Measurement of Orthogonal Stress Gradients Due to ImpactLoad on a Transparent Sheet using Digital GradientSensing Method

C. Periasamy & H.V. Tippur

Received: 14 March 2012 /Accepted: 25 June 2012 /Published online: 17 August 2012# Society for Experimental Mechanics 2012

Abstract A full-field optical method called Digital Gradi-ent Sensing (DGS) for measuring stress gradients due to animpact load on a planar transparent sheet is presented. Thetechnique is based on the elasto-optic effect exhibited bytransparent solids due to an imposed stress field causingangular deflections of light rays quantified using 2D digitalimage correlation method. The measured angular deflec-tions are proportional to the in-plane gradients of stressesunder plane stress conditions. The method is relatively sim-ple to implement and is capable of measuring stress gra-dients in two orthogonal directions simultaneously. Thefeasibility of this method to study material failure/damageis demonstrated on transparent planar sheets of PMMAsubjected to both quasi-static and dynamic line load actingon an edge. In the latter case, ultra high-speed digital pho-tography is used to perform time-resolved measurements.The quasi-static measurements are successfully comparedwith those based on the Flamant solution for a line-loadacting on a half-space in regions where plane stress con-ditions prevail. The dynamic measurements, prior to mate-rial failure, are also successfully compared with finiteelement computations. The measured stress gradients nearthe impact point after damage initiation are also presentedand failure behavior is discussed.

Keywords Optical metrology . Transparent solids .

Elasto-optic effect . Stress concentration . Contact stresses .

Stress waves . Dynamic failure . Digital image correlation

Introduction

The optical transparency requirement is an essential charac-teristic of solids used in many engineering applications suchas automotive windshields, electronic displays, aircraft win-dows and canopies, hurricane resistant windows, bullet re-sistant enclosures, personnel helmet visors, and transparentarmor materials used by the military [1, 2]. In applicationsinvolving military armor, helmet visors and personnel enclo-sures, the ability of a material to continue to remain trans-parent and bear load after impact is critical for personnelsafety. Over the years, there has also been a great deal ofinterest in developing novel transparent composites for avariety of other engineering applications [3–6] as well. In allthese situations, understanding the mechanical failure char-acteristics of transparent materials in general and understress wave loading conditions in particular is critical.

Performing full-field, non-contacting, real-time measure-ment of deformations, strains, and stresses during stress wavedominant events in solids, transparent or otherwise, is ratherchallenging due to stringent spatio-temporal resolution require-ments. Over the years, a few optical methods have been suc-cessfully used for understanding failure mechanisms involvedand for quantifying the associated engineering parameters. Forexample, de Graaf [7] used photoelasticity to witness stresswaves around a dynamically growing crack in steel. Photo-elasticity continues to be popular in the study of fast fracture/failure events [8–10]. The method of caustics was popular forstudying dynamic fracture mechanics and contact mechanicsproblems in the 1980s [11, 12] in view of a relatively simpleoptical setup and stress intensity factor extraction procedure.The presence of triaxial stress zone near a crack tip and theassociated difficulty to position the initial curve (that producesthe optical caustic) outside the zone of triaxiality during stress-wave loading event led to a shift away from this method [13]. Inits wake, a full-field lateral shearing interferometer called

C. Periasamy :H.V. Tippur (*)Department of Mechanical Engineering, Auburn University,Auburn, AL 36849, USAe-mail: htippur@eng.auburn.edu

Experimental Mechanics (2013) 53:97–111DOI 10.1007/s11340-012-9653-x

Coherent Gradient Sensing (CGS) attained popularity. Its ap-plicability to transparent and opaque solids and insensitivity torigid body motions/vibrations [14–17] contributed to its viabil-ity to experimental fracture mechanics. Moiré interferometryhas also been used in dynamic fracture studies to measure in-plane displacement fields with sub-micron sensitivity [18]. Theneed for coherent optics and an elaborate specimen preparationinvolving high-frequency diffraction gratings on engineeringsubstrates to implement the method has limited its usage to afew dynamic studies. There are also a few early reports of laserspeckle photography and holographic interferometry to studydynamic problems [19, 20] as well.

The above mentioned techniques are generally based onoptical interference principles and require special material(optical) characteristics (e.g., optical birefringence for photo-elasticity), coherent optics and/or special surface preparation(e.g., moiré interferometry and CGS) to be implemented suc-cessfully. In recent years, however, due to advances in digitalhigh-speed photography and image processing, and ubiqui-tous computational power, digital image correlation (DIC)method [21] has become popular for studying transient prob-lems. The work of Chao, et al., [22] to study deformationsaround a propagating crack with the aid of a Cranz-Schardinfilm camera is an early attempt in this direction. They digitizedanalog film recordings using a scanner to correlate successiveimages and estimate displacements. With the advent of mod-ern digital high-speed cameras offering recording rates from afew thousand to a few million frames per second, there hasbeen a significant interest in this method for studying highlytransient problems [23–26]. The fact that this method (i) needslittle or no surface preparation, (ii) uses white light illumina-tion, and (iii) can be fashioned to measure 2D (planar) or 3Ddisplacement components are indeed very attractive. In thiscontext, it is worth noting that the state-of-the-art digital imagecorrelation method can measure macro- and micro-scale dis-placements very accurately [27]. The measured strains fromDIC, however, tend to be of a lower accuracy due to reasonssuch as first order representation of displacement gradients ornumerical differentiation of noisy displacement data [28]. Thisis an issue of significance particularly for investigating me-chanical failures caused by stress risers producing steep de-formation gradients. From this perspective, it is attractive tohave a method that offers all the advantages of DIC yetcapable of directly measuring stress gradients in the wholefield near stress concentrators during dynamic events. Accord-ingly, a full-field optical method called Digital Gradient Sens-ing (DGS) based on stress-optic effect and 2D DIC principleshas been proposed recently by the authors for measuring smallangular deflections of light rays caused by stresses in trans-parent planar solids [29]. In mechanically loaded planarobjects, the angular deflections can be related to spatial gra-dients of stresses when plane stress conditions hold. In thispaper, the feasibility of this new method to study evolution of

