Residual Stress Determination

Residual Stress Determination in the Aluminium Alloy 7075 Using the Strain-Gage Hole-Drilling Method.

Mr Barry Mooney

Submitted for the award of: Bachelor of Engineering in Design and Manufacture

Supervisor: Dr David Tanner Final Year Project report submitted to the University of Limerick, March 20, 2017. I declare that this is my work and that all contributions from other persons have been identified acknowledged.

Residual Stress Determination in the Aluminium Alloy 7075 Using the Strain-Gage Hole-Drilling Method. Barry Mooney Abstract The residual stresses in four heat-treated aluminium alloy 7075 rectilinear blocks have been determined using the ASTM standardized Hole-Drilling Strain-Gage technique. Stress variation as a result of quenching, ageing, and stress-relieving cycles have been monitored and compared with values determined by x-ray and neutron diffraction techniques. For each condition examined, complimentary and harmonious results were observed across all three measurement techniques. For the geometry of block used, the effect of uni-axial cold compression (1 − 2% plastic strain) has been shown to dramatically reduce quench induced near-surface residual stresses. For blocks having 30 minutes, and 4 hour post quench delays, the hole-drilling method was inconclusive in discerning a difference in the residual stress state. The effect of the delay was exclusively exposed by neutron diffraction, which had a resolution capable of detecting the diminutive differences in residual stresses. The data provided sufficient statistical evidence to conclude that an increase in post quench delay results in a higher residual stress state in the 7075 alloy.

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Contents Abstract

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List of Figures

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List of Tables

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List of Abbreviations and Symbols

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1 Introduction

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2 Background 2.1 Heat Treating of Aluminium Alloys . . . . . . . . . . . . . . . . . . . 2.2 Residual Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Literature Review 3.1 Residual stress relief . . . . . . . . . . . . . . . 3.2 Residual stress measurement . . . . . . . . . . . 3.2.1 The Hole-drilling Strain-gage technique. 3.2.2 X-ray diffraction. . . . . . . . . . . . . . 3.2.3 Neutron diffraction. . . . . . . . . . . . . 3.2.4 Ultrasonic. . . . . . . . . . . . . . . . . . 3.2.5 Summary of methods . . . . . . . . . . . 4 Theoretical Analysis 4.1 Computation of Non-Uniform Stress. 4.1.1 Strain conditioning . . . . . . 4.1.2 Stress calculation . . . . . . . 4.1.3 Regularization . . . . . . . . .

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5 Stress Calculation

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6 Experimental Apparatus 6.1 Strain-gage and instrumentation 6.1.1 Strain-gage . . . . . . . 6.1.2 Strain recording . . . . . 6.2 Hole-drilling . . . . . . . . . . .

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7 Experimental Proceedure 38 7.1 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.1.1 Material and heat-treatment. . . . . . . . . . . . . . . . . . . 38 iii

7.2

7.1.2 Hole-drilling Strain-gage procedure. . . . . . . . . . . . . . . . 40 X-ray and Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . 42

8 Results 8.1 Relaxed strain. . . . . . . . . . 8.2 Cartesian stresses. . . . . . . . 8.3 Principal Stresses . . . . . . . . 8.4 Diffraction results. . . . . . . . 8.5 AA7075 natural ageing curve. . 8.6 Post quench delay comparison. .

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9 Discussion 9.1 Strain recordings. . . . . . . . . . . . . . . . 9.2 Residual stress in quenched and aged blocks. 9.3 Residual stress due to cold compression. . . 9.4 Residual stress due to post quench delay. . .

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10 Conclusions

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A Computer model 65 A.1 MATLAB programming code . . . . . . . . . . . . . . . . . . . . . . 65 B Tabulated strain and stress values B.1 Tabulation of relaxed strain readings. . . . . . . . . . . . . . . . . . . B.2 Tabulation of Cartesian stresses. . . . . . . . . . . . . . . . . . . . . . B.3 Tabulation of Principal stresses. . . . . . . . . . . . . . . . . . . . . .

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C Statistical model 87 C.1 R programming code . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 D Turnitin Originality Report

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List of Figures 2.1

2.2 2.3

Ashby chart showing the topmost materials in terms of Specific Modulus vs. Specific Strength, including design guidelines for material selection of a beam of minimum weight (loaded in bending), where the objective is to maximize strength and stiffness. The plot shows how 7xxx series alloys are a competing choice with composites for structural aerospace applications. .

5

Illustration of the effects of slow vs. fast cooling from SHT temperature, on aluminium alloy’s microstructure. Source: (Croucher 1982a). . . . . . .

6

Graph showing the effect of section thickness on mid-plane cooling rate in AA7075 blocks (12in. × 12in. ×0 thickness0 ). The quench media are: cold water, boiling water and still air. Source: (see Totten et al. 2003, p.6) and/or (Vruggink 1968) . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.4

C-17 cargo ramp warped by the relese of residual stresses from material removed during the manufacturing process. Courtesy of D. Bowden (Boeing Company) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1

Ideal quench path for high strength aluminium alloys. Source: (Totten et al. 2003). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2

Comparison of residual stresses in AA7075-T6 plate before and after stress relief. 3.2a High residual stresses in the SHT and quenched alloy. 3.2b Reduction in residual stresses after stretching by 2%. and 3.2c Composite micrograph showing the highly directional grain structure of a 38mm thick AA7075 T6 plate. Source: (see Davis et al. 1993, p.588). . . . . . . . . . 14

3.3

Post-yield (0.2% proof stress) work-hardening behaviour from true-stress-strain curves for CWQ AA7050. Source: (Robinson et al. 2014). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4

Plot of Measurement Penetration vs. Spatial Resolution for residual stress measurement techniques that are used typically in industry and/or by this investigation. Adapted from: (Schajer 2013). . . . . . . . . . . . . . . . 20

3.5

(a) HDM where the shallow hole is drilled at the geometric centre of a specialised three element strain gage rosette. (b) Image showing non-uniform stresses for the incremental depth steps of HDM. Hole geometry labelling shown also. Source: Schajer (2013) and ASTM (2013) . 21

3.6

(a) XD can be observed in θ-direction if n λ = 2 d sinθ. (b) Diffraction pattern obtained with fine grain structure. Source: SEM (1996) . . . . . . 24

v

Residual Stress Determination 3.7

(a) Showing the orientation of laboratory coordinate system in relation to a rectilinear sample xyz. Automated orientation of the sample through the laboratory coordinate system allow for a thorough characterization of lattice spacing. (b) Image showing the gage volume definition of ND. The beam is generated and collected through narrow slits in neutron absorbing (cadmium) sheets. Source: SEM (1996) . . . . . . . . . . . . . . . . . . 25

3.8

Representation of the rectilinear blocks used in this investigation, showing the block section (removed) within which the neurton diffraction measurement was conducted. Points (a) and (b) denote the start and endpoints of the line-scan which is of interest to this investigation. Source: Robinson et al. (2014) . . . . . . . . . . . . . . 26

3.9

Various ultrasonic acoustoelastic configurations. Stress averaging takes place through the region in which the ultrasonic waves propagate, as indicated by cross-hatched regions. 3.9 Through thickness (pulse-echo config.). 3.9b through thickness (pitch-catch config.) and 3.9c surface (pitch-catch config.) Source: SEM (1996). . . . . . . . . . . . . . . . . . 27

4.1

Three element clock-wise, Type A strain gage rosette which was specifically designed for the HDM. Source: (ASTM 2013). . . . . . . . . . . . . . . . 30

6.1

Measurements Group strain reading instrumentation (a) P-3500 Strain Indicator. (b) SB-10 Switch and Balance Unit. Source: Vishay (2000). . . 36

6.2

Micro Measurements RS-200 Hole-drilling apparatus(a) Optical device configuration for allignment and inspection. (b) Hole-drilling configutration used for machineing the percicely located flat bottom hole. Source: ASTM (2013) and Vishay (1992) . . . . . . . . . . . . . . . . . 37

6.3

Hole-drilling reccomendations for increased acuracy (a) Orbiting technique. Source: Vishay (1992) (b) Tungsten-Carbide inverted cone end-burr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7.1

(a) Image of the AA7075 block measured by the Strain Gage Hole-Drilling (A10 shown). (b) Schematic showing the block’s Cartesian dimensions and location of strain gage rosette. Identifiers: L (Longitudinal), ST (Short Transverse) and LT (Long Transverse), match the hot-rolled plate directions from which the blocks were cut. . . . . . . . . . . . . . . . . . 39

7.2

(a) Experimental set-up consisting of RS200 milling guide assembly and strain indicator equipment. The RS-200 is attached to a purpose-built support jig as shown. (b) Size 18 inch nom. strain gage rosette installed to the aluminium block prior to drilling. (c) Rosette on completion of drilling. 41

8.1

Plots of hole-drilling relaxed strain vs. depth from surface. The data points are measured values for: (a) Block B8, (b) block A5, (c) block A7 and (d) block A10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8.2

Plot of calculated Cartesian stress results (by Uniform, Power Series and Integral methods) vs. depth from surface for block B8. Plots show stresses in Cartesian directions (L, L-ST, ST) together (top-left), and individually (others). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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Residual Stress Determination 8.3

8.4

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8.10

8.11 8.12

8.13

Plot of the calculated Principal Stress results (σM AX , σM IN ), by Uniform, Power Series and Integral methods, along with Principal stress direction (β) vs. depth from surface for block B8. The Integral method calculation values are plotted on top left (Max) and right (Min). The Power-series and Uniform calculations are plotted together on bottom left (Max) and right (Min). β is measured clockwise of strain gage 1 (L-direction). . . . Plot of calculated Cartesian stress results (by Uniform, Power Series and Integral methods) vs. depth from surface for block A5. Plots show stresses in Cartesian directions (L, L-ST, ST) together (top-left), and individually (others). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the calculated Principal Stress results (σM AX , σM IN ), by Uniform, Power Series and Integral methods, along with Principal stress direction (β) vs. depth from surface for block A5. The Integral method calculation values are plotted on top left (Max) and right (Min). The Power-series and Uniform calculations are plotted together on bottom left (Max) and right (Min). β is measured clockwise of strain gage 1 (L-direction). . . . Plot of calculated Cartesian stress results (by Uniform, Power Series and Integral methods) vs. depth from surface for block A7. Plots show stresses in Cartesian directions (L, L-ST, ST) together (top-left), and individually (others). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the calculated Principal Stress results (σM AX , σM IN ), by Uniform, Power Series and Integral methods, along with Principal stress direction (β) vs. depth from surface for block A7. The Integral method calculation values are plotted on top left (Max) and right (Min). The Power-series and Uniform calculations are plotted together on bottom left (Max) and right (Min). β is measured clockwise of strain gage 1 (L-direction). . . . Plot of calculated Cartesian stress results (by Uniform, Power Series and Integral methods) vs. depth from surface for block A10. Plots show stresses in Cartesian directions (L, L-ST, ST) together (top-left), and individually (others). . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot of the calculated Principal Stress results (σM AX , σM IN ), by Uniform, Power Series and Integral methods, along with Principal stress direction (β) vs. depth from surface for block A10. The Integral method calculation values are plotted on top left (Max) and right (Min). The Power-series and Uniform calculations are plotted together on bottom left (Max) and right (Min). β is measured clockwise of strain gage 1 (L-direction). . . . . . . Neutron diffraction determined residual stress magnitudes for blocks B8, A5, A7 and A10. Showing scan taken from block’s core to the surface in the LT direction. The x-ray diffraction determined stress magnitude for block B8, and the average determined principal stress magnitudes from the hole-drilling method are included on the graphs also. . . . . . . . . Plot of the natural ageing response for block A9. Source: Robinson et al. (2017) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Box plots comparing the range of residual stress in CC blocks having different post quench delays. Stress results have been determined by HDM (Average principal stress: calculated by the integral method-regularized). Box plots comparing the range of residual stress in CC blocks having different post quench delays. Stress results have been determined by ND.

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List of Tables 2.1

Table containing the design yield strength (MPa) at various thickness for aluminium alloy 7075 plate, and die-forgings. Source: (see AMSC 1998, p.662). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.1

Effect of Delay in Uphill Quenching from the Initial Quench on Stress Reduction (AA7049) (Totten et al. 2003) . . . . . . . . . . . . . . . . 17

3.2

General comparison of the different techniques used in the determination of residual stresses. Source: Adopted from Schajer (2013) and SEM (1996). 28

6.1

Geometrical dimensions and schematic of a ’Type A’ three-element Clockwise (CW) Hole-Drilling Rosette, (a) Rosette Layout, (b) Detail of Strain Gage (ASTM 2013) . . . . . . . . . . . . . . . . . . . . . . . . . 35

8.1

Calculated strain standard error values

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B.1 Normalized strain readings (µ), for each of blocks B8 and A5 . . . . . . . 81 B.1 Normalized strain readings (µ), for each of blocks A7 and A10 . . . . . . 82 B.2 Tabulation of the calculated Cartesian stress results (by Uniform, Power Series and Integral methods) vs. depth from surface. Data for blocks B8 and A5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Tabulation of the calculated Cartesian stress results (by Uniform, Power Series and Integral methods) vs. depth from surface. Data for blocks A7 and A10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Calculated Principal stresses (σM AX , σM IN ), and Principal stress direction (β) for Uniform, Power-Series and Integral calculation methods. β is measured clockwise of strain gage 1 (L-direction). Data for blocks B8 and A5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Calculated Principal stresses (σM AX , σM IN ), and Principal stress direction (β) for Uniform, Power-Series and Integral calculation methods. β is measured clockwise of strain gage 1 (L-direction).Data for blocks A7 and A10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Abbreviations and Symbols Abbreviations AA SHT HDM PQD CWQ In. GP W NA

Aluminium Alloy Solution heat treatment Hole-drilling strain-gage method Post quench delay Cold water quench Inch Guinier–Preston zone Temper designation Natural age

PAG L LT ST FEA XD ND

Polyalkylene Glycol Longitudinal direction Long-transverse direction Short=-transverse direction Finite element analysis X-ray diffraction Neutron diffraction

Symbols a ¯ ¯b a ¯jk ¯bjk ◦C d do D Do E ◦F j k kg K M pa m n Pk Pk qk Qk ρ SS s t tk

Calibration constant for isotropic stresses Calibration constant for shear stresses Calibration matrix for isotropic stresses Calibration matrix for shear stresses Degree Celsius Measured lattice spacing Lattice spacing in stress free reference Diameter of the gage circle, Diameter of the drilled hole Young’s modulus Degree Fahrenheit Number of hole depth steps so far Sequence number for hole depth steps Kilogram Material dependant acoustic constant Mega pascal Meter Number of observations Isotropic stress within hole depth step k Isotropic strain after hole depth step k 45◦ shear strain after hole depth step k 45◦ shear stress within hole depth step k kg Density m 3 Sample statistic Second x-y shear strain, time x-y shear strain after hole depth step k

t0 tsurf ace tprof ile T Tk Tw TS V Vo αP αq αt β j λ µ ø v θ σmax σmin (σx )k (σy )k (τxy )k Ys

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Time zero Estimated time to take surface measurement Estimated time to complete a full profile measurement (superscript) matrix transpose x-y shear stress within hole depth step k Water temperature Test statistic Ultrasonic wave velocity Ultrasonic wave velocity in a stress free reference Regularization factor for P stresses Regularization factor for Q stresses Regularization factor for T stresses Clockwise angle from the x-axis (gage 1) to the max. principal stress direction Relieved strain for “uniform” stress case Relieved strain measured after j hole depth steps have been drilled Wavelength micro Diameter Poisson ratio Angle of strain gage from the x-axis Maximum principal stress Minimum principal stress Normal x-stress within hole depth step k Normal y-stress within hole depth step k Shear xy-stress within hole depth step k Yield strength

