Young Stress Analyst Competition 2015

10th International Conference on Experimental Mechanics

Heriot-Watt University, Edinburgh, UK.

CONTENTS

Young Stress Analyst Competition Finalists

Determination of micro-residual stress distribution in multi phase ceramic composites

using Raman spectroscopy

Matthew DeVries, University of Florida (USA) …………..……………………………………………………….. 1

A new white light imaging approach for intralaminar fatigue characterisation of GFRP

Jens Glud, Aalborg University (Denmark) ………………………………………………………….…………....... 7

Three dimensional in situ quantification of localised non-linear deformations within

heterogeneous materials

Matthew S. L. Jordon, University of Oxford (UK) …………………………………………………..………….. 13

Measurement of highly non-uniform residual stress fields with reduced plastic error

Ho Kyeom Kim, University of Bristol (UK) ……………………………………………………….………………… 21

Competition overview:

Entrants to the Young Stress Analyst (YSA) competition were invited to submit 1000 word

summaries of their work. The submissions were reviewed by an expert panel and four

finalists were selected and invited to attend the 10th International Conference on

Experimental Mechanics.

The YSA 2015 finalists will present their papers in a plenary session on Wednesday 2nd

September 2015. The finalists will be giving 10 minute presentations on their work with a

further 5 minutes for questions. A panel of judges will mark the presentations on technical

content, quality of presentation and response to questions. All conference delegates are

invited to attend these presentations and are welcome to ask questions.

We look forwards to welcoming you all to this session.

Determination of Micro-Residual Stress Distribution in Multi-Phase Ceramic

Composites Using Raman Spectroscopy

Matthew DeVries, Phillip Jannotti, Ghatu Subhash

Department of Mechanical and Aerospace Engineering, University of Florida,

Gainesville, FL 32611 USA

Introduction

Boron carbide (B4C) is a lightweight ceramic with excellent mechanical properties and

is an attractive material for structural applications where superior strength and lightweight are

fundamental requirements, e.g., military armor. Reaction bonding is a promising alternative to

traditional sintering of B4C, offering reduced processing temperatures and times and the ability

to produce complex, near net-shape parts. In reaction bonding, a porous B4C powder-compact

is infiltrated with molten silicon (Si). The reaction between Si and C in the compact forms

silicon carbide (SiC), bonding the microstructure together. The final ceramic is composed of

B4C, SiC, and Si. Thus, evaluating the presence and spatial distribution of residual

microstresses originating from coefficient of thermal expansion (CTE) mismatch is crucial to

understanding the mechanical properties and failure behavior of these composites. Because Si

is the weakest mechanical constituent of the microstructure, it is the focus of this study.

Residual stress determination within a material is crucial to the material design process.

X-ray diffraction is a popular method for residual stress determination, but due to limited

spatial resolution only bulk stress values can be determined. Raman spectroscopy possesses

finer resolution and is used to determine the microscale distribution of residual stresses in

electronics. In this method, characteristic vibrational frequencies of a material manifest as

characteristic peaks in the Raman spectrum for a given (laser) excitation wavelength. When a

crystal is deformed, the induced strain changes the equilibrium spacing and force constants

1

between atoms, shifting the characteristic Raman peaks which can be correlated with a tensile

or compressive residual stress [1].

Aims

The current study employs Raman spectroscopic mapping to assess the magnitude and

spatial distribution of residual microstresses due to CTE mismatch in B4C-based ceramic

composites. These values are compared to theoretical residual stresses in two-phase composites

calculated with Eq. (1) [2],

𝜎 =∆𝛼∗∆𝑇

0.5(1+𝑣𝑚)+𝑓𝑝(1−2𝑣𝑚)

𝐸𝑚(1−𝑓𝑝)+1−2𝑣𝑝

𝐸𝑝

(1)

where ∆α is the CTE difference, E is elastic modulus, v is Poisson’s ratio, and fp is Si particle

volume fraction. Raman spectroscopy results will verify the validity of this equation.

