Displacement Field Data Research Articles

Determining Stress Intensity Factors at Varied Inclination Angles through Innovative Discontinuous Digital Image Correlation

An innovative method is proposed for determining stress intensity factors at crack tips with different inclination angles, based on the Adaptive Digital Image Correlation (AD-DIC) method. This method involves fitting a set of nonlinear equations with unknown parameters using displacement field data. The initial values for crack tip coordinates and inclination angles, computed through the AD-DIC method, are employed in this process to obtain stress intensity factors at the crack tips. The results demonstrate that this approach accurately captures the crack tip positions and calculates stress intensity factors for cracks at various inclination angles. This addresses the challenge of computing stress intensity factors during the crack propagation stage, where cracks may deviate, making conventional calculations impractical.

Landslide susceptibility evaluation in Alpine environment: 2. Thermo-hydro-mechanical modeling for the response to climate-related variables

This paper is Part 2 of two companion papers, proposing a multidisciplinary approach to assess stability and velocity evolution of a large landslide located in the Central Italian Alps (upper Valtellina region): the Ruinon landslide. Part 1 of this work presented a 3D stress–strain finite element analysis, which assessed the morphological and geomechanical predisposition of the slope to gravitational instabilities and defined the current stress state along the slope. In this paper, a thermo-hydro-mechanical (THM) numerical analysis is applied to the landslide shear zone, to assess the link between landslide driving factors and the shear band material response. Data used as input for the model were pore pressure, reference stresses and initial temperature at the sliding surface, as well as the monitored velocity of the landslide body, assumed to move as a rigid block. The shear band material was modeled as a visco-plastic medium with thermal softening and velocity hardening, thus thermal- and load-rate sensitivity of the material were estimated through laboratory testing. To this end, triaxial compression tests with thermal control were performed on rock samples representative of the shear band. To constrain the model, results of the analysis presented in Part 1 were used to define the stress state at the sliding surface and the relationship between pore pressure and shear stresses. Then, pore pressure data from in-situ piezometers relevant to the period 2014–2018 were introduced and a best fitting between modeled and monitored landslide velocities was obtained. Finally, velocities were forecasted for the period 2018–2020​ and a process of validation was performed using field displacement data. The outputs of the model adequately simulate the measured landslide velocity, reproducing the sliding behavior and its relationship with pore pressure. The presented approach may be applied to further case studies, aimed at defining a novel physics based early warning strategy for landslides.

Study on Temperature Field Distribution Law and Mechanical Properties of Hydraulic Tunnel-Surrounding Rock under the Action of Large Temperature Differences

This study aims to explore the temperature field and mechanical characteristics of hydraulic tunnels during operation in an area with large temperature differences throughout a year. By relying on the measured displacement field change results of a water transmission tunnel in Xinjiang, a finite element calculation model applicable to the hydraulic tunnel-surrounding rock under the action of large temperature differences was established. Then, the FEM model was used to analyze the temperature field distribution and stress field variation of hydraulic tunnel-surrounding rock under the effect of large temperature differences. The results showed that, under the action of large temperature differences, the displacement of the cave top was the largest, followed by the cave bottom, and finally the cave waist. The obtained displacement field data using the calculation model in this study were basically consistent with the on-site measured data, and the displacement field change law was the same, indicating that the established finite element calculation model is reliable. When the ambient temperature was below zero, the freezing front gradually expanded to the depth of the surrounding rock along the tunnel radial direction, while the frozen rock at the intersection of the lining and the surrounding rock melted first when the ambient temperature was above zero. The stress in various parts of the surrounding rock of the hydraulic tunnel increased continuously. To be specific, the growth rate of the tensile stress at the cave bottom, the tensile stress at the cave top, and the compressive stress at the cave waist of the tunnel reached 0.052 MPa/year, 0.053 MPa/year, and 0.096 MPa/year, respectively, in the first year, but the increasing rate subsequently tended to slow down continuously. Therefore, in actual projects, attention should be focused on the effect of large temperature differences on the stress growth of tunnel-surrounding rocks at an early stage, especially the growth of compressive stress at the cave waist. Notably, there are a large number of symmetrical and asymmetric phenomena related to engineering in this study.

