Discontinuous Displacement Field Research Articles

A new digital image correlation method for discontinuous measurement in fracture analysis

The Digital Image Correlation (DIC) method is commonly used in fracture analysis, but it has limitations in measuring discontinuities, which is crucial in fracture analysis. To address this issue, this paper proposes the Adaptive Discontinuous Digital Image Correlation (AD-DIC) method, which is explicitly designed to measure discontinuous displacement fields effectively and accurately in fracture analysis. The AD-DIC method both overcomes the limitations of previous methods by incorporating an adaptive subset tracking technique, which adaptively adjusts the binarization threshold, and improves the traditional Reliability Guided-Digital Image Correlation algorithm to reconstruct the displacement field near cracks. According to high-precision discontinuous displacement fields measured by AD-DIC method, we can accurately identify cracks, measure crack opening, and calculate inclination angles. To validate the effectiveness of the AD-DIC method, theoretical images of type I and mixed-mode I/II cracks are analyzed, as well as four actual experiments. The results show that the AD-DIC method achieves a high level of accuracy in calculating crack opening, with an error of less than 0.1 pixels. The actual experiments presented in this study demonstrate the wide-ranging applications of the AD-DIC method in fracture analysis, including identifying cracks that are difficult to detect with the naked eye, measuring crack opening and inclination angle, and tracking crack propagation processes over time. This study provides an effective tool for brittle fracture analysis, and the relevant algorithms can serve as a reference for discontinuous measurement.

A finite-volume implementation of the phase-field model for brittle fracture with adaptive mesh refinement

The phase-field method (PFM) has advantages in modeling crack propagation in rock-like materials by treating the crack surface as a continuous function, therefore avoiding dealing with the sharp discontinuous displacement field. This study presents an implementation of PFM using the finite volume method (FVM) to discretize the governing equations for the conservation of linear momentum and the evolution of phase field. The coupled equations are solved by an iterative staggered scheme. The adaptive mesh refinement (AMR) technique is further employed to improve computational efficiency, which is relatively easy to achieve in FVM as the method can naturally handle unstructured meshes with hanging nodes in both two- and three-dimensional problems, as opposed to the finite element method (FEM). Several classical crack propagation problems, including the single-edge notched tension and shear tests, the L-shaped panel test, and the test on a notched plate with a hole, are simulated and compared with available experimental and FEM results, which demonstrate that the FVM-based PFM can model crack propagation accurately and effectively and could be more efficient than the FEM-based PFM.

Element-removal global digital image correlation for accurate discontinuous deformation field measurement in fracture mechanics

We propose an element-removal (ER) global digital image correlation (DIC) method to improve the measurement accuracy of discontinuous deformation fields, such as crack propagation. The occurrence of cracks in materials or structures inevitably deteriorates the tracking accuracy, and, consequently, the strain field accuracy obtained by regular subset and global DIC. The proposed ER-global-DIC algorithm iteratively identifies and removes all the elements covering the crack, during the updating of displacement fields. In the remaining elements, the continuous shape function is applicable for accurate deformation measurement. In principle, although elements that contain the cracks are removed, the algorithm preserves the same number of nodes since the nodes are retained by the remaining elements. Synthetically deformed images based on analytical discontinuous displacement fields validate the effectiveness and accuracy of the proposed method. The ER-global-DIC is further applied to measure the discontinuous displacement fields containing a crack deflection, generated from a finite element model with a cohesive zone model. The results demonstrate the potential of the proposed method for discontinuous deformation measurement on advanced materials, e.g., fiber-reinforced composites.

Meshless numerical method for the contact problems of joint surface

PurposeThe partition of unity of the standard meshless Galerkin method is used as basis in expressing the discontinuity of the contact surface displacement, particularly by adding discontinuous terms into the displacement mode, and constructing the discontinuous meshless displacement field function. In this study the contact surface equation is aimed to derive from the improved Coulomb friction contact model.Design/methodology/approachIn this paper based on the basic idea of meshless method, an improved moving least squares approximation function (expansion method based on out of unit division) is applied to the analysis of two-dimensional contact problems.FindingsOn the basis of this equation after discrete processing, it is combined with the discrete form of the virtual work equation with added contact conditions, and eventually transformed into a standard linear complementary problem. Moreover, it is solved by using the Lemke algorithm, and a corresponding example is provided in this research.Originality/valueThe proposed method can effectively control the mutual embedding of the contact surface, and the stress distribution that is the same as the actual situation can be obtained on the contact surface.