Experimental Procedure

The reaction bonded B4C-SiC-Si ceramic examined was obtained from M-Cubed

Technologies (Trumbull, CT, USA) with a reported composition of 75% B4C, 8% SiC, and

17% residual Si. Figure 1 shows the typical ceramic microstructure using optical microscopy

(Fig. 1(a)) and Raman mapping (Fig. 1(b)) of various phases based on their characteristic

Raman peaks (Fig. 1(c)). We will focus on residual stresses in Si because Si is the weakest

mechanical phase and also has the lowest CTE. The CTE for B4C, SiC, and Si are 6.0x10-6/°C,

4.8x10-6/°C, and 3.7x10-6/°C, respectively. Thus, after reaction bonding, the carbide phases

shrink more than the Si phase, leading to residual compression in the Si. Assuming equibiaxial

residual stresses in the Si pockets, Eq. (1) predicts -589 MPa and -127 MPa for B4C-Si and

SiC-Si mismatch, respectively. Taking a weighted average of the thermal stresses based on

relative fraction of carbide phases, -545 MPa of residual compression is expected over the

2

entire Si region. Note that this predicted value assumes spherical Si phases surrounded by a

semi-infinite carbide matrix.

Fig. 1. (a) Optical micrograph of the reaction bonded ceramic. (b) Raman micrograph showing

spatial phase distribution. (c) Characteristic Raman spectra for each component phase.

Raman spectroscopic mapping was performed using a Raman spectrometer with a 532

nm laser (800 nm spatial resolution) at 10 mW power. Note that a stress-free Si Raman peak

appears at a position of 520 cm-1 (Fig. 2(a)). Shifts in peak position (∆𝑤) can be correlated with

a residual stress value based on (2) [1],

𝜎𝑥 = 𝜎𝑦 = −250∆𝑤 (2)

where peak shifts to greater wavenumbers (+∆𝑤) indicate compressive stresses and peak shifts

to lower wave numbers (−∆𝑤) indicate tensile stresses.

Results and Discussion

Figure 2(a) shows the typical Raman spectra of Si in its pre-processed and post-

processed states. The Si material used to infiltrate the B4C compact (lump and powder) exhibits

a peak centered at 520 cm-1, indicating that it is stress-free before processing. The residual Si

regions in the processed ceramics showed significant peak shifting (1-2 cm-1), indicating

residual stress. In the periphery of the Si region, the peak shifted towards higher wavenumbers,

indicative of residual compressive stress. In the interior, lower wavenumbers or residual tensile

stresses were observed.

3

Fig. 2. (a) Raman spectra for various Si regions. (b-d) Raman maps of stress distribution in Si

regions of various size and shape. (e) Color bar indicating magnitude and stress state in b-d.

Raman maps (Fig. 2(b-d)) illustrate the wide distribution of residual stresses in Si, in

contrast to the uniform stress predicted by Eq. (1). Stress distributions became more non-

uniform as the Si area increased. Small Si regions (Fig. 2(b)) revealed primarily compressive

residual stresses, while large silicon regions (>5 µm) exhibited large interior tensile stress close

to 500 MPa due to quicker peripheral cooling. Residual compression along the periphery (up

to -495 MPa) is in reasonable agreement to theoretical predictions using Eq. (1). Additionally,

it was observed that greater stress non-uniformity occurred in irregularly-shaped regions and

regions with high aspect ratios.

Conclusions

Raman spectroscopic mapping was used to generate spatial distribution map of residual

microstresses due to CTE mismatch in a reaction-bonded ceramic composite. The mapping

technique matches the classical model near the periphery of Si regions but showed significant

non-uniformity and tensile stress in interior regions not predicted by the conventional

relationship. These results demonstrate the need to measure residual stresses at finer scales than

4

the classical equation which predicts global residual stress values. Such refinement is necessary

to understand the influences of processing on thermal stress generation which are crucial to

correlating the microstructure to the resulting properties and mechanical behavior.

References

[1] A. Tiberj and J. Camassel. "Raman imaging in semiconductor physics: Applications to

microelectronic materials and devices," in Raman Imaging 2012.

