The full response of a brittle structure containing multiple cracks to loading under the servo control is of vital importance in the evaluation of properties of the structure. During crack growth, ...the fracture toughness condition at all the crack tips as well as the equilibrium condition should be obeyed, leading to a nonlinear complementarity problem (NCP). The vector-valued function in the NCP depends implicitly on the cracking increments which in turn determine the stress field. The stress field can be obtained through solving a mixed variational problem. Since the degrees of freedom in the discrete variational problem vary with cracking not only in magnitude but also in number, the Jacobian matrix of the NCP is hard to compute and, it cannot be expected to be solved by the Newton methods that involve the calculation of Jacobian matrices. Therefore, a well-scaled projection-contraction algorithm is designed. The proposed procedure is able to simulate growth of multiple cracks in a natural way, allowing the crack tips to stop anywhere with no sensitivity to node configuration or cracking increments. Through the analysis of some examples that have been widely tested, many interesting and profound phenomena are found which have never been revealed.
•The growth of multiple cracks can be reduced to a nonlinear complementarity problem (NCP).•The NCP can be solved by the projection-contraction algorithm PCA.•A consensus in fracture mechanics on brittle fracture is subverted.•Some interesting and profound phenomena in brittle fracture are revealed.•The collapse load of a brittle structure is unique and independent of the cracking paths.
In this paper, a combination of the high-order numerical manifold method with material interpolation is established with the goal of improving the optimization of cracked structures. Complex Fourier ...shape functions, known for their inherent advantages, are utilized as weight functions in the numerical manifold method. By implementing high-order analysis, the occurrence of checkerboard patterns is effectively avoided. In addition, a modified sensitivity filtering technique is introduced to address the issue of mesh dependency and minimum length scale in the final topology. This technique is based on the notion of a manifold element. Furthermore, a methodology is introduced to represent void regions within the design domain without the need for passive elements. This methodology involves describing an algorithm to accurately determine the manifold element and the corresponding element area within the final topology. Numerical problems of topology optimization are presented to examine all the aforementioned advantages. The findings indicate that the proposed algorithm for topology optimization does not exhibit any instances of checkerboard patterns. Moreover, the final topology of the curved void regions does not show any zigzag patterns. The above method demonstrates a commendable rate of convergence, and the final topology of the cracked regions can be conveniently simulated.
•Crack problems in 2-D orthotropic composites are studied.•Numerical manifold method and interaction integral technique are combined for SIFs extraction.•Our solutions match well with the reference ...ones.•Influences of material orientations and crack geometries on SIFs are examined.•New SIFs for a three-branched and a snowflake-shaped crack are reported.
Orthotropic composites are ubiquitous in engineering design, while the existence of cracks may significantly affect their reliability and lifetime. In this work, the numerical manifold method (NMM) is explored to investigate the fracture behavior of 2-D orthotropic solids with both non-intersecting and multi-branched cracks. Due to the use of dual-cover system (namely, mathematical cover and physical cover), the NMM is superior for crack modeling. The displacement jump across crack faces is naturally portrayed by the unique two covers. Further by injecting the base functions extracted from the crack-tip asymptotic fields of orthotropic materials into the associated NMM local approximation, the near-tip physical properties are well captured. The stress intensity factors (SIFs) are evaluated through the domain-form interaction integral. Through several numerical examples with increasing challenges using mathematical covers totally inconsistent with both external boundaries and internal discontinuities, the validity of the proposed approach is inspected. Besides, the influences of the material orientations and the crack geometries on the SIFs are also examined. Our results demonstrate that the proposed method can model cracked orthotropic bodies with satisfying convