# Replacing strain gages by line camera DIC in Hopkinson bar experiments

Christian C Roth, Foulques Legrelle, Thomas Tancogne-Dejean, Vincent Grolleau, Dirk Mohr

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Christian C Roth, Foulques Legrelle, Thomas Tancogne-Dejean, Vincent Grolleau, Dirk Mohr. Replacing strain gages by line camera DIC in Hopkinson bar experiments. International Journal of Impact Engineering, 2026, 212, pp.105620. 10.1016/j.ijimpeng.2025.105620. hal-05450423

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Short Communication

Replacing strain gages by line camera DIC in Hopkinson bar experiments

Christian C. Roth<sup>a,†</sup>, Foulques LeGrelle<sup>a</sup>, Thomas Tancogne-Dejean<sup>a</sup>, Vincent Grolleau<sup>a,b †</sup>, Dirk Mohr<sup>a,*</sup>

ARTICLE INFO

Keywords:

Hopkinson bars

SHPB

Direct impact

Strain gage

Optical measurement

Line camera

Digital image correlation

A B S T R A C T

Strain gages are widely used to acquire the signals in dynamic experiments with Hopkinson bars. Here, we explore the potential of displacement history measurements with line camera based digital image correlation (DIC) to substitute the role of strain gages and directly obtain particle velocity. After outlining the fundamental equations for deriving stress-strain curves, the technique is applied and validated through split-Hopkinson bar compression and tension tests, as well as direct impact experiments. In direct impact tests, the line camera enables simultaneous measurement of input and output forces, facilitating the verification of quasi-static equilibrium. Moreover, in cases where quasi-static equilibrium is clearly satisfied, a single line camera measurement on the striker bar is sufficient to determine the entire stress-strain curve. Compared to laser interferometry and photon Doppler velocimetry, the line camera DIC system demonstrates superior capability in measuring large displacements of Hopkinson bars. It also offers a reliable non-contact measurement alternative to strain gages, which are prone to delamination under high-impact conditions.

1. Introduction

Since the seminal works of Hopkinson [1] and Kolsky [2], the thor-
ough understanding of the wave propagation in bars has been exploited
to develop experimental systems that are suitable for measuring the
stress-strain response of engineering materials at high strain rates. The
central idea of a “Hopkinson bar” is to measure the displacement-, ve-
locity- or strain-history at a single location along a bar’s lateral surface
 Nurture
 provides access to the force- and displacement-histories at the bound-
aries of a specimen (secondary measurement) that is mechanically
connected to the base faces of the Hopkinson bar. In other words, a
Hopkinson bar serves as force transducer and position sensor in dynamic
experiments. Today, split-Hopkinson bar systems are widely used for
characterizing the rate-dependent response of materials [3].

There are many technical details related to the primary measurement

that define the accuracy of Hopkinson bars. Early systems relied on
measuring the displacement history at a specific location on a
compressively loaded bar [4]. With the advent of strain gages, the SHPB
technique saw a significant shift away from displacement towards strain
measurements [5–8]. In particular, progress on amplifier technology,
signal conditioners, high-speed acquisition cards, and semi-conductor
strain gages (e.g. [9]) improved the sensitivity and accuracy of the
strain measurements under dynamic loading conditions.

A variety of different non-contact measurement techniques have also
been developed. Inspired by work on polymer specimens [10], a laser
extensometer was used on a transparent PMMA bar in [11] to measure
the displacement history and to derive the particle velocity. The same
authors also investigated Photon Doppler Velocimetry (PDV), pointing
out the difficulty in observing the bar plane away from the free ends
[12]. Advances in high-speed photography enabled the measurement of
displacements on bar and specimen surfaces. For example, Jakkula et al.
[13] used a line-scan camera to monitor the displacement of the

**Dedication:** This note is dedicated to the memory of our friend, mentor, and close collaborator, Professor Gérard Gary of École Polytechnique, who unexpectedly passed away during the development of this manuscript. Professor Gary played a pivotal role in shaping this work, contributing the foundational idea of using line cameras to overcome some limitations of strain gages. He also foresaw benefits of integrating line camera measurements with wave deconvolution techniques for long-duration Hopkinson bar experiments. In addition, Professor Gary served as a co-advisor to the research of F. LeGrelle, offering invaluable guidance and support. His vision, expertise, and passion for Hopkinson bar experiments will continue to serve as an inspiration for advancing our field.

