Problem Of Brittle Fracture Research Articles

Continuum Model of Peridynamics for Brittle Fracture Problems

The article investigates the nonlocal method of peridynamics, which makes it possible to simulate the brittle fracture of a solid body without using spatial derivatives. The basic motion equation of a particle with a given volume is written in integral form. A model combining the key features of continuum mechanics and of the nonlocal method is considered. To determine the forces of pair interaction, the dependence of the Cauchy stress tensor on the rate-of strain tensor was used. This formulation correctly describes the behavior of the material during damage and allows to get rid of the limitations inherent to simple bond-based model and ordinary state-based model. The maximum value of the tensile stress is used as a criterion of fracture, which describes the process of nucleation and evolution of damage. To test the implemented model, tasks in a two-dimensional formulation were used. Using the example of the elastic problem about uniaxial tension of a thin rod, the convergence of the numerical solution is shown with a decrease of interaction horizon and an increase of particles number. The second task demonstrates the capabilities of the implemented model to describe the nucleation and evolution of a crack under uniaxial load on a plate with an initial horizontal defect.

Assessment of properties of bamboo fiber and EPDM reinforced polypropylene microcellular foam composites

Plant fiber reinforced composite has broad industrial application prospects, and has the trend of gradually replacing traditional materials, but there are also problems such as not obvious advantages of lightweight and poor toughness. In this paper, alkali modified bamboo fiber reinforced polypropylene (BF/PP) composite was prepared by supercritical foam injection molding. The effect of fiber content on the formability of BF/PP composite was studied. Furthermore, the strength and toughness of BF/PP composites were balanced by blending Ethylene Propylene Diene Monomer (EPDM) elastomers with different content under the optimal fiber content of 20 %. The physical and mechanical properties of BF/PP composites were studied. The results showed that the addition of bamboo fiber improved the structure of the cell and significantly enhanced the mechanical properties of the composite. The addition of elastomer EPDM effectively alleviates the problem of poor toughness and brittle fracture of bamboo fiber composites. The foam material density decreased by about 15 %, the addition of bamboo fiber brought the maximum 57.37 % tensile strength and 79.96 % flexural strength improvement to the polypropylene matrix. When the EPDM content was 10 %, the foaming effect of the material was the best, and the impact toughness increased by 34.42 % compared with pure polypropylene. This study expands the application prospect of the bamboo fiber reinforced polypropylene microcellular foam materials.

Analysis of mechanical property degradation of fracture in phase field model

In the AT2 model, there is a problem of premature damage when describing brittle fracture, as well as the length dependence. In this paper, some new degradation function and a parameter are used to characterize fracture in AT2 model. Firstly, new trigonometric and polynomial degradation function are proposed. The mechanical properties of these two functions in describing brittle fracture are theoretically analyzed, the results indicate that the new degradation function can maintain the linear elastic response of the model before fracture. Secondly, the influence of degradation function on fracture characteristic is discussed. A parameter is introduced to characterize the difference in failure performance for degradation function. In addition, a new rational degradation function is proposed based on the polynomial function, which couples the failure strength into the function and attenuates the length dependence of the AT2 model. Finally, the brittle fracture problem under different conditions is studied and compared with the results of classical quadratic degradation function through some representative examples. The results show that the new trigonometric and polynomial degradation functions can reduce the overall performance degradation of material before brittle fracture by nearly 3 orders of magnitude, verifying that the presented degradation functions can better study the brittle fracture, and the effectiveness of the introduced parameter. At the same time, when the rational degradation function is adopted, the model is length-independent, and the results are consistent with experiment and previous numerical results.

The phase-field fracture model enriched by interpolation cover functions for brittle fracture problems

In this study, interpolation cover functions were introduced into the phase-field fracture model (PFM) to improve its convergence and efficiency. The traditional PFM has proven to be successful in simulating brittle fracture problems. However, one disadvantage is that it requires a relatively dense mesh. The PFM model enriched by interpolation cover functions (EPFM) was used for numerical fracture analysis under a relatively coarse mesh. The rank deficiency problem caused by the introduction of interpolation cover functions was resolved using the rank deficiency repair technique. Higher gradients were captured by introducing interpolation cover functions without changing the mesh structure. The numerical examples showed the improvement in convergence. The high efficiency of the proposed EPFM was demonstrated through a series of numerical examples. The numerical results of the EPFM and PFM under different length-scale parameters and meshes were also compared. The results showed that the proposed EPFM is an excellent alternative for evaluating engineering fracture problems.

An effective correspondence-based peridynamics-FEM coupling model for brittle fracture

Correspondence-based peridynamics (CBPD) is attractive and promising because of its outstanding ability to employ the traditional constitutive relations for material modeling. However, application of CBPD to brittle fracture problems is somewhat limited due to the numerical instability issue and lack of a rational bond failure criterion. To enable CBPD to accurately and stably predict brittle fracture problems, an improved energy-based bond failure criterion is firstly established in this work, in which the expression of bond potential is re-derived, and its simplified expression for isotropic linear elastic solids is clearly defined. The critical bond potential is determined by assuming it is directly proportional to the bond length, and the discretization error for the critical energy release rate in the discretized model is corrected with a correction factor. Besides the zero-energy mode control method, additional stabilization technique to reduce the influence of ill-posed or singular reference shape tensor on brittle fracture simulations is also explored to avoid the unexpected irregular damages during the fracture process. In addition, a direct coupling model of CBPD and FEM is established to improve the computational efficiency, and the ghost forces at the interface are verified to be negligible. Finally, several numerical examples involving typical quasi-static and dynamic brittle fracture problems are investigated, and fracture behaviors including the crack paths, load-displacement curves, and crack propagation speeds are all well predicted, which sufficiently demonstrates the accuracy and stability of the proposed method.

Computation of the crack opening displacement in the phase-field model

The phase-field model treats discrete cracks in a smeared sense by a regularisation technique. It holds attractive properties: there is no need to consider cracks as geometric discontinuities, and it avoids remeshing around crack tips. The method has been employed in the analysis of brittle and cohesive fracture problems. In the brittle fracture model, a Griffith-like energy functional is used in the simulation, while in the cohesive fracture model, the fracturing problem exploits a displacement jump governed energy functional. Obviously, the displacement jump is crucial in the cohesive fracture model and in certain other applications, e.g. for hydraulic fracturing. In the current study, the approximated form of the crack opening displacement is derived for the brittle and cohesive fracture models. In both models, the crack opening displacement is associated with a line integral normal to the crack, but different factors in front of the integral apply. The derived integrals are verified analytically in a one-dimensional setting and numerically in multi-dimensional set-ups, featuring straight and curved cracks.

A micromorphic phase-field model for brittle and quasi-brittle fracture

In this manuscript, a robust and variationally consistent technique is proposed for local treatment of the phase-field fracture irreversibility. This technique involves an extension of the phase-field fracture energy functional through a micromorphic approach. Consequently, the phase-field is transformed into a local variable, while a micromorphic variable regularizes the problem. The local nature of the phase-field variable enables an easier implementation of its irreversibility using a pointwise ‘max’ with system level precision. Unlike the popular history variable approach, which also enforces local fracture irreversibility, the micromorphic approach yields a variationally consistent framework. The efficacy of the micromorphic approach in phase-field fracture modelling is demonstrated in this work with numerical experiments on benchmark brittle and quasi-brittle fracture problems in linear elastic media. Furthermore, the extensibility of the micromorphic phase-field fracture model towards multiphysics problems is demonstrated. To that end, a theoretical extension is carried out for modelling hydraulic fracture, and relevant numerical experiments exhibiting crack merging are presented. The source code as well as the data set accompanying this work would be made available on GitHub (https://github.com/ritukeshbharali/falcon).

Realizing excellent tribological properties of BCC/FCC gradient high -entropy alloy coating via an in-situ interface reaction

The simultaneous improvement of strength and toughness has been a long-standing problem in the pursuit of high-performance materials. Using a gradient-structure design that exploits the migration and recrystallization processes of elements in the molten pool, this study generates an FeCoCrNiAl intermediate interfacial layer from FeCoCrNi and FeCoCrAl powders in situ. Further, a strong and tough integrated composite high–entropy alloy (HEA) coating with an outer layer of high-strength body centered–cubic (BCC) HEA, an interface of BCC and face centered–cubic (FCC) dual-phase HEA, and an inner layer of high-toughness FCC HEA is innovatively designed and prepared. The Vickers hardness of the composite exhibited a stepped gradient of 594.4Hv0.5, 311.8Hv0.5, and 182.9Hv0.5 from top to bottom. Subsequently, the surface and cross-sectional tribological behaviors of the gradient coating are studied in various air environments. The composite HEA coating overcomes the brittle fracture problem of single FeCoCrAl coatings and the adhesion problem of single FeCoCrNi coatings. The composite HEA coating exhibits a low friction coefficient, a small width/depth, and a low wear rate (0.374 × 10−5 mm3/(N·m)). The cross-section wear is a dynamic friction process coupled with multiple wear mechanisms, and the wear marks are characterized by ladders, width changes, and debris accumulation. This new approach can guide the future development and design of HEA wear-resistant gradient coatings and HEA composite coatings.

