Xavier Fayolle

François Hild
email: hild@lmt.ens-cachan.fr

Controlling Stress Intensity Factor Histories with Digital Images

Keywords: Digital image correlation, fracture mechanics, full eld measurements, measurement resolution

It is shown that stress intensity factors (SIF) can be prescribed in mechanical tests by using digital image correlation. A priori resolutions are validated with a posteriori results. A cascade controller is implemented to prescribe SIF histories. A proof of concept is shown when testing a cracked elastomer. A tensile experiment is nally analyzed with a center cracked sheet made of commercially pure titanium. The linearity error is shown to be less than one order of magnitude higher for the closedloop system when compared to the SIF resolution. However, it is still small compared to the SIF ranges investigated herein.

Introduction

The study of cracks is part of the theories describing the degradation of materials and failure of structures [START_REF] Lemaitre | Mechanics of Solid Materials[END_REF]. Crack initiation, propagation and arrest are concepts that are important to master to design and monitor actual structures such as airplanes, cars, trains, or power plants [START_REF] Kanninen | Advanced Fracture Mechanics[END_REF]. In many cases, elastic approaches are still used to assess the criticality of cracks. One way of mechanically characterizing cracks is to consider stress intensity factors (SIFs) that describe stress singularities in the vicinity of the crack tip, or equivalently, displacement discontinuities across crack faces [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF]. It is the latter property that will be used hereafter in conjunction with full-eld measurements.

Performing experiments on cracked samples or structures is a dicult task since crack propagation may become unstable. This is particularly true when load-controlled experiments are carried out since the SIF is generally increasing with the crack size.

Conversely, when displacement controlled, the tests are usually easier to perform since crack propagation is likely to be stable. However, propagation may stop. The reason is that the parameters that control crack propagation are not necessarily the applied load or displacement, but crack opening displacements (CODs) [START_REF] Wells | Unstable crack propagation in metals: cleavage and fast fracture[END_REF][START_REF] Rice | Mechanics of crack tip deformation and extension by fatigue[END_REF], SIFs [START_REF] Irwin | Analysis of the Stresses and Strains near the End of a Crack Traversing a Plate[END_REF], energy release rates [START_REF] Grith | The Phenomenon of Rupture and Flow in Solids[END_REF] or J-integral [810].

Except for CODs for which displacement transducers (so-called crack gauges) are used on large structures (e.g., bridges, buildings) or COD (clip) gauges on smaller ones, the other quantities are only indirectly evaluated when using point data, provided the crack length is known or monitored. Load shedding techniques whereby the applied load is gradually decreased as the crack propagates [START_REF] Zuidema | A model for predicting slant fatigue crack growth in A1 2024[END_REF] have been proposed to keep the nonlinearities conned in the vicinity of the crack tip. However, the crack tip position needs to be determined to relate the applied load to the SIF level. Conversely, ! the sample geometry itself can be adapted to partially compensate for the load vs. SIF increase (e.g., edge-cracked tapered plate [START_REF] Srawley | Stress Intensity Factors for Crackline-Loaded Edge-Crack Specimens[END_REF]). Yet another route consists of a linearly coupled load and displacement control loop that does not require the exact crack tip position to be determined [START_REF] Pulos | Constant stress intensity factors through closed-loop control[END_REF]. One key parameter to tune is the so-called mixing ratio between displacement and remote load.

An alternative route is provided by full-eld measurement techniques, in particular Digital Image Correlation (DIC [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF][START_REF] Hild | Digital Image Correlation[END_REF]). Besides the qualitative aspect of detecting a crack, quantitative estimates of mechanically signicant properties have been reported such as opening loads in fatigue experiments [START_REF] Riddel | Determining fatigue crack opening loads from near-crack-tip displacement measurements[END_REF][START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF] and/or SIFs [1822] by least squares t using closed-form solutions [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF]. However, all these analyses were performed a posteriori and not during the experiment itself. Pictures have already been used to drive experiments. For instance, by tracking points, average strains are controlled [START_REF] Sell | Video-Controlled Tensile Testing of Polymers and Metals Beyond the Necking Point[END_REF]. This is also possible by resorting to DIC [START_REF] Fayolle | Controlling testing machines with digital image correlation[END_REF]. In both cases, experiments could be performed in the presence of localized strains. The advantage of DIC is that it is, by construction, a full-eld measurement technique that was validated to analyze cracked media. In particular, its resolution may reach small values (typically 1 cpixel or even lower), which was shown to be compatible with the uncertainty level of strain gauges [START_REF] Fayolle | Controlling testing machines with digital image correlation[END_REF].

DIC is appealing when trying to control experiments, in particular when the input signals cannot be given by a strain gauge, an extensometer or a transducer (e.g., measuring a displacement or a load). DIC has already been used to control cracking experiments [START_REF] Fayolle | IC : programme de pilotage d'essais asservis sur un facteur d'intensité des contraintes[END_REF][START_REF] Durif | Controlling Stress Intensity Factors During a Fatigue Crack Propagation Using Digital Image Correlation and a Load Shedding Procedure[END_REF]. Either post-processed DIC measurements [START_REF] Fayolle | IC : programme de pilotage d'essais asservis sur un facteur d'intensité des contraintes[END_REF] or integrated DIC [START_REF] Durif | Controlling Stress Intensity Factors During a Fatigue Crack Propagation Using Digital Image Correlation and a Load Shedding Procedure[END_REF] have been used to evaluate the SIF history. In the present paper, the former approach is used to control experiments in which the crack does not propagate. The goal is to assess the resolution of the closed-loop system and to prove the feasibility " on two very dierent materials (i.e., a very soft elastomer and a metallic material). To design the closed-loop software, the SIF resolution has to be assessed. This procedure is performed by rst evaluating the displacement resolution of the implemented DIC code. Thereafter, it is possible to assess the resolution of SIFs. The performance of the controller is analyzed when an edge-cracked plate made of a soft material (i.e., elastomer) is tested. Last, experimental results are analyzed when an SIF history is prescribed on a center cracked sheet made of commercially pure titanium. In both cases, the a priori resolutions are compared with linearity errors for the open-loop and closed loop congurations.

SIF Resolution

When designing a closed-loop system, one key quantity is the resolution of the measuring device of the controlled variable (i.e., the SIF in the present case). To evaluate the SIF resolution, the displacement resolution is needed. It is based upon the analysis of the noise sensitivity of DIC [START_REF]ISO, International Vocabulary of Basic and General Terms in Metrology (VIM)[END_REF][START_REF]ISO, Guide to the Expression of Uncertainty in Measurements (GUM)[END_REF]. The SIF resolution is then obtained by propagating the uncertainty induced by the least squares procedure to evaluate the mode I SIF from the measured displacements.

Displacement Resolution

In practice, the region of interest (ROI) is selected for the SIF gauge. Within the ROI, smaller correlation windows or zones of interest (ZOIs) are chosen. For each ZOI in the reference image, its corresponding position is sought in subsequent pictures.

The pattern-matching algorithm is based on cross-correlation. The cross-correlation is a function giving the maximum likelihood between two ZOIs shifted by a constant displacement. Its computation is performed via fast Fourier transforms [2932]. The location of this maximum gives the most likely displacement between the ZOI in the reference and deformed congurations. For a sub-pixel estimate, a parabolic interpolation is performed around the maximum pixel value and the displacement corresponds to the maximum of the interpolated correlation function. Then the ZOI in the deformed conguration is shifted by an amount equal to the evaluated displacement using the shift/modulation property in Fourier space. Since interpolation errors occur, these three steps need to be iterated until convergence is reached. In the present case, since very small displacement increments are sought, a constant displacement assumption for each ZOI is deemed to be sucient [START_REF] Fayolle | Controlling testing machines with digital image correlation[END_REF].

An estimate of the displacement resolution of the correlation algorithm is needed.

The following procedure is applied to the actual picture of the experiment with the elastomeric material. The CCD digital camera used herein has a 1280 × 1024-pixel denition and a 12-bit dynamic range. Telecentric lenses are used to minimize the (spurious) eects of out-of-plane motion. To have independent estimates of the displacements in the following analyses, the shift δ between consecutive ZOIs is at least equal to the ZOI size ℓ. The noise level of the camera is of the order of σ = 20 gray levels. By following the derivations of Ref. [START_REF] Hild | Comparison of local and global approaches to digital image correlation[END_REF], the corresponding standard displacement resolution is expressed as

σu = √ 2σp G f ℓ (1)
where G f is the root mean square (RMS) value of the picture gradient (i.e., its mean contrast), p the physical size of one pixel.

