Y O U N G S T R E S S A N A L Y S T

C O M P E T I T I O N

BSSM In te rna t iona l Confe rence on Advances i n Exper imenta l Mechan ics

T h e N a t i o n a l P h y s i c a l L a b o r a t o r y ( N P L ) , L o n d o n , U K .

2008

Sponsors:

Young Stress Analyst Final

Special Conference Session, Day 2, 15.00hrs. List of Finalists’ Papers NAME TITLE

PAGE

Paulo F.P. de Matos Investigation of Thickness Effects in Plasticity –

Induced Fatigue Closure

2

Oscar Portillo Experimental Investigation of the Fracture Mechanics of

Bituminous Mixes

5

Shamala Sambasivam

Observations on the Thermoelastic Response of Fibre

Reinforced Orthotropic Laminated Composites

11

Geoff Wilde Lens Stiffness Measurement 16

Young Stress Analyst Poster Competition

EMEX Exhibition, Day 2, 12.40hrs.

List of Posters NAME TITLE

PAGE

Jonathan G. Fry Photoelastic Materials for Biomedical Applications

21

Arin Jumpasut Photogrammetry for Impact Engineering

28

Xu Song

Strain Gradient Crystal Plasticity Modeling and

Synchrotron X-Ray Diffraction Experiments: Keys to

Stress Analysis at the Nanoscale

32

Investigation of thickness effects in plasticity‐induced fatigue crack closure 

Paulo F. P. de Matos

Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, United Kingdom

ABSTRACT

The effect of the specimen (or component) thickness has been shown to have a significant effect on closure behaviour and this seems to be related to the relative size of the plastic zone. Real cracks are inherently three-dimensional; plane stress-like behaviour is found close to the region where the crack front intersects the free surface, whereas most of the crack front will experience something close to plane strain. The aim of the present work is to investigate the influence of specimen thickness on closure behaviour (both close to and remote from the surface) and on fatigue crack propagation. Results from a simple experimental program, which consists of fatigue testing CT specimens with different thicknesses are presented. Fatigue crack propagation is measured optically; crack closure is assessed using traditional compliance techniques (clip gauge and back face strain gauge) and Digital Image Correlation methods. Experimental results are compared with two and three-dimensional finite element simulations of plasticity-induced fatigue crack closure. The implications of thickness effects for predicting the propagation of three-dimensional fatigue cracks are discussed.

MOTIVATION AND GENERAL DESCRIPTION OF THE WORK

The prediction of the fatigue life of engineering components and structures is of primary concern for designers in a wide range of applications. Major advances in this area have, of course, been made over the last 50 years. However, major challenges still remain. In particular, it can often be difficult to predict the life of structure or component subject to a practical loading cycle, starting from available materials data, which is very often for the case of uniaxial loading at constant amplitude. Elber's discovery of fatigue crack closure, nearly 40 years ago, held out the prospect of significant advances in this area. Closure can be difficult to measure experimentally, and different measurement techniques can give apparently different results. Modelling of crack closure is computationally demanding and models frequently have to be simplified with respect to the experimental conditions. Focusing more specifically on plasticity induced closure, which is thought to be the most significant effect over the majority of the propagation life, some useful progress might be made by combining high quality experimental measurements with detailed finite element modelling. A particularly promising technique is that of image correlation. This offers the advantages of sophisticated techniques such as moiré interferometry without the requirement for complex specimen preparation and specialised optical equipment. In the current work the application of full field digital image correlation to the measurement of plasticity induced crack closure will be described. A range of experiments are undertaken, using different thickness compact tension specimens. These enable an evaluation of crack closure, which is often thought to be a surface phenomenon, in determining the crack propagation rate. The

experimental results will be compared to three-dimensional finite element models of the experimental configuration. The material chosen for the investigation was the aluminium alloy 6082 T6, which is widely used in the aerospace industry. Compact tension specimens with three different specimen thicknesses, t, of 3, 10, and 25 mm were fatigue tested. The main pieces of instrumentation employed were a load cell, a ‘back face’ strain gauge, a crack mouth opening displacement (CMOD) and a ‘Questar’ QM 1 long distance microscope, mounted on a translational stage with digital readout. This could be used for observing the specimen, both to measure the crack growth rate optically and to capture sequences of images during load cycles, which could be used for Digital Image Correlation (DIC). In order to restrict the amount of data processing required, displacement measurements were taken at five locations along the crack, corresponding to different distances, Li, from the crack tip (see Figure 1a). Figure 1 b) shows a typical variation of this quantity with frame number (i.e. time). Also shown on this plot is the variation of load with time. It can be seen that whilst the load varies sinusoidally, the relative displacement does not. The flat region at the bottom of the displacement cycle corresponds to crack closure, where the relative displacement is constant even though the load is varying.

a)

b)

Figure 1 – Digital Image Correlation: a) Points on surface of a fatigue crack; b) Relative displacement between two points and load - 3 load cycles.

As well as digital image correlation, closure was also assessed using conventional compliance

approaches, a back face strain gauge and a crack mouth opening displacement gauge. Figures

2 a) and b) show crack opening load data obtained for constant amplitude loading in specimens

with different thicknesses. It will be seen that the opening load obtained with the compliance

gauges are very similar to those found using the digital image correlation approach for thin

specimens (t=3mm). A different picture emerges when data from the thick (25 mm) specimens

is considered.

Figure 2 – Opening stresses: a) Digital image correlation; b) Back face strain gauge.

These results therefore suggest that surface closure levels are almost independent of specimen thickness, whereas measurements obtained by the compliance technique, which effectively average over the thickness, do vary. This is probably because relatively high levels of closure exist at the surface, with lower levels close to the centre of the specimen. The lower proportion of the specimen affected by surface closure leads to lower overall closure levels in the thick specimens, as measured by the compliance technique. A key practical question is which of these closure measurements is more appropriate for determining crack growth rate. Figures 3 a) and b) show the rate of crack propagation as a function of effective stress intensity factor range, ΔKeff, calculated using DIC and back face strain gauge. These results suggest that an average closure measurement, such as that obtained by compliance technique, may be more appropriate for determining crack growth rate and a pure surface measurement, such as that obtained by DIC.  

 Figure 3 – Fatigue crack growth rate for CT specimens with different thicknesses: a) FCG rate as a function of ΔKeff (DIC); b) FCG rate as a function of ΔKeff (Back face strain gauge).

a) b)

a) b)

1 INTRODUCTION

Fatigue cracking is a common form of structural failure in asphalt pavements. As a result of repeated wheel loads and/or environmentally-induced loads, cracks may initiate and propagate through the surface layers of the road. The fracture mechanism is controlled by the mechanical properties of the pavement material and by the loading geometry. Therefore understanding the micromechanics of cracking of pavements relies on a knowledge of the fracture of bituminous mixes, which ultimately requires understanding the fundamental fracture properties of pure bi-tumen.