transient stress gradients in the vicinity of an impact loadededge of an elastic sheet is examined.

In the following, first the experimental method of DGS, itsworking principle and the governing equations are presented.After discussing issues related to material homogeneity andmeasurement accuracy of DGS method, stress gradients in theload point vicinity of a statically applied edge load on a trans-parent PMMA sheet are presented and discussed. Subsequent-ly, the extension of the method to transient impact loadingexperiments using digital high-speed photography is described.The time-resolved stress gradients and the estimated load his-tories in the impact point vicinity are presented. These resultsare compared to those from a complementary finite elementanalysis of the problem. Stress gradient data pertaining to post-failure initiation are also presented and discussed. Lastly, theresults are summarized and conclusions are drawn.

Experimental Procedure

Figure 1 shows the experimental schematic for the DGS meth-od. A transparent planar specimen to be investigated is placedbetween a digital camera and a target plane coated with randomspeckles. Ordinary white light sources are used for illuminatingthe speckle/target plane uniformly. The recording camera fittedwith a long focal length lens is used to image speckles from arelatively large distance from the specimen (L) and target(L+Δ) [28] planes. (The distance between the mid-plane ofthe specimen and the target is Δ.) By using a relatively smalllens aperture (or a high F#) a good depth of focus is achieved.This is also helpful in recognizing the specimen features (e.g.,specimen edges) in the recorded image. An image of thespeckle pattern on the target plane is captured under no-loadcondition and is used as the reference or ‘undeformed’ image.The ‘deformed’ image of the speckle target is captured duringloading. The speckles in the deformed image are displacedrelative to the undeformed ones due to local changes in thick-ness and refractive index of the specimen [29]. If a point in thereference image is displaced by δx and δy in the x- and y-directions, respectively, due to deformation the angular deflec-tion fields, ϕx and ϕy can be obtained sinceΔ is known. Theseangular deflections can be related to in-plane stress gradients,as discussed in the following section.

Working Principle

In Fig. 1, let the in-plane Cartesian coordinates of the speci-men and target planes be (x, y) and (x0, y0), respectively, andthe optical axis of the setup coincide with the z-axis. Let thespeckles on the target plate be photographed normally throughthe transparent specimen of nominal thickness B and refrac-tive index n in its reference (undeformed) state. That is, a

98 Exp Mech (2013) 53:97–111

generic point P on the target plane, corresponding to point Oon the specimen (object) plane, is recorded by the camera inthe reference state. Upon imposing the load (say, force Facting on the edge of the specimen in Fig. 1), both refractiveindex and thickness changes occur throughout the specimen.A combination of these changes causes light rays to deflect.That is, the light ray OP in the reference/undeformed statenow corresponds to OQ after the specimen deforms. Byquantifying the vector PQ and knowing the separation dis-tanceΔ between the mid-plane of the specimen and the target,the angular deflection of the light ray relative to the opticalaxis can be determined.

Let bi; bj and bk denote unit vectors in the Cartesiancoordinates defined with point O as the origin. When the

specimen is undeformed, the unit vector bk is collinearwith OP bringing point P(x0, y0) to focus when imagedby the camera via point O(x, y). Upon deformation, theoptical path is locally perturbed, thereby bringing aneighboring point Q x0 þ dx; y0 þ dy

� �to focus. Here δx

and δy denote components of the vector PQ in the x- andy-directions. Let the unit vector corresponding to theperturbed optical path OQ be,

bd ¼ abiþ bbjþ gbk; ð1Þwhere α, β and γ are the direction cosines of bd, and ϕxand ϕy are the components of angular deflection ϕ in thex-z and y-z planes, respectively, as shown in Fig. 2.

If the initial thickness and refractive index of thespecimen are B and n, respectively, the optical pathchange, δS, for symmetric deformation of the specimenabout the mid-plane in the z-direction, is given by theelasto-optical equation [30],

dS x; yð Þ ¼ 2B n� 1ð ÞZ

0

1 2="zzd z B=ð Þ þ 2B

Z0

1 2=

dn d z B=ð Þ

ð2ÞThe two terms on the right hand side of the aboveequation represent the contribution of thickness-wisenormal strain, εzz, and the change in refractive index,δn, to the overall optical path change, respectively. Therefractive index change in an optically isotropic mediumcaused by the local normal stresses is given by the wellknown Maxwell-Neumann relation [31],

dn x; y ¼ D1 σxx þ σyy þ σzz

� �; ð3Þ

where D1 is the stress-optic constant and σxx, σyy, andσzz are normal stresses in the x-, y- and z-directions,respectively. Using the generalized Hooke’s laws for anisotropic, linear elastic solids, the normal strain compo-nent ε z z can be re la ted to normal s t resses

"zz ¼ 1E σzz � u σxx þ σyy

� �� �� �: That is, equation (2) can

be written as,

dS ¼ 2B

�D1 � u

En� 1ð Þ

�Z0

1 2=

σxx þ σyy

� �1þ D2

σzz

u σxx þ σyy

� � !" #( )

dðz B= Þ; ð4Þ

Q

P

δy

δx

specimen

OF

B

L

Δ

x

y

camera

x0

y0

zspeckle target

Fig. 1 Schematic of the experi-mental set up for Digital Gradi-ent Sensing (DGS) method todetermine planar stress gradientsin phase objects

Exp Mech (2013) 53:97–111 99

the elastic modulus and υ is the Poisson’s ratio of thespecimen. In equation (4), the term D2

σzzuðσxxþσyyÞ�

representsthe degree of plane strain, which can be neglected forapplications where plane stress assumptions (in-planedimensions >> thickness of the specimen and σzz00) arereasonable. Thus, for plane stress conditions, equation (4)reduces to,

dS x; yð Þ � CσB σxx þ σyy

� �; ð5Þ

where Cσ ¼ D1 � u E=ð Þ n� 1ð Þ is the elasto-optic constantof the transparent material. In equation (5), stresses σxx andσyy denote the average values over the specimen thickness.

The angular deflection of a generic light ray is caused bythe change in the optical path due to elasto-optic effects.Hence, the propagation vector can be related to the opticalpath change. That is, using eikonal equations of optics [11,12, 31], the propagation vector can be expressed as,

bd � @ dSÞð@x

biþ @ dSÞð@y

bjþ bk: ð6Þ

From equations (1), (5) and (6), for small angular deflec-tions, the direction cosines α and β are proportional to in-plane stress gradients as,

a ¼ @ dSÞð@x

¼ CσB@ σxx þ σyy

� �@x

; and

b ¼ @ ðdSÞ@y

¼ CσB@ ðσxx þ σyyÞ

@y:

ð7Þ

By performing a geometric analysis for the perturbed rayOQ, the relationship between the direction cosines α, β andangular deflection components ϕx, ϕy, respectively, can beestablished. Referring to Fig. 2, the perturbed ray subtendssolid angles θx and θy with the x- and y-axes. The angulardeflections ϕx and ϕy as defined earlier are also shown inFig. 2. With reference to the planes defined by OQC, OQA,OPE and OPD,

cos θx ¼ dxR; cos θy ¼ dy

R; tan fx ¼

dx$

and tan fy ¼dy$;

ð8Þ

where R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi$2 þ d2x þ d2y

q� is the distance between O and

Q. From the above, expressions for the angular deflectioncomponents can be obtained as,

tan fx ¼R

$cos θx and tan fy ¼

R

$cos θy: ð9Þ

After recognizing that R$ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ d2xþd2y

$2

qin equation (9), for

δx, δy<<Δ, fx � cos θx ¼ a and fy � cos θy ¼ b. Thus, forthe case of small angular deflections (tanϕx,y≈ϕx,y) of lightrays, equation (7) reduce to

fx � a ¼ CσB@ ðσxxþσyyÞ

@x ;

fy � b ¼ CσB@ ðσxxþσyyÞ

@y ;ð10Þ

Fig. 2 Schematic of the work-ing principle of DGS

100 Exp Mech (2013) 53:97–111

where, D2 ¼ uD1 þ u n� 1ð Þ E=½ � D1 � u n� 1ð Þ E=½ �= E is,

which serve as the governing equations for the method andcan be used to obtain stress gradients when specimenparameters Cσ and B are known.

The above governing equations reveal that the angu-lar deflections ϕx and ϕy, and hence the stress gradientsin the x- and y-directions can be obtained by quantify-ing local displacements δx, δy when the separation dis-tance Δ is known. The displacements can be evaluatedby carrying out a standard 2D digital image correlationof speckle images recorded in the reference and de-formed states of the specimen. A subtle but importantpoint to note here is that displacements δx, δy areevaluated on the target plane whose coordinates are(x0, y0), but can be replaced with the specimen planecoordinates (x, y) for Δ<<L [29]. Alternatively, a map-ping function can be used to rectify the coordinatesusing paraxial approximation. It is also interesting tonote that equation (10) shows that DGS method meas-ures quantities identical to the ones measured by theCoherent Gradient Sensing (CGS) method [14, 23, 32].However, unlike CGS, DGS can be used to measure twoorthogonal stress gradients simultaneously and does notrequire any coherent optics. Furthermore, from equation(8) it can be noted that the measurement sensitivity dependson Δ and it can be chosen suitably during an experiment.Further, the measurement sensitivity for δx and δy isdictated by a number of parameters that affect 2Ddigital image correlation methods including specklecharacteristics/size, parameters of the recording device(pixel size, sensor resolution, optical magnification) andimage processing algorithm employed. For the sake ofbrevity, discussion of those issues is avoided here andcan be readily found in the literature [27]. For thespeckle and camera parameters used in this study, thein-plane displacement resolution is approximately 3 μmas demonstrated in the previous works [23, 24].

Optical Homogeneity and Measurement Accuracy

The DGS method assumes optical homogeneity of refractiveindex and specimen thickness in the unloaded/referencestate. Mass produced materials such as PMMA sheets usedin this work could have spatial variations of these parame-ters or an initial residual stress state when machined from alarger sheet stock. To examine this aspect and the resultingmeasurement errors, an experiment was carried out on aPMMA specimen using a setup shown schematically inFig. 3 but without applying any mechanical load.

A target plane decorated with a speckle pattern wasplaced at a relatively large distance of ~1575 mm from arecording camera (Nikon D3000 digital camera fitted with a70–300 mm lens; aperture setting #11) and an extensiontube. Then, the specimen, a clear, 9.3 mm thick PMMAplate, was mounted on a multi-axis translation stage withmicrometer adjustments and introduced between the cameraand the speckle plane. A 40 mm square window/aperture,placed in front of the specimen, was used to fix the region ofobservation during the test. The distance from the mid-planeof the specimen to the speckle plane, Δ, was 30 mm. Thecamera was focused on the speckle plane through the spec-imen and a reference image of the speckle pattern wasrecorded. Then, while the target plate was held stationary,the specimen alone was translated horizontally in steps of1 mm and speckles were recorded at each step. (The choiceof these translational steps was based on the maximumanticipated total displacement due to mechanically imposedloads, expected not to exceed a couple of mm.) The samewas repeated in the vertical direction as well. A pixel reso-lution of 1936×1296 pixels (1 pixel038 μm on the targetplane) was used for recording the speckles. In these trans-lated positions, the speckles were recorded through points ofthe specimen different (shifted) from the correspondingpoints in the reference position. The presence of any non-

camera speckletarget

PMMA

P

θ

yo

z

L

O

Δ

aper

ture

tran

slat

ion

yFig. 3 Schematic of the experi-mental set up used to check forresidual stresses in transparentplanar specimens

Exp Mech (2013) 53:97–111 101

homogeneity in the specimen between these shifted posi-tions would cause the light rays to deflect. Hence, theimages recorded at each translational step were considered‘perturbed’ or ‘deformed’ images relative to the initial re-cording. By correlating the speckle images from the refer-ence and perturbed states, the optical uniformity of thespecimen was assessed. The horizontal and vertical an-gular deflection fields (ϕx and ϕy) for one casecorresponding to a horizontal translation of 2 mm isshown in Fig. 4. (Contour levels measured for all othertranslation distances were similar and are not presentedhere for brevity.) Clearly, the resulting field shows ran-dom angular deflections variation with the largest angu-lar deflection magnitude of less than 1×10−4 radians.Thus, the accuracy of angular deflection measurementsbased on optical homogeneity is limited to values abovethis threshold. Further, the displacement measurementaccuracy based on the type of speckles, the recordingparameters, and the correlation algorithm used in thisstudy is 2–3 μm [23, 24]. This also translates into anangular deflection measurement accuracy of 0.5×10−4 to1×10−4 radians when the separation distance (Δ) be-tween the target and the specimen is in the 20–30 mmrange. Thus, for the measurements to be credible, theload induced angular deflections during experimentsshould exceed this value.

Quasi-static Line-load on an Edge of a Sheet

Experimental Details

A stress concentration problem of quasi-statically appliedline-load on the edge of a large planar sheet was first studiedusing the DGS method. A 180×69.5 mm2 rectangular sheetof clear PMMA specimen (elastic modulus03.3 GPa, Pois-son’s ratio00.35 and Cσ0−1×10−10 m2/N) of thickness (B)

9.4 mm was used for this experiment. A photograph of theexperimental setup is shown in Fig. 5. The specimen wasplaced on a flat rigid platform and subjected to line-loadingusing a cylindrical steel pin (diameter 7.7 mm). An Instron4465 universal testing machine was used for loading thespecimen in displacement control mode (cross-head speed00.005 mm/sec). A target plate painted with random blackand white speckles was placed at a distance Δ030 mmaway from the specimen mid-plane (Fig. 1). A few heavyblack dots (see, Fig. 6) were marked on the speckleplane to relate the image dimensions to the actual spec-imen/target dimensions. A Nikon D100 digital SLR cam-era with a 28–300 mm focal length lens (aperture setting#11) and an extension tube were used to record specklesthrough the specimen in the load point vicinity. Thecamera was situated at a distance (L) of approximately1040 mm from the specimen.

A reference image of the target was recorded through thetransparent specimen in the region of interest at a small loadof a few Newtons (<5N). As the load was increased gradu-ally, speckle images were recorded using time-lapse photog-raphy (12 frames per minute). One of the speckle images inthe load point vicinity corresponding to a 3520N load isshown in Fig. 6. It shows that due to deformation of thespecimen, the speckles are noticeably smeared very close tothe loading point whereas they appear relatively unaffectedat far-away locations. The digitized speckle images (1504×1000 pixels) recorded at different load levels were correlatedwith the one corresponding to the reference condition usinga 2D digital image correlation software ARAMIS®. Asdescribed previously, an array of in-plane speckle displace-ments on the target plane (and hence the specimen plane)was evaluated and converted into local angular deflectionsof light rays ϕx and ϕy. A facet/sub-image size of 15×15pixels (1 pixel036.5 μm on the target plane) without anyoverlap was used in the image analysis for extracting dis-placement components.