Chapter 1 Introduction It is desirable that thick sections of aluminium alloy, which is intended as material feed for machining, display a level of residual stress as low as possible. Deformation of the work-piece during machining is a consequence of excessive stresses, leading to non-conforming product, and increased economic costs within the manufacturing industry (Rhenalu 2003; Zhang et al. 2014; Wei-Ming 2011). 7xxx series heat-treatable aluminium alloys possess mechanical properties that place them in competition with composites as a light-weight and cost-effective solution to components used in the aerospace industry (Robinson et al. 2011). Aluminium alloy (AA) 7075 is the outstanding member of the group in terms of strength, and is amoungst the highest strength aluminium alloys available (see Aluminium Association 2008, p. 105). Forgings or thick plate sections (up to ≈ 100mm) have been used widely in air-frame structures and for highly stressed parts. (Aluminium Association 2008; Wei-Ming 2009). AA7075 however, has a major drawback amoungst other 7xxx series alloys. Its advantageous mechanical properties are dependant on high quench rates which maximise the precipitation hardening response by freezing in place the hardening atoms that have been arranged during the solution heat treatment (SHT) process (Croucher 2011a; Mitchel 2004; Robinson et al. 2015a). In thick sections of the alloy, the high quench rates cause large thermal gradients which can exceed the instantaneous local yield strength (Tanner et al. 2003). The resulting inhomogeneous plastic flow is tensile in nature at the material surface where thermal gradients are highest. Upon full cooling, the stress distrubution invariably consists of compressive surface stresses, balanced by tensile core stresses (Robinson et al. 2002). Such stresses have the potential to cause severe dimensional instability as they re-distribute and equilibrate during downstream machining operations (Robinson et al. 2014)(see ASM 1991a, p. 113). 1

Residual Stress Determination

Since the early 1960’s primary mills producing plate and forgings have applied stress reliving tempers to high strength AA after quenching and prior to aging to reduce the impact of residual stress (Stickley et al. 1964). This is achieved by mechanically working the material (1 − 5% perminant set) through stretching (ANSI temper designation Txx511 ) or compression (Txx52). Forged products are often re-struck in the die when cold (Txx54) (ASTM 2009; SAE 2002). This stress-relieving proceedure, however, is not a standardized practice and the methods by which it is administered, along with the results achieved, are subject to variation between alloy producers (see Croucher 2011a, p.25). Moreover, as per Mordfin et al. (1988), publications on the topic are a scarcity. The effectiveness of the cold-working procedure can only be measured by the extent to which the treated component distorts during machining (Altschuler et al. ibid., p.23). Typical mechanical stress-relief operations have shown to reduce stress magintudes by up to 80% at best, the 20% that remain is more than enough to cause distortion during machining (Robinson et al. 2014; Mordfin et al. 1988). Thus, it is the purchasers of the raw materials who have since been coping with the technical and financial consequences of residual stresses in their raw material (Wei-Ming 2012; Prime et al. 2002). In order to solve residual stress problems of high strength aluminium alloys, it is important that the scarcity of publication in the area of stress relieving is addressed in order to develop a standardized approach that would benefit all parties concerned. Key to this is the quantification of residual stress levels to determine the effectiveness of a given treatment (Croucher 2011a). Research into the area is currently under way at the University of Limerick, with the objective of developing process improvements in the production of heat-treatable alloys, hence reducing the problem of machining distortion. This final year study has limited scope with respect to the overall field of research being undertaken. However, it intends to compliment current research through application of the ASTM Hole-Drilling Strain-Gage method (HDM) in order to determine residual stresses in the rectilinear AA7075 blocks which are being used in the investigations. The main aim of the work presented is to apply the HDM for the determination of near-surface residual stresses in blocks that have been quenched and stress relieved by cold compression. The blocks examined have undergone contrasting 1

xx represents the temper treatment

2

Residual Stress Determination heat-treatment and quench plans to allow for wide-scale investigation of the residual stress variation. The experimental variable investigated is the delay between quenching, and the application of stress relieving cold-compression. This variable is termed post quench delay (PQD). PQD is considered to influence the final residual stress magnitudes because of spontaneous ageing (strengthening) which occurs when the alloy is removed from the quench bath. The primary objective is to assess the effectiveness of the HDM in its ability to identify the effects of PQD on residual stresses. With the results obtained, a comparison with other state-of-the-art experimental methods is conducted (X-ray and Neutron diffraction methods) to identify the advantages/disadvantages, and limitations of the methods.

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Chapter 2 Background Aluminium alloys can be classified as heat-treatable or non-heat-treatable. The terms are linked to the operations used to increase mechanical properties (strength and hardness)(see ASM 1991b, p.1861). Heat-treatable alloys, achieve strength by means of a three-step process: 1. Solution Heat Treatment, 2. Quenching, and 3. Ageing (ASM 1991a). Non-heat-treatable alloys rely upon cold-working operations to gain their strength. The focus herein is on the 7xxx series alloys, and in particular 7075, whose strength can be significantly increased by heat-treatment (see ASM 1991b, p.1865). Considered the high-strength aerospace family, 7xxx alloys provide designers with materials that have the necessary high specific strength ( Yρs ) and specific modulus ( Eρ ) for structural aerospace applications. This is illustrated in the Ashby chart in Figure 2.1.

2.1

Heat Treating of Aluminium Alloys

The following description of the heat-treatment process is a gross over-simplification of the substantive physical changes that occur in reality, however, it is believed to be an adequate explanation in order to clarify the work undertaken by this investigation. The goal of solution heat treatment (SHT), whereby the alloy is heated and held close to its liquidus temperature (470−480◦ C for AA7075 (ibid.)), is to maximize the concentration of hardening elements by fully dissolving them into a super-saturated solid solution. Rapid-cooling by quenching follows, and is considered to be the most critical stage of the heat-treatment of high-strength aluminium alloys (Robinson et al. 2012a). 4


Figure 2.1: Ashby chart showing the topmost materials in terms of Specific Modulus vs. Specific Strength, including design guidelines for material selection of a beam of minimum weight (loaded in bending), where the objective is to maximize strength and stiffness. The plot shows how 7xxx series alloys are a competing choice with composites for structural aerospace applications.

The quench attempts to prevent precipitation and diffusion by freezing the finely dispersed solution of hardening atoms in place (Croucher 2011a). The optimum condition for ageing is to reproduce the super-saturated solid solution at room temperature, which requires sufficiently fast cooling-rates to accomplish the freezing process (see ASM 1991b, p.1892)(Robinson et al. 2012a). The required severity depends on the alloy type, its quench sensitivity, and the component’s thickness and configuration (Croucher 2011a). Slow cooling from SHT temperature facilitates the precipitation of alloying elements, which diffuse from solid solution and accumulate particularly at grain boundaries, near large voids, or un-dissolved particles (Croucher 1982a). The effect of slow vs. rapid-cooling of the alloys micro-structure is illustrated in Figure 2.2. For the alloy 7075, ASM (1991b) prescribes immersion quenching into water at room temperature (Tw ' 20◦ C) for plate products, referred to Cold Water Quench (CWQ), and at a slightly elevated temperature (Tw = 60 : 80◦ C) for die forgings. Agitation during quenching is recommended to assist with surface contact between the quenchant and the alloy (ASM 1991b; Croucher 1982a; ASM 1984). Practical convienience in mills sees the quench being administered by sheet or spray-quench with plate stock (Robinson et al. 2017). Aside from the obvious economic advantages, water can provide the rapid cooling rates necessary, and has a versatility through which the quench characteristics can be altered by adjusting the water temperature (Croucher et al. 2010; AZoM 2004). 5


Figure 2.2: Illustration of the effects of slow vs. fast cooling from SHT temperature, on aluminium alloy’s microstructure. Source: (Croucher 1982a).

In order to achieve near-maximal strength in the alloy 7075, cooling rates in excess of 300◦ C/s are required (see ASM 1991b, p.1892). Increasing component thickness has the same effect of reducing the cooling rate, thus impacting on the alloy’s mechanical properties (see Table 2.1) (AMSC 1998). AA7075 is highly quench sensitive in this regard. Vruggink (1968) proved this when he examined the effect of cooling rates on the variation of yield stress in aluminium alloys.

He measured the mid-plane

cooling-rates in 7075 blocks of varying thickness using quench media of cold-water, boiling-water and still-air (5000 : 3 ◦ F/s), as shown in Figure 2.3 (ibid.)(see Totten et al. 2003, p.6).

6


Figure 2.3: Graph showing the effect of section thickness on mid-plane cooling rate in AA7075 blocks (12in. × 12in. ×0 thickness0 ). The quench media are: cold water, boiling water and still air. Source: (see Totten et al. 2003, p.6) and/or (Vruggink 1968)

When cooled sufficiently quickly, the ’before’ and ’after’ micro-structure remain very much alike. The outcome of this process paves the way for the ensuing ageing (precipitation hardening) response, which causes the alloying elements to precipitate locally. In the as-quenched condition, atomic vacancies amongst the finely dispersed solution, provide vacant lattice sites for homogeneous precipitation (Mitchel 2004). Solute atoms diffuse to these sites, known as Guinier-Preston (GP) Zones, and nucleate to develop precipitate particles. The developing precipitates cause large strain fields amid solvent and solute atoms (Kaçara et al. 2015). It is the combination of the precipitate particles development, and culmination in strain fields within the alloys Table 2.1: Table containing the design yield strength (MPa) at various thickness for aluminium alloy 7075 plate, and die-forgings. Source: (see AMSC 1998, p.662). Plate Thickness (mm)

T6 (Mpa)

T73 (Mpa)

6.35-12.7 12.7-25.4

482

393

50.8-63.5 63.5-76.2

475 455

76.2-88.9 88.9-101.6

413 386

Die Forgings Thickness (mm)

T6 (Mpa)

T73 (Mpa)

≤25.4

441

434

379 348

25.4-76.2

434

386

338 331

76.2-101.6

427

379

7

Residual Stress Determination matrix, which give higher strength through obstructing and retarding the movement of dislocations (see ASM 1991b, p.1863-1865). The strengthening process is termed ageing, age-hardening, or precipitation hardening. 7xxx alloys age naturally (at room temperature), and continue to do so indefinitely which makes them highly unstable in this condition (see ASM 1984, p.176,184)(AZoM 2004). The naturally aged temper designation for AA7075 is the suffix letter W. Artificial ageing, by means of slowly heating to an elevated temperature, expedites the ageing process by allowing GP-zones to grow to a size that will not melt at higher temperatures. The process secures stable properties, higher strengths, improved corrosion resistance and better fatigue performance (see ASM 1984, p.184). Artificial ageing can be performed in a single or two-step treatment, and timing and temperature parameters are selected based on the components section thickness. The artificially aged temper designation is given as T6, which represents the highest practical1 strength temper produced industrially (ASM 1991b).

Re-ageing, also known as over-ageing, refers to a further heat-treatment step at higher temperature.

GP-zones, which at this point are stable, transform with

elevated temperature to allow an equilibrium between solvent and solute atoms. This alters the metallurgical structure, and hence reduces interstitial strains. The micro-structural modification reduces the alloy’s resistance to deformation. This consequential loss in strength however, is traded-off by an improved corrosion and fatigue resistance. The reduction of interstitial strain has the added benefit of reducing the alloy’s residual stress state (see ibid., p.1923). This stabilised temper has the T7 designation and is favoured by designers where maximal strength is not required. The forfeit in strength (10-15%)(see ASM 1991a, p. 115) is replaced with far superior fatigue and corrosion resistance performance than the T6 temper (see ibid., p. 115)(Mordfin et al. 1988; Van-Horn 1967; Tanner et al. 2008). Tests have confirmed an 80% increase in resistance to SCC over the T6 temper (see Davis et al. 1993, p.591). Moreover, the annealing gives a thermal stress-relief (10-20%), counteracting potential distortion in components that are subject to high in-service temperatures (Mordfin et al. 1988). Such conditions have the potential to cause dimensional change by facilitating further precipitation (Nordmark et al. 1970). 1

i.e. without sacrificing other material properties and characteristics which are desired in engineering applications.

8


2.2

Residual Stress

This thesis pertains to the circumvention of the problems relating to residual stresses which effect dimensional instability during manufacture. Since residual stresses which arise through the heat-treatment process account for 90% of the total, only these will be considered (Croucher 2011a). Residual stresses in 7xxx alloys are the unfortunate and unavoidable consequence of the cooling rates required to achieve tempers which conform to minimum strength limits (ibid.). Because of the alloys high coefficient of expansion, thermal and hence residual stresses arise during quenching due to the volume changes induced (see ASM 1991b, p.1339). At liquidus temperature, just prior to quenching, the alloy’s yield strength is very low.

Thermally induced stresses at the quench, caused by differential

through-thickness contraction, far exceed the yield strength. The resulting plastic flow eventuates as residual stresses at room temperature (see Davis et al. 1993, p.108)(see ASM 1991b, p.1949)(Mitchel 2004; Tanner et al. 2008) Robinson et al. (2012b) precisely explain the chronological events which lead to this stress state in rectilinear blocks such as those investigated here. They write: "immediately after quenching tensile plastic strains occur initially at the rapidly cooling edges of the material. The plastic zone then expands to cover all the rapidly cooling surfaces. The block at this point consists of a soft hot interior surrounded by a harder and cooler exterior stretched shell. As the central region starts to cool, it tries to contract but is constrained by the hard outer-shell and also undergoes tensile plastic deformation. As the block cools further, the magnitude of surface plastic strains diminish as a compressive stress is developed, finally resulting in a surface stressed into compression and a centre into tension". For a given alloy in the as-quenched state, its surface stress increases with section thickness (Robinson et al. 2014). Many articles have reported surface quenching stresses for thick sections of 7xxx alloy in excess of -200MPa upon cooling Mordfin et al. (1988), Robinson et al. (2012a), Robinson et al. (2017), Tanner et al. (2008, 2004), and Robinson et al. (2009). Unlike sheet-product, or components with abrupt sectional changes, thicker components of uniform geometry (e.g.

plate) are better capable of internally

enduring quench-induced stresses, and self-equilibrate without significant warping or distortion. Forgings which generally have complex shapes, undergo a less severe quench to lessen the effects of warping or distortion arising from cooling-rate non-uniformity (ASM 1991b).

It is only during downstream machining, or 9


Figure 2.4: C-17 cargo ramp warped by the relese of residual stresses from material removed during the manufacturing process. Courtesy of D. Bowden (Boeing Company)

during service, that residual stresses become problematic in such components (Croucher 2011a). Material removal during machining causes disruption to the parts static-equilibrium, thus invoking part distortion. This is a major obstacle among components destined for the aerospace industry, which have rigid requirements & specifications, and a strict precision is necessary for their manufacture. The challenges include very tight tolerances, multiple machined features, high volumes of removed material during machining, and numerous quality specifications (CNC 2013). The distortion potentially leads to re-work, or scrappage, both of which add complexity and/or cost to the manufacturing process (Croucher 2011a; Becker et al. 1996; Withers et al. 2001; Yoshihara et al. 1992). Figure 2.4 shows an example of such distortion in an aircraft cargo ramp produced from a forging which had a large volume of material removed to reduce structural weight. The excessively large residual stresses, which were relieved during machining caused severe distortion. Desirable surface compressive stresses, which are the by-product of quenching, are removed during machining thus exposing unwanted tensile stresses. Tensile surface residual stresses contributed to circumstances which caused major disruption to most aerospace companies in the early 1960’s. The situation arose where residual, assembly, and service stresses combined to produce a sustained tensile surface 10

Residual Stress Determination stress, exposing 7xxxT6 aluminium in-service parts to stress-corrosion cracking and corrosion-fatigue (Croucher 2011a; Tanner et al. 2003; ASM 1991a; Tanner et al. 2008; Nordmark et al. 1970; Wanhill et al. 2010). The problem was addressed in the mills when they incorporated a post-quench mechanical stress-relieving step whereby controlled amount of plastic strain (≈ 1 − 5%) was applied by stretching and/or compression (Rhenalu 2003; Haszler et al. 2000, 2002). Alongside this, new less sensitive alloys (7050 and 7449), and new tempers (T7 ) were developed, and the residual stress problems subsided even though they had not been completely eradicated. Croucher (2011a) explains how few alloy producers have any type of a quality procedure to determine if their stress relieving process is effective.

ASM

International do not specify limits, or require residual stress evaluation under their quality assurance guidelines (see ASM 1991b, p.1959). The onus for residual stress management has shifted from the alloy producer, to the manufacturer, who have since been coping with the issues. Since those early days, concepts of residual stress relief by quenching control have been explored, but they have not gained widespread acceptance among the aerospace industry in the way that mechanical methods have.

Such methods

include uphill quenching, and use of glycol-water quenchants. Croucher (2011a) advocates the success of such methods which he practised successfully at his job-shop style heat-treatment company in the US. In his articles he describes a tailored heat-treatment approach was he carried out to components during/after production. This differs to the traditional approach whereby plate/forgings are supplied from the mills in their heat-treated temper. The development of larger modern-day aircraft, has roused the need for larger and stronger structural components (Wei-Ming 2009). The scaling-up in size has amplified the problems associated with residual stresses. Problems not only relating to production stages, but also to SCC of in-service components (Croucher 2011a). Current research (Robinson et al. 2017) looks towards improving the already established processes that are performed in mills for the production of high-strength aluminium alloys.

