Experimental estimation of the heat energy dissipated in a volume surrounding the tip of a fatigue crack

Fatigue crack initiation and propagation involve plastic strains that require some work to be done on the material. Most of this irreversible energy is dissipated as heat and consequently the material temperature increases. The heat being an indicator of the intense plastic strains occurring at the tip of a propagating fatigue crack, when combined with the Neuber’s structural volume concept, it might be used as an experimentally measurable parameter to assess the fatigue damage accumulation rate of cracked components. On the basis of a theoretical model published previously, in this work the heat energy dissipated in a volume surrounding the crack tip is estimated experimentally on the basis of the radial temperature profiles measured by means of an infrared camera. The definition of the structural volume in a fatigue sense is beyond the scope of the present paper. The experimental crack propagation tests were carried out on hot-rolled, 6-mm-thick AISI 304L stainless steel specimens subject to completely reversed axial fatigue loading.


INTRODUCTION
umerical or experimental evaluation of plastic dissipation at the tip of fatigue cracks have attracted the attention of several researchers, who investigated, just as few examples, crack propagation assessment criteria [1,2], the thermal effects on stress intensity factors [3,4], the plastic zone size and energy dissipation [5][6][7].In the field of the experimental approaches, the development of infrared cameras having increased performances (for example in terms of thermal sensitivity, spatial resolution and frame rate) has given impulse to temperature-related fatigue studies.In a previous paper, dealing with fatigue assessment of notches, the heat energy dissipated in a unit volume of material per cycle, Q, has been assumed as a fatigue damage index and a proper experimental procedure has been put forward to estimate the Q parameter at any point of a specimen or a component undergoing fatigue loadings [8].Such experimental technique is based on temperature measurements performed by means of an infrared camera or a thermocouple glued at the point of a component where the fatigue assessment is to be performed and it has the advantage that thermal boundary conditions do not need to be controlled during experimental tests.The Q parameter has been applied to correlate fatigue test results obtained on smooth and bluntly notched specimens made of an AISI 304L stainless steel subjected either to constant amplitude [9,10] and two load level [11] fatigue tests.Being a point-related quantity, Q can hardly correlate fatigue test results generated from severely notched specimens, because the well known notch support effect makes questionable the use of peak quantities (stress-, strain-or energy-based) evaluated at the notch tip in order to assess fatigue life.In particular, the use of peak quantities evaluated at the apex of stress concentrators fails by a large amount in the case N of fatigue cracks.To account for the notch support effect, Peterson postulated that the controlling factor is the stress at the distance of the structural size ahead the notch tip [12]; on the other hand, Neuber introduced the structural volume concept, inside which stresses are to be averaged [13].In view of the extension of the heat energy-based approach to severely notched specimens, a theoretical frame and an experimental procedure have been established in the present paper, by considering a specimen containing a propagating fatigue crack.In particular, the specific heat loss Q has been averaged over a volume V surrounding the tip of the propagating crack, leading to the definition of the averaged energy parameter Q*.The volume V, even though on the order of the size of the structural volume for construction steels, has been chosen arbitrarily, since the focus of the present paper is the thermal problem and not yet the validation of a fatigue assessment method.Q* has been estimated starting from the temperature field measured close to the fatigue crack tip.Experimental temperature distributions have been compared with an analytical solution available in the literature.

THEORETICAL BACKGROUND
n order to derive the energy per cycle dissipated in a volume V surrounding a crack tip, a previous theoretical model [8] has been adopted.Let us consider a material undergoing a fatigue test and consider a control volume V surrounding the crack tip, as shown in Fig. 1.The external surface S of the control volume V can be divided into three parts, namely Scv, Scd and Sir through which the heat Q is transferred to the surroundings by convection, conduction and radiation, respectively.The first law of thermodynamics states: where W is the input mechanical energy and U the variation of the internal energy.All quantities are referred to a unit volume of material per cycle.Eq. ( 1) can be written in terms of mean power exchanged over one loading cycle as: where f L is the frequency of the applied mechanical load, H=H cd +H cv +H ir is the thermal power dissipated by conduction, convention and radiation, respectively,  the material density, c the specific heat and p E  the rate of accumulation of damaging energy in a unit volume of material.Let us consider a plane problem and assume that the temperature of a material undergoing a constant amplitude, sinusoidal fatigue loading is given by:   ) where Ta is the amplitude of temperature oscillations due to the thermoelastic effect and Tm is the mean temperature evolution.T a and T m depend on the position (r,) considered in the component.It is worth noting that the thermoelastic effect consists of a reversible exchange between mechanical and thermal energy, that does not produce a net energy dissipation or absorption over one loading cycle [14][15][16][17].Since Eq. ( 2) considers the rate of energy contributions averaged over one cycle, then only the mean temperature evolution T m (t) appears on the right hand side of Eq. ( 2).The specific net heat generation Hgen is given by: Therefore Eq. ( 2) can be written in order to put into evidence only the thermal problem:

I
Typically, the mean temperature Tm increases at the beginning of a fatigue tests and, after some time, it stabilises as soon as thermal equilibrium is reached with the surroundings.At thermal stabilisation, Tm is constant with time, therefore the last term on the right hand side of Eq. ( 5) disappears.As it will be demonstrated later on, H cd can be supposed much greater than Hcv and Hir, therefore it can be assumed H Hcd.The thermal power extracted from volume V of Fig. 1 by conduction can be calculated starting from the thermal flux through its boundary: ( , ) where gradT m is the gradient of the temperature field T(r,),  is the material thermal conductivity, z is the specimen's constant thickness, dS cd =z•ds (see Fig. 1) and R the radius of the circular volume shown in Fig. 1.In Eq. ( 6), Energy balance for a volume of material V surrounding a crack tip subject to fatigue loadings.
After having calculated the thermal power dissipated by conduction by means of Eq. ( 6), it is possible to estimate the energy per cycle averaged in the volume V: Making use of Eq. ( 6), it is obtained: In previous papers, the specific heat loss Q=H/fL was used as a fatigue damage indicator to perform fatigue assessments [8][9][10][11].Being a point-related quantity, its use was limited to the analysis of the fatigue strength of bluntly notched specimens.Following Neuber's structural volume concept [13], in the present paper the energy term Q is averaged over a control volume V according to Eq. ( 7), in view of the application of the energy-based approach to severely notched or cracked components.However, it should be noted that the aim of the present paper is to formulate the fatigue related thermal problem and validate an approach able to estimate experimentally Q*.Therefore, the control volume surrounding the crack tip of Fig. 1 was fixed arbitrarily and has not a precise relation to the fatigue properties of the material.As it will be shown later on, in order to apply Eq. (7b) the surface material temperature was monitored by means of an infrared camera operating at a sample frequency f acq .In order to estimate T m (r,) at a given time t s during the fatigue test, a trigger signal was given to the infrared camera at t=ts and a number of infrared images nmax were acquired with a sampling rate f acq .By recalling Eq. ( 3), the mean temperature field was estimated as pixel-by-pixel average value: where ts is the time when the temperature acquisition starts.A error index can be defined between the estimated (Tm,estimate) and the actual (Tm) mean temperature field: By using Eq.(8b) into Eq.( 9), the error index results: Eq. ( 10) says that for typical testing conditions adopted in the present work, i.e. f L =37Hz, f acq =200Hz, n max =1000, the relative error  in the estimation of the mean temperature is lower than 0.1%.

MATERIAL, SPECIMENS' GEOMETRY AND TEST PROCEDURE
ingle edge V-notched specimens were machined from a 6-mm-thick hot rolled AISI 304L stainless steel plate (elastic modulus E=194700 MPa, engineering proof stress R p0.2 =327 MPa, engineering tensile strength R m =690 MPa [9]), according to the geometry shown in Fig. 2. Constant amplitude, push-pull stress-controlled fatigue tests were carried out by using a servo-hydraulic Schenck Hydropuls PSA 100 machine equipped with a 100 kN load cell and a Trio Sistemi RT3 digital controller.Load test frequencies between 30 and 37 Hz were adopted.Crack propagation was monitored by using a travelling optical stereo-macroscope operating with a magnification of 40x.The material surface temperature was monitored by means of a FLIR SC7600 infrared camera, having a 1.5-5.1 m spectral response range, 50 mm focal lens, a noise equivalent temperature difference (NETD) < 25 mK, an overall accuracy of 0.05°C, operating at a frame rate, f acq , equal to 200 Hz and equipped with an analog input interface, that was used to sample synchronously the force signal coming from the load cell.The infrared camera and the travelling microscope monitored the opposite surfaces of the specimens, respectively.To increase the infrared camera spatial resolution, a 30 mm extender ring was adopted, which allowed a spatial resolution ranging from 20 to 23 m/pixel, depending on the distance between the specimen's surface and the focal lens.Due to the extender ring, the Field of View (FoV) was reduced to 320x256 pixels, which corresponds to a minimum of 6.4 mm x 5.1 mm and a maximum of 7.4 mm x 5.9 mm.The specimens' surface were polished by using progressively finer emery papers, namely starting from grade 100 up to grade 1000, and after that the surface was polished with a diamond abrasive powder.Finally, a black paint was applied to the specimens' surface to increase the emissivity.The acquired temperature maps were processed first by using the FLIR MotionByInterpolation tool to correct the relative motion between the fixed camera lens and the moving specimen subject to cyclic loads, whose displacements ranged from 6 to 14 pixels, depending on the crack length.The infrared images were analysed by means of the dedicated ALTAIR 5.90.002 software, in order to calculate the mean temperature distribution Tm(r,) at a given time t=ts during the fatigue S test according to Eq. (8a).By using Eq. ( 10), being typically fL=37 Hz, nmax=1000, facq=200Hz and 0< ts < 0.5/fL, it is obtained 10 -3 ; such a reduced error was considered acceptable from an engineering point of view.After having determined the distribution of the mean temperature Tm(r,), the heat power dissipated by conduction was calculated by solving Eq. ( 6) numerically on the basis of a finite number of radial temperature profiles: in particular, 7 radial paths were considered emanating from the crack tip at different  angles (see Fig. 1), namely 0°, 45°, 90°, 135°, -45°,-90° and-135°.In the present paper, the radius R of the volume surrounding the crack tip was assumed equal to 3•10 -4 m.Even though such a radius may be on the order of the Neuber's structural volume size of a construction steel, it has been assumed as a pure reference example in the present paper, in order to demonstrate the applicability of the proposed approach.

FATIGUE TEST RESULTS AND TEMPERATURE PROFILES CLOSE TO THE CRACK TIP
hree specimens were tested and the relevant crack growth data are shown in Fig. 3. Linear elastic, twodimensional, plane stress finite element analyses were performed to evaluate the Mode I stress intensity factor range, K=K max -K min , for different crack lengths.To account for the machine grip effect, displacements were applied in the numerical model to the lines shown in Fig. 2. The Paris curve relating the crack growth rate to K was evaluated and plotted in Fig. 3b.The crack length and the temperature field at different K values were measured at several times t=ts, regularly distributed during each fatigue test.As stated above, 1000 infrared images were acquired at each time ts and then processed according to Eq. (8a).Some typical radial temperature profiles evaluated at =0° are shown in Fig. 4a and 4b, in the case of K=26 MPa•m 0.5 (i.e.K max =13 MPa•m 0.5 ) and K=60 MPa•m 0.5 (i.e.K max =30 MPa•m 0.5 ), respectively.In the former case a temperature drop equal to about 0.8 K within a distance of 2.5 mm from the crack tip can be observed; conversely, in the latter case, the temperature decreases much more, being the drop about 3 K.Therefore, the signal-to-noise ratio is T significantly higher in Fig. 4b than in Fig. 4a.To evaluate the first derivative at r=R (Eq.( 6)), the temperature-distance data along the seven considered paths were fitted using a proper polynomial function, shown with a continuous line in Fig. 4.

ENERGY PER CYCLE AVERAGED IN A VOLUME AT THE CRACK TIP
ig. 5a, 5b and 5c show the specific thermal flux h at the different points along the boundary of V (Fig. 1) for specimen V_3, V_4 and V_5, respectively, using =16 W/(m•K) [8].Finally, Fig. 5d shows, as an example, the specific energy flux per cycle q, obtained simply dividing h by the load test frequency.In the authors' opinion, for the material and the experimental conditions analysed in the present paper, a reasonably accurate evaluation of the heat power can be achieved by considering K values higher than 25 MPa•m 0.5 (K max >12.5 MPa•m 0.5 ).Having in hand the specific thermal flux h evaluated at different angles  of the boundary of V, numerical integration was performed according to Eq. ( 6).To evaluate the errors due to the discretisation, Eq. ( 6) was solved by dividing the 360° angle starting from a minimum of 4 intervals (=90°) to a maximum of 24 (=15°).A 0.51% variation on results was found by using 8 as compared to 24 intervals.Therefore, 8 intervals (=45°) were adopted in numerical calculations.Finally, the energy per cycle averaged in the volume V, Q*, was evaluated by means of Eq. (7b).Results are listed in Tab. 1 and it can be seen that Q* increases as K increases.It should be noted that K are elastically calculated, independently on plastic zone size evaluations.

COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL TEMPERATURES CLOSE TO THE CRACK TIP
n analytical solution is available in order to evaluate the time-dependent temperature field in the case of a homogeneous and isotropic infinite plate with a time-independent heat generation hL distributed along a line in the thickness direction [18].At the time t=0 when the heat generation starts, the temperature is supposed homogeneous and equal to T0. Between time t=0 and t, the temperature variation T(r,t)=T(r,t)-T0 can be expressed by Eq. ( 11) [18]: 2 ( , ) 4 4 where Ei is the integral exponential function given by Since the major source of heat power is the cyclic plastic zone, the linear heat generation hL was applied in its centre, according to [3].Fig. 6a shows the cyclic plastic zone idealised as a circle having radius rp.According to Irwin [20], the cyclic plastic zone radius in the plane stress condition is equal to: A where '   ,02 p  is the material cyclic proof stress.For the AISI 304L steel material analysed in this paper, =7940 kg/m 3 , c=507 J/(kg•K) [19], '   ,02 p  =290 MPa [9].
With the aim to compare experimental results with the analytical solution Eq. ( 11), a dedicated fatigue tests was performed.A specimen containing a crack as long as half the width (i.e. the ligament length was about 23 mm according to Fig. 2) was installed in the fatigue machine to allow for thermal equilibrium with the surroundings so that the homogeneous temperature T 0 could be measured.Then the fatigue test was started with f L =37 Hz and the load was adjusted to apply a linear elastic stress intensity factor range equal to K=36.9MPa•m 0.5 ; therefore rp is equal to 6.44•10 -4 m according to Eq. ( 12).The temperature field as well as the signal coming from the load cell were registered synchronously by the infrared camera using a sampling rate f acq =200 Hz.To disregard the thermoelastic temperature oscillations superimposed to the mean temperature evolution Tm, which are not taken into account in Eq. ( 11), an infrared image captured at a time t=t* was considered, when the applied force was close to zero.Fig. 7 shows the evolution of the temperature averaged inside the plastic zone Vp, T*(t), defined as: where n pixel is the number of pixel inside the cyclic plastic zone size V p .Fig. 7b shows the enlarged view of the "detail A" of Fig. 7a, where the thermoelastic effect superimposed to the mean temperature evolution T* m can be appreciated.Considering the radial temperature profiles that have been measured at t*=1.835 s (the applied force was approximately zero at this time), the total heat generated inside the cyclic plastic zone V p was calculated according to Eqs ( 5) and (6).To evaluate the last contribution on the right hand side of Eq. ( 5) (the internal energy contribution), a linear fit of T* in a time window equal to 1s (Fig. 7b) was done and the slope of T*m(t) was considered.After that, the constant heat generation per unit thickness h L to input in Eq. ( 11) was calculated as: and resulted equal to 63.0+0.4=63.4W/m, where the first contribution is the conduction and the second is the internal energy term.Fig. 6b shows a comparison between the temperature field evaluated according to Eq. ( 11) and the experimental data for =0°.According to [2], it can be seen that outside the cyclic plastic zone the measured temperature field is in good agreement with that evaluated under the hypothesis of linear heat generation, which Eq. ( 11) is based on.
Inside the cyclic plastic zone differences are more pronounced because heat generation is actually distributed inside Vp, while in Eq. ( 11) it has been lumped to the centre of the cyclic plastic zone.where  is the heat transfer coefficient by convection,  is the surface emissivity, n the Stephan-Boltzmann constant and T  the room temperature.By considering, as an example, the experimental conditions of V_5 specimen and K=40.6MPa•m 0.5 , the mean temperature averaged inside V, T*m, was equal to 28.1°C, T  =19°C, =0.92 [8] and fL=35 Hz.By assuming a reasonable heat transfer coefficient under the hypothesis of natural convection on the order of 10 W/(m 2 K) and considering that S cv =S ir =2•R 2 , Q* cv and Q* ir can be calculated by means of Eqs 15 and 7a: The result is Q*cv=867 J/(m 3 •cycle) and Q*ir=472 J/(m 3 •cycle), respectively, that are four order of magnitude lower than the relevant Q* values reported in Tab. 1, that take into account only the conduction term.

CONCLUSIONS
theoretical framework has been defined to estimate the specific heat energy per cycle averaged in a defined volume surrounding the tip of a propagating crack, Q*.A two-dimensional thermal and structural problem has been considered.Experimental tests were executed on AISI 304L stainless steel cracked specimens subjected to push-pull fatigue loads and the Q* parameter has been determined starting from temperature measurements performed in the vicinity of the crack tip by means of an infrared camera.With reference to the material and experimental equipment available in the present paper, a reasonably accurate estimation of Q* was possible only for K max >13 MPa•m 0.5 .The experimental temperatures close to the crack tip were compared successfully with an analytical solution available in the literature.
A A

Figure 3 :
Figure 3: Tension-compression a) crack propagation curves and b) Paris curve of AISI 304 L stainless steel specimens.

Figure 4 :
Figure 4: Typical radial temperature profiles measured during the tension-compression fatigue tests in the case of (a) K=26 MPa•m 0.5 and (b) K =60 MPa•m 0.5 .

FK=30. 3 Figure 5 :
Figure 5: Distribution of the thermal flux h along the boundary of the control volume for different  angles for (a) V_3, (b) V_4 (c) V_5 specimen and (d) and corresponding energy flux per cycle q of V_5 specimen.

Figure 6 :
Figure 6: a) Cyclic plastic zone Vp and (b) comparison between experimental and theoretical radial temperature profile.

Figure 7
Figure 7: a) Temperature evolution during a fatigue test and b) enlarged view of detail A

Table 1 :
Q* values calculated for different specimens at different K values.