Simulated Displacement Fields Research Articles

An iterative crack tip correction algorithm discovered by physical deep symbolic regression

Digital image correlation is a widely used technique in the field of experimental mechanics. In fracture mechanics, determining the precise location of the crack tip is crucial. In this paper, we introduce a novel crack tip detection algorithm based on displacement and strain fields obtained by digital image correlation. Iterative crack tip correction formulas are discovered by applying deep symbolic regression guided by physical unit constraints to a dataset of simulated cracks under mode I, II and mixed-mode conditions with variable T-stress. For the training dataset, we fit the Williams series expansion with super-singular terms to the simulated displacement fields at randomly chosen origins around the actual crack tip. We analyse the discovered formulas and apply the most promising one to digital image correlation data obtained from uniaxial and biaxial fatigue crack growth experiments of AA2024-T3 sheet material. Throughout the experiments, the crack tip positions are reliably detected leading to improved stability of the crack propagation curves.

Quantitative prediction of surrounding rock deformation via an energy-based damage model combining with microseismic monitoring

In this study, combined with microseismic (MS) monitoring and numerical simulation, a novel energy-based damage model is proposed for quantitatively predicting the surrounding rock deformation for the stability analysis of underground engineering. First, the rock damage range is delineated based on MS event location and source radius. Second, seismic efficiency is introduced to calculate the actual released energy, and the total releasable energy density and total stored energy density are derived from element stress information and stress–strain curves in numerical simulation. Finally, a novel energy-based damage model with definite physical significance is developed based on the above-mentioned three forms of energy. Our new model introduces MS source parameters into the classical damage variable based on dissipated energy, accurately reflecting the qualitative relationship between them, which is highly applicable to practical engineering. Moreover, this model comprehensively reflects the damage degree variation of surrounding rocks from in-situ stress state to different excavation stages and describes real-time deterioration of rock mechanical parameters. In addition, a new approach based on seismology formulation is applied to solve seismic efficiency, without requiring reference to the destruction situation. To validate our proposed model, a three-dimensional (3D) numerical simulation embedding the damage model is conducted to investigate the surrounding rock deformation behavior of underground caverns in the Lianghekou hydropower station. A reasonable consistency is observed between the simulated displacement field and source parameters distribution obtained from MS monitoring and field failure. Furthermore, compared to the original model without a damage variable, the energy-based damage model greatly improves the deformation prediction precision.

Determination of notch stress intensity factors under mode I loading using the 3D-DIC and finite element over-deterministic methods

In this paper, the fracture behaviour of Single Edge Notched Tension Specimens (SENT) is investigated experimentally and numerically using the three-dimensional digital image correlation (3D-DIC) experimental method and the Finite Element Analysis (FEA) technique, respectively. The mode I notch stress intensity factors (NSIFs) for the SENT specimens are derived from the measured and simulated displacement fields using the Finite Element Over-Deterministic (FEOD) method. The step-by-step process of a 3D-DIC experimental system for the SENT specimens is described. The SENT specimens were tested with pinned and clamped boundary conditions (BCs) and different notch initial opening angles. There are three main contributions of the paper: (1) the derivation of the NSIFs under Mode I Loading for thin plates subjected not only to the common pinned BCs, but also to the rare, clamped BCs; (2) the application of the FEOD method to derive the NSIFs from the displacement fields measured using the 3D DIC technique and predicted using FEA; (3) the validation of the NSIFs derived numerically via experimentally-determined NSIFs for notched thin plates subjected to both pinned and clamped BCs. Thus, four sets of NSIF data are provided in the paper and enable confident application of the NSIF values for the conservative (pinned BCs) and light-weight (clamped BCs) design of notched thin plates.