These a priori results are compared with actual (i.e., a posteriori) measurements for which a set of 10 pictures is acquired for the same conguration. It was checked that the mean displacement was very small for all the analyzed cases. Figure 1 shows the standard displacement resolution σu as a function of the ZOI size ℓ; the larger the ZOI size, the smaller the resolution σu. However, if the ZOI size increases, the spatial resolution is degraded, i.e., less independent measurement points are obtained. The ZOI size also needs to be adapted to the speckle size (here of the order of ≈ 2 pixels when characterized by an average correlation radius [START_REF] Bergonnier | Digital image correlation used for mechanical tests on crimped glass wool samples[END_REF], i.e., suciently small with respect to ℓ). Furthermore, as the ZOI size increases, so does the computation time. A good compromise for the present study is given by a ZOI size of 32 pixels for which a standard displacement resolution of the order of 2 × 10 -3 pixel is achieved.

Last, when interpolating the displacement resolution σu by A/ℓ, a value A = 7.4 × 10 -2 is found, to be compared to

√ 2σ/G f = 8.0 × 10 -2 (according to Equation (1)
when σu and ℓ are expressed in pixels). These two levels being close, the present results validate the a priori estimate of the standard displacement resolution σu. The low level of the displacement resolution is obtained thanks to a mean contrast G f of the order of 355 gray levels / pixel.

SIF Resolution: A Priori Estimate

The displacement eld u in an innite solid containing a semi-innite crack along the

x-axis (x < 0) is written as a combination of a double semi-innite series [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF]. In the following, only the dominant term associated with the mode I SIF is considered under a plane stress hypothesis (i.e., the considered samples have small thicknesses compared with the other dimensions)

ux = K I E √ r 2π cos θ 2 [3 -ν -(1 + ν) cos θ] uy = K I E √ r 2π sin θ 2 [3 -ν -(1 + ν) cos θ] (2)
where r, θ denote the polar coordinates with respect to the crack tip, x, y the cartesian coordinates with respect to the crack tip, E Young's modulus, and ν Poisson's ratio.

This displacement eld is rewritten as

u = K I υ K (3)
where υ K corresponds to the displacement eld of a unitary mode I SIF.

Let us now assume that the displacement is known at a number of discrete points xn

∑ i a i υ i (xn)∥ 2 (4)
where {a} gathers the unknown translations u 0x , u 0y , rotation ω, and SIF K I . A linear system is solved

[T ]{a} = {b} (5)
with

T ij = ∑ n υ i (xn) • υ j (xn) , b i = ∑ n υ i (xn) • um(xn) (6)
& It was mentioned that displacements are corrupted by measurement noise, ηu, which is considered as white in the present case since each ZOI is independent of the other ones [START_REF] Bergonnier | Digital image correlation used for mechanical tests on crimped glass wool samples[END_REF]. Consequently, the optimal SIF extractor is given by the least squares procedure [START_REF] Réthoré | Noise-robust Stress Intensity Factor Determination from Kinematic Field Measurements[END_REF]. Moreover, it was checked that the acquisition noise was a Gaussian distribution. It is of interest to investigate the eect of uncertainties on the resulting evaluation of {a}. By considering the linear system to be solved for {a}, it is observed that no systematic error (or bias) will result from the noise, and the covariance matrix

[C] reads [C] = [T ] -1 σ 2 u (7)
When the ROI is centered about the crack tip, the only o diagonal term of [T ] is associated with υx • υ K (xn). A rst order estimate of the diagonal terms of the covariance matrix is given by (no index summation)

C ii ≈ σ 2 u /T ii (8)
which consists of neglecting any cross correlation between the various amplitudes (i.e.,

C ij ≪ C ii , when i ̸ = j). Further, the following approximation is used to assess C ii , which corresponds to the variance σ 2 i of amplitude a i , the other parameters being xed,

1 nxny ∑ n ∥υ i (xn)∥ 2 ≈ 1 πR 2 max ∫ Rmax 0 ∫ 2π 0 [υ 2 ix (r, θ) + υ 2 iy (r, θ)] rdrdθ (9)
where Rmax is determined by assuming a surface equivalence of the ROI πR 2 max = LxLy [START_REF] Rice | A Path Independent Integral and Approximate Analysis of Strain Concentrations by Notches and Cracks[END_REF] where Lx and Ly denote the width and height of the ROI. The centers of the ZOIs correspond to the collocation points of the approximation of the previous integral. The standard SIF resolution σ K is normalized to become

σ K √ nxnyRmax Eσu ≈ √ 6π 19 -10ν + 3ν 2 (11)
When the Poisson's ratio ν varies from -1 to 0.5, the approximation to the dimensionless standard SIF resolution ranges from ≈ 0.77 to ≈ 1.13 (see Figure 2), i.e., of the order of unity (since √ 6π/19 ≈ 1). Figure 2 shows that for small numbers of ZOIs (ranging from nx = ny = 2 to 6), the proposed approximation is a good rst order estimate. It can be shown that when the number of ZOIs becomes very large, the present approximations lead to an underestimation of the order of 20% of the standard SIF resolution.

Equation ( 11) also shows that the higher the Young's modulus E, the larger σ K .

Further, the larger Rmax, the lower σ K . However, some care should be exercised since increasing Rmax may induce a ROI that no longer belongs to the K-dominant region so that the chosen kinematic basis is no longer relevant. This can be checked by assessing the identication error

η = √ T ({a}) nxny (12)
The closer η to σu, the better the identication. It is also possible to decrease σ K by increasing nxny, which is accompanied by an increase in computation time. A nal way is to decrease the displacement resolution σu. This is achieved by considering larger ZOIs, again inducing a higher computation cost. This last discussion shows that the SIF resolution σ K is the result of a trade-o between the resolution level and the computation time. This is particularly important for a closed-loop system in which the reference signal is obtained by the analysis of digital images. By using the results of the previous section, the SIF resolution becomes

σ K √ nxnyRmax E ∝ √ 2σp G f ℓ (13)
Equation ( 13) summarizes the dierent compromises that have to be considered when designing a controller with a DIC-based evaluation of SIFs. The left hand side term

σ K √
nxnyRmax/E is related to the sought quantity itself, which is associated with analysis of nxny points in a ROI of characteristic radius Rmax of a material with Young's modulus E. The second parameter, √ 2σp/G f ℓ, is the displacement resolution, which is related to the analyzed texture (i.e., its mean contrast G f ) by a given camera (of acquisition noise σ) with a chosen magnication (i.e., p), and to the spatial resolution ℓ. It is worth noting that the SIF resolution is valid when the crack tip is centered in the ROI. If this hypothesis is not satised, there are additional correlations that need to be accounted for. However, for the sake of simplicity, they are not considered in the following analyses.

Last, the fact that only the very rst mode I displacement eld associated with the presence of a crack is chosen is related to the limited number of measurement points, themselves linked to speed of the controller. This choice implicitly assumes that the ROI belongs to the K-dominant area and that a pure mode I loading is achieved. These hypotheses may be checked a posteriori with a more complete analysis considering more terms of Williams' series [START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF][START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF][START_REF] Mathieu | Identication of a crack propagation law by digital image correlation[END_REF] and more measurement points.

Furthermore, the estimate of the stress intensity value may be biased as additional terms of Williams' series are needed to describe the measured displacement eld. The level of the identication error is one means of checking whether the considered basis is sucient when compared to the displacement resolution.

Controlling SIF Histories via DIC

The general setting of the controller is identical to the one that was used to prescribe strain histories to a sample [START_REF] Fayolle | Controlling testing machines with digital image correlation[END_REF], namely, a two-loop system is implemented to allow a slow outer loop to compute the current value of the SIF to be compared to the prescribed level. One cycle consists of an image acquisition, image saving (on the hard disk) and the evaluation time of the SIF through DIC calculations. It requires a total time of 2.4 s. This is not a limitation per se as the use of GPUs may signicantly reduce this time [START_REF] Leclerc | Integrated digital image correlation for the identication of mechanical properties[END_REF]. The aim here is to show the feasibility of SIF control, more than `real time' issues. The inner loop consists of a displacement prescribed signal with a frequency of 20 Hz compatible with electromechanical (and servohydraulic) testing machines.

In the present application, the input signal is an SIF. In the following, the loading history is given by linear ramps loading and unloading the crack. To cancel out the steady state error between the input signal (i.e., a ramp) and the measured SIF (feedback), a double integral controller is implemented in a cascade system (Figure 3).