Improved Staggered Algorithm for Phase-Field Brittle Fracture with the Local Arc-Length Method

The local arc-length method is employed to control the incremental loading procedure for phase-field brittle fracture modeling. An improved staggered algorithm with energy and damage iterative tolerance convergence criteria is developed based on the residuals of displacement and phase-field. The improved staggered solution scheme is implemented in the commercial software ABAQUS with user-defined element subroutines. The layered system of finite elements is utilized to solve the coupled elastic displacement and phase-field fracture problem. A one-element benchmark test compared with the analytical solution was conducted to validate the feasibility and accuracy of the developed method. Our study shows that the result calculated with the developed method does not depend on the selected size of loading increments. The results of several numerical experiments show that the improved staggered algorithm is efficient for solving the more complex brittle fracture problems.

Dimensionality reduction through convolutional autoencoders for fracture patterns prediction

Dimensionality reduction methods construct a low dimensional or reduced space. Complex structural mechanics problems can be approximated on a low dimensional space with a Reduced-Order Model or surrogate model. For instance, Principal Components Analysis can compute a reduced basis from a database of full-order snapshots. Principal Components Analysis, however, does not reconstruct low dimensional spaces efficiently for highly nonlinear problems, mainly because it is a linear dimensionality reduction method. In this paper, an original approach based on an autoencoder neural network is developed to construct a nonlinear Reduced-Order Model for a highly nonlinear brittle fracture problem. We show on a set of simulations how the autoencoder can be efficient for dimensionality reduction or compression of highly nonlinear data. A complete deep learning framework is proposed to predict crack propagation patterns directly from the loading conditions. Finally, we validate our nonlinear Reduced-Order Model with data sets generated for two problems using proportional and non-proportional loading conditions. In the first problem, the crack always initiates at the same location but propagates in various directions, which may even vary during the simulation if loading conditions are non-proportional. In the second problem, the crack may additionally initiate at various locations. Results for the two problems evaluate the capabilities of the proposed approach.

A hybrid meshfree discretization to improve the numerical performance of peridynamic models

Efficient and accurate calculation of spatial integrals is of major interest in the numerical implementation of peridynamics (PD). The standard way to perform this calculation is a particle-based approach that discretizes the strong form of the PD governing equation. This approach has rapidly been adopted by the PD community since it offers some advantages. It is computationally cheaper than other available schemes, can conveniently handle material separation, and effectively deals with nonlinear PD models. Nevertheless, PD models are still computationally very expensive compared with those based on the classical continuum mechanics theory, particularly for large-scale problems in three dimensions. This results from the nonlocal nature of the PD theory which leads to interactions of each node of a discretized body with multiple surrounding nodes. Here, we propose a new approach to significantly boost the numerical efficiency of PD models. We propose a discretization scheme that employs a simple collocation procedure and is truly meshfree; i.e., it does not depend on any background integration cells. In contrast to the standard scheme, the proposed scheme requires a much smaller set of neighboring nodes (keeping the same physical length scale) to achieve a specific accuracy and is thus computationally more efficient. Our new scheme is applicable to the case of linear PD models and within neighborhoods where the solution can be approximated by smooth basis functions. Therefore, to fully exploit the advantages of both the standard and the proposed schemes, a hybrid discretization is presented that combines both approaches within an adaptive framework. The high performance of the developed framework is illustrated by several numerical examples, including brittle fracture and corrosion problems in two and three dimensions.

Investigation of crack propagation in plain concrete using Phase-field model

Phase-field approach is adopted to investigate numerically the fracture behavior of the plain concrete beams under static load. Simply supported beams under three-point loads are examined. The implementation of the phase-field model is conducted within the commercial finite element software ABAQUS for two-dimensional brittle mode-I fracture problems. The phase-field model is built on the rate-independent variational principle of diffuse fracture. The implementation is based on both subroutines user element (UEL) and user material (UMAT). The elastic displacement and the fracture problem are decoupled and solved separately as a staggered solution. The main variables considered in this study are the length scale parameter, the fracture energy Gf and stability parameter K. The method is verified by comparing the results with a previous experimental study. The results have shown that the length scale parameter have high effect on the ultimate load and it depends strongly on the mesh size. It is also found that the ultimate load increases with increasing the fracture energy Gf, and the change of the parameter K seems not to affect the ultimate load value and the behavior of beams. To investigate the benefits and defects of the phase-field method, the analysis is compared with another numerical method using the eXtended Finite Element Method.

