Determination of Residual Stress Fields Beneath aVickers Indentation Using Photoelasticity

by R. J. Anton, I. Miskioglu and G. Subhash

ABSTRACT—Static Vickers indentation tests were performedon Homalite specimens with an intent to obtain the resid-ual stress distribution beneath the indentation. The indentedspecimens were placed in a circular polariscope to view thefringe patterns corresponding to the induced residual stress.Similitude analysis was later employed to identify the func-tional relationship between the various parameters related toan indentation test. The analysis resulted in a unified relation-ship that can assist in the determination of residual stress innontransparent materials subjected to similar geometric andloading conditions. The shear stress contours provided herecan also be used as guidelines to verify constitutive modelsunder complex three-dimensional loads.

KEY WORDS—Residual stress, indentation, photoelasticity

Static indentation fracture mechanics has been widelyused for the determination of mechanical and fracture char-acteristics of brittle materials.1−3 Indentation by a sharpindenter has been successfully used to model the mate-rial removal mechanisms during surface grinding of brittlematerials.4−6 The technique was also extended for determin-ing the residual stress in surface coatings on brittle substrates7

and thin surface layers.8 Evaluation of residual stress due toan indentation has been the focus of several recent investi-gations because residual stress plays a key role in dictatingthe fracture characteristics of machined ceramics. Yoffe9

developed a theoretical model for elastic stress due to inden-tation by modeling the indentation as an expanding cavity inan infinite medium. Chiang, Marshall and Evans10,11 foundthat the residual stress is a function of the ratio of elasticmodulus to hardness. Zeng and colleagues12−14 used the in-dentation technique itself to determine residual stress arounda macroindentation in soda-lime glass by subjecting the pro-cess zone to microindentations. The above experimentalinvestigations have focused their attention on the determina-tion of residual stress around the indentation but not beneathit, which poses a greater challenge. Efforts to access thiszone by sectioning the specimen close to the indentation cancause the release of stress and may lead to further damage ina brittle specimen. One typically has to resort to nondestruc-

R. J. Anton is Product Engineer, Ford Motor Company, Dearborn, MI 48121.I. Miskioglu (SEM Member) and G. Subhash (SEM Member) are AssociateProfessors, Mechanical Engineering–Engineering Mechanics Department,Michigan Technological University, Houghton, MI 49931.

Original manuscript submitted: December 4, 1998.Final manuscript received: February 8, 1999.

tive methods for accurate analysis. Accordingly, this workfocuses on the determination of residual stress beneath theindentation of a transparent birefringent material using pho-toelasticity, whose principle of operation is described brieflyin the following.

Photoelasticity is the study of stress induced in an opti-cally isotropic transparent material. The material must ex-hibit optical anisotropy while stressed and return to isotropyas the stress is relieved. This behavior is known as temporarydouble refraction or birefringence. The influence of stress isviewed in the form of fringes in the specimen material throughan optical device called a polariscope (see Ref. 15). The pa-rameter that relates the induced stress to the fringes developedin a birefringent material is called the material fringe value(fσ), which is a measure of the load required per unit lengthof a material to induce a single fringe order. A common wayto determinefσ is by measuring the fringe orders induced ina disk specimen subjected to diametral compressive loads.15

Homalite-100, a thermoset polyester, was used as a modelmaterial in the current investigation to represent the responseof brittle materials. The brittle nature of Homalite-100 hasbeen confirmed in a uniaxial compressive test. Figure 1 il-lustrates the linear nature of the quasi-static stress-strain re-sponse of Homalite until failure. Figure 2 reveals a plot ofload (P ) versus fringe order (N ) measured at the center of theHomalite-100 circular disk during the diametral compressiontest used for the calibration offσ. Note that the slope of thelinear fit is proportional tofσ, which is given by

fσ = 8

π(D)

(P

N

), (1)

whereD is the diameter of the disk. The linear nature of thecurve indicates a constant material fringe order for Homalitein the range of loads used in this investigation. Thefσ forHomalite was found to be 21.76 kN/m which is slightly lowerthan 23.6 kN/m quoted by Dally and Riley.15 This differenceis possibly due to the slight variations in the processing con-ditions, such as casting temperature, resin batch composition,age of the material and so on, which is common in polymericmaterials.

Experimental Method

The Homalite specimens were rough cut from a large sheetand then routed along the edges at 30,000 rpm to obtain astress-free surface. One of these surfaces was carefully pol-ished to a 1-micron finish so that the fringes could be seenclearly when viewed in a polariscope. The final specimen

Experimental Mechanics • 227

Fig. 1—Quasi-static stress-strain curve for Homalite-100

dimensions were of approximately 6×6×6 mm. Static Vick-ers indentation experiments were performed on the clear flatsurface at three different loads (98.1 N, 196 N, 294 N) on threeseparate specimens. The indentation loads were kept low toavoid cracking at the corners of the indentation and subse-quent release of stress. Once a sample had been indented,it was quickly brought to an optical microscope to minimizethe stress relaxation that may occur in Homalite. An opticalmicroscope, fitted with a condensed circular polariscope anda monochromatic filter, was used to observe the isochromaticfringe patterns. The photographs of the fringe patterns weredigitized using an image-processing system. The digitizedfringe pattern was then sharpened and multiplied to obtainboth whole- and half-order fringes. A total of 40 data pointswere collected from each digitized fringe pattern. A digitalimage, along with a trace of points, is shown in Fig. 3, wherethe indentation impression is indicated by a thick line. Afterdetermining the fringe orders that were developed beneathan indentation, the shear stress along a fringe contour wasdetermined as per the equation

τ = σ1 − σ2

2= Nfσ

2h, (2)

wheret is the shear stress,N is the isochromatic fringe orderandh is the thickness of the sample.