Strain fields measurement using frequency domain Savitzky–Golay filters in digital image correlation

Savitzky–Golay (SG) filter is an effective and convenient gradient calculation method employed for full-field strain measurement in digital image correlation (DIC). Currently, the strain field can be conveniently obtained by moving smoothing filtering with SG filters. This is a method of spatial filtering, which offers the advantages of easy implementation and high accuracy. This study proposed a strain calculation method based on frequency domain SG (FD-SG) filtering. Prior to performing FD-SG filtering, data extension operations involving outer-boundary padding and zero-padding were performed on the displacement field data. Similarly, the SG filters template was extended with zero-padding and circularly shift operations. Subsequently, FD-SG filters were generated by applying the fast Fourier transform (FFT) to the expanded SG template. Thereafter, FD-SG filtering was implemented through the multiplication of the displacement in the Fourier-domain by an FD-SG filter. Finally, the strain field was obtained via inverse FFT and valid data interception operation. The simulation and practical experiments confirmed the equivalent accuracy exhibited by FD-SG and spatial domain SG filtering. Thus, the proposed FD-SG filtering method has great potential for real-time strain measurement in DIC.

A Deep Learning Framework to Estimate Elastic Modulus from Ultrasound Measured Displacement Fields.

Ultrasound (US) elastography is a technique that enables non-invasive quantification of material properties, such as stiffness, from ultrasound images of deforming tissue. The displacement field is measured from the US images using image matching algorithms, and then a parameter, often the elastic modulus, is inferred or subsequently measured to identify potential tissue pathologies, such as cancerous tissues. Several traditional inverse problem approaches, loosely grouped as either direct or iterative, have been explored to estimate the elastic modulus. Nevertheless, the iterative techniques are typically slow and computationally intensive, while the direct techniques, although more computationally efficient, are very sensitive to measurement noise and require the full displacement field data (i.e., both vector components). In this work, we propose a deep learning approach to solve the inverse problem and recover the spatial distribution of the elastic modulus from one component of the US measured displacement field. The neural network used here is trained using only simulated data obtained via a forward finite element (FE) model with known variations in the modulus field, thus avoiding the reliance on large measurement data sets that may be challenging to acquire. A U-net based neural network is then used to predict the modulus distribution (i.e., solve the inverse problem) using the simulated forward data as input. We quantitatively evaluated our trained model with a simulated test dataset and observed a 0.0018 mean squared error (MSE) and a 1.14% mean absolute percent error (MAPE) between the reconstructed and ground truth elastic modulus. Moreover, we also qualitatively compared the output of our U-net model to experimentally measured displacement data acquired using a US elastography tissue-mimicking calibration phantom.

A wavelet neural operator based elastography for localization and quantification of tumors

Background and ObjectivesThe application of intelligent imaging techniques and deep learning in the field of computer-aided diagnosis and medical imaging have improved and accelerated the early diagnosis of many diseases. Elastography is an imaging modality where an inverse problem is solved to extract the elastic properties of tissues and subsequently mapped to anatomical images for diagnostic purposes. In the present work, we propose a wavelet neural operator-based approach for correctly learning the non-linear mapping of elastic properties directly from measured displacement field data. MethodsThe proposed framework learns the underlying operator behind the elastic mapping and thus can map any displacement data from a family to the elastic properties. The displacement fields are first uplifted to a high-dimensional space using a fully connected neural network. On the lifted data, certain iterations are performed using wavelet neural blocks. In each wavelet neural block, the lifted data are decomposed into low, and high-frequency components using wavelet decomposition. To learn the most relevant patterns and structural information from the input, the neural network kernels are directly convoluted with the outputs of the wavelet decomposition. Thereafter the elasticity field is reconstructed from the outputs from convolution. The mapping between the displacement and the elasticity using wavelets is unique and remains stable during training. ResultsThe proposed framework is tested on several artificially fabricated numerical examples, including a benign-cum-malignant tumor prediction problem. The trained model was also tested on real Ultrasound-based elastography data to demonstrate the applicability of the proposed scheme in clinical usage. The proposed framework reproduces the highly accurate elasticity field directly from the displacement inputs. ConclusionsThe proposed framework circumvents different data pre-processing and intermediate steps utilized in traditional methods, hence providing an accurate elasticity map. The computationally efficient framework requires fewer epochs for training, which bodes well for its clinical usability for real-time predictions. The weights and biases from pre-trained models can also be employed for transfer learning, which reduces the effective training time with random initialization.

Characterization of 3D Displacement and Stress Fields in Coal Based on CT Scans

Computed tomography (CT) scans were performed on samples of an outburst-prone coal seam at different loading stages. The area and roundness of the CT images were used to quantify the degree of the coal macroscopic deformation under different loads. A spatial matching algorithm was used to calculate the three-dimensional (3D) displacement fields of different regions of interest (ROIs, containing primary fractures, minerals, and only coal) under different loads. The presence of fractures and minerals were found to promote and inhibit displacement, respectively, and the 3D displacement field data followed a normal distribution. A meshfree numerical simulation was used to determine the 3D maximum principal stress, shear stress and displacement fields under different loads. The following results were obtained: fractures and minerals significantly affect the stress state and displacement field distribution features, the maximum principal stresses and shear stresses in different matrices differ significantly, and the presence of minerals and fractures induce a prevalent shear stress in coal and make coal prone to stress concentration.

