Displacement Field Research Articles

A quadratic optimization program for the inverse elastography problem

In this work we focus on the development of a numerical algorithm for the inverse elastography problem. The goal is to perform an efficient material parameter identification knowing the elastic displacement field induced by a mechanical load. We propose to define the inverse problem through a quadratic optimization program which uses the direct problem formulation to define the objective function. In this way, we end up with a convex minimization problem which attains its minimum at the solution of a linear system. The effectiveness of our method is illustrated through numeral examples.

3D analysis of crack propagation in nuclear graphite using DVC coupled with finite element analysis

The initiation and propagation of cracks within nuclear graphite components compromise the lifespan of advanced gas-cooled reactors and even lead to safety incidents. A thorough understanding of fracture properties in nuclear graphite is crucial for the effective management and design of graphite structural components. In this work, an in-situ test of a double cleavage drilled compression (DCDC) specimen was conducted using X-ray tomographic imaging to fully analyze the 3D crack propagation behavior in IG-11 nuclear graphite. “Subvolume Splitting” digital volume correlation (SS-DVC) was employed to accurately quantify the displacement fields surrounding the crack. The entire 3D crack propagation and crack surface around the crack front were precisely captured through 3D imaging processing of the gray level residual (GLR) field. Additionally, crack opening displacements (CODs) and stress intensity factors (SIFs) were determined using finite element (FE) analysis with extended finite element method (XFEM). Experimental results indicate that although mode I crack remains predominant throughout the whole process, complex crack propagation exists around the crack front. The utilization of irregular crack surfaces in FE analysis helps to obtain refined displacement fields and accurate SIFs of the crack front. This work provides comprehensive insights into crack propagation and reveals the influence of complex crack morphology on SIF calculations, facilitating accurate prediction of structural integrity and rational design of graphite components.

Bending of bidirectional functionally graded nonlocal stress-driven beam

This paper provides a comprehensive analysis, using nonlocal stress-driven integral theory, of the static behavior of a nanoscale beam of bidirectionally graded materials. After a brief explanation of the mathematical formulation of BDFGMs, the work done and strain energy expressions derived from the displacement field are discussed. Variational formulations and Hamilton's principle are used to develop the equilibrium equation. An analytical development of the nonlocal kernel for stress-driven integral theory and formulated governing equation which was nondimensionalized later. Explicit equations for displacement and moment are obtained by solving this equation using the Laplace transformation. Three different boundary conditions are examined, and differences in the maximum displacement with respect to the nonlocal parameter and the two material FGM parameters are displayed both visually and in table form. The results exhibit excellent agreement and provide a standard for further research when they are closely compared to the existing numerical data. This work contributes to the knowledge of BDFGMs under nonlocal effects generated by stress-driven integral theory and offers solutions that have been confirmed for further investigation.

Towards a reduced order model for EVAR planning and intra-operative navigation

IntroductionThe pre-operative planning and intra-operative navigation of the endovascular aneurysm repair (EVAR) procedure are currently challenged by the aortic deformations that occur due to the insertion of a stiff guidewire. Hence, a fast and accurate predictive tool may help clinicians in the decision-making process and during surgical navigation, potentially reducing the radiations and contrast dose. To this aim, we generated a reduced order model (ROM) trained on parametric finite element simulations of the aortic wall-guidewire interaction. MethodA Design of Experiments (DOE) consisting of 300 scenarios was created spanning over seven parameters. Radial basis functions were used to achieve a morphological parametrization of the aortic geometry. The ROM was built using 200 scenarios for training and the remaining 100 for validation. ResultsThe developed ROM estimated the displacement of aortic nodes with a relative error below 5.5% for all the considered validation cases. From a preliminary analysis, the aortic elasticity, the stiffness of the guidewire and the tortuosity of the cannulated iliac artery proved to be the most influential parameters. ConclusionsOnce built, the ROM provided almost real-time and accurate estimations of the guidewire-induced aortic displacement field, thus potentially being a promising pre- and intra-operative tool for clinicians.

