’sratio will be used as the reference values in this paper. How-ever, for a unidirectional layup, the strain rate dependence ofthe material properties proved quite significant, especiallyon the transverse and shear moduli [24, 25]. The extension

of the present work to unidirectional specimens is currentlyunderway and will be reported in the near future.

The experimental set-up used to perform the impact testsis shown in Fig. 1. It is composed of a gun connected to anair pump and a 9 mm diameter steel ball used as the projec-tile. The specimen was positioned in front of the gun withthe help of a foam support. A thin steel tab of thickness1 mm was bonded onto the impact end of the specimen soas to mitigate the stress concentration. This has the unfortu-nate effect of also absorbing some of the impact energy asthe tab deforms plastically but tests without the tabs haveproved worse with too much damage to the composite spec-imens at the impact end, which is detrimental to the presentanalysis. This is clearly a disadvantage of the ball impactwhen compared to the planar impact used in [21]. The firedball and specimen were both enclosed in a chamber withtransparent perspex walls at its top and front faces. Threequasi-isotropic specimens with different lengths but samewidth were tested in this study. The dimensions and theimpact speeds are shown in Table 1.

Ultra-high Speed Imaging

To record the impact event, a Shimadzu HPV-X UHS cam-era was used in this work. The imaging configuration isshown in Table 2. This camera uses a dedicated sensorcalled FTCMOS which is a special type of CMOS sensorwith on-board solid-state memory storage. More detailsabout this camera can be found in [26, 27].

The images of the deforming specimens were acquiredby the camera through a transparent PMMA window. Theselected frame rate was 5 million frames per second (fps).Due to the short inter-frame time, two strong flash lights(Bowens Prolite 60, here) proved necessary. The outputpower of each flash light is 300 Joules. The rising time is

Fig. 1 Schematic and picture of the impact test set-up

6 Exp Mech (2016) 56:3–23

Table 1 Dimensions, impact speeds and region of interests for thedifferent specimens

Length: mm Impact speed: Region of Interests

m.s−1

Specimen 1 40 ≈ 35 Column 6 to 54

Specimen 2 60 ≈ 45 Column 5 to 56

Specimen 3 80 ≈ 45 Column 4 to 66

around 50 microseconds and the flash duration is 1 mil-lisecond. In order to acquire high quality images and tomake full use of the recording capacity of the camera, theimages should be acquired with full flash intensity, and thecamera should be triggered at the instant the steel ball con-tacts with the specimen. Thus, two independent triggeringsystems are necessary. The triggering system used here con-sisted in two closing circuits. One was positioned betweenthe exit of the gun barrel and the specimen and consisted ofa piece of metallic wire wound around a cardboard frameand a thin metal strip fixed at the back of the frame. Whenthe ball passed through the frame, it pushed at least oneof the wires against the metal strip, closing the circuit andtriggering the flash lights. Since the distance between theframe and the front end of the specimen was about 15 mm,this ensured that the imaging took place roughly midwaythrough the lighting event. The other triggering system con-sisted of two small pieces of wire attached to the frontend of the specimen but separated by about 1 mm. Whenthe ball reached the specimens, it made contact with bothwires, closing the circuit and triggering the camera. Thiswas not 100 % accurate since before impacting the speci-men, the ball had first to crush the wires but thanks to thebuilt-in post-triggering of the camera, satisfactory triggeringwas achieved.

The Grid Method

Although Digital Image Correlation (DIC) is currently themost popular full-field strain measurement technique, anattractive alternative for small heterogeneous strain distri-butions is the grid method, which has recently been used inhigh rate dynamics testing [20, 21]. This technique presentsa slightly better compromise between spatial resolution andstrain resolution at the cost of bonding a grid onto the testpiece. This is the technique that has been selected here.

The grid method is a full-field optical technique measur-ing displacement [7, 28]. A grid pattern consisting of a setof horizontal and vertical contrasted lines was transferredonto the surface of the specimen using an appropriate proce-dure [29]. The details for the grid method is not introducedhere in order not to overload the manuscript. The reader isreferred to [20, 30]. During the tests, the grid did not peel orflake off. Comparison with strain gauge data in [20] showedgood fidelity of the grid measurements in dynamic.

From the displacement vector, strain components can becomputed by spatial differentiation. Correspondingly, forfully time-resolved displacements in dynamics, accelerationfields can be calculated through second order differentia-tion over time from the displacements. Similarly, strain ratefields can be obtained through first order differentiationover time from the time-resolved strains. However, thesenumerical differentiations amplify the noise contained in themeasured displacement [31]. Therefore, spatial and tempo-ral smoothing are usually necessary for strain, strain rate andacceleration calculation.

Resolution

Before moving to the dynamic tests, it is necessary toassess the accuracy of the deformation measurements. Forfull-field measurements, the resolution can be evaluated as

Table 2 Imaging andmeasurement performanceinformation (specimen 1)

Pixel array size 400×250

Inter-frame time (microsecond) 0.2

Number of images (FP mode) 128

Exposure time (nanosecond) 110

Pitch of the grid (mm) 0.6

Sampling (pixel per period) 5

Field of view (mm) 33.6 ×26.4

Raw displacement resolution 0.17 % of grid pitch (0.0085 pixel)

