Some improvements in the analysis of fatigue cracks using thermoelasticity

International Journal of Fatigue 26 (2004) 365–376 www.elsevier.com/locate/ijfatigue

Some improvements in the analysis of fatigue cracks using thermoelasticity F.A. Dı´az ∗, J.R. Yates, E.A. Patterson Department of Mechanical Engineering, University of Sheffield, Sheffield S10 3JD, UK Received 7 January 2003; received in revised form 28 July 2003; accepted 11 August 2003

Abstract Thermoelastic stress analysis (TSA) has been developed in recent years as a technique for the direct measurement of crack tip stresses in structural components. The technique provides full-field stress maps from the surface of a component subjected to a cyclic load. An improvement in the methodology for monitoring fatigue crack growth and inferring the stress intensity factor from thermoelastic data is presented. The approach is based on a multipoint over-deterministic method (MPODM) [Sanford JR. A general method for determining mix-mode stress intensity factors from isochromatic fringe pattern. Engineering Fracture Mechanics 1979;11:621– 33], where experimental data collected from thermoelastic images are fitted to Muskhelishvili’s equations describing the stress field around the crack tip. The fitting algorithm employed is based on the Downhill Simplex Method and includes the crack tip position as a variable to be optimized. Results obtained from artificially generated images and from tests performed using stress relieved and as-welded single edge notched specimens tested at different R-ratios are reported and compared. It is shown that TSA is a sensitive technique for examining the influence of phenomena such as crack closure and residual stresses during fatigue crack growth. Information obtained from the map of phase difference between the forcing signal and the thermal response is presented and discussed. It appears that the phase map may contain information about heat generation at the crack tip due to plastic work and contact of the crack forces.  2003 Elsevier Ltd. All rights reserved. Keywords: Thermoelastic stress analysis (TSA); Fatigue cracks; Stress intensity factor (SIF); Deltatherm; Differential thermography

1. Introduction Thermoelastic stress analysis (TSA) has been developed in recent years as a technique for the direct measurement of crack tip stresses in cracked components and structures. The technique is non-contacting and provides full field data from the surface of the component. Its fundamentals are based on the thermoelastic effect [1], in which small temperature changes are induced in a solid component when a cyclic load is applied. When adiabatic and reversible conditions are achieved, these temperature variations are proportional to the sum of the principal stresses. Consequently, by measuring these small temperature changes, it is possible to obtain infor-

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Corresponding author. Tel.: +44 114 222 7764. E-mail address: [email protected] (F.A. Dı´az).

0142-1123/$ - see front matter  2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2003.08.018

mation about the stress distribution ahead of a crack tip (see Fig. 1). For the analysis of the crack tip stresses using TSA, it is convenient to have a load cycle small enough to ensure that the crack is not growing during the period of time when the image is being collected. Early systems, like SPATE, were limited by the speed of data collection. The SPATE system is based on a single point detector using a set of movable mirrors to scan the component. This system requires thousand of cycles to acquire a full-field thermoelastic image of a fatigue crack and during the data collection period the crack can grow, making it more difficult to interpret and analyse the data. Nevertheless, some work related to the study of fatigue crack growth using SPATE has been published [2–4]. The Deltatherm 1500 system employed in the studies presented in this paper is based on an array of detectors, making it possible to collect information from a growing fatigue crack in just a few tens of cycles over a period

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Nomenclature e Surface emissivity i Square root of –1 z Complex coordinate in the physical plane A Calibration constant for thermoelastic equipment AN, BN Fourier series coefficients B Constant that depends on the detector operating wavelength Specific heat at constant strain Ce Specific heat at constant pressure Cp E Elastic modulus Mode I stress intensity factor KI Mode II stress intensity factor KII Q Heat input R Ratio of smin to smax S Thermoelastic signal Maximum thermoelastic signal per row of pixels Smax T Absolute temperature of the material a Linear coefficient of thermal expansion ex, ey Strains in x and y directions n Poisson’s ratio r Density s1, s2 Principal stresses Stress change tensor sij sx, sy Stresses in x and y directions Shear stress txy w Mapping function z Complex coordinate in the mapping plane ⌬T Temperature change ⌽ Spectral photon emittance ⌽(z), ⌿(z) Complex stress functions