Correlation-Based Mutual Information Model for Analysis of Lung Cancer CT Image.

Most of the people all over the world pass away from complications related to lung cancer every single day. It is a deadly form of the disease. To improve a person's chances of survival, an early diagnosis is a necessary prerequisite. In this regard, the existing methods of tumour detection, such as CT scans, are most commonly used to recognize infected regions. Despite this, there are certain obstacles presented by CT imaging, so this paper proposes a novel model which is a correlation-based model designed for analysis of lung cancer. When registering pictures of thoracic and abdominal organs with slider motion, the total variation regularization term may correct the border discontinuous displacement field, but it cannot maintain the local characteristics of the image and loses the registration accuracy. The thin-plate spline energy operator and the total variation operator are spatially weighted via the spatial position weight of the pixel points to construct an adaptive thin-plate spline total variation regular term for lung image CT single-mode registration and CT/PET dual-mode registration. The regular term is then combined with the CRMI similarity measure and the L-BFGS optimization approach to create a nonrigid registration procedure. The proposed method assures the smoothness of interior of the picture while ensuring the discontinuous motion of the border and has greater registration accuracy, according to the experimental findings on the DIR-Lab 4D-CT public dataset and the CT/PET clinical dataset.

Splitting proximate algorithm for deformable image registration based on functions of bounded deformation.

Deformable image registration is a crucial task in the field of medical image analysis. Functions of bounded deformation (BD) have been proved effective for modeling the displacement fields between medical images since they can capture the discontinuity of displacement fields along edges of organs andtissues. However, we find that at the same time, BD functions-based models tend to obtain discontinuous displacement fields inside the regions of organs and tissues due to image noises in some cases and the presented gradient descent algorithm is time-consuming. To alleviate these problems, we propose a faster algorithm named SPA: splitting proximatealgorithm. In the framework of variable-splitting scheme, we incorporate a proximal term in the deformable registration energy based on functions ofBD. The proposed algorithm can efficiently solve the original model and obtain displacement fields, which look more natural and plausible. Numerical experiments show the effectiveness and stability of the proposed algorithm. The proposed SPA is able to drastically register the images with a plausible deformation field and not sensitive tonoise.

An explicit representation of cracks in the variational phase field method for brittle fractures

The phase field method (PFM) by means of regularization fractures has become a promising and popular method for complex crack propagation, especially in geotechnical applications. Attractive as it is, a challenge exists for the PFM to obtain explicit representation of cracks and jump displacement field across cracks that are essential and crucial for some applications, e.g., hydraulic fracturing, leakage problems, and contact problems. In the present paper, an algorithm in which the values of phase field are regarded as the indicator to identify crack paths is proposed. Then physical patches are cut by the reconstructed crack paths to obtain explicit cracks and jump displacement field in the context of the numerical manifold method (NMM). The numerical examples investigate that various explicit cracks, including a straight crack, a curve crack, the branching cracks, and the merging cracks, predicted by the proposed method are in good agreement with the literature. Besides, the Sneddon’s example verifies that the discontinuous displacement field across explicit cracks is in line with the closed form solution. The proposed numerical model deserves a further investigation.

Three-dimensional crack propagation and inclusion-crack interaction based on IGABEM

The isogeometric boundary element method (IGABEM) is developed to simulate the crack propagation and the inclusion-crack interaction in 3D infinite isotropic medium. The influence of complex shape inclusions on the stress intensity factors (SIFs) along the crack front is studied from the aspects of shape, stiffness, size and position. The non-uniform rational B-spline (NURBS) basis functions can be used to accurately describe the geometric shapes of inclusions and cracks, and the displacement, traction, and discontinuous displacement fields also can be approximated by the same NURBS basis functions. During crack propagation, the normal and tangential vectors of the crack boundary can be uniquely solved. Three examples verify the accuracy and effectiveness of the proposed method. The results show that the SIFs can be calculated accurately even using the single point formula, and the crack propagation process is stable and the path is smooth.

