## Abstract

This paper investigates the feasibility of an ultrasonic method for measuring internal displacements and strains in engineering components at depths of tens of millimetres. The principle is to use an ultrasonic array to generate images of the speckle pattern produced by the material microstructure before and after the application of load. Under the assumption of constant ultrasonic velocity, a block-matching method is used to find the relative displacement of small portions of the images between the two loading states, and hence the strain. Experiments performed using a 5 MHz ultrasonic array show that good displacement measurement results are obtained from the speckle directly below the array at depths of up to 45 mm. The results demonstrate that the technique can be used to identify the onset of plasticity and non-uniformity in strain across the field of view. However, while the actual values of strain obtained are correct in some directions, they are systematically incorrect in other directions by a factor between five and six. It is shown that this is because the change in ultrasonic velocity owing to load (the acousto-elastic effect) has, in some cases, a more significant effect on ultrasonic propagation time than the change in distance owing to strain.

## 1. Introduction

Traditionally, strain is measured using strain gauges but these are limited to measuring surface strain over a gauge length of a few millimetres and provide point rather than full-field information. Techniques for measuring internal strains are limited. For example, internal strains can be measured with neutron diffraction (up to depths of 100 mm in aluminium) and synchrotron X-rays (depths of 70 mm are easily achieved in aluminium) with small gauge volumes (1 mm×1 mm×25 μm; Reimers *et al.* 1998). Full-field information can potentially be generated by scanning over the surface of the material. However, portable equipment is not readily available for any of these techniques. Another method to measure three-dimensional internal strains is to cross-correlate digitized high-resolution X-ray tomographic images (Bay *et al.* 1999). The technique, termed digital volume correlation (DVC), is limited to use on samples with microarchitectural detail, such as advanced synthetic foams (Smith *et al.* 2002) and bio-materials like trabecular bone (Bay *et al.* 1999).

Various full-field surface strain measurement techniques exist, and these are summarized in table 1. Full-field two-dimensional surface strain measurements can be made using optical techniques. For example, Moiré interferometry (Sciammarella 1982) involves measuring in-plane displacement fields by applying a grating to the surface of a component. Non-contact full-field surface deformation can be made with laser speckle patterns. Electronic speckle pattern interferometry (ESPI) (Yamaguchi 1985) and shearography (Steinchen *et al.* 1998) are methods that use the digitized interference fringes between speckle patterns obtained before and after component deformation to calculate surface displacements. Digital image correlation (DIC) uses correlation of two-dimensional digitized images of the surface of a component before and after deformation (Chu *et al.* 1985). Digitized images can be generated from laser speckle patterns, by spray painting the surface of the component and illuminating with white light (Chu *et al.* 1985). For high-resolution applications scanning electron micrographs (SEMs; Vendroux *et al.* 1998) can be used.

Ultrasonic waves can penetrate deep into materials and therefore have potential for internal strain measurement. Rigid body displacements have been measured in solids by cross-correlating ultrasonic speckle in pitch-catch scans of rough surfaces within simple components (Schaeffel *et al.* 1980). However, this technique has been limited to measuring motion of rough interfaces and surfaces. In the medical field, static elastography uses ultrasonic speckle images to measure strain and the elasticity of tissues deep (up to 40 mm) within the body (Ophir *et al.* 1999). In some cases, particularly cancer tumours, the elasticity reflectivity contrast (the difference in elastic modulus between soft and hard tissue) is higher than conventional ultrasonic contrast. The technique is based on measuring the strain distribution by cross-correlation of segments of pulse-echo scans of the pre- and post-compressed tissue along the axis of the elements of an ultrasonic array transducer. Another technique is dynamic elastography where the tissue is compressed dynamically at a certain frequency. Typical strains measured in elastography range from 5×10^{−3} (5000 μ*ϵ*) to 3×10^{−2} (30 000 μ*ϵ*) (Ophir *et al.* 1999).

This paper investigates the feasibility cross-correlation of ultrasonic speckle images of solid metallic components in unloaded and loaded states to measure internal strain fields. In terms of image processing, this is similar to DIC except that the internal displacements are mapped, as opposed to surface displacements. It is comparable with static elastography except that it concerns the strain fields in metallic components rather than soft biological tissue, and hence the measurement of much smaller applied strains is required. Though the work described in this paper only considers metals, other materials exhibiting ultrasonic speckle such as carbon-fibre and glass-fibre composites and polymers could also be used. Only one-dimensional arrays and two-dimensional displacement and strain fields are considered but in principle three-dimensional internal displacement and strain fields could be obtained using three-dimensional speckle images generated from two-dimensional arrays. The key principles of the technique are stated in §2. The method of obtaining the ultrasonic speckle images and the numerical procedure for tracking the speckle and obtaining the strain are described in §3. Section 4 reports experiments which characterize the performance of the technique and an application of the technique is given in §5. Section 6 explains the experimental results using theory, and §7 gives a model to show expected gains in performance of the technique by changing experimental parameters.