0

0

0

-5e-5

-5e-5

-5e-5

-1e-4

-5e-5

-1e-4

-1e-4

-1e-4 -1e-4

5 mm 5 mm

Fig. 4 Contour plots of ϕx (left)and ϕy (right) fieldscorresponding to a horizontalspecimen translation of 2 mm ofthe PMMA specimen. Contourlabels are in radians

102 Exp Mech (2013) 53:97–111

Comparison of Experimental and Analytical Fields

Figure 7 shows the resulting contours of ϕx and ϕy forthree representative load levels in a reduced squareregion around the loading point. It is important to notethat, accounting for rigid body motions and imposingappropriate boundary conditions of the problem toquantify the contour levels for further analysis is need-ed. That is, in the current problem, the boundary con-ditions such as asymmetric stress gradients in the y-direction relative to the x-axis, symmetric stress gra-dients in the x-direction about the x-axis, and vanishingstress gradients far away from the loading point couldall be used.

Knowing that the plane stress field near the line-loadacting on an elastic half-space is described by the Flamantproblem [33] for which,

σxx þ σyyÞ� ¼ σrr ¼ � 2F

pBcos θÞð

r; σθθ ¼ 0; σrθ ¼ 0; ð11Þ

where F is the load, B is the thickness of the half-spaceand (r, θ) are the polar coordinates, as shown in Fig. 6.Note that the hoop and shear stresses vanish and

σxx þ σyy

� � ¼ σrrð Þ for plane stress. Furthermore, theradial stress σrr becomes singular/unbounded as theloading point (r→0) is approached. From equations(10) and (11),

fx ¼ CσB@ ðσrrÞ@x

and fy ¼ CσB@ ðσrrÞ@y

: ð12Þ

69.5 mm

10 mm

specimen edge

speckle target

y

x

r

5 mm

F

B = 9.4 mm

PMMA

F

y

x

r

180

mm θ

θ

Fig. 6 Schematic of the quasi-static line-load problem and arepresentative deformed image

camera

lamp

transparent specimen

speckle target

loading pin

rigid base

Fig. 5 Experimental setup used to measure angular deflections of lightrays caused by a deformed PMMA sheet subjected to a quasi-static,compressive line-load

Exp Mech (2013) 53:97–111 103

Using equations (11) and (12), the expressions for ϕx

and ϕy fields become,

fx ¼ CσB2F

pBcos ð2θÞ

r2; fy ¼ CσB

2F

pBsin ð2θÞ

r2: ð13Þ

For comparison, the experimental and analytical angu-lar deflection contours for a representative load case ofF02022N are shown in Fig. 8. The dominant stresstriaxiality where plane stress assumptions are violated isexpected in regions close to the loading point. In crackedbodies where a stress singularity of r−1/2 occurs, a regionof dominant stress triaxiality is shown to exist near thecrack tip (0 � r B= � 1 2= ) [30]. Based on that observa-tion, it is reasonable to expect that stress triaxiality todominate over a region of similar size in the current caseas well. Hence agreement between analytical solutionsand experimental measurements is not expected at leastup to r/B00.5. This region is shown in Fig. 8 as the onebounded by a semi-circle centered at the origin. In theregion outside the zone of dominant triaxiality, a goodqualitative and quantitative agreement between experi-mental and analytical contours is apparent.

The ϕx and ϕy data corresponding to a particular appliedload can be used to back calculate the load using equation(13). Figure 9 shows the plot of calculated load at eachmeasurement point (sub-image) as a function of r/B alongθ00° and θ045° from ϕx and ϕy fields, respectively, for thecase of F02022N. From the graph, it can be seen that, afteran initial non-conformity up to r/B~0.5–0.6, the extractedload values agree with the applied load quite well.

Dynamic Line-load on an Edge of a Sheet

Experimental Details

A photograph of the experimental setup employed for study-ing the problem of dynamic line-load acting on the edge of aplanar sheet using DGS is shown in Fig. 10. The loadingdevice consisted of an Al 7075-T6 long-bar (2 m long,25.4 mm diameter) with a cylindrical (bull-nose) head, agas-gun and a high-speed digital image acquisition system.The long-bar was aligned with the gas-gun barrel containinga 305 mm long, 25.4 mm diameter cylindrical striker also

s

lo

F = 1077

F = 1077

F = 1077

pecimen edg

oading point

7 N

N

2 mm

2 mm

N

2 mm

e

F

F

F = 1552 N

F = 1552 N

F = 1552 N

2 mm

2 mm

2 mm

y

F = 2

F = 2

F = 2

x

( ),r θ

2022 N

2022 N

2 m

2 m

2022 N

2 m

m

mm

mm

Fig. 7 Measured ϕx (row 2) andϕy (row 3) contours near theloading point for different loadlevels. Contour interval01×10−3

radian. (The left vertical edge ofeach image corresponds to theloading edge where F acts at theorigin)

104 Exp Mech (2013) 53:97–111

made of aluminum. A Cordin model-550 ultra high-speeddigital camera equipped with 32 CCD sensors and a five-facet rotating mirror, and two high-energy flash lamps wasused for recording speckles. A computer connected to thecamera was used to control parameters such as trigger delay,flash duration, framing rate and image storage. The speci-men, a 129×67.5×9.4 mm3 clear PMMA plate, was placedon an adjustable platform and its long edge was registeredagainst the cylindrical head of the long-bar as shown inFig. 10. The loading was initiated by suddenly releasing

the compressed air in the gas-gun cylinder using a solenoidvalve to propel the striker placed inside the barrel. Theaccelerating striker impacted the long-bar and initiated acompressive stress wave that traveled along the length ofthe bar before imparting a transient line-load to the edge ofthe specimen. An electrical circuit, closed when the strikercontacted the long-bar, was used to trigger a delay generator