Displacement spatial smoothing Gaussian, 3 × 3 then 10 × 10 data points

Displacement temporal smoothing 3rd order polynomial over 25 images

Strain temporal smoothing 3rd order polynomial over 25 images

Strain resolution (microstrain) 34

Acceleration resolution (m.s−2) 1.4 × 104

Exp Mech (2016) 56:3–23 7

the standard deviation of deformation maps obtained fromstationary images. In this work, it has been qualified bycapturing images of the stationary specimen just before thetests, using the same imaging conditions as in the impacttests. Theoretically, the displacement, strain and accelera-tion between two nominally stationary images should beuniformly null, but in practice, this is not the case becauseof the digital noise. Therefore, the displacement should bespatially smoothed before differentiation. To perform spatialsmoothing, different methods are available, e.g. polynomialfitting [32, 33], diffuse approximation [34, 35], Gaussianfilter [36, 37], etc. For temporal smoothing, a local least-squares algorithm can be applied on a sliding windowwith fixed number of images to reconstruct the displace-ment and strain in the form of a polynomial function, forinstance. In this work, spatial smoothing was performedwith a Gaussian filter, see Table 2. The acceleration andstrain rate fields were calculated from the displacement andstrain fields respectively using temporal smoothing. Thesmoothing parameters (size of sliding window and Gaussiansmoothing kernel) have been selected after a convergencestudy to find an appropriate compromise between resolu-tion and spatial resolution. It is beyond the scope of thepresent paper to investigate the effect of these parameters.This will be performed in the future using the simulationsimulator developed in [30, 38]. Average standard devia-tions of displacement, strain and acceleration over all framesfor specimen 1 are presented in Table 2. The performanceswere similar for the other two specimens.

Comparing the information in Table 2 with that in [21],it can be seen that the measurement performances are verysimilar, except for a slightly better acceleration resolutionin the present work. This shows that the measurement per-formance is quite robust for this particular camera. Thedynamic images have been smoothed using the same con-figuration to extract full-field strain and acceleration maps.Finally, these full-field data are processed by the VFMwhich is recalled in the next section.

The Virtual Fields Method

The Virtual Fields Method is based on the principle of vir-tual work which in dynamics and in absence of body forcescan be expressed as [21]:

−∫

Vm

σ : ε∗dV +∫

∂Vm

�T .�u∗dS =∫

Vm

ρ∂2 �u∂t2

.�u∗dV (1)

where σ represents the Cauchy stress tensor, �T the Cauchy

stress vector acting at the boundary surface ∂Vm, �u∗ a C0

vectorial function referred to as ‘virtual displacement field’,ε∗ the virtual strain tensor derived from �u∗ and ρ the density

of the material. ‘.’ denotes the scalar product between vec-tors whereas ‘:’ represents the contracted product betweenmatrices (or scalar product for matrices). In case of an in-plane test on a thin test piece, if h is the thickness of thevolume Vm and S the specimen planar surface, equation (1)reduces to a 2D situation with:

−h

∫S

σ : ε∗dS + h

∫∂S

�T .�u∗dL = h

∫S

ρ∂2 �u∂t2

.�u∗dS (2)

To identify the constitutive parameters, a non-parametricapproach was proposed by Othman et al. [39, 40] consistingin calculating the average stress in any transverse slice of arectangular specimen loaded uniaxially in a Kolsky bar froma measured force at the boundary, the full-field accelera-tion maps and the cross-sectional area of the specimen. Thestress-strain curve can then be obtained using the full-fieldstrain maps. In fact, the internal virtual work in equation (2)can be zeroed out through a rigid virtual field (see [21] forthe details of the derivation), leading to:

σx(x, t) = ρxax(x, t) (3)

where t is the time, σx the average longitudinal stress overthe transverse slice at a given x coordinate, and ax the spa-tial average of the longitudinal acceleration component overthe area between the free end and the slice at coordinate x.

With the reconstructed stress above and the correspond-ing average strain calculated from the measured strain overthe same transverse slice, local stress-strain curves can beplotted for each frame. In this way, the elastic modulus canbe extracted from these curves without the need for any apriori constitutive model, hence the term ‘non-parametric’.Another two sets of virtual fields were proposed in [21] toreconstruct shear stress and moment. This non-parametricapproach is simple but may not necessarily work in all situa-tions. In the more general case, a parametric VFM approachneeds to be employed.

In equation (2), �T represents the stresses applied at theboundary of the solid. In the present case, if the wholespecimen is considered, this vector is only non zero at theright-hand side boundary and represents the impact load-ing. It is not easy to measure external forces at high ratesbecause of inertial effects. However, it is possible to definea particular virtual field to cancel out the virtual work ofthe impact forces. To do so, the virtual displacement com-ponents along the right-hand side boundary of the region ofinterest in Fig. 1 can be set to zero. Thus, equation (2) canbe rewritten as:

−∫

S

σ : ε∗dS =∫

S

ρ∂2 �u∂t2

.�u∗dS (4)

This equation is valid for any virtual fields provided thatthe virtual displacement vector is continuous and null at theboundary where unknown impact forces are acting. Since

8 Exp Mech (2016) 56:3–23

there are two unknown parameters here, only two inde-pendent virtual fields are required to identify them. Thevirtual fields can be defined manually as described in [22].However, this paper will make use of the special optimisedvirtual fields proposed in [41, 42]. This is an automatedprocedure to define virtual fields minimizing the effect ofnoise on the identified parameters. Moreover, this procedureprovides ratios between the standard deviations and meanvalues of the identified stiffness components. These ratiosindicate the noise sensitivity of each identified stiffnesscomponent. In the present work, bilinear finite elementshave been used to expand the virtual fields. The bilinearelements ensure displacement continuity between elements,as needed for the principle of virtual work. More details canbe found in [13, 42].