## 2. Principle of method

### (a) Background

If multiple scattering is assumed to be negligible, the ultrasonic array imaging process may be described by a linear system that obeys the imaging equation (Meunier & Bertrand 1994; Goodman 2005):
2.1where *f*(*x*,*z*) is the (RF) image amplitude, *h*(*x*,*z*) is the impulse response of the imaging system (or system point spread function (PSF)), *f*′(*x*,*z*) is the scatterer distribution, *n*(*x*,*z*) is the electronic noise and *x* and *z* are coordinates in the image plane. This equation implies that a change in the scatterer distribution results in a change in the image. This forms the basis of the technique described in the paper. An ultrasonic array is used to collect a set of time-traces of reflected echoes from the grain structure of a metallic component. An image is reconstructed from these time-traces, called a speckle image. Motion of the grains owing to, for example, a change in the loading state of the component causes movement of the speckles in the image. By tracking the movement of the speckles in a local region of the image, the internal displacements at that point in the material can be estimated. A strain field within the component can be calculated from the gradient of displacements measured at different locations in the image. Figure 1 shows the key stages in the method that involve collecting ultrasonic array data from the component in reference and deformed states, reconstructing speckle images from the array datasets, tracking speckles between reference and deformed images and strain calculation.

### (b) Speckle imaging

Ultrasonic speckle images of the test piece are generated in unloaded and loaded states using the total focusing method (TFM; Holmes *et al.* 2005). This algorithm is adopted as opposed to standard beamforming techniques as it provides better resolution and noise suppression. The TFM forms an image *f*(*x*,*z*) of the test piece in post-processing by focusing the full matrix of time-traces from the array at every pixel in the image by delay-and-sum beamforming:
2.2where *x* is the distance parallel to the array, *z* is the distance perpendicular to the array, *e*(*t*) is the time (*t*) domain signal when the *m*th array element is transmitting and *n*th element is receiving. The total propagation time from the transmitting element to the pixel and back to the receiving element, *t*_{m,n}, is given by
2.3where *c* is the velocity of propagation of longitudinal waves and *r*_{m} and *r*_{n} are the distances from the pixel to the *m*th and *n*th array elements given by
2.4where *x*′_{m} and *x*′_{n} are the *x*-coordinates of the relevant elements.

### (c) Speckle tracking and strain estimation

A block-matching method is implemented to track the motion between speckle images of the test piece in unloaded and loaded states, referred to as the reference image, *f*_{1}, and the deformed image, *f*_{2}, respectively. Consider a point (*x*_{e},*z*_{e}) in the reference image that is displaced by (unknown) amounts and in the *x*- and *z*-directions in the deformed image. In the vicinity of (*x*_{e},*z*_{e}), the relationship between coordinates in the reference image (*x*,*z*) and deformed image (*x**,*z**) is approximated by a zeroth-order Taylor series (Vendroux *et al.* 1998; i.e. the analysis assumes rigid body motion only),
2.5

The similarity between the reference image in the vicinity of (*x*_{e},*z*_{e}) and the deformed image in the vicinity of is assessed using a normalized coefficient, *C*, defined as (Vendroux *et al.* 1998):
2.6where the quantities *X* and *Z* define the dimensions of a rectangular region centred on (*x*_{e},*z*_{e}) in the reference image, which is referred to as a sub-image. In order to estimate the actual displacements, *u* and *v*, at (*x*_{e},*z*_{e}), the coefficient *C* is minimized with respect to two parameters:
2.7

The strains are obtained by using the forward difference formulae applied to the displacements obtained at adjacent points:
2.8where Δ*x* and Δ*z* are the distances between the displacement measurements in the *x*- and *z*-directions, respectively.

Note that the minimization of the correlation function in equation (2.7) can also be performed with respect to stretching and skewing parameters. This improves strain estimation for highly deformed (strain greater than 1%) low-noise images typically encountered in DIC and elastography. However, for the lower strains and low signal-to-noise ratios (SNRs) encountered in this work, the use of stretching and skewing parameters was found to lead to multiple peaks in the correlation functions and an increased likelihood of erroneous results (Bowler 2010).