F = 2022 N

F = 2022 N

2 mm

2 mm

Fig. 8 Comparison of experimental and analytical angular deflection(top: ϕx and bottom: ϕy) contours for F02022N. Contour levels are in1×10−3 radians

1000

1250

1500

1750

2000

2250

2500

0.2 0.5 0.8 1.1 1.4 1.7 2

Load

(N

)

r/B

Applied LoadLoad fromLoad from

triaxial zone

plane stresszone

x

y

φφ

Fig. 9 Comparison of experimentally extracted load to the knownapplied load; quasi-static line-load problem

camera lens

gas-gun

cylindricalimpact hea

s

lad

speci

striker l

imentarget

long bar

plane

Fig. 10 Experimental setup used to measure angular deflections oflight rays caused by a deforming transparent specimen when subjectedto dynamic line-load (top). Close-up of the specimen, speckle target,and the long-bar (bottom)

Exp Mech (2013) 53:97–111 105

which in turn activated the camera with a user-specifieddelay. A strain gage (CEA-13-062UW-350 from Vishaymicro-measurements) affixed to the long-bar, and connectedto a LeCroy digital oscilloscope via an Ectron signal condi-tioner was used to measure the strain history (Fig. 11) in thelong-bar during loading [34]. The measured strain historywas used to calculate the particle velocity history in thespecimen at the specimen/bar interface.

The distance between the specimen and the camera lensplane (L) was ~1000 mm and the one between the specimenmid-plane and the target plane (Δ) was 30 mm. Using thehigh-speed camera, a set of 32 reference (undeformed)images, one for each sensor, were first captured under no-load condition at 200,000 frames per second. Next, thespecimen was subjected to a dynamic line-load using thelong-bar setup. During loading, a set of 32 consecutiveimages of the deforming specimen were captured at thesame framing rate. The deformed-undeformed image pairsfor each of the 32 CCD sensors were then correlated toobtain the in-plane displacement fields, δx and δy. A facet/sub-image size of 15×15 pixels (1 pixel029.2 μm on thetarget plane) without any overlap was used in the imageanalysis for extracting displacement components. The dis-placement fields were then used to compute the angulardeflection fields (ϕx and ϕy), and are shown in Fig. 12. Alsoshown in Fig. 12 are the corresponding speckle imagesrecorded during the dynamic event. Appropriate boundaryconditions were imposed as discussed previously for quan-tifying the contour levels.

Comparison of Experimental and Numerical AngularDeflection Fields

The dynamic angular deflection fields were also numericallyobtained by performing an elasto-dynamic finite element(FE) analysis to compare with the measured fields. Theelastic modulus and Poisson’s ratio of PMMA used in thesimulation were from ultrasonic pulse echo measurement oflongitudinal and shear wave speeds [35] and mass density.

Using the measured incident strain history, εI(t), in thebar during loading, and the measured parameters listedin Table 1, the transmitted particle velocity, VT, wascalculated using [36],

VT ¼ VI2ρICIAI

ρICIAI þ ρTCTAT; ð14Þ

where VI0CIεI is the particle velocity in the incidentbar, and ρ, C, and A denote the mass density, bar wavespeed and area, respectively, and subscripts I and Tdenote the incident and transmitted values. (The areavalues used in this work were assumed to be propor-tional to length over which stress wave transmissionoccurred.) The transmitted particle velocity historyshown in Fig. 13 was used as the input for FE compu-tations using the structural analysis software packageABAQUS. The FE model consisted of 17,004 four-node quadrilateral elements (total DOF033964) withthe smallest element of size~0.5 mm. The numericalmodel was solved using explicit time integration schemeand the instantaneous in-plane stress invariant (σxx+σyy)field near the loading point was obtained. The represen-tative (σxx+σyy) contours corresponding to the timeinstants in Fig. 12 are as shown in Fig. 14. From thisfield, the data along θ00°, 30° and 60° were extractedand the in-built differentiation scheme embedded inABAQUS was used to compute spatial derivatives of(σxx+σyy) in the x- and y-directions. For comparisonwith measurements, these numerically obtained stressgradient data along with the experimental onescorresponding to a time instant 30 μs after the arrivalof the stress waves at the specimen/bar interface areshown in Fig. 15. From the graphs, a rather goodagreement between the computational and experimentalresults is evident beyond the triaxial zone (r/B~0.5).

The (σxx+σyy) contours in Fig. 14 are generally circular(except very close to the loading point) and are similar to theones expected from the Flamant solution (equation (11)). Thissuggests that the functional form of the instantaneous defor-mation fields near the transient line-load can be approximatedby the Flamant’s equations.1 Accordingly, an attempt wasmade to use equation (13) for a dynamic line-load problem toextract the load history. Three sets of discrete values of angulardeflection fields at various (r, θ) locations, excluding theimmediate vicinity of the loading point (within r/B00.4, 0.5and 0.6) where triaxial deformations dominate, were collected

-0.18

-0.12

-0.06

0

0.06

0.12

0.18

0 150 300 450 600 750

Str

ain

(%)

Time (μs)

Incident pulse, εI

Reflected pulse, εR

Fig. 11 Measured strain history in the long-bar used to deliver adynamic line-load on the edge of the transparent PMMA specimen.(time00 corresponds to the start of data acquisition and not impactloading)

1 For simplicity, this approach was used instead of the solution to theLamb problem [40]. Lamb solutions are available for a line pulse or asuddenly imposed constant line load on an elastic half-space; for largevalues of time, the latter case reduces to the elastostatic solution. Theapplied load history in this case being approximately a ramp transition-ing to a constant value, elastostatic behavior is to be expected behindthe shear wave front.