These processes, as identified by Robinson et al. (2009),

do not completely remove residual stresses, and the surplus stresses are more than adequate to burden producers of aerospace components during downstream machining operations. It is believed that even small reductions in the residual stress state of raw materials will realise significant cost reductions at the manufacturing stages.

11

Chapter 3 Literature Review Two main topics are covered in this review: 1. Residual stress relief methods, and 2. Residual stress measurement techniques. While many alternatives to each topic exist, the spotlight herein is on the methods and techniques that are most closely related to this investigation.

3.1

Residual stress relief

Residual stresses can be managed in two ways, firstly through controlling the quench parameters, and secondly by performing additional operation(s) after the stresses have been imparted. Adjusting the quench parameters by increasing quench water temperature is the simplest approach, and is effective for stress reduction.

However, because the

mechanical properties of 7xxx series alloys are so strongly dependant on maintaining a supersaturated solid solution after quenching, the cooling rates necessary preclude this as an option. The cooling rates must always be high enough to meet the minimum property requirements. Tanner et al. (2008) demonstrated this experimentally on quench sensitive 2xxx series forgings using water at 60 & 100◦ C. The experiment evaluated residual stresses at the specimen’s surface for each of the quench temperatures.

After

undergoing a two-step ageing treatment, the boiling water quench returned 50% lower residual stresses, however had unacceptably low mechanical properties. On the other hand, Dolan et al. (2004) demonstrated how slowly cooling 20mm thick AA7175 test-pieces from SHT to an intermediate temperature prior to CWQ, achieved a 15% surface stress reduction with negligible change to mechanical properties. They used quench factor analysis in conjunction with time-temperature 12

Residual Stress Determination property curves to select the appropriate temperature, which remained above the determined critical temperature region. While the method is effective, it may be difficult to avoid appreciable precipitation on a large-scale basis, as in an industrial setting the material could potentially pre-cool to a temperature below the critical range. As noted previously, large thermal stresses at the beginning of the quench are replaced by residual stresses upon cooling. And, it is the cooling rate of the alloy that ultimately dictates the strengthening response. From the ideal quench path shown in Figure 3.1 taken from Totten et al. (2003), the largest thermal gradients take place at t0 when yield strength is at its lowest. The stage responsible for the development of mechanical properties however, occurs at a time after this (t0 + t). Water-soluble polymers, most commonly polyalkalene-glycol (PAG), have proven to be very effective in minimizing part distortion in sheet products by exploiting this characteristic (Croucher 2011a). The technique is also very effective in parts with non-uniform cross section as it permits a more uniform heat extraction rate (see ASM 1991b, p.1339). The method has been used to solve long established distortional problems on components that have been heat-treated post machining, with significant cost savings (Croucher 2008, 2011b). The glycol essentially forms a film around the part, allowing it to cool slowly at first, and then breaks down at a predicted temperature. The quench speed is controlled by adjusting the water-polymer concentration (Croucher 1982b). Concentrations of 12 − 16% are approved for use with AA7075 parts of thickness ≤ 25.4mm (see ASM 1991b, p.1902). The limits of the technology however, have not yet been expanded to thicker sections. Croucher (2011b) discusses how these water-based polymers are effective in reducing residual stresses in thick sections of high strength aluminium alloys. In his paper, he reviews numerous test programs conducted on high strength aluminium forgings using the glycol quenchant. The conclusions state that while large reductions of distortion and residual stresses are achievable, the technique is only considered as an effective alternative to the hot-water quenching method due to the mechanical properties attained.

Uphill quenching is a thermomechanical stress relief technique that was also developed in the 1960’s to help solve residual stress issues (Pellman et al. 1990). It can be applied to alloys of substantial cross-section, and involves cooling to temperatures as low as liquid nitrogen temperature, which are then rapidly heated 13


Figure 3.1: Ideal quench path for high strength aluminium alloys. Source: (Totten et al. 2003).

(a)

(c) (b)

Figure 3.2: Comparison of residual stresses in AA7075-T6 plate before and after stress relief. 3.2a High residual stresses in the SHT and quenched alloy. 3.2b Reduction in residual stresses after stretching by 2%. and 3.2c Composite micrograph showing the highly directional grain structure of a 38mm thick AA7075 T6 plate. Source: (see Davis et al. 1993, p.588).

14

Residual Stress Determination in a steam blast. The cycle causes mechanical plastic deformation which reverses (cancels out) residual stresses present from water quenching (see ASM 1991b, p.1339). Difficulties associated with the technology, for example, the targeting of steam nozzles, mean that any non-uniform heating will result in subsequent machining distortion, which nullifies the methods effectiveness.

The process is

prohibitively costly to install and run, and is time consuming, therefore does not lend itself to continuous processing (Pellman et al. 1990). Further, thermal stress relief methods such as uphill quenching are not well suited to 7xxx alloys, as the temperatures required for effective stress relief are greater than artificial ageing temperatures. Figure 3.2c shows a typical AA mill product with constant cross-section, of which the processing history results in a highly directional grain structure (see Davis et al. 1993, p.588). Results from literature by (Robinson et al. 2014; Prime et al. 2002) showed residual stress magnitudes in the longitudinal rolling direction (L) to be ≈ 40% greater than long transverse (LT) and short transverse (ST) directions in non-stress-relieved hot-rolled AA7050 plate. The product’s anisotropy is influential to the alloys susceptibility to SCC, thus, tensile residual stresses, especially in the short-transverse (ST) direction are undesirable (see Davis et al. 1993, p.591). A classic residual stress distribution and magnitude after SHT and CWQ for 7xxx plate is shown in Figure 3.2a (ibid., p.588). Conventionally such stresses are relieved through controlled mechanical stretching, whereby the plate is clamped between the gripping jaws of a hydraulic machine and stretched to between 1 12 − 3% permanent set (Aluminium Association 2008)(see Davis et al. 1993, p.558)(Prime et al. 2002; Proulx 2011). Stress-relief of complex forgings is a rarity, but if required, receive compression due to their geometry. This is typically done by re-striking them in a second die (which will fit the component after cooling). The associated costs make this a rare occurance (Robinson et al. 2014). ASM (1991b) provide for compressive stress relieving in the order of 1 − 5% permanent set. Residual stresses can be released by inducing plastic flow through the application of mechanical forces which exceed the material’s yield strength. Upon unloading, much of the pre-existing residual stresses will have been relaxed as per Figure 3.2b (see Mordfin et al. 1988, p.112). The direction of load application is usually geometry dependant. Alexander (1959) and Robinson et al. (2009) have shown that that uni-axial loads applied in the 15

Residual Stress Determination longitudinal direction relieve stresses in the transverse directions. The strains transmitted through stretching/compression have hardly any effect on the mechanical properties of 7xxx alloys in the T6 temper, however the contrary is true for T7 tempers. T7 tempers have lower property specifications thus (see ASM 1991b, p.1927). A report for the US Defence was performed by Bergstedt et al. (1959) to examine the effect of varying degrees of stretching using AA7075 extruded bars in the as-quenched condition. The metric under investigation was the distortion after machining. It was found that there was no difference between minimum (0.78%) and maximum (2.65%) amounts of stretching employed by the test. What was discovered, however, was that longitudinal mechanical properties (tensile, yield and ultimate strength) were diminished (5%-10%), the greater losses occurring in the more highly stretched bars. The compression yield strength was also reduced by stretching in parallel with the other losses. The losses are due to the Bauschinger Effect which is broadly defined as the antistrophic modification of the plastic properties of a metal that has undergone previous plastic deformation (Newton 1961). While comparing the effectiveness of stretching vs. compression, Altschuler (see Mordfin et al. 1988, p.22) found that plastic deformation by compression caused reductions in residual stress which were similar, but not identical, with those observed in tension. He also stated that stress relief by tension was more effective by 10%. Conclusions from a finite element analysis (FEA) model created by Tanner et al. (2003) agree. Tanner et al. (ibid.) found that compression, due to friction forces, resulted in higher (tensile) surface stress magnitudes. In his experiment with AA7075 rectilinear specimens, Altschuler showed that it was not possible to obtain complete stress relief from either method, achieving at best a 79% reduction (-165 MPa to -35 MPa) which was measured on the specimen’s (un-constrained) ST face. He mentions that similar results (unpublished) were obtained by the Aluminium Company of America (ALCOA) when they characterized AA7079 forgings in the same manner. Van-Horn (1967), in an investigation involving AA2014-T6 stated that the extent of the stretching technique’s effectiveness was complete after 2% strain, which was confirmed later with the FEA model created by Tanner et al. (2003). The evidence suggests that mechanical methods are undoubtedly both efficient and 16

Residual Stress Determination effective in reducing residual stresses by up to 80%. However, aluminium’s low modulus of elasticity (≈ 70GP a) lends to the fact that the stress ranges that remain afterwards in thicker sections are more than sufficient to cause movement during machining. It has been shown that quenching and stress relieving are important processes to be controlled. Nevertheless, what happens in-between these processes can also impact on the residual stress state, as precipitation hardening will occur at room temperature.

Refinement of process parameters during this period affords the

heat-treater with an opportunity to reduce the magnitude of residual stresses which will reflect in a better-quality product for the customer. Tanner et al. (ibid.) used the FEA model mentioned earlier to simulate residual stress development during quenching. They predicted amongst other things, the effect that natural ageing has on the final (relieved) stress distribution. With AA7010 stress-strain data, this pioneering research demonstrated that allowing a forging to naturally age prior to compression will result in larger stresses for the same degree of compression than those compressed immediately after quenching. The report signalled the importance of performing mechanical stress reduction within 2h of quenching, as precipitation hardening leads to a requirement for higher deformation forces. Similarly, delay after quenching significantly influences the effectiveness of stress-relief by uphill quenching as shown in Table 3.1 (Totten et al. 2003). Table 3.1: Effect of Delay in Uphill Quenching from the Initial Quench on Stress Reduction (AA7049) (Totten et al. 2003) Post Quench Delay (hour)

Measured Residual Stress (MPa)

Stress Reduction(%) (MPa)

28 48 69 97

83 71 58 42

1 3 8 24

The quench delay time, the time to transfer from SHT furnace to full immersion in the quench tank, is well defined in the literature as it is known to alter properties (ASM 1991a,b; Totten et al. 2003). The post quench delay (PQD) however, the time between quenching and the application stress relief, is not. 17


Figure

3.3:

Post-yield (0.2% proof stress) work-hardening behaviour true-stress-strain curves for CWQ AA7050. Source: (Robinson et al. 2014).

from

ASM (1991b) advises that production operations must be aligned so that most of the plastic deformation is accomplished before an appreciable amount of precipitation hardening takes place. (see ibid., p.1908). The general belief is that industrially, this delay is kept below 2h as previously advised by Tanner et al. (2003). Robinson et al. (2015b) demonstrated that a reduction in PQD resulted in lower magnitude residual stresses after cold compression, and suggests that the PQD should be reduced as far as practically possible. The reason given is that before natural ageing, the material is more likely to behave uniformly with respect to yield and post-yield work hardening. Post-yield work hardening behaviour presented in Robinson et al. (2014) shows the effect for CWQ AA7050 samples at various PQD’s, and is shown in Figure 3.3. Thick sections will work harden at different rates from surface through to core, which potentially leads to a larger residual stress state after deformation (ibid.). Directly after quenching, at the early stages of precipitation hardening, the material is in its most homogeneous state. While in this condition, the material responds uniformly to mechanical work. This is true both from a power-capacity and from a stress-reduction point of view (Tanner et al. 2003; Robinson et al. 2015b). As can be seen from Figure 3.3, there is significant disparity in work hardening and yield stress for the various PQD’s. It is also of note for the proceeding section that Robinson et al. (2015b) found that the stress variation due to PQD was not detectable through certain measurement techniques (single peak neutron diffraction and surface X-ray diffraction). 18


3.2

Residual stress measurement

Residual stress systems are self-equilibrating, i.e. the resultant force and moment that they produce equals zero (Totten et al. 2003). The distance over which residual stresses act has classically been a metric by which to group them. In increasing resolution, these groups are: Type I stresses which act over distances of mm (termed macro-structural) Type II acting over micro-structural distances, typically 1-100µm, and Type III stresses over distances of atomic scale (Totten et al. 2003; Schajer 2013). Measurement techniques can be broadly categorized as (i) mechanical, and (ii) physical. It is important that the appropriate type is used to allow engineers to form a technical understanding of the gradients and their magnitudes that exist. All mechanical approaches are termed destructive, and involve to some degree, the removal of material. By measuring strains or deformations released through material removal, the stresses that pre-existed can be determined. Physical methods in contrast are non-destructive, and typically use acoustic and magnetic radiation to measure atomic spacing’s in crystalline or polycrystalline materials (see Totten et al. 2003, p.315). Method selection very much depends on the measurement objective, and among the available options lies a broad range of capabilities in terms of sensitivity, resolution and depth penetration. For academic investigations like this, a combination of stress characterization techniques is merited, and three methods have been employed: (a) Hole-drilling Strain-gage, (b) X-ray diffraction, and (c) Neutron diffraction. However, in a large-scale industrial setting such as the aluminium mill, the requirement for cost-effectiveness, practical speed, and ease of application make methods such as the ultrasonic technique prevalent for in-situ measurement (Aluminium Association 2008). Industrial off-line measurements are also frequently accomplished using mechanical or X-ray diffraction methods (Bray 2010). The searches failed to un-earth any further information regarding the stress-relieving practices of primary producers of alloys. Since the process is such an important one, the dearth of available information relating to the techniques employed industrially is possibly due to the safe-guarding of intellectual property. In addition to the three above-listed techniques, some attention will be given to the ultrasonic measurement 19

Residual Stress Determination technique also, as this is practised industrially (Aluminium Association 2008). Figure 3.4 shows where each of these techniques lie in terms of spatial resolution and depth of measurement penetration. The following descriptions provide the basic operating principals of each.

Figure 3.4: Plot of Measurement Penetration vs. Spatial Resolution for residual stress measurement techniques that are used typically in industry and/or by this investigation. Adapted from: (Schajer 2013).

3.2.1

The Hole-drilling Strain-gage technique.

The Hole-drilling Strain-gage method (HDM), around which this thesis is centred, is a well-established and widely accepted technique (Grant et al. 2006). This standardised procedure, fully defined in ASTM (2013), is performed by measuring surface strains using a strain gage rosette. The strains are caused by localized stress relaxation on the outside of a shallow hole as it is introduced incrementally into the stressed material. These strains can then be used for the back-calculation of residual stress (Schajer 2013; SEM 1996). The finished hole is small (max: ø4.25mm × 2mm), thus the damage caused to the test-piece is often insignificant and repairable. The method can measure in-plane (2D) stresses, while incremental depth steps allow the development of a stress profile (ASTM 2013; OU 2016). The hole drilling method (HDM) is applicable to isotropic materials which follow 20


(a)

(b)

Figure 3.5: (a) HDM where the shallow hole is drilled at the geometric centre of a specialised three element strain gage rosette. (b) Image showing non-uniform stresses for the incremental depth steps of HDM. Hole geometry labelling shown also. Source: Schajer (2013) and ASTM (2013)

linear-elastic behaviour. Stresses may remain approximately constant with depth (Uniform), or in thick sections, they may vary somewhat with depth (Non-uniform) providing the stresses remain below 80% of the yield strength (ASTM 2013). The technique has been shown to produce excellent quantitative data, providing due appreciation of the factors that contribute to uncertainties in measurement, meticulous experimental practice, and appropriate data analysis is practised (Kandil et al. 2001).

The HDM technique has been developed and improved significantly since it was first introduced by Mathar in 1934 (SEM 1996). On the experimental side, Rendler and Vigness contributed greatly towards making the procedure reproducible and systematic when they developed the ASTM standard strain gage rosette in 1966 (Rendler et al. 1966). Theoretically, Schajer is the most noteworthy contributor to the calculation procedures used. In 1981 he provided the first generalised finite element analysis of the method, creating calibration coefficients which account for the loss in strain sensitivity as the hole progresses deeper from the surface. These coefficients have been updated since to account for the most accurate calculation method (integral method) which was introduced in 1988 (Schajer 1988a). Summaries of the three calculation methods that will be used in this investigation are given below. 21

Residual Stress Determination Calculation variants. One of the fundamental limitations of the HDM is that small experimental errors in strain measurement translate into larger calculated residual stresses.

Three

calculation variants which can be applied to the HDM have been used in this report, each of which having a different spatial resolution and hence sensitivity to measurement error.

The three calculation methods are listed in order of

increasing spatial resolution and sensitivity to measurement error, and are as follows: 1. Uniform method 2. Power Series method, and 3. Integral method. • The Uniform method assumes that the residual stresses are uniform with depth from the specimen surface. The HDM method is only valid when the residual stresses are in-fact uniform. This calculation method is the least sensitive to measurement error 1 . • The Power Series Method provides a limited amount of spatial resolution by assuming that the residual stresses vary linearly with depth from the specimen surface. This calculation method is most informative when the measured residual stresses vary smoothly with depth1 .

This calculation

procedure has been superseded by the Integral method and is not detailed in the most recent ASTM test standard. • Integral Method: The spatial resolution and sensitivity to measurement error is highest for the integral calculation method, especially for large depth increments. It provides a separate evaluation of residual stress within each increment of depth used during the HDM measurements, and assumes the stresses within each increment are uniform. This method is suitable when measuring residual stresses that vary rapidly with depth1 .

The Integral

method equations address the relationship between the measured deformations (strains) arising from a depth of removed material, and the ’inverse’ calculation involves solving a set of linear equations regarding matrices that contain geometrical factors that depend on the drilled hole depth and diameter. Such inverse solutions are characteristically noisy due to numeric ill-conditioned matrices, which has proportional consequences between small strain errors and the calculated stress result (Schajer 2007). Tikhonov regularization is therefore applied to the practical data, which effectively smoothes the stress solution by diminishing the adverse effects of the measurement noise, without significantly distorting the stress solution corresponding to the ’true data’ (Schajer et al. 2006). 1

(Schajer 2001)

22

Residual Stress Determination In summary, for the blocks under investigation, the Integral method values are considered to have the most significance. It gives a stepped approximation to the actual stress variation with depth, and is best suited to cases where the residual stress field varies abruptly with depth (Schajer 1988b). However, due to the test procedure’s practical challenges of obtaining accurate measurement for the initial depth increment, the method’s high sensitivity to very small measurement error has the potential to exaggerate the stress result near the surface. The Power Series method can be used to assist with judgements on near the surface stress magnitudes for this reason. The method gives a close straight-line fit of the stress variation with depth (ibid.), but its assumption of linearity limits the method’s credibility for stresses determined further from the surface, especially when the stress field does not vary smoothly with depth. The Uniform method is not considered to have any significance in this investigation, however, is calculated to provide a representative average value of the pre-existing stresses in the total volume of removed material.

3.2.2

X-ray diffraction.

X-ray diffraction (XD) methods has been traditionally used for residual stress measurement. Of the physical measurement types, XD is most commonly used, and that over which we have most control (SEM 1996). XD determines the arithmetic average macroscopic stress in a volume of diffracting material defined by: the irradiated area × the x-ray beam depth-penetration. The maximum feasible irradiated area is approximately 13mm × 8mm, and x-rays are limited to low penetration depths which does not exceed 100µm. Thus, XD requires a good surface condition, and the technique’s gauge volume is small in comparison to the HDM. (Schajer 2013; OU 2016; Prevéy 1986). A characteristic of each metallic material is that they have their own well defined crystalline structure with periodic lattice spacing (d). The x-ray beam, when targeted at the material, is scattered upon contacting the comprising atoms. The scattered rays cause an optical interference pattern through diffraction (SEM 1996). When the concentration of scattered waves summates to cause constructive interference, a condition known as Bragg’s law is met. This law is: 2d sinθ = n λ, and a schema is shown in Figure 3.6a. Each grain’s regular crystalline structure is orientated randomly. When an incident beam is targeted upon an area covering multiple grains, cones of diffracted beams, 23


(a)

(b)

Figure 3.6: (a) XD can be observed in θ-direction if n λ = 2 d sinθ. (b) Diffraction pattern obtained with fine grain structure. Source: SEM (1996)

with the incident beam as the cones axis, are observed. Each cone belongs to an individual lattice plain, and the angles relate to the spacing (d) as shown in Figure 3.6b (see SEM 1996, p.72). The lattice spacing for a stress free polycrystalline sample (d0 ) is constant irrespective of the grain orientation. If the specimen is stressed the spacing is altered, i.e. for a tensile stress, the spacing increases. Hence, the elastic strain is simply the change in lattice spacing (d − d0 ) whereby the crystal lattice acts as an internal strain gage (Prevéy 1988). As x-rays are scattered off many atoms from many grains, diffraction peaks are created. Changes in the peak spacing (peak shift) is what allows the measurement of lattice strain. Stress magnitudes can then be determined through the established theory (SEM 1996; Prevéy 1988). For larger grained materials, the irradiated volume can be increased to allow a sufficient quantity of diffraction peaks. This is achieved by conducting scans at a series of angles. The XD techniques have a precision to within ≈ 20MPa, and the accuracy is dependent on the resolution with which the stress distribution is measured. Accuracy and resolution are often conflicting objectives, with one having a diminishing effect to the other (Zhang et al. 2014; Schajer 2013).

3.2.3

Neutron diffraction.

Neutron diffraction (ND) works off the same principals as XD, except neutrons feature much higher penetration depths. This allows thick sections of aluminium (up to ≈ 300mm) to be characterized through the thickness. The deep penetration 24

Residual Stress Determination allows for measurement of bulk stresses by using the lattice structure as a strain gage. For accurate results the lattice spacing in the work-piece are compared with an unstressed material sample since unlike XD, the stress state is tri-axial (x,y,z) when measuring at depth (Schajer 2013). Strain measurements can be obtained by orientating the sample through the incident beam. To determine the stress field, it may be necessary to take measurements through many orientations, hence the technology has been developed to be highly automated. Figure 3.7b shows the laboratory coordinate system with respect to the work-piece (SEM 1996). The incident beam is narrow, and the diffracted neutrons from the scatter are collected through fine slits as part of the evaluation. The gage area is described by the incident and diffracted beams overlap as shown in Figure 3.7a. When this area is moved linearly, it describes a volume. It is over this region that the strain recordings are averaged (ibid.). Figure 3.8 shows a representation of the aluminium blocks under investigation here. Tri-axial stresses have been characterized in these blocks by performing line scans in the x (L), y (LT) and z (ST) directions.

(a) (b)

Figure 3.7: (a) Showing the orientation of laboratory coordinate system in relation to a rectilinear sample xyz. Automated orientation of the sample through the laboratory coordinate system allow for a thorough characterization of lattice spacing. (b) Image showing the gage volume definition of ND. The beam is generated and collected through narrow slits in neutron absorbing (cadmium) sheets. Source: SEM (1996)

25


Figure 3.8: Representation of the rectilinear blocks used in this investigation, showing the block section (removed) within which the neurton diffraction measurement was conducted. Points (a) and (b) denote the start and endpoints of the line-scan which is of interest to this investigation. Source: Robinson et al. (2014)

3.2.4

Ultrasonic.

As previously noted, it is known that aluminium mills employ ultrasonic techniques. It can be used to detect flaws in the material, but in the context of residual stress determination, it does so based on the measurement of variation in sound waves. The technology can be configured to inspect surface stress and through thickness stress (see Figure 3.9), and does so much faster (within minutes) than the other methods (HDM, XD and ND) making it highly applicable industrially (SEM 1996).

A material dependant parameter (K ) known as the acoustic constant can be used to describe the stress state by the relationship: V = V0 + Kσ Where V is the ultrasonic wave velocity, and Vo is its velocity in a stress-free reference sample. 26


(a)

(b)

(c)

Figure 3.9: Various ultrasonic acoustoelastic configurations. Stress averaging takes place through the region in which the ultrasonic waves propagate, as indicated by cross-hatched regions. 3.9 Through thickness (pulse-echo config.). 3.9b through thickness (pitch-catch config.) and 3.9c surface (pitch-catch config.) Source: SEM (1996).

A caveat of the method is that since the wave velocity shift is proportional to the stress present in the area through which the waves propagate, only an average stress can be determined (ibid.). A cancelling of compressive vs. tensile stresses through the material thickness does not assist with solving distortional problems in heat-treated aluminium parts, but instead potentially masks them.

3.2.5

Summary of methods

Both the HDM and XD provide localized stress quantification.

The HDM

(semi-destructive) is used to gather a stress-profile based on stress relaxation in a series of small gage-volumes of material to a max depth of 2mm from the surface. XD (non-destructive) provides the average stress over a much smaller gage volume at the materials surface. ND has the ability of penetrating deep into the sample, and can determine a global stress map by conducting a series of 2D line scans to obtain a series of gage volume measurements. Lattice spacing can therefore be obtained through the entire thickness if necessary. Through automated line-scan sequences, ND provides an all-encompassing evaluation, as opposed to localised XD and HDM methods. To do so, a source of neutrons is required, and scanning time is lengthy making the equipment and technique extremely expensive. ND is thus suited mainly to specialised laboratories rather than industrial settings. 27

Residual Stress Determination Table 3.2: General comparison of the different techniques used in the determination of residual stresses. Source: Adopted from Schajer (2013) and SEM (1996). HDM σ analysed Parameter Min. Depth Cost of Equip. Precision Min. zone tsurf ace tprof ile

Type I Surface strain 20 µm 10-50k USD ±20 MPa 0.5mm2 40 min. 2 hours

XD

ND

Ultrasonic

Type I Type II Type III δ Lattice spacing 5-40 µm 100-200k USD ±20 MPa 0.5mm2 20 min. 8 hours

Type I

Type I Type II Type III δ Wave speed 100 µm 40-200k USD 10-20 MPa ≤ 100mm2 Several min. 20 min.

Type II δ Lattice spacing 1mm 100’s millions ±30 MPa 4mm3 2 hours 1 week

When combined, XD, HDM and ND measurement techniques complement each-other giving a broad range of penetration and spatial resolution (refer to Figure 3.4). The rapid evaluation properties of the ultrasonic method justify its use in aluminium mills (see Aluminium Association 2008, p.71), however the method should be reserved for detecting material defects rather than determining residual stress states. The efficacy of the method in characterizing the true residual stress state is a cause for concern.

28

Chapter 4 Theoretical Analysis 4.1

Computation of Non-Uniform Stress.

Quench induced thermal stresses indicate that the blocks under examination will possess a non-uniform stress state. As noted previously, the integral method is the most applicable calculation variant where stress gradients are strong. Only the Integral method calculation procedure, including regularization ’smoothing’ calculations, will be defined in this section. References ASTM (2013) and Schajer (1988a,b) provide details regarding the calculation procedures for the alternative Uniform and Power Series variants.

The HDM employs a specially designed strain gage rosette (three elements) such as the example shown in Figure 4.1. The relaxation of stresses through hole-drilling is recorded in the form of strains which are averaged over the three elements of the gage. For non-uniform stress calculation, as shown in Figure 3.5b, calibration coefficients which were introduced in section 3.2.1 are used. The constants, denoted a¯jk and b¯jk , are unit stresses which have been pre-determined through FEA to account for loss of strain sensitivity at increasing hole depths. The subscript j in the case of the calibration constants indicates the relieved strains at j steps deep, due to the unit stresses at step k. i.e. the bottom of the hole, as k is the total number of depth steps. The published list of calibration constants is available in the test standard ASTM (2013). The values that these coefficients take are test specific, and must be adjusted to account for the size of strain gage used, and the final diameter of the drilled hole.

29


Figure 4.1: Three element clock-wise, Type A strain gage rosette which was specifically designed for the HDM. Source: (ASTM 2013).

4.1.1

Strain conditioning

The relationship for the surface strain relief at each depth step is given by: j 1+v X σx + σy j = a¯jk E k=1 2 k j 1 X σx − σy ¯ bjk cos 2θ + E k=1 2 k

+

j 1 X b¯jk (τxy )k sin 2θ (4.1) E k=1

Combination strain vectors p, q, and t are computed for each hole depth by: (3 + 1 ) 2 (3 − 1 ) qj = 2 (3 + 1 − 22 ) tj = 2 pj =

(4.2) (4.3) (4.4)

An estimate of the standard error in the strain measurements is performed. It is done to allow for optimal regularization (smoothing) of the stress solution using Tikhonov regularization which is a mathematical technique commonly used to stabilize inverse calculations (discussed in section 4.1.3 below). All practical strain data have an element of measurement noise. Tikhonov regularization reduces sensitivity to this noise, and has been proven to be effective in moderating the otherwise severe sensitivity of calculated HDM residual stresses to small errors in the measured strain data (Schajer 2007). 30

Residual Stress Determination The combination strain standard errors are computed by the following formulae1 , where n is the number of depth increments. The summation is carried out over the range: 1 ≤ j ≤ n − 3.

p2std =

n−3 X j=1

2 qstd

=

n−3 X j=1

t2std =

(pj − 3pj+1 + 3pj+2 − pj+3 )2 20(n − 3)

(4.5)

(qj − 3qj+1 + 3qj+2 − qj+3 )2 20(n − 3)

(4.6)

n−3 X j=1

4.1.2

(tj − 3tj+1 + 3tj+2 − tj+3 )2 20(n − 3)

(4.7)

Stress calculation

The adjusted calibration constants a ¯ and ¯b, and material properties E and v are used as inputs to the next calculation. With these, the residual stress combination vectors (P, Q and T) for each hole-depth-step can be calculated from the measured strains, by solving the following matrix equations:

E p 1+v ¯b Q = Eq

(4.9)

¯b T = Et

(4.10)

a ¯P =

(4.8)

whereby: ((σy )k + (σx )k ) 2 ((σy )k − (σx )k ) Qk = 2 Tk = (τxy )k Pk =

4.1.3

(4.11) (4.12) (4.13)

Regularization

As there are many depth steps, the effect of the FEA calibration constants diminish, which in turn leads to errors in the stress calculation. The compensating technique of Tikhonov regularization, can be applied at this point to equations 4.8, 4.9 and 4.10 (i.e. stress matrices) to control this error. This is 1

There is an error in the ASTM E837-13a standard, equation 21 (section 10.1.3). It has been corrected here in Eqn. 4.6.

31

Residual Stress Determination achieved by use of a tri-diagonal matrix c (below) with equations 4.15, 4.16 and 4.17. Regularization factors (αP , αQ and αT ) can subsequently be used to control the level of smoothing. Zero values of these factors are equivalent to no smoothing. 

c=

0

  −1       



0 2

−1

−1

2

−1

−1

2 0

        −1 

(4.14)

0

E a ¯T p 1+v (¯bT ¯b + αQ cT c) Q = E ¯bT q

(¯ aT a ¯ + αP cT c) P =

(¯bT ¯b + αT cT c) T = E ¯bT t

(4.15) (4.16) (4.17)

Solving equations 4.15, 4.16 and 4.17 gives the three unknown stresses P, Q, and T.

The strain values, unlike the stress values, have not been regularized, therefore do not correspond to the strains that could be derived from the new regularized stress values (P, Q and T). The misfit vectors (i.e. the difference between original vs. strains derived from new regularized stresses) can be calculated using the following:

1+v E 1 = q − E 1 = t − E

pmisf it = p − pmisf it pmisf it

a ¯P

(4.18)

¯b Q

(4.19)

¯b T

(4.20)

By calculating the mean square of each of the misfit vectors, the regularization can be controlled, hence allowing for optimum smoothing. p2rms =

n 1 X (pmisf it )2j n j=1

(4.21)

2 qrms =

n 1 X (qmisf it )2j n j=1

(4.22)

t2rms =

n 1 X (tmisf it )2j n j=1

(4.23)

32

Residual Stress Determination Smoothing can be moderated using a statistical approach known as the ’Morozov criterion’, which controls the amount of regularization (modelling error) to be within ±5% of the data error calculated statistically from the strain readings. The outcome is optimal regularization which minimizes distortion of the stress solution whilst removing noise (Schajer 2007). Schajer (ibid.) states that Morozov criterion is applicable in practice when the values of the standard strain errors are reliable. The standard strain errors are measurement properties (data errors) and ideally would equal zero. 2 The optimum level of smoothing occurs when p2rms , qrms and t2rms are within (±5%) 2 of the p2std , qstd and t2std as per Schajer (1988a).