ParticLS: Object-oriented software for discrete element methods and peridynamics

ParticLS (\emph{Partic}le \emph{L}evel \emph{S}ets) is a software library that implements the discrete element method (DEM) and meshfree methods. ParticLS tracks the interaction between individual particles whose geometries are defined by level sets capable of capturing complex shapes. These particles either represent rigid bodies or material points within a continuum. Particle-particle interactions using various contact laws numerically approximate solutions to energy and mass conservation equations, simulating rigid body dynamics or deformation/fracture. By leveraging multiple contact laws, ParticLS can simulate interacting bodies that deform, fracture, and are composed of many particles. In the continuum setting, we numerically solve the peridynamic equations -- integro-differential equations capable of modeling objects with discontinuous displacement fields and complex fracture dynamics. We show that the discretized peridynamic equations can be solved using the same software infrastructure that implements the DEM. Therefore, we design a unique software library where users can easily add particles with arbitrary geometries and new contact laws that model either rigid-body interaction or peridynamic constitutive relationships. We demonstrate ParticLS' versatility on test problems meant to showcase features applicable to a broad selection of fields such as tectonics, granular media, multiscale simulations, glacier calving, and sea ice.

Image‐based simulation of complex fracture networks by numerical manifold method

AbstractComplex geometric features of fracture networks contained in rock masses can be captured by digital images quickly and accurately. However, the continuous and discontinuous deformation simulation of rock masses directly based on digital images is still challenging. Based on the Numerical Manifold Method (NMM), a simple and efficient image‐based simulation method is developed for rock masses containing complex fracture networks. A new crack model, named as “voxel crack model,” is generated based on pixel values of the digital image and a user‐defined resolution. With the voxel crack model, physical cover and manifold elements in NMM are then efficiently generated for complex fracture networks. After this, the continuous and discontinuous displacement fields of rock masses are simulated straightforwardly. In the present work, conventionally burdensome tasks, like boundary detections to transform digital images into computer aided design (CAD) models, generating conforming meshes, and explicitly tracking of fracture surfaces, are avoided. Several benchmarks validate the developed method. The calculated continuous and discontinuous deformation field of the San Leo rock slope containing a complex fracture network demonstrates the capability of the proposed method for practical engineering application.

Deformable Image Registration Based on Functions of Bounded Generalized Deformation

Functions of bounded deformation (BD) are widely used in the theory of elastoplasticity to describe the possibly discontinuous displacement fields inside elastoplastic bodies. BD functions have been proved suitable for deformable image registration, the goal of which is to find the displacement field between a moving image and a fixed image. Recently BD functions have been generalized to symmetric tensor fields of bounded generalized variation. In this paper, we focus on the first-order symmetric tensor fields, i.e., vector-valued functions, of bounded generalized variation. We specify these functions as functions of bounded generalized deformation (BGD) since BGD functions are natural generalizations of BD functions. We propose a BGD model for deformable image registration problems by regarding concerned displacement fields as BGD functions. BGD model employs not only the first-order but also higher-order coupling information of components of the displacement field. It turns out that BGD model allows for jump discontinuities of displacements while, in contrast to BD model, at the same time is able to employ higher-order derivatives of displacements in smooth regions. As a result, BGD model tends to capture possible discontinuities of displacements appeared around edges of the target objects while keep the smoothness of displacements inside the target objects as well. This characteristic enables BGD model to obtain better registration results than BD model and other variational models. To our knowledge, it is the first time in literature to use BGD functions for image registration. A first-order adaptive primal–dual algorithm is adopted to solve the proposed BGD model. Numerical experiments on 2D and 3D images show both effectiveness and advantages of BGD model.

Mechanisms for the control of the complex propagation behaviour of hydraulic fractures in shale

The complex propagation behaviours of hydraulic fracture (HF) at bedding planes (BPs) and the produced complicated fracture geometry are essential to enhance the production of shale gas reservoirs. To better understand the complex propagation behaviour of HF at BPs with different bond strengths, the propagation behaviour, including arrest, crossing, and deviation, was identified first from post-fracturing shale specimens. The discontinuous stress and displacement fields on both sides of a BP ahead of an HF were then determined using the numerical simulation method. Finally, three mechanisms—principal stress jump, Cook–Gordon debonding or Poisson effect, and elastic dissimilarity—were explored in detail to interpret the complex propagation behaviour. The results revealed that HF is arrested/deviated only at extremely weakly cemented or fully opened BPs, whereas HF crosses strongly cemented BPs. The high heterogeneity of the BPs in cementing strength is responsible for the complex propagation behaviour of HF. The principal stress jump at a BP is caused by the difference in stiffness between the BP and the rock matrix. The maximum tensile principal stress ahead of the HF cannot be transmitted across the weakly cemented or fully opened BPs, suggesting that the HF cannot cross BPs. The principal stresses may rotate at a weakly cemented BP, and the rotated principal stresses tend to terminate or deviate an HF. Because of Cook–Gordon debonding and the Poisson effect, if a weakly cemented BP is present and is roughly normal to an advancing HF, the BP may break at some distance ahead of the fracture tip and induce a secondary fracture along the BP. The HF then reaches the opened BP and deviates towards the BP. The propagation process of L- or T-shaped fractures can be interpreted both by the Cook–Gordon debonding and Poisson effect from the viewpoints of stress and displacement. The three mechanisms often operate together when an HF deviates towards a weak BP; while for a special case, there may be only one dominant mechanism.