[2] Chawla K. K. Ceramic Matrix Composites, Kluwer Academic, Boston, MA, USA, 2003

5

6

A NEW WHITE LIGHT IMAGING APPROACH FOR INTRALAMINAR

FATIGUE CHARACTERISATION OF GFRP

J.A. Glud1, J.M. Dulieu-Barton

2, O.T. Thomsen

1,2 and L.C.T. Overgaard

1

1Department of Mechanical and Manufacturing Engineering, Aalborg University

Fibigerstraede 16, DK-9220 Aalborg East, Denmark

E-mail: {jag,ott,lcto}@m-tech.aau.dk, web page: http://www.m-tech.aau.dk

2Faculty of Engineering and the Environment, University of Southampton

Highfield, SO17 1BJ Southampton, United Kingdom

Email: {janice,o.thomsen}@soton.ac.uk, web page: www.southampton.ac.uk/engineering

Keywords: Multiaxial Fatigue, Composites, Intralaminar, Digital Image Correlation

Research Objective:

For more than four decades, experimental characterisation of fatigue damage in laminated composites has

been a subject of great interest. The experimental characterisation has benefitted from major advances in

non-destructive measurement techniques. However, the characterisation process from sampling test results to

post processing data is, to a large extent, still manual. Therefore, accurate fatigue characterisation becomes a

labour intensive task since many sampling points are required, which in turn generates large amounts of data

for post-processing. The consequence of the manual characterisation process is often experimental

campaigns of limited scope and size, and/or significant data parameterisation, which often makes it difficult

to draw firm conclusions. The objective of the current research is to develop an automated sampling and

post-processing method capable of quantifying the fatigue damage in laminated composites in a fast,

accurate and reproducible way. The damage mode of interest is the most commonly occurring damage mode,

namely intralaminar damage represented by transverse through-the-thickness cracks, and its influence on the

overall laminate and structural behaviour. In order to monitor the evolution of transverse cracks a novel

automatic crack counting method was developed. The method uses Trans-Illuminated White Light Imaging

(TWLI) to monitor crack evolution and state of the art digital image processing to automatically count the

cracks. The complete automated experimental method also relies on lock-in DIC [1] to detect stiffness

degradation. This summary outlines the development of the automated experimental method, and in

particular the development of the novel automatic crack counting method. Furthermore, the summary

7

presents damage evolution results from an experimental campaign of fatigue tests on GFRP laminates with a

[0/-60/0/60]s layup.

Automating the Experimental Fatigue Characterisation:

Figure 1 shows the experimental setup used for capturing images during tests on the GFRP coupon

specimens. One camera is for lock-in DIC and the other is for TWLI. Uniform intensive light is applied on

both sides of the specimen. Lock-in DIC is a method for calculating strain amplitudes from an undersampled

signal. This means that a high resolution, low frame rate camera can be used to capture images from a

specimen subjected to fatigue loading without the need to synchronise image capture at the maximum load or

interrupt the fatigue test to capture images. The non-contact nature of DIC provides robust measurements

since difficulties such as detachment of strain gauges and clip gauges slipping are avoided. However, the

resolution of the DIC is such that it cannot monitor individual crack evolution, just global reductions in

stiffness. Therefore it was necessary to devise a new means of imaging individual crack evolution (known as

TWLI). This exploits the transparent nature of the GFRP material and uses transmitted light to monitor the

evolution of off-axis cracks. The combination of lock-in DIC and TWLI constitutes the fully automated

acquisition system.

Figure 2a shows an in-situ image acquired using TWLI. The “texture” seen in the material is due to stitching

of the fiber mats that constitute the tested laminate. Cracks appear as dark straight lines. The stitching makes

it difficult to distinguish cracks and stitches. The stitching pattern remains unchanged throughout the fatigue

test, whereas the crack pattern changes as more cracks develop. Therefore, a motion compensation procedure

was developed so the in-situ image of the damaged laminate could be compared with the image of the

laminate acquired before commencing the fatigue test. Comparing the images through pixel-wise division

gives the image shown in Figure 2b. It is seen that the texture from the stitching is removed and a clean

image of the transverse cracks is obtained.

Figure 1: Experimental setup for in-situ measurement of strain field and intralaminar damage.

8

(a) (b)

Figure 2: (a) transverse cracks detected using trans-illuminated white light imaging and (b) filtered image

making transverse cracks more visible.

Combining the motion compensation with an image filtering technique called Gabor filtering [2] along with

the application of a heuristic counting procedure enabled an automatic counting algorithm capable of

counting cracks in each layer of the laminate to be developed. In Figure 3, the cracks counted by the

automatic counting algorithm are overlaid on the motion compensated image from Figure 2b. As can be

seen, virtually all cracks are captured over their full length.