Nonlinear dynamic study of FG-GFRC-NPR structures

The functionally graded glass fibre-reinforced polymer composite(FG-GFRC) is one of the contemporary sophisticated materials that may be employed for a variety of technical purposes. The shock absorption capacity of FG-GFRP laminates can be improved by incorporating auxetic property (i.e., negative Poisson's ratio NPR) in the structures, which can be attained by a particular stacking sequence of glass fibres (GF) and a different percentage of GF distribution along the thickness direction within the FG-GFRP structure. The nonlinear dynamic behaviour of such structures must be investigated. The clamped-clamped FG-GFRP-NPR structure is stimulated under a uniform transverse load distribution in the current work. Based on the higher-order shear deformation theory (SDT), the displacement field and nonlinear stress–strain relationship are derived. The nonlinear differential equation with cubic non-linear components were derived using Hamilton’s principle and transformed using Galerkin’s technique and obtained governing equation of motion. MATLAB has been used to conduct all numerical simulations. A time series, a phase picture and a Poincare diagram (PCM) were generated to characterize the vibration behavior of such auxetic structures.

Valley landau level crossings in weakly coupled bilayer WSe2

Abstract WSe2 is a p-type 2D-semiconductor that can be mechanically exfoliated down to atomic layers. Unlike the Bernal stacked bilayer graphene, the two layers in a bilayer WSe2 flake are weakly coupled. The electric displacement field can easily break the layer degeneracy of the bilayer WSe2. Together with the strong spin–orbit coupling, it exhibits many novel quantum physical properties. In this work, we fabricate high quality dual-gated bilayer WSe2 devices and observe twofold degenerate Landau levels(LLs) and density-dependent quantum Hall states which show transitions between even and odd filling. When introducing carriers into the system from the valence band edge by gating, two WSe2 layers are filled independently and the bottom layer WSe2 shows negative compressibility at the crossover point. Above 9 T, we observe the degeneracy of LLs is completely broken and there are two sets of LL crossings in the top WSe2 layer at Zeeman-to-cyclotron energy ratio E Z / E N ≈ 4 and E Z / E N ≈ 5. The interplay between two LLs from the two WSe2 layers confirms the weak coupling between them. Our results show that the bilayer WSe2 behave like two closely packed independent electronic systems under electric displacement fields.

Non-standard interface conditions in flexure of mixture unified gradient Nanobeams

Structural schemes of applicative interests in Engineering Science frequently encounter the intricate phenomenon of discontinuity. The present study intends to address the discontinuity in the flexure of elastic nanobeam by adopting an abstract variational scheme. The mixture unified gradient theory of elasticity is invoked to realize the size-effects at the ultra-small scale. The consistent form of the interface conditions, stemming from the established stationary variational principle, is meticulously set forth. The boundary-value problem of equilibrium is properly closed and the analytical solution of the transverse displacement field of the elastic nanobeam is addressed. As an alternative approach, the eigenfunction expansion method is also utilized to scrutinize the efficacy of the presented variational formulation in tackling the flexure of elastic nanobeams with discontinuity. The flexural characteristic of mixture unified gradient beams with diverse kinematic constraints is numerically illustrated and thoroughly discussed. The anticipated nanoscopic features of the characteristic length-scale parameters are confirmed. The demonstrated numerical results can advantageously serve as a benchmark for the analysis and design of pioneering ultra-sensitive nano-sensors. The established variationally consistent size-dependent framework paves the way ahead in nanomechanics and inspires further research contributing to fracture mechanics of ultra-small scale elastic beams.

Superposition of Substrate Deformation Fields Induced by Molecular Clutches Explains Cell Spatial Sensing of Ligands.

