Measured Displacement Field Research Articles

Recovery of Tractions Exerted by Single Cells in Three-Dimensional Nonlinear Matrices.

Cell-generated tractions play an important role in various physiological and pathological processes such as stem-cell differentiation, cell migration, wound healing, and cancer metastasis. Traction force microscopy (TFM) is a technique for quantifying cellular tractions during cell-matrix interactions. Most applications of this technique have heretofore assumed that the matrix surrounding the cells is linear elastic and undergoes infinitesimal strains, but recent experiments have shown that the traction-induced strains can be large (e.g., more than 50%). In this paper, we propose a novel three-dimensional (3D) TFM approach that consistently accounts for both the geometric nonlinearity introduced by large strains in the matrix, and the material nonlinearity due to strain-stiffening of the matrix. In particular, we pose the TFM problem as a nonlinear inverse hyperelasticity problem in the stressed configuration of the matrix, with the objective of determining the cellular tractions that are consistent with the measured displacement field in the matrix. We formulate the inverse problem as a constrained minimization problem and develop an efficient adjoint-based minimization procedure to solve it. We first validate our approach using simulated data, and quantify its sensitivity to noise. We then employ the new approach to recover tractions exerted by NIH 3T3 cells fully encapsulated in hydrogel matrices of varying stiffness. We find that neglecting nonlinear effects can induce significant errors in traction reconstructions. We also find that cellular tractions roughly increase with gel stiffness, while the strain energy appears to saturate.

Robust Multiscale Identification of Apparent Elastic Properties at Mesoscale for Random Heterogeneous Materials with Multiscale Field Measurements.

The aim of this work is to efficiently and robustly solve the statistical inverse problem related to the identification of the elastic properties at both macroscopic and mesoscopic scales of heterogeneous anisotropic materials with a complex microstructure that usually cannot be properly described in terms of their mechanical constituents at microscale. Within the context of linear elasticity theory, the apparent elasticity tensor field at a given mesoscale is modeled by a prior non-Gaussian tensor-valued random field. A general methodology using multiscale displacement field measurements simultaneously made at both macroscale and mesoscale has been recently proposed for the identification the hyperparameters of such a prior stochastic model by solving a multiscale statistical inverse problem using a stochastic computational model and some information from displacement fields at both macroscale and mesoscale. This paper contributes to the improvement of the computational efficiency, accuracy and robustness of such a method by introducing (i) a mesoscopic numerical indicator related to the spatial correlation length(s) of kinematic fields, allowing the time-consuming global optimization algorithm (genetic algorithm) used in a previous work to be replaced with a more efficient algorithm and (ii) an ad hoc stochastic representation of the hyperparameters involved in the prior stochastic model in order to enhance both the robustness and the precision of the statistical inverse identification method. Finally, the proposed improved method is first validated on in silico materials within the framework of 2D plane stress and 3D linear elasticity (using multiscale simulated data obtained through numerical computations) and then exemplified on a real heterogeneous biological material (beef cortical bone) within the framework of 2D plane stress linear elasticity (using multiscale experimental data obtained through mechanical testing monitored by digital image correlation).

3D DIC-assisted residual stress measurement in 316 LVM steel processed by HE and HPT

A method has been developed for determining residual stress based on displacement fields near drilled holes analyzed using 3D digital image correlation. Finite element modeling was used to determine corrections for analytical equations describing displacement fields near the blind holes, which made it possible to determine the residual stress distribution over a wide range of hole depth-to-hole diameter ratios and various areas of displacement field measurements using inverse method iterative calculations. The proposed method eliminates many drawbacks of traditional procedure based on strain gauges as hole eccentricity sensitivity and requirement of the relatively large span between holes. The method and testing setup, build-up of generally available components, were used to determine the residual stress distribution for 316 LVM samples processed by two methods from the large deformation group: hydrostatic extrusion (HE) and high-pressure torsion (HPT), by drilling 1.75 and 0.58-mm-diameter blind holes, respectively. In the case of the measurements performed on the surface of a HE-processed 16 mm bar cut along its diameter, a gradual change was revealed—from a compressive to a tensile residual stress distribution (from ~ − 300 MPa in the center to 400 MPa in 4 mm distance from the edge) in the longitudinal direction, with near-zero values in the radial direction. Moreover, the method was also adapted to perform measurements on the outside surface of the bar, which gave results consistent with those taken along the radius profile (~ 600 MPa longitudinal stress). Measurements on the top surface of a cylinder 10 mm in diameter and 1 mm high processed by HPT showed a high compressive residual stress in the center and a dominant shear component for the holes drilled at different distances from the center.