of a few seconds. This makes the system ideal for studying the behaviour of fatigue cracks and the associated stress field in a much more detailed way. Temperature changes associated with the thermoelastic effect are usually of the order of tens of mK. Nowadays, these temperature variations can be readily measured using modern high precision infrared detectors. Since the concept of crack closure was introduced by Elber in 1970 [5,6], a large amount of research has been performed on the measurement and interpretation of crack closure mechanisms using different methods (direct observation methods, compliance methods and indirect methods) [7]. With TSA, stresses at the crack tip stresses are derived from the small temperature changes in the material around the crack tip. Consequently, the stress range derived from the thermoelastic measurements is the effective stress range rather than the one inferred from the applied load. So, the TSA measurement is sensitive to the local effects of crack shielding and growth retardation mechanisms; whereas measurements inferred from the applied load are insensitive to these local mechanisms. This feature makes the

technique quite promising in the evaluation of crack closure and the influence of residual stresses during fatigue crack growth. However, little research has been done on the application of TSA to the quantification of crack closure and other crack shielding mechanisms [8]. Fulton et al. [9] were the first to report that some form of closure could be observed using TSA during a fatigue test. It was noted that the range of the stress intensity factor (SIF) decreased with respect to the expected theoretical value when the stress range increased. After that, some work was performed by Batchelor et al. [10] to measure the crack closure effect by cutting out the wake of a fatigue crack and measuring the change in the SIF using TSA. Results showed a reduction in the closure effect as more of the crack wake was cut away. More experiments, with the same purpose, were performed by Tomlinson et al. [11] using Deltatherm 1000 on an aluminium alloy panel. The specimen used was a plate with a central hole with two spark-eroded slots loaded at a R-ratio of 0.33. The results from cracks growing from slots showed that some form of closure was present and it had the effect of decreasing the value of ⌬KI when

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crack tip. Initially, results related to the quality of the fitting algorithm are presented from a set of artificial images generated from Westergaard’s stress field equations. Subsequently, the algorithm is employed to monitor the variation of the SIF with crack length for tests performed at different R-ratios. Two kinds of single edge notched (SEN) specimens have been employed; welded specimens with a known residual stress distribution in the welded region, and otherwise identical, stressrelieved specimens. At the same time the variation of fatigue crack growth rate with SIF range is presented and compared for the different tests. Finally, the maps of phase difference between the applied load and the material surface temperature are presented and discussed. The phase maps appear to indicate the presence of heat conduction due to local plastic work.

2. Principles of thermoelastic stress analysis

Fig. 1. (A) Typical thermoelastic image of a 18 mm crack captured during a fatigue test. (B) Distribution of the sum of principal stresses surrounding the crack tip derived from a thermoelastic image.

compared with the nominal value calculated from the applied load. Further work using thermoelasticity was started by Dı´az et al. [12,13] when welded steel plates were tested to study how the presence of residual stresses due to welding could affect fatigue cracks. In those tests it was observed that the difference between the theoretical and the experimental SIF range decreased as the Rratio increases, which was consistent with most of the work published by previous authors. It was also noted that the closure was not eliminated at a high R-ratio of 0.6 and this was attributed to the substantial compressive residual stresses present in the specimens. In this paper, a novel methodology for the calculation of mode I and mode II SIF from thermoelastic images is presented. The principle behind the new algorithm is to fit a theoretical model describing the elastic stress field around the crack tip to the experimental data collected from the crack tip region. This approach not only makes it possible to determine the SIF but also to locate the

TSA is based on the principles of the thermoelastic effect. The phenomenon is that the temperature of any substance changes slightly when its volume is changed by a force. This effect can be induced either by a compressive load which heats the stressed substance or by a tensile load which causes a cooling. The thermoelastic effect is a reversible conversion between mechanical and thermal forms of energy, since higher and lower temperatures will revert when the load is withdrawn. However, this energy conversion is reversible only if the elastic properties of the object are not exceeded and there is no significant transport of heat during loading and unloading of the object. When adiabatic and reversible conditions are achieved, the temperature variations experienced by a cyclically loaded component are proportional to the sum of the principal stresses. The relationship between the change in temperature due to the application of a cyclic load and the change in the stress state of a linear elastic and homogeneous material can be expressed as: ⌬T ⫽ ⫺

冘

T rCe

Q ∂sij eij ⫹ ∂T rCe

(1)

If the component is cyclically loaded at such rate that no heat conduction takes place, the second term of the equation is usually neglected. The following expression is obtained by manipulating Eq. (1): ⌬T ⫽ ⫺

a T⌬(s1 ⫹ s2) rCp

(2)

To ensure that the above relationship is linear, the time variation of the load must be fast enough to prevent heat transport. Heat transfer within any specimen is dependent on the thermal conductivity of the material, the stress gradients and temperature gradients in the speci-

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men and the loading frequency. Adiabatic conditions may be achieved if the thermal conductivity of the material is zero or if there are no stress gradients in the specimen. However, if the frequency is increased the thermal diffusion length is reduced and consequently, the non-adiabatic effects are minimized. The variation in temperature due to the thermoelastic effect can be derived from the output voltage of an infrared detector. Under these conditions, the spectral radiant photon emittance from the surface of the material is related to its temperature by the following expression:

Where A denotes a calibration constant that depends on material properties such as the coefficient of thermal expansion, density, the specific heat capacity at constant pressure, surface temperature and the detector parameters.

movement of the specimen when it is being cycled. It is also possible to get a high level of precision by using a special set of lenses available with the equipment that allows focusing on the position of interest with a thermal resolution of 2 mK full field with a 30 s acquisition time, and a maximum spatial resolution of 25 µm. The array of detectors in the DelthaTherm 1500 is cooled by a Stirling cooling engine system rather than liquid nitrogen, as in earlier systems. The thermoelastic signal is normally very noisy and signal processing is required to reduce such noise. The way of reducing this noise is by correlating the thermoelastic signal to the cyclic loading reference signal. In the correlation process the period of the reference signal is used to extract the information due to the thermoelastic effect, since this phenomenon happens at the same frequency as the applied load. Thermoelastic information is presented as a vector, where the modulus denotes the magnitude that is proportional to the variation in temperature experienced by the specimen and the phase denotes the angular shift between the thermoelastic and reference signals. The magnitude of the phase is normally a constant value over the whole phase map unless phenomena like heat conduction are taking place.

3. Description of the thermoelastic equipment

4. Calibration

⌽ ⫽ eBT3

(3)

Combining Eqs. (2) and (3) and assuming that thermoelastic signal is linearly proportional to the change in photon emittance, the typical expression used in TSA is obtained: ⌬(s1 ⫹ s2) ⫽ AS

(4)

Certain materials have the property that incident radiant energy may change their electrical characteristics. Such materials can be used as a transducer to convert radiant energy into an electrical signal. The bandwidth in the electromagnetic spectrum between 3 and 14 µm contains the maximum photon and energy emission for bodies at room temperature. This band lies in the infrared region, and these detectors are called infrared detectors. In recent years, the development of infrared cameras based on array detectors accompanied by improved software and hardware has changed the use of thermoelasticity in fracture mechanics experiments. The Deltatherm system, produced by Stress Photonics Inc, incorporates a 256×320 indium antinomide focal plane array detector, which obviates the necessity of scanning mirrors as in previous equipment such as SPATE. The photon emission from the specimen surface is captured by the infrared head. Each sensor reports the temperature that is observed once per image frame. This information, at a rate of more than 400 images per second, is used to produce thermoelastic images. The signal processing is digital, which considerably reduces the time needed to generate full-field images, making it possible to observe real fatigue cracks and damage accumulation. Moreover, the system is more portable and less sensitive to mechanical vibrations than earlier scanning systems. Motion compensation, of the specimen is also possible with Deltatherm equipment, which takes into account the

The calibration process for a set of thermoelastic images consists of defining the stress at a point in the image for a given load on the structure. When the loading of a structure is maintained in the elastic region and changes in load happen fast enough, little conduction takes place and thermoelastic data are assumed to be linearly proportional to the principal stress sum. Different methods have been developed to derive the calibration constant A in Eq. (4). These methods are described in [14]. The calibration constant, A, is dependent on the radiometric properties of the detector, the material properties, the absolute temperature of the specimen and the emissivity of the surface. The method of performing the calibration is to generate an independent measure of the stress responsible for the thermoelastic signal using strain gauges. Hence, a rosette consisting of two orthogonal gauges is bonded to the specimen in a region of uniform stress, where the thermoelastic signal is constant (see Fig. 2B). From the rosette the strain invariant is obtained, thus the principal stress sum can be derived by application of Hooke’s law ⌬(s1 ⫹ s2) ⫽ ⌬(sx ⫹ sy) ⫽

E ⌬(e ⫹ ey) ⫽ AS (5) 1⫺m x

By correlating the thermoelastic signal from the same location as the gauge with the principal stress sum derived, the calibration constant, A, can be calculated.