The aim of this experimental investigation is to study the fracture behaviour of bitumen and bituminous mixes subjected to a wide range of temperatures and load rates, and to develop a failure mechanism map to characterize the response of the materials similar to the failure mechanism map for bitumen films reported by Harvey and Cebon [1].

The experimental part of this study consists of three-point bending tests on pure bitumen and asphalt mix, using test methods derived from relevant ASTM standards eg [2]. Fracture pa-rameters such as the Mode I fracture toughness, KIC,; fracture energy, GIC,; crack mouth opening displacement, ∆; and J integral, JIC; were derived from the tests, Table 1.

Table 1. Fracture characterization parameters and formulas for the single edge notch SE(B) specimen.

EXPERIMENTAL INVESTIGATION OF THE FRACTURE MECHANICS OF BITUMINOUS MIXES Portillo Oscar Doctoral Research Student at the Engineering Department, Cambridge University, Cambridge, UK

ABSTRACT: Systematic 3-point bend tests were conducted on bitumen and idealised bituminous mix specimens in order to develop fracture mechanism maps classifying the brittle, ductile and transition response of the materials as a function of temperature and load rate. Evaluation of fracture toughness of test specimens was conducted in terms of the stress intensity factor, KIC, fracture energy, GIC, and J-integral, JIC. Three different failure regimes were identified for both bitumen and bituminous mix.

PARAMETER APPLIES TO FORMULA NOTES

Stress Intensity Factor KIC

Linear Elastic Materials

Fracture EnergyGIC

Linear Elastic Materials

JIC- Integral Elastic-Plastic Materials

Uc = Total critical potential energy3-Point SE(B) S= 4W and 0.45 ≤ a/W ≤ 0.65

( )2/3

22/1

1212

7.293.315.2199.13

)/(

⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ +

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+−⎟

⎠⎞

⎜⎝⎛ −−

=

Wa

Wa

Wa

Wa

Wa

Wa

Wa

WaY

)(21)(

max max

0 0max

aWB

dFMgdPG

s

IC −

∆++=∫ ∫

∆δ

δδδ

( )( ) ( )aWB

UaWB

UUJ cpe

IC −=

−+

= 22

Deflection δ

Load

P

GF

M= Mass (specimen)Fs=Force (clip gauge)

Deflection δ

Load

P

Failure( )WaYBW

PSK IC /2/3=

2 EXPERIMENTAL TESTING

A 50 pen pure bitumen was used to prepare the SE(B) specimens. The same 50 pen bitumen was used for the mix preparation. The mix consisted of pure bitumen and 64% volume fraction of subspherical sand particles between 1.8 and 2.36 mm in size.

A three-point bend fixture was designed in order to carry out fracture tests, see Figure 1. An environmental chamber fitted to the testing machine was used to control the test temperature over a range of –40°C to +40°C. A 2 kN load cell was used to measure the force and an LVDT fitted to the testing machine measured load-line displacement.

Figure 1. 3-Point bend experimental set up. Specimen dimensions meet the requirements of the ASTM E399 standard, [2].

A spring steel chip gauge was manufactured to measure the crack mouth opening displacement of the SE(B) specimens, see Figure 2. This novel clip-gauge was designed for measuring large displacements that are beyond the measurement range of commercial clip gauges. The clip gauge displayed a linear relationship between the voltage and displacement for a range of temperatures and over the desire range of crack mouth opening measurements.

Measuring the crack length in these experiments was a particular challenge. For various reasons, it was found necessary to use an optical system to measure crack lengths, rather than any kind of electrical crack length gauge. When an asphalt specimen is fractured in the regime of viscous behaviour, the test may take tens of minutes. When a specimen is tested in the brittle regime, failure may occur in fractions of a second. In the transition regime, either type of behaviour may occur, and it is not normally possible to predict which. In order to cope with the necessary widely differing data acquisition rates, an analogue video recording was made of each test. The video tape was subsequently post-processed to generate images of the crack at appropriate times through the fracture event, and these images were loaded into a database.

Camera

The measurement system comprised:

(i) A high-speed video camera, with special optics, located outside the environmental chamber; with a lighting system located inside the chamber to illuminate the side of the specimen. The specimen was photographed through a window in the front of the environmental chamber. Before the start of the test, the face of the specimen was painted white to help distinguish the tip of the crack and a square grid was painted on the specimen to provide calibration reference.

(ii) A flashing LED that was captured on the video image, in order to synchronize the time signal on the video images with the clock of the testing machine;

(iii) A digital image data acquisition system that converted the analog video signal from the

video tape to a set of digital images at a chosen sampling rate;

(iv) A special-purpose image analysis program, written in MATLAB. This program digitized the crack profile from each image and saved it in a text file, along with summary information including the instantaneous crack length. This file was then automatically loaded into the database along with the image itself.

Figure 2. Clip-gauge used to measure crack mouth opening displacements and its calibration curve.

3 EXPERIMENTAL RESULTS

In order to compare experimental results from tests at different temperatures and load rates, a temperature-compensated crack mouth opening strain rate is defined in similar manner to the temperature-compensated strain rate proposed by Harvey and Cebon [1]. is expressed as:

∆∆ (1)

Clip Gauge Calibration Curve

y = 0.3369x + 0.0256R2 = 0.9998

012345678

0 5 10 15 20 25

Displacement (mm)

Volta

ge (V

)

where ∆ is the crack mouth opening displacement rate and ∆ is the initial notch width, Q represents the thermal activation energy, R is the universal gas constant, TO and T1 stand for the reference temperature and test temperature respectively. In this study the ∆ was computed from the ∆ vs. time test characteristic curve, specimens were manufactured with a of 3mm, Q is 228 x 103 J mol-1 as per Cheung and Cebon [3] for 50pen bitumen, R is equal to 8.314 Jmol-1K-1 and the reference temperature was chosen to be 273K.

Fracture 3-point bend tests on pure bitumen specimens were conducted at temperatures ranging from -30 ˚C to 0 ˚C. It was not possible to perform tests at higher temperatures because bitumen beams exhibited excessive creep deformation due to self-weight. Fracture energy results are presented in the form of a failure mechanism map similar to the failure mechanism maps in viscoelastic films reported by Harvey and Cebon [1]. Ductile, brittle and transition regimes of behaviour are observed in the map, Figure 3-b. The failure mechanisms are dependent on temperature and crack mouth opening rate. In the ductile region, the fracture energy increases with in a power law relationship until a critical value. It then decreases rapidly through to the transition region for temperature compensated crack mouth opening rates ranging from 0.03 s-1 to 0.35 s-1. In the brittle region, the fracture energy is largely independent of and is considerable lower than in the ductile region, indicating that the energy needed to initiate fracture is less at high displacement rates and lower temperatures, as expected.

A failure mechanism map plotting strain energy release rate as a function of the temperature-compensated crack mouth opening strain rate (Eq. 1) in asphalt mix specimens is presented in Figure 3-c. It again displays three main regimes of failure behaviour: ductile, brittle and transition. In the ductile regime, the fracture energy increases as the temperature decreases from 30°C to 0°C. Conversely, the fracture energy of the mix decreased as the temperature decreases from -10°C to -30°C in the brittle region, a softening mechanism can also be observed. The ductile to brittle transition is located at temperatures between 10°C and 0°C or 0.002 s-1 < < 0.04 s-1, although there is some evidence of strain rate sensitivity up to =1 s-1 . A similar description of the failure behaviour in asphalt was found using the J-integral as a fracture characterization parameter.

Log[1/s] )ELog( T&

Figure 3. Mechanism map for pure bitumen. (a) Fracture energy per unit volume G/2h of DCB bitumen specimens, Harvey and Cebon [1]. (b) Mechanism map for bitumen specimens. (c) Mechanism map for asphalt mix specimens.

Ductile Transition Brittle

[ ]1/sLog)εLog( T&

Ductile Transition Brittle

Log[1/s] )ELog( T&

(a)

(b)

(c)

1

10

100

1000

10000

-6 -4 -2 0 2 4 6 8

20 C

-30 C

0 C10 C

-10 C

Ductile Transition Brittle

G /

2h, k

Jm-3

l Ý

range of G /2h for film thicknesses testedIC

1

10

100

1000

10000

-3 -2 -1 0 1 2 3

GIC

(N/m

)

T=-30C

T=-20C

T=-10C

T= 0C

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5

GIC

(N/m

)

T=-30CT= -20CT=-10CT= 0CT= 10CT=20CT=30C

Crack length measurements were conducted using the optical system described in the previous section. Figure 4 depicts some images of the crack growth as a function of time superimposed in a typical Force and ∆ vs. δ curve for an asphalt specimen. It is interesting to observe that the strains are small at the peak load when fracture occurs (t~4s.) and the crack is not visible until the deformation is larger and the force is low (t~10s.).

Figure 4. Force and CMOD vs. Time with superimposed photographs of the SE(B) specimen, the sample was tested at a temperature of -20 °C and displacement rate of 0.05 mm/s. 4 CONCLUSIONS

3-point bend tests were performed on pure bitumen and asphalt mix to characterize the frac-ture behaviour over a wide range of temperatures and loading rates. A wide range of crack be-haviour was observed including brittle, ductile and transition. These regimes can be seen clearly on the failure mechanism maps. A special clip-gauge was developed to measure the large crack mouth opening displacements of the SE(B) specimens, the data were used to calculate tempera-ture-compensated crack mouth opening strain rates. Crack length analysis was conducted by us-ing an optical system and new software tools were set-up to handle the special data require-ments of the experimental programme.

5 REFERENCES

[1] Harvey, J. A. F. and D. Cebon (2005). "Fracture Tests on Bitumen Films." Journal of Materials in Civil Engineering 17(1): 99-106.

[2] ASTM Standard E 399 (2005). "Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials."

[3] Cheung, C. Y., Cebon, D.; (1997a). "Deformation Mechanisms of Pure Bitumen." Journal of Materials in Civil Engineering 9(3): 117-129

Time (s)

0

0.5

1

1.5

2

2.5

3

0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30 35 40 45

Forc

e(N

)

Cra

ck M

outh

Ope

ning

Dis

plac

emen

t-C

MO

D (m

m)

CMOD

Force

Crack Length =0.25 mm

Crack Length =4.44 mm

Crack Length =17.98 mm

Crack Length =23.74 mm

Crack Length =24.26 mm

OBSERVATIONS ON THE THERMOELASTIC RESPONSE OF

GLASS FIBRE REINFORCED ORTHOTROPIC LAMINATED

COMPOSITES

Shamala Sambasivam

Fluid Structure Interactions Research Group, School of Engineering Sciences,

University of Southampton, UK

Objective

Designing components using composite materials is challenging because of the

anisotropic nature of the mechanical and physical properties of the material, and its

limited ability to plastically deform. Thus justifying the need for experimental

verification in the design process. In this work, the prospect of using thermoelastic

stress analysis to provide full-field strain measurements in composite components is

investigated. Thermoelastic stress analysis (TSA) has advantages over other

experimental techniques through its ability to provide full-field, non-contact stress

details from structures under dynamic load in practically real time. The technique is

based on infra-red thermography, where the small temperature changes resulting from

a change in elastic stress are obtained by measuring the change in infra-red photon

emission.

Previous research has shown that the interpretation of the thermoelastic response from

orthotropic composite materials requires special consideration. This is because the

mechanical and thermal properties result in a different response that is dependent on

the loading direction. A series of theoretical considerations have been made aimed at

identifying the source and assessing prominent factors influencing the thermoelastic

signal from laminated composites; these are crucial in implementing a routine that can

extract full-field strain data from the measurements. To provide a more detailed

assessment of the origin of the thermoelastic response this work focuses on the

thermoelastic temperature changes that occur on the surface of cross-ply and

quadriaxial quasi-isotropic laminates. The experimental results are compared with

those computed using a variety of propositions. To do this all relevant material

properties are measured.

Options for defining the thermoelastic response of an orthotropic composite

laminate

In this work a strain based approach is devised for investigating the thermoelastic

response for orthotropic materials. For an orthotropic material the temperature change

as a function of stress increment, is given by

)(C

TT

2211

p

σ∆α+σ∆αρ

−=∆ (1)

where ∆T is the small thermoelastic temperature change, ∆σ1 and ∆σ2 are the

principal stresses in the surface ply, T is the absolute surface temperature, ρ is the

density, Cp is the specific heat at constant pressure and α is the coefficient of thermal

expansion (CTE) in the principal material directions.

For laminated composites, it is more convenient to work in terms of strain rather than

stress. This is because for components loaded in uniaxial tension the strain field in the

surface layer can be assumed to be constant across the laminate thickness. Therefore,

reformulation of Eq. (1) in terms of strain is given as:

( ) ( ){ }T222112L212111

p

QQQQC

TT ε∆α+α+ε∆α+α

ρ−=∆ (2)

where ∆ε is the change in the strain, Q is the material stiffness, the subscripts 1 and 2

denote the principal material directions in a lamina and the subscripts L and T denote

the principal axes of the laminate.