Figure 3: Cracks detected by image processing algorithm overlaid on filtered image of cracks.

Experimental Results:

The automatic crack counting algorithm allows for in-situ crack counting during the fatigue tests. In Figure

4a crack density evolution curves for one-block Constant Amplitude (CA) tests are seen. The crack density is

a standard quantification of the crack state in a given layer and represents the number of cracks counted

9

perpendicular to the propagation direction. Based on the results from the CA tests it was possible to conclude

that crack density evolution rate and saturation value increases with increased load amplitude. Variable

Amplitude (VA) tests were carried out to study the load history dependency of the crack density evolution.

The VA tests consisted of two CA blocks. The first and second block was chosen to be equal to the low and

medium amplitude one-block CA tests. In Figure 4b the crack density evolution in the VA tests is seen along

with the crack density evolution for the medium amplitude CA tests. The CA results have been shifted such

that the average crack density of the CA and VA tests coincides at the number of cycles where the second

block loading of the VA test is initiated. From Figure 4b it can be observed that the crack density evolution

rate and saturation value only depend on the current crack density state and the current loading experienced

by the GFRP specimen.

(a) (b)

Figure 4: (a) Crack density evolution for three different constant amplitudes and (b) comparison of evolution

rate observed in variable amplitude test with the crack density evolution observed in a constant amplitude

test.

The result of increasing crack density is an increasing stiffness degradation of the material. Several models

exist for predicting the stiffness degradation as function of the crack density. In Figure 5 predictions from

one of the most recent stiffness degradation models, the so-called GLOB-LOC model [3] which requires

crack density measurements as input, is compared to stiffness degradation measurements obtained using

lock-in DIC. It is observed that the predicted and measured stiffness reductions correlate well. This strongly

supports the hypothesis that the predictive capabilities of fatigue models can only be enhanced significantly

if the understanding of the crack evolution process is improved and if the crack density is included in the

models. The research presented here shows that both of these aspects are now feasible paths due to the

developed automated experimental fatigue characterisation method. Future work will include the utilisation

of the developed experimental methodologies to further enhance the predictive capabilities of fatigue

10

modelling, and in particular to establish a combined experimental and modelling framework for taking

fatigue life predictions for GFRP composites from the coupon level to the component/sub-structure level.

Figure 5: GLOB-LOC model stiffness degradation prediction compared to stiffness measurements using

lock-in DIC and extensometer.

Acknowledgements:

The work presented was conducted as a part of a Ph.D. project at the Department of Mechanical and

Manufacturing Engineering, Aalborg University, Denmark. The project has received sponsorship from the

Danish Center for Composite Structures and Materials for Wind Turbines which is supported financially by

Innovation Fund Denmark. The support received is gratefully acknowledged.

References:

[1] R. K. Fruehmann, J. M. Dulieu-Barton, S. Quinn, and J. P. Tyler, "The use of a lock-in amplifier to apply

digital image correlation to cyclically loaded components," Optics and Lasers in Engineering, vol. 68,

pp. 149-159, may 2015. [Online]. http://dx.doi.org/10.1016/j.optlaseng.2014.12.021

[2] John G. Daugman, "Uncertainty relation for resolution in space, spatial frequency, and orientation

optimized by two-dimensional visual cortical filters," JOSA A, vol. 2, no. 7, pp. 1160-1169, 1985.

[3] Janis Varna, "Modelling mechanical performance of damaged laminates," Journal of Composite

Materials, vol. 47, no. 20-21, pp. 2443-2474, sep 2013. [Online].

http://dx.doi.org/10.1177/0021998312469241

11

12

THREE DIMENSIONAL IN SITU QUANTIFICATION OF

LOCALISED NON-LINEAR DEFORMATIONS WITHIN

HETEROGENEOUS MATERIALS

Matthew S.L. Jordan1a, D. Nowell2 and T. James Marrow1

BACKGROUND

The measurement of non-linear deformations is a common problem in engineering and material science. Non-

linear strains may result from various mechanisms: dislocation movement; crystal twinning; micro-cracking;

creep and fatigue damage are a few examples. At the onset of the non-linearity, the effect is typically localised

at ‘hot spots’ in the structure. How the strain is distributed, and its evolution with increasing load, is often critical

in the safe and economical design of components. Stresses will concentrate about inhomogeneities (pores,

cracks, particles or interfaces); nearly all materials have some form of intrinsic stress concentrator, or the

extrinsic geometry may concentrate stresses on a small region. The strain behaviour within a material may be

explored by mechanics testing with in situ non-destructive imaging techniques, such as X-ray computed

tomography, and quantitatively analysed by Digital Volume Correlation (DVC) [1-3].

