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Monte Carlo bond switch algorithm; (a) pristine hexagonal lattice; (b) underlying dual lattice; (c) selected dual bond; (d) switch of the dual bond; (e) minimization of the dual lattice and evaluation of the Si atoms in the triangle centers; (f) overlying atomic network structure; (g)-(i) three different random networks representing a 2D model glass after 2 ⋅ 10 3 dual bond switches.
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Although modeling of fractures in solid materials has been within the focus of researchers for decades, a generally applicable and reliable numerical description is still an open topic. The complexity of fracture description hides within its multiscale nature, whereby the nano- and macroscale material behavior often vary significantly, and the tran...
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... time periods. Secondly, one can investigate the mechanical influence of the atomic structure only. The atomic samples of the model network glass are obtained by random sequences of topological flip transformations. We start from an almost quadratic hexagonal sample containing 1350 atoms with periodic boundary conditions. As demonstrated in Figs. 1a-1f, the network structure is transformed into an underlying dual lattice of tessellating triangles. In this configuration, the flip transformation is performed by switching the dual bond from the blue to the red node pair, as depicted in Figs. 1c and 1d. The dual lattice is minimized using a harmonic potential function and transformed ...
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... hexagonal sample containing 1350 atoms with periodic boundary conditions. As demonstrated in Figs. 1a-1f, the network structure is transformed into an underlying dual lattice of tessellating triangles. In this configuration, the flip transformation is performed by switching the dual bond from the blue to the red node pair, as depicted in Figs. 1c and 1d. The dual lattice is minimized using a harmonic potential function and transformed back to the overlying network structure, as shown in Fig. 1f. As the final step within one flip transformation, the potential energy of the network structure is minimized using the Yukawa-type potential function in Eq. (1). Three samples after a series ...
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... into an underlying dual lattice of tessellating triangles. In this configuration, the flip transformation is performed by switching the dual bond from the blue to the red node pair, as depicted in Figs. 1c and 1d. The dual lattice is minimized using a harmonic potential function and transformed back to the overlying network structure, as shown in Fig. 1f. As the final step within one flip transformation, the potential energy of the network structure is minimized using the Yukawa-type potential function in Eq. (1). Three samples after a series of 2 ⋅ 10 3 topological flip transformations are shown in Figs. 1g-1i. Intriguingly, such a network structure has experimentally been imaged ...
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... potential function and transformed back to the overlying network structure, as shown in Fig. 1f. As the final step within one flip transformation, the potential energy of the network structure is minimized using the Yukawa-type potential function in Eq. (1). Three samples after a series of 2 ⋅ 10 3 topological flip transformations are shown in Figs. 1g-1i. Intriguingly, such a network structure has experimentally been imaged using transmission electron microscopy [14] and is seen as a two-dimensional benchmark model of network materials such as silica glass. Using the Monte Carlo bond switch algorithm, we prepared 100 samples, each of which reveal a unique network ...
Citations
... Many systems have the characteristics of SOC, with the scales of systems changing from nanometers such as kinds of materials crack (Heider et al., 2022;Jensen and Magnasco, 1999), to the whole ecosystem (Da Cruz and Lind, 2012;Dupoyet et al., 2011) or solar system (Paczuski and Hughes, 2004). Table 1 has summarized the SOC research based on different systems. ...
... In addition, some natural disasters (Turcotte and Malamud, 2004) such as landslides and earthquakes (Bhattacharya and Manna, 2007), including earthquake magnitudes distribution, have their own possibility and size models, which fit the SOC characteristics. The SOC theory is also applied in the material crack modelling (Bernardes and Moreira, 1995;Ebrahem et al., 2020;Heider et al., 2022;Huang et al., 2013), and prove the presence in both macroscopic and microscopic. For the social science research, the workplace accidents (Mauro et al., 2018;Nielsen, 2014;Shannon et al., 1997;Turcotte and Rundle, 2002) fit a power law with their severity as measured by the number of days from lost work. ...
... Fractals are explained in a mathematically rigorous manner, with an emphasis on examples and fundamental concepts such as Hausdorff dimension, self-similar sets, and Brownian motion [6]. Fractal structures are seen and used in many branches of physical science and art engineering [7][8][9][10][11][12][13]. Fractal spacetime was considered in the different processes to obtain more general models to explain experimental data [14][15][16][17]. ...
In this paper, we extend Nambu mechanics by incorporating fractal calculus. We introduce Hamiltonian and Lagrangian mechanics involving fractal derivatives. This generalization allows us to analyze the dynamics of fractal systems, capturing their intricate and self-similar properties. It opens up new possibilities for understanding and modeling complex fractal structures.
... The starting point in these studies is the definition of a scalar-valued and space-time-dependent phase-field variable in the range [0, 1], which allows for differentiation between the cracked and the intact regions of the domain. With the PFM, the sharp edges of the crack are approximated by diffusive ones, whereas the width of this region is controlled via an internal length scale, see, e.g., [6,12,16,17,21,49,53,58,62,63,70,73,74] among others. In this context, it is worth mentioning that although the PFM is intensively applied for fracture modeling, it is also widely used in the simulation of many other engineering applications, as phase-change materials, see, e.g., [4,[94][95][96]104], for references. ...
This research aims to extend the isothermal continuum mechanical modeling framework of hydraulic fracturing in porous materials to account for the non-isothermal processes. Whereas the theory of porous media is used for the macroscopic material description, the phase-field method is utilized for modeling the crack initiation and propagation. We proceed in this study from a two-phase porous material consisting of thermomechanically interacting pore fluid and solid matrix. The heat exchange between the fluid in the crack and the surrounding porous environment through the diffusive fracture edges is carefully studied, and new formulations here are proposed. Besides, temperature-dependent solid and fluid material parameters are taken into account, which is of particular importance in connection with fluid viscosity and its effect on post-cracking pressure behavior. This continuum mechanical treatment results in strongly coupled partial differential equations of the mass, the momentum, and the energy balance of the thermally non-equilibrated constituents. Using the finite element method, two-dimensional initial-boundary-value problems are presented to show, on the one hand, the stability and robustness of the applied numerical algorithm in solving the emerged strongly coupled problem in the convection-dominated heat transport state. On the other hand, they show the capability of the modeling scheme in predicting important instances related to hydraulic fracturing and the role of the temperature field in this process. Additionally, they show the importance of using stabilization techniques, such as adding an artificial thermo-diffusivity term, to mitigate temperature fluctuations at high flow velocity.
... Several brittle [50,[56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73] and ductile [74][75][76][77][78][79][80][81][82][83][84][85][86][87][88] phase-field fracture formulations have been proposed for the modeling of small and large strain deformations, and multi-scale/physics problems. These phase-field models are also devoted to many applications in engineering, including the brittle failure of metals based on elastic framework [69][70][71][72], the low-cycle fatigue of engineering components based on elastic-plastic framework [15,16,79,85,87], and the adiabatic shear failure of parts under large impact loads based on thermos-elastic-plastic framework [54,[89][90][91][92][93][94]. ...
The thermal-induced failure mechanism of the bearing outer-ring guiding-surface is investigated within this work when subjected to cyclic impact and sliding actions. The paper combines numerical simulations and experi- mental analysis. A high-speed bearing oil interruption experiment is carried out for testing the severe damage of the bearing steel at high-speed impact-sliding contacts. A coupled thermo-elasto-plastic phase-field model is established and validated by experimental results. It then allows, by simulating the multi-physics problem, the predictions of damage propagation and failure for ductile materials at cyclic impact-sliding contacts. To this end, a temperature-dependent isotropic-kinematic hardening model combined with thermal softening, cyclic strain hardening, and damage degradation is employed. The results show that under high-speed cyclic impact-sliding conditions, the damage initiated and accumulated at the contact near-surface is accompanied by instantaneous high temperature and plastic deformation. The failure of bearing is induced by a strong thermal softening effect at high-speed sliding and rapidly propagated under cyclic impact loading. In addition, the impact velocity, impact frequency, and friction coefficient have significant effects on damage initiation and accumulation.
... The main driving force for these developments is the possibility to handle complex fracture phenomena within numerical methods in two and three dimensions. In recent years, several brittle [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42] and ductile [43,44,45,7,46,47,48,49,50,51,52,53,54,55,56,57,58,59] phase-field fracture formulations have been proposed in the literature. These studies range from the modeling of 2D/3D small and large strain deformations, variational formulations, multi-scale/physics problems, mathematical analysis, different decompositions, and discretization techniques with many applications in science and engineering. ...
A probabilistic approach to phase-field brittle and ductile fracture with random material and geometric properties is proposed within this work. In the macroscopic failure mechanics, materials properties and spatial quantities (of different phases in the geometrical domain) are assumed to be homogeneous and deterministic. This is unlike the lower scale with strong fluctuation in the material and geometrical properties. Such a response is approximated through some uncertainty in the model problem. The presented contribution is devoted to providing a mathematical framework for modeling uncertainty through stochastic analysis of a microstructure undergoing brittle/ductile failure. Hereby, the proposed model employs various representative volume elements with random distribution of stiff inclusions and voids within the composite structure. We develop an allocating strategy to allocate the heterogeneities and generate the corresponding meshes in two- and three-dimensional cases. Then the Monte Carlo Finite Element Method (MC-FEM) is employed for solving the stochastic PDE-based model and approximate the expectation and the variance of the solution field of brittle/ductile failure by evaluating a large number of samples. For the prediction of failure mechanisms, we rely on the phase-field approach which is a widely adopted framework for modeling and computing the fracture phenomena in solids. Incremental perturbed minimization principles for a class of gradient-type dissipative materials are used to derive the perturbed governing equations. This analysis enables us to study the highly heterogeneous microstructure and monitor the uncertainty in failure mechanics. Several numerical examples are given to examine the efficiency of the proposed method.