Finite Element Simulation

FE Simulation of Offset Contact

Series of 3D finite element models of the impact tests havebeen built up. The objective of these simulations was toinvestigate the influence of 3D wave propagation on theidentification of parameters and to search for an effec-tive way mitigate these effects. It is worth emphasisingthat only the full-field data at the top and bottom surfaceswere processed, because this is the information availableexperimentally. However, the average fields over the thick-ness were output as well and used as reference in thefollowing. This 3D model is composed of an isotropic spec-imen of dimensions 40 × 30 × 4 mm (length × width× thickness) impacted by a steel ball of diameter 9 mm.The input Young’s modulus was 47.5 GPa, and 0.3 forthe Poisson’s ratio. The dynamic response was simulatedusing ABAQUS/EXPLICIT to produce full-field strain andacceleration maps which were then processed by the VFM.The details of the FE simulations are shown in Table 3.Both temporal and spatial convergence has been checked.From this simulation, it was found that the contact timebetween the two solids was about 20 microseconds. More-over, damping proved necessary in the computations to

Table 3 Details of the ABAQUS/EXPLICIT model

Mesh size (mm) 0.5

Element type C3D8R∗

Time step (microsecond) Automatic (around 0.01)

Inter-frame time (microsecond) 1

Contact type Hard contact

Impact speed (m.s−1) 30

*CPS8R: 8-node linear brick, reduced integration, hourglass control.

improve numerical stability. In FE simulations, classicalRayleigh damping was considered [43]. It combines themass and stiffness damping through the mass-proportionaland stiffness-proportional coefficients α and β, respectively.For instance, Fig. 2 plots the stress-strain curves at x =20.75 mm with different damping parameters, from equa-tion (3). One can clearly see that, without damping (α =0 s−1 and β = 0 s), the curve is linear only when the twosolids are in contact, whereas after the contact phase, thedata are inconsistent. With large damping (e.g. α = 0 s−1

and β = 1.10−6 s), this model experiences significantlynon-linear behaviour, as expected for a visco-elastic mate-rial. Therefore, the damping coefficient should be smallenough to respect the condition of elastic material behaviourbut large enough to damp the numerical instabilities dur-ing computation. It can be seen from Fig. 2(c) that dampingcoefficients of α = 0 s−1 and β = 2.10−8 s are appropri-ate, because the curve throughout the impact simulation islinear. Therefore, this set of values will be kept for the sim-ulations presented in this paper. It is also worth noting thatthe current approach enabling to calculate stress from accel-eration is also a relevant method to check for the quality ofdynamic explicit computations.

A first item to investigate is the effect of the point load.Indeed, this generates stress and strain states that are notuniform through the thickness, as opposed to the cylindri-cal impactor configuration used in [21]. This will generatean error when transforming the volume integrals in equa-tion (1) into surface integrals in equation (2). Moreover, inpractice, perfect centring of the point contact in the cross-section is not easy to achieve. For instance, Fig. 3 shows thephotos of the tested specimens and tabs in the experimentaltests. One can clearly see that the contact point is not alwaysperfectly centred. Aside from small discrepancies of the testspecimen position in the impact chamber, it is thought thatthe wires used for triggering may also cause some slightdeflection of the ball at the moment of impact. This mayresult in an even larger error when estimating the volumeintegrals from surface measurements. In order to investigatethese issues, three contact models have been considered.Contact model (1) describes an ideal case where the full-field data at the top and bottom surfaces are symmetricalabout the mid-plane. Contact model (2) represents a smallmisalignment of 0.5 mm (12.5 % of the thickness) from themid-plane in the cross-section, and a larger misalignmentof 1.5 mm (37.5 % of the thickness) is considered in con-tact model (3). Due to this misalignment, the stress wavepropagation in the specimen tends to be three-dimensional.It is thought however that the metal tab used to protect thespecimen can already act as a load spreader to mitigatethis effect. This was considered in the present simulationsbut only with elastic behaviour for the tab in order to savecomputation time.

Exp Mech (2016) 56:3–23 9

Fig. 2 Stress-strain curve fromdata at the top surface atx =20.75 mm with differentdamping parameters. Contactduration: 20 microseconds. (a)α = 0 s−1, β = 0 s. (b)α = 0 s−1, β = 1.10−6 s. (c)α = 0 s−1, β = 2.10−8 s

The ideal contact model (1) without the steel tab wassimulated first. The strain and acceleration fields at the topand bottom surfaces were output. The average fields overthe thickness were also output and used as the referencefield as this is the data that will provide exact estimationsof the surface integrals in equation (2) from the volumeintegrals in equation (1). The difference between the topsurface strain and the average strain through the thicknessin the longitudinal direction at 14 microseconds is shownin Fig. 4(a). The strain difference represents about 10 %of the global strain values. One can clearly see a wavepattern produced by the multiple reflections of the waveson the top and bottom surfaces. The strain concentrationscaused by the point load appear very clearly. The hetero-geneity of the mechanical fields is far more pronouncedthan for the quasi-uniaxial case in [21]. Figure 4(b) showsthe same data but with the tab. It is interesting to see thatthe curvature of the strain ripples is less pronounced withthe tab, showing the load spreading effect of the tab. Thetop to average difference is also slightly smaller than with-out the tab, as expected. Finally, one can see that awayfrom the loading point, the error fades away, illustrating akind of Saint-Venant effect [44, 45] in dynamics. In practicehowever, the metal tab exhibits local yielding as seen on

Fig. 3, thus absorbing some of the impact energy that willnot be available to deform the specimen. Nevertheless, thetab will prevent local indentation damage in the materialwhich is beneficial for the analysis presented here. There-fore, 1 mm thick steel tabs will systematically be used in therest of the paper, for both simulations and experiments.

Contact model (2) with the tab was then simulated. Thedifferences in longitudinal strain at the top and bottomsurfaces at 14 microseconds are shown in Fig. 5(a). It can beseen that the strains are much more heterogeneous throughthe thickness. This problem will significantly disrupt theidentification results as the volume integrals will be falselyevaluated from the surface ones. In addition to the tabs,another interesting idea to mitigate these 3D effects is tolengthen the specimen in order to be in a better situation tobenefit from Saint-Venant’s principle. In this case, the fieldof view will be restricted to 40 mm from the free-end sideof the specimen to be consistent with experimental condi-tions as imaging a longer field of view would compromisethe spatial resolution of the measurements. Two models withspecimen lengths of 60 and 80 mm for contact models (2)and (3) respectively were simulated with the same FE con-figuration as for the 40 mm specimen. Figure 5(b) and (c)presents the differences in longitudinal strains at the top and

10 Exp Mech (2016) 56:3–23

Fig. 3 Pictures of the testedspecimens and associated steeltabs. (a) Specimen 1,misalignment: 1 mm (b)Specimen 2, misalignment:0.5 mm. (c) Specimen 3,misalignment: 1.5 mm

bottom surfaces for the two longer specimens at differenttime steps. Comparing three sets of maps in Fig. 5, one canclearly see that for the two longer specimens the strain dif-ferences at both surfaces are significantly lower than that forthe specimen of length 40 mm, although the average strainsthrough the thickness of the three specimens are compara-ble. This means that lengthening the specimen mitigates thethrough-thickness strain heterogeneity in the specimen.

Identification from Simulated Data

This section presents three different procedures to extractthe elastic stiffness parameters from the simulated data, withincreasing complexity. The first one assumes Poisson’s ratioas known and investigates the identification of E throughthe plots of stress-strain curves. The second one brings thisone step further by using over-determined systems arisingfrom the application of equation (3). Finally, the full VirtualFields Method approach is considered.

Stress-strain curves from simulated data

The influence of misalignment can also be quantita-tively verified by the non-parametric method described inSection “The Virtual Fields Method”. Figure 6 presentsthe average longitudinal stress as a function of the averageεx +νεy (ν = 0.3, here) at x = 20.75 mm from the free end.The slope of the stress-strain curve provides the stiffnesscomponent Qxx which related to E through:

{E = Qxx(1 − ν2)

ν = Qxy/Qxx(5)

It is clear that the stress-strain curves from the top andbottom surfaces for contact model (1) are consistent witheach other and linear, and the estimated Young’s modu-lus for model (1) is around 47.2 GPa, very close to theinput value of 47.5 GPa. For contact model (2) however, thecurves from the top and bottom surfaces diverge, although

Exp Mech (2016) 56:3–23 11

Fig. 4 Differences between thetop surface and through-thickness average strains in thelongitudinal direction at 14microseconds. (a) Without tab.(b) With tab. L=40 mm

the curve from the average data through the thicknessmatches that of contact model (1) very well, as expected.

To validate the Saint-Venant effect in the dynamic sim-ulation, the stress-strain curves at x = 20.75 mm fromthe free end for all offset contact models mentioned pre-viously are presented in Fig. 7. It can clearly be seenthat longer specimens lead to reduced discrepancies in thestress-strain curves from the top and bottom surfaces, eventhough larger misalignment was considered for the spec-imen of length 80 mm. The estimated E through fittingthe first 20 points of the stress-strain curve on the top sur-face of the longest specimen is about 47.7 GPa. Young’smodulus can also be identified for all transverse slices of the40 mm long field of view along the longitudinal axis of thespecimen. Figure 8 presents the identified Young’s mod-ulus for all offset contact models. It seems that, for allspecimens, the identified results from the top and bot-tom surfaces are symmetric about the reference value andtend to converge at the free (left) end. Moreover, with theincrease of specimen length, the discrepancies of the identi-fied Young’s modulus from the top and bottom surfaces are

reduced, even though a larger offset of the point contact wasconsidered for the longest specimen. This means thatlengthening the specimen reduces the stress/strain hetero-geneity through the thickness at slices away from theimpact. This solution was implemented experimentally tocheck for its practical validity.

Over-determined system solution from simulated data

According to Hooke’s law, the relationship between stressand strain in the longitudinal direction can be expressed as:

σx = Qxxεx + Qxyεy (6)

In this work, Qxx and Qxy in equation (6) are unknown.The stress and strain components in this equation can alsobe expressed in the form of average values. The averagelongitudinal stress can be constructed by the acceleration ateach transverse slice as in equation (3), and the strains aver-aged over the same slice. Experimentally, the accelerationand strain fields can be calculated from the displacementthrough temporal and spatial numerical differentiations

12 Exp Mech (2016) 56:3–23

Fig. 5 Differences inlongitudinal strain at the top andbottom surfaces with tab. (a)L = 40 mm, contact model (2)at 14 microseconds. (b)L = 60 mm, contact model (2)at 19 microseconds. (c)L = 80 mm, contact model (3)at 23 microseconds

respectively. In this FE simulation, they were output fromABAQUS directly (here, 50 data frames were output). Com-bining equations (3) and (6), the following relationship isobtained:

ρxax(x, t) = Qxxεx + Qxyεy (7)

This equation can be used for each transverse slice at alltimes when strain and acceleration maps are available. So,at each section, an over-determined system consisting of 50equations (from the 50 data frames) with two unknownsQxx and Qxy can be built up. This can be solved for Qxx

and Qxy by a least-squares solution. However, for all off-set contact models, the linearity of the stress-strain curvesis only restricted to the loading stage. The reason for thisis currently unknown but might be caused by erroneous FEcalculation. In any case, in the rest of this paper, all of theover-determined system solutions only consist of data fromthe loading stage.