However, a double integral controller, by itself, is unstable. A zero is added in the low frequency range to raise the phase by up to 45 • , thereby yielding a sucient phase margin (from the 180 • limit). Last, a unitary open loop gain is sought so that SIF signals are directly controlled.

Contrary to what was performed previously (i.e., using a servohydraulic testing machine [START_REF] Fayolle | Controlling testing machines with digital image correlation[END_REF]), an electromechanical testing machine is chosen. The machine manufacturer gives a rise time of 50 ms for 10 µm displacement increments. To avoid accounting for the frequency response of the testing machine itself the computer process is delayed by 300-ms hold time (i.e., only the steady state response is needed). The consequence is that the total cycle time of the outer loop is 2.7 s (rather than 2.4 s). The testing machine is modeled as a sti equation (i.e., no Laplace function). To check various aspects of the controller depicted in Figure 3 and to estimate the zero value of the controller Simulink [START_REF] Mathworks | Simulation and Model-Based Simulation, version R2012b[END_REF] is used. Figure 4 shows the error (E) and reference (R) signals.

Apart from the changes of slope that induce high (unavoidable) errors, the system has a behavior that is consistent with what was expected (e.g., zero steady state error for ramps).

One important aspect of the controller is related to gain G 0 that links the prescribed SIF (outer loop) with the displacement of the testing machine crosshead (inner loop).

To compute G 0 , the total displacement associated with a sample of initial length L 0 , cracked at its mid-height is needed. The edge crack size is denoted by a and the width of the sample by b. The relative displacement ∆up of the two L 0 -apart sections reads [START_REF] Tada | The Stress Analysis of Cracks Handbook[END_REF] ∆up

= L 0 Σ E ( 1 + 4av 2 L 0 ) ( 14
)
where Σ is the remote stress, and v 2 a dimensionless parameter dependent upon a/b.

The applied stress Σ is related to the SIF K I by

K I = Y Σ √ πa ( 15
)
where Y is a dimensionless parameter dependent upon a/b [START_REF] Tada | The Stress Analysis of Cracks Handbook[END_REF]. By combining these two expressions, gain G 0 becomes

G 0 = ∆up K I = L 0 Y E √ πa ( 1 + 4av 2 L 0 ) (16)
All the parameters dening the SIF controller have now been introduced. The following section is a proof of concept of the implemented controller in a LabVIEW TM environment [START_REF]LabVIEW T M , user's manual[END_REF].

4 Proof of Concept

Studied Material

Being a feasibility study, the implemented controller is used to perform a test on a cracked elastomer. The displacement range was kept suciently low to avoid various ! non linear aspects associated with elastomers [START_REF] Ward | An introduction to the mechanical properties of solid polymers[END_REF]. To evaluate the elastic parameters of the studied material, a tensile test on a dogbone sample has been carried out. Digital images were acquired during the test and the same DIC code was used to evaluate the mean longitudinal and transverse strains. The strain resolution σϵ is determined by following the same type of derivation as for the SIF resolution. The only dierence is given by the interpolating displacement eld, which is linear (i.e., three unknowns per direction)

σϵ xx = σu Lx √ nxny √ 12(nx -1) nx + 1 , σϵ yy = σu Ly √ nxny √ 12(ny -1) ny + 1 (17)
with Lx = δ(nx -1) and Ly = δ(ny -1). With the chosen parameters (i.e., Ly > Lx = 350 pixels, ℓ = 32 pixels, and nx = ny = 4) the strain resolution is less than 5 × 10 -6 .

Figure 5 shows the stress/strain response of the material. As explained above, the stress and strain levels remain very low. By using a least squares t of the measured data, a Young's modulus E = 7.6 MPa and Poisson's ratio ν = 0.4 are obtained.

Further, most of the uctuations observed in Figure 5 are due to the load cell (whose resolution is of the order of 1 N or 0.02 MPa) and not the DIC code since the strain range is ten thousand times the corresponding resolution.

A Posteriori Validation of Strain and SIF Resolutions

With the experimental conguration, the strain resolution is less than 5 × 10 -6 and the SIF resolution is evaluated as σ K = 2 × 10 -6 MPa √ m, since the physical size of one pixel is p = 53 µm, Rmax = 275 pixels, and ℓ = 32 pixels. When performing the strain and SIF analyses on the same set of pictures as those used in the displacement resolution analysis, it is found that the strain resolution is of the order of 4 × 10 -6 and the SIF resolution σ K = 2.1 × 10 -6 MPa √ m. These two values are very close to the a " priori estimates proposed herein. Consequently, the a priori displacement, strain and SIF resolutions are validated a posteriori.

Open-Loop Response

The response of the open-loop controller is shown in Figure 6. 137 pictures were acquired. Except for the changes of slope, the RMS dierence with a piecewise linear response is of the order of 1.4 × 10 -5 MPa √ m. This value can be compared to the SIF resolution given above. There is a factor 7 between the two results.

To explain this dierence, there are three types of cause to be listed. First, the resolution analysis performed herein does not account for sub-pixel interpolation errors [START_REF] Bergonnier | Digital image correlation used for mechanical tests on crimped glass wool samples[END_REF]. In the present case, it is expected that they are non negligible since the displacement resolution is found to be very small. Second, the linearity error is also due to the crosshead displacement, which itself has its own resolution and errors. Last, the choice of only 4 displacement elds induces identication and model errors of the SIF level. It is expected that the model error be very small because of the low level of SIF. Conversely, the identication error is expected to increase with the SIF level. This trend was experimentally observed when the change of η against the SIF was analyzed.

These results show that the resolution analysis provides a lower bound to the actual resolution of the open-loop system. The controller itself induces its own error, which will be assessed in the next section.

Closed-Loop Response

Figure 7 shows the input, output and error signals of the implemented feedback system.

From these results it is concluded that such a system is able to drive an SIF history with a very small error (Figure 7(b)) for SIF amplitudes ranging from 0.008 to 0.04 MPa √ m (Figure 7(a)). The errors are the highest at slope changes, as expected from the simulated system (Figure 4). Their absolute level is less than 5 × 10 -4 MPa √ m, i.e., one hundred times less the maximum SIF level, the SIF history was indeed controlled in a very satisfactory manner. 1437 pictures were acquired during this experiment.

On a more quantitative basis, the RMS error is equal to 8×10 the controller contributes to about 60 % of the total error. This level is to be expected from a standard closed-loop system.

Test on Commercially Pure Titanium

Having shown that the designed controller is able to accurately prescribe SIF histories on a soft material, the nal step of the present study is to perform the same type of experiment on a stier material (i.e., with a Young's modulus in the 100-GPa range).

In the present case, commercially pure titanium (i.e., T35) is chosen. This type of material has already been studied to estimate crack propagation parameters [START_REF] Mathieu | Identication of a crack propagation law by digital image correlation[END_REF][START_REF] Mathieu | Image-based Identication procedure of a crack propagation law[END_REF].

The sample has a CCT geometry with a width of 30 mm, a height of 150 mm, and thickness of 0.3 mm. The initial crack size is equal to 6 mm. In the present case, it is believed that the plane stress solution is a good approximation. ). This is due to the fact that the overall level of the latter is signicantly higher. The RMS error with respect to linearity is found to be equal to 9.2 × 10 -2 MPa √ m. Being more than 60 times lower than the minimum prescribed amplitude, the implemented system can be used to control cyclic tests with prescribed SIF histories For the chosen (T35) grade, the Young's modulus is equal to 100 GPa and Poisson's ratio ν = 0.33 [START_REF] Mathieu | Identication of a crack propagation law by digital image correlation[END_REF]. With these elastic parameters, the a priori SIF resolution becomes

σ K = 1.4 × 10 -2 MPa √ m (18)
since the physical size of one pixel is p = 13.4 µm, Rmax = 400 pixels, and ℓ = 32 pixels.

For the open-loop system, an RMS error with respect to linearity is found to be equal to

4.1×10 -2
MPa √ m (to be compared to 9.2×10 -2 MPa √ m for the closed-loop system).

In the present case, there is a factor 3 dierence between the a priori SIF resolution and the RMS linearity error of the open-loop response, and a factor 7 dierence between the SIF resolution and the RMS error of the SIF controller. With the chosen SIF amplitudes whose level is close to SIF threshold in high cycle fatigue [START_REF] Mathieu | Identication of a crack propagation law by digital image correlation[END_REF], it can be concluded that the controller can be used to monitor crack opening during fatigue cycles with SIF amplitudes greater than a few MPa √ m.