Measurement of 1\u201310\xa0Hz 3D vibration modes with a CT-scanner

A new in situ vibration mode measurement method within a tomograph is proposed based on Projection-based Digital Volume Correlation techniques. Several projection angles are selected and a large number of radiographs of the vibrating sample are acquired at random instants with a small exposure time in order to ‘freeze out’ the displacement and avoid motion blurring. Based on an initial reconstruction acquired in a static configuration, the displacement field measurement is performed using a Proper Generalized Decomposition technique. All projections are analyzed as being due to a few vibration modes deforming the known reference volume. The different projection directions are related to each other assuming that the modal amplitude probability distribution functions are statistically similar. A synthetic test case, mock-up of a liver, is used to illustrate and validate the approach. In this case, 5 projection angles were chosen, 300 radiographs per angle, and the first three vibration modes could be recovered with a good accuracy.

Effect of reference region size on strain measurements using geometrical phase analysis.

Geometrical phase analysis (GPA) is typically a powerful tool to investigate the deformation in high resolution transmission electron microscopy images and has been used in various fields. During GPA, strain components are calculated relative to an undistorted reference region. In the present work, the effect of reference region size on strain measurements has been investigated. Experimental measurements on a locally distorted gold nanoparticle exhibited that a small reference region below the GPA spatial resolution can introduce an inaccuracy in the measured displacement field, which appears as a significant increase in measured strains and severe fluctuation in phase images. The inaccuracy may be ascribed to an error of insufficient sampling. Our results suggest that a small reference region below the GPA spatial resolution should be avoided during GPA. This prerequisite should be paid more attention to during strain measurement on nanoparticles.

Three-dimensional traction microscopy accounting for cell-induced matrix degradation

Tractions exerted by cells on the extracellular matrix (ECM) are critical in many important physiological and pathological processes such as embryonic morphogenesis, wound healing, and cancer metastasis. Three-dimensional Traction Microscopy (3DTM) is a tool to quantify cellular tractions by first measuring the displacement field in the ECM in response to these tractions, and then using this measurement to infer tractions. Most applications of 3DTM have assumed that the ECM has spatially-uniform mechanical properties, but cells secrete enzymes that can locally degrade the ECM. In this work, a novel computational method is developed to quantify both cellular tractions and ECM degradation. In particular, the ECM is modeled as a hyperelastic, Neo-Hookean solid, whose material parameters are corrupted by a single degradation parameter. The feasibility of determining both the traction and the degradation parameter is first demonstrated by showing the existence and uniqueness of the solution. An inverse problem is then formulated to determine the nodal values of the traction vector and the degradation parameter, with the objective of minimizing the difference between a predicted and measured displacement field, under the constraint that the predicted displacement field satisfies the equation of equilibrium. The inverse problem is solved by means of a gradient-based optimization approach, and the gradient is computed efficiently using appropriately derived adjoint fields. The computational method is validated in-silico using a geometrically realistic neuronal cell model and synthetic traction and degradation fields. It is found that the method accurately recovers both the traction and degradation fields. Moreover, it is found that neglecting ECM degradation can yield significant errors in traction measurements. Our method can extend the range of context where tractions can be appropriately measured.

Coupling between topology optimization and digital image correlation for the design of specimen dedicated to selected material parameters identification

The work addresses the optimization of specimen geometry in the context of parameter identification from full-field measurements. For this purpose, we propose to use topology optimization tools in order to maximize the sensitivity of the measured displacement field to sought parameters, under volume fraction constraints and without any a priori information on the specimen shape. The associated cost function is conveniently defined from sensitivity fields which are directly available from the identification procedure. In addition, further analyzes are conducted in order to fully set the theoretical and numerical aspects of the methodology, and enhance both stability and convergence properties. Numerical experiments illustrate the interest of the proposed approach, while its validation is performed from a real identification test using a specimen with optimized topology.