Quadtree-polygonal smoothed finite element method for adaptive brittle fracture problems

In this work, a quadtree-polygonal smoothed finite element method is proposed for adaptive consistent framework of phase field model on brittle fracture problems. A Smoothed Galerkin Weak form aided with the gradient smoothing technique is formulated to construct the variational formulations for both displacement and phase field. Staggered scheme is employed to solve the coupled phase and displacement field, in which the displacement field is obtained by Newton iterating and central difference method for implicit and explicit dynamic, respectively, while the phase field is solved directly with a linear equation. The critical history energy in the phase field fracture model is obtained with spectral decomposition. In order to acquire high efficiency without accuracy loss, a novel quadtree-based adaptive algorithm is developed for phase field fracture model, and the nodal phase field value is adopted as the direct indicator for mesh refinement. In this way, mesh local refinement is implemented with quadtree subdivision when the nodal phase field value is achieved the given threshold. Meanwhile, arbitrary sided polygonal elements provide an effective way to connect different mesh regions with different sizes. In other words, there is no hanging node but the connecting node on polygonal elements. Several numerical examples are performed for validating the feasibility of the proposed approach, in which the adaptive quadtree-polygonal method can save much computational costs without accuracy loss.

A vector form conjugated-shear bond-based peridynamic model for crack initiation and propagation in linear elastic solids

A vector form conjugated-shear bond-based peridynamic model is proposed to overcome the limitation of the fixed Poisson’s ratio based on the assumption of central force between material points in bond-based peridynamic theory. In the virtue of this proposed approach, the angle variation of a pair of conjugated-shear bonds is derived as a vector in terms of the displacements at three points located at the conjugated-shear bonds instead of the direct use of the angle, which improves the computational efficiency in both two and three dimensional conditions. Emphasis is placed on deriving the relationship between micro moduli in the proposed model and macro constants by equating peridynamic strain energy to the macroscopic strain energy. It is found that the micro moduli in this proposed approach can reduce to classical bond-based peridynamic parameters when the rotation effect of conjugated-shear bonds is not considered. A comparative study of the proposed approach and the FEM or analytical solutions on a square isotropic plate and a three-dimensional rectangular block under uniaxial tension shows that this proposed approach can well investigate the behavior of elastic solids under both two and three dimensional conditions. We test the numerical convergence behavior of the proposed model to classical solutions in the varying horizon/grid spacing ratios or horizon sizes. Moreover, the improvement of computational efficiency of the proposed model is verified by comparison of the CPU time computed from the scalar form conjugated bond-based peridynamic model. The capability of the proposed model in simulating dynamic brittle fracture problems is demonstrated by comparing the results obtained by the proposed model with the previous numerical results in the case of Kalthoff-Winkler experiment.

Behavior of an Internally Confined Hollow Reinforced Concrete Column with a Polygonal Cross-Section

The new supporting structure, internally confined hollow reinforced concrete (ICH RC), was suggested by former researchers. It maintains the material saving effect, which is the advantage of the hollow concrete structure, and it solves the brittle fracture problem of the inner wall by the inner steel pipe to make it into the 3-axis confinement state. However, until now, its design and analysis model has been limited to a circular cross-section. In this study, to expand the applicability, research and development of an ICH RC structure with a polygonal cross-section were performed. The material model was developed by defining the constraint stress in the members of the concrete and deriving a reasonable stress-strain relationship. For the column model, it was developed to predict the behavior of the polygonal ICH RC columns by analyzing the axial force-moment correlation, moment-curvature, and lateral force-displacement relationship. Each model was verified not only by comparing with the results of previous experiments but also by analyzing the results according to parameters. The maximum load and ultimate displacement values through the developed model showed the difference with the experimental results within 6% of mean error. It was verified that the proposed analytical model reasonably reflects the behavior of actual columns.