Figure 9 shows the change of the measured load with respect to the prescribed SIF. There is a linear relationship between these two quantities, thereby validating the small scale yielding hypothesis. When measuring the RMS error to linearity, it is equal to 9.5 × 10 -2 MPa √ m when expressed in terms of SIF, or 1.7 N in terms of load. The former is very close to the RMS error observed for the controller itself. The latter is of the order of what is expected from the used load cell (see Figure 5). This last result validates the whole procedure developed herein.

Summary

It was shown that it is possible to control in a mechanical test a stress intensity factor (SIF) history by resorting to digital images. In the present case, a local approach to digital image correlation is considered to measure the displacement eld. The latter is then post-processed to evaluate the rigid body motions of the crack tip and the mode I stress intensity factor (SIF). Since this procedure requires almost 3 seconds per control cycle, a two-loop system is implemented. The inner loop is displacement-controlled and the outer loop is SIF-controlled.

Resolution analyses were performed to design the controller. A priori estimates of displacement, strain and SIF resolutions were validated a posteriori by analyzing a series of ten pictures for which no displacements were prescribed. The SIF resolution was then compared with the linearity error of the open-loop and closed-loop system.

There is almost an order of magnitude dierence when the former is compared with the latter ones. This result is to be expected as it includes not only measurement uncertainties but also other aspects of the controller. With the two studied cases, it was shown that the SIF range is signicantly larger (i.e., almost two orders of magnitude) compared with the linearity error of the controller. It is concluded that the SIF histories prescribed herein are suciently accurate to monitor gradual crack openings and closures in cyclic tests.

Having validated the implemented controller, the next step is to consider crack propagation. This step requires an accurate estimation of the crack tip position, which

was not discussed herein. The same procedure as that proposed by Hamam et al. [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF],

which is based on canceling out the amplitude of the rst supersingular displacement eld, will be considered. As already mentioned, additional displacement elds accounting for, say the T-stress or other subsingular contributions [START_REF] Henninger | Enriched kinematic elds of cracked structures[END_REF][START_REF] Mathieu | Identication of a crack propagation law by digital image correlation[END_REF], may also be added.

They will yield a more robust estimate of the stress intensity factor. These issues being solved, gradual propagation (e.g., in high cycle fatigue) will enable the controller to be used in a `real time' conguration since the crack tip motion per cycle is expected to be very small so that the determination of its position will be needed only at large time increments.

  (i.e., the center of each ZOI) with n = 1, . . . , nxny close to the crack tip, where nx, ny denote the number of ZOIs in the x and y directions. The measured displacements are denoted by um. The identication basis consists of elds associated with rigid body motions (i.e., translations v 1 , v 2 , rotation v 3 (xn)) and the mode I SIF eld v 4 (xn) = υ K (xn). The corresponding amplitudes are determined by minimization of the quadratic cost function T T ({a}) = ∑ n ∥um(xn) -

- 5

 5 MPa √ m when the whole history is analyzed. The latter includes the 11 slope changes of the SIF history (Figure 7(a)). When these changes are not considered, the RMS error decreases by a factor of 3 to reach 2.7 × 10 -5 MPa √ m. This last value includes the resolution of the DIC measurements, the SIF extraction and the performance of the controller itself. When compared with the a priori SIF resolution (σ K = 2 × 10 -6 MPa √ m), and that obtained with an open-loop conguration (σ K = 10 -5 MPa √ m)), it is concluded that

Figure 8 (

 8 Figure 8(a) shows the input and output signals for the four applied cycles when the SIF range varies between 6 and 9 MPa √ m.In the present case, 563 pictures are acquired. The overall response of the controller is deemed satisfactory as no particular deviations are observed. Contrary to the previous case, the change of slope is not accompanied by a higher error (Figure8(b)). This is due to the fact that the overall

List of Figures 1 24 2 26 4 27 5 28 6 29 7 31 9Fig. 1 Fig. 2 1 Fig. 3 BlockFig. 4 Fig. 5 2 )Fig. 6 Fig. 7 Fig. 8 (Fig. 9

 124262728293112134526789 Fig.1Standard displacement resolution σu as a function of the ZOI size ℓ. The symbols correspond to the analysis with the reference picture of the SIF-controlled test on the studied elastomer. The solid line is the interpolation given by Equation[START_REF] Lemaitre | Mechanics of Solid Materials[END_REF]

Acknowledgments

This work was partially supported by a grant from Région Île-de-France (plateforme francilienne d'expérimentation mécanique de troisième génération). The authors thank Dr. Roux for useful discussions during the project.


Wahyoe Soedarmono
email: wahyoe.soedarmono@ssb.ac.id

Amine Tarazi
email: amine.tarazi@unilim.fr

Market structure, financial intermediation and riskiness of banks: Evidence from Asia Pacific

Keywords: Bank competition, loan growth, risk, market discipline, Asia Pacific JEL Classification: G21, G28

From a sample of commercial banks in Asia Pacific over the 1994-2009 period, this study highlights that banks in less competitive markets exhibit lower loan growth and higher instability. Such instability is further followed by a decline in deposit growth, suggesting that Asian banks are also subject to indirect market discipline mechanisms through bank market structure. This study therefore sheds light on the importance of enhancing bank competition to overcome bank risk and strengthen financial intermediation. This study also advocates greater reliance on market discipline to promote bank stability.

Introduction

In the wake of the global financial crisis of 2008 that has resulted in a cyclical downturn around the world, bank regulators are concerned with two major conflicting objectives. These include boosting financial intermediation that spurs economic growth or enhancing bank stability that may reduce banks' incentive to grant new loans.

Meanwhile, some studies argue that banks in competitive markets tend to behave imprudently by taking too much risk due to a decline in their charter value as a selfdisciplining factor. However, it is also admitted that higher bank competition may enhance bank efficiency and reduce intermediation costs which in turn contribute to enhance economic growth (e.g. [START_REF] Brissimis | Exploring the nexus between banking sector reform and performance: Evidence from newly acceded EU countries[END_REF][START_REF] Demirgüç-Kunt | Market discipline and deposit insurance[END_REF][START_REF] Fernandez | How institutions and regulation shape the influence of bank concentration on economic growth: International evidence[END_REF].

In the meantime, to our best knowledge, there is no study that examines the implication of bank market structure on bank stability and financial intermediation simultaneously. Accordingly, financial stability and economic growth are often considered as two separate dimensions. As a matter of fact, bank stability plays an important role in economic development because higher bank riskiness can deteriorate banks' ability to grant loans, for example when losses are too important and lead to bank capital shortfalls. This problem is often referred to as "capital crunch" (e.g. [START_REF] Bernanke | Credit crunch[END_REF]Peek and Rosengren, 1995). Given that bank stability and financial intermediation are interrelated, this paper examines how bank competition affects both lending behaviour and stability in banking.

In this paper, we focus on commercial banks in Asia Pacific for several reasons.

First, the banking sector remains a major source of financing for the real sector in Asia Pacific [START_REF] Adams | Emerging East Asian banking system: Ten years after the 1997/98 crisis[END_REF]. Second, substantial consolidations in the banking industry also occurred in Asia Pacific after the 1997 Asian crisis. Specifically, bank consolidations have grown significantly reaching 25% per year as of 2003 [START_REF] Santoso | Comment on "The Determinants of Cross-Border Merger and Acquisition Activity in the Banking Sector in Asia: Did the Asian Financial Crisis Change Them?[END_REF][START_REF] Soedarmono | Bank competition, crisis and risk taking: Evidence from emerging markets in Asia[END_REF]. In addition, foreign direct investment (FDI) in banking also has increased from US$ 2.5 billion over the 1991-1995 period to US$ 67.5 billion over the [2001][2002][2003][2004][2005] period and hence, Asian countries have become the second largest recipient of FDI in banking after Latin American countries [START_REF] Domanski | Foreign banks in emerging economies: Changing players changing issues[END_REF][START_REF] Moshirian | Financial services in an increasingly integrated global financial market[END_REF][START_REF] Soedarmono | Bank competition, crisis and risk taking: Evidence from emerging markets in Asia[END_REF]. These financial landscape changes in Asia Pacific are more likely to affect bank market structure which can in turn alter the behavior of banks in terms of lending activities and risk taking.