Starting with zero values for the regularization factors, the misfit and rms vectors will most likely not equate. Their values can however be used in an iterative fashion to hone in towards meeting the ±5% criterion. New values can be calculated using the following equations: p2std (αP )old p2rms q2 = 2std (αQ )old qrms t2std = 2 (αT )old trms

(αP )new =

(4.24)

(αQ )new

(4.25)

(αT )new

(4.26)

When the ±5% criteria is met for each stress P, Q and T, the regularized stress values can be accepted, and used to determine the Cartesian stresses (i.e. the stresses in the directions of 1 , 2 and 3 . Cartesian stresses can be computed by the following: (σx )j = Pj − Qj

(4.27)

(σy )j = Pj + Qj

(4.28)

(τxy )j = Tj

(4.29)

Finally, the principal stresses and principal stress angles (βk ) can be calculated by: (σmax )k , (σmin )k = Pk ±

q

βk = arctan

33

Q2k + Tk2 −Tk −QK

(4.30)

!

(4.31)

Chapter 5 Stress Calculation Integral method calculations as outlined in E837-13a were performed on a software R by MathWorks. This software tool has also been used for tool using MATLAB

the combined plotting of Uniform, Power Series and Integral calculation values. Uniform and Power Series stress results are determined herein using a commercially available software model called H-Drill (version 2.22) (Schajer 2001). The H-Drill residual stress calculation tool was developed for use with the ASTM Test Method E837-99. This revision has since been superseded by E837-13a, which now caters for the calculation of non-uniform residual stress by the Integral method. The MATLAB script created caters for the latest version of integral calculation, along with the associated optimum Tikhonov regularization. The software program includes a graphical user interface (GUI) for strain input, and automatically performs residual stress calculation, then outputs all the associated graphs. The basic code script which can be further extended is transcribed into Appendix A.1. It can be accessed, along with the support files required at the following link1 : https://www.dropbox.com/sh/nn6ixsv4ar3x1mw/AACmqdoMaYl6yIdzzpiU45HGa?dl=0

1

MATLAB software is required (2011a or later). Save all files contained in the link folder, R (2) Point to the saved folder, (3) locally to your PC. To run the script: (1) Open MATLAB , Open the file named IntegralCalc and (4) Select ’Run’. Pre-populated cells for the measured strain values (zeros) will appear in the GUI. These can be manually changed in the script file if desired.

34

Chapter 6 Experimental Apparatus 6.1 6.1.1

Strain-gage and instrumentation Strain-gage

Various specialized (application-dependant) strain-gage configurations and sizes are available for use with the HDM procedure. After a trial-measurement, the type A ’clockwise’ EA-06-125RE-120 strain gage rosette from Vishay Measurements Group of nominal size

1 8

inch was selected for use with this investigation. This option has

the largest gage circle diameter (D =5.13mm) providing the highest attainable strain sensitivity, and permitted the HDM’s maximum recommended depth range (2mm) (see Group 2010, p.27). This investigation requires a high level of strain sensitivity to enable the accurate detection and measurement of residual strains from blocks which have been stress-relieved. The schematic geometry of the rosette is detailed below in Table 6.1, where D is the diameter of the gage circle (see section for a full listing of symbols). The Type A pattern is as shown in Figure 4.1.

Table 6.1: Geometrical dimensions and schematic of a ’Type A’ three-element Clockwise (CW) Hole-Drilling Rosette, (a) Rosette Layout, (b) Detail of Strain Gage (ASTM 2013) Rosette Dimensions (mm) Rosette Type A

D

GL

GW

R1

R2

Conceptual in. nominal

D 10.26

0.309D 3.18

0.309D 13.18

0.3455D 3.54

0.6545D 6.72

1 8

35


(a)

(b)

Figure 6.1: Measurements Group strain reading instrumentation (a) P-3500 Strain Indicator. (b) SB-10 Switch and Balance Unit. Source: Vishay (2000).

6.1.2

Strain recording

Connection of the gage elements is achieved through a Measurements Group P-3500 Strain Indicator (6.1a) in conjunction with SB-10 Switch and Balance Unit (6.1b). The three-wire quarter-bridge temperature-compensation circuitry is chosen for increased measurement accuracy (see Group 2015). The unit has a sensitivity (resolution) of ±1µ, and an accuracy of ±0.5% of reading ±3µ which meets with the requirements outlined in the test standard. The strain reading equipment is shown in Figure 6.1

6.2

Hole-drilling

A high precision Model RS-200 milling guide, developed by Vishay Measurements Group specifically for the measurement of residual stress is shown in Figure 6.2. The RS-200 assembly can be configured for inspection and alignment (6.2a), and for the drilling operation (6.3) as shown. The hole is created using a high-speed low-torque air-turbine drilling system. A geometrically well-defined hole is critical to the accuracy of stress evaluation for which the RS200 milling-guide provides precise alignment. Consideration towards the tendency of the technique to induce additional stresses during drilling must be given. The RS-200 is equipped with high-speed low-torque air-turbine drilling system to reduce this risk. Risk can be further reduced by use of a technique known as orbiting in conjunction with the appropriate cutting tool (Grant et al. 2006; Vishay 1992). The orbital technique, whereby the cutter is radially offset as shown in Figure 6.3a is 36


(a)

(b)

Figure 6.2: Micro Measurements RS-200 Hole-drilling apparatus(a) Optical device configuration for allignment and inspection. (b) Hole-drilling configutration used for machineing the percicely located flat bottom hole. Source: ASTM (2013) and Vishay (1992)

reccomended. It has the effect of reducing force and heat output during the drilling operation, hence providing more stable strain readings (Grant et al. 2006). A Tungsten Carbide Inverted Cone end-burr (shown alongside in Figure 6.3b) is also recommended. It provides a clean, flat-bottom and straight-sided hole with a sharp corner, and is less inclined to wear then the alternatives (ibid.).

(a)

(b)

Figure 6.3: Hole-drilling reccomendations for increased acuracy (a) Orbiting technique. Source: Vishay (1992) (b) Tungsten-Carbide inverted cone end-burr.

37

Chapter 7 Experimental Proceedure Principal of Operation The procedure for determining residual stresses by HDM consists of the following steps (Vishay 1992): • Bonding of the specialised strain gage rosette (three grids) at the location where residual stresses are to be determined. • Connection of each grid to the switch and balance unit and strain indicator. • Alignment of the RS-200 milling guide over the geometric centre of the rosette. • Introduction of the precision drilled hole through the centre of the rosette. • Recording of residual strains. • Computation of residual stresses.

7.1 7.1.1

Experimental details Material and heat-treatment.

Blocks B8, A5, A9, A7 and A10 were cut from AA7075 hot-rolled plate of thickness approx. 82mm (ST-direction, y), which was supplied from Mettis Aerospace UK. They were machined to length 120mm (rolling direction-L, x), and width 44mm (LT-direction), as shown in Figure 7.1. The pertinent mechanical properties of Young’s Modulus (E) = 70 GPa, and Poisson ratio (v) = 0.3, have been used for all calculations. These values have been selected to allow a direct comparison with results obtained by Robinson et al. (2017) who had previously characterized the stress in these blocks by XD and ND methods. Four years prior to testing, all four blocks had undergone SHT for two hours at 470◦ C, followed immediately by quenching vertically in ST-direction into a bath of 38


(a)

(b)

Figure 7.1: (a) Image of the AA7075 block measured by the Strain Gage Hole-Drilling (A10 shown). (b) Schematic showing the block’s Cartesian dimensions and location of strain gage rosette. Identifiers: L (Longitudinal), ST (Short Transverse) and LT (Long Transverse), match the hot-rolled plate directions from which the blocks were cut.

water with rigorous agitation. B8 received a CWQ (Tw = 20◦ C) to induce maximum residual stresses, while the remaining blocks were quenched at 60◦ C. The quench and soak time correspond to a ’W’ temper designation as per ASM (1991b). The 60◦ C water temperature is less severe, but is sufficient for obtaining the T6 temper strength requirements.

Each block was subjected to differing post-quench treatment as follows: Block B8 remand in the as-quenched condition and was naturally aged (NA) only. Block A5 did not undergo any post-quench stress relief and was artificially aged for seven hours (7h) at 105◦ C. Block A9 , was treated in the same way as B8. It was taken aside to monitor the natural ageing response in terms of its Vickers hardness (HV 20), and electrical conductivity (%IACS), with respect to time. Block A7 was held for four hours (4h), i.e. experienced PQD at room temperature (Tambient ≈ 20◦ C), which was followed by stress-relief through the application of cold compression in the ST-direction (see Figure 7.1a) inducing a controlled amount of plastic strain (≈ 1 − 2%). Block A10 experienced a shorter PQD of thirty minutes (0.5h) at room temperature, followed again by cold compression in the ST-direction.

39


7.1.2

Hole-drilling Strain-gage procedure.

The ASTM standardized Hole-Drilling Strain-Gage method (HDM) (ASTM 2013) was applied to determine the residual stresses in the centre of the L-ST face on each rectilinear block (one measurement per block). The location of measurement: the surface at the centre of the L-ST face, was chosen (a) to correspond with the region characterized previously by XD, and (b) as the axis of the drilled hole would be aligned with the ND measurements LT-direction scan (Robinson et al. 2017). A Measurements Group Type A, EA-06-125RE-120 strain gage rosette of nominal size

1 8

inch. was used in each case having gage factor of 2.12 ± 2% nominal.

The strain gage rosettes were installed in accordance with Measurements Group Instruction Bulletins B-129-8, and B-127-14 with M-Bond 200 adhesive (Vishay 2014, 2005). This involved some light surface preparation which was kept to a minimum, and is not believed to have had any influence on the accuracy of the first increment measurement (Grant et al. 2006; Prevéy 1988). In each case, gage elements 1, and 3 were orientated in the block’s L and ST directions respectively. These bi-axial directions, along with the 135◦ bisector, form the Cartesian stress directions which are discussed in more detail in section 8.2 In line with the recommendations given by (Grant et al. 2006), the drilling was performed using the orbital technique with a Tungsten Carbide Inverted Cone Ø1.8mm dental drill burr, replacement followed each measurement. After establishing the measurement’s ’Zero’ point, incremental volumes of material were removed, and the relaxed strains at the block’s surface were measured and recorded. The cutter was advanced for a total of 20 × 0.1mm depth increments, giving a final blind hole depth of 2mm. The Measurements Group RS200 milling guide and assembly was used for the procedure in accordance with the guidelines provided in the test standard, and by Measurements Group (Group 2010). Strain measurements were taken from a Measurements Group P-3500 Strain Indicator in conjunction with SB-10 Switch and Balance Unit, using Quarter-Bridge temperature-compensation circuitry. The unit had a sensitivity (resolution) of ±1µ, and an accuracy of ±0.5% of reading ±3µ which meets with the requirements outlined in the test standard. An image of the experimental set-up, is shown in Figure 7.2. The diameter of the drilled hole, shown in Figure 7.2c, was measured on completion of each strain evaluation using a microscope with graduated scale. A value of Ø3.847mm was measured in each case giving a total gage volume of 7.4mm3 for the 2mm measurement depth. The hole diameter represents ≈ 90% of the 40


(a)

(b)

(c)

Figure 7.2: (a) Experimental set-up consisting of RS200 milling guide assembly and strain indicator equipment. The RS-200 is attached to a purpose-built support jig as shown. (b) Size 81 inch nom. strain gage rosette installed to the aluminium block prior to drilling. (c) Rosette on completion of drilling.

41

Residual Stress Determination recommended maximum allowable diameter (the upper-level value allows increased strain sensitivity (Grant et al. 2006)). The drilled hole diameter to mean gage diameter ratio was calculated to be ≈ 0.38 in each case (maximum = 0.4)(see Group 2010, p.24).

7.2

X-ray and Neutron diffraction

Measurement data for XD and ND have been provided to allow for a wide-scale comparison of techniques. XD measurements were performed on a Phillips X’Pert x-ray diffractometer at the University of Limerick (UL) laboratories. It was calibrated using a sample of known residual stress, and used to measure the surface stress of block B8 in the centre of the L-ST face. The irradiated area was ’line-shaped’, measuring 2mm thick and 12mm long. Penetration was in the order of 100µm. ND measurements were performed using the Time-of-Flight Pulse Overlap neutron diffractometer (POLDI), which is located at the Paul Scherrer Institut in Villigen, Switzerland. A gage volume of 2mm3 was defined by the incident beam overlap and line scan length, which was orientated in line with the rectilinear blocks orthogonal directions. These directions are assumed to be principal stress directions as they are concurring with the direction of maximum heat flow from the blocks during quenching (Robinson et al. 2017). Diffraction and ageing results, along with each of the test-pieces, were provided by Dr. J.S. Robinson, School of Engineering, UL.

42

Chapter 8 Results 8.1

Relaxed strain.

Relaxed strain data 1 (L-direction), 2 , (L-ST) at 135 deg, and 3 (ST-direction) are plotted vs. hole depth from the surface are shown in Figure 8.1. The data points indicate the experimentally measured strains. The recorded strain data has been normalized against the Zero point strain reading, and are tabulated for each of the three blocks in Table B.1 which can be found in Appendix B.1. There were no corrections or smoothing applied to the strain data. Standard strain errors, calculated by the application of equations 4.5, 4.6 and 4.7 are tabulated in Table 8.1. The values provide a measure of the statistical accuracy of the strain readings taken for each strain-gage element.

Table 8.1: Calculated strain standard error values p2std. (1 )

2 qstd. (2 )

t2std. (3 )

Block A8 P Std. err

5.65×10−13

2.89×10−13

2.22×10−13

Block A5 P Std. err

7.03×10−13

6.65×10−13

2.39×10−12

Block A7 P Std. err

1.26×10−13

3.53×10−14

5.18×10−13

Block A10 P Std. err

9.26×10−14

2.5×10−14

2.01×10−13

43


(a)

(b)

(c)

(d)

Figure 8.1: Plots of hole-drilling relaxed strain vs. depth from surface. The data points are measured values for: (a) Block B8, (b) block A5, (c) block A7 and (d) block A10.

8.2

Cartesian stresses.

HDM results are determined for a series of in-plane, bi-axial (L and ST directions) stresses, along with their shear-stress component (τL−ST ). These three directions are denoted as Cartesian stress components. As noted previously, the HDM’s total gage volume is divided into 20 depth increments. The average stress within each incremental volume of removed material, for each stress component are plotted in Figures 8.2 (B8 ), 8.4 (A5 ), 8.6 (A7 ) and 8.8 (A10 ). Their values are tabulated in Table B.2 which can be found in Appendix B.2. The plots present stresses determined by all three calculation variants.

The

characteristic noise sensitivity of the fine increment Integral calculation have been regularized using Tikhonov regularization. Both non-regularized and regularized forms are plotted. 44


8.3

Principal Stresses

The principal stresses, and the directions in which they act have been derived from the Cartesian stresses using equations 4.30 and 4.31. They are plotted in in Figures 8.3 (B8 ), 8.5 (A5 ), 8.7 (A7 ) and 8.9 (A10 ). Principal stress values for each block are tabulated in Table B.4 which can be found in Appendix B.3. As before, the plots present stresses determined by all three calculation variants.

8.4

Diffraction results.

Stresses determined from ND measurements are presented in Figure 8.10. The measurement of concern begun at the block’s core, labelled (a) in Figure 3.8, and was moved out to the block’s face in the LT direction, labelled (b). Lattice strain recordings were taken at discrete points along this path, and the tri-axial stresses were calculated using established theory (Robinson et al. 2017). Each series of data points represents the average (determined) stress for each tri-axial stress component (L, ST, LT) in the LT direction. The XD result (block B8 only) which determines an average stress within the shallow gage volume (≈ 100µm), was taken at one location (centre of L-ST face), thus, is a single data point at the surface. The XD and HDM stress results have been combined on the ND plots (Figure 8.10) to allow for comparison of the determined magnitudes, and to give perspective towards the resolution of each measurement technique. It is worth noting that the stresses determined by diffraction are not directly comparable with those determined by the HDM, since HDM is capable of only determining bi-axial stress. To provide the closest representation, HDM’s average principal stresses vs. depth values have been plotted.

8.5

AA7075 natural ageing curve.

Since this investigation is interested in the effects of PQD on residual stresses, the precipitation response of the alloy after quenching is of interest. The hardening precipitation which occurs by natural ageing during the period between quenching and the application of the stress-relieving plastic deformation is considered to influence the final residual stress magnitudes (ibid.). The precipitation in AA7075 45

Residual Stress Determination can be monitored by measuring changes in hardness, and electrical conductivity with respect to time (ASM 1991b). The natural ageing response for block A9 was recorded at intervals over a 58-hour period as shown in Figure 8.11.