<footnote><sup>a</sup> Department of Mechanical and Process Engineering, ETH Zurich, Switzerland</footnote>

<footnote><sup>b</sup> Univ. Bretagne Sud, FRE CNRS 6027, IRDL, Lorient F-56100, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: mohr@mit.edu (D. Mohr).</footnote>
specimen/bar interfaces. Laser interferometry was used in [14] on the specimen free end, while an ultra-high speed optical camera observed a grid pattern applied to the specimen. Ultra-high-speed photography was employed in [15] to obtain the strain and acceleration fields, before identifying constitutive models with the help of the virtual fields method. These technological developments also revived the interest for direct impact experiments [16], with further improvement achieved by instrumenting the striker bar with standard strain gauges [13,17] or polyvinylidene fluoride (PVDF) stress sensors [18,19] to verify equilibrium or directly measure the force, respectively.

In this short communication, we propose a contactless measurement technique that uses an industrial grade high-speed line camera to monitor points on the surface of Hopkinson bars. With the help of digital image correlation (DIC), a Lagrangian measure of the surface displacement is obtained. After outlining the technical details and processing formulas, the proposed technique is applied to compression and tension split-Hopkinson bar systems. Furthermore, it is integrated into a direct impact bar system for compression testing. For the latter case in particular, the technique is very promising as it facilitates measuring both forces and displacements solely with the striker bar, thereby eliminating the need for an input bar.

## 2. Preliminaries

### 2.1. Waves in a Hopkinson bar

In 1D theory, the axial displacement of a material point of a Hopkinson bar (elastic wave propagation speed c, cross-sectional area A, Young's modulus E, mass density ρ) at an initial position x and at a time t may be described by the superposition of two waves, $u[x, t] = u_p[x - ct] + u_n[x + ct]$, with $u_p[x - ct]$ and $u_n[x + ct]$ corresponding to waves travelling in the positive and negative x-direction, respectively. The strain ε and particle velocity v are then obtained by spatial and temporal differentiation.

### 2.2. Standard SHPB configuration

In a standard split-Hopkinson pressure bar (SHPB) experiment, the specimen is positioned between an input and an output bar, while a striker bar is used to generate a rightward traveling compression wave in the input bar. With the x-axis pointing from the left to the right, we introduce the subscripts I (incident), R (reflected) and T (transmitted) to denote the rightward and leftward traveling waves in the input bar as well as the rightward traveling wave in the output bar. The strain histories $ε_I[t]$, $ε_R[t]$ and $ε_T[t]$, and similarly the velocities $v_I[t]$, $v_R[t]$ and $v_T[t]$, may be computed by evaluating the waves at the position of the specimen-bar boundaries. The widely-used Kolsky formulas are then derived from the strain, or the displacement and velocity measurements

$$ε_S[t] = -2 \frac{c}{L_S} \int_0^t ε_R[τ] dτ = -\frac{2}{L_S} u_R[t], \qquad (1)$$

$$σ_S[t] = \frac{A}{A_S} Eε_T[t] = -\frac{A}{A_S} ρcν_T[t]. \qquad (2)$$

When performing the experiment with Lagrangian displacement measurements (rather than conventional strain measurements), the last two formulas are particularly useful. Aside from the displacement history $u_R[t]$, the velocity history $v_T[t]$ is readily available by numerical differentiation of the corresponding displacement history $u_T[t]$.