For such reasons, focusing on Asia Pacific countries is particularly relevant to study the implication of bank competition on both bank stability and financial intermediation. To assess these issues, we retrieve a sample of commercial banks from 12 countries in Asia Pacific that have been affected by the 1997 crisis. These countries consist of Indonesia, Malaysia, South Korea and Thailand that have been severely affected by the 1997 crisis; as well as India, Pakistan, Philippines, Sri Lanka, China, Hong Kong, Taiwan and Vietnam that were less affected. This study covers the 1994-2009 period and emphasizes on the use of dynamic panel data techniques.

Our empirical findings show that higher bank competition strengthens both bank loan growth and stability. Furthermore, we also embrace discussions on the role of market discipline, since Shaeck and Cihak (2007) document that market discipline is likely to play a role in banks in competitive markets. In other words, competing banks tend to behave prudently by holding higher capital ratios to gain competitive advantage.

For a similar reason, in order to gain competitive advantage, banks in competitive markets can also be more transparent that those in less competitive markets. Hence, higher bank transparency facilitates ex-post market discipline by bank depositors whenever bank risk increases via changes in the market structure of banks. Specifically, our empirical results show that higher market power in banking increases bank risk, while banks in less competitive markets also experience lower deposit growth as a negative reaction of bank depositors. Such findings are consistent with the effectiveness of market discipline mechanisms in commercial banks in Asia Pacific.

The rest of this paper is structured as follows. Aside from highlighting our research contribution, Section 2 describes prior literature on the bank competitionstability nexus, as well as the bank competition-growth nexus. Section 3 presents our data, variables of interest and econometric models. Section 4 discuses our empirical findings and provides some robustness checks, while Section 5 concludes the paper.

Related literature and research contribution

Bank market structure plays an important role in affecting bank stability and financial intermediation. However, the impact of bank market structure on bank stability and financial intermediation remains an open research area. With regards to the link between bank market structure and stability, two major hypotheses have emerged in the literature, i.e. the "charter value hypothesis" and the "competition-stability hypothesis".

Regarding the impact of bank market structure on economic development, there are also two conflicting hypotheses: the "perfect information hypothesis" and the "asymmetric information hypothesis".

The charter value hypothesis argues that bank charter value is a self-disciplining factor that prevents banks from behaving imprudently and thus, higher bank competition eroding bank charter value can contribute to higher instability due to higher bank risk taking. [START_REF] Keeley | Deposit insurance, risk and market power in banking[END_REF] documents that after financial liberalization in the nineties, US banks exhibited a decline in their charter value which in turn increased bank risk.

Similarly, [START_REF] Demsetz | Banks with something to lose: the disciplinary role of franchise value[END_REF] document that US banks with higher charter value tend to have better asset quality. [START_REF] Levy-Yeyati | Concentration and foreign penetration in Latin American banking sectors: Impact on competition and risk[END_REF] also support the charter value hypothesis in the case of Latin America. [START_REF] Beck | Bank concentration, competition and crises: First results[END_REF] also examine the effect of bank concentration on the probability of banking crises in 69 countries during the 1980-1997 period. Their findings show that the probability of banking crises in countries with a concentrated banking system is lower than in countries with a competitive banking system. More recently, [START_REF] Ariss | On the implications of market power in banking: Evidence from developing coutnries[END_REF] has focused on the case of developing countries and supports the charter value hypothesis.

In contrast, recent theoretical and empirical studies demonstrate that higher bank competition has a stabilizing effect when asymmetric information in loan markets is taken into consideration [START_REF] Boyd | The theory of bank risk taking and competition revisited[END_REF][START_REF] Boyd | Bank risk-taking and competition revisited: new theory and new evidence[END_REF]. This is often referred to as the competition-stability hypothesis. As bank market power increases, banks can be riskier due to higher interest rate on loans. This is because the presence of asymmetric information in loan markets exacerbates entrepreneurial moral hazard to undertake risky projects to offset higher interest rate charged by banks with higher market power. Higher borrowers' risk can in turn negatively affect bank stability through a risk-shifting mechanism [START_REF] Stiglitz | Credit rationing with imperfect information[END_REF]. In the case of European banks, Udhe and Heimeshoff (2009) support the competition-stability hypothesis through the link between bank concentration and several risk measures. [START_REF] Liu | Competition and risk in South East Asian commercial banking[END_REF] also find the similar results for Southeast Asian banks in which higher bank competition derived from the Panzar-Rosse approach is not associated with higher bank risk taking. As well, [START_REF] Soedarmono | Bank competition, crisis and risk taking: Evidence from emerging markets in Asia[END_REF] indicate that higher market power in Asian banking is associated with higher bank moral hazard, although banks hold a higher capital ratio. This finding is however reversed during the 1997 crisis period indicating that market power is necessary to prevent an increase in bank default risk in times of crisis.

While the link between bank competition and financial stability remains inconclusive, a related important issue is how bank competition affects financial intermediation and economic growth. On the one hand, the perfect information hypothesis argues that higher bank competition leads to higher economic growth because banks with higher market power tend to charge higher loan prices which may in turn preclude financial intermediation activities [START_REF] Black | Entrepreneurship and bank credit availability[END_REF][START_REF] Degryse | Distance, lending relationships and competition[END_REF]Cetorelli and Gamberra, 2001). On the other hand, the asymmetric information hypothesis states that in the presence of asymmetric information, banks with higher charter value can erode asymmetric information by investing in relationship lending activities. Relationship lending can reduce firms' financing constraints, spur loan availability and enhance economic growth (Petersen andRajan, 1994, 1995;[START_REF] Berlin | On the Profitability and Cost of Relationship Lending[END_REF].

Building on these previous studies, a potential trade-off between bank stability and financial intermediation can occur along with changes in bank market structure.

Higher bank competition is indeed expected to boost loan growth because banks in more competitive markets tend to misprice their loan interest rates in expanding intermediation activities [START_REF] Ogura | Learning from a rival bank and lending boom[END_REF]. However, [START_REF] Foos | Loan growth and riskiness of banks[END_REF] provide evidence in which higher loan growth can lead to higher bank risk. In this paper we attempt to contribute to the ongoing debate regarding the impact of bank market structure on both bank stability and financial intermediation proxied by bank loan growth. Our paper extends the literature in several directions. First, a large number of studies rely on bank concentration or Herfindahl index measures to assess the degree of bank competition, while our present study employs the non-structural measure of bank competition derived from the new industrial organization approach following [START_REF] Soedarmono | Bank market power, economic growth and financial stability: Evidence from Asian banks[END_REF][START_REF] Soedarmono | Bank competition, crisis and risk taking: Evidence from emerging markets in Asia[END_REF]. Second, prior studies focus on the immediate impact of bank market structure, while this study also explores whether today's bank risk, and loan growth is determined by last year's bank market structure. Third, we extend the competition-stability-growth nexus by examining whether bank market structure affects depositors' reaction to bank risk and hence, we embrace issues on the role of market discipline exerted by bank depositors.

As a matter of fact, the association between bank competition and market discipline cannot be separated. [START_REF] Schaeck | Banking competition and capital ratios[END_REF] emphasize that banks in competitive markets tend to behave prudently due to the presence of "peer market discipline" that induces competing banks to hold more capital to gain competitive advantages. Similarly, banks in competitive markets can be more motivated than those in concentrated market to strengthen information transparency for gaining further competitive advantages. Eventually, competitive markets can strengthen bank depositors' ability to monitor bank behaviour.

In the meantime, prior studies demonstrate that bank depositors can strongly react to bank risk. In the presence of market discipline, bank depositors require higher interest rates on their deposits or withdraw their deposits when banks take on more risk [START_REF] Demirgüç-Kunt | Market discipline and deposit insurance[END_REF]Martinez-Peria and Schumkler, 2001). In this paper, our investigation regarding market discipline differs from previous studies, since we assess bank depositors' reaction to bank risk indirectly via bank market structure. As the market structure of banks clearly affects bank risk, we expect that bank depositors' reaction to bank risk is also dependent on bank market structure.

On the whole, this paper provides several policy implications in order to enhance both financial intermediation and bank stability. Specifically, what type of bank market structure will strengthen both financial intermediation and bank stability? Could bank regulators rely on the role played by depositors to discipline banks' behavior instead of preserving a banking market structure that precludes financial intermediation? We believe that these questions are of particular interest in both academic and policy circles to ensure better bank resilience and macroeconomic performance.