From 0.5 hour post-quench onwards, clear

inverse changes can be observed. The alloy’s hardness increases while its electrical conductivity reduces.

8.6

Post quench delay comparison.

The two blocks subjected to the process variable of PQD were A7 (4h) and A10 (0.5h). Box-plots have been used to communicate the effect of PQD on the range of residual stress present after artificial ageing. The plots compare (i) the stress range observed in both blocks, and (ii) the measurement techniques ability (HDM and ND) to detect changes in stress state due to PQD. The HDM determined stresses are shown in Figure 8.12, and ND determined stresses are shown in Figure 8.13. In the case of HDM, the average principal stress values for each depth increment are used as inputs, hence can be shown on a single plot (Figure 8.12). Stress values for each orthogonal (tri-axial) component are obtained through the ND measurement technique. These are therefore assessed individually, and are presented in three separate box-plots (Figures 8.13a, 8.13b and 8.13c). This exploratory analysis has been further developed in section 9.4, and Appendix ??.

46

47

show stresses in Cartesian directions (L, L-ST, ST) together (top-left), and individually (others).

Figure 8.2: Plot of calculated Cartesian stress results (by Uniform, Power Series and Integral methods) vs. depth from surface for block B8. Plots


48

stress direction (β) vs. depth from surface for block B8. The Integral method calculation values are plotted on top left (Max) and right (Min). The Power-series and Uniform calculations are plotted together on bottom left (Max) and right (Min). β is measured clockwise of strain gage 1 (L-direction).

Figure 8.3: Plot of the calculated Principal Stress results (σM AX , σM IN ), by Uniform, Power Series and Integral methods, along with Principal


49


Figure 8.4: Plot of calculated Cartesian stress results (by Uniform, Power Series and Integral methods) vs. depth from surface for block A5. Plots


50

stress direction (β) vs. depth from surface for block A5. The Integral method calculation values are plotted on top left (Max) and right (Min). The Power-series and Uniform calculations are plotted together on bottom left (Max) and right (Min). β is measured clockwise of strain gage 1 (L-direction).



51




52




53




54




55

to the surface in the LT direction. The x-ray diffraction determined stress magnitude for block B8, and the average determined principal stress magnitudes from the hole-drilling method are included on the graphs also.

Figure 8.10: Neutron diffraction determined residual stress magnitudes for blocks B8, A5, A7 and A10. Showing scan taken from block’s core



Figure 8.11: Plot of the natural ageing response for block A9. Source: Robinson et al. (2017)

Figure 8.12: Box plots comparing the range of residual stress in CC blocks having different post quench delays. Stress results have been determined by HDM (Average principal stress: calculated by the integral method-regularized).

56


(a)

(b)

(c)

Figure 8.13: Box plots comparing the range of residual stress in CC blocks having different post quench delays. Stress results have been determined by ND.

57

Chapter 9 Discussion 9.1

Strain recordings.

The success of the HDM results relies on the quality of strain readings obtained, poor strain readings emerge as a highly erroneous stress solution. The first step towards obtaining good quality readings is accurate and precise sample preparation (strain-gage installation etc.). The second step is scrupulous measurement practice during the hole drilling procedure itself. A metric upon which the combined success of these two steps can be assessed, is the quality of the strain readings obtained. The relaxed strain vs. depth graphs shown in Figure 8.1, along with the standard strain errors tabulated in 8.1, provide both a visual and a quantitative assessment of the variability (scatter) in the strain data. Blocks B8 and A5 in Figures 8.1a & 8.1b, which had no stress-relief, produced significantly larger strain readings (by a factor of 10) than the stress-relieved blocks A7 and A10 (Figures 8.1c & 8.1d). The strain readings for B5 and A5 showed distinct strong positive trends. Much weaker trends were observed for the low-stressed blocks (A7 and A10 ) and the extent of the strain reading equipment’s sensitivity (±1µ) is evident in plots 8.1c & 8.1d. Fluctuation due to instruments inability to measure small changes in relieved strain can be clearly seen. Despite this, the plots exhibit relatively smooth trends. There were no substantial irregularities or obvious outliers. The scatter in the data indicates some experimental error exists. The standard strain errors (Table 8.1) represent these measurement properties (data errors) which ideally would equal zero. All of the standard strain errors are predominantly in the order of 10−13 . The values are more than reasonable for the instrumentation used (see Schajer 1988a, 58

Residual Stress Determination p.443).

9.2

Residual stress in quenched and aged blocks.

The HDM determined large magnitude compressive surface stress in blocks B8 (Figures 8.2 & 8.3), and A5 (Figures 8.4 & 8.5), which did not receive cold compression. B8 (CW Q + N A) displayed large compressive surface stresses within the region of -200:-225 MPa (minimum principal stress-integral method). The XD surface stress result for this block was determined to be -185MPa. To compare the XD stress (which is averaged over the gage volume) with those determined by HDM, the average principal stress for the initial HDM increment was calculated. A value of -195MPa was obtained, hence both methods are in reasonable agreement, plus their values are within the range of uncertainty for each technique (±20M P a). The determined stress magnitudes are comparable to blocks having similar treatments in studies by Tanner et al. (2003) and Robinson et al. (2012b). Block A5 (60◦ C Quench + 7h) on average had lower residual stress magnitudes than B8 by ≈ 10%, indicating that the less severe quench, coupled with artificial ageing, results in lower stresses when compared to CWQ’d and naturally aged B8. Recent analysis on AA7075 blocks, reported that x-ray diffraction did not detect a reduction in residual stress due to the initial ageing treatment at 105◦ C (Robinson et al. 2017). The HDM results are interesting in this respect. Both sets of HDM stress graphs show a distinct reduction in compressive stress as the measurement progresses deeper from the surface. This depiction is consistent with the usually reported pattern of surface compression balanced by core tension in blocks that have been SHT quenched (Robinson et al. 2014; Robinson et al. 2012b, 2009). This distinct profile is confirmed without doubt by the ND measurement technique in Figure 8.10. The largest reported stresses are ≈ 60% of the material’s T73 temper yield strength (Ys ≥

338 MPa)(Table 2.1). It is only when the stresses exceed 80% of the

materials yield strength that plasticity around the drilled hole becomes a concern (Grant et al. 2006; ASTM 2013). The Cartesian shear-stress component approximated zero in all blocks, hence the clear majority of stress is aligned in the L and ST directions. This finding in part confirms the assumption mentioned previously regarding the tri-axial directions 59

Residual Stress Determination (3.8) being principal stress directions. The HDM was effective in determining stresses in quenched and aged blocks. Parallels drawn between it, and XD and ND measurement results in the plots (shown in Figure 8.10) reinforce this point. Stress results calculated by the integral method (regularized), in comparison to the power-series method, appear to provide data that are more consistent with previously published results (Robinson et al. 2012a; Robinson et al. 2012b). Power series calculations, while complimentary to the integral method, tended to over-estimate the surface stress solution in highly stressed blocks. The power-series model is reported to give its most accurate values at the initial two depth increments, however it is best used when stresses vary smoothly with depth (Schajer 1988b). In the highly-stressed blocks assessed here, integral method calculations received a more favourable weighting of judgement.

9.3

Residual stress due to cold compression.

In assessment of HDM results for blocks A7 (60◦ C Quench+4h P QD +7h) and A10 (60◦ C Quench + 0.5h P QD + AA), shown in Figures (8.6, 8.7) and (8.8, 8.9) both exhibited very low magnitude residual stresses due to the effects of cold compression. The reduction, if assessed by HDM results alone, would suggest that the blocks were essentially stress free, however it must be recalled that the calculations are based on a 2mm depth range. ND results confirm that the stress reduction due to cold compression is significant, but the blocks are not fully stress free. The effect of uni-axial compression in the ST-direction provoked a drastic reduction of residual stress not only in the direction of application, but also in the transverse direction. HDM measurements calculated surface stresses ≈ +14M P a, tensile in nature in each case. A10’s regularized integral result showed peak in stress magnitude at the block’s surface which is believed to be an anomaly pertaining to the technique’s high sensitivity which can often effect the initial measurement value (Grant et al. 2006). Block A7 had stresses that showed slight negative linear trend, whereby they dip back into compression with depth from the surface. HDM stress results ranged from −21 : +16M P a within the 2mm depth. The stresses in block A10 were flatter in trend than A7 which supported expectation as this block had a shorter PQD. A10 ’s Cartesian components are more tightly grouped over the interval than those 60

Residual Stress Determination of A7. This pattern is clearly observable over much greater distances in the ND results shown in Figure 8.10. A negative linear trend in the stress profile, albeit less pronounced, is clearly visible through HDM results in this block also, transitioning from tension into compression within the short 2mm distance from the surface. If the suspected anomalies from the initial two measurements were discarded, the stress range for the now ~1.8mm depth is −12 : +13M P a, which translates to a 30% reduction from stresses measured in A7. Despite observing significantly weaker trends in the strain recordings, the HDM can provide good results in low stressed blocks. Aluminium’s low modulus helped in this regard, and strains were detectable throughout the procedure. However, stresses at these magnitudes are believed to be at the lower limits of ’mechanical’ type residual stress measurement methods’ ability. For low stressed materials, the HDM and other mechanical stress-relaxation methods face challenges in terms of the resolution of sensory equipment, sensitivity of calculation procedures, and external effects such as stress imparting during material removal. Mechanical methods are simply too crude for obtaining accurate data at these levels. Nonetheless, the results obtained here have verified the technique as a viable option in aluminium alloys. Reassuringly, HDM results1 for each of the four blocks are in harmony with both XD & ND values as shown in Figure 8.10. Diffraction techniques are believed to be more effective for evaluation of stress relieved materials. There are fewer external influences effecting the final solution, thus are considered more accurate. Further, ND has the capacity to characterize deeper into low stressed material, which allows for complete rationalization.

9.4

Residual stress due to post quench delay.

The ageing curve in Figure 8.11 gives a strong indication that the alloy’s strengthening behaviour at room temperature takes noticeable effect after 30 minutes. Per the data presented, during the period between 0.5 hours and 4 hours the alloy has hardened by ≈ 11%, and its electrical conductivity has reduced by ≈ 3% due to precipitation. As a consequence of these ageing effects, it is believed that increasing PQD has the effect of increasing residual stress state (Robinson et al. 2017). Based on the ND results presented, this belief has been shown to be statistically factual with a 99% confidence level. To assess the process variable of PQD initially, an exploratory analysis was 61

Residual Stress Determination conducted using box-plots (see Figures 8.12 and 8.13). The plots show the range of residual stresses observed by both HDM and ND measurement methods. HDM data for blocks A7 and A10 when presented in this way do not suggest that PQD increases residual stress, in-fact they suggest the opposite. The total gage volume of 7.4mm3 is too small, and the 2mm depth range is insufficient for assessing PQD effects using HDM. Moreover, the sensitivity of the method suggests that the HDM has inadequate repeatability, hence results that have been obtained are unsuitable for statistical analysis. What can be concluded from the data however, is that bi-axial stresses due to longer PQD have a more dispersed distribution. Box-plots based on ND stress data are shown in Figure 8.13. They compare blocks A7 vs.

A10 by assessing each tri-axial stress component in the LT-direction.

There is an observable difference in all three plots. Residual stresses are higher in block A7 having 4 hour PQD, than A10 having 0.5 hour PQD. The symmetric shape of the plots indicate that the stresses are normally distributed over this measurement distance (20mm), and that the spread looks reasonably similar in each case, indicating that the variability of each group is approximately the same. Two outliers are observed in block A7 (Lxx) 18Mpa, and (LTyy) 10MPa. Both these values relate to the initial measurement taken at the block’s core, and are not considered to be significant for this reason. A formal hypothesis test has determined that there is sufficient evidence in the ND data (tri-axial components for the LT-line scan) to infer that longer durations of PQD result in higher stresses. This finding applies to the entire population of AA7075 blocks under these conditions, not just to the data presented in this report. The level of significance chosen was 99%, and the analysis is described more comprehensibly in Appendix ??. Neutron diffraction is an appropriate measurement practice for evaluation of the effects of PQD with a resolution capable of detecting small stress magnitude differences. Of the comparisons made, the stress component with the largest stress magnitude variation is LTyy (Figure 8.13b).

This is the stress component that is acting

perpendicular to the centre of the L-ST face (along the axis of the HDM drilled hole). This face has the largest surface area, and the shortest linear distance to the block’s core. It is expected that the ageing response is most appreciable in this direction due to larger heat-loss during quenching, and thus results in a larger difference in stress magnitude after PQD. HDM and XD methods have not contributed to the findings on PQD. The advantage of ND in terms of penetration is obvious from Figure 8.10.

62

Chapter 10 Conclusions Combining all of the discussed results the following conclusions have been drawn: • The hole-drilling strain-gage method was in-conclusive in evaluation of residual stresses as a function of PQD in rectilinear aluminium alloy 7075 blocks. • The neutron diffraction method provided sufficient evidence to infer that increasing post quench delay duration has the effect of increasing the residual stress state. • When monitored by neutron diffraction, the effect of post quench delay is greatest along the shortest path between the blocks core and the surface. • When stress as a function of post quench delay was assessed by the hole-drilling, the method identified a more compact stress distribution for blocks having shorter delay. • Collectively,

x-ray

diffraction,

neutron

diffraction

and

hole-drilling

measurement techniques provide complimentary data with harmonious trends acting over a broad range of penetration and spatial resolution. • The as-quenched level of near-surface principal residual stress have been determined by hole-drilling to be compressive and in the order of 200:225 MPa. • When monitored by the hole-drilling method, quenching at 60◦ C and applying an initial ageing treatment (7h

105◦ C), results in a 10% residual stress

reduction when compared with 20◦ C quench and natural age. • In as-quenched highly-stressed rectilinear blocks, the hole drilling method confirmed that L and ST orthogonal directions are principal stress directions. 63

Residual Stress Determination • Application of uni-axial stress relieving cold compression (1-2%) greatly reduces residual stresses in both the direction of application and the transverse directions. • Following application of cold compression, near-surface stresses determined by hole-drilling were generally in the range of -21 : +16 MPa. • The cold compression of blocks (ST-direction) leads to a tensile surface condition on the un-constrained face (L-ST) which is in the order of +10MPa. Stresses transposed into compression within the hole drilling method’s 2mm measurement range. • Poor strain sensitivity becomes an issue when attempting to profile residual stresses in stress-relieved aluminium blocks.

64

Appendix A Computer model A.1

MATLAB programming code

clc clear all close all format long g x3=linspace(0,2,21); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 1. CREATE Pop-up box to enter name of project etc. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% prompt = {’Test name:’,’Material & treatment’,... ’Date of test (dd/mm/yy):’,’Young”s Modulus (MPa)’,’Poisson’’s Ratio’}; dlg_title = ’Details’; num_lines = [1 30;1 50;1 30;1 30;1 30]; defaultans = {’Block A5’,... ’AA7075, SHT, As Quenched @ 60degC + 7h @105degC’,... ’Feb. 3, 2017’,’70000’,’0.30’}; answer = inputdlg(prompt,dlg_title,num_lines,defaultans); Name=answer{1}; Treatment=answer{2}; date_of_test=answer{3}; E=answer{4};% Youngs Modulus (N/mm^2) E = str2double(E); v=answer{5} ;%poisson ratio v=str2double(v);

65


clear defaultatans clear num_lines clear answer clear dlg_title clear prompt %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 3. ENTER STRAIN data collected during experiment %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Strain is first in-putted by the user as it is % read from the strain reading instrument, % then converted into micro strain by the code below)

%_____________Zero Strain input___________________________________ prompt = {’Enter ’’Zero point’’ micro-strains (use spacebar between)’}; dlg_title = ’Zero Point’; num_lines = [1 40]; defaultans = {’9 6 10’}; answer2 = inputdlg(prompt,dlg_title,num_lines,defaultans); initial_strain_read=str2num(answer2{1});

clear ’prompt’ clear ’dlg_title’ clear ’num_lines’ clear ’defaultans’ clear answer2 %__________Strain Readings_________________________________________ %incremental_strain_read is measured strains for %depth increments (in order %10 (gauge 1,sigmaX),9 (gauge 2,Tau),8 (gauge 3,sigmaY) %Strain inputs are split into two user input % blocks then combined as one matrix for i = 1:10 line_name = sprintf(’Step %d’, i); prompt{i} = line_name; end dlg_title = ’Strain Readings’;

66


num_lines = [1 50]; defaultans = {’32 27 33’,... ’58 53 59’,... ’85 84 92’,... ’111 120 132’,... ’137 158 174’,... ’161 188 215’,... ’184 211 261’,... ’208 237 301’,... ’232 261 344’,... ’255 285 379’}; answer3 = inputdlg(prompt,dlg_title,num_lines,defaultans); read1=str2num(answer3{1}); read2=str2num(answer3{2}); read3=str2num(answer3{3}); read4=str2num(answer3{4}); read5=str2num(answer3{5}); read6=str2num(answer3{6}); read7=str2num(answer3{7}); read8=str2num(answer3{8}); read9=str2num(answer3{9}); read10=str2num(answer3{10}); clear prompt clear dlg_title clear num_lines clear defaultans clear line_name clear i for i = 1:10 line_name = sprintf(’Step %d’, i+10); prompt{i} = line_name; end dlg_title = ’Strain Readings ctd...’; num_lines = [1 50]; defaultans = {’274 308 407’,... ’290 333 436’,... ’301 355 471’,...

67


’308 378 498’,... ’315 395 527’,... ’322 408 554’,... ’326 417 580’,... ’332 425 603’,... ’335 433 621’,... ’339 443 636’}; answer4 = inputdlg(prompt,dlg_title,num_lines,defaultans); %incremental_strain_read=str2num(answer4{1}); clear prompt clear num_lines clear line_name clear i clear dlg_title read11=str2num(answer4{1}); read12=str2num(answer4{2}); read13=str2num(answer4{3}); read14=str2num(answer4{4}); read15=str2num(answer4{5}); read16=str2num(answer4{6}); read17=str2num(answer4{7}); read18=str2num(answer4{8}); read19=str2num(answer4{9}); read20=str2num(answer4{10}); incremental_strain_read=[read1;read2;read3;read4;read5;read6;read7; read8;read9;read10;read11;read12;read13;read14;read15;read16; read17;read18;read19;read20]; clear(’answer3’,’answer4’,’defaultans’,’read1’,’read2’,’read3’,... ’read4’,’read5’,’read6’,’read7’,’read8’,’read9’,’read10’,... ’read11’,’read12’,’read13’,’read14’,... ’read15’,’read16’,’read17’,’read18’,’read19’,’read20’) %zero the data (using initial strain read) and change %from micro-strain to strain Strains = bsxfun(@minus,initial_strain_read,incremental_strain_read); Strains= Strains*-10^-6; %clear initial_strain_read; %clear incremental_strain_read;

68


%Code to insert 0,0,0, into top row of matrix top_row=[0,0,0] ; row_no=1; %%matrix row where want to insert Strains(1:row_no-1,:) = Strains(1:row_no-1,:) ; tp =Strains(row_no:end,:); Strains(row_no,:)=top_row; Strains(row_no+1:end+1,:) =tp; clear top_row; clear tp; clear row_no; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %(See E837-13A 10.1.1) plot of the strains with depth %increments to confirm that data follow generally smooth trends. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x=linspace(0,2,21); %need to include zero point in x so 21 points e1=Strains(:,1); e2=Strains(:,2); e3=Strains(:,3); % (DATA SMOOTH TREND CHECK)plot: plot1=’RELAXED STRAIN vs. HOLE DEPTH’; plot(x,e1,’b--d’,x,e2,’--gv’,x,e3,’r--s’); grid on %axis auto %axis manual axis tight %axis fill title({plot1,Name,date_of_test,Treatment}); %title({plot1,Name,Treatment,date_of_test}); xlabel(’Depth (mm)’); ylabel(’Strain’); legend(’\epsilon_1 (X) (L)’,’\epsilon_2 (\tau) (L-ST)’,... ’\epsilon_3 (Y) (ST)’,’Location’,’best’); Strains(1,:)=[]; e1=Strains(:,1); e2=Strains(:,2); e3=Strains(:,3);

69


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %(See E837-13A 10.1.2) COMPUTE the combination strain % vectors for each set of measured strains e1,e2,e3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p=(e3+e1)/2; q=(e3-e1)/2; t=(e3+e1-2*e2)/2; clear x clear e1; clear e2; clear e3; clear plot1; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Calculate combination strain vectors (p,q,t) %for Un-reguralised stress calc. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p_of_E=p*(E/(1+v)); %i.e. (E/1+v)(p) q_of_E=q*E; % i.e. (E)(q) t_of_E=t*E; %i.e. (E)(t) %correct %now need to multiply these by calibration coefficients a_bar or b_bar) %Coefficients depend on 1. Strain gauge rosette type (A,B,C), 2. Strain %gauge rosette size (1/16, 1/32 etc) and 3. Hole diameter

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Generate the correct values for calibration matrices %(a_bar and b_bar) %based on hole and rosette size/type etc. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% load(’coefficients.mat’); clc gauge=menu(’Select Rosette Type’,’Type A (1/32)’, ’Type A (1/16)’,... ’Type A (1/8)’); if gauge ==1 x2=[0.00825 0.0375 0.0625 0.0875 0.1125 0.1375 0.1625 0.1875

70


0.2125 0.2375 0.2625 0.2875 0.3125 0.3375 0.3625 0.3875 0.4125 0.4375 0.4625 0.49175]; elseif gauge ==2 x2=[0.0165 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 0.475 0.525 0.575 0.625 0.675 0.725 0.775 0.825 0.875 0.925 0.9835]; elseif gauge==3 x2=[0.033 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.967]; end % The actual diameter of hole is used to recalculate %coefficients based on a nominal 2mm diameter %______Ask user for actual drilled hole diameter________________ prompt = {’Measured Hole Diameter(mm):’}; dlg_title = ’Drilled Hole’; num_lines = [1 30]; defaultans = {’3.846990291’}; answer5 = inputdlg(prompt,dlg_title,num_lines,defaultans); hole_dia=answer5{1}; % takes hole diameter from user input hole_dia=str2double(hole_dia); %Converts that input to a readable form

switch gauge %see ASTM E-837-13a section 10.2.1 case{1} hole_adj_factor=(hole_dia/(2*0.5))^2; %See ASTM E-837-13a 10.2.3 a_bar=(a_bar*hole_adj_factor); b_bar=(b_bar*hole_adj_factor); case{2} hole_adj_factor=(hole_dia/(2*1))^2;

% See ASTM E-837-13a 10.2.3

a_bar=(a_bar*hole_adj_factor); b_bar=(b_bar*hole_adj_factor); case{3} hole_adj_factor=(hole_dia/(2*2))^2; % See ASTM E-837-13a 10.2.3 a_bar=(a_bar*hole_adj_factor); b_bar=(b_bar*hole_adj_factor); otherwise %if menu is closed without any selection it returns {0}, error(’No selection made’);

71


end

clear defaultatans clear num_lines clear answer5 clear dlg_title clear prompt clear gauge; clear hole_adj_factor; clear hole_dia; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculate Combination vectors P,Q and T (non regularisation) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P_non_reg = pinv(a_bar)*p_of_E; %pinv(a_bar) = a_bar inverse Q_non_reg = pinv(b_bar)*q_of_E; %pinv(b_bar) = b_bar inverse T_non_reg = pinv(b_bar)*t_of_E;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Calculate stresses sigma x (Axial), sigma y (Hoop), and Tau (Shear) %(non-regularisation) results are in MPa or N/(mm)^2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma_x_non_reg=P_non_reg-Q_non_reg; sigma_y_non_reg=P_non_reg+Q_non_reg; Tau_non_reg=T_non_reg; %only writing again for completeness sake max_principal_non_reg = P_non_reg+(((Q_non_reg.^2)+(T_non_reg.^2)).^0.5); min_principal_non_reg = P_non_reg-(((Q_non_reg.^2)+(T_non_reg.^2)).^0.5); Beta_non_reg=0.5*atan2(-T_non_reg,-Q_non_reg); % in radians Beta_non_reg=radtodeg(Beta_non_reg); %converts to degrees clear P_non_reg; clear Q_non_reg; clear T_non_reg; %Next stage is to calculate reglarised stresses-need std errors

%============================================================== %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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%(See E837-13A 10.1.3) ESTIMATE the standard errors in the combination %strains (p,q,t). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %length is the number of strain readings = 20(not including zero depth) %(20-degrees of freedom) is same as: length(p)-3 p_std_sq=0; for j = 1:length(p)-3; p_loop=((p(j)-(3*p(j+1))+3*p(j+2)-p(j+3))^2/(20*(length(p)-3))); p_std_sq=p_std_sq+p_loop; end clear p_loop; clear j; q_std_sq=0; for j = 1:length(q)-3 q_loop=((q(j)-(3*q(j+1))+3*q(j+2)-q(j+3))^2/(20*(length(q)-3))); q_std_sq=q_std_sq+q_loop; end clear q_loop; clear j; t_std_sq=0; for j = 1:length(t)-3 t_loop=((t(j)-(3*t(j+1))+3*t(j+2)-t(j+3))^2/(20*(length(t)-3))); t_std_sq=t_std_sq+t_loop; end clear t_loop; clear j; %correct %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Now Calculate combination strain vectors (for regularised) %p_reg,q_reg,t_reg %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p_reg=a_bar’*p_of_E; % i.e. (E/1+v)(p)(a_bar’) q_reg=b_bar’*q_of_E; % i.e. (E)(q)(b_bar’) t_reg=b_bar’*t_of_E; % i.e. (E)(t)(b_bar’) %correct

73


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Regularisation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %initially set regularisation factors to 10^-6 for p,q and t alpha_p=10^-6; alpha_q=10^-6; alpha_t=10^-6; % Need to run iterations of the calculation of vectors P_reg, % Q_reg and T_reg, % adjusting the regularisation factors (alphas) each time %Calculate the percentage difference %between standard errors (p_std, q_std,and t_std)... %vs( p_misfit, q_misfit) and t_misfit) based on this value %Initial ’fictional’ alpha values are entered %which are outside the range p_diff= 100000; while p_diff < -5 || p_diff > 5; P_reg = (inv((a_bar’*a_bar)+(alpha_p*(c’*c))))*p_reg; %solving eqn.30 for P %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %CALCULATE the Mean Squares of the misfit Stress vectors: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p_misfit=((1+v)/E)*a_bar*P_reg; %i.e. eqn(33) p_rms_sq=mean((p_misfit-p).^2); %i.e. eqn(36) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Calculate % difference between mean squares and %standard errors for stress vectors %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p_diff=((p_rms_sq-p_std_sq)/p_std_sq)*100; %overwrites p_diff to correct value alpha_p=(p_std_sq/p_rms_sq)*alpha_p; %updates apha_p to correct value based on calc end clear p_misfit; clear p_diff;

74


clear p_rms_sq; %clear p_std_sq; clear p_reg; clear p; clear p_of_E; q_diff=100000; while q_diff < -5 || q_diff > 5; Q_reg = (inv((b_bar’*b_bar)+(alpha_q*(c’*c))))*q_reg; %for Q q_misfit=(1/E)*b_bar*Q_reg;

%i.e. eqn (34)

q_rms_sq=mean((q_misfit-q).^2); %i.e. eqn (37) q_diff=((q_rms_sq-q_std_sq)/q_std_sq)*100; alpha_q=(q_std_sq/q_rms_sq)*alpha_q; end clear q_misfit; clear q_diff; clear q_rms_sq; %clear q_std_sq; clear q_reg; clear q; clear q_of_E; t_diff=100000; while t_diff < -5 || t_diff > 5; T_reg = (inv((b_bar’*b_bar)+(alpha_t*(c’*c))))*t_reg; %for T t_misfit=(1/E)*b_bar*T_reg;

%i.e. eqn (35)

t_rms_sq=mean((t_misfit-t).^2); %i.e. eqn. (38) t_diff=((t_rms_sq-t_std_sq)/t_std_sq)*100; alpha_t=(t_std_sq/t_rms_sq)*alpha_t; end clear t_misfit; clear t_diff; clear t_rms_sq; %clear t_std_sq; clear t_reg; clear t;

75


clear t_of_E; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Calculate stresses sigma x (Axial), sigma y (Hoop), and Tau (Shear) %(regularisation) results are in MPa or N/(mm)^2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sigma_x_reg=P_reg-Q_reg; sigma_y_reg=P_reg+Q_reg; Tau_reg=T_reg; %only writing again for completeness sake max_principal_reg =... P_reg+(((Q_reg.^2)+(T_reg.^2)).^0.5); min_principal_reg =... P_reg-(((Q_reg.^2)+(T_reg.^2)).^0.5); Beta_reg=... 0.5*atan2(-T_reg,-Q_reg); % in radians Beta_reg=radtodeg(Beta_reg); %converts to degrees clear (’P_reg’,’Q_reg’,’T_reg’,... ’a_bar’,’b_bar’,’c’,’alpha_p’,’alpha_q’,’alpha_t’,’v’,’E’);

%____________________________________________________ % Cartesian Stress REGULARISED VS NON-REGULARISED STRESS % (INTEGRAL METHOD) %_____________________________________________________ y1=sigma_x_reg; y2=sigma_x_non_reg; y3=sigma_y_reg; y4=sigma_y_non_reg; y5=Tau_reg; y6=Tau_non_reg; figure(’units’,’normalized’,’outer-position’,[0 0 1 1]) %sets it to open full screen hold on plot2=’CARTESIAN STRESS’; plot(x2,y1,’b-s’,

x2,y2,’b:x’,...

x2,y3,’r-s’,

x2,y4,’r:x’,...

x2,y5,’g-v’,

x2,y6,’g:*’,...

’LineWidth’,2,

’MarkerSize’,5);

76


axis tight grid on %title({plot2,strcat(Name,date_of_test),Treatment}); title({plot2,Treatment,date_of_test}); %title({plot2,Name,Treatment,date_of_test}); xlabel(’Depth from Surface (mm)’); ylabel(’Residual Stress (MPa)’); legend(’L-direction (Regularised)’,’L-direction (Non-Reg)’,... ’ST-direction (Regularised)’,’ST-direction (Non-Reg)’,... ’\tau_{L-ST} (Regularised)’,’\tau_{L-ST} (Non-Reg)’,... ’Location’,’best’); hold off clear(’y1’,’y2’,’y3’,’y4’,’y5’,’y6’,’plot2’) %_______________________________________________________________ %

MAX PRINCIPAL STRESS & PRINCIPAL STRESS DIRECTION PLOT

%_______________________________________________________________ plot7=’MAXIMUM PRINCIPAL STRESS & PRINCIPAL STRESS DIRECTION’; y1 = max_principal_reg; y2 = max_principal_non_reg; y3 = Beta_reg; y4 = Beta_non_reg; figure(’units’,’normalized’,’outerposition’,[0 0 1 1]) hold on plot(x2,y2,’k:s’,x2,y1,’k-x’,’LineWidth’,2,’MarkerSize’,8), addaxis(x2,y4,’k:^’,’LineWidth’,2,’MarkerSize’,8); addaxisplot(x2,y3,2,’k-.v’,’linewidth’,2,’MarkerSize’,8); grid on title({plot7}) xlabel(’Depth from Surface (mm)’) addaxislabel(1,’Residual Stress (MPa)’); addaxislabel(2,’Principal Stress Direction, \beta (degrees)’); legend(’Maximum Principal Stress (non-regularised)’,... ’Maximum Principal Stress (regularised)’,... ’Principal Stress Direction (non-regularised)’,... ’Principal Stress Direction (regularised)’,... ’Location’,’south’); hold off

77


%{ set(gcf,’PaperUnits’,’inches’,’PaperPosition’,[0 0 12 6]) %sets aspect ratio print(gcf,’-dpng’,’-r150’,’./Max_principal_stress.png’); %} clear(’y1’,’y2’,’y3’,’y4’,’plot7’) %_________________________________________________________________ %

MAX AND MINIMUM PRINCIPAL STRESS PLOT

%_________________________________________________________________ plot8=’MAXIMUM & MINIMUM PRINCIPAL STRESS’; y1 = max_principal_reg; y2 = max_principal_non_reg; y3 = min_principal_reg; y4 = min_principal_non_reg; figure(’units’,’normalized’,’outerposition’,[0 0 1 1]) hold on plot(x2,y1,’k-x’,x2,y2,’k:s’,x2,y3,’k-v’,x2,y4,’k:^’,... ’LineWidth’,2,’MarkerSize’,8), grid on title({plot8}) xlabel(’Depth from Surface (mm)’) addaxislabel(1,’Residual Stress (MPa)’); legend(’Maximum Principal Stress (regularised)’,... ’Maximum Principal Stress (non-regularised)’,... ’Minimum Principal Stress(regularised)’,... ’Minimum Principal Stress (non-regularised)’,... ’Location’,’south’); hold off %{ set(gcf,’PaperUnits’,’inches’,’PaperPosition’,[0 0 12 6]) print(gcf,’-dpng’,’-r150’,’./MAX&MIN_principal_stress.png’); %} clear(’y1’,’y2’,’y3’,’y4’,’plot8’) %removed von mises plot 27-01-17 %_________________________________________________________

78


%

MIN PRINCIPAL STRESS & PRINCIPAL STRESS DIRECTION PLOT

%_________________________________________________________ plot9=’MINIMUM PRINCIPAL STRESS & PRINCIPAL STRESS DIRECTION’; y1 = min_principal_reg; y2 = min_principal_non_reg; y3 = Beta_reg; y4 = Beta_non_reg; figure(’units’,’normalized’,’outerposition’,[0 0 1 1]) hold on plot(x2,y2,’k:s’,x2,y1,’k-x’,’LineWidth’,2,’MarkerSize’,8), addaxis(x2,y4,’k:^’,’LineWidth’,2,’MarkerSize’,8); addaxisplot(x2,y3,2,’k-.v’,’linewidth’,2,’MarkerSize’,8); grid on title({plot9}) xlabel(’Depth from Surface (mm)’) addaxislabel(1,’Residual Stress (MPa)’); addaxislabel(2,’Principal Stress Direction, \beta (degrees)’); legend(’Minimum Principal Stress (non-regularised)’,... ’Minimum Principal Stress (regularised)’,... ’Principal Stress Direction (non-regularised)’,... ’Principal Stress Direction (regularised)’,... ’Location’,’south’); hold off %{ set(gcf,’PaperUnits’,’inches’,’PaperPosition’,[0 0 12 6]) print(gcf,’-dpng’,’-r150’,’./Min_principal_stress.png’); %} clear(’y1’,’y2’,’y3’,’y4’,’plot9’)

79

Appendix B Tabulated strain and stress values B.1

Tabulation of relaxed strain readings.