Mechanical stiffness and anisotropy measured by MRE during brain development in the minipig

The relationship between brain development and mechanical properties of brain tissue is important, but remains incompletely understood, in part due to the challenges in measuring these properties longitudinally over time. In addition, white matter, which is composed of aligned, myelinated, axonal fibers, may be mechanically anisotropic. Here we use data from magnetic resonance elastography (MRE) and diffusion tensor imaging (DTI) to estimate anisotropic mechanical properties in six female Yucatan minipigs at ages from 3 to 6 months. Fiber direction was estimated from the principal axis of the diffusion tensor in each voxel. Harmonic shear waves in the brain were excited by three different configurations of a jaw actuator and measured using a motion-sensitive MR imaging sequence. Anisotropic mechanical properties are estimated from displacement field and fiber direction data with a finite element- based, transversely-isotropic nonlinear inversion (TI-NLI) algorithm. TI-NLI finds spatially resolved TI material properties that minimize the error between measured and simulated displacement fields. Maps of anisotropic mechanical properties in the minipig brain were generated for each animal at all four ages. These maps show that white matter is more dissipative and anisotropic than gray matter, and reveal significant effects of brain development on brain stiffness and structural anisotropy. Changes in brain mechanical properties may be a fundamental biophysical signature of brain development.

Spatial distribution of mechanical parameters along a fracture interface

Many theoretical and numerical approaches for wave propagation in fractured media use normal and shear fracture specific stiffnesses to represent the complexity of fracture topology during deformation under stress. In this paper, XLiFE++ is used to simulate the interaction between a fracture and elastic wave on a 2D plane, the simulated displacement field around the fracture is converted into traction on the fracture interface, and the jump in displacement on the fracture interface is calculated relative to displacements on both sides of the fracture. Based on the linear slippage model, the specific value between jump displacement and traction is calculated at each fracture point to obtain the fracture stiffnesses and then the stiffness curves along the fracture surface are plotted using the stiffnesses at each fracture point. Using XLiFE++, 144 orthogonally designed parameter combinations are simulated, the simulated displacement field around the fracture with material parameters and fracture stiffnesses is used as a training sample, and a neural network method for estimating the fracture stiffnesses is provided. The verification for the training results indicates that the ANN model can effectively represent the non-linear relationships between Young’s modulus, Poisson’s ratio, excitation frequency of materials, shear stiffness and normal stiffness of the fracture, and displacement field around the fracture and estimate the fracture stiffnesses relative to displacement field around the fracture surface. Therefore, the proposed method is an effective tool to rapidly determine the fracture interface stiffnesses. • Spatial distribution of elastic modulus around fracture •Simulate the interaction between elastic waves and fracture with different stiffness •Calculation of fracture stiffness based on displacement discontinuity theory

Assessing Urban Landslide Dynamics through Multi-Temporal InSAR Techniques and Slope Numerical Modeling

Landslides threaten more than before the urbanized areas and are a worldwide growing problem for the already affected communities and the local authorities committed to landslide risk management and mitigation. For this reason, it is essential to analyze landslide dynamics and environmental conditioning factors. Various techniques and instruments exist for landslide investigation and monitoring. Out of these, Multi-temporal Synthetic Aperture Radar Interferometry (MT-InSAR) techniques have been widely used in the last decades. Their capabilities are enhanced by the availability of the active Sentinel-1 mission, whose 6-day revisiting time enables near real-time monitoring of landslides. Interferometric results, coupled with ground measurements or other approaches such as numerical modeling, significantly improve the knowledge of the investigated surface processes. In this work, we processed the C-band SAR images of the available European Space Agency (ESA) satellite missions, using MT-InSAR methods to identify the surface deformations related to landslides affecting the Iași Municipality (Eastern Romania). The results (i.e., velocity maps) point out the most active landslides with velocities of up to 20 mm/year measured along the satellite Line of Sight (LOS). Following, we focused on the most problematic landslide that affects the Țicău neighborhood and is well-known for its significant implications that it had. To better understand its behavior and the sensitivity of the displacements to the environmental factors (i.e., rainfall), we carried out 2D numerical modeling using a finite difference code. The simulated displacement field is consistent with the InSAR displacements and reveals the most active sectors of the landslide and insights about its mechanism.