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In the method derived by Tomlinson et al. and by Nurse and Patterson, the stress field around the crack tip is represented by two analytical functions ⌽(z) and ⌿(z) of a complex variable, such that: (s1 ⫹ s2) ⫽ 2⌽(z) ⫹ ⌽(z) with

z⫽x

(7)

⫹ iysy⫺sx ⫹ 2txy ⫽ 2(z¯ ⌽’(z) ⫹ ⌿(z)) The requirements of internal equilibrium and compatibility are automatically satisfied and only the boundary conditions have to be satisfied. The boundary conditions for the crack surface are satisfied by the use of conformal mapping in which the crack geometry in the physical plane is mapped onto a circle in the geometrical plane (see Fig. 3). The stress equations are generated from a Fourier series in complex form:

冘 ⬁

⌽(z) ⫽

冘 ⬁

ANz2N ⫹

N⫽0

冘

am with z ⫽ rj ⫽ r(cosq ⫹ isinq) 2n z m⫽1

⬁

Fig. 2. Illustration of the preparation of the specimen for TSA. (A) Detail of the matt black paint for increasing the surface emissivity. (B) Detail of location of the rosette strain gauges employed for calibration.

A⫽

[E / (1⫺m)]⌬(ex⫺ey) S

(6)

5. Calculation of the stress intensity factor A number of investigators have used TSA in the study of crack propagation behaviour and the determination of the stress field around a crack tip in different materials. As a result of this activity, several different methods have been developed to determine SIFs. One such example was developed by Stanley [4], who developed a method based on the first two terms in Westergaard’s equations for the elastic stresses in the vicinity of the crack tip. Stanley and Dulieu-Smith [14, 15] introduced a technique based on the fact that isopachic contours in the crack tip region generally take the form of a cardioid curve. By determining the orientation and the area of the cardioid it is possible to obtain mode I and mode II SIFs. Lesniak [16] developed another method based on the fitting of thermoelastic data to Airy stress functions using a least squares method. Lin [17] presented a hybrid method based on the equilibrium, compatibility and the J-integral method. Tomlinson [18] proposed a procedure based on a Newton-Raphson iterative method combined with a least squares correlation to fit thermoelastic data to the equations describing the stress field around the crack tip using Muskhelishvili’s approach [19]. This methodology is based on previous work by Nurse [20] for the determination of SIFs associated with a mixedmode crack from photoelastic data.

⌿(z) ⫽

冘 ⬁

BNz2N ⫹

N⫽0

bm 2n z m⫽1

(8)

where the coefficients of each term (AN, BN, am and bm) are complex variables that allow different states to be described. As a result, it is possible to obtain an expression for the stress function that depends exclusively on the coefficients of the Fourier series and the coordinates of the points surrounding the crack [20]. ⌽(z) ⫽ A0 ⫹

冘 冋冉 冊冉 ⬁

⫹

A0 ⫹ A0 ⫹ B0 (z2⫺1)

N⫽1

册

⫹ A2N Nz

2N z2N

with

冊

z2 ⫹ 1 AN BN A ⫺ ⫺ z2⫺1 N z2N z2N z(z) ⫽

z ⫹ a

(9)

冪冉 冊 z2 ⫺1 a2

Where z is a complex variable in the mapping plane and A and B are the coefficients of the Fourier series previously mentioned. Consequently, an analytical expression for the sum of principal stresses at the crack tip can be inferred, by substitution in Eq. (7). At the same time, it is possible to obtain experimental information about the distribution of the sum of principal stresses around the crack tip from thermoelastic images (see Fig. 1). By collecting data from a few points in the region ahead of the crack tip this information could be used to evaluate the unknown coefficients describing the stress field. Now, it is proposed that the crack tip location could be included as an unknown variable since the coordinates of the points are always defined with respect to the crack tip. As a result, an optimization function can be derived: g ⫽ 2[⌽(z) ⫹ ⌽(z)]⫺AS

(10)

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Fig. 3. Scheme of the mathematical approach implemented in the computer algorithm for SIF calculation, including diagrams of the coordinate systems for the physical and the mapping plane.