B.2

Tabulation of Cartesian stresses.

B.3

Tabulation of Principal stresses.

80


Table B.1: Normalized strain readings (µ), for each of blocks B8 and A5 Block B8

Block A5

Hole Depth (mm)

µ1 (L)

µ2 (L-ST)

µ3 (ST)

µ1 (L)

µ2 (L-ST)

µ3 (ST)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

20 50 85 127 166 198 232 263 294 322 348 374 400 422 440 454 467 477 487 495

24 62 103 152 199 241 285 330 379 427 464 507 542 577 605 629 652 673 698 718

23 54 91 130 174 216 261 302 340 378 409 440 470 498 525 550 572 589 605 617

23 49 76 102 128 152 175 199 223 246 265 281 292 299 306 313 317 323 326 330

21 47 78 114 152 182 205 231 255 279 302 327 349 372 389 402 411 419 427 437

23 49 82 122 164 205 251 291 334 369 397 426 461 488 517 544 570 593 611 626

81


Table B.1: Normalized strain readings (µ), for each of blocks A7 and A10 Block A7

Block A10

Hole Depth (mm)

µ1 (L)

µ2 (L-ST)

µ3 (ST)

µ1 (L)

µ2 (L-ST)

µ3 (ST)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

-2 -3 -5 -7 -10 -12 -14 -17 -19 -22 -25 -26 -28 -28 -28 -28 -28 -29 -30 -31

-1 0 1 0 -2 0 1 1 1 0 0 1 0 2 2 4 4 6 7 8

-1 -2 -2 -1 -2 -2 0 1 2 3 3 5 6 7 8 8 10 12 13 14

-4 -6 -7 -8 -10 -11 -12 -14 -16 -19 -20 -22 -23 -24 -26 -27 -28 -29 -30 -30

-3 -3 -3 -4 -5 -4 -4 -4 -4 -4 -4 -4 -5 -7 -8 -9 -10 -11 -12 -11

-2 -2 -2 -2 -2 -2 -1 -1 -1 -1 0 0 1 2 2 3 2 2 3 4

82


Table B.2: Tabulation of the calculated Cartesian stress results (by Uniform, Power Series and Integral methods) vs. depth from surface. Data for blocks B8 and A5.

Block B8 Hole Depth (mm)

σX (MPa) (L)

Block A5

σY (MPa) (ST)

στ (MPa) (L-ST)

σX (MPa) (L)

σY (MPa) (ST)

στ (MPa) (L-ST)

-212

32

-144

-200

-8

-233 -229 -225 -221 -217 -213 -209 -205 -201 -197 -193 -189 -185 -181 -177 -173 -169 -165 -161 -157 -153

23 25 26 27 29 30 31 33 34 35 37 38 39 41 52 43 45 46 47 49 50

-190 -182 -174 -166 -159 -151 -143 -135 -127 -120 -112 -104 -96 -88 -81 -73 -65 -57 -49 -42 -34

-214 -211 -209 -206 -203 -200 -197 -195 -192 -189 -186 -183 -181 -178 -175 -172 -169 -166 -164 -161 -158

-4 -5 -5 -6 -7 -7 -8 -8 -9 -9 -10 -10 -11 -12 -12 -13 -13 -14 -14 -15 -16

-187 -189 -193 -198 -201 -202 -202 -196 -183 -164 -151 -153 -168 -185 -201 -211 -213 -206 -195 -181

-11 2 12 14 3 -19 -38 -43 -35 -15 6 20 23 14 -6 -27 -39 -39 -29 -14

1. Uniform Method na

-188

2. Power Series Method 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

-223 -217 -211 -205 -199 -193 -187 -181 -175 -169 -163 -157 -151 -145 -139 -133 -127 -121 -115 -108 -102

3. Integral Method (Tikhinov Regularization) 0.033 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55 1.65 1.75 1.85 1.967

-188 -205 -215 -211 -192 -170 -155 -148 -148 -152 -160 -167 -170 -165 -154 -140 -127 -116 -106 -97

-202 -209 -214 -215 -213 -208 -201 -191 -182 -177 -175 -181 -190 -199 -203 -199 -185 -165 -142 -118

26 27 25 22 18 18 25 38 50 56 53 45 35 26 20 23 34 51 72 93

83

-197 -176 -156 -140 -128 -122 -125 -131 -133 -125 -108 -90 -77 -72 -76 -84 -93 -100 -107 -113


Table B.2: Tabulation of the calculated Cartesian stress results (by Uniform, Power Series and Integral methods) vs. depth from surface. Data for blocks A7 and A10.

Block A7 Hole Depth (mm)

σX (MPa) (L)

Block A10

σY (MPa) (ST)

στ (MPa) (L-ST)

σX (MPa) (L)

σY (MPa) (ST)

στ (MPa) (L-ST)

0

4

9

2

1

6 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -11 -12 -13 -14 -15 -16

5 5 5 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0 0

11 11 10 10 9 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2

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

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

15 5 -1 -2 -2 -2 -1 1 2 1 -2 -4 -4 -2 1 3 3 0 -6 -12

2 1 1 1 1 2 3 4 4 3 1 -2 -5 -6 -7 -6 -5 -3 -2 0


9


15 14 13 12 11 10 9 8 7 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6


12 10 10 10 8 8 10 11 11 10 8 5 0 -6 -10 -10 -6 -3 -1 0

9 5 2 0 -1 -2 -4 -4 -3 -4 -5 -6 -7 -7 -8 -11 -15 -19 -20 -20

6 5 5 4 4 3 3 2 2 2 2 2 2 2 2 2 3 3 3 3

84

26 13 4 1 2 3 5 8 10 10 7 5 4 6 8 9 8 5 1 -4


Table B.3: Calculated Principal stresses (σM AX , σM IN ), and Principal stress direction (β) for Uniform, Power-Series and Integral calculation methods. β is measured clockwise of strain gage 1 (L-direction). Data for blocks B8 and A5

Block B8 Hole Depth (mm)

Block A5

σM AX

σM IN

τM AX

β

σM AX

σM IN

τM AX

β

(MPa)

(MPa)

(MPa)

(deg.)

(MPa)

(MPa)

(MPa)

(deg.)

-234

34

-39

-143

-201

29

8

24 25 27 29 30 32 33 35 37 38 40 41 43 45 46 48 50 51 53 55 56

-39 -38 -37 -36 -36 -35 -35 -34 -34 -34 -33 -33 -33 -33 -33 -32 -32 -32 -32 -32 -32

-189 -181 -173 -166 -158 -150 -142 -134 -126 -118 -111 -103 -95 -87 -79 -71 -63 -55 -48 -40 -32

-215 -212 -209 -207 -204 -201 -198 -196 -193 -190 -187 -185 -182 -179 -176 -174 -171 -168 -165 -163 -160

13 15 18 21 23 26 28 31 33 36 38 41 44 46 49 51 54 56 59 62 64

10 9 9 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7 7 7

-204 -189 -196 -201 -201 -206 -218 -218 -200 -169 -152 -160 -173 -186 -201 -216 -225 -219 -203 -184

-11 2 12 14 3 -19 -38 -43 -35 -15 6 20 23 14 -6 -27 -39 -39 -29 -14

57 -9 -17 -13 -2 13 22 27 27 19 -8 -16 -13 -7 3 11 17 18 16 11


-166


-204 -198 -191 -185 -178 -171 -165 -158 -152 -145 -138 -132 -125 -118 -112 -105 -98 -92 -85 -78 -72

-252 -249 -245 -242 -238 -235 -231 -228 -225 -221 -218 -214 -211 -208 -204 -201 -198 -194 -191 -188 -184


-168 -181 -190 -191 -182 -162 -143 -126 -112 -108 -114 -128 -144 -152 -146 -132 -112 -84 -50 -13

-222 -234 -240 -235 -223 -215 -212 -213 -218 -222 -221 -220 -217 -213 -211 -206 -200 -197 -198 -201

26 27 25 22 18 18 25 38 50 56 53 45 35 26 20 23 34 51 72 93

-38 -43 -45 -42 -30 -22 -24 -30 -36 -39 -41 -41 -37 -28 -20 -19 -25 -32 -38 -42

85

-180 -175 -153 -137 -128 -118 -109 -110 -115 -120 -107 -84 -72 -70 -76 -79 -81 -87 -98 -110


Table B.4: Calculated Principal stresses (σM AX , σM IN ), and Principal stress direction (β) for Uniform, Power-Series and Integral calculation methods. β is measured clockwise of strain gage 1 (L-direction).Data for blocks A7 and A10 Block A7 Hole Depth (mm)

Block A10

σM AX

σM IN

τM AX

β

σM AX

σM IN

τM AX

β

(MPa)

(MPa)

(MPa)

(deg.)

(MPa)

(MPa)

(MPa)

(deg.)

-2

6

-18

9

1

4

-8

7 7 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5

-24 -23 -23 -22 -21 -20 -19 -18 -17 -15 -14 -13 -12 -10 -9 -8 -6 -5 -4 -2 -1

13 12 11 11 10 9 9 8 8 7 7 6 6 6 5 5 5 5 4 4 4

3 3 2 2 2 1 1 1 0 0 -1 -1 -2 -3 -3 -4 -5 -6 -7 -7 -8

5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 6 6

-24 -22 -20 -17 -15 -12 -9 -6 -3 1 4 7 10 13 15 17 19 21 23 24 25

14 5 -1 -3 -2 -3 -3 -1 0 0 -2 -4 -6 -6 -3 0 0 -2 -6 -12

2 1 1 1 1 2 3 4 4 3 1 -2 -5 -6 -7 -6 -5 -3 -2 0

-10 -10 -10 -10 -16 -22 -24 -24 -22 -17 -5 13 24 29 31 32 32 26 14 1


10


17 16 15 13 12 11 10 9 7 6 5 4 3 2 1 0 -1 -3 -4 -5 -6

3 2 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16


16 13 12 11 10 9 10 11 11 10 8 5 0 -5 -7 -8 -6 -3 -1 0

4 2 0 -2 -2 -3 -4 -4 -4 -4 -5 -7 -8 -9 -11 -13 -16 -19 -21 -21

6 5 5 4 4 3 3 2 2 2 2 2 2 2 2 2 3 3 3 3

-37 -33 -24 -20 -19 -16 -11 -9 -8 -8 -8 -9 -14 -36 -54 -36 -15 -10 -9 -9

86

26 13 4 1 2 4 6 10 12 11 7 6 7 9 12 13 11 7 1 -4

Appendix C Statistical model C.1

R programming code

The following script can be used to in further hypothesis tests. R-package is a free statistical software.

The

It can be downloaded from:

https://www.r-project.org/. R-Studio provides a pleasant operating environment; therefore, it is recommended to download and install this also. Download R-Studio from: https://www.rstudio.com/. The code below can be used to create box-plots of the data, and conduct hypothesis testing as it is defined in Appendix ??. To execute each line of code individually (recommended), highlight it using the curser, then select ’CTRL+ R’ on your keyboard. #Download R from: https://www.r-project.org/ #R-Studio is a good operating environment, therefore #it is recommended to download and install this also. #Download R-Studio from: https://www.rstudio.com/ ####################################################### #TWO SAMPLE INFERENCE ####################################################### #This code allows the comparison of the stress state #in two blocks by conducting a 99% confidence interval #on the stress values data set. ####################################################### #STRESS DATA 87


####################################################### #Save stress data in a .csv file. #The code below will look for two columns of data: #The first must have the heading ’Block’ #(e.g. cells A2:12=’A10’; A13:23=’A7’) #where ’A10’ and ’A13’ are the ID’s of the #blocks being analysed. # NOTE the first set of entries should have the #block with shortest #PQD to test the hypothesis as it is stated below. #The first cell in the second column must be called ’Stress’. #NOTE, the test is conducted under the understanding that the #blocks are matched pairs. i.e. same material and #processing (except for PQD). ####################################################### #To perform the analysis, adapt and execute the following code: ###################################################### getwd() #set path on your PC to where the raw stress data have been saved: #For example: setwd("D:/University of Limerick/4.FYP/Thesis/R/") #To read a file within the working directory, enter #file name here (at end): stress < read.csv("D:/myfolder/LTscan_Lxx.csv", header=TRUE) #The ’header =True tells the code that the data have #headings (e.g. Block,Stress) ###################################################### #EXPLORITORY ANALYSIS ####################################################### boxplot(Stress~Block,data=stress) #both boxplots show that the data should be normally #dist with no obvious outliers which warrant #investigation. The t-dist. used assumes a normal #dist. of data.

88


#The spread for each boxplot should look similar #which would suggest that the variability between #the two groups in approx. equal. #summary statistics using the "by" command which #applies a function to the ’Stress column, split by #the levels of ’Block’ factor. by(stress$Stress, stress$Block, sd,na.rm=TRUE) #the na.rm command tells R to include missing #values from the dataset. #This is required if data that are collected have #some voids, or if you wish to remove an outlier #from the set. #Draw a conclusion on box plot: e.g. there seems #to be a clear difference between the mean no. of #Stress for the two block types. The data for each #appear to be normally dist. The SD’s for each #group appear to be similar #indicating that we may be able to assume equal #var. Next, we do some formal analysis. #HYPOTHESIS: Ho: mean difference less than or equal to 0 #(i.e. higher PQD = higher stress) #Ha: mean diff greater than zero.

#ttest with equal variance (from result of test above) t.test(Stress~Block,data=stress,alternative=’greater’, var.equal=TRUE,paired=TRUE, mu=0, conf.level=0.99) # var.equal=TRUE: assume equal variances # paired=TRUE: these are paired groups # mu=0: assume the difference between the pop. means is 0

89

Appendix D Turnitin Originality Report

90

Turnitin Originality Report Residual Stress Determination

by Barry Mooney

From Final Year Project (BE Design and Manufacture projects 2017) Processed on 20Mar2017 2:10 AM GMT ID: 786288540 Word Count: 27033 Similarity Index 9% Similarity by Source Internet Sources: 5% Publications: 6% Student Papers: 2%

sources: 1

1% match (publications)

Robinson, J. S., P. J. Tiernan, and J. F. Kelleher. "Effect of postquench delay on stress relieving by cold compression for the aluminium alloy 7050", Materials Science and Technology, 2015.

2

1% match (Internet from 03Mar2017) http://documents.mx/download/link/handbookaluminium

3