Results and Discussions

The low loads applied to Homalite caused an observablefringe pattern on the order of 1.5 at a load of 294 N (30 Kg).A typical plot of fringe contours and the interpolated shearstress values in Homalite for the 294-N load are shown inFig. 4. In this figure, circles indicate the position wherethe data points were extracted from the photographs of theisochromatic fringe patterns, and the continuous lines are thelinear-interpolated value of the residual shear stress that wasobtained using Matlab software.16 It can be observed thatat a depth of about 0.02 mm beneath the indentation andfor an indentation load of 294 N, the maximum shear stressdeveloped was found to be around 1.25 MPa.

The indentation tests on Homalite were conducted accord-ing to the ASTM specification17 that recommends that thedimensions of the specimen be such that the distance be-tween a free surface and the center of an indentation be no

Fig. 2—Photoelastic calibration curve for determining the ma-terial fringe value in Homalite-100

Fig. 3—Digital trace of photoelastic fringe pattern. Note thelocation of points (indicated by +) where the data were ex-tracted for mapping the contours

Fig. 4—Contours of residual shear stress distribution beneaththe tip of the indentation in Homalite for a load of 294 N

228 • Vol. 39, No. 3, September 1999

closer than 2.5 times the indentation diagonal. Additionally,the depth of the specimen must be sufficiently long such thatno impression of the indentation is left on the opposite side,as this can also lead to incorrect hardness measurements.Hence, the geometry of the Homalite specimens was simi-lar to any other brittle material used in a typical indentationtest. Moreover, the load distribution on Homalite specimenswould also be similar to the load distribution used in testing ofother brittle materials where similar indenter geometry wouldbe used. Hence, according to the laws of similarity (simili-tude analysis),18 the results obtained from Homalite can beused to estimate the residual stress distribution in nontrans-parent brittle materials under similar loading conditions. Inthe following analysis, the problem was considered to betwo-dimensional, where the effect of Poisson’s ratio is negli-gible except at the loading point, where the stress distributionwould be three-dimensional. Since the goal was to estimatethe residual stress beneath the indentation, the effect of theloading point will be negligible as one moves farther awayfrom it (as per St. Venant’s principle).

The similitude analysis requires the determination of thepertinent variables that affect the phenomenon under study.These variables for the Vickers indentation tests conductedon Homalite are taken as follows:

Variable DimensionP Indentation load Fd Mean diagonal of Vickers

indentation Lz z-coordinate as measured from

the tip of the indentation Lτ Residual shear stress FL−2

For the four pertinent variables above and two basic dimen-sions (F, L), the following nondimensionalπ terms can bewritten:

π1 = τ

(P/d2)π2 = z

d.

These twoπ terms suggest a functional relationship of theform

τ

(P/d2)= f

( z

d

). (3)

Note thatP/d2 is proportional to the Vickers indentationhardness, which is given by17 1.8544P/d2, whereP is inNewtons andd is in millimeters.

The nature of the functional relationship mentioned abovewas studied using data from the photoelastic fringe patternsof residual stress obtained from the static Vickers indentationtests at three different loads. The results are shown in Fig. 5.This figure reveals that the relationship betweenπ1 andπ2can be approximated by a straight line. Since for similitudethe terms must be the same in the model and the prototype,that is,

(τ

(P/d2)

)model

=(

τ

(P/d2)

)prototype

(4)

Fig. 5 can be used to estimate the residual stresses beneaththe tip of the Vickers indenter in other brittle materials.

Fig. 5—Plot of normalized residual stress versus the depthbeneath the indentation in Homalite

There are several limitations to the current method whenapplied to real materials. First, a typical ceramic consistsof microstructure with grain boundaries, second-phase parti-cles, triple points, compositional variations and so on, whichcan significantly influence local variations in the inducedstress states. Since Homalite is an amorphous material with-out such microstructural variations, the current photoelas-tic analysis fails to capture the influence of local variationsin the stress distribution. Second, the current analysis fo-cuses on obtaining only the maximum shear stress contoursdue to the simplicity of the analysis. In principle, the fail-ure of brittle materials is governed by the maximum normalstress criterion. However, with additional photoelastic data,the principal stresses along the isochromatic fringes can beobtained18 beneath the indentation. This requires collectionof isochromatic and isoclinic (principal stress direction) data,and if automated systems are not used, the data reductionand analysis becomes a little more complex. Third, the cur-rent method does not allow for the determination of residualstresses closest to the tip of the indentation due to the three-dimensional state of stress in this area. Also, the fringe den-sity at the close proximity of the tip of the indentation be-comes extremely high because of the stress concentration,and the resolution of these fringes is then limited by the reso-lution of the optics used. However, the analysis can still givea reasonably good estimate of the overall stress magnitudebeneath the indentation away from the tip.

Conclusions

The merging of indentation testing and photoelasticityprovides a useful analysis tool for determining the residualstress beneath an indentation in a model material. From thesimilitude analysis, if the indentation loadP and the meandiagonald of the Vickers indentation is known for other ma-terials, the magnitude of the residual shear stress and its lo-cation below the indenter tip can be estimated by the use ofFig. 5. This is an important feature of this technique, since anestablished experimental method for determining the resid-ual stress beneath the indentation is not currently available.The residual stress distribution determined here can also beused to verify the validity of the constitutive models undercomplex three-dimensional loads. Knowledge of residualstress fields under an indentation can also provide valuableinsight into the magnitude of stress that causes the crack

Experimental Mechanics • 229

systems to propagate under complex loading conditions (suchas machining).

Acknowledgments

The authors acknowledge the support of the National Sci-ence Foundation under Grant No. DMI 9610454. The sup-port of the Michigan Space Grant Consortium in the form ofa research seed grant is also gratefully acknowledged.

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