106 Exp Mech (2013) 53:97–111

for each time instant. These values were then used along withequation (13) to extract three values of the instantaneous load(F(t)) using an overdeterministic least-squares analysis [37] forevery time instant. For each time instant, the intermediate valueis plotted in Fig. 16 and the error bars show the upper andlower bounds of F(t) resulting from the usage of three differentdata sets. The load history was also assessed from the straingage measurements on the long-bar as, FbðtÞ ¼ EbAb

"I þ "Rð Þb where Eb (072 GPa) and Ab are the elastic modulusand cross-section area of the bar, respectively, and εI and εRdenote incident and reflected strain signals. Figure 16 shows aplot of measured load histories from the strain gage along with

s

lo

0

0

0

t = 15 μs

t = 15 μs

t = 15 μs

specimen edg

oading point

5 mm

5 mm

5 mm

ge

t = 25 μs

t = 25 μs

t

0

0

0

t = 25 μs

5 mm

5 mm

y

5 mm

t = 35 μs

0

0

x

( ),r θ

t = 35 μs

t = 35 μs

0

5 m

5 m

5 m

m

m

mm

Fig. 12 Measured ϕx (row 2)and ϕy (row 3) contours near theloading point for different timeinstants. Contour interval01×10−3 radian. (The left verticaledge corresponds to the loadingedge with impact load F(t) act-ing at the origin along the x-axis)

Table 1 Material properties of long-bar and PMMA specimen used inthe dynamic line-load experiment

Parameter Value

Density of bar material (Al 7075-T6) 2730 kg/m3

Longitudinal wave speed in bar 5700 m/s

Width of cylindrical head 25.4 mm

Density of specimen material (PMMA) 1010 kg/m3

Longitudinal wave speed in PMMA 2657 m/s

Thickness of PMMA specimen 9.4 mm

0

5

10

15

20

0

V(t)

50 100Time(μs)

150)

200

Fig. 13 Particle velocity history (bottom) used in the elasto-dynamicfinite element model (top)

Exp Mech (2013) 53:97–111 107

the ones from optical data analysis. The load historyextracted using ϕx and ϕy fields and equation (13) is ingood agreement with the strain gage measurements untildamage/failure initiation (at ~40 μs after impact) occursat the loading point. Beyond that, the two results no-ticeably deviate from each other. It is evident from thegraph that just after the start of local crushing/pulveri-zation of PMMA at ~40 μs, the measured load historystarts to decrease for about 20 μs before rising again.

These differences are to be expected as the closed formsolutions used for optical data extractions do not hold inthe post-failure regime. Further, the local crushing andmicro-cracking events of PMMA are sensed by thestrain gage data as a drop in the signal whereas theload extracted by analyzing the optical data from un-damaged regions away from the load point/s lag inrevealing this information. Furthermore, the opticallyextracted loads based on over-deterministic analysis

t = 15 s t = 25 s t = 35 s

Fig. 14 Contour plots of in-plane stress invariant at various time instants during the dynamic loading of PMMA from finite element analysis.Dotted lines correspond to paths along 0°, 30° and 60° relative to the loading direction used for computing stress gradients

0

0.005

0.01

0.015

0.02

φ x(r

adia

ns)

r/B

Flamant -0deg.FEA -0deg.Exp -0deg.

0

0.005

0.01

0.015

0.02φy

(rad

ians

)

r/B

Flamant -0deg.FEA -0deg.Exp -0deg.

0

0.005

0.01

0.015

0.02

φx(r

adia

ns)

r/B

Flamant -30deg.FEA -30deg.Exp -30deg.

0

0.005

0.01

0.015

0.02

φ y(r

adia

ns)

r/B

Flamant -30deg.FEA -30deg.Exp -30deg.

0

0.005

0.01

0.015

0.02

φ x(r

adia

ns)

r/B

Flamant -60deg.FEA -60deg.Exp -60deg.

0

0.005

0.01

0.015

0.02

0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5

0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5

0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5

φ y(r

adia

ns)

r/B

Flamant -60deg.FEA -60deg.Exp -60deg.

Fig. 15 Comparison of analyti-cal, finite element and experi-mental angular deflections(column 1: ϕx and column 2: ϕy)in PMMA plate subjected to adynamic line load at t030 μs

108 Exp Mech (2013) 53:97–111

provide an averaged response over the region. The large errorbars in the post-failure regime are due to the expanding zoneof triaxiality as the failure front progresses (Fig. 17) during thedynamic event. Therefore, the optical data used for least-squares analysis had to be collected well beyond r/B00.5,based on the position of the instantaneous damage front.

These load estimates thus correspond to optical data analyzedusing measurements from 1.0<r/B<1.8. The correspondingerrors in the post-failure regime were based on data analysisfor different inner radii used during the analysis.