In all fracture mechanics experiments, the gross deformations and movements of the sample dominate the

measured displacement field; these displacements are a combination of rigid body motions (RBM) and the

elastic strains of the sample in response to the load. The localised microstructure-dependent deformations are

much smaller. To study the damage, and relate it to the microstructure, it is necessary to separate out these

deformations sequentially. Coarse heterogeneous microstructures are particularly difficult to analyse for stress

concentrating behaviour, as the inherently variant elastic properties of its constituent phases can create

significant background variations, which may obscure the effects of damage.

OBJECTIVES

A novel analysis methodology has been developed to decouple and correct for the two dominant contributions

(RBM and imposed strains) to aid the quantitative study of heterogeneous microstructure deformations in

engineering materials. This is done by the application of a 3-D linear-elastic finite element simulation that

accurately accounts for the experimentally-measured boundary displacement conditions, to obtain the

background reference state of the homogeneous displacement field. The analysis presented is part of work that

considers fracture initiation criterion at a blunt stress-concentrating feature in polygranular graphite.

METHOD AND RESULTS

Polygranular graphite (Gilsocarbon), used as a neutron moderator in the UK Advanced Gas Reactors, has a

heterogeneous microstructure (Figure 1), of approximately spherical ‘filler’ particles of diameters up to

approximately 1 mm with large lenticular pores in a matrix comprising finer equi-axed pores. Specimens with

keyhole notches that simulate a blunt stress concentrating features (diameter 4 mm, inset Figure 2) were quasi-

statically wedge-loaded until a load drop was recorded (‘pop-in’), using a loading jig that permitted sample-

wedge self-adjustment with two degrees of freedom to minimise misalignment (Figure 2). Conducted at the

1 Department of Materials, University of Oxford2 Department of Engineering, University of Oxforda matthew.jordan@materials.ox.ac.uk

13

Diamond Light Source (DLS), in situ monochromatic X-ray imaging recorded radiographs (absorption projections,

parallel to the slot direction) during loading (Figure 3a), and X-ray computed tomographs in the loaded and

unloaded states. The radiograph field-of-view was 12 x 16 mm (4.8 μm.pixel-1); the tomographs (volume 7 x 7 x

3.5 mm, concentrated at the notch) with a voxel, or 3-D pixel, dimension of 1.8 µm.

Digital image correlation of the radiographs (DIC) and the tomographs (DVC), using LaVision’s DaVis, measured

the displacement fields. For a rigid body translation, the DVC measured displacement standard deviation was

0.2 µm with a spatial resolution of 30 µm. Data were manipulated as displacements rather than derived strains,

to preserve data integrity. Anomalous border values were excluded and the RBM of displacement and rotation

were removed: displacement by calculating the mean movement of the body, and the rotation correction with

a development of the efficient algorithm by Mostafavi et al.[2].

The homogeneous background 3-D displacement field was simulated using a half-symmetry, linear elastic finite

element (FE) model (Figure 3d). An explicit ab initio simulation would have been inaccurate and inefficient to

produce a complete description of forces, friction and crushing damage at the wedge contact would have been

required. DIC measurements of the radiographs (Figure 3b) indicated that adjacent to the notch the horizontal

and vertical displacement components varied in a linear fashion; linear fits to these displacements (Figure 3c),

which were slightly asymmetric (up to 15 % difference in gradient), were injected directly into the FE model as

boundary conditions on the notch faces. The ‘Deviation’ field was calculated as a simple subtraction of the DVC

and FE displacement fields; the FE field was spatially registered and then interpolated to the DVC nodes with a

neural network interpolator.