Some previous work [1,2] have shown that the resin rich surface layer, that occurs in

composite components due to the manufacturing process, acts as a strain witness.

When adiabatic conditions prevail the resin layer, which has a low thermal

conductivity, insulates the orthotropic material beneath and the measured response is

that resulting from the strain transferred into the resin rich surface layer, so that the

temperature change resulting from the thermoelastic effect can be expressed as

follows:

( )

ε+ε∆

ν−

α

ρ−=∆

TL

r

rr

p1

E

C

TT (3)

where the subscript r denotes the resin properties.

When assessing the behaviour of a general multidirectional composite laminate

(consisting of lamina with arbitrary orientations) classical laminate plate theory is

used so that the material can be treated as a homogeneous anisotropic plate. Here the

mechanical (e.g. stiffness) and thermoelastic (e.g. CTE) properties are considered ply

by ply and then brought together relative to (say) the laminate axes to provide a

‘global’ stiffness and CTE. For quasi-isotropic laminates (e.g [0, 90]s and [0, ±45,

90]s) it is evident that the global CTE is equal in the L and T directions (i.e. αL = αT)

because of the stacking sequence. Therefore, for composite laminates with a quasi-

isotropic nature, it might be pertinent to express the thermoelastic temperature change

in the following manner:

( ) ( ){ }T2212L1211

p

AAAAC

TT ε∆++ε∆+

ρα−=∆ (4)

t

k

k

N

1k

ijijh

h)Q(A ∑

=

= (5)

where Aij is the global stiffness of the laminate and hk and ht are the thicknesses of the

kth

ply and the overall laminate.

Equation (4) simply assumes that the material response is that of a homogeneous

mechanically orthotropic material and discounts the inclusion of the response of the

surface ply. This does not comply with any previously described treatment but is

included here for completeness.

A further and as yet unexplored idea is that the CTE is coupled in the stack with the

surrounding layers and this also might have an effect on the response. To explore this

orthotropic nature of the surface ply is retained in the treatment but the CTE is treated

as a global property, giving the following equation:

( ) ( ){ }T2212L1211

p

QQQQC

TT ε∆++ε∆+

ρα−=∆ (6)

The different scale of idealization in each model aims to provide an insight into

thermoelastic behaviour of composite material. This is achieved by computing the

thermoelastic temperature change based on each treatment and comparing it with

measured temperature change for quasi-isotropic laminates, which accommodates all

the assumptions considered in the models.

Thermoelastic work

The material used for manufacturing the test specimens was a unidirectional

glass/epoxy pre-impregnated (E-glass and Novalac epoxy resin) material. The

mechanical properties and physical properties such as density, specific heat capacity

and (CTE) were determined according to respective ASTM standards from

unidirectional composite as provided in Table 1.The global mechanical and physical

properties are given in Table 2. To validate the theoretical assumptions cross-ply (CP)

and quasi-isotropic (QI) panels were manufactured, with the stacking sequences given

above and consolidated in an autoclave. Two different sets of CP laminates were

obtained from same panel; one with a 0o

surface ply (CP0) and another with a 90o

surface ply (CP90) in order to evaluate the effect of surface ply on the thermoelastic

signal.

Table 1: Mechanical and physical properties of unidirectional E-glass/epoxy pre-

impregnated composite and epoxy

Young’s

Modulus

(GPa)

Poisson’s

ratio

CTE,

(x 10-

6/oC) Specimen

E1 E2 νννν12 νννν21

Density,

ρ (kg/m3)

α1 α2

Specific

heat

capacity,

Cp

(J/(kgoC))

UD 34.2 10 0.325 0.1 1230 9 31 843

Epoxy 4.2 n/a 0.413 n/a 1207 52 n/a 1230

Table 2: Global mechanical and physical properties of cross-ply and quadriaxial

laminate

Specimen

Young’s

Modulus

(GPa), EL

Poisson’s

ratio, ννννLT

Density, ρ

(kg/m3)

CTE, α

(x 10-6

/oC)

Specific

heat

capacity,

Cp,

(J/(kgoC))

CP 20 0.15 1880 10.59 843

QI 20 0.29 1880 9.25 843

The test specimens were mounted in an Instron servo-hydraulic test machine and a

cyclic tensile load was applied at a loading frequency of 10 Hz. A strain gauge rosette

was attached to the specimens to measure the strain in the laminate principal

directions. The thermoelastic temperature change (averaged over a uniform area) from

each specimen were collected (as shown in Figure 1) and clearly a large point wise

variation can be observed in the images. The measured (with standard deviation) and

calculated temperature change are summarised in Table 3. It is apparent from the

images and measurements that there are significant differences in the thermoelastic

temperature change obtained from the CP specimens due to the differences in the

surface ply orientations. This clearly violates assumptions made for developing Eq.3

and it can be observed from Table 3 that the measured and predicted temperature

values do not correspond well. However, by accounting for the scatter in the data, the

predictions based on Eq. 2 and Eq. 6 seem to be valid. However, due to the close

proximity of the predicated values it is not possible to establish if the response is a

function of surface ply or a global laminate property and further investigation is

needed to distinguish this.

Conclusion

This study has shown that for material considered, E-glass/ epoxy pre-preg laminates,

consideration of the surface resin layer does not give the best agreement to measured

thermoelastic data. A new approach, in which CTE is coupled in the stack, gives a

much better agreement to the measured data similar to consideration of the surface ply

only. This work highlights the importance of careful consideration of the source of the

thermoelastic signal while working with composite materials.

a) b) c)

Table 3: Thermoelastic response and individual strain values from the calibration

specimens

Figure 1: Thermoelastic images obtained from composite specimen at a loading

frequency of 10 Hz, a) CP(0), b) QI c) CP(90)

Reference

[1] G. Pitarresi, M.S. Found and E.A. Patterson: Composites Science and Technology

Vol. 65 (2005), p. 269.

[2] T.R. Emery, J.M. Dulieu-Barton, J.S. Earl and P.R., Cunningham, Composites

Science and Technology, Vol. 68 (2008), p. 743.

Specimen ∆∆∆∆T

measured

(std.)