It must be emphasised that for contact model (1), Young’smodulus is calculated through equation (5) with the identi-fied Poisson’s ratio, whereas for contact models (2) and (3),E is calculated with ν = 0.3 due to the inaccurate iden-tification of Poisson’s ratio. Firstly, the identified E and ν

at each slice for the short specimen with contact model (1)are plotted in Fig. 9. For the good contact model, it is clearthat the identification of Young’s modulus is very good,except for the identification at slices close to the free andimpact ends. This is not surprising for the free end as thestress values become very small. For the impact end, onewould expect errors for the top and bottom data but notfor the average. This is probably caused by errors in theFE data arising from the impedance difference between thesteel tab and the tested material. As for the identificationof Poisson’s ratio, it is not as good as Young’s modulus.For the offset contact models, the identified results are pre-sented in Fig. 10. It can be seen that the identification is

Exp Mech (2016) 56:3–23 13

Fig. 6 Stress-strain curves during the loading stage for contact mod-els (1) and (2) at x = 20.75 mm. L = 40 mm, ν = 0.3

significantly worse than that for contact model (1). For thesame specimen, the identified Young’s modulus from thetop and bottom surfaces converges at the left part of thespecimen. This tendency is consistent with that in Fig. 8,however, the identification from the over-determined sys-tem is worse than that in Fig. 8. The exact reason for thishas not been established yet but it might be that at sometime during the loading, each slice contains a low stressand low strain situation which provides unstable lest-squareinversion. This will need to be investigated in the future asthis identification approach is very appealing because of itssimplicity.

Fig. 7 Stress-strain curves during the loading stage for offset contactmodels with different specimen lengths at x = 20.75 mm, ν = 0.3

It is also possible to consider all slices at the same timeto identify one overall value of E and ν. Since it has beenshown that the data close to the impact and free ends are notreliable, only a central section away from the ends has beenkept by discarding 4 and 12 columns of data from the freeand impact ends respectively over a total of 80 columns ofdata. The identified overall Young’s modulus and Poisson’sratio are shown in Table 4. The percentages in brackets rep-resent the errors from the reference. It is clear that for allcontact models the identified results from the average fieldsthrough the thickness are good. Moreover, the identificationof Poisson’s ratio is worse than that of Young’s modulus.For the offset contact models (2) and (3), the identificationis unsatisfactory, however, with increasing specimen length,the errors reduce, even for the larger offset considered forthe longest specimen.

VFM identification from simulated data

The same simulated sets of data were processed by theVFM. Firstly, the data from the whole field of view wereconsidered. In this procedure, the virtual mesh used toexpand the virtual fields is composed of 4 elements in thex-direction and 3 elements in the y-direction. The virtualdisplacement along the right-hand side boundary of the fieldof view is set to 0 so as to cancel out the virtual work ofthe impact forces at the right end. In the VFM, Qxx andQxy are first identified. Young’s modulus and Poisson’sratio are then calculated through equation (5). The resultsfor contact model (1) are shown in Fig. 11. It can be seenthat the identified results from the top and bottom surfacesmatch, even though they slightly diverge from that from theaverage data through the thickness, even for the good con-tact model. Moreover, the identified Young’s modulus from

Fig. 8 Identified Young’s modulus from the stress-strain curves ofspecimens with different lengths with offset contact

14 Exp Mech (2016) 56:3–23

Fig. 9 Identification of Young’s modulus and Poisson’s ratio from theover-determined system for the good contact model. L = 40 mm. (a)Young’s modulus. (b) Poisson’s ratio

the top and bottom surfaces are significantly higher thanthat for the average data during the first 20 microseconds(within the contact stage), thereafter, they gradually con-verge to the reference value. This was expected because,for the good contact model, during the contact stage thestrain levels on the top and bottom surfaces are always lowerthan the average value through the thickness, which leadsto higher Young’s modulus on the top and bottom surfaces.When contact is lost at around 20 microseconds, the strainstate through the thickness tends to be more uniform, hencethe converged values. In addition, as seen in Fig. 11, someoscillations can clearly be observed at around 20 and 30microseconds, especially for Poisson’s ratio. It is still notclear what is causing this.

Fig. 10 Identification of Young’s modulus and Poisson’s ratio fromthe over-determined system for the offset contact models. (a) Young’smodulus. (b) Poisson’s ratio

It has been shown that the identification at slices closeto the free and impact ends of the specimen is not reliable,as seen in Figs. 8 and 10. In the VFM processing, if onlydata from the central area is considered (the same data forlarge over-determined system in Section “Over-determinedsystem solution from simulated data”), the VFM results inFig. 12 are obtained. It is worth noting that in this case thevirtual displacement vector along the left and right bound-aries of the region of interest is necessarily set to zero soas to cancel out the virtual work of the unknown forces

applied at both boundaries. As seen in Fig. 12, it is clearthat the identification errors are significantly reduced, andthe results from the surfaces and the through-thickness aver-age match very well. This is mainly because the through-thickness stress and strain heterogeneities are concentratedat the impact end, as already illustrated previously. How-ever, some oscillations at 20, 30 and 40 microseconds persistand cannot be interpreted at the present time.