An asynchronous and task-based implementation of peridynamics utilizing HPX—the C++ standard library for parallelism and concurrency

On modern supercomputers, asynchronous many task systems are emerging to address the new architecture of computational nodes. Through this shift of increasing cores per node, a new programming model with the focus on handle the fine-grain parallelism of this increasing amount of cores per computational node is needed. Asynchronous Many Task (AMT) run time systems represent an emerging paradigm for addressing fine-grain parallelism since they handle the increasing amount of threads per node and concurrency. HPX, a open source C++ standard library for parallelism and concurrency, is one AMT which is confirm with the C++ standard. Which means that HPX's Application Programming Interface (API) is confirm with its definition by the C++ standard committee. For example for the concept of futurization the hpx:future can be replaced by std::future without breaking the API. Peridynamics is a non-local generalization of continuum mechanics tailored to address discontinuous displacement fields arising in fracture mechanics. As many non-local approaches, peridynamics requires considerable computing resources to solve practical problems. This paper investigates the implementation of a peridynamics EMU nodal discretization in an asynchronous task-based fashion. The scalability of asynchronous task-based implementation is to be in agreement with theoretical estimations. In addition, to the scalabilty the code is convergent for implicit time integration and recovers theoretical solutions. Explicit time integration, convergence results are presented to showcase the agreement of results with theoretical claims in previous works.

A mechanical model to interpret distributed fiber optic strain measurement at displacement discontinuities

Distributed fiber optic (strain) sensing, which provides the unique advantage of sensing damage (e.g. cracking) at locations that are not known a priori, has been increasingly used in civil engineering. Quantitative crack measurement requires the translation of a discontinuous displacement field at the crack to a continuous strain deformation in the fiber. The main purpose of this article is to develop a mechanical model to explain the fiber deformation in the presence of a displacement discontinuity. The proposed mechanical model is validated with experimental results from cable calibration tests and concrete cracking tests. The model is extended to simulate the effects of multiple closely spaced cracks on fiber optic strain measurement, and this model is used to create an algorithm to automatically distinguish multiple cracks in distributed fiber optic (strain) sensing strain distributions. Using the model and two shape parameters, kurtosis and standard variation, the effects of cable properties (i.e. shear stiffness between cable and fiber, cable radius, elastic modulus, and interface cohesion) on the shape of fiber optic strain distributions across cracks are also quantified. The results provide an indication of beneficial cable properties for various measurement objectives.

Upper bound finite element limit analysis method with discontinuous quadratic displacement fields and remeshing in non-homogeneous clays

As is known, limit analysis is a common way to study stability and failure mechanism of geotechnical engineering. This paper presents an upper bound finite element limit analysis (UB-FELA) formulation for non-homogeneous clays using discontinuous quadratic displacement fields and second-order cone programming (SOCP). The details are provided about the application of SOCP to the UB-FELA formulation for non-homogeneous clays. To reduce the calculation error, the analytic expressions are proposed concerning the rate of external work done by body forces and power dissipation caused by the plastic deformation following discontinuous quadratic displacement fields. With SOCP solver MOSEK, large-scale optimization problems of UB-FELA can be achieved in minutes via a desktop computer. The method is then applied to analyzing the stability of an undrained square tunnel and an undrained plane strain heading. The numerical results are compared with those reported in related studies. This method helps to obtain highly precise upper bound solution and clear failure mechanism of the computational domain.