... The energetic phase-field method (PFM), which is widely applied in the modeling of phase change evolution of phasechange materials, see, for example, [16][17][18][19][20][21] has witnessed in recent years increasing popularity in the application of simulating crack propagation. ...
... Furthermore, the relative permeability has been calculated as a function of . Based on the LB simulation results, the parameters of the van Genuchten relations in Equation (19) and (20) have been tuned to fit the results. The outcomes of these simulations and post-processes are summarized in Figure 10 and Table 3. Regarding the − curves in Figure 10, the retention curve of the crack zone demonstrates more sensitivity than that of the intact zone due to the flow enhancement caused by the presence of the crack. ...
In this paper, we present a reliable micro‐to‐macroscale framework to model multiphase fluid flow through fractured porous media. This is based on utilizing the capabilities of the lattice Boltzmann method (LBM) within the phase‐field modeling (PFM) of fractures in multiphase porous media. In this, we propose new physically motivated phase‐field‐dependent relationships for the residual saturation, the intrinsic as well as relative permeabilities. In addition, an anisotropic, phase‐field‐dependent intrinsic permeability tensor for the fractured porous domains is formulated, which relies on the single‐ and multiphasic LBM flow simulations. Based on these results, new relationships for the variation of the macroscopic theory of porous media (TPM)–PFM model parameters in the transition zone are proposed. Whereby, a multiscale concept for the coupling between the multiphasic flow through the crack on one hand and the porous ambient, on the other hand, is achieved. The hybrid model is numerically applied on a real microgeometry of fractured porous media, extracted via X‐ray microcomputed tomography data of fractured Berea Sandstone. Moreover, the model is utilized for the calculation of the fluid leak‐off from the crack to the intact zones. Additionally, the effects of the depth of the transition zone and the orientation of the crack channels on the amount of leakage flow rates are studied. The outcomes of the numerical model proved the reliability of the multiscale model to simulate multiphasic fluid flow through fractured porous media.
... The material parameters presented in Table 1 show a clear difference in the stiffness between the machine direction (MD) and the cross direction (CD). Therefore, we adopt in this work a transversely isotropic model, discussed in, e.g., [27,35,36], for small strain problems. In this, the symmetry group MG T with the preferred fiber direction a that fulfills a = 1, can be expressed as ...
Nonwovens are a type of textile that possess a wide range of unique properties, such as lightweight and damping characteristics, which make them suitable for many applications as in medicine and engineering. In this study, the focus lies on the mechanical response of nonwovens as a multiphase porous layer excited by an underlying vibrating plate. The material properties of the nonwovens are characterized via laboratory measurements applied to different samples. In particular, a dynamic analysis of the underlying thin plate is carried out to obtain its eigenmodes and, thus, the maximum response. These eigenmodes are then utilized in the boundary conditions in an advanced numerical porous media model to simulate the dynamic response of the anisotropic fibrous material. To understand the coupled processes in the fibrous textile material, a three-dimensional initial-boundary-value problem of porous media dynamics is introduced. The numerical results demonstrate the capability of the proposed model to realize the interplay between the pore-air pressure and the effective stresses during nonwovens vibration and, thus, the role of the pore air in vibration-induced fiber-fiber friction reduction as well as the effectiveness of the nonwovens in the dissipation of the kinetic energy, i.e., damping propagating acoustic waves.
... (2) A phase-field model for capturing the freezing as a phase-change process [20,38,4,32,2]. (3) A phase-field model for capturing the possible onset of fracturing and ice lenses formation [1,29,11,18,31,19,30,26,14,17,15]. (4) A constitutive model that captures the cryo-suction effects [39,37]. ...
The focus of this contribution is laid on different aspects and instances related to porous media fracture under non-isothermal conditions. This includes the extreme case of fracturing due to pore-fluid freezing, where the micro-cryo-suction plays an important role in generating the required stresses for crack onset. This also includes studying the instances related to hydraulic fracturing and heat transfer under non-isothermal conditions. In all cases, the continuum mechanical modeling of the induced fractures is based on macroscopic porous media mechanics together with the phase-field method (PFM) for fracture modeling. For the micro-cryo-suction in saturated porous media, the water freezing is treated as a phase-change process. This is modeled using a different phase-field approach, in which the thermal energy derives the phase change and, thus, leads to the occurrence of micro-cryo-suction. Two numerical examples are presented to show the effectiveness of the proposed modeling frameworks.