Data, variables and methodology

Data

An unbalanced panel of annual bank-level data are retrieved from BankScope Fitch IBCA. Our dataset initially consists of 686 commercial banks from 12 countries in Asia Pacific from 1994 to 2009. These countries include China (103),Hong Kong (68),India (84),Indonesia (108),Malaysia (63),Sri Lanka (14),South Korea (50) All country-specific data other than the ratio of stock market capitalization to gross domestic product (STOCK) and the economic freedom index (ECOFREE) come from the IFS. Meanwhile, STOCK and ECOFREE come from Beck and Demirgüç-Kunt (2009) and Heritage Foundation, respectively.

Variables

Dependent variables

Bank loan growth is used to assess financial intermediation. Let i and t represent bank index and time index, respectively. We define bank loan growth as follows:

    1 , , 1 , , , 5 . 0      t i t i t i t i t i TA TA L L DLOAN
L is banks' total loans in million US dollar, while TA is banks' total assets in million US dollar. Moreover, for robustness considerations, we also consider another measure of bank loan growth. Specifically, we use the annual growth rate of total loans (LOANG).

In the meantime, we use credit risk and income volatility measures to assess bank stability. Credit risk is defined as the ratio of loan loss provisions to total loans (LLP), while income volatility is the standard deviation of the return on assets computed from a three-period rolling window (SDROA). In other words, the standard deviation of return on assets at time t is calculated based on the return on assets from time t to 2  t . Finally, in order to measure how bank depositors react to bank risk, we use the following measure where D is bank total deposits in million US dollar.

    1 , , 1 , , , 5 . 0      t i t i t i t i t i TA TA D D DDEPO
Alternatively, we consider the annual growth rate of total deposits (DEPOG) as in [START_REF] Martinez-Peria | Do depositors punish banks for bad behavior? Market discipline, deposit insurance and banking crises[END_REF].

Bank market structure

In order to assess bank market structure, we construct the Lerner index derived from the industrial organization approach. Indeed, there are various methods to assess the degree of competition in banking markets. As a straightforward method, previous papers use the Herfindahl index (e.g. [START_REF] Berger | Bank competition and financial stability[END_REF]Schaeck and Cihak, 2009;[START_REF] Behr | Market structure, capital regulation, and bank risk taking[END_REF], or bank concentration ratios as in [START_REF] Uhde | Consolidation in banking and financial stability in Europe: Empirical evidence[END_REF]. However, these two indicators are rather crude measures which do not appropriately infer the degree of competition among banks, since they do not take into account different characteristics of banks, such as asset size, business type, ownership type, and so on [START_REF] Beck | Bank competition and financial stability: friends or foes? Working paper series[END_REF].

To better infer the degree of bank competition, [START_REF] Uchida | Has competition in the Japanese banking sector improved[END_REF] is the first to modify the non-structural measure of competition derived from [START_REF] Bresnahan | The oligopoly solution concept is identified[END_REF] and applied to Japanese banks. The merit of this measure is that it does neither require any assumption on the equilibrium of the banking market as in [START_REF] Panzar | Testing for 'monopoly' equilibrium[END_REF], nor the market structure of each bank which can be different across banks, even for similar types of banks. Using a panel data methodology, [START_REF] Uchida | Has competition in the Japanese banking sector improved[END_REF] method can again capture the degree of bank competition over time.

The aggregate index of bank market power derived by [START_REF] Uchida | Has competition in the Japanese banking sector improved[END_REF] is already used in previous studies related to Asian banks. These include [START_REF] Soedarmono | Bank market power, economic growth and financial stability: Evidence from Asian banks[END_REF] and [START_REF] Soedarmono | Bank competition, crisis and risk taking: Evidence from emerging markets in Asia[END_REF]. Specifically, the degree of bank competition is computed using the system of equations as follows: C is defined by total expenses, it q by total earning assets, it d by total deposits and short-term funding, it w by the ratio of operating expenses to total assets, it R by total revenue, it r by the ratio of interest expenses to total deposits, it p by the ratio of total revenue to total earning assets. the ratio of stock market capitalization to GDP, the short-term interest rate, and bank total assets, respectively.

                                 D it it t t t it it S it it it it it it it it it it it it it it t t it C it it it it it it it it it it it it it it e TA g IR g STOCK g GDPG g q g p e w b q b d b b d q C d b w b q b b c q r R R e w d b d q b w q b w b w b d b d b q b q b b C                             
To estimate System (1), country-level estimations are conducted and we use the Seemingly Unrelated Regression (SUR) method, while t  is estimated by annual time dummy variables and  by bi-annual time dummy variables (every two years), since  values are linearly dependent on the time-specific variables (GDPG, STOCK and IR). In the subsequent analyses, t  denotes the Lerner index of the banking industry in each country over time (LERNER).

Control variables

Several bank-specific control variables are incorporated in this study. First, bank size (SIZE) measured by the logarithm of banks' total assets is considered as one of the control variables because larger bank size tends to exacerbate bank risk taking [START_REF] Mishkin | How big a problem is too big to fail?[END_REF] and to some extents, larger banks are subject to greater market discipline [START_REF] Demirgüç-Kunt | Do we need large banks? Evidence on performance, strategy and market discipline[END_REF]. Bank size is also a determinant of bank loan growth as in [START_REF] Foos | Loan growth and riskiness of banks[END_REF].

Second, we incorporate the ratio of total deposits to total assets (DEPO) as a control variable which directly affects deposit growth (DDEPO or DEPOG). Bank deposits are also the source of withdrawal risk and thus, DEPO is expected to have a significant relationship with bank risk. Bank deposits can also have an impact on loan growth because of the nature of intermediation activities in banking (deposits funded loans). Third, we consider the banks' total capital adequacy ratio (CAR), since bank capital ratios admittedly affect lending behaviour which may lead to "capital crunch"

problems. The capital adequacy ratio may also affect risk and in turn affect market discipline through its influence on bank risk.

Another source of loan growth and bank risk is the extent to which banks engage in non-interest income activities. In particular, banks engaging in fee and commissionbased income can underprice loans and hence, granting loans at a lower cost in order to sell other fee-based products to their customers. This in turn fosters loan growth, but underestimated credit risk may erode bank stability. To account for non-interest income activities, we incorporate the ratio of non-interest income to total revenue (NNI) as a control variable.

Aside from bank-specific factors, we also consider country-specific factors as control variables. We consider the inflation rate (INF), where higher inflation rate can increase the bank lending rate which deepens adverse selection problems and might increase bank instability. Meanwhile, bank loan growth and risk are also strongly correlated with economic growth. As economic growth increases, banks tend to foster loan activities and underestimate risk which can result in higher bank fragility. However, economic growth can also neutralize bank moral hazard and risk, particularly when economic growth contributes to real sector profitability and bank stability [START_REF] Soedarmono | Bank market power, economic growth and financial stability: Evidence from Asian banks[END_REF]. To account for economic growth, we use the growth rate of real per capita gross domestic product (GDPG).

In the next turn, we also consider the degree of economic freedoms (ECOFREE)

and the foreign exchange reserves growth rate (FOREXG) to control for institutional development and macroeconomic stability. Higher institutional development indeed affects bank risk and banks' incentives to grant loans (Houston et al, 2011;[START_REF] Fernandez | How institutions and regulation shape the influence of bank concentration on economic growth: International evidence[END_REF], while Mohanty and Turner (2006) argue that foreign exchange reserves may affect bank risk exposure because banks are sensitive to foreign exchange fluctuations.

Methodology

To accurately conduct our empirical investigation, we eliminate values which are higher than 1 and smaller than 0 for DEPO, CAR and NNI. Table 1 presents the descriptive statistics of all the "clean" variables used in this study.

[Insert Table 1 here]

In terms of econometric methodology, we use a dynamic panel data model. In reality, loan growth, risk and deposit growth in banking vary over time, while their current values at time t are more likely to be dependent on their value at time 1  t . Based on the nature of these dependent variables, the use of dynamic panel data methodology is relevant. Dynamic panel data models that account for the one year-lagged value of dependent variable as control variable, and that are able to endogenize several explanatory variables, are again appropriate to cope with reverse causality problems which may occur between dependent and explanatory variables. Dynamic panel data models are also appropriate to analyze both the short-run and long-run effects of explanatory variables on dependent variables.

In this study, our explanatory variable of interest is the degree of bank competition (LERNER). Given the possibility that banks do not immediately respond to changes in market structure, it is also worth examining whether bank market structure after a one year time lag or LERNER(-1) has an impact on dependent variables. Furthermore, we use the two-step Generalized Method of Moments (GMM) estimations following [START_REF] Arellano | Another look at the instrumental variable estimation of error-component models[END_REF] and [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]. This method is also referred to as the system GMM.