A comprehensive study of single-flawed granite hydraulically fracturing with laboratory experiments and flat-jointed bonded particle modeling

Understanding the mechanisms of crack initiation and development is important for utilizing the hydraulic fracturing in practical engineering applications. In this study, hydraulic fracturing experiments were conducted on granite specimens containing a single pre-existing flaw, and the failure processes were analyzed using the digital image correlation method. A flat-jointed bonded particle model was built with a modified fluid flow algorithm to better understand the fracture behavior under the experimental conditions. Combining the DIC strain fields and the simulated displacement fields, we found that shear behavior with a small value also emerges ahead of the tips of the microcracks and was covered by the subsequent tensile behavior, as a result, shear strain could not be observed in the experiments. A comparison between the strain localization and the microcracks revealed that the fracturing process observed in the experiments had two stages: the isolated strain localization formed during Stage I, mainly resulting from the increase in microcracks, and the continuous high tensile strain path formed during Stage II. Additionally, the effect of the different flaw inclination angles was discussed from two perspectives: the different types of macrocracks and the change in the fracturing liquid pressure.

Mechanical Metrics of Virtual Polycrystals (MechMet)

The details of polycrystalline microstructure often influence the early stages of yielding and strain localization under monotonic and cyclic loading, particularly in elastically anisotropic materials. A new software package, MechMet (mechanical metrics) provides a convenient finite element tool for solving field equations for elasticity in polycrystals in conjunction with investigations of microstructure-induced heterogeneity. The simulated displacement field is used to compute several mechanical metrics, such as the strain and stress tensors, directional stiffness, relative Schmid factor, and the directional strength-to-stiffness ratio. The virtual polycrystal finite element meshes needed by MechMet can be created with the Neper package or any other method that produces a 10-node, tetrahedral, serendipity element. Formatted output files are automatically generated for visualization with Paraview or VisIt. This paper presents an overview of the MechMet package and its application to polycrystalline materials of both cubic and hexagonal structures.

Diffusive subsurface propagation of Rayleigh mode induced by embedded ultrasonic waveguide

Surface degradation in the wood surface has generated unwanted variation in the ultrasound and reduced the efficacy of ultrasonic based non-destructive evaluation (NDE). This study proposes a novel alternative by inserting a waveguide into the wooden medium to eliminate the impact of surface variation on the characteristics of the ultrasonic propagation. This technique creates a small circular insertion of 2 cm depth with 5 mm in diameter, which has been accepted by many utility inspection firms for structural health monitoring (SHM) of wooden utility grids. This work develops a closed-form analytical solution of the elastodynamic problem using the proposed embedded waveguide technique. The resulted solution based on the Navier's formulation discovered two interfering wave modes embedded in the displacement field. These two modes termed fast and slow induce diffusive displacement propagation as a function of the Poisson's ratio of the medium. The analytical result also shows a greater elastic energy penetration as a function of the insertion depth. Resulted wave characteristics are different from the traditional contact-based Rayleigh excitation. A developed numerical model using the semi-explicit differential-algebraic equation (DAE) technique with both steady-state and transient load conditions is demonstrated. The simulated displacement fields are analyzed and compared with the analytical result.

Numerical simulations of magnetic resonance elastography using finite element analysis with a linear heterogeneous viscoelastic model

Magnetic resonance elastography (MRE) is a technique to identify the viscoelastic moduli of biological tissues by solving the inverse problem from the displacement field of viscoelastic wave propagation in a tissue measured by MRI. Because finite element analysis (FEA) of MRE evaluates not only the viscoelastic model for a tissue but also the efficiency of the inversion algorithm, we developed FEA for MRE using commercial software called ANSYS, the Zener model for displacement field of a wave inside tissue, and an inversion algorithm called the modified integral method. The profile of the simulated displacement field by FEA agrees well with the experimental data measured by MRE for gel phantoms. Similarly, the value of storage modulus (i.e., stiffness) recovered using the modified integral method with the simulation data is consistent with the value given in FEA. Furthermore, applying the suggested FEA to a human liver demonstrates the effectiveness of the present simulation scheme.Graphical abstract