The algorithm used to fit the analytical solution to the experimental data is based on the Downhill Simplex method [21]. This method only requires evaluation of the function, not the derivatives. It is more robust than the Newton-Raphson method, making it possible to increase the number of variables to be optimized simultaneously and in particular, the crack tip location. The Downhill Simplex Method was successfully used by Pacey [22], when evaluating crack closure in polycarbonate compact tension specimens using photoelastic stress analysis. All these ideas have been implemented in a Matlab computer routine called from a graphical interface that allows the selection of data points to perform the fitting. To evaluate the accuracy of the algorithm when calculating the SIF and finding the crack tip, a set of artificial thermoelastic images were generated from Westergaard’s stress field equations (Fig. 4). Artificial images corresponding to a pure mode I stress field through a range of mixities to a pure mode II field were generated and analysed (see Table 1). Results related to the accuracy of the algorithm when finding the crack tip location from an initial crack tip location in a test specimen are presented in Table 2. Results for the crack length obtained using an optical microscope are compared with those from the algorithm in Fig. 5.

6. Processing thermoelastic information In order to process the thermoelastic images, a software interface was developed. By using this interface approximately 400 data points were collected from every thermoelastic image captured during a fatigue test from the region surrounding the crack tip. In the data collection process, care was taken when collecting the points. In a crack with a small plastic zone three regions can be identified (see Fig. 6A): a region of large strain and plasticity, where the linear elastic fracture mechanics

Fig. 4. (A) Thermoelastic image of an 8 mm crack generated artificially using Westergaard’s equations. (B) Distribution of the sum of principal stresses surrounding the crack.

model adopted is not valid; a region dominated by the elastic crack tip stress field, that is described by the model; and a region far away from the crack tip that is dominated by the remote stress distribution which the model does not describe. Points were collected from the

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Table 1 Results for mode I and mode II SIF generated by processing thermoelastic images generated artificially using Westergaard’s equations Values used to generate artificial images from Westergaard’s equations

Results from fitting algorithm

Difference (%)

KI

KII

KI

KI

10 10 10 10 10 10 8 6 4 2 0

0 2 4 6 8 10 10 10 10 10 10

10.06 10.04 10.10 9.99 9.96 9.94 7.92 5.91 3.89 1.88 1.72

KII 0.02 1.97 3.76 5.55 7.33 9.12 9.08 9.04 9.00 8.97 9.91

KII

0.63 0.35 1.02 ⫺0.09 ⫺0.41 ⫺0.65 ⫺1.04 ⫺1.47 ⫺2.69 ⫺5.99 –

– ⫺1.32 ⫺6.03 ⫺7.57 ⫺8.38 ⫺8.81 ⫺9.20 ⫺9.59 ⫺9.96 ⫺10.34 ⫺0.89

Table 2 Results showing the location of the crack tip and the SIF obtained using the fitting algorithm for pure mode I and mixed mode artificial images Number of data points

Initial KI (MPa m1/2)

Initial KII (MPa m1/2)

Calculated KI (MPa m1/2)

Calculated KII (MPa m1/2)

Initial x (pixels)

Initial y (pixels)

Calculated x (pixels)

Calculated y (pixels)

91 119 91 119

10 10 10 10

0 0 10 10

10.0003 10.0002 10.0012 10.0002

0.0078 0.0033 9.9997 10.005

85 85 85 85

135 135 135 135

80 80 79.995 80.0005

127.01 127.005 127.005 127.005

Artificial images were generated with the crack tip positioned at x = 80 pixels and y = 127 pixels, the initial crack tip input coordinates given to the algorithm to perform the fitting and calculate the real crack tip position were x = 85 pixels and y = 135 pixels.

regions previously mentioned can be found easily. Data were collected in the images just from the region where linear behaviour was observed (diamonds in Fig. 6B). Subsequently the coordinates of the collected points, referred to a first estimate of the location of the crack tip, were employed as an input for the computer routine to fit the experimental points to Muskhelishvili’s equations. Finally, the crack tip location was recalculated in successive iterations with the aim of minimizing the fitting error.