Extended finite volume method with enriched HPCK shape functions for dynamic crack propagation modeling

In the present work, an extended finite volume method (EFVM) is incorporated with the enriched hp-cloud approximate function which has Kronecker delta property (EHPCK) for modeling crack propagation in 2D media. In this approximation function, appropriate crack path enrichment and crack tip enrichment functions are considered for modeling the discontinuous displacement field of cracked bodies. Unlike the conventional FVM approach, in the proposed EFVM, excessive control volumes (CVs) are embedded inside the main CVs which are called here as enrichment control volumes (ECVs). The EFVM method doesn't restrict the crack propagation path to obey any predefined path or edges of control volumes. To increase the prediction accuracies, the subtriangular technique is applied for the integration on the cracked control volumes. Capability of introduced EFVM is investigated by solving different problems such as dynamic stress intensity factor calculations and dynamic crack propagation.

A peridynamic Reissner‐Mindlin shell theory

SummaryIn this work, we have developed a state‐based peridynamics theory for nonlinear Reissener‐Mindlin shells to model and predict large deformation of shell structures with thick wall. The nonlocal peridynamic theory of solids offers an integral formulation that is an alternative to traditional local continuum mechanics models based on partial differential equations. This formulation is applicable for solving the material failure problems involved in discontinuous displacement fields. The governing equations of the state‐based peridynamic shell theory are derived based on the nonlocal balance laws by adopting the kinematic assumption of the Reissner and Mindlin plate and shell theories. In the numerical calculations, the stress points are employed to ensure the numerical stability. Several numerical examples are conducted to validate the nonlocal structure mechanics model and to verify the accuracy as well as the convergence of the proposed shell theory.

Crack development assessment using modal analysis in peridynamic theory

AbstractIf structural damage remains undetected and is allowed to grow, structure's load-bearing capacity deteriorates, which can lead to costly repairs or in extreme cases its collapse. Modal analysis is widely used to detect structural damage because, when damage, such as cracks, is introduced, structure's geometrical and/or mechanical properties change, and these changes can be used for damage detection. Peridynamics is a non-local alternative to the continuum mechanics theory that represents forces and displacements using integral equations, which are defined even with discontinuous displacement fields, thus making this theory an attractive option for damage modeling. In this paper, authors verify peridynamic (PD) modal analysis against finite-element (FE) results, and validate it against experimental modal analysis results. The modal solver was implemented in the open-source program Peridigm and four different damage configurations were considered for verification and validation. The results show close agreement between the PD and the FE results, and the PD and the experimental results. Moreover, PD modal frequencies are shown to have similar accuracy to experimental data as the FE results. It is also shown that the frequency shifts are comparable between all three types of modal analysis. The PD mode shapes agreed well with both the FE and the experimental mode shapes at all considered damage configurations. Furthermore, the change in mode shapes from the introduced damage is similar in all three analyses.

Embedded discontinuity approach for coupled hydromechanical analysis of fractured porous media

SummaryIn this paper, a new continuum approach for the coupled hydromechanical analysis of fractured porous media is proposed. The methodology for describing the hydraulic characteristics invokes an enriched form of Darcy's law formulated in the presence of an embedded discontinuity. The constitutive relations governing the hydromechanical response are derived by averaging the fluid pressure gradient and the discontinuous displacement fields over a selected referential volume of the material, subject to some physical constraints. The framework incorporates an internal length scale which is explicitly embedded in the definition of gradient operators. The respective field equations are derived following the general form of balance equations in interacting continua. The conventional finite element method is then employed for the spatial discretization, and the generalized Newmark scheme is used for the temporal discretization. The proposed methodology is verified by some numerical examples dealing with a steady‐state flow through fractured media as well as a time‐dependent consolidation in the presence of a discontinuity.

Nonlinear augmented finite element method for arbitrary cracking in large deformation plates and shells

AbstractThis article presents a nonlinear augmented finite element method (N‐AFEM) for the analysis of arbitrary crack initiation and propagation in large deformation plates and shells. The FE formulations for plate/shell elements and a shell‐like cohesive zone element, both with explicit consideration of geometric nonlinearity, have been derived in detail. The geometrically nonlinear shell‐like cohesive element has the essential feature of 3D but with crack displacements directly extracted from midplane shell element nodes, which enables an accurate description of crack propagation in shells and plates under large deformation. Furthermore, a novel augmentation process that can explicitly account for the discontinuous displacement fields of cracked elements without the need of extra nodes or nodal DoFs has been develop based on a nonlinear Newton‐Raphson method. The numerical performance of the N‐AFEM in modeling a number of benchmark shell/plate fracture problems demonstrates that the method is efficient, accurate, and robust.