Magnetic resonance elastography of the brain: An in silico study to determine the influence of cranial anatomy

Magnetic resonance elastography (MRE) of the brain has demonstrated potential as a biomarker of neurodegenerative disease such as dementia but requires further evaluation. Cranial anatomical features such as the falx cerebri and tentorium cerebelli membranes may influence MRE measurements through wave reflection and interference and tissue heterogeneity at their boundaries. We sought to determine the influence of these effects via simulation. MRE-associated mechanical stimulation of the brain was simulated using steady state harmonic finite element analysis. Simulations of geometrical models and anthropomorphic brain models derived from anatomical MRI data of healthy individuals were compared. Constitutive parameters were taken from MRE measurements for healthy brain. Viscoelastic moduli were reconstructed from the simulated displacement fields and compared with ground truth. Interference patterns from reflections and heterogeneity resulted in artifacts in the reconstructions of viscoelastic moduli. Artifacts typically occurred in the vicinity of boundaries between different tissues within the cranium, with a magnitude of 10%-20%. Given that MRE studies for neurodegenerative disease have reported only marginal variations in brain elasticity between controls and patients (e.g., 7% for Alzheimer's disease), the predicted errors are a potential confound to the development of MRE as a biomarker of dementia and other neurodegenerative diseases. Magn Reson Med 76:645-662, 2016. © 2015 Wiley Periodicals, Inc.

Displacement Fields of Sedimentary Layers Controlled by Fault Parameters: The Discrete Element Method of Controlling Basement Motions by Dislocation Solutions

In the two-dimensional discrete element modeling of displacement of sedimentary layers caused by faulting within the basement, we attempted to move a rigid basement as if it were an elastic basement by controlling its motion through application of dislocation solutions. An advantage of our modeling procedure is that we can discuss displacement fields of sedimentary layers in connection with fault parameters. We simulated displacement fields of the sedimentary layers by means of our modeling procedure and found that our simulated fields are different from the fields obtained in rigid basement models and are dependent on the selected fault parameters.

Quantifying Hepatic Shear Modulus In Vivo Using Acoustic Radiation Force

The speed at which shear waves propagate in tissue can be used to quantify the shear modulus of the tissue. As many groups have shown, shear waves can be generated within tissues using focused, impulsive, acoustic radiation force excitations, and the resulting displacement response can be ultrasonically tracked through time. The goals of the work herein are twofold: (i) to develop and validate an algorithm to quantify shear wave speed from radiation force-induced, ultrasonically-detected displacement data that is robust in the presence of poor displacement signal-to-noise ratio and (ii) to apply this algorithm to in vivo datasets acquired in human volunteers to demonstrate the clinical feasibility of using this method to quantify the shear modulus of liver tissue in longitudinal studies. The ultimate clinical application of this work is noninvasive quantification of liver stiffness in the setting of fibrosis and steatosis. In the proposed algorithm, time-to-peak displacement data in response to impulsive acoustic radiation force outside the region of excitation are used to characterize the shear wave speed of a material, which is used to reconstruct the material’s shear modulus. The algorithm is developed and validated using finite element method simulations. By using this algorithm on simulated displacement fields, reconstructions for materials with shear moduli (μ) ranging from 1.3–5 kPa are accurate to within 0.3 kPa, whereas stiffer shear moduli ranging from 10–16 kPa are accurate to within 1.0 kPa. Ultrasonically tracking the displacement data, which introduces jitter in the displacement estimates, does not impede the use of this algorithm to reconstruct accurate shear moduli. By using in vivo data acquired intercostally in 20 volunteers with body mass indices ranging from normal to obese, liver shear moduli have been reconstructed between 0.9 and 3.0 kPa, with an average precision of ±0.4 kPa. These reconstructed liver moduli are consistent with those reported in the literature (μ = 0.75–2.5 kPa) with a similar precision (±0.3 kPa). Repeated intercostal liver shear modulus reconstructions were performed on nine different days in two volunteers over a 105-day period, yielding an average shear modulus of 1.9 ± 0.50 kPa (1.3–2.5 kPa) in the first volunteer and 1.8 ± 0.4 kPa (1.1–3.0 kPa) in the second volunteer. The simulation and in vivo data to date demonstrate that this method is capable of generating accurate and repeatable liver stiffness measurements and appears promising as a clinical tool for quantifying liver stiffness. (E-mail: mark.palmeri@duke.edu)