7. Phase information Fig. 5. Crack length measured from the reverse of a test specimen during a fatigue test using a microscope against the crack length inferred after processing the thermoelastic images.

second region, that is dominated by the crack elastic stress field. To perform this analysis a plot representing the inverse square of the maximum thermoelastic signal per pixel row versus the distance from the crack line was calculated [23] (see Fig. 6B). Using this graph, the three

Thermoelastic information is presented as a vector, where its modulus is related to the magnitude of the thermoelastic signal (normally visualized as the R-image) and the phase is referred to the phase difference between the thermoelastic and the reference signal. If adiabatic conditions are achieved, the phase always takes a constant value, and this should actually happen in regions of uniform stress. However, the presence of high stress gradients or local plasticity leads to a loss of adiabatic conditions, and consequently conduction through the

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elastic stresses ahead of the crack is the region where y is linearly proportional to 1 / S2max (see Fig. 6). This defines the domain over which data may be collected for fitting Muskhelishvili’s equations, since the conditions outside this area are neither elastic nor adiabatic. In Fig. 6, there are also a few points marked with a circular dotted line that do not follow the trends exhibited by the remainder of the data. These data values are due to the presence of imperfections or ‘dead pixels’ in the infrared array detector. Moreover, the loss of adiabatic conditions is also observable when looking at the phase map obtained with Deltatherm. For image processing purposes, the phase is always set to zero. This means, that where adiabatic conditions are achieved, the reference and the thermoelastic signal are in-phase. Consequently, any deviation of the phase from zero is due to the presence of non-adiabatic conditions. When thermoelastic images were captured during a fatigue test the same pattern was always observed in the phase map (Fig. 7A). There was a region surrounding the crack tip and ahead of the crack where

Fig. 6. (A) Schematic illustration of the data point selection used to reconstruct the stress field and calculate the SIF from thermoelastic images. (B) Plot showing the variation of the squared inverse of the maximum thermoelastic signal versus the vertical distance from the crack tip.

specimen takes place. This fact has been investigated to some extent over the last ten years [24,25]. The loss of adiabatic conditions normally happens at the crack tip and it can be easily observed when plotting the inverse square of the maximum thermoelastic signal versus the vertical distance from the crack line (Fig. 6B). This information arises from Stanley’s methodology [23] for the calculation of the SIF. According to this approach, the vertical distance from the crack line, y, is proportional to the squared inverse of the maximum thermoelastic signal per pixel row, 1 / S2max. The constant of proportionality depends on the SIF and the calibration constant. y⫽

冉 冊 3冑3K2I 4pA

2

1 2 max

S

(11)

The usefulness of this relationship is that the region of

Fig. 7. (A) Typical phase map captured with Deltatherm 1500 during a fatigue test. (B) Phase profile along the line of the crack.

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the phase becomes negative, that is the thermal response lags behind the loading cycle. This fact can be seen in the phase profile along the crack line (Fig. 7B). This behaviour is due to the lack of adiabatic conditions and could be due to heat generation and conduction as a consequence of plastic work at the crack tip. To the left of the region of negative phase in Fig. 7B, there is an area where the phase is continuously oscillating from positive to negative. This region corresponds to the crack faces in the specimen. Such behaviour could be attributed to contact of the crack faces. This distinctive phase shift pattern has been observed in all the fatigue tests performed using TSA and although quantitative results cannot be reported at the moment, more work in associating the phase shift to the size of the plastic zone is being performed. This information could be used to estimate the size of the plastic zone in cracked components. So far, this information has been used to find the location of the crack tip that is employed as the initial value in the calculation of the SIF. The initial estimate of the crack tip is taken to be at the point where the phase changes from positive to negative. Subsequently, this location is optimized in the calculation procedure. Using this approach, it is possible to ensure convergence of the fitting algorithm.

8. Experimental work The experiments consisted of growing fatigue cracks at different R-ratios in SEN tension specimens. Two sets of specimens were used: one with an initial crack in an as-received weld; the other with the crack in a stressrelieved weld. Specimens were obtained from an initial test plate which was fabricated by welding together two 40 mm thick plates of length 4 m and width 0.5 m each, which were then sliced into 12 mm thick sections from which test specimens were made. The steel used for the test plate was BS 1501 490 LT50 and the technique used for welding was multi-pass submerged arc. The preparation used was an asymmetric double V preparation with the weld root positioned at 2/3 of the plate thickness from the surface that was filled first. Specimens used for the study of the crack growth had a 4 mm initial edge notch 0.12 mm wide inserted by electric discharge machining in the mid point of the weld. Specimen dimensions are shown in Fig. 8A. Details of the weld and the notch are shown in Figs. 8B and C. Information about the material properties of the weld and parent material is presented in reference [26]. Residual stresses due to welding were largely removed in one of the specimens tested by employing a stress-relieving heat treatment. The welded specimen was heated to 400 °C and held at that temperature for 3 h, before cooling at approximately 100 °C per hour. In order to increase the emissivity and to have a uni-

373

Fig. 8. (A) Photograph of the test specimen with dimensions. (B) Detail of the welded region. C) Detail of the notch.

form signal from the surface where thermoelastic images were to be taken, the specimen was sprayed on one face with a black matt paint (black matt paint RS 496-782) (see Fig. 2A). On the reverse face, two rosette strain gauges (2 mm length, 120±0.5 ⍀) were bonded for the calibration procedure [15]. One of the strain gauges was far away from the crack line, and the other was along the line of intended crack growth (Fig. 2B). The specimen was placed in ESH 100 kN uniaxial servo-hydraulic machine using an assembly consisting of two grips and two pins. Subsequently, a constant amplitude cyclic tensile load was applied along the longitudinal axis of the specimen at a frequency of 12 Hz. The mean load and the cyclic load range were set for different test specimens at different R-ratios (see Table 3). During the test, thermoelastic images were recorded from the region around the crack tip at nominal intervals of 10 000 cycles. Each data map was integrated over 40 s. The spatial resolution obtained for each map was 15 pixel/mm. A typical thermoelastic map is shown in Fig. 1A. Simultaneously, in order to monitor the crack growth, the crack length was measured from the reverse side of the specimen using a Vernier microscope. Table 3 Specifications of the fatigue tests performed Specimen characteristics

smax (MPa)

smin (MPa)

smean (MPa)

R

Initial ⌬K (MPa m1/2)

Stress relieved Weld Weld

85 107 187.9

7.7 30.3 109.7

46.3 68.7 148.8

0.1 0.3 0.6

10.3 10.3 10.3

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Once the thermoelastic images were obtained at increments of numbers of cycles and crack lengths, the data was processed using the Matlab routine implementing the methodology previously described, and the SIF ranges and the crack tip locations were determined. This procedure was repeated for tests at different Rratios. Experimental data for the SIF range versus crack length were compared to the SIF range deduced from the applied load using a published calibration expressions [27]. The stress range during the different tests was kept constant to give an initial ⌬K of 10 MPa m1/2, and the mean load was changed in order to perform tests at different R-values. Consequently, the theoretical SIF range, obtained from the applied load at increments of crack length was the same for all tests, since the theoretical value is a function of the stress range, the crack length and the geometry of the specimen. The results of the tests performed for the first of these two cases are shown in Fig. 9. The theoretical value is presented as a continuous line, and experimental values for each test are shown with different symbols. It can be seen that the mode II SIF range is very close to zero because the tests were performed under nominally pure mode I conditions, and crack paths were essentially straight across the specimen. Fig. 10 shows the variation of the crack growth rate with the SIF range (theoretical and experimental) for a fatigue test performed with a stress-relieved specimen by processing thermoelastic images captured during the test. Results are compared with those obtained by an industrial collaborator [26] from a three-point bend specimen manufactured from the parent material.

9. Discussion In this paper a new approach for calculating the SIF from a fitting equation is presented. Initially, to check

Fig. 9. Theoretical and experimental SIF ranges against the crack length for tests performed at different R-ratios.