## 3. Experimental method

### (a) Equipment

The equipment consisted of a 5 MHz ultrasonic array probe (Imasonic, 4154-A101, Voray-sur-l’Ognon, France) with 128 rectangular elements. The element width and pitch were 0.2 and 0.3 mm, respectively, and the element length was 15 mm. The natural focal point of the array in the elevation direction (i.e. the start of the near field and the point of maximum field intensity) is 51 mm. Although not investigated here, it may be desirable to tailor the element length such that the natural focus of the array in the elevation direction is close to the depth of interest. The array was connected to an array controller (Peak NDT, Micropulse 5PA, Derby, UK) and controlled from Matlab on a PC via an Ethernet cable. Each of the array elements were excited with a 100 V, 100 ns rectangular pulse at a pulse repetition frequency of 1 kHz. The full matrix of time-traces from every combination of the transmitter and the receiver of the array was captured after 16 averages, at 16 bit resolution over a range ±1 V and at a sampling frequency of 25 MHz.

The test specimens were made from stainless steel 304 plate. Experiments involving load application used dog bone-shaped samples as shown in figure 2*a*,*b*, where the backface was 65 mm from the array. For all other experiments, the samples were 200×100×30 mm rectangular blocks, where the distance to the backface was 100 mm. The array was ultrasonically coupled to the specimens with standard ultrasonic coupling gel (Chemetall, UCA-2M, Frankfurt, Germany).

A sample of the stainless steel was polished, etched and viewed under a microscope, and is shown in figure 3. The grain sizes range from 10 to 100 μm and they are randomly orientated. The velocity of longitudinal bulk waves in the stainless steel, *c*, was measured and found to be 5740 m s^{−1}, hence the wavelength is 1.1 mm at 5 MHz. From figure 3, it is evident that there are a large number of grains per wavelength and so an ultrasonic image of the grains is an unresolved random interference pattern of the scattering from multiple grains. The image can therefore be regarded as speckle (Dainty 1975).

Figure 4 shows a representative subset of the full matrix of time-traces collected when the array was coupled to one of the stainless steel specimens. The time-traces are shown for transmitter element number 1 and odd-numbered receiver elements from 1 to 127. Signals from the backface, surface waves, post-excitation ringing and inter-element cross-talk can be distinguished. The grain signals are present but are much smaller in amplitude than the other signals.

### (b) Array signal processing

Signals in the time-traces from the relatively high-amplitude backface reflection, surface waves, post-excitation ringing and associated cross-talk cause imaging artefacts. These artefacts obscure the speckle pattern and have to be suppressed. Therefore, it has been found that some pre-processing of the raw time-domain data is essential prior to image reconstruction. First, the signals in the time-domain data corresponding to post-excitation ringing, the backface reflection and its associated cross-talk are all suppressed using a window function *w*(*t*,*u*,*v*),
3.1where
3.2and *t*_{1}=1, *t*_{2}=2 and *t*_{3}=13 μs.

In order to obtain a final image from which the envelope intensity can be easily extracted, the echoes are converted to complex signals by the Hilbert transform before further processing. The signals are then filtered to remove low- and high-frequency noise using a Hanning window function, with a centre frequency of 5.5 MHz and a bandwidth of 5.45 MHz. The signals are also filtered in the wavenumber domain to suppress shear waves using a Hanning window function centred at 0 m^{−1} and with a bandwidth of , where *k* is the wavenumber and *θ*=*π*/2.

Figure 5 shows the resulting image of the rectangular sample when the above pre-processing is performed prior to image formation. The image is plotted on a decibel (dB) scale with respect to the backface reflection (0 dB) at *z*=100 mm in the original image (i.e. that formed from *e*_{m,n} rather than *e*′_{m,n}) and has a 0.1 mm pixel size. The coordinate origin of the image is the centre of the array. The image shows the speckle from the grains at around −65 dB relative to the amplitude of the backface reflection in the original image.

### (c) Speckle tracking

The Newton–Raphson method is used to converge on the minimum in equation (2.6) with a small number of iterations (Vendroux *et al.* 1998). Iterations are continued until there is no further decrease in the coefficient, *C*. Typically, this is seven iterations. Note that in practice the integrals in equation (2.6) are evaluated at discrete points. Finer resolution than that of the original image pixels is achieved using a bicubic spline interpolator. This interpolation scheme assumes continuity in the first-order gradients between adjacent splines and zero second-order gradients at the endpoints. In the following results, equal sub-image sizes in the *x*- and *z*-directions, *X* and *Z*, are used and the distances between displacement measurements, Δ*x* and Δ*z*, are equal to the sub-image size.