∆∆∆∆T

Surface ply

(Eq. 2)

∆∆∆∆T

Resin layer

(Eq. 3)

∆∆∆∆T

Global

(Eq. 4)

∆∆∆∆T

Mixed

(Eq. 6)

CP(0) 0.149

(±0.0089) 0.135 0.127 0.175 0.141

CP(90) 0.129

(±0.0021) 0.106 0.130 0.171 0.104

QI 0.138

(±0.0012) 0.136 0.123 0.144 0.136

Lens Stiffness Measurement Geoffrey Wilde, Engineering Science, University of Oxford

Objective and Background: The aim of my work is to measure the stiffness

of the isolated human crystalline lens for ages up to 60 years.

The human crystalline lens is the component of the eye which allows it to focus at a

range of distances – that is to accommodate. It effects this by altering shape in

response to changes in radial tension. The capacity to change focus declines with age

until it is essentially lost by fifty years. At this stage the eye is termed presbyopic, and

reading glasses are generally required for close work. A large increase in the stiffness of the lens material with age is understood to be a major contributor to the

development of presbyopia[1], though other causes are also advanced[2, 3].

The lens is made up of long cells arranged in orderly concentric shells and surrounded by a relatively stiff extracellular membrane, or capsule. The lens is very delicate,

especially when the capsule is removed making testing difficult, and any test which disrupts the arrangement of the cells may not give information relevant to the in vivo

state of the lens.

The information gathered in this work on the stiffness of the lens is intended for use in

the simulation of the in vivo accommodation apparatus. This will be used to assess

and refine the effectiveness of existing proposals to reverse presbyopia through

surgical intervention.

Figure 1 – the lens spinning rig and image acquisition system.

DC motor

microprocessor flash controller

optical sensors

lens in humid box

flashgun synchronized flash signal camera

flash signal

position information

USB link

DSLR camera with macro lens

spinning rig speed

display laptop

Methods: The approach used to measure the lens stiffness is an updated version

of Fisher’s spinning lens test[4] in which the lens is spun about its axis of symmetry at

a known angular speed, and the resulting deformations recorded photographically.

The lens is spun upon a custom-built rig driven by a variable-speed direct-current

motor. The rotor shaft terminates with a 6.5 mm plastic ring to support the lens. The

speed of rotation and position of the shaft are monitored by optical sensors. A

Perspex box with a cover-slip window surrounds the lens to limit tissue drying and

potential contamination. A 6 megapixel digital SLR camera equipped with a macro

lens and extension rings is used to photograph the lens, providing a resolution of

about 4 µm per pixel. The lens is illuminated by a flashgun set to its lowest

illumination, giving an exposure time of 24 µs which is brief enough to avoid motion

blur. The timing of the flash is controlled by a microprocessor taking the position

signals as inputs, allowing the lens to be photographed repeatedly when it is at eight

equally-spaced orientations.

Figure 2 – A forty- year decapsulated lens spinning at 1000 rpm.

Each lens is carefully placed on the support so its axis of symmetry lies close to the

axis of rotation. It is then photographed while spinning at 700 rpm, 1000 rpm, and for

old lenses 1400 rpm. Reference photographs are taken while the lens is essentially

stationary before and after each test. The lens is then taken from the apparatus, its

capsule removed, and the tests repeated on the decapsulated lens.

Figure 3 – An axisymmetric lens mesh for a 40 year old lens.

To calculate the shear modulus of the decapsulated lens, an inverse analysis is

performed using a axisymmetric hyperelastic finite element model of the test. The

outline of the lens is obtained from the reference photographs and used to generate a

mesh for the simulation. The shear modulus is assumed to vary linearly from the

centre to the surface of the lens, with the specific profile taken to be that which most

closely reproduces the experimental displacements at the pole and equator of the lens.

The current version of the test provides advantages over the original methods used.

The synchronized flash and its much shorter exposure avoid random and systematic

errors in the equatorial measurement, while the numerical simulation avoids the need

for substantial simplifying assumptions in the analysis.

Discussion: The tests depend on postmortem donor lenses which are only

available intermittently, so only six decapsulated lenses have been analyzed to date.

The computed stiffness profiles for this limited set of samples displayed a large increase in stiffness with age at the centre of the lens and no clear change at the

surface, though the small number of lenses and large variability of biological specimens make it impossible to draw firm conclusions at this stage.

fixed support

axis of

rotation

Figure 4 – Shear modulus profiles calculated for 6 lenses.

The current test does provide some insight into the effect of the capsule on the lens.

In the original test the capsule was removed from only one lens, and it was reported

that this had no effect on the equatorial strain[4]. However, a naïve finite element

simulation of the test predicts a 300% increase in strain once the restriction of the

capsule is removed[5]. In the current test the lenses have shown an average increase

in equatorial strain of only 25% after decapsulation. This indicates that the simple

capsule model used in the finite element simulation is not appropriate; this extends to

in vivo modelling. Comparison of simulations with the results of the tests performed

on lenses with capsule intact will provide information on how the capsule should be

modelled, aided by the information already obtained for each lens after decapsulation.

G1

(surface) G0

(centre)

0

1

2

3

4

5

6

sh

ea

r m

od

ulu

s (

kP

a)

22-year lens

40-year pair

47-year pair

54-year lens

References: 1. Glasser, A. and M. C.W. Campbell, Biometric, optical and physical changes

in the isolated human crystalline lens with age in relation to presbyopia. Vision Research, 1999. 39(11): p. 1991-2015.

2. Weale, R.A., Letter to Editor - On potential causes of presbyopia. Vision

Research, 1999. 39(7): p. 1263-1265.

3. Strenk, S.A., L.M. Strenk, and J.F. Koretz, The mechanism of presbyopia.

Progress in Retinal and Eye Research, 2005. 24(3): p. 379-393.

4. Fisher, R.F., The elastic constants of the human lens. J Physiol, 1971. 212(1):

p. 147-180.

5. Burd, H.J., G.S. Wilde, and S.J. Judge, Can reliable values of Young's

modulus be deduced from Fisher's (1971) spinning lens measurements? Vision

Research, 2006. 46(8-9): p. 1346-1360.

Photoelastic Materials for Biomechanical Applications

Jonathan Fry 1. INTRODUCTION AND OBJECTIVES

Photoelastic materials provide an excellent means of modelling ‘real-world’ structures in a

variety of applications. The doubly refractive (birefringent) nature of these materials allows

the optical assessment of stress developments within structures giving useful information for

optimisation and improvements. This project involves finding and testing suitable

birefringent materials with a range of photoelastic and mechanical properties to enable their

use in biomechanical applications. Orthodontic models made from these materials can be

analysed by different methods of compensation and photoelastic techniques. 2. MATERIAL CALIBRATION

The various materials examined are indicated below:

. • Mechanical and photoelastic properties of the cooking gelatine with various

heating and mixture combinations were investigated. Three mixtures were used:

10/15/20 % gelatine to water. Each of these was heated to three different

temperatures; 40/50/60 oC and poured into cylindrical moulds for mechanical

calibration or circular moulds for photoelastic calibration and then refrigerated

for 36 hours prior to testing.