Exp Mech (2016) 56:3–23 15

Table 4 Identification of E

and ν from the largeover-determined system withdata from the central area.C (*) means contact model (*).Reference: E = 47.5 GPa.ν = 0.3

E: GPa ν

Top Bottom Average field Top Bottom Average field

40 mm, C (1) 47.9 (0.8 %) 48.0 (1.1 %) 46.9 (1.3 %) 0.31 (3.3 %) 0.31 (3.3 %) 0.29 (3.3 %)

40 mm, C (2) 38.0 (20.0 %) 41.5 (12.6 %) 46.9 (1.3 %) 0.02 (93.3 %) 0.10 (66.7 %) 0.29 (3.3 %)

60 mm, C (2) 46.2 (2.7 %) 48.6 (2.3 %) 47.3 (0.4 %) 0.22 (26.7 %) 0.35 (16.7 %) 0.30 (0.0 %)

80 mm, C (3) 48.3 (1.7 %) 49.4 (4.0 %) 47.4 (0.2 %) 0.34 (13.3 %) 0.44 (46.7 %) 0.30 (0.0 %)

The simulated data for the offset contact models havealso been processed with the VFM. The data over the com-plete field of view was kept here, since as the results were

Fig. 11 Identification of Young’s modulus and Poisson’s ratio fromthe whole field of view with the VFM. Data points: 80 by 60. Virtualmesh: 4 by 3. L =40 mm. Contact Model (1). (a) Young’s modulus.(b) Poisson’s ratio

much worse than for contact model (1), removing the enddata did not significantly change the trends of the results. Asmentioned previously, for the offset contact models, good

Fig. 12 Identification of Young’s modulus and Poisson’s ratio fromthe central area with the VFM. Data points: 64 by 60. Virtual mesh:4 by 3. L = 40 mm. Contact Model (1). (a) Young’s modulus. (b)Poisson’s ratio

16 Exp Mech (2016) 56:3–23

linearity of stress-strain curves is only restricted to the load-ing stage. Moreover, in practice, with the current imagingconfiguration, the recording duration is only 25.6 microsec-onds. Therefore, for all offset contact models, only theidentification within the first 25 microseconds is presentedin the following. The identification for the short specimen(L = 40 mm) proved quite unsatisfactory, as shown inFig. 13. It is clear that the modulus identified from the topsurface data is systematically lower than that from the bot-tom surface. The reason is the same as for the results inFig. 11. On the top surface, strains are larger than on the bot-tom surface within the first 25 microseconds for the shortspecimen, hence the identified Young’s modulus at the top

Fig. 13 Identification of Young’s modulus and Poisson’s ratio withthe VFM. Data points: 80 by 60. Virtual mesh: 4 by 3. L = 40 mm.Contact Model (2). (a) Young’s modulus. (b) Poisson’s ratio

surface is lower. Figure 14 shows the VFM identificationresults for the longer specimens. It can be seen that thediscrepancies of the identified Young’s Modulus and Pois-son’s ratio are significantly reduced through lengthening ofthe specimen, even though a larger offset has been con-sidered for the longest specimen. However, for these longspecimens, the identification from the top surface seemsto be higher than that from the bottom surface, which isopposed to the trend seen in Fig. 13. This is thought to becaused by an inversion of the through-thickness strain dis-tribution pattern as the wave propagates further down thelonger specimens.

As a conclusion to this section on numerical simu-lations, all three identification strategies have confirmedthat increasing the specimen length mitigates the effect ofthrough-thickness strain heterogeneity and provides results

Fig. 14 Identification of Young’s modulus and Poisson’s ratio withthe VFM. Data points: 80 by 60. Virtual mesh: 4 by 3. Offset contactmodels & longer specimens. (a) Young’s modulus. (b) Poisson’s ratio

Exp Mech (2016) 56:3–23 17

closer to the reference. The next section is dedicated to anexperimental investigation of this effect.

Experimental Results

Full-field Deformation Results

Three tests with different quasi-isotropic composite speci-mens were performed on the impact rig as shown in Fig. 1.The dimensions of the specimens and the impact speedsare presented in Table 1. The acquired images were pro-cessed according to the configurations presented in Table 2.For specimen 1 (L = 40 mm), the maps of strain and

acceleration at 9 microseconds with respect to the triggeringof the camera are shown in Fig. 15. It is worth emphasisingthat for this specimen, the surface used for data processinghas been reduced from the image field of view by remov-ing 6 mm from the impact end of the specimen in orderto avoid the strain distortion due to the permanent dam-age caused by the localized impact point. It can be seenthat these fields are symmetric or antisymmetric (for theshear components) with respect to the horizontal axis ofthe specimen, as expected. This confirms that the impactpoint is positioned at the middle of the impacted surface inthe y direction. Only elastic deformation was considered inthe FE analysis whereas experimentally, tab plasticity andcomposite damage occur at the loading point. As shown in

Fig. 15 Strain and accelerationmaps for specimen 1 at 9microseconds. Data points: 56by 44. (a) Strain. (b) Strain rate(in s−1). (c) Acceleration (inm.s−2)

18 Exp Mech (2016) 56:3–23

Fig. 16 Strain and accelerationmaps for specimen 2 at 13microseconds. Data points: 68by 47. (a) Strain. (b)Acceleration (in m.s−2)

Fig. 3, significant delamination at the impact end of speci-men 3 was observed, in spite of the steel tab. This dissipatesa significant amount of energy and leads to lower strain andacceleration levels.

The full-field maps for specimens 2 and 3 are shown at 13microseconds in Fig. 16 and at 17 microseconds in Fig. 17respectively. Different times were selected for the threespecimens to show comparable strain and acceleration maps

Fig. 17 Strain and accelerationmaps for specimen 3 at 17microseconds. Data points: 76by 46. (a) Strain. (b)Acceleration (in m.s−2)

Exp Mech (2016) 56:3–23 19

as the triggering time varied between tested specimens. Asseen in these figures, longer specimens lead to more uni-axial and unidirectional strain and acceleration maps. Thiswas expected as part of Saint-Venant’s principle acting inthe (x, y) plane.