## 4. Rigid body motion measurement and optimization of experimental procedure

### (a) Demonstration of rigid body motion measurement

The array was uniformly translated by 100 μm in the *x*-direction parallel to the surface of the rectangular specimen using the set-up shown in figure 6. The translation of the array was achieved using a micrometer (OptoSigma, 122-0410/0415, CA, USA) and a digital piezoelectric transducer (DPT) (Queensgate Instruments, AX301-IEEE, Torquay, UK). The array was held between two flexible spring-steel shims and the length of each shim (150 mm) was long in relation to the maximum translation of the array (4 mm) to reduce displacement in the *y*-direction and ensure translational motion only in the *x*-direction without rotation. At each displacement increment, the time-traces were captured from the array and a speckle image generated. The translation of the array, , was measured using a linear variable differential transformer (LVDT) (RDP Electronics, DCTH100AG, Wolverhampton, UK) interfaced to a PC via an analogue-to-digital converter (InstruNet, Model 100B, MA, USA) to an accuracy of 0.1 μm. This experiment provided a benchmark, enabling the performance of the ultrasonic grain speckle technique to be quantified.

Displacement maps were generated from multiple sub-images taken from different positions throughout the images. Figure 7*a* shows arrows representing the displacements measured from the speckle image using a 5 mm sub-image when the translation of the array was 100 μm. The displacements are plotted as arrows, with the beginning of each arrow at a sub-image centre and the length and angle showing the magnitude and direction of the displacement of that sub-image. A 100 μm reference arrow is plotted on the same axes to show the expected magnitude and direction of the arrows. It can be seen from figure 7*a* that the majority of the arrows are of similar length and orientation to the reference arrow. There is good agreement in the central region, with the agreement deteriorating towards the top and bottom of the image.

### (b) Definition of region of interest

Figure 7*b* shows the mean absolute errors on the *x*-component of the measured displacements when the array was translated by 100 μm. The figure was obtained by averaging data from six realizations. Errors in the central region between −10 and 10 mm in the *x*-direction and between 25 and 45 mm in the *z*-direction are low (8 μm or less). Further experimental work only considered this central region (unless otherwise stated), which is henceforth referred to as the region of interest (ROI) and indicated by the dashed box in figure 7*b*. Displacement errors near the top of the image (*z*<25 mm) are high because the artefacts of the ringing and surface waves reduces the quality of the speckle in this region. The figure shows that the displacement errors steadily increase when the distance from the array increases. This is because the amplitude of the speckle decreases further from the array owing to spreading losses and attenuation. The speckles are also spatially larger further from the array as the PSF becomes broader. This means the speckle tracking is less accurate because the SNR is lower and there are fewer trackable speckles per sub-image.

### (c) Precision and range of displacement measurement

The array was translated up to a displacment of 16 μm using the DPT and up to a maximum of 4 mm using the micrometer, as shown in figure 6. The mean and standard deviation of the displacements were monitored in the ROI over a range of translations of the array. The mean displacement in the ROI was defined as,
4.1where *A* is area of the ROI. The standard deviation of the displacement in the ROI was defined as
4.2Note that these equations are evaluated in discrete form by integrating over the points where the displacements have been calculated.

Figure 8*a* shows that the mean displacement of the speckle in the *x*-direction in the ROI agrees well with the displacement measured by the LVDT over the full range of translations tested. The maximum measured displacement of 4 mm was limited by the equipment used to translate the array. In principle, larger displacements may be measured using the ultrasonic technique. Figure 8*b* shows the standard deviation of the displacements, , in the ROI is 4 μm near zero displacement. As the translation increased, the standard deviations increased to a maximum of 44 μm at a 4 mm translation. The trend was similar in both the *x*- and *z*-directions.

### (d) Factors limiting performance

A major factor limiting the measurement of small displacements is the electronic noise. Random electronic noise is present in any electrical system, and is therefore present in the ultrasonic array images. The electronic noise causes errors in the measured displacements and strains and may be reduced by further averaging the signals during the data capture. As averaging takes a finite amount of time, there comes a point where performing more averages cannot improve measurement accuracy because time-varying effects become significant. Time-varying effects include temperature changes in the sample and evaporation from the gel.