. • Sigma-Aldrich bovine and porcine gelatines were used in addition to the above

due to there pharmaceutical quality. Photoelastic properties were determined with

and without glycerine. The temperature used in their fabrication was 50oC.

. • Photoelastic testing of ‘Dog-bone’ specimens of EVA (Ethylene Vinyl Acetate),

PET-G (Poly-Ethylene Terephthalate), and Polycarbonate was also carried out.

2.1 Mechanical Testing

Sample cylinders of cooking gelatine were tested in compression apparatus with a dial gauge

to determine the applied load. From measurements taken stresses and strains could be

determined with incremental loads. Young’s modulus (E) and Poisson’s ratio (ν) were

calculated.

2.2 Photoelastic Testing

2.2.1 Brazilian Disc Method

All photoelastic testing took place in the stress laboratory, using a circular polariscope. In

order to achieve small loads, a special configuration was set up to create a load on the disc.

Two compensation methods, Null-balance and Tardy, were employed to analyse the fringe

order across the centre of the gelatine disc. From this, stress fringe constants, fσ were

determined.

2.3 Tensile Testing

For materials which could not be moulded into discs (EVA, PET-G and Polycarbonate) a

‘Dog-bone’ tensile test was used for calibration. Each specimen was clamped, fixed in the

testing rig and placed in the polariscope. A Null compensator was used to determine the

fringe order at the centre of the specimen at each load increment.

3. RESULTS 3.1 Mechanical Testing

Figure 3.1 shows the results of mechanical testing for 10 % cooking gelatine specimens.

Stress vs. Strain - 10% Gelatine

Strain

Figure 3.1: 10 % gelatines stress vs. strain graph

The table below shows the Young’s modulus for all the cooking gelatines.

Mass percentage of Gelatine to water

Temperature heated to (oC) Young’s modulus (kPa)

10 40 34.3 10 50 41.7 10 60 46.1 15 40 53.7 15 50 55.1 15 60 53.4 20 40 61.7 20 50 56.2 20 60 70.3

Table 3.1: Young’s modulus for different mixture and heats of gelatine

The Poisson’s ratio for all specimens tested is approximately 0.6. This was calculated

Stress (K

Pa)

18

16

14

12

10

8

6

4

2

from lateral strain vs. axial strain graph gradient

3.2 Photoelastic Testing

3.2.1 Brazilian Disc Method

Figure 3.2 shows the results of photoelastic testing for 10 % porcine gelatine specimen

compared with theory (blue curve).

Gelatine Composition

10 % 15 % 20 %

Cooking Gel (40 oC) 32.3 N/m 33.3 N/m 28.3 N/m

Cooking Gel (50 oC) 37.2 N/m 35.2 N/m 34.2 N/m

Cooking Gel (60 oC) 33.7 N/m 31.8 N/m 38.2 N/m

Porcine Gel 32.5 N/m 46.8 N/m 46.4 N/m

Porcine Gel with Glycerine 31.4 N/m 46.5 N/m 52.0 N/m

Bovine Gel 29.9 N/m 43.0 N/m 49.3 N/m Bovine Gel with Glycerine 31.9 N/m 42.6 N/m 52.6 N/m

Table 3.2: Fringe constants for gelatine specimens

3.2.2 Tensile Method

All three ‘Dog-bone’ specimens produced graphs with a linear relationship between load and fringe order. The gradient of these graphs represents the stress fringe constant, fσ multiplied by the width of the specimen, b. The stress fringe constants were

calculated for each specimen.

3.2.1 Orthodontic Model

Various orthodontic models were fabricated and a final solution was arrived at (figure 3.3).

4. DISCUSSION

For all of the nine cooking gelatine specimens, there is a linear relationship between stress

and strain. This enabled the calculation of the Young’s modulus for each specimen (table

3.1). It can be seen that as the ratio of gelatine to water is increased the Young’s modulus

increases. This is independent of temperature. Higher temperatures would increase the

mobility of molecules and therefore the propensity to form bonds. One might expect this to

have a greater effect at lower concentrations, i.e. 10 % gelatine. Generally, composition is a

more important factor than fabrication temperature when influencing the stiffness of the

material.

Table 3.2 shows a clear trend in the results achieved from photoelastic testing with the

Sigma-Aldrich gelatine. For the bovine gel with and without glycerine and the porcine gel

with glycerine; as the concentration of gelatine is increased the stress fringe constant is also

increased.

A linear relationship was obtained for load against fringe order from all three materials in

tensile photoelastic testing. With no load the EVA specimen showed a zero order dark

fringe in the polariscope meaning that there were no residual stresses present from either

manufacturing or machining.

Orthodontic models have been developed and a final model produced. This allowed

experiments to be conducted to test various sports, mouth guard designs. 5. CONCLUSIONS

This work has given calibrations of various materials that have been used in biomechanical

applications. From the photoelastic testing of the Sigma-Aldrich gelatine there is a clear

trend; as the concentration of gelatine is increased, the stress fringe constant, fσ increases.

This means the material is less photoelastically sensitive as concentration is increased. 15 %

porcine and bovine gelatine mixtures with glycerine are desirable in orthodontic models as

they exhibit good photoelastic properties as well as allowing significant tooth movement to

analyse stresses and strains. Glycerine, although it may improve the gelatines resistance to

tear, only has a small effect on the stress-optical sensitivity of gelatine.

From the work and experimentation presented in this study, a journal publication can be

completed.

PHOTOGRAMMETRY FOR IMPACT ENGINEERING

Arin Jumpasut

Introduction This summary describes research that has provided insight into utilising photogrammetry on high speed events. There are two processes, the acquisition of the images and their analysis to derive the required measurements [1]. Consideration of these tasks has allowed the development of bespoke photogrammetric techniques for advanced quantification of geometry and motion in scientifically and industrially important experiments. Advances in numerical modelling of material response to loading, particularly in the case of heterogeneous materials such as composites, as well as the use of data hungry inverse parameter identification strategies have led to an increasing requirement for full field measurements of specimen deformation during experimental characterisation.