0

1500

3000

4500

6000

7500

0 20 40 60 80 100

Load

(N

)

Time (microseconds)

Start of material crushing/fracture

Load from x

Load from y

Strain gage

φφ

Fig. 16 Load histories measured using DGS (symbols) and strain gage(broken line). The agreement between the two is good up to the start ofthe material crushing/cracking at the impact point

Fig. 17 Failure progression(row 1) and corresponding an-gular deflection contours, ϕx

(row 2) and ϕy (row 3) contoursin PMMA plate subjected to adynamic line load. Contourinterval01×10−3 radian

mirror re

point of i

egion

impact

cam

avity formedmaterial rem

crushed regloading poi

hackle re

d due to moval

gion at int

egion

Fig. 18 Failed PMMA specimen after experiencing a dynamic lineload on the edge (top). Close-up of the damaged/cracked region(bottom)

Exp Mech (2013) 53:97–111 109

Stress Gradients in the Post-failure Regime

The analytical and/or numerical solutions are generally validonly until the onset of failure—crushing and/or micro-cracking. In addition, they are dependent on assumptionsassociated with failure modes and post-failure material be-havior. In such scenarios, only direct experimental measure-ments offer reliable data for structural analyses/design. Inview of this, stress gradients even after the material hasundergone cracking and crushing near the loading point isalso presented. The first row in Fig. 17 shows failure pro-gression in the specimen after the initiation of failure at theloading point. This causes the image correlation to fail in theregion of intense damage. The highly saturated (or, decorre-lated) gray scale around the loading point represents thedamaged zone. The second and third rows in Fig. 17 showcontours of ϕx and ϕy. The observed changes in contourdensity and their shapes relative to the ones prior to damageinitiation qualitatively indicate contributions of a smearedloading front as well as the arrival of the reflected stresswaves from the far edges of the specimen to the load pointvicinity. Figure 18 shows photographs of the failed speci-men as well as a close-up of the load point vicinity. Due tothe dynamic load, PMMA is symmetrically chipped offfrom the plate relative to the mid-plane. A close-up of thefailure surface reveals the typical mirror and hackle regions.References [38, 39] describe the mechanisms involved inthe formation of conical chips in glass produced due tospherical indenters near the specimen edge. Irrespective ofthe indentation distance from the edge, early material failurehas been shown to be characterized by a sub-surface mediancrack directly under the contact point. However, highermaterial compliance in the near-edge region and the bendingmoment experienced by a growing crack have been sug-gested as sources of chip formation when the indentationdistance from the specimen edge is small (less than0.05 mm) [38]. In the current case, the entire specimenthickness makes a line contact with the cylindrical impactorhead. Given the symmetric nature of the contact, it is rea-sonable to expect symmetric chipping about the specimen’smid-plane as in Fig. 18. Chai et al [39] add to the aboveinferences by attributing chip initiation and the radial stria-tions in the hackle region to stress enhancement in thespecimen edge and multiple stress-wave reflections at theedge respectively.

Conclusions

A full-field optical method called Digital Gradient Sensing(DGS) to measure angular deflections of light rays causedby in-plane stresses in transparent planar sheets under staticand dynamic loading conditions has been developed and

demonstrated. The method uses 2D digital image correlationtechnique to quantify deflections of light rays caused bynon-uniform elasto-optical variations in planar phase objectssubjected to mechanical loads. The governing equations tointerpret the optical measurements are described. The anal-ysis suggests that the angular deflection of light rays isrelated to the in-plane gradients of (σxx+σyy) under planestress conditions. The method is demonstrated by measuringcontact stress gradients due to statically and dynamicallyapplied line loads acting on the edge of a PMMA sheet. Thedetails of the experimental setup including those on highstrain-rate loading and high-speed imaging are presented.

A test for optical homogeneity of PMMA sheet is firstcarried out to evaluate the reliability of measurements underimposed loads. The results indicate that the inherent opticalanomalies in no-load conditions produce angular deflectionsthat are at least an order of magnitude lower than the onesdue to the applied loads. The reported angular deflectionsare significantly higher than the accuracy of the method(0.5×10−4–1×10−4 radians) assessed based on the smallestmeasurable displacement using image correlation for thechosen experimental/optical parameters.

The full-field angular deflections (ϕx, ϕy) with respect tothe two orthogonal in-plane (x, y) directions are successfullymeasured for specimens subjected to both quasi-static anddynamic line-loads. In regions outside the dominant zone oftriaxiality, the measured angular deflections are in goodagreement with the ones based on the Flamant solution fora line load on the edge of a half-space. For the impactloading case, the angular deflections are compared withnumerical results obtained by performing a complementaryelasto-dynamic finite element analysis of the problem usingmeasured strain history on the long-bar as input. Based onthe observed similarities in the static and dynamic stressinvariant contours, the Flamant equations are used to esti-mate the load history from dynamic angular deflection fieldsthrough regression analysis of optical data in the impactpoint vicinity outside the zone of dominant triaxiality. Theload history thus obtained agrees well with the one fromstrain gage measurements on the loading bar until the initi-ation of material failure. The angular deflectionscorresponding to post failure duration are also measuredand reported.

Acknowledgments Partial support for this research through grantsW911NF-12-1-0317 from the U.S. Army Research Office and NSF-CMMI-1232821 from the National Science Foundation is gratefullyacknowledged.

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