To optimize the data-acquisition procedure, extended experiments were performed. Initial results showed that steady-state results were not recorded until around 48 h after the application of the array to the sample and it was speculated that this was due to slow compression of the gel-coupling layer. In another experiment, multiple datasets, each recorded with 16 averages, were acquired consecutively for a period of 10 h, following a settling period of 48 h. By averaging data from multiple consecutive datasets, it was found that the lowest displacement and strain standard deviations were obtained after averaging 32 datasets, equivalent to performing 32×16=512 averages of the raw array data. With the available equipment, the time taken to perform 512 averages was around 2 h and the resulting minimum displacement and strain standard deviations are 1 μm and 3×10^{−4}, respectively, in both the *x*- and *z*-directions. All further work in this paper uses data captured after an initial settling time of 48 h with 512 averages, unless otherwise stated.

## 5. Strain measurement

### (a) Uniform strain field

The bar with a constant cross section in figure 2*a* was subjected to uniaxial tensile force, *F*, of 150 kN (applied in the *x*-direction in the figure). The array was fixed to the bar using a specially designed magnetic clamp. A schematic of this device is shown in figure 9, which, without contacting the bar, enabled the array to slide freely on the surface of the bar. Array data were captured during the loading and displacements and strains of the material within the bar were measured from the ultrasonic speckle images, using the image at zero load as the reference image. Surface strains were measured by strain gauges (TML, FLA-5-17, Tokyo, Japan), bonded to the bars at positions shown by the small grey rectangles in figure 2*a* and connected to a control box (TC Electronic, System 6000, Risskov, Denmark). The strain gauges had a working area of 5×2 mm.

Figure 10*a* shows the measured displacement field when the bar was subjected to a tensile force of 150 kN. The displacement field is shown to a depth of 55 mm as the backface of the sample is at 65 mm. The arrows agree qualitatively with the expected deformation of the material; the *x*-components of the arrows increase towards the left and right of the image. Figure 10*b* shows the *x*-direction (axial) strain field, . The strain field shows fluctuations about a mean value of 2×10^{−3} owing to electronic noise.

The bar with a constant cross section was then loaded incrementally from zero up to a maximum tensile force of 605 kN, subjecting the bar to elastic and plastic deformations. Array data were captured at each increment using 16 averages (rather than 512), which enabled the data to be captured quickly before any significant strain redistribution occurred when in the plastic region. Figure 11*a* shows the relationship between the axial stress, *σ*, and the positive *x*-direction axial strains measured in the ROI (shown by the dashed rectangle in figure 10*b*) over the range of loads applied to the bar. This figure is the axial stress–strain curve (elongation). The stress was calculated from
5.1where *A*_{0} is the cross-sectional area of the bar at zero load (65×20 mm). Figure 11*b* shows the negative *z*-direction transverse stress–strain curve (owing to Poisson’s ratio contraction). Both stress–strain curves show the elastic region, where the relationship between strain and stress is linear, and the plastic region. The transition between the elastic and plastic regions (the yield point) is estimated from the figure to start at around 450 MPa. A typical textbook value for the yield stress of stainless steel 304 is 215 MPa (Boyer & Gall 1987), but this is based on the point where the residual strain after unloading is 0.2 per cent and consequently occurs at a much lower stress level than that required to cause the gross plasticity seen in figure 11*a*,*b*.

In figure 11*a*,*b*, the error bars represent 1 s.d. away from the mean, and show how the errors increase for larger strains. This is due to a number of factors such as grain distortion and image decorrelation.

The figures also show the strains measured with the strain gauges, plotted as single values at each load ( and ). There is very good agreement between the strain gauge measurements and the ultrasonic strain measurements in the *z*-direction. However, figure 11*a* shows the agreement in the *x*-direction is poor, with the mean ultrasonic strain being on average 4.62 times larger than that measured with the strain gauges for the elastic region. It is speculated that this is due to the acousto-elastic effect and arguments supporting this thesis are given in §6. Note that the maximum measured axial strain was (), as this was the limit of the strain gauge. In principle, higher strains can be measured with the ultrasonic technique.