By subtracting the background elastic displacement field from the DVC data, it is possible to visualise and

quantify variations due to heterogeneity from microstructure or damage. The Deviation maps from 50N to 85N

(Figure 4) and associated line profiles (Figure 5) confirm that the FE simulation correctly describes the

experiment; the comparison with the FE simulation becomes invalid at the onset of macroscopic cracking. Prior

to this, the baseline Deviation variation is comparable to the DVC error (± 0.2 µm). Localised regions with larger

displacements (up to 0.7 µm), are observed. These are associated with filler particles (Figure 6); containing

larger lenticular pores, these have lower stiffness than the matrix. Previous 2-D surface observations with ESPI

also observed greater deformations associated with larger pores [4]. This methodology is now being used to

examine for damage associated with the stress concentration of the notch.

CONCLUSION

This novel methodology combines digital image correlation techniques with finite element simulation to correct

for the dominant displacement fields due to sample loading. This permits quantification of deformation at the

microstructure scale.

ACKNOWLEDGEMENTS

This work is sponsored by EPSRC Industrial CASE studentship 11220486, with support and materials supplied by

EDF Energy. The opinions expressed in this paper are the Authors’, and are not necessarily those of EDF Energy.

Diamond Light Source provided beam time as experiment EE8519-1 on beam line I12.

1. Mostafavi, M., et al., Observation and quantification of three-dimensional crack propagation in poly-granular graphite.Engineering Fracture Mechanics, 2013. 110(0): p. 410-420.

2. Mostafavi, M., et al., Yield behavior beneath hardness indentations in ductile metals, measured by three-dimensional computedX-ray tomography and digital volume correlation. Acta Materialia, 2015. 82(0): p. 468-482.

3. Vertyagina, Y., et al., In situ quantitative three-dimensional characterisation of sub-indentation cracking in polycrystallinealumina. Journal of the European Ceramic Society, 2014. 34(12): p. 3127-3132.

4. Joyce, M.R., et al., Observation of microstructure deformation and damage in nuclear graphite. Engineering Fracture Mechanics,2008. 75(12): p. 3633-3645.

14

FIGURES

Figure 1: Gilsocarbon microstructure imaged by X-ray computed tomography, consisting of filler particles of diameters up to 1 mm and

containing lenticular pores in a matrix phase containing interconnected equi-axed pores. Resolution is 1.8 µm.pixel-1.

Figure 2: Experimental loading jig and sample geometry (inset). The jig incorporates two degrees of freedom to promote wedge-sample

alignment. All dimensions are given in millimetre.

15

Figure 3: Methodology to accurately capture experimental loading conditions: (a) large field of view radiographs, focused on the notch,

imaged the general sample deformation, from which DIC measured displacements were calculated (b). Considering the regions of interest

(ROI) indicated adjacent to the notch, the displacements were linear in both the horizontal (c) and vertical (not shown) directions. The

fitted behaviour was directly injected as boundary conditions on the notch faces in the half-symmetry, linear elastic finite element

simulation (d); the FE model therefore deformed according to the experimental loading conditions. (b-d) are displaying the horizontal

opening displacement (Uy in the simulation) in millimetres.

16

Figure 4: Maps of horizontal displacement for the loading progression presented as (a) DVC and (b) Deviation. The horizontal lines indicate the position of profiles in Figure 5, and the region of interest highlighted

on the 85 N Deviation map is expanded in Figure 6.

Load /N (a) DVC displacements (b) Deviations displacements

50

75

85

95

17

Figure 5: Line profiles for the horizontal displacements extracted from the (a) DVC and (b) Deviation fields, for the lines indicated on Figure 4.

The labelled filler particle is highlighted in Figure 6.

a) DVC displacement

b) Deviation displacement

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Figure 6: Magnified region of the microstructure overlaid 85 N Deviation map (a) from Figure 4b, composed of the microstructure (b) and the

horizontal component of the Deviation field (c). Spatial localisation of the Deviations are observed in the (c), which are typically associated with

filler particles. The horizontal line in (a) is part of the line profile path plotted in Figure 5.

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20

Measurement of Highly Non-Uniform Residual StressFields with Reduced Plastic Error

H. K. Kim, M. J. Pavier and A. Shterenlikht

1 Introduction

Most mechanical strain relaxation (MSR) techniques for the measurement of residual stressinvolves three general steps: (1) removing some stressed material of the body, (2) measuringthe deformations produced by the removal of that material and then (3) the measured strain ordisplacement fields are converted to stress using inverse elastic models [1]. Examples of MSRtechniques are: slitting and elastic crack solutions [2], hole drilling and a 2D elastic solutionfor a hole under uniform stress at infinity [3].