Investigation of twin boundary thickness and energy in CuAlNi shape memory alloy

The nanometer-scale thickness and energy of twin boundaries govern the martensitic twin structure and its dynamics, which are responsible for the actuation and superelasticity of shape memory alloys. In this letter the authors apply a method, which has been recently developed for investigating twin wall structures in ferroelectric crystals, on a shape memory alloy crystal. Fittings of simulated displacement fields to atomic force microscopy measurements allow them to extract the thickness of type-I twin boundaries in CuAlNi. Further, a relation between the twin boundary thickness and energy is developed and used for evaluating the twin boundary energy.

Finite Element Modeling of the Head Skeleton With a New Local Quantitative Assessment Approach

The present study was undertaken to build a finite element model of the head skeleton and to perform a new assessment approach in order to validate it. The application fields for such an improved model are injury risk prediction as well as surgical planning. The geometrical reconstruction was performed using computed tomography scans and a total of 4680 shell elements were meshed on the median surface of the head skeleton with the particular characteristic of adapted mesh density and real element thickness. The assessment protocol of the finite element model was achieved using a quasi-static experimental compression test performed on the zygomatic bone area of a defleshed isolated head. Mechanical behavior of the finite element model was compared to the real one and the assessment approach was divided into two steps. First, the mechanical properties of the anatomical structure were identified using the simulation and then the simulated displacement field was compared to local displacement measurement performed during test using a digital correlation method. The assessment showed that the head skeleton model behaved qualitatively like the real structure. Quantitatively, the local relative error varied from 8% up to 70%.

Current shortening across the Himalayas of Nepal

SUMMARY Underthrusting of the Indian lithosphere beneath the Himalayas occurs during the Quaternary period along a gently north-dipping main basal detachment (main Himalayan thrust: MHT), from which the southernmost emergent ramp (main frontal thrust: MFT) branches. Historical seismicity shows that slip on the MHT is frequently accommodated through M > 8 shallow earthquakes, but shows a seismic gap in western Nepal. This absence of major historical earthquakes in western Nepal can be explained either by an aseismic slip on the MHT or a longlived elastic strain accumulation. To test these hypotheses, the present-day displacement field has been measured for a GPS network formed of 35 sites. The updated solution presented in this paper combines data from 1995, 1997, 1998 and 2000 measurements. The lack of deformation (less than 3 · 10 −8 yr −1 ) through the outer belt does not fit with a regional aseismic slip along the southern part of MHT. A less than 3 mm yr −1 aseismic slip could nonetheless affect restricted areas of the outer belt. In contrast, a strain accumulation of more than 30 · 10 −8 yr −1 is measured south of the Higher Himalayas, in a zone where an intense microseismicity reflects a stress build-up. It is presumably generated by locking of the aseismic creep that occurs along the MHT beneath the Higher Himalayas and Tibet. The displacement field is simulated by a dual-dislocation model that takes into account the pattern of microseismicity, and particularly a segmentation between central and western Nepal. The best fit between the measured and simulated displacement fields is obtained with 19 mm yr −1 thrust and 0‐1 mm yr −1 dextral strike-slip components along a 117 ◦ NE dislocation locked to a depth of 20‐21 km beneath western Nepal, and 19‐20 mm yr −1 thrust and 0‐2 mm yr −1 dextral strike-slip components along a 108 ◦ NE dislocation locked to a depth of 17‐21 km beneath central Nepal. The width of the locked zone between the main frontal thrust and the creeping zone is of the same order, but rather greater, in western Nepal than in central Nepal. Therefore it is expected that M > 8 earthquakes could occur in western Nepal.