Impact engineering photogrammetric framework High speed photography in impact engineering is characterised by these features: o Motion in the experiment

o High kinetic energy involved results in large deformations, fracture, fragmentation and interpenetration, labelled A in Figure 1.

o Debris often obscures the components of most interest in the experiment: labelled B in Figure 1 and present around the projectile in Figure 2.

o Filming at high frame rates o Light is the limiting factor: the competing requirements for large areas of

interest (and thus depth of field requirements that benefit from small lens apertures) at high speeds (and thus light requirements that benefit from large apertures) results in images that are a compromise.

o The light level usually varies within an experiment, e.g. as flashes warm up and discharge. This is shown in Figure 1: the earliest frame is overexposed; the last shows underexposure.

o There can be large translational and rotational movement of objects between frames due to the limited frame rates of cameras.

o It is hard to account for the full range of material behaviour, e.g. in the bulge test shown in Figure 3: the bulged plate may rise at a rate of 50ms-1 but once the plate bursts, a crack may propagate through the specimen at a rate in the order of 5000ms-1.

o Images often have more noise present than in quasi static filming, due to the required amount of amplification of the signal from the detector.

o Specialised high speed cameras are necessary to record impact experiments. These are often of much lower resolution than standard cameras. High speed cameras often have a trade off between increasing frame rate and decreasing resolution. It is uncommon to have more than one camera, which poses problems if the goal is to track motion in more than one plane. Even with multiple cameras, synchronisation at high speeds is difficult.

t0+2ms t0+4ms t0+6.4ms Figure 1: A sample sequence from an impact experiment. The time between frames was

83µs (around 12 000 frames per second) and the projectile had a mass of approximately 250g and was fired at a speed of 400ms-1.

If perfect images could be produced from impact experiments, then there would be no difference between analysing a quasi static experiment and a high speed experiment. It is due to the complex nature of impact experiments, that the analysis is more challenging. This research used a series of error analyses on different combinations of targets and target detection methods to consider conditions present in impact experiments. Subsequently, controlled experiments were analysed to confirm the methods derived. This experience was then used to design large scale experiments with photogrammetric measurements providing the primary interpretation of the experiment.

Error analysis on targets and target detection for impact engineering At the simplest level, the error analyses consist of artificially generated images that are complemented by small scale experiments with a real camera. This process investigates the effect of camera conditions, lighting conditions and motion in experiments, on the accuracy of different target detection methods and using a choice of targets. In order to perform image analysis a ‘target’ must be applied to the specimen. The most basic role of an artificial target on a specimen is to provide ‘a high contrast object that can be differentiated from the background illumination’ [2]. The choice of target to apply to a specimen is important when designing an impact experiment. There are two main categories of targets: random (e.g. speckle [3]); and regular (e.g. grids of corners). There are advantages and disadvantages of both. Using Figure 2 as an example, one advantage of regular targets is that there is separation of scale between the targets on the specimen and the noise and debris.

Figure 2: A projectile being fired from the left between into two symmetric panels

A A B B

If a speckle pattern were used to mark the projectile, the pattern would have to be large enough to be distinguished in such a noisy environment. Since speckle analysis relies on the random nature of the speckle pattern, when this random pattern has random noise added onto it, it complicates analysis greatly. Consequently, the projectile is marked with stripes because in this situation that is easier to track. The main disadvantage of using a regular target is that the association of targets between frames can become a problem; since each target is not unique, it is easy for one to be confused with another. These error analyses have allowed the development of a protocol of best practice on how to mark and detect targets in various types of impact experiments [4-7] that is in use within the team.

Figure 3: An example of a high speed bulge test showing how line fitting was used to overcome low quality images from the experiment (left) and the 3D displacement of a

corner from the analysis (right)

Applications Photogrammetry is an established field but technology is only now becoming available that allows digital images of impact experiments to be taken at a sufficient quality for accurate measurements. Experiments were designed and performed to test the results of the error analyses in controlled conditions. Figure 3 shows a biaxial tension (bulge) test on Ti64 as one example of this. Since the error analyses were done on single targets, the analysis of real experimental data allows other techniques to be employed, such as using geometric constraints on targets through line fitting as shown in Figure 3. Figure 3 also shows how the 3D displacement was calculated in this example (only one corner is shown) from which strains could be calculated should this be required. Finally, there are large scale experiments for which photogrammetric measurements will be used as one of the primary quantitative ways of analysing the experiment. This type of experiment has been completed in conjunction with an industrial partner, Rolls Royce Plc., and includes symmetric impact on panels for a gas turbine engine and a full scale bird strike test on a single blade. The analysis of the symmetric impact on panels has already brought important new insight into the behaviour of materials inside aero-engines.

References 1. Mikhail, E.M., J.S. Bethel, and J.C. McGlone, Intoduction to Modern

Photogrammetry. 2001: Wiley. 496. 2. Clarke, T.A., An analysis of the properties of targets used in digital close range

photogrammetric measurement. Videometrics III, 1994. SPIE Vol. 2350: p. pp 251-262.

3. Sjodahl, M. and L.R. Benckert, Electronic Speckle Photography - Analysis of an Algorithm Giving the Displacement with Subpixel Accuracy. Applied Optics, 1993. 32(13): p. 2278-2284.

4. Jumpasut, A., C.R. Siviour, and N. Petrinic. The Use of Photogrammetry in Aiding Finite Element Modelling of Impact Engineering Experiments. in Joint 8th. World Congress on Computational Mechanics and 5th. European Congress on Computational Methods in Applied Sciences and Engineering. June 30 – July 5, 2008. Venice, Italy.

5. Jumpasut, A., et al., Regular targets and target detection in image analysis for impact engineering. UTC Technical Report, 2008(tbd).

6. Jumpasut, A., et al., Analysis of dynamic fast crack propagation experiments and comparison with a finite element model. UTC Technical Report, 2008(231).

7. Jumpasut, A., et al., Analysis of bulge test experimental data at quasi static rates and dynamic rates and comparison with a finite element model. UTC Technical Report, 2008(232).

YSA08 Xu Song

Strain gradient crystal plasticity modeling and synchrotron X-ray diffraction experiments: keys to stress analysis at the nanoscale

Xu Song*, Alexander M. Korsunsky* * Department of Engineering Science

University of Oxford Parks Road, Oxford OX1 3PJ, UK

Objective: The creation of nano-science is one of the most exciting developments taking place in the beginning of XXI century. It is driven by the realization that material properties (including deformation behaviour and strength) change significantly at the nano-scale; by the advent of hitherto unavailable tools for manipulation and characterization of objects at the length scale of nanometers; and the attendant development of new theoretical and modeling approaches that allow faithful description of these nano-scale effects. In the present project we use the X-ray diffraction method for deformation characterization, since it offers unique possibilities for sub-micro-structure analysis, providing information nondestructively about both the near-surface and bulk regions. By analyzing the peak intensity, position and shape from single grains or polycrystals, one can obtain information about grain orientation, grain group stresses and dislocation density. Furthermore, analysis of peak shape dependence on sample tilts (rocking curve) provides insight into orientation-dependence of reflection intensity, and thus crystal domain rotation due to dislocation structures – a very important issue in the analysis of strain gradient effects. Note that by post-processing the predictions of our length scale dependent model, we can construct diffraction peak patterns; hence give better insight and perspective on the use of diffraction for microstructure and deformation analysis. Therefore, the central thrust of the present project is the linked, parallel development of modelling and experimental tools for the study of SSD and GND effects on deformation behaviour of (poly) crystals. Modelling & FE implementation: The present model has been implemented by the authors on the basis of the lengthscale-dependent, rate-dependent formulation presented by K.S.Cheong and E.P.Busso (2004)[1]. A brief summary of the formulation is given below.