Fig. 10. Fatigue crack growth rate against the SIF range for fatigue tests performed on a parent plate [26] and an stress relieved specimen tested at R = 0.1 using thermoelastic images to infer the fatigue crack growth rates and the SIF ranges.

the quality of the algorithm, a set of artificial images were generated for various levels of SIF mixities using Westergaard’s stress field equations. Values for the mode I and mode II SIF obtained using the fitting algorithm are shown in Table 1. Results related to the accuracy of the algorithm when finding the crack tip are presented in Table 2. For the stress fields dominated by the mode I component, the new algorithm yielded mode I results within 1% of the value used to generate the image. For the mixed mode cases, the difference between the inferred SIF and the one employed in generating the images is ⬍6% for the mode I SIF and 10 % for mode II. The reduction in the quality of the fit of Muskhelishvili’s stress field equation to the experimental data when the mode II component increases may be due to the nature of the changing stress field singularity. In pure mode I loading there is only a positive stress singularity at the crack tip. Mode II loading introduces asymmetry to the stress field, with both positive and negative singularities. A continuous function such as Muskhelishvili’s equations can experience difficulties in fitting to such a rapidly changing near crack stress field if the experimental data are not ideally placed. This effect is exacerbated in this case by the use of the perfectly elastic stress filed data in the artificial images. Subsequently, the fitting algorithm was employed to infer the SIF for various crack lengths using real thermo-

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elastic images captured during uniaxial fatigue tests (Table 3). The results are presented in Fig. 9. The continuous line represents the theoretical mode I SIF, the solid and hollow symbols represent the mode I and II SIFs obtained from thermoelastic images. Fatigue tests at R = 0.3 and R = 0.6 were performed with as welded received specimens. In those specimens, the presence of residual stresses along the weld line had been previously determined [23]. The test at R = 0.1 was performed with a stress relieved-specimen. The results from Fig. 9 clearly show a deviation of the experimental results with respect to the theoretical line. This happens in the case of as-welded specimens when the crack is longer than 10 mm. At this distance substantial compressive residual stresses due to the welding start to be present. The reduction in the discrepancy when the R-ratio is increased from 0.3 to 0.6 demonstrates the sensitivity of the TSA technique and the new fitting algorithm to changes in crack closure. In the case of the stress relieved specimen, experimental results for the SIF follow the theoretical line until the crack length reaches 14 mm, after this point a deviation from the theoretical line is observed. It is thought that this is due to the build up of closure in the wake of the fatigue crack. It can be also observed that for the case of the mode II SIF, the values are always close to zero as expected for mode I test conditions. In Fig. 5 the crack length measured from the reverse side of the specimen with a microscope against the crack length inferred by processing the thermoelastic images. The agreement seen in Fig. 5 between the optical measurement of the crack length and the results of the processing of the thermoelastic image is good. The deviation at long crack lengths is associated with the difficulty in identifying the crack tip in the optical image because of distortion and surface finish. It is also influenced by the fact that the optical and the thermoelastic images are taken from the opposite faces of the specimen and the crack front is not perfectly straight. In Fig. 10 the fatigue crack growth rate against the SIF range is presented. The continuous line represents the results corresponding to a fatigue test performed on a plate manufactured from the parent material [26]. The circles correspond to a test performed using a stressrelieved welded specimen at R = 0.1. Thermoelastic images were captured during this test and subsequently automatically processed to monitor the fatigue crack growth rate and the SIF range. From this graph it is clearly observed that results obtained using the computer algorithm previously discussed, are consistent with the data provided by the manufacturer for the parent material.

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10. Conclusion A new computer routine has been successfully developed for evaluating the SIF from thermoelastic images captured during fatigue tests performed at different R-ratios. The experimental results lead to the conclusion that modern thermoelastic equipment, like Deltatherm 1500, is a very suitable tool for crack growth analysis or for visualizing experimentally phenomenon like the influence of residual stress fields or crack closure mechanisms on fatigue crack propagation. Tests have been performed on as-received welded and stressrelieved steel specimens. Experimental results for the mode I SIF show a deviation with respect to ideal results that is reduced when the increasing the R-ratio and when using stress-relieved specimens. Results for the fatigue crack growth rate and the SIF range using the computer algorithm have been presented for a stress-relieved specimen, and good agreement has been shown when compared with those results obtained from the original parent material. The phase map obtained with TSA has been also introduced and discussed. Work is also in progress to find a relation between phase information and plastic deformation ahead of the crack tip.

Acknowledgements Some of this work was undertaken as part of a collaborative project with AEA Technology, now Serco Assurance, and M R Goldthorpe and Associates for the Health and Safety Executive. Their assistance is gratefully acknowledged. Other parts of this work were supported by an EPSRC grant (GR/M57712/01) for which the authors are also grateful.

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