Bayesian-EUCLID: Discovering hyperelastic material laws with uncertainties

Within the scope of our recent approach for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID), we propose an unsupervised Bayesian learning framework for discovery of parsimonious and interpretable constitutive laws with quantifiable uncertainties. As in deterministic EUCLID, we do not resort to stress data, but only to realistically measurable full-field displacement and global reaction force data; as opposed to calibration of an a priori assumed model, we start with a constitutive model ansatz based on a large catalog of candidate functional features; we leverage domain knowledge by including features based on existing, both physics-based and phenomenological, constitutive models. In the new Bayesian-EUCLID approach, we use a hierarchical Bayesian model with sparsity-promoting priors and Monte Carlo sampling to efficiently solve the parsimonious model selection task and discover physically consistent constitutive equations in the form of multivariate multi-modal probabilistic distributions. We demonstrate and validate the ability to accurately and efficiently recover isotropic and anisotropic hyperelastic models like the Neo-Hookean, Isihara, Gent–Thomas, Arruda–Boyce, Ogden, and Holzapfel models in both elastostatics and elastodynamics. The discovered constitutive models are reliable under both epistemic uncertainties — i.e. uncertainties on the true features of the constitutive catalog – and aleatoric uncertainties – which arise from the noise in the displacement field data, and are automatically estimated by the hierarchical Bayesian model.

A new method for extrapolating the load-displacement response of drilled shaft foundations for interpretation of side capacity

In this study, the side capacity of drilled shaft foundations is estimated from partially-mobilized load–displacement field data using a new method in the extrapolation of load–displacement response. A dataset of 138 bi-directional load tests is used to evaluate the degree of mobilization of unit side resistance. A total of 612 unit side-resistance curves obtained from measured strain gauge recordings are utilized in this study. The proposed extrapolation approach is based on a new technique, the Double Tangent method, characterizing the extent of mobilization for each unit side-resistance curve. Roughly, 12% of the dataset exhibits a fully-mobilized load–displacement response, with the remainder exhibiting varying degrees of a partially-mobilized response. Fully-mobilized records are further characterized using the Double Tangent method over different ranges of mobilization, resulting in four regression models based on predominant soil types. Each model is assessed statistically, and a global regression model is found suitable to predict maximum unit side resistance. The global model is further validated using two independent load test datasets, comparing measured values of unit side resistance against predicted values. The model is then used to predict maximum unit side resistance for all partially-mobilized data within the dataset, and the results are compared to two extrapolation techniques currently used in practice. The corresponding resistance-displacement response is extrapolated using a proposed asymptotic curve-fitting function for side resistance, and an example extrapolation is illustrated to showcase how the proposed method can be used in engineering practice.

Experimental Implementation of Energy Closure Analysis for Reduced Order Modeling

Abstract Reduced order models (ROMs) provide an efficient, kinematically condensed representation of computationally expensive high-dimensional dynamical systems; however, their accuracy depends crucially on the accurate estimation of their dimension. We here demonstrate how the energy closure criterion, developed in our prior work, can be experimentally implemented to accurately estimate the dimension of ROMs obtained using the proper orthogonal decomposition (POD). We examine the effect of using discrete data with and without measurement noise, as will typically be gathered in an experiment or numerical simulation, on estimating the degree of energy closure on a candidate reduced subspace. To this end, we used a periodically kicked Euler–Bernoulli beam with Rayleigh damping as the model system and studied ROMs obtained by applying POD to discrete displacement field data obtained from simulated numerical experiments. An improved method for quantifying the degree of energy closure is presented: the convergence of energy input to or dissipated from the system is obtained as a function of the subspace dimension, and the dimension capturing a predefined percentage of either energy is selected as the ROM dimension. This method was found to be more robust to data discretization error and measurement noise. The data-processing necessary for the experimental application of energy closure analysis is discussed in detail. We show how ROMs formulated from the simulated data using our approach accurately capture the dynamics of the beam for different sets of parameter values.

Combined evaluation of Young modulus and fracture toughness in small specimens of fine grained nuclear graphite using 3D image analysis

The fracture toughness of the fine-grained nuclear graphite SNG742 has been investigated by observation of stable crack propagation in double cleavage drilled compression specimens. The three-dimensional displacement fields were obtained by digital volume correlation (DVC) of in situ laboratory X-ray computed tomographs. The crack tip location and crack opening displacements were determined using an image edge detection algorithm based on the wavelet modulus maxima. The Young modulus was estimated by fitting a finite element model to DVC displacement field data measured before crack initiation. Using the 3D crack geometry and the surrounding full-field 3D displacement fields as boundary conditions, the elastic strain energy release rate J and the three-dimensional stress intensity factors KI to KIII were then evaluated via the contour integral method. Constant mode I critical stress intensity factor was obtained along the curved crack fronts, with negligible shearing modes. This method allows evaluation of the fracture toughness without prior knowledge of elastic properties, and has potential applications to assess the effects of high temperature, oxidation and irradiation in small specimens of nuclear graphite.

Energy balance of rapidly deforming foam filled cylindrical shells in a high pressure fluid environment

The pressure differential between an air-filled structure and the surrounding fluid for underwater applications can potentially lead to instabilities and collapse of the structure. When air-filled structures collapse underwater, large pressure pulses are emitted into the surrounding water, which can damage nearby living organisms and structures. The present study aims to mitigate the high-pressure spikes generated in underwater implosions. Experiments were performed underwater in a pressure vessel. Thin-walled aluminum 6061 cylindrical shells were placed in the water filled pressure vessel, and the hydrostatic pressure was increased until the tubes reached instability and subsequently collapsed. To mitigate the pressure pulses radiated into the fluid by the implosion of the tubes, closed cell polyvinyl chloride foam rods of varying densities were placed inside of the aluminum tubes and assembled concentrically. The densities of the solid foam rods were 45 and 100 kg/m3, which were selected to investigate the effect of foam density on the mitigation capabilities of the foam rods. Three-dimensional digital image correlation was coupled with high-speed photography to obtain the full-field displacements and velocities of the aluminum tubes during implosion. Moreover, the pressure fields created during the structural collapse of the tubes were recorded with tourmaline pressure transducers. The 45 kg/m3 foam reduced the peak pressure pulse generated during the implosion event by 30%, whereas the 100 kg/m3 foam rods reduced the peak pressure pulse by 50%. Furthermore, the 45 kg/m3 foam decreased the energy imparted onto the fluid by the implosion of the tubes by 16%, whereas the 100 kg/m3 foam decreased the energy imparted onto the fluid by the implosion by 43%. Full field displacement data showed that both foam densities reduced the radial velocity of the tube during implosion, as well as reducing the peak centerpoint decelerations of the tube prior to wall contact by 45% and 62% for the 45 kg/m3 and the 100 kg/m3 foam, respectively. Fully coupled fluid–structure interaction finite element models were created using the Abaqus software to obtain the structural deformation energy in the aluminum tubes after implosion, as well as the energy absorbed by the foam rods. The models were in good agreement with the experiments in terms of radial displacements during implosion, as well as the final foam rod volume.

Using Full Field Data to Produce a Single Indentation Test for Fully Characterising the Mooney Rivlin Material Model

A theoretical testing method for fully characterising the Mooney-Rivlin three-parameter hyper-elastic material model is proposed by capturing full-field digital image correlation (DIC) data, namely displacement field and indentation force data. A finite element model with known parameters will act as the experimental model against which all data will be referenced (a preliminary test case). Going forward this stand-in model will be replaced with physical test data. This paper also introduces a new concept, the concept of hyperplanes. These hyperplanes represent regions in the force and displacement field data where all the objective function values are equal. The paper concludes that the Mooney-Rivlin material model can theoretically be fully characterised in a single indentation test. By applying the methods discussed in the paper when using full-field data operating under the assumption of hyper-planes.

Study on fracture mechanics of granite based on digital speckle correlation method

Crack propagation length (CPL) and stress intensity factor (SIF) are key parameters to solve rock failure issue when using fracture mechanics. In order to study the evolution law of the two factors during the mode Ⅰ fracture evolution, a series of three-point bending tests of granite specimens with central cracks were carried out. During the loading process, digital speckle correlation method (DSCM) was used to monitor the full field deformation. The crack opening displacement is first determined through the full field displacement data. Then, this value is compared with those calculated by ASTM and Tada equations. Additionally, a virtual extensometer is arranged at the crack tip. The crack tip position and CPL are determined by the measurement hybrid method. Results show that the crack length evolution can be divided into three stages: crack initiation, crack tip formation and crack propagation. At the 90% peak load, the crack propagation rate increases rapidly until the rock failure. During this period, the crack growth consists of 87% of the total length. While the increasing rate of SIF presents quick-slow-quick evolution law, and SIF reaches the maximum at the peak load. When the crack forms, CPL and SIF all experience rapid increase at the deformation point and reach the maximum during the instable failure of rock.