Cells can sense the physical properties of the extracellular matrices (ECMs), such as stiffness and ligand density, through cell adhesions to actively regulate their behaviors. Recent studies have shown that varying ligand spacing of ECMs can influence adhesion size, cell spreading, and even stem cell differentiation, indicating that cells have the spatial sensing ability of ECM ligands. However, the mechanism of the cells' spatial sensing remains unclear. In this study, we have developed a lattice-spring motor-clutch model by integrating cell membrane deformation, the talin unfolding mechanism, and the lattice spring for substrate ligand distribution to explore how the spatial distribution of integrin ligands and substrate stiffness influence cell spreading and adhesion dynamics. By applying the Gillespie algorithm, we found that large ligand spacing reduces the superposition effect of the substrate's displacement fields generated by pulling force from motor-clutch units, increasing the effective stiffness probed by the force-sensitive receptors; this finding explains a series of previous experiments. Furthermore, using the mean-field theory, we obtain the effective stiffness sensed by bound clutches analytically; our analysis shows that the bound clutch number and ligand spacing are the two key factors that affect the superposition effects of deformation fields and, hence, the effective stiffness. Overall, our study reveals the mechanism of cells' spatial sensing, i.e., ligand spacing changes the effective stiffness sensed by cells due to the superposition effect of deformation fields, which provides a physical clue for designing and developing biological materials that effectively control cell behavior and function.

Real-Time Simulation of Tube Hydroforming by Integrating Finite-Element Method and Machine Learning

The real-time, full-field simulation of the tube hydroforming process is crucial for deformation monitoring and the timely prediction of defects. However, this is rather difficult for finite-element simulation due to its time-consuming nature. To overcome this drawback, in this paper, a surrogate model framework was proposed by integrating the finite-element method (FEM) and machine learning (ML), in which the basic methodology involved interrupting the computational workflow of the FEM and reassembling it with ML. Specifically, the displacement field, as the primary unknown quantity to be solved using the FEM, was mapped onto the displacement boundary conditions of the tube component with ML. To this end, the titanium tube material as well as the hydroforming process was investigated, and a fairly accurate FEM model was developed based on the CPB06 yield criterion coupled with a simplified Kim–Tuan hardening model. Numerous FEM simulations were performed by varying the loading conditions to generate the training database for ML. Then, a random forest algorithm was applied and trained to develop the surrogate model, in which the grid search method was employed to obtain the optimal combination of the hyperparameters. Sequentially, the principal strain, the effective strain/stress, as well as the wall thickness was derived according to continuum mechanics theories. Although further improvements were required in certain aspects, the developed FEM-ML surrogate model delivered extraordinary accuracy and instantaneity in reproducing multi-physical fields, especially the displacement field and wall-thickness distribution, manifesting its feasibility in the real-time, full-field simulation and monitoring of deformation states.

Overall constitutive properties of stratified lattices with alternating chirality.

The present research focuses on a continuum description of stratified metamaterials achieved through the superposition of layers with alternating chirality. Each layer is constructed as a periodic assembly of centre-symmetric periodic cells, formed by a recurring arrangement of rigid circular discs connected by elastic ligaments. The layers are interconnected through elastic pins passing through the centres of aligned discs, allowing for either restrained or free relative rotation. A micropolar continuum model is used to describe each individual layer. The overall response of the metamaterials to in-plane forces is derived using a multi-field non-local model, expressed in terms of the average and difference of displacement and rotational fields. The overall micropolar and standard (Cauchy) constitutive tensors have been determined in closed form. The validity of the equivalent generalized micropolar model has been confirmed through comparison with discrete Lagrangian solutions of representative examples. In addition, a detailed analysis of a pseudo-indentation test has been carried out. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

Fracture Evolution during CO2 Fracturing in Unconventional Formations: A Simulation Study Using the Phase Field Method