### 2.3. Direct impact configuration

In a direct impact system, the striker bar applies the loading onto the specimen without any input bar. After impact at time t = 0, we then need to account for the effects of the initial velocity $V_0$ and a leftward

propagating wave $u_n[x + ct]$ in the striker bar. The axial displacement and velocity of a material point comprised in the striker bar reads $u[x, t] = V_0t + u_n[x + ct]$. Denoting the velocity history at the striker-specimen interface as $v_{in}[t]$, the strain in the striker bar at the same position reads $ε_{str}[t] = (v_{in}[t] - V_0)/c$. For identical bars and under the assumption of quasi-static equilibrium, the formulas for extracting the stress-strain curve from a direct impact experiment read:

$$ε_S[t] = \frac{1}{L_S} (V_0t - 2u_{in}[t]), \qquad (3)$$

$$σ_S[t] = \frac{A}{A_S} Eε_{str}[t] = \frac{A}{A_S} ρc(v_{in}[t] - V_0). \qquad (4)$$

The displacement $u_{in}[t]$ is obtained by wave transport (here without dispersion correction) from a displacement measurement $u_{LC}[t]$ at a distance $Δl_{LC}$ from the striker-specimen interface:

$$u_{in}[t] = u_{LC} \left[ t + \frac{Δl_{LC}}{c} \right] - V_0 \frac{Δl_{LC}}{c}. \qquad (5)$$

We emphasize that measuring the striker velocity is sufficient to derive the full stress-strain curve from a direct impact experiment. Conversely, a single strain history measurement (in the striker or the output bar) is insufficient as the initial striker velocity $V_0$ must be measured as well. An additional strain gauge mounted on the striker is used solely to verify equilibrium, a task that can equally be accomplished using the line camera approach. As an alternative to measuring $V_0$, the specimen strain could be measured directly with a high-speed camera.

## 3. Experimental procedures

### 3.1. Hopkinson bar setups

Three different experimental configurations with Hopkinson bars are used to demonstrate the capabilities of the line camera measurement technique:

I. A classic split-Hopkinson pressure bar (SHPB) system [2] with steel bars (density ρ = 8017kg/m³, celerity c = 4634m/s) (Fig. 1a). Opposing strain gages are glued at the bar mid-point on both bars.

II. A modified split-Hopkinson pressure bar system with purpose-built load inversion device for tensile testing [20] (Fig. 1b). Strain gages are positioned at 430 mm from the specimen-output bar interface (ρ = 7753kg/m³, c = 5138m/s) to calculate the axial force histories.

III. A Direct Impact Hopkinson Pressure Bar (DIHPB) system with aluminum bars (ρ = 2703kg/m³, c = 4984m/s) (Fig. 1c). Strain gages are positioned at 250 mm from the specimen-bar interface.

All strain gages are connected either as full Wheatstone bridges (setup I, III) or half Wheatstone bridges (setup II) to an amplifier (Micro Measurements 2210B, excited at 6V), resulting in a strain resolution of the order of $10^{-5}$ strain. Together with a data acquisition card (National Instruments NI USB-6366), the signals from the strain gages are recorded at 400 kHz. In setup II (resp. III), a Photron SA-Z high-speed camera is used to observe the specimen surface (resp. the specimen-bar interface) at 100 kHz to enable surface DIC and measure the displacement.

### 3.2. Particle velocity measurement with line camera DIC

The particle velocity of a point on a Hopkinson bar is determined by performing DIC on line camera images. A line camera (JAI model SW-4000M-PMCL) with a 1 × 4096 pixel sensor and a 1:1 macro lens (Tamron f3.5 180 mm) is used in our demonstration experiments. Line images featuring a resolution of about 7.2 µm/pix are acquired at a
<figure><img src="image_2.png" /><figcaption>Fig. 1. Hopkinson bars used in this work (a) System I: Kolsky setup with line camera (red) mounted on input bar (b) System II: Modified SHPB bar setup with load inversion device. The line camera (red) observes the output bar. An optical high-speed camera (black) observes the specimen surface. (c) System III: Direct Impact Hopkinson Pressure Bar setup. The line camera (red) observes the striker just before and during impact. The optical high-speed camera (black) observes the specimen-bar interface.</figcaption></figure>

frequency of 200 kHz. An f8 aperture is chosen to minimize optical aberrations such as distortion and vignetting. The approximately 30 mm long bar-code like pattern is applied onto the respective bar surfaces using a black permanent marker that is held in a purpose-built fixture allowing for bar rotation. The patterns feature a high contrast with a minimum line width of about 0.5 mm, which corresponds to about 70 pixels (Fig. 3c). Uniform lighting of the pattern is achieved with a line light (LNSP2–200SW, CSS Inc.). Fig. 2a illustrates the system as mounted on setup I.