The System GMM is essentially an extension of the standard GMM developed by [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF]. [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF] only consider the first difference of each variable in the regressions, while the lagged levels of explanatory variables are used as instruments. The use of the lagged levels as instruments may be inappropriate, particularly when variables are close to a random walk. To overcome this shortcoming, [START_REF] Arellano | Another look at the instrumental variable estimation of error-component models[END_REF] and [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] develop the system GMM to improve the standard GMM by introducing the levels equation to produce a system of two equations consisting of the levels equation itself and the first-differenced equation.

Hence, lagged differences of the explanatory and dependent variables can be valid instruments for the levels equation. To this end, [START_REF] Baltagi | Econometric Analysis of Panel Data[END_REF] proves that the System GMM produces more efficient and precise estimates than the standard GMM. Likewise, we introduce several endogenous variables in our dynamic panel data models. In all dynamic panel data models we run, we consider LERNER and LERNER(-1) as endogenous variables because both variables are estimated using the SUR method based on a system of three equations as shown in System (1). Hence, LERNER and LERNER(1) can contain measurement error and thus, need to be instrumented. We also consider CAR as an endogenous variable, because bank capital management is dependent on banks' managerial discretion, as well as bank-specific and country-specific characteristics including bank size, ownership type, bank market structure, business cycle, and so on. Finally, we also endogenize GDPG, since the real per capita income growth is admittedly dependent on various factors, including bank market structure [START_REF] Fernandez | How institutions and regulation shape the influence of bank concentration on economic growth: International evidence[END_REF]. Endogenizing GDPG can thus overcome simultaneity bias in which GDPG can be influenced by other explanatory variables, including LERNER.

Given that our models contains an autoregressive variable, we then use the orthogonal deviations transformation of instrumental variables in order to take crosssection fixed effects into account. Considering bank-level fixed effects is an important dimension because our bank sample is heterogeneous and comes from different countries with different macroeconomic, institution and regulatory environments. We further include time dummies in our models to account for time fixed-effects. Yet, we also take into account [START_REF] Windmeijer | A finite sample correction for the variance of linear efficient twostep GMM estimators[END_REF] finite sample correction to ensure that our estimates are robust. Finally, we verify the validity of our dynamic panel data models using AR (2) test and Hansen-J test. The GMM model is valid when AR (2) and Hansen-J test are both not significant, which respectively show that there is no second order serial correlation among errors of first-differenced equation, and that our identifying restrictions are valid.

Empirical results

Bank market structure and loan growth

When banks operate in a more competitive market, they tend to boost loans by setting lower loan prices to attract new borrowers [START_REF] Ogura | Learning from a rival bank and lending boom[END_REF]. Prior literature also suggests that higher degree of bank competition boosts small businesses development, as borrowing costs decline [START_REF] Degryse | Distance, lending relationships and competition[END_REF]Cetorelli and Gamberra, 2001).

Building on such findings, we focus our investigation on the impact of bank market structure on loan growth as a proxy of financial intermediation.

As shown in Table 2, we highlight that banks in less competitive markets tend to reduce loan growth. In other words, our findings are consistent with the nexus between LERNER and bank loan growth (DLOAN or LOANG). We further notice that DLOAN and LOANG equations are both valid, as both AR (2) test and Hansen-J test are not significant.

[Insert Table 2 here] Several insights can be drawn from these findings. Banks in less competitive markets tend to have higher market power which enables them to increase their lending rates. However, Banks in less competitive markets might also be more prudent than those in competitive markets, and therefore higher credit risk might be priced appropriately which unfortunately leads to lower loan growth.

Moreover, in the presence of asymmetric information in loan markets, entrepreneurs who obtain loans can respond to higher loan pricing by running riskier projects to offset higher lending rates. Entrepreneurs' moral hazard can therefore exacerbate bank riskiness through risk-shifting mechanisms [START_REF] Stiglitz | Credit rationing with imperfect information[END_REF]).

On the contrary, in the absence of asymmetric information, bank riskiness will decline with higher LERNER, as long as banks in less competitive markets do not underestimate risk and set up lending rates appropriately, albeit reducing loan growth afterward. In the following section, we investigate the impact of LERNER on bank riskiness in order to verify whether the relation between LERNER and bank loan growth (DLOAN or LOANG) is presumably characterized by the presence of asymmetric information.

Bank market structure and risk

In Table 2, we document our empirical results on the nexus between LERNER and bank risk measured by credit risk (LLP) and income volatility (SDROA). We find that higher LERNER is associated with higher LLP and SDROA, suggesting that banks in less competitive markets tend to have higher credit risk and income volatility. The positive link between LERNER and bank riskiness (LLP and SDROA) is also unaltered when we consider the impact of LERNER at time t-1 on bank riskiness at time t, although such a positive link is tempered. Hence, higher LERNER clearly results in higher bank riskiness. Moreover, both LLP and SDROA equations in Table 2 are valid because AR (2) test and Hansen-J test are not significant. This result remains consistent with [START_REF] Soedarmono | Bank market power, economic growth and financial stability: Evidence from Asian banks[END_REF][START_REF] Soedarmono | Bank competition, crisis and risk taking: Evidence from emerging markets in Asia[END_REF], although in their study they apply a static panel data model using both fixed effect and two-stage least squares methods.

Combining the results regarding the competition-growth nexus, as well as the competition-stability nexus, we show that although banks in less competitive markets reduce loan growth, such a reduction is not sufficient to improve their stability. Hence, as mentioned earlier, asymmetric information problems are more likely to exist exacerbating entrepreneurs' moral hazard that in turn might increase the riskiness of banks in less competitive markets.

Bank market structure and deposit growth

Since higher market power in banking leads to lower loan growth but higher instability in Asian banks, it is worth examining whether such instability can be mitigated by market discipline. In this section, we are particularly interested in the role played by bank depositors in facing higher bank riskiness. Market discipline is indeed of particular interest for bank regulators and it has been further emphasized in the new Basel accords.

From Table 2, we observe that the link between LERNER and deposit growth (DDEPO and DEPOG) is negative and significant. This indicates that banks in less competitive markets experience lower deposit growth. Given that banks in less competitive markets (higher LERNER) tend to exhibit higher credit risk and income volatility as discussed in the previous section, our results show that bank depositors are indeed sensitive to bank risk along with an increase in market power in banking. Overall, our GMM models for DDEPO and DEPOG equations are valid because AR (2) test and Hansen-J test are both insignificant.

Robustness checks 4.5.1. Alternative model specifications

For robustness3 , we now provide several sensitivity analyses. First, we do not consider GDPG and CAR as endogenous variables in the estimations using a dynamic panel data methodology. Using this specification, the impact of LERNER on loan growth, risk and deposit growth in banking remains unaltered. This new specification also remains valid because both AR(2) test and Hansen-J are not significant. Second, we eliminate time dummies from our dynamic panel data models and again, this consideration does not change our findings.

Static panel data methodology

To further ensure the robustness of the impact of market power in banking on bank loan growth, risk and deposit growth, we re-estimate the link using a static panel model by taking into account individual and time-fixed effects. Table 3 presents our empirical results using the panel fixed effects regressions. Our previous findings remain consistent and robust.

[Insert Table 3 here]

Bank market structure and loan pricing: an additional analysis

We also perform an additional analysis to highlight whether a decline in bank loan growth and an increase in risk along with higher market power in banking are indeed due to the fact that banks in less competitive markets set higher loan pricing. Hence, we now examine how market power in banking affects bank loan pricing.

In order to assess bank loan pricing, we use the ratio of net interest income to total earning assets or net interest margin (NIM). Alternatively, we also use the ratio of net interest income to total assets (NIMTA). Initially, NIM and NIMTA are cleaned to ensure that they neither contain outliers nor exhibit a skewed distribution. Specifically, we impose a restriction to NIM and NIMTA by eliminating both their 2.5% lowest values and 2.5% highest values. Table 4 presents the empirical results using the two-step GMM dynamic panel data model following [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF].

[Insert Table 4 here] From Table 4, we document that higher LERNER is significantly associated with higher NIM, particularly when we exclude LERNER(-1) from the regression, as shown in Model 2. In fact, incorporating LERNER(-1) in the regression does not alter the relationship sign between LERNER and NIM, albeit not statistically significant, as shown in Model 1 (Table 4).

Our model that specifically uses NIM as the dependent variable is valid because AR(2) tests are not rejected. Meanwhile, Hansen-J tests are not significant and hence, the validity of our instrumental variables remains ensured at least at the 5% level. However, when we use NIMTA as an alternative measure of net interest margin, the results are less robust than those when NIM is used as a proxy of net interest margin. Although we still find a positive relationship between LERNER and NIMTA.