With the introduction of China’s “dual carbon” goals, CO2 is increasingly valued as a resource and is being utilized in unconventional oil and gas development. Its application in fracturing operations shows promising prospects, enabling efficient extraction of oil and gas while facilitating carbon sequestration. The process of reservoir stimulation using CO2 fracturing is complex, involving coupled phenomena such as temperature variations, fluid behavior, and rock mechanics. Currently, numerous scholars have conducted fracturing experiments to explore the mechanisms of supercritical CO2 (SC-CO2)-induced fractures in relatively deep formations. However, there is relatively limited numerical simulation research on the coupling processes involved in CO2 fracturing. Some simulation studies have simplified reservoir and operational parameters, indicating a need for further exploration into the multi-field coupling mechanisms of CO2 fracturing. In this study, a coupled thermo-hydro-mechanical fracturing model considering the CO2 properties and heat transfer characteristics was developed using the phase field method. The multi-field coupling characteristics of hydraulic fracturing with water and SC-CO2 are compared, and the effects of different geological parameters (such as in situ stress) and engineering parameters (such as the injection rate) on fracturing performance in tight reservoirs were investigated. The simulation results validate the conclusion that CO2, especially in its supercritical state, effectively reduces reservoir breakdown pressures and induces relatively complex fractures compared with water fracturing. During CO2 injection, heat transfer between the fluid and rock creates a thermal transition zone near the wellbore, beyond which the reservoir temperature remains relatively unchanged. Larger temperature differentials between the injected CO2 fluid and the formation result in more complicated fracture patterns due to thermal stress effects. With a CO2 injection, the displacement field of the formation deviated asymmetrically and changed abruptly when the fracture formed. As the in situ stress difference increased, the morphology of the SC-CO2-induced fractures tended to become simpler, and conversely, the fracture presented a complicated distribution. Furthermore, with an increasing injection rate of CO2, the fractures exhibited a greater width and extended over longer distances, which are more conducive to reservoir volumetric enhancement. The findings of this study validate the authenticity of previous experimental results, and it analyzed fracture evolution through the multi-field coupling process of CO2 fracturing, thereby enhancing theoretical understanding and laying a foundational basis for the application of this technology.

Atomistic and continuum Ascertainment of The crack tip stress fields in anisotropic elastic cubic media

The article provides a comparative analysis of stresses associated with the crack tip in an anisotropic elastic medium with cubic crystalline symmetry of elastic properties under mixed loading conditions obtained by two fundamentally different approaches: atomistic simulations and continuum fracture mechanics methods. The atomistic modelling approach is premised on the application of the molecular dynamics (MD) method implemented in the open source code program Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). The continuum mechanics approach is based on the classical elasticity theory of anisotropic media, in which the mechanical fields associated with the crack tip are represented by the crack tip stress and displacement series expansions generalizing the well-known classical asymptotical presentation of M. Williams to the case of anisotropic media. Using the MD method, a large series of computations of the loading of monocrystalline copper and aluminum plates with a face-centered cubic crystal lattice weakened by a central crack applying the embedded atom potential (EAM) was implemented. MD modeling is aimed at determining the atomic stresses and strains near the crack tip. The computed atomic stresses were compared with the stress field determined by the continuum elasticity theory of anisotropic media for crystal lattices with cubic symmetry of elastic properties. A comparative analysis was carried out for the angular dependencies of the stress and strain tensor components at various selected distances from the crack tip for the entire range of mixed deformation forms starting from pure tensile loading and ending with forms close to pure transverse shear. It is ascertained that the crack tip fields determined on the basis of two fundamentally different approaches, precisely, discrete and continuum, are completely consistent with each other. It is established that mathematical methods of continuum fracture mechanics can be applied to describe stress, strain and displacement fields at the atomic level.

(Invited) Bragg Coherent Diffraction Imaging Studies of Composition Distribution and Strain in Catalytic Alloy Nanoparticles

Compositional distribution, strain, and defects in alloy nanoparticles significantly affect the catalytic properties for a wide range of applications such as gas reforming, fuel cells, electrolysis, etc. while various alloy compositions of nanoparticles have been investigated to improve performance and reduce the amount of precious metals used. It was reported that the composition distribution of the alloy particles, expectedly homogeneous in the whole particle, changed dynamically depending on the operating environment, resulting in altering the catalytic activity [1]. Furthermore, defects such as dislocations, grain boundaries, and elastic strain of the crystalline lattice also significantly affect catalytic properties. Since these factors dynamically change and evolve under operating environments, in situ measurements are essential to reveal these factors and ultimately to exploit them to enhance the catalytic properties. Bragg coherent diffraction imaging (BCDI) is a lens-less imaging method that measures speckle patterns near Bragg diffraction from a crystalline sample and reconstructs the 3D particle images. Since Bragg diffraction is inherently sensitive to displacements of the crystal lattice, BCDI yields the displacement field within the sample as well as the electron density, i.e., the “shape” of the particle. This displacement field information further allows us to evaluate strain distribution, dislocations, and composition distribution in the particle. BCDI is also advantageous for in situ measurements because the 3D image can be obtained by rotating the sample by only a few degrees, which eases the geometrical constraint of an electrochemical cell for the measurement. This is in contrast to conventional 3D imaging techniques such as X-ray CT, which usually requires 180 degrees of sample rotation.Furthermore, it is expected that the measurement capabilities of BCDI will be rapidly advanced by upgrading synchrotron radiation facilities such as APS-U, which delivers high-flux coherent X-rays. This should expand the applications of BCDI to wider electrochemical materials. In this presentation, we will briefly overview the principle of BCDI followed by specific examples of in situ measurements of catalytic alloy particles [2-4] conducted at 34-ID-C at Advanced Photon Source.[1] F. Tao et al., Science 322, 932 (2008).[2] T. Kawaguchi et al., Phys. Rev. Lett. 123, 246001 (2019).[3] T. Kawaguchi et al., Nano Lett. 21, 5945 (2021).[4] D. Sheyfer, R. G. Mariano, T. Kawaguchi, et al., Nano Lett. 23, 1 (2023).