At a given time $t_n$, we identify the axial displacement $u$ of a point with the initial pixel coordinate $X_0$ (in the line camera image) through digital image correlation. For this, we define a pixel subset of length $L$ (from the pixel position $X_0 - L/2$ to the position $X_0 + L/2$) within the pattern. Subsequently, the displacement $u$ is obtained with a local search optimization algorithm, minimizing the cost function

$$ \min_{u} \left( \sum_{i=1}^{L} \left( g[X_0 - L/2 + i, 0] - g[X_0 - L/2 + i - u, t_n] \right)^2 \right). \quad (6) $$

<figure><img src="image_3.png" /><figcaption>Fig. 2. Line camera system: (a) 3D-visualization of the experimental set up on setup I. (b) Compression experiment signal as observed by the line camera on the incident bar. (c) Lagrangian displacement and speed history obtained for the image in Fig. 2b (d) Superposed strain gage signals (transported) and line camera measurements of the compression experiment.</figcaption></figure>
In (6), the function $[x, t]$ corresponds to a linear interpolation of the normalized gray level values (affine transformation to interval [20]). A convergence criterion of 0.01 pixels is selected.

To increase performance, the algorithm is bounded by an estimate of the maximum attainable particle velocity for the setup used (about 20m/s for our system). Gray-level normalization is performed to further minimize vignetting and changes in lighting. The numerical time derivative of the obtained discrete displacement history $u[t_n]$ provides the particle velocity $v[t_n]$ of the bar at the center location of the subset. To improve numerical derivative accuracy, the displacement is spatially averaged after repeating the correlation for other points in the vicinity of $X_0$. Due to the space-time analogy in wave signals, the spatial averaging may also be interpreted as time averaging. For our specific camera-pattern configuration and convergence criterion, the lowest attainable speed sensitivity is about 0.014 m/s, corresponding to a strain sensitivity of $3.11 \times 10^{-6}$. The simple least square difference of normalized gray levels approach is chosen, as only minimum straining and no significant changes in contrast are observed in the region of interest, that would warrant more complex correlation criteria, e.g. zero-mean normalized sum of squared difference.

# 4. Results

## 4.1. Line camera sensitivity and uncertainty estimation

The sensitivity and accuracy of the line camera measurement technique are influenced by several factors, including (i) acquisition-related aspects such as camera positioning, alignment, lighting conditions, and pattern design, and (ii) post-processing factors such as the choice of pattern-tracking algorithm as well as the selecting of its processing parameters. They are assessed by exciting the 20 mm diameter steel incident bar of the SHPB setup I (Fig. 1a) with a spring-mounted 15 mm diameter steel ball. Fig. 3a shows the strain histories in the as-recorded

absolute time frame, while the transported line camera signal is shown along with the (standard) strain gage signal in Fig. 3b. The line camera velocity measurement is converted into a strain signal by applying the identity $e[t] = v[t]/c$ (neglecting the negative sign to convert the signal for the leftward traveling wave). Both the strain gage and line camera signals show good agreement, resolving strain levels of the order of $10^{-5}$ accurately.

Similar to strain gages, correct alignment of the line camera with respect to the bar axis is crucial to obtain accurate measurements. As the camera can only observe a very thin line, i.e. about 30 mm x 7.5 µm in the present case, we make use of a laser level to detect any misalignment around the Z axis directly on the image. In a second step, the reflection of the aligned line light on an unpainted section of the bar is used to align the X and Z axes with the bar (Fig. 2a). Trigonometric considerations suggest that a misalignment of 5 degrees around the X or Z axis would result in axial displacement measurement errors of about ±0.4 %, respectively. In addition to misalignment, errors in the displacement measurement can also originate from the scale of the line camera. In the present case, the measured scale of the setup is 7.2 ± 0.06 µm/pixel, which implies an uncertainty in the reported displacements and velocities of up to 0.9 %.

Lighting and the resulting pattern contrast also affect the measurement quality. While the average signal remains largely unaffected, limited contrast can significantly impact noise levels. For the highly reflective steel of setup I, gray levels from 12–235 (out of 255) are observed, resulting in signal noise levels of about 0.7 % ($9 \times 10^{-6}$ in strain). In contrast, for the much more matte tensile output bar, only gray levels between 20–120 are achieved (inset Fig. 4b), which, despite gray level normalization, result in up to 14 % noise ($1.7 \times 10^{-5}$ in strain). Fig. 3c shows the gray value distribution for a reference subset (red line) and the same subset 5 ms (1000 frames) and 7.7 mm later, transported with the pattern tracking algorithm.

Besides lighting, the choice of the subset size for the pattern tracking

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Results of the sensitivity analysis of the line camera system: (a) Overall and (b) close-up signal as recorded by the strain gage (black line) and the line camera (transported, red line) for setup I. (c) Subset tracking: 7.7 mm in displacement and 1000 frames (=5 ms) lie between the two subsets. (d) Evolution of DIC floor noise and signal noise for different subset sizes.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Results from demonstration experiments: (a) Engineering stress-strain curve for compression of a cylindrical AA6082 specimen computed with Kolsky equations. (b) Tensile experiment on a 1.5 mm thick DC01 notched tension specimen. Strain gage (black line) and line camera (red line) are used to obtain redundant output force measurements. The optical high-speed camera is used to measure specimen displacement (at blue dots shown in inset). (c) Strain-time results for a direct impact experiment on a cylindrical plywood specimen. The line camera measures the leftward traveling wave in the striker bar (red line), while the strain gage records the rightward traveling wave in the output bar (black line). The agreement of both strain histories validates the equilibrium assumption. (d) Engineering stress-strain curves extracted from (c). Direct impact results fully computed with Kolsky equations (green line), using a high-speed camera to obtain the engineering strain on the bar ends (black line) and completely based on the striker displacement obtained from the line camera (orange line).</figcaption></figure>

has a significant impact on signal noise. For the chosen convergence
criterion of 0.01 pixel, pattern tracking is performed for 30 subset sizes
at 26 different positions with a spacing of 200 pixels within the pattern
without motion over 100 images. For each subset size, the spatial
displacement average is obtained via pattern tracking and its standard
deviation computed over all images. Furthermore, the speed is
computed by taking the finite difference of the average displacement
along with its standard deviation. Fig. 3d shows the resulting noise on
the displacement and the speed. It follows that particularly small subsets
may introduce very large errors. The best compromise between sensi-
tivity and computational time is found for a subset size of 400 pixels,
resulting in a floor noise of 0.005 pixels in displacement and 0.008 m/s
in speed.

## 4.2. Application I: Measuring input bar waves in SHPB experiments

A cylindrical specimen (aluminum alloy AA6082, diameter 10 mm, length 12.8 mm) is subjected to compressive loading at a strain rate of about 400/s on the SHPB system (setup I, Fig. 1a). The line camera pattern is applied at a distance of 2050 mm from the bar-specimen interface (near the input bar center). The recorded line images are shown in Fig. 2b, while Fig. 2c shows the DIC extracted displacement history (black curve) along with the corresponding particle speed history (blue curve). Fig. 2d shows the line camera strain estimate (red curve) against the corresponding strain gage measurements (black curve). The good agreement confirms the validity of the proposed line camera measurement technique. Note that the Fourier transform of the

incident signal is in excellent agreement up to 25 kHz, with minor de-
viations for higher frequencies. Due to the lower acquisition frequency,
the line camera signal shows less ringing.

Using standard Kolsky equations the engineering stress is computed
from the transmitted signal acquired by the strain gage on the output bar
(Eq. (2)), while the engineering strain is obtained either: (i) from the
reflected strain signal of the strain gage on the input bar (Eq. (1)) (black
solid line, Fig. 4a) or (ii) from the displacement signal of the line camera
(red solid line, Fig. 4a). The results of the two measurements agree well,
only showing slight difference at the end of the elastic range.