These findings indicate that banks in less competitive markets tend to charge higher interest rates on loans which increases net interest margin. By extension, as discussed from Section 4.1 to 4.2, such a behavior leads to a decline in loan growth and an increase in bank risk because of the presence of entrepreneurial moral hazard in loan markets.

Conclusion

This paper aims to highlight the implication of bank market structure on financial intermediation and bank riskiness. Our findings strongly emphasize that market power in banking is negatively related to bank loan growth as a proxy of financial intermediation, and is positively associated with bank instability measured by credit risk and income volatility. Banks in less competitive markets are indeed more unstable and less beneficial than those in competitive markets in strengthening financial intermediation. In addition to that, this study further highlights the role of market discipline exerted by bank depositors.

As market power in banking results in higher instability, bank depositors in less competitive markets tend to deposit lower amounts. Nevertheless, there are also other factors affecting bank market discipline in the case of Asian banks.

  , Taiwan (50), Thailand (40), Pakistan (34), Philippines (39), and Vietnam (33) 2 . Moreover, country-specific data are also taken from various sources. These include the International Financial Statistics (IFS) established by the International Monetary Fund, the World Bank Financial Structure Dataset established by Beck and Demirgüç-Kunt (2009), as well as the economic freedom index at the country level coming from Heritage Foundation.

  that affect the demand for loans, defined as the growth of real gross domestic product (GDP),

Table 1 .

 1 Descriptive statistics

	Variable	Definition

Table 2 .

 2 Bank market structure, loan growth, risk and market discipline. All variable definitions follow Table1. Models are estimated using the two-step Generalized Method of Moments (GMM) estimator following[START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] with[START_REF] Windmeijer | A finite sample correction for the variance of linear efficient twostep GMM estimators[END_REF] finite sample corrections, orthogonal deviations transformation of instruments and period dummies. Models are valid when AR(2) tests and Hansen-J tests are not rejected are not significant. (***), (**), and (*) indicate significance at the 1%, 5% and 10% levels, respectively.

			Loan Growth		Risk		Deposit Growth
		DLOAN	LOANG	LLP	SDROA	DDEPO	DEPOG
	Explanatory Variables
	DEP.VAR(-1)	-0.0672**	0.1871***	0.1365**	0.6519***	0.0438	0.0900***
		(0.029)	(0.029)	(0.043)	(0.031)	(0.034)	(0.034)
	LERNER	-0.0402*	-0.0895**	0.0241***	0.0016*	-0.0895***	-0.1244***
		(0.023)	(0.035)	(0.008)	(0.001)	(0.025)	(0.047)
	LERNER(-1)	0.0240	0.0480	-0.0129**	0.0006	0.0542**	0.0941**
		(0.021)	(0.035)	(0.007)	(0.001)	(0.024)	(0.045)
	SIZE	-0.0036	0.0025	-0.0012**	-0.0006***	-0.0000	-0.0059
		(0.003)	(0.004)	(0.001)	(0.000)	(0.002)	(0.005)
	DEPO	-0.0077	0.0378	0.0040	-0.0001	0.2412***	0.1380**
		(0.031)	(0.044)	(0.006)	(0.001)	(0.026)	(0.055)
	CAR	-0.1107	-0.0067	0.0370	-0.0092**	0.0532	-0.0134
		(0.074)	(0.120)	(0.030)	(0.003)	(0.074)	(0.153)
	NNI	-0.0038	0.0263	0.0133**	0.0048***	0.0644**	0.1015
		(0.035)	(0.055)	(0.007)	(0.001)	(0.032)	(0.070)
	INF	0.1859***	0.4070***	-0.0205**	-0.0033*	0.2231***	0.4385***
		(0.054)	(0.102)	(0.010)	(0.002)	(0.063)	(0.129)
	GDPG	0.5003***	0.8482***	-0.0907***	-0.0064	0.6102***	0.8357***
		(0.123)	(0.203)	(0.023)	(0.005)	(0.124)	(0.236)
	ECOFREE	-0.1688***	-0.2812***	-0.0132**	0.0026**	-0.1135***	-0.1638**
		(0.000)	(0.001)	(0.000)	(0.000)	(0.000)	(0.001)
	FOREXG	0.0905***	0.1639***	0.0080***	-0.0002	0.1734***	0.2625***
		(0.017)	(0.027)	(0.003)	(0.001)	(0.017)	(0.034)
	Observations	3,268	3,000	2,609	3,023	3,037	3,006
	Z-score for AR(1) test.	-4.36***	-6.72***	-2.78***	-5.42***	-7.24***	-5.15***
	Z-score for AR(2) test	-1.72	0.04	0.21	-0.54	-0.87	-0.43
	p-value for Hansen-J test	0.516	0.871	0.963	0.795	0.842	0.867
	Endogenous variables	DLOAN(-1)	LOANG(-1)	LLP(-1)	SDROA(-1)	DDEPO(-1)	DEPOG(-1)
		LERNER	LERNER	LERNER	LERNER	LERNER	LERNER
		LERNER(-1)	LERNER(-1)	LERNER(-1)	LERNER(-1)	LERNER(-1)	LERNER(-1)
		CAR	CAR	CAR	CAR	CAR	CAR
		GDPG	GDPG	GDPG	GDPG	GDPG	GDPG

Table 3 .

 3 Bank market structure, loan growth, risk and market discipline. All variable definitions follow Table1. Models are estimated using the fixed-effect panel regressions, accounting for both individual and time-fixed effects. (***), (**), and (*) indicate significance at the 1%, 5% and 10% levels, respectively. Constants are included, but not reported.

			Loan Growth		Risk		Deposit Growth
		DLOAN	LOANG	LLP	SDROA	DDEPO	DEPOG
	Explanatory Variables
	LERNER	-0.1225***	-0.1202***	0.0049*	-0.0036	-0.0776***	-0.1184***
		(0.022)	(0.026)	(0.003)	(0.001)	(0.015)	(0.029)
	SIZE	0.0010	0.0259**	-0.0053***	-0.0013***	0.0387***	0.0519***
		(0.010)	(0.011)	(0.001)	(0.000)	(0.006)	(0.012)
	DEPO	-0.0467	-0.0540	0.0117*	0.0065***	0.4567***	0.5511***
		(0.051)	(0.059)	(0.006)	(0.002)	(0.033)	(0.067)
	CAR	-0.1229*	-0.2560***	0.0315***	-0.0016	0.0970**	0.1020
		(0.069)	(0.081)	(0.009)	(0.002)	(0.045)	(0.090)
	NNI	0.0980***	0.0340	-0.0085**	-0.0014	0.1104***	0.1762***
		(0.036)	(0.041)	(0.004)	(0.001)	(0.023)	(0.046)
	INF	0.2931***	0.3381***	-0.0193*	-0.0046	0.2357***	0.3837***
		(0.104)	(0.116)	(0.011)	(0.003)	(0.067)	(0.131)
	GDPG	0.3235*	0.6730***	-0.0928***	-0.0041	0.5069***	0.8964***
		(0.177)	(0.199)	(0.020)	(0.006)	(0.114)	(0.223)
	ECOFREE	0.0033*	-0.0044**	0.0009***	0.0001**	-0.0006	-0.0008
		(0.002)	(0.002)	(0.000)	(0.000)	(0.001)	(0.002)
	FOREXG	0.0869***	0.1867***	0.0098***	-0.0006	0.1878***	0.3038***
		(0.021)	(0.024)	(0.002)	(0.001)	(0.014)	(0.027)
	Observations	3,532	3,342	3,087	3,353	3,371	3,351
	R-square	0.019	0.039	0.048	0.066	0.141	0.081

Table 4 .

 4 Bank market structure and net interest margin. All variable definitions follow Table1. Models are estimated using the two-step Generalized Method of Moments (GMM) estimator following[START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] with[START_REF] Windmeijer | A finite sample correction for the variance of linear efficient twostep GMM estimators[END_REF] finite sample corrections, orthogonal deviations transformation of instruments and period dummies. Models are valid when AR(2) tests and Hansen-J tests are not rejected are not significant. (***), (**), and (*) indicate significance at the 1%, 5% and 10% levels, respectively.

		Model 1	Model 2
	NIM	NIMTA	NIM	NIMTA

The number of banks in each country is shown in parentheses.

The results are not presented in the paper, but are available upon request to the authors