Nonlocal elastodynamics and fracture

A nonlocal field theory of peridynamic type is applied to model the brittle fracture problem. The elastic fields obtained from the nonlocal model are shown to converge in the limit of vanishing non-locality to solutions of classic plane elastodynamics associated with a running crack. We carry out our analysis for a plate subject to mode one loading. The length of the crack is prescribed a priori and is an increasing function of time.

Thermomechanical phase‐field fracture modeling of fluid‐saturated porous media

AbstractIn this work, the problem of brittle fracture in a fluid‐saturated porous material is extended by considering the non‐isothermal states of the sample. The temperature field will affect the problem in two aspects: 1) Temperature‐dependent material parameters, such as elasticity modulus (E) and critical energy release rate (Gc). 2) Thermal expansion due to thermo‐mechanical volume coupling. In hydraulic fracturing, we further study the effect of the temperature difference between the injected fluid and the surrounding porous media ambient on the crack behavior. The modeling of the porous media domain is based on the macroscopic theory of porous media (TPM), whereas the phase‐field method (PFM) is applied to approximate the sharp crack edges by diffusive ones. In the numerical implementation, the coupled system of partial differential equations will be solved using the FEM in order to simulate the heat transition in the crack and non‐crack regions.

КОМПЬЮТЕРНЫЙ АНАЛИЗ РЕСУРСА ТОНКОСТЕННЫХ КОНСТРУКЦИЙ, РАБОТАЮЩИХ В АГРЕССИВНЫХ СРЕДАХ

The problem of brittle fracture of thin-walled shell structures operating under conditions of combined thermomechanical and corrosive effects is considered. The task solution is based on linear fracture mechanics taking into account plastic deformations in the vicinity of the tip of a developing crack and the impact of an aggressive environment. A mathematical model of the corrosion process was built, an integral part of which is a mathematical model of the corrosion rate of structural material, based on experimental data using Lagrange interpolation polynomials. A numerical method for solving the problem is proposed. To determine the available resource and the parameters of the state of the structures under consideration, a step-by-step method is used, followed by refinement using the dichotomy method. At each step of the computational process, the Cauchy problem is solved for the main system of differential equations. The proposed method for solving the problem is implemented in the form of software. The results of a computer study of the bearing capacity and resource of a typical design of a chemical apparatus body are presented.

Stochastic phase-field modeling of brittle fracture: Computing multiple crack patterns and their probabilities

In variational phase-field modeling of brittle fracture, the functional to be minimized is not convex, so that the necessary stationarity conditions of the functional may admit multiple solutions. The solution obtained in an actual computation is typically one out of several local minimizers. Evidence of multiple solutions induced by small perturbations of numerical or physical parameters was occasionally recorded but not explicitly investigated in the literature. In this work, we focus on this issue and advocate a paradigm shift, away from the search for one particular solution towards the simultaneous description of all possible solutions (local minimizers), along with the probabilities of their occurrence. Inspired by recent approaches advocating measure-valued solutions (Young measures as well as their generalization to statistical solutions) and their numerical approximations in fluid mechanics, we propose the stochastic relaxation of the variational brittle fracture problem through random perturbations of the functional. We introduce the concept of stochastic solution, with the main advantage that point-to-point correlations of the crack phase fields in the underlying domain can be captured. These stochastic solutions are represented by random fields or random variables with values in the classical deterministic solution spaces. In the numerical experiments, we use a simple Monte Carlo approach to compute approximations to such stochastic solutions. The final result of the computation is not a single crack pattern, but rather several possible crack patterns and their probabilities. The stochastic solution framework using evolving random fields allows additionally the interesting possibility of conditioning the probabilities of further crack paths on intermediate crack patterns.

Coupling XFEM and Peridynamics for brittle fracture simulation: part II\u2014adaptive relocation strategy

An adaptive relocation strategy for a coupled XFEM–Peridynamic (PD) model is introduced. The motivation is to enhance the efficiency of the coupled model and demonstrate its applicability to complex brittle fracture problems. The XFEM and PD approximation domains can be redefined during the simulation, to ensure that the computationally expensive PD model is applied only where needed. To this end a two-step expansion/contraction process, allowing the PD patch to adaptively change its shape, size and location, following the propagation of the crack, is employed. No a priori knowledge of the crack path or re-meshing is required, and the methodology can automatically switch between PD and XFEM. Three 2D fracture examples are presented to highlight the performance of the methodology and the ability to follow multiple crack tips. Results indicate significant computational savings. Furthermore, the characteristic length scale of PD theory bestows a nonlocal and multiscale component to the methodology.