On the Optimal Pattern for Displacement Field Measurement: Random Speckle and DIC, or Checkerboard and LSA?

This paper deals with the optimal pattern that can be used to retrieve displacement fields by minimizing the optical residual calculated over small regions of contrasted images. This minimization is generally performed in the spatial domain by processing speckle patterns with DIC. Another option is also considered here. It consists in switching this minimization to the Fourier domain. The benefit is that periodic patterns can be processed, which is generally not possible with DIC. It turns out that the optimal pattern in terms of sensor noise propagation is theoretically the checkerboard if it is correctly sampled, and this pattern is periodic. The reason why checkerboard is optimal is that the image gradient is maximum in this case. In addition, the minimization of the image residual in this case has a quasi-direct solution, which considerably speeds up the calculations. We first recall the basics of the different techniques used in the paper, namely classic subset-based DIC, and a spectral method called Localized Spectrum Analysis (LSA). A recent deconvolution procedure introduced to enhance the metrological performance of DIC and LSA is also briefly recalled and used in this study. Synthetic images are considered to assess in different cases the displacement resolution, as well as other sources of spurious spatial fluctuations observed in the displacement fields such as the pattern-induced bias with DIC. The main conclusion is that using checkerboards instead of random speckles leads to measurements featuring a better compromise between spatial resolution and measurement resolution.

Slowness curve surface acoustic wave transducers for optimized acoustic streaming.

Surface acoustic waves can induce force gradients on the length scales of micro- and nanoparticles, allowing precise manipulation for particle capture, alignment and sorting activities. These waves typically occupy a spatial region much larger than a single particle, resulting in batch manipulation. Circular arc transducers can focus a SAW into a narrow beam on the order of the particle diameter for highly localised, single-particle manipulation by exciting wavelets which propagate to a common focal point. The anisotropic nature of SAW substrates, however, elongates and shifts the focal region. Acousto-microfluidic applications are highly dependent on the morphology of the underlying substrate displacement and, thus, become dependent on the microchannel position relative to the circular arc transducer. This requires either direct measurement or computational modelling of the SAW displacement field. We show that the directly measured elongation and shift in the focal region are recapitulated by an analytical model of beam steering, derived from a simulated slowness curve for 128° Y-cut lithium niobate. We show how the negative effects of beam steering can be negated by adjusting the curvature of arced transducers according to the slowness curve of the substrate, for which we present a simple function for convenient implementation in computational design software. Slowness-curve adjusted transducers do not require direct measurement of the SAW displacement field for microchannel placement and can capture smaller particles within the streaming vortices than can circular arc IDTs.

Initial Displacement Estimation Based on ESA in Improved Digital Volume Correlation and Its Application

In order to improve the algorithm efficiency in point matching and reduce the error of displacement field in deformation calculation, an Improved Digital Volume Correlation (IDVC) method is proposed. Aimed at the problem that the accuracy of initial displacement estimation of DVC has a great influence on the convergence calculation accuracy by the sub-voxel measurement algorithm, the coordinates of the estimated points in deformation are quickly matched by the Explosion Search Algorithm (ESA), with the displacement vectors obtained. Then, the non-physical fluctuation of displacement field and the wrong strain concentration are suppressed in regularization. Finally, the deformation information obtained by correlation search is substituted into the iterative least squares algorithm to attain the sub-voxel precision displacement. The results of the simulation experiment and concrete uni-axial compression experiment show that the computational efficiency by the proposed method is 20.44% higher than that by the traditional DVC method, and that the accuracy level of displacement field measurement reaches 0.001 voxels.