Model order reduction of thermal-dynamic coupled flexible multibody system with multiple varying parameters

Spacecrafts usually suffer from the thermal shock during operation, leading to large control error or even failure. At the same time, spacecrafts are often rigid-flexible coupled multibody systems with large degrees of freedom and multiple varying parameters. The real-time control and condition monitoring require effective order reduction methods. In this investigation, the displacement and temperature fields are discretized by the Absolute Node Coordinate Formulation (ANCF) to achieve unified description. The coupled thermal-dynamic reduced-order model (ROM) is established based on the Proper Orthogonal Decomposition (POD) method. For the purpose of further increase efficiency, a linearization method is introduced so that the calculation times of the elastic force can be significantly reduced. In order to predict the response of the thermal-dynamic coupled system with multiple varying parameters, the interpolation on Grassmann manifold is introduced to maintain the orthogonality of the basis. As verification and validation, three numerical examples are presented to show the feasibility and efficacy of the proposed method.

К теории рентгеновской Лауэ дифракции в термомиграционном кристаллическом канале с легирующей примесью

X-ray Laue diffraction in a silicon crystal with Si(Al) thermomigration channels has been theoretically considered. Based on the model of elastic fields of atomic displacements in the channel, expressions for the distribution of strains have been obtained to describe diffraction in the Laue geometry. A numerical calculation of the X-ray scattering intensity distribution near a reciprocal lattice point has been performed. The difference between diffraction in a perfect and strained crystal has been shown.

Experimental Characterization of Anisotropic Mechanical Behaviour and Failure Mechanisms of Hardened Printed Concrete.

Extrusion-based printing of cementitious materials represents an innovative technology in civil engineering. The additive manufacturing process significantly influences the material properties in the hardened state, leading to anisotropic behaviour in terms of stiffness and strength compared to conventionally cast concrete. This experimental study aims to deepen the understanding of the mechanical behaviour of hardened printed concrete. Beam-like specimens with varying printing patterns, loading orientations and lengths are investigated within three-point bending tests (3PBT) and uniaxial compression tests (UCT). Homogenized material parameters such as Young's modulus, compressive and flexural tensile strength and density are statistically evaluated using optically measured displacement and strain fields on the specimen surface. The qualitative and quantitative results demonstrate a strong dependency of material properties and failure mechanisms on the printing pattern. The interfilamental and interlayer areas with weak adhesion are identified as the main reason for anisotropy.

Modeling and Estimation of Continuous Flexible Structure Using Theory of Functional Connections

This paper presents a novel method for modeling and estimating the dynamics of a continuous structure based on a limited number of noisy measurements. The goal is reached using a Kalman filter in synergy with the recently developed mathematical framework known as the Theory of Functional Connections (TFC). The TFC allows deriving a functional expression capable of representing the entire space of the functions that satisfy a given set of linear and, in some cases, nonlinear constraints. The proposed approach exploits the possibilities offered by the TFC to derive an approximated dynamical model for the flexible system using the Lagrangian mechanics. The result is a representation of the structural dynamics using a finite number of states, in contrast to the infinite-dimensional model that would be obtained by application of the traditional continuum mechanics models that are based on sets of partial differential equations. The limited number of states enables the application of the well-known Kalman filter framework to improve the estimation of the displacements and displacement velocities. In addition, the continuous displacement field of the structure can be reconstructed with high fidelity. The theoretical development of the method is presented in relation to the case of an Euler–Bernoulli beam. Finally, the obtained model is used to carry out a simulation campaign aimed at assessing the accuracy, efficiency, and robustness of the proposed method.