Although this general scheme has been used with great success over many decades to measureresidual stress in a wide range of specimen geometries and engineering material, it’s successfulimplementation depends on two key assumptions. Firstly, residual stress state itself in majorityof analytical models is assumed to be uniform. For example, for the hole drilling method, thetypical analytical model is a hole in a 2D plate under uniform far field stress [3]. Secondly,the relaxation process is assumed to be purely elastic. This is known not to be true in manypractical applications, in particular, when the magnitude of residual stress is close to yield[4–6]. These two assumptions are focused in this work.

(a) (b)

Figure 1: Schematic diagram of (a) the problem geometry for a 2D semi-infinite rectangularstrip, of width 2c, with an arbitrary self-equilibrated end load. The boundary conditions are:x2 = ±c : σ22 = σ12 = 0 and x1 = 0 :

∫ c−c σ22dx2 =

∫ c−c σ12dx2 = 0 and (b) of a four-point

bend test used in this work. The beam was plastically bent and then unloaded for controlledresidual stress creation between inner rollers. The specimen is cut with wire EDM on the x1 = 0symmetry plane. This releases the residual stresses.

2 Analytical model

To remove the stress uniformity assumption, we have employed Mathieu series solution for a2D semi-infinite strip with arbitrary self-equilibrated loading at the end [7, p.61], see Fig. 1(a).

21

The solution uses the two even, f , and odd, g, stress functions which satisfy the bi-harmonicequation:

f = e−γx1/c(ξ cos

γx2

c+γx2

csin

γx2

c

); g = e−φx1/c

(ψ sin

φx2

c+φx2

ccos

φx2

c

)(1)

With satisfying the boundary conditions, see Fig. 1(a), unknown dimensionless parameters ofthe stress function, γ, ξ, φ, ψ have infinite number of solutions. At the heart of the method isthe problem of minimising the standard Linear Least Square (LLS):

minx||Ax− u||2 (2)

where x is the vector of 4n unknown coefficients for the terms in the mathieu series, u is a vectorof 2m displacement values, A is a 2m × n matrix of integral functions taken at the locations ofthe measurement points. For the stability of LLS, the number of measured displacement points(M) is considerably greater than the terms (N) in the Mathieu series. Therefore, in practice,the method requires a full-field measurement technique.

3 Experiment

The analytical model was employed for measuring residual stress in a 25 × 25 × 250 mm3 barsmachined of Al 7075-T6(σY = 506 MPa, E = 71.7 GPa) under four-point bending. The zigzagresidual stress profile with the peak residual stress to yield ratio is 0.75 was introduced by 9mm vertical displacement on the inner rollers and unloading, see Fig. 1(b). Then, the specimenwas cut in the midsection of the longitudinal direction of the bar, with φ0.1 mm wire EDM.The residual stress profile on the symmetry line, x1 = 0, can be reconstructed from measuredrelaxation displacement on free surface.

The required resolution of the displacement was estimated by preliminary FE relaxationstudy. As a result, the maximum displacement, at the cutting boundary, is around 20 µmand this decays exponentially away from the cut. Therefore, collecting the data as close tothe cut as possible is critical. This requires high displacement resolution which leads to highmagnification and short working distances. In this work, we use 2D DIC system due to thepractical limitation of 3D DIC system, such as fitting all the optical devices within a veryconfined space.

The DIC system consisted of a 10-bit 1392 × 1040 pixel CCD from Dantec dynamics [9].Fixed focus optics and ring light illumination, both from Navitar [10], were used. The workingdistance was 175 mm, giving 7.6 µm/pixel spatial resolution. With sub-pixel resolution, thesystem resolve displacements between 0.08 and 4 µm [11]. Random surface pattern was achievedfrom light scratching of Aluminium surface with P180 sandpaper for the preservation of thesurface during the EDM cutting. Due to the small size of the field of view, four verticaloverlapping images were recorded along the both side of the cut to complete whole width ofthe specimen.

4 Result

A large tail at the upper edge is likely due to lost displacement data at the very edge andoscillations are caused by the noise in the measured displacement field 2(a). However, thereconstructed residual stress field obtained from measured relaxation displacement shows agood overall agreement with that predicted by FE and neutron diffraction results, see Fig.2(b).