Figure 18(a) shows the temporal evolution of the aver-age of the longitudinal strain over the field of view. Asseen in this figure, the strain profiles are consistent withthe impact speeds reported in Table 1, as the 40 mm spec-imen was impacted at a lower ball speed. In this work,the strain levels are only about a tenth of that reportedin [21]. Complete wave rebounds are captured except forspecimen 3 due to late triggering. The complete set ofdynamic maps are provided as ‘Supplementary material’to this article. The next part of this section is dedicatedto the processing of these data for stiffness identificationpurposes.

Fig. 18 Strain levels and stress-strain curves for three specimens. (a)Longitudinal strain levels for the three tests. (b) Stress-strain curvesfor the three specimens at x = 17.7 mm. ν = 0.3

Simplified VFM Identification

The approaches used in Section “Finite Element Simu-lation” for the simulated data are applied to the identi-fication from experimental data as well. Firstly, Young’smodulus is extracted from stress-strain curves at each trans-verse slice along the longitudinal axis of the specimen.For instance, stress-strain curves from different specimensat x = 17.7 mm are shown in Fig. 18(b). The impactspeed for specimen 1 was lower than for the other twospecimens, as already mentioned previously, explaining the‘truncated’ stress-strain curve. The stress-strain curves forthe three specimens are reasonably linear and consistentwith each other. It is worth noting that there are signif-icant oscillations for specimen 2 and 3 marked by blackcircles. Figure 19(a) presents the profiles of displacement,velocity and acceleration for specimen 2 during the test.Even though the displacement curve is very smooth, afirst differentiation to obtain velocity shows slight distur-bances (magnification window). This is further enhancedby the next differentiation to obtain acceleration. The dis-turbance is now clearly visible, and will translate to thestresses obtained from equation (3). This can be traced

Fig. 19 Profiles of displacement, velocity, acceleration and inten-sity for specimen 2. (a) Longitudinal displacement, velocity andacceleration. (b) Average grey level intensity of the raw images

20 Exp Mech (2016) 56:3–23

back to some artefacts in the imaging. The mean inten-sity profile of the raw images of specimen 2 during thisdynamic test is shown in Fig. 19(b). One can see an increaseof the mean intensity as a function of time. This is acharacteristic of the camera sensor and happens systemat-ically. Fortunately, the phase extraction is not sensitive tothe mean image intensity so this did not affect the dis-placement measurements. However, some oscillations canbe observed there, especially early in the image series, asmarked by the red circle. This is similar to the problemreported for the earlier version of this camera, the ShimadzuHPV-1 [46], although on a much smaller scale. Clearly here,the temporal smoothing used to derive acceleration reducesthis effect but does not cancel it. Nevertheless, its impacton the global stress-strain curve is rather limited, as seenon Fig. 18(b).

Fig. 20 Identification of Young’s modulus for three specimens. (a)Identified Young’s modulus. (b) Correlation coefficients

Young’s modulus for each transverse slice along the lon-gitudinal axis of the specimen is shown in Fig. 20(a). Onecan clearly see that at slices close to both specimen ends,the results are not good. This trend is similar to that fromthe simulated data in Section “Finite Element Simulation”.The estimated Young’s modulus over the central part of thefield of view is comparatively good. Figure 20(b) reportsthe goodness of fit (R-square values) for all slices for eachspecimen. This figure first confirms that only a certain dis-tance away from both ends is the behaviour linear, as alreadyevidenced before. Moreover, the plot for specimen 1 showsoscillations certainly related to the significantly 3D natureof wave propagation, resulting in more perturbed stress-strain curves. Specimens 2 and 3 both exhibit excellentfit at a higher level than specimen 1, confirming the rel-evance of using longer specimens to mitigate for the 3Dstress wave propagation. It is also interesting to note thatfor both specimens 2 and 3, the identified Young’s mod-ulus is systematically larger than the value identified inquasi-static tests. This was not expected after the numericalresults which showed unbiased estimation for E. However,these numerical simulations used the strains and accelera-tions directly from the FE model whereas as noted before,the full-field measurements provide significant low-pass fil-tering (both temporally and spatially) of the data. Temporalsmoothing reduces the acceleration levels whereas spatialsmoothing decreases the strain peaks. Since E results froma balance between two terms containing these data, positiveor negative systematic errors can certainly be obtained. Theonly way to investigate this further would be to simulate theimage processing as in the simulator presented in [30] butthis is beyond the scope of the present paper.

As seen in Fig. 20, the identification in the central sectionof the field of view proved more stable. Therefore, a largeover-determined system consisting of data in the centralsection has been built up frame by frame and used to identifythe overall Young’s modulus and Poisson’s ratio. Accordingto the identification in Fig. 20, different regions of inter-ests were selected. The details are shown in Table 1. Theidentified results are shown in Table 5. It is clear that theidentified ν is very bad. Thus, Young’s modulus is calcu-lated with ν = 0.3. The modulus from specimens 2 and 3 iscloser to the expected reference, in spite of the large contactpoint offset for specimen 3, as seen in Fig. 3. This illustratesagain the benefit of longer specimens.