### (b) Non-uniform strain field

The bar with a change in cross section in figure 2*b* was subjected to a uniaxial tensile force in the *x*-direction of 150 kN. Figure 12*a* shows the non-uniform axial strain field computed using the ultrasonic technique. This can be compared with the strain field for the bar with a constant cross section in figure 10*b*. In figure 12*a* the strain increases from 1×10^{−3} at *x*=15 mm to 3.5×10^{−3} at *x*=−10 mm, whereas the strain field in figure 10*b* is relatively uniform. In figure 12*a*, the increase in strain across the image is a result of the thickness of the bar decreasing from 40 mm for *x*>0 to 20 mm for *x*<0 mm. The figure also shows the strains increase as the *z*-coordinate increases.

Strain gauges, which were bonded to the surface of the 20 mm thick section (figure 2*b*), gave . Averaging the strains measured from the ultrasonic images in the 20 mm section gives 2.7×10^{−3}. It can be seen that the sub-surface ultrasonic strains are 3.9 times larger than the true surface strain. This is similar to the level of overestimation of axial strain observed in the previous section on the bar with a constant cross section. A static finite element (FE) simulation was performed using ABAQUS (Dassault Systèmes, Providence, RI, USA) to give an indication of the expected variation of strain within the sample. The FE simulation was validated by comparing surface strain data with the strain gauges and found to be in good agreement (the surface strain in the *x*-direction predicted by the FE simulation was 6.1×10^{−4} compared with 6.8×10^{−4} measured experimentally using strain gauges). The internal strain, , from the simulation of the same region as figure 12*a* and averaged over the element length in the *y*-direction (15 mm), are shown in figure 12*b*. The average strain over the whole area of the ultrasonic image is 4.66 times larger than the average strain over the same area predicted by the FE model, which is consistent with the results of §5*a*. The ratio of average strain between the *x*<0 and the *x*>0 portions of the images that correspond to the two different cross sections of the sample is similar in both the FE simulation and ultrasonic images (1.33 and 1.24, respectively).

## 6. Significance of the acousto-elastic effect

The axial strains measured in the uniform bar in §5 showed a significant systematic difference from those measured with strain gauges, and it was speculated that these results could be caused by the change in acoustic velocity in the material owing to the acousto-elastic effect (Hughes & Kelly 1953). In this section, a simple qualitative analysis for the case of hydrostatic loading is performed to assess whether the acousto-elastic effect is significant in the experimental ultrasonic array data.

The time taken for a pulse to travel between the transducer and a scatterer is given by
6.1where *d* is the propagation distance. The change in arrival time due to small changes in propagation distance, *δd*, and velocity, *δc*, is given by
6.2where
6.3

Assuming a hydrostatic pressure Δ*σ* is applied to the sample (i.e. resulting in the hydrostatic strain *ϵ*_{xx}=*ϵ*_{yy}=*ϵ*_{zz}=*ϵ*), the resulting strain is given by
6.4where *ν* is Poisson’s ratio and *E* is Young’s modulus. The change in the velocity of propagation of acoustic waves is given by
6.5Hence, the change in the arrival time of a reflected pulse from a scatterer for a pitch-catch time trace is given by
6.6where *r*_{m} and *r*_{n} are the distances to the scatterer from the transmitting and receiving elements and *σ* is the hydrostatic pressure. Term *A* represents the change in arrival owing to the movement of the scatterer position caused by the strain and is given by
6.7where *ν* is Poisson’s ratio and *E* is Young’s modulus. Term *B* is given by
6.8and is the change in arrival time owing to the acousto-elastic effect. The expression for the relationship between the rate of change of velocity of the acoustic waves with hydrostatic pressure (d*c*/d*σ*) for longitudinal waves is given by
6.9where *λ* and *μ* are the Lamé constants, *K*_{0}=*λ*+(2/3)*μ*, *l*, *m* and *n* are the Murnaghan constants and *ρ*_{0} is the density at zero pressure (Hughes & Kelly 1953). Table 2 gives values for *ν*, *E*, Lamé and Murnaghan constants for various engineering metals.

The basis of the strain measurement technique described so far is that the dominant change in *δt* in equation (6.6) is due to *A*, the movement of the scatterer, and not *B*, the acousto-elastic effect. The ratio of *A* to *B*, |*A*/*B*|, determines the significance of the acousto-elastic effect and values are given in table 2 for the different metals. For rail steel, which is thought to be similar to the stainless steel material used in the experiments, |*A*/*B*|=0.25. This means that term *B* dominates equation (6.6) and the acousto-elastic effect is larger than the direct strain effect. The table shows the ratio of |*A*/*B*| for the other engineering metals. The acousto-elastic effect is significant for many other engineering materials. However, the ratio of |*A*/*B*| for aluminium is significantly higher than all other metals, and this metal may give better consistency between the strains from the images and strain gauges. However, at the start of the experimental campaign, the significance of the acousto-elastic effect was not realized and samples were chosen instead based on the quality of the speckle images generated. In addition, had aluminium been used from the outset, the problems caused by the acousto-elastic effect in many other materials may not have been recognized at all.