22

Length, x1 (mm)

Wid

th, x

2 (m

m)

Displacement, u1 (mm)

0 1 2 3 4 5

−10

−8

−6

−4

−2

0

2

4

6

8

10

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

-12-10.5

-9-7.5

-6-4.5

-3-1.5

0 1.5

3 4.5

6 7.5

9 10.5

12

-400 -200 0 200 400

Wid

th o

f Sp

ecim

en (

mm

)

Stress (MPa)

FE surface FE interiorExp. result

Neutron Dffraction

(a) (b)

Figure 2: (a) The measured horizontal displacement field, u1. Horizontal discontinuities arelikely due to imperfect stitching. The DIC subset size was 25 × 25 pixels with no overlap. (b)One of four DIC experimental results is compared to the FE prediction and is validated againstthe neutron diffraction measurement [8].

5 Plastic flow on relaxation

During cutting residual stress redistributes to satisfy force and momentum equilibrium. Thisredistribution might cause plastic deformation where the magnitude of residual stress is close toyield [4–6]. However, it is possible to reduce plastic strain on relaxation by carefully choosingorientation and direction of the cut [8]. To prove this, a detailed FE study of plastic flow on

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3 4.5

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0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

-600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600

Wid

th o

f Sp

ecim

en(m

m)

Equivalent plastic strain

Residual stress (MPa)

3D Plastic top 3D Plastic front

FE residual stress

(a) (b)

Figure 3: (a) Plastic flow resulting from different cutting methods. Superimposed is theresidual stress field. The equivalent plastic strain profiles through thickness at cutting edge(x1 = 0.09 mm) are shown in the 3D FE model. Only cuts from the top toward the bottomface produce substantial stress redistribution in the region of the topmost residual stress peak.(b) Surface displacement fields in the region from x1 = 0.09 (cutting edge) to 10 mm obtainedfrom in the 2D model with the cut from the top face model.

different cutting schedules was conducted with the same zigzag residual stress profile from the

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previous experiment. It was found that when the cut is advanced from the top towards thebottom face, there is substantial stress redistribution that causes plastic strain in the regionof the topmost residual stress peak, see Fig. 3(a) and this causes distortion in relaxationdisplacement field at the peak tensile displacement region, see Fig. 3(b). However, when thecut is advanced from the front to the rear face, there is no detectable plastic flow on the frontface.

6 Concluding Remarks

It is shown that different cutting schedules, which produce the same cut, cause a differentamount of plastic flow. It means that if there is some prior knowledge of the expected residualstress, then methods based on different cutting schedules might have different results.

The use of free surface displacement data implies that uniform stress field is still assumedthrough the thickness. However, it is shown in Fig. 2(b) that residual stress profile on thesurface differs from that in the interior. This problem was particularly prominent in thiswork because the thickness was equal to height. Therefore, the method should work better onrelatively thinner components.

Two practical imaging problem: an inability to distinguish out-of-plane from in-plane mo-tion and a need for accurate stitching up can be resolved with the use of a high resolution rigidpositioner or introducing a rig that 3D DIC system can fit.

References

[1] GS Schajer and MB Prime. J. Eng. Mater. Technol., 128:375–382, 2006.[2] W. Cheng and I. Finnie. Residual Stress Measurement and the Slitting Method. Springer, 2007.[3] G. S. Schajer. J. Eng. Mater. Technol., 110:338–343, 1988.[4] M. Beghini, L. Bertini, and P. Raffaelli. J. Testing Eval., 22(6):522–529, 1994.[5] Y. C. Lin and C. P. Chou. Mater. Sci. Technol., 11:600–604, 1995.[6] A. H. Mahmoudi et. al. Int. J. Mech. Sci., 53:978–988, 2011.[7] S. Timoshenko and J. N. Goodier. Theory of Elasticity. McGraw-Hill, 3 edition, 1970.[8] H K Kim, H E Coules, M J Pavier, and A Shterenlikht. Exp. Mech., pages 1–14, 2015.[9] Dantec Dynamics. ISTRA 4D Software Manual Q-400 system, 2012.

[10] Navitar. Precise Eye Performance Specifications, 2013. http://www.navitar.com (HTML).[11] Bing Pan, Kemao Qian, Huimin Xie, and Anand Asundi. Measur. Sci. Technol., 20:062001, 2009.

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