Table 5 Identification from the large over-determined system withexperimental data for the three specimens. EQS = 47.1 GPa

E: GPa ν

Specimen 1 (L = 40 mm) 70.6 0.26

Specimen 2 (L = 60 mm) 50.8 −0.14

Specimen 3 (L = 80 mm) 51.9 −0.10

Exp Mech (2016) 56:3–23 21

Full VFM Identification

The same data used to build up the large over-determinedsystem was also processed by the VFM. In this case, thevirtual displacement components along the left and rightboundaries of the region of interest are necessarily set tozero so that the virtual work of the unknown forces appliedat both boundaries can be zeroed out. Concerning the vir-tual mesh, in FE simulations it is not critical as the data are(nearly) exact. However, the experimental data is noisy anddifferent virtual fields provide different results. Followinga convergence study on the virtual mesh density, the vir-tual mesh selected here is composed of 13 elements in thex-direction and 3 elements in the y-direction. The identi-fied Young’s modulus and Poisson’s ratio are presented inFig. 21. Here again, Young’s modulus has been calculatedwith ν = 0.3. The identified values for E are consistent withthe results from the over-determined system. The values aresystematically lower for the 40 mm specimens and system-atically higher for the other two, to about the same levels.Poisson’s ratio was also reasonably estimated, particularlyfor the 60 mm specimen.

Finally, it is interesting to look at the ratios between thestandard deviations and mean values of the identified stiff-ness components, denoted ηxx/Qxx and ηxy/Qxy in this

Fig. 21 Identified Young’s modulus and Poisson’s ratios with theVFM for the three specimens. Virtual mesh: m = 13, n = 3. (a)Young’s modulus. (b) Poisson’s ratio

paper and provided by the optimized virtual field procedure.The lower the η/Q values, the better the identification. Assuch, these parameters provide an indication of the qual-ity of the identification. High values of these ratios indicatepoor signal to noise ratios for the identification [42].

The values of these parameters are shown in Fig. 22.As can be seen, the values of ηxx/Qxx and ηxy/Qxy forspecimen 1 are systematically higher than for the other twospecimens. This is consistent with the lower strain levels forthis test, as seen previously. Also, the abnormally high val-ues at the early and late stages are because of low signalto noise ratios, as explained before. Finally, it is clear thatthe levels of ηxy/Qxy are higher than that of ηxx/Qxx forthe three specimens. This is not surprising as Poisson’s ratiohas less influence on the actual strain field than Young’smodulus and is always going to be more difficult to identify.

Generally, the identified results are close to the quasi-isotropic reference values and the previous results reportedin [21]. This is even more remarkable if one considers thevery low strain levels involved here, systematically lessthan 1000 microstrains, together with the more complex2D strain distribution arising from a point load and the3D parasitic effects coming from the non-uniform through-thickness load distribution. This strengthens the fact thatthis new inertial testing approach already presented in a few

Fig. 22 Coefficients of sensitivity to noise for the three specimens.Virtual mesh: m = 13, n = 3. (a) ηxx . (b) ηxy

22 Exp Mech (2016) 56:3–23

papers in the past has great potential to become a practi-cal tool for high strain rate identification in the future, eventhough plenty of work is still to be done to realize thispotential.

Conclusion and Perspectives

This paper has presented a new contribution towards thelong term goal of defining novel high strain rate materialtesting procedures based on high speed full-field kinematicmeasurements and inverse identification with the virtualfields method. The underpinning novelty is the use of thefull-field acceleration maps as a volume distributed loadcell, avoiding the need for external impact force measure-ment. Compared to a previous feasibility study published in[21], much smaller strain levels were recorded in the presentwork, providing a good test case for the robustness of theidentification method. The main conclusions of the presentpaper are as follows.

• The new point load impact configuration gives rise toheterogeneous distributions of stress and strain throughthe thickness which evolve in time and space as thewaves propagate and bounce off the different specimenfaces.

• The presence of a thin 1 mm steel tab slightly mitigatesthis problem, though its plastic deformation absorbs asignificant amount of the impact energy which led tosmall experimental strain levels in the specimen.

• Using longer specimens and a field of view containingthe free end away from the impact point, the 3D effectscan be sufficiently mitigated to obtain good mechani-cal identification. As such, this study has confirmed theexistence of a Saint-Venant effect in high rate dynamics,as already established by previous authors and reportedin [47], with a fade away distance of around one to twotimes the width.

• Experimental results on three different specimenlengths confirmed the findings of the simulations:longer specimens provided more stable and preciseidentification.

• Finally, the new results presented here confirmed thatthe present alternative to classical Split HopkinsonPressure Bar has potential to become a standard tech-nique in the future.

Clearly, this is still early days for this new procedureand much more work is needed to make it fully operationalas a routine testing technique. First, the identification isonly elastic here. While this is justified at these early stagesto validate the technique and test its robustness, the realinterest lies in non-linear behaviour. Such inertial impacttests have already been conducted on off-axis unidirectional

specimens and the results will be reported soon. Extensionto elasto-visco-plasticity for metals is also underway and inthe near future, more materials and constitutive models willbe considered to widen the applicability of the technique.Finally, it will be necessary to delve into test configurationdesign by adapting recent tools developed for quasi-staticsituations [30] to such inertial impact tests.

Acknowledgments This work is supported by China ScholarshipCouncil (CSC) through the government grant of Haibin Zhu. ProfessorPierron gratefully acknowledges support from the Royal Society andthe Wolfson Foundation through a Royal Society Wolfson ResearchMerit Award. The authors would like to acknowledge Mr Brian Speyerfrom Speyer Photonics Ltd and Dr. Markus Ortlieb from ShimadzuEuropa GmbH for lending the camera and helping out with the exper-iments. They would also like to thank Dr. Nicola Symonds and Dr.Liam Goodes from the nCATS group at the University of Southamptonfor their help with the impact rig.

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