The formulation is built on the expression for shear strain rate on a particular slip system α. This shearing rate is thermally dependent via a Boltzmann type exponential thermal activation expression, and also contains a dependence on the system-specific critical resolved shear stress (slip resistance):

αγ

000

0 0

/e x p [ {1 } ] s g n ( )

ˆ /p q

SFk

α αα α

τ μ μγ γ τ

θ τ μ μ

−= − − < > , (1)

1

YSA08 Xu Song

The central element of the present length scale dependent, dislocation-based deformation modeling approach is the system of evolution laws for the densities of edge and screw types of Statistically Stored Dislocations (SSD) and three types of Geometrically Necessary Dislocations (GND) and their contribution to the slip resistances on the corresponding planes. It is, specifically, the GND contribution that is closely linked to plastic strain gradient effects, and thus causes size and scale dependent effects. However, it is also important to note that in practice SSD and GND co-exist in deformed metallic (poly) crystals, so the separation of their contributions to overall deformation (hardening) behaviour is not at all a trivial task. For the analysis of model predictions we employed the near-α titanium alloy Ti-6Al-4V with the dominant HCP crystal structure having a total of 30 slip systems: 3 basal systems <a>, 3 prismatic systems <a>, 6 pyramidal systems <a>, 12 1st-order pyramidal systems <c+a>, and 6 2nd-order pyramidal systems <c+a>. In the finite element model implementation, the representative volume element (RVE) of polycrystal HCP material Ti-6Al-4V was implemented as a cubic-shaped unit volume discretised into a ‘static’ 3D regular hexahedral 10×10×10 mesh (C3D20R element type) with crystal lattice orientation assigned at each integration point (IP). A MathCAD pre-processor was used to generate the positions and crystal orientation at grain ‘seeds’. A population of 600 grains was ‘grown’ from ‘seeds’ distributed in space via the Voronoi tessellation method. Fully reversed (R = −1) strain-controlled loading was considered, and macroscopic stress-strain history, as well as meso-scale (grain-orientation specific) information was extracted. Results and discussion: 1) Model calibration against the monotonic stress-strain curve Crystal elastic stiffness matrix obtained from the literature[2] was used (with minor scaling). The critical resolved slip resistances were found in [3] and converted to the equivalent initial dislocation densities. All other parameters were adopted from [1] with modifications to fit Ti macroscopic plastic deformation behaviour. Figure 1 shows the macroscopic tensile test result and the model simulation after ‘tuning’ the parameters. 2) Model post-processing and comparison against XRD data We have developed several post-processors to enable us to obtain orientation and strain parameters for the so-called grain families. “Grain family” here refers to the subset of grains which contribute to the same particular diffraction reflection along a certain scattering direction. The post-processing is realized by labeling diffraction planes prior to the simulation. The orientation could be considered fixed (rotation ignored), or adjusted throughout the simulation. The post-processor identifies grains as belonging to certain groups, allowing averages and standard deviations (strains and strain spreads/peak widths) to be estimated. Figure 2 shows the comparison between FE model stress predictions and measurements

from diffraction experiments for the 0110 and 2110 grains. The vertical axis represents

2

YSA08 Xu Song

the macrostress applied on the sample/model and the horizontal one is the hkl lattice (elastic) strain response. Smooth curves without markers illustrate FE predictions, while the curves with markers correspond to the data obtained from diffraction experiments. It is apparent that the calibrated model is good at predicting general trends, although discrepancies are seen in the detailed magnitudes of the deviation from elastic lines (caused by plasticity). The post-processing procedure was then used for diffraction peak reconstruction. Peak centres can be readily calculated from the deviation of the lattice parameter from its original (unstrained) value. To predict the peak shape function we employ the framework introduced by G. Ribarik and T. Ungar (2001)[4]. The formulation relies on the use of Fourier coefficient to reflect the broadening effects due to size and distortion, respectively, which in turn depend on 2

Lε , the mean square strain depending on the displacement of atoms relative to their ideal positions, and ρ the dislocation density. Figure 3 demonstrates the pattern prediction capability of the post-processor. [5] Conclusion: It’s well known that the peak profiles are determined by several factors such as scattering planes, peak positions, peak shapes and intensities. Given known metal polycrystalline grain orientation information plus normalized peak intensities, we can argue that there’s a way to re-construct the whole XRD profile and give reasonable prediction, which provides a unique perspective to the stress analysis of advanced engineering materials. Reference: [1] K.S.Cheong and E.P.Busso, "A study of microstructural length scale effects on

the behaviour of FCC polycrystals using strain gradient concepts " international journal of plasticity, vol. 21, pp. 1797-1814, 2004.

[2] E.R.Naimon, W.F.Weston, and H.M.Ledbetter, "Elastic properties of two titanium alloys at low temperatures," CRYOGENICS, vol. 14, pp. 246-249, 1974.

[3] X. Song, S. Y. Zhang, D. Dini, and A. M. Korsunsky, "Finite element modelling and diffraction measurement of elastic strains during tensile deformation of HCP polycrystals," computational materials science, article in press.

[4] G.Ribarik, T.Ungar, and J.Gubicza, "MWP-fit: a program for multiple whole-profile fitting of diffraction peak profiles by ab initio theoretical functions," Journal of Applied Crystallography, vol. 34, pp. 669-676, 2001.

[5] S. y. Zhang, "High energy white beam X-ray diffraction studies of strains in engineering materials and components," in engineering science. oxford: university of oxford, 2008.

3

YSA08 Xu Song

Figures:

0

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0 5000 10000 15000 20000 25000 30000 35000 Strain (microstrain)

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Figure 1: Comparison of stress-strain response between FE model prediction and Instron tensile test

Figure 2: Comparison of FE simulation of Ti-6Al-4V alloy with the experimental (diffraction) data

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YSA08 Xu Song

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Figure 3: Comparison of post-processor constructed peaks with the data collected from the

diffraction in different orientations

5