Inverse coefficient problem of the variable properties reconstruction for the viscoelastic cylindrical waveguide

AbstractAn effective technique for inverse coefficient problem solving of the shear modulus reconstruction for both inhomogeneous elastic and viscoelastic cylindrical waveguide is proposed. The identification is carried out using the displacement field data measured on the finite part of the outer boundary of the cylindrical waveguide in the torsional vibrations mode. An iterative scheme for solving the inverse problem is developed. At each iteration step, the correction to the shear modulus distribution function is calculated using the Fredholm integral equation of the first kind with a smooth kernel. Numerical results are obtained for both direct and inverse problems for different laws of the shear modulus variation.

Identification of the Lamé parameters of an inhomogeneous pipe based on the displacement field data

The present paper considers the direct and inverse problems on steady-state oscillations of functionally graded elastic pipe. The direct problem is reduced to solving a numerical set of canonical systems of first-order differential equations with variable coefficients. The influence of the variable Lamé parameters on the displacement functions and amplitude–frequency characteristics is estimated. The solution of the inverse problem on determination of the functions describing the change in the Lamé parameters is obtained numerically by inverting differential operators on the basis of the displacement field data gathered in a finite set of points at a fixed frequency. The results of computational experiments are discussed.

Correction of Scanning Electron Microscope Imaging Artifacts in a Novel Digital Image Correlation Framework

The combination of digital image correlation (DIC) and scanning electron microscopy (SEM) enables to extract high resolution full field displacement data, based on the high spatial resolution of SEM and the sub-pixel accuracy of DIC. However, SEM images may exhibit a considerable amount of imaging artifacts, which may seriously compromise the accuracy of the displacements and strains measured from these images. The current study proposes a unified general framework to correct for the three dominant types of SEM artifacts, i.e. spatial distortion, drift distortion and scan line shifts. The artifact fields are measured alongside the mechanical deformations to minimize the artifact induced errors in the latter. To this purpose, Integrated DIC (IDIC) is extended with a series of hierarchical mapping functions that describe the interaction of the imaging process with the mechanics. A new IDIC formulation based on these mapping functions is derived and the potential of the framework is tested by a number of virtual experiments. The effect of noise in the images and different regularization options for the artifact fields are studied. The error in the mechanical displacement fields measured for noise levels up to 5% is within the usual DIC accuracy range for all the cases studied, while it is more than 4 pixels if artifacts are ignored. A validation on real SEM images at three different magnifications confirms that all three distortion fields are accurately captured. The results of all virtual and real experiments demonstrate the accuracy of the methodology proposed, as well as its robustness in terms of convergence.

MFS fading regularization method for the identification of boundary conditions from partial elastic displacement field data

ABSTRACTA method is proposed to solve an inverse problem in two-dimensional linear isotropic elasticity. The inverse problem consists of the determination of both the entire displacement field and ...

MFS fading regularization method for the identification of boundary conditions from partial elastic displacement field data

A method is proposed to solve an inverse problem in twodimensional linear isotropic elasticity. The inverse problem consists of the determination of both the entire displacement field and the boundary conditions inaccessible to the measurement from the partial knowledge of the displacement field. The algorithm is based on a fading regularization method (FRM) and is numerically implemented using the method of fundamental solutions (MFS). The inverse technique is first validated with synthetic data and is then applied to the interpretation of experimental measurements obtained by digital image correlation (DIC).

Physical modeling experimental study on failure mechanism of surrounding rock of deep-buried soft tunnel based on digital image correlation technology

In this paper, a physical model experiment was conducted to study the deformation and failure mechanism of surrounding rock of deep-buried soft tunnel. With the help of noncontact displacement monitoring system, strain data acquisition system, and high-definition camera, the displacement field data, the strain field data, and the deformation and failure process of surrounding rock were obtained during the experiment. The results of study showed that the excavation footage and horizontal stress had a significant influence on the deformation of surrounding rock of deep-buried soft rock tunnel; the surrounding rock had obvious asymmetrical deformation, and the key position of failure was located at the top and right side of tunnel; the uncoordinated deformation areas were the potential dangerous destruction areas of surrounding rock; and the surrounding rock of tunnel was destroyed under the combined action of tension and compression stress.