When the acousto-elastic effect is significant, the imaging velocity must be known for accurate strain estimation from the images. In the experiments, a fixed value for velocity was used, and it is believed that the bias in experimental results in §5 is due to the errors caused by this.

For illustrative purposes, the hydrostatic case has been discussed as this enables a simple estimation of the relative severity of the acousto-elastic effect in different materials. For uniaxial loading, the acousto-elastic effect depends on the direction of the load relative to the direction of the propagating wave (Hughes & Kelly 1953). Using the equations in Hughes & Kelly (1953) and the Lamé and Murnaghan constants for steel (Egle & Bray 1976), the relationship between the velocity of longitudinal waves, *c*_{L}, and the angle of the direction of the propagating wave in the *x*−*z* plane relative to the *z*-axis, *θ*, is shown in figure 13. The relationship is plotted for different values of applied stress in the *x*-direction. This shows that when there is an applied stress, there is almost no change in the velocity in the *z*-direction, but a large change in the *x*-direction. Strained speckle images were simulated numerically for the experimental configuration considered in §5 using a collection of omnidirectional point scatterers in the single scattering regime and incorporating the acousto-elastic effect. This showed that when the true strains were and assuming *ν*=0.3, the measured strains were and . This shows that the strains in the *x*-direction are significantly overestimated whereas the strains in the *z*-direction are actually underestimated by a lesser amount. The overestimation by a factor of 6.4 predicted in the *x*-direction is of similar magnitude to the overestimation observed in the experimental results in §5*a*. The difference can reasonably be attributed to uncertainty of the Lamé and Murnaghan constants in the experimental samples.

## 7. Theoretical analysis of strain estimation errors

### (a) Lower bound on the strain estimation error

In this section of the paper, the theoretical lower bound on the strain estimation error (LBSE) is investigated when a single component of the strain is estimated from reference and deformed two-dimensional speckle sub-images corrupted by electronic noise. An unbiased estimator that attains the theoretical lower bound of the estimation error does not always exist. However, even if the lower bound is not achieved, it can provide a benchmark for evaluating the performance of the strain estimation techniques and motivate future research. The LBSE is shown to be a function of the image characteristics (bandwidth, SNR and sub-image size) and the spatial distance between displacement estimates that determines the resolution of the strain estimation technique.

### (b) Bounds on strain estimate errors

From equation (2.8), the strain estimate is a function of the two displacement estimates. Assuming the errors on the displacement estimates are random variables, the sub-images are stationary and the strain is constant throughout the material, the lower bound on strain variance is given by (Céspedes *et al.* 1995)
7.1where *σ*_{CRLB}{*u*_{x}} is the Cramér–Rao lower bound (CRLB) of the displacement estimate. Following the approach in Walker & Trahey (1995), the CRLB for the standard deviation of displacements estimated from correlated envelope signals with flat bandlimited spectra, and with additive electronic noise is:
7.2where *k*_{x} and *k*_{z} are the bandwidths of the flat spectra of the enveloped images and electronic noise in the *x*- and *z*-directions, and SNR is the ratio of the root mean square (r.m.s.) amplitudes of the noise and speckle (the SNR).

It is common to approximate the PSF of the ultrasonic imaging system in equation (2.1) as a two-dimensional Gaussian envelope modulated with a cosine in the *z*-direction (Meunier & Bertrand 1994):
7.3where the spatial frequency of the modulation is *k*_{0} (*k*_{0}=2*ω*/*c* taking into account the two-way propagation). *L*_{x} and *L*_{z} define the lengths of the principle axes of the PSF and are related to the pulse width and length of the aperture and proportional to the size of the speckles. This means the speckle signals can be defined in the frequency domain as Gaussian-shaped spectra. The bandwidth of an equivalent flat spectrum with the same mean square amplitude can be derived (Varghese & Ophir 1997):
7.4

Substituting equations (7.4) and (7.2) into equation (7.1) gives a useful relationship between the sub-image size to speckle size ratio, SNR and the lower bound on the standard deviation on the measured strain: 7.5

Typical speckle tracking and image parameters for enveloped speckle images of stainless steel 304 generated with a 5 MHz 128-element array and 16 averages are *X*=*Z*=5 mm, *L*_{j}=0.4 mm and SNR=12.5. The SNR was found by measuring the r.m.s. amplitudes of noise and speckle images at 16 averages. The noise was obtained by subtracting the speckle image at 16 averages with the same image at 512 averages (noise was assumed to be averaged to zero in this image). Using these parameters gives *σ*{*ϵ*_{xx}}=5×10^{−3}. This is an encouraging result as it is the same order of magnitude as the standard deviations estimated at zero strain in figure 11*a*. In the next section of the paper, equation (7.5) is used to deduce general trends about speckle tracking using ultrasonic array images.