Dynamic Behavior of Track Bridge on Serviced Long-Span Road/Railway Bridge

In this study, we predicted the structural behavior of a track-bridge and a bridge based on the expected increases in future train speeds by modeling the Yeongjong Bridge (a part of the Incheon International Airport Expressway in Korea). To verify the train’s safe operation, we performed a three-dimensional (3D) numerical analysis using full-scale bridge modeling. The rail-girder interaction force generated at one end of the direct fixation track of Yeongjong Bridge during train operations was evaluated by taking field measurements of the vertical displacements of the rail and girder at the center and end of a track girder. We further compared our predictions with various field measurements to evaluate the dynamic behavior of the entire Yeongjong Bridge.

Optimisation based material parameter identification using full field displacement and temperature measurements

A material parameter identification is presented for a fully thermo-mechanically coupled material model based on full field displacement and temperature measurements. The basic theory of the inverse problem is recapitulated, focusing on the choice of the objective function, proposing a new formulation, and explaining in detail the necessary numerical treatment of experimental data during the pre-processing of an identification. This includes the handling of the intrinsically different data sets of displacement (Lagrangian type) and temperature (Eulerian type). Experimental data is obtained by means of a Digital-Image-Correlation (DIC) as well as by a thermography system and three algorithmic boxes are provided for the necessary pre-processing. The experimental setup is discussed, measured data presented and analysed. From this setup, a successive approach to the identification process is motivated. Based on the experimental observations, a thermo-mechanically coupled material model is chosen, the required constitutive relations summarised and the material parameters interpreted. For the fixed choice of model and experiments, the inverse problem is solved. A very good fit was obtained for both the displacement and the temperature field. Results are interpreted and remaining errors discussed.

The role of stiffness variation in switches and crossings: Comparison of vehicle–track interaction models with field measurements

The performance of switches and crossings compared with plain line is complicated by the presence of movable parts, changing rail geometry and non-uniformities in the composite and/or trackbed stiffness. These features lead to complex vehicle–track interactions and higher maintenance costs. The trackbed stiffness is the least well-controlled engineering property. A greater variability in trackbed stiffness leads to higher differential trackbed settlement and associated poorer track quality. At switches and crossings, changes in trackbed stiffness are exacerbated by changing rail properties which also contribute to changes in the overall composite track stiffness. This work focuses on the role of variations in stiffness on the performance of switches and crossings. Field measurements of bearer displacement were carried out using geophones at a switch and crossing equipped with under sleeper pads. The vehicle–switch and crossing interaction was modelled using a multi-body system and finite element method. The trackbed stiffness along the whole of the switch and crossing was inferred using the measurements of track deflections in an iterative back-calculation taking account of changing rail properties. It is shown that not including the variation in trackbed/composite stiffness leads to significant under/overestimates of the wheel–rail contact forces. Under sleeper pads are shown to reduce absolute maximum loads, but may increase the variation in deflection.

Unipolar MR elastography: Theory, numerical analysis and implementation.

In MR elastography (MRE), zeroth moment balanced motion‐encoding gradients (MEGs) are incorporated into MRI sequences to induce a phase shift proportional to the local displacement caused by external actuation. To maximize the signal‐to‐noise ratio (SNR), fractional encoding is employed, i.e., the MEG duration is reduced below the wave period. Here, gradients encode primarily the velocity of the motion‐reducing encoding efficiency. Thus, in GRE‐MRE, T2* decay and motion sensitivity have to be balanced, imposing a lower limit on repetition times (TRs).We propose to use a single trapezoidal gradient, a “unipolar gradient”, to directly encode spin displacement. Such gradients cannot be used in conventional sequences as they exhibit a large zeroth moment and dephase magnetization. By time‐reversing a spoiled SSFP sequence, the spoiling gradient becomes an efficient unipolar MEG. The proposed “unipolar MRE” technique benefits from this approach in three ways: first, displacement encoding is split over multiple TRs increasing motion sensitivity; second, spoiler and MEG coincide, allowing a reduction in TR; third, motion sensitivity of a typical unipolar lobe is of an order of magnitude higher than a bipolar MEG of equal duration.In this work, motion encoding using unipolar MRE is analyzed using the extended phase graph (EPG) formalism with a periodic motion propagator. As an approximation, the two‐transverse TR approximation for diffusion‐weighted SSFP is extended to incorporate cyclic motion. A complex encoding efficiency metric is introduced to compare the displacement fields of unipolar and conventional GRE‐MRE sequences in both magnitude and phase. The derived theoretical encoding equations are used to characterize the proposed sequence using an extensive parameter study. Unipolar MRE is validated against conventional GRE‐MRE in a phantom study showing excellent agreement between measured displacement fields. In addition, unipolar MRE yields significantly increased octahedral shear strain‐SNR relative to conventional GRE‐MRE and allows for the recovery of high stiffness inclusions, where conventional GRE‐MRE fails.