MRI-based quantification of cardiac-driven brain biomechanics for early detection of neurological disorders.

We present a pipeline to quantify biomechanical environment of the brain using solely MRI-derived data in order to elucidate the role of biomechanical factors in neurodegenerative disorders. Neurological disorders, like Alzheimer's and Parkinson's diseases, are associated with physical changes, including the accumulation of amyloid-β and tau proteins, damage to the cerebral vasculature, hypertension, atrophy of the cortical gray matter, and lesions of the periventricular white matter. Alterations in the external mechanical environment of cells can trigger pathological processes, and it is known that AD causes reduced stiffness in the brain tissue during degeneration. However, there appears to be a significant lag time between microscale changes and macroscale obstruction of neurological function in the brain. Here, we present a pipeline to quantify the whole brain biomechanical environment to bridge the gap in understanding how underlying brain changes affect macroscale brain biomechanics. This pipeline enables image-based quantification of subject-specific displacement field of the whole brain to subject-specific strain, strain rate, and stress across 133 labeled functional brain regions. We have focused our development efforts on utilizing solely MRI-derived data to facilitate clinical applicability of our approach and have emphasized automation in all aspects of our methods to reduce operator dependance. Our pipeline has the potential to improve early detection of neurological disorders and facilitate the identification of disease before widespread, irreversible damage has occurred.

Clustering of negative topological charges precedes plastic failure in 3D glasses.

The deformation mechanism in amorphous solids subjected to external shear remains poorly understood because of the absence of well-defined topological defects mediating the plastic deformation. The notion of soft spots has emerged as a useful tool to characterize the onset of irreversible rearrangements and plastic flow, but these entities are not clearly defined in terms of geometry and topology. In this study, we unveil the phenomenology of recently discovered, precisely defined topological defects governing the microscopic mechanical and yielding behavior of a model 3D glass under shear deformation. We identify the existence of vortex-like and antivortex-like topological defects within the 3D nonaffine displacement field. The number density of these defects exhibits a significant anticorrelation with the plastic events, with defect proliferation-annihilation cycles matching the alternation of elastic-like segments and catastrophic plastic drops, respectively. Furthermore, we observe collective annihilation of these point-like defects via plastic events, with large local topological charge fluctuations in the vicinity of regions that feature strong nonaffine displacements. We reveal that plastic yielding is driven by several large sized clusters of net negative topological charge, the massive annihilation of which triggers the onset of plastic flow. These findings suggest a geometric and topological characterization of soft spots and pave the way for the mechanistic understanding of topological defects as mediators of plastic deformation in glassy materials.

Investigation of fatigue crack growth behavior and crack tip plastic zone characteristics in welded structures based on local displacement fields

The fatigue crack growth behavior in welds is difficult to evaluate due to complex internal stresses and microstructural differences. This paper uses the CJP model based on displacement field variations to describe the fatigue crack growth behavior and the size of the crack tip plastic zone in welds. First, fatigue crack growth rate tests were conducted on base metal and as-welded specimens. The digital image correlation technique was used to capture the speckle pattern variations on the specimen surface during crack growth, obtaining the crack tip displacement field required by the CJP model. Based on this, by calculating the stress intensity factor ranges ΔK and ΔKCJP, two da/dN–ΔK(ΔKCJP) curves defined by the traditional and CJP models were obtained, proving the model’s applicability for describing fatigue crack growth behavior in welded joints. Finally, the CJP model was used to calculate and compare the crack tip plastic zones of base metal and as-welded specimens. It was found that the maximum error for the as-welded specimen was 37.84%, occurring at the initial stage of crack propagation and gradually decreasing with the fatigue process.