### (c) Effect of increasing frequency

An indication of the size of strains of interest in engineering applications is given by the yield strain, which is typically around 1×10^{−3} for metals. The lowest strain standard deviation obtained in this paper was 3×10^{−4} (§4*d*). This suggests that at yield, a strain signal-to-noise ratio, SNR_{e}, of only around 3.3 is currently achievable with the proposed technique. One way to improve this is to obtain more speckles per unit area in the image by increasing the frequency and bandwidth of the ultrasonic pulse. Array technology is currently available for frequencies up to approximately 15 MHz. The model in §7 can be used to investigate the effect of frequency on the precision of strain estimation.

In this paper, the sub-images sizes in the *x*- and *z*-directions were equal (*X*=*Z*). Figure 5 shows the speckle sizes are approximately equal in both directions in the ROI (*L*_{x}=*L*_{z}). Therefore, from equation (7.5),
7.6

Assuming the frequency is increased from 5 to 15 MHz and the array geometry is scaled by the same amount, the speckle size, which is proportional to *L*_{x}, should be reduced by a factor of three. In this case, equation (7.6) shows the strain standard deviation decreases by a factor of nine. This suggests that strain signal-to-noise ratios as high as 9×3.3≈30 may be obtained using a 15 MHz array. However, increasing the frequency of the array may result in higher amounts of multiple scattering between the grains, because the amplitude of the scattering increases with the fourth power of the frequency for Rayleigh scattering (Saniie & Wang 1988), leading to speckle decorrelation. At higher frequencies, the aperture of an array with a given number of elements that satisfies the spatial sampling criterion is necessarily smaller and the ultrasonic attenuation is higher. The combination of these effects means that the working area of the technique is reduced and the maximum depth of measurement is limited.

## 8. Conclusions

This paper has investigated the feasibility of mapping internal displacement and strains in metals using speckle from ultrasonic array images. The technique has potential to work on any engineering material that exhibits speckle, such as metals and composites. The main advantage of the technique over other strain measurement techniques is that internal strains can be measured deep within the component. In contrast to radiographic techniques that also allow sub-surface strain measurement, the necessary ultrasonic equipment is relatively low cost and portable.

The standard deviations on the displacement and strain estimates were 1 μm and 3×10^{−4}, respectively, at a resolution of 5 mm and measured in a 20×20 mm area in the centre of the images (the ROI). The largest measured displacement was 4 mm and the highest strain detected was 9.6×10^{−3}, although in principle larger values can potentially be measured. The elastic–plastic stress–strain behaviour was measured in the ROI and the onset of plasticity was observed. The experiments showed that the strain measurements were accurate in the *z*-direction but axial strains were overestimated by a factor of 4.62. The technique was used to measure the strain in a bar with a stepped cross-section in uniaxial tension. The change in strain at the cross section was observed. An FE model of the same sample, validated using experimental surface strain gauge data, showed that the axial strains were overestimated by a factor of 4.66.

A brief theoretical analysis using the case of a material in hydrostatic load indicated that the large overestimation in the measured axial strain was due to the acousto-elastic effect, whereby the presence of load affects (anisotropically) the ultrasonic velocity. Hence, the motion of speckles in ultrasonic images is caused not just by the spatial movement of the material but also by the loading-induced changes in velocity. Strained speckle images were simulated for the case of uniaxial loading, using a point scatterer model and incorporating the acousto-elastic effect. Results from the simulation were in quantitative agreement with the experiments. This indicates a rather important limitation of applying the technique to quantitative strain measurement; extraction of the true strain in the presence of significant acousto-elasticity requires both the solution of a nonlinear inverse problem and prior knowledge of the third-order elastic constants of the material under examination.

## Acknowledgements

The authors are grateful to the EPSRC for research funding.

- Received September 19, 2010.
- Accepted January 17, 2011.

- This journal is © 2011 The Royal Society