A benchmark testing technique to characterize the stress–strain relationship in materials based on the spalling test and a photomechanical method

The experimental testing of concrete and other brittle materials under high-strain rate tensile loading remains a major issue in many research fields and industrial applications. Among the experimental methods, the spalling technique provides a major advantage as it does not rely on a static mechanical balance of the sample. A short compressive pulse transmitted to the sample reflects from the rear free surface as a tensile pulse leading to sample tensile fracturing. However, the classic processing (i.e. Novikov’s acoustic approximation) based on the measurement of the sample rear face velocity profile, relies on strong assumptions that can lead to an over-estimation of the material tensile strength. A new processing technique was proposed by Pierron and Forquin (2012 Strain 48 388–405) to derive the stress field and apparent Young’s modulus using only data from an ultra-high speed camera. This purely inertial method is based on the time resolved measurement of axial displacement fields and the virtual fields methods (VFM). However, the measurement accuracy as well as uncertainty remains to be studied in more detail and better understood. In the present work, a series of spalling tests have been performed with a sample made of aluminium alloy behaving linearly elastic. Such test allows investigating a possible influence of several testing parameters (e.g. interframe time, lens focusing, number of pixels per period…), processing parameters (mainly the levels of spatial and temporal smoothing for regularization) as well as the influence of the camera sensor technology. The results are compared to data from strain gauges and to the expected Young’s modulus and stress–strain response of the tested aluminium alloy, so the accuracy of measurements is evaluated. Finally, the spalling set-up applied to a sample of well-known mechanical properties appears to be a good benchmark to assess the quality of stress versus strain measurements based on photomechanical methods.

Characterizing fracture response of cracked transversely graded materials

Property gradation along a crack front makes conventional fracture mechanics analyses challenging. This issue is addressed in the present study through a combination of full-field measurements and modeling. We develop a novel experimental setup that consists of a synchronous dual Digital Image Correlation (DIC) system, facilitating in situ measurement of displacement and strain fields on both sides of a transversely precracked multi-layered structure subjected to uniaxial tensile load. By measuring the evolution of stress intensity factors developed on the opposite sides of the sample, we explore the mechanisms of crack initiation and propagation in the examined graded sample. Our experimental measurements are supplemented by a finite element analysis that elucidates the deformation and fracture response of the internal layers for which surface measurement is not possible. Finally, based on experimental and numerical observations, we develop a simple model that allows for prediction of critical far-field loads at which transversely graded structures fail. The capability of our proposed model in predicting tensile failure loads is demonstrated through a brief study of the influence of gradient function on the load bearing and fracture resistance in various graded structures.

Quantifying the anisotropic linear elastic behavior of solids

In this work, we present a non-destructive approach to characterize the heterogeneous mechanical behavior of anisotropic elastic solids. This approach is based on iterative inverse algorithms using the framework of finite element discretization schemes. We test the proposed approach with several theoretical studies and observe that for a low noise level in measured displacement fields, it is possible to map the anisotropic linear elastic parameter distributions with high accuracy. We also observe that with additional displacement field measurements, the solution to the inverse problem becomes more unique. However, for higher noise levels (3%), the quality of the reconstructions deteriorates for the parameter C12 of the orthotropic elasticity tensor. A theoretical analysis is performed and we demonstrate that this may be due to small gradients of the objective function with respect to C12. Overall, the proposed approach has potential to characterize the anisotropic mechanical behavior of polymer based materials or tissues for pathology.

The Coupled Adjoint-State Equation in forward and inverse linear elasticity: Incompressible plane stress

A persistent challenge present in inverse or parameter estimation problems with interior data is how to deal with uncertainty in the boundary conditions employed in the forward or state model. In this work we focus on a linear plane stress inverse elasticity problem with measured displacement data where one component of the measured displacement field is known with considerably greater precision than the other. This situation is commonly encountered when the displacement field is measured using ultrasound or optical coherence tomography. We present a novel computational formulation in which no displacement or traction boundary conditions are assumed. The formulation results in coupling the state and adjoint equations, that are typically uncoupled when a well-posed state model is available. Two variants of residual-based stabilization are added. Our approach is applied to a simulated data set and experimental data from an ultrasound phantom.

Three-dimensional traction microscopy with a fiber-based constitutive model

Tractions exerted by cells on their surroundings play an important role in many biological processes including stem cell differentiation, tumorigenesis, cell migration, cancer metastasis, and angiogenesis. The ability to quantify these tractions is important in understanding and manipulating these processes. Three-dimensional traction force microscopy (3DTFM) provides reliable means of evaluating cellular tractions by first measuring the displacement of fluorescent beads in response to these tractions in the surrounding matrix, and then using this measurement to compute the tractions. However, most applications of 3DTFM assume that the surrounding extra-cellular matrix (ECM) is non-fibrous, despite the fact that in many natural and synthetic environments the ECM contains a significant proportion of fibrous components. Motivated by this, we develop a computational approach for determining tractions, while accounting for the fibrous nature of the ECM. In particular, we make use of a fiber-based constitutive model in which the stress contains contributions from a distribution of nonlinear elastic fibers and a hyperelastic matrix. We solve an inverse problem with the nodal values of the traction vector as unknowns, and minimize the difference between a predicted displacement field, obtained by solving the equations of equilibrium in conjunction with the fiber-based constitutive model, and the measured displacement field at the bead locations. We employ a gradient-based minimization method to solve this problem and determine the gradient efficiently by solving for the appropriate adjoint field. We apply this algorithm to problems with experimentally observed cell geometries and synthetic, albeit realistic, traction fields to gauge its sensitivity to noise, and quantify the impact of using an incorrect constitutive model: the so-called model error. We conclude that the approach is robust to noise, yielding about 10% error in tractions for 5% displacement noise. We also conclude that the impact of model error is significant, where using a nonlinear exponential hyperelastic model instead of the fiber-based model, can lead to more than 100% error in the traction field. These results underline the importance of using appropriate constitutive models in 3DTFM, especially in fibrous ECM constructs.

RGB-D-DIC technique for low-cost 3D displacement fields measurements

Optical displacement measurements can provide data on the behavior of structural elements without altering key parameters such as damping, stiffness, and mass. Optical techniques can also substantially simplify the equipment that is required for acquiring data effectively and can map displacements at specified locations of elements with complex geometries, such as steel connections. In addition, these techniques can provide simultaneous data at multiple points using a single imaging device (for plane measurements) or multiple devices (for three-dimensional measurements); however, these systems rely on specialized imaging sensors that are coupled with expensive optics and high-cost software licenses. In contrast, range/depth cameras (RGB-D) can provide a color and 3D imaging solution for capturing motion at an affordable cost. These cameras are widely available and applied in the video game industry using a mass production scheme; hence, they are of low cost. This paper proposes a methodology, along with an experimental validation, of a large three-dimensional displacement field measurement approach that uses artificial vision techniques. This methodology uses digital image correlation (DIC) principles and an RGB-D camera (RGB-D-DIC) for a variety of structural applications, which range from experimental modal analysis, cyclic loading tests, and creep tests to static displacement tests. A low-cost Kinect V2 camera was single- and stereo-calibrated, evaluated and implemented in a comparison test with a commercial three-dimensional DIC system. With the RGB-D camera, the proposed methodology can provide accurate displacement measurements that are comparable to those that are obtained via precise displacement measurement techniques and/or by using sensors; further, this methodology has the advantages of low cost, ease of implementation, and contactless measurement and provides three-dimensional information.