4.3. Application II: Force measurement in SHB tension experiments

A notched tension specimen made from 1.5 mm thick DC01 deep
drawing steel is subjected to tensile loading at about 3.3 m/s with setup
II (Fig. 1b). The optical high-speed camera is used to measure the
displacement on the specimen surface (relative displacement of the two
blue dots shown in Fig. 4b). In this application, the line camera is
positioned on the output bar, approximately 500 mm from the
specimen-bar interface to obtain the force history. The line camera raw
signal is filtered with a lowpass filter at 50 kHz. For validation purposes,
strain gages are used near the same location. A good agreement in the
force signal from the strain gage (black solid line) and the line camera
(red solid line) is obtained. However, it is noted that the noise in the line
camera signal is higher in the present application due to the lower strain
levels in the output (as compared to the strains in the input bar of
application I).
## 4.4. Application III: Direct impact experiments

A direct impact experiment is conducted on a cylindrical plywood specimen (diameter 18.9 mm, thickness 12.1 mm) with an initial striker speed of 6.29 m/s using setup III (Fig. 1c). The line camera begins to capture the bar-code pattern on the striker, located ~50 mm from its right end, just before impact, allowing a Lagrangian measurement of the striker's displacement and speed profile $v_{in}$ [t]. The strain in the striker bar is calculated from the strain-velocity relationship of a leftward traveling wave (striker impact velocity $V_0$), while a strain gage measures the rightward traveling wave in the output bar. The line camera signal (striker bar) and the strain gage signal (output bar) are then transported to the corresponding specimen-bar interfaces. Fig. 4c shows the unfiltered strain histories from the strain gage on the output bar (black line) and the line camera on the striker bar (red line). The difference between the two curves is limited to the measurement noise, and their good agreement further validates the line camera technique. It is also worth noting that it is technically difficult to perform striker bar measurements with conventional strain gages (due to the inertia of the required wiring and its interference with striker bearings). Furthermore, the line camera technique provides both an impact velocity and strain measurement which is another advantage over strain gages in direct impact experiments. For this setup and lighting conditions, the standard deviation on the speed before impact is 0.14m/s.

Fig. 4d shows the engineering stress-strain curve calculated using three methods: (i) output bar strain gage for the stress recording along with the measured engineering strain from the bar ends with high-speed camera DIC (black line), (ii) combining the strain gage and line camera displacement measurement (green line) and (iii) using the line camera displacement and velocity measurement only (Eqs. (3) and (4), orange line). As in the previous examples, we observe good agreement which confirms the validity of proposed line camera measurements.

## 5. Conclusions

This short communication proposes using an industrial-grade high-speed line camera, combined with line digital image correlation (DIC) to measure the displacement history on the surface of Hopkinson bars during dynamic experiments. After presenting the fundamental processing equations for extracting stress-strain curves, the line camera technique is applied and validated through split-Hopkinson bar compression and tension experiments, as well as direct impact tests. In the latter, the line camera enables measuring the input force in addition to the output force, allowing verification of quasi-static equilibrium. Additionally, when quasi-static equilibrium is readily achieved, a single line camera measurement on the striker bar suffices to determine the complete stress-strain curve. Compared to laser interferometry or photon Doppler velocimetry, the line camera DIC system offers the advantage of measuring large displacements of Hopkinson bars. It also provides a robust alternative to strain gauges that may delaminate at high impact speeds.

### CRediT authorship contribution statement

**Christian C. Roth:** Writing – original draft, Visualization, Project administration, Methodology, Investigation. **Foulques LeGrelle:** Writing – original draft, Visualization, Software, Investigation, Data curation. **Thomas Tancogne-Dejean:** Writing – original draft, Visualization, Methodology, Investigation. **Vincent Grolleau:** Writing – original draft, Resources. **Dirk Mohr:** Writing – review & editing, Writing – original draft, Supervision, Funding acquisition, Formal analysis.

### Declaration of competing interest

Given his role as Associate Editor, Dirk Mohr had no involvement in

the peer review of this article and had no access to information regarding its peer review. Full responsibility for the editorial process for this article was delegated to another journal editor. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

The authors would like to thank Mr. Erik de Best for his help with the line camera installation.

### Data availability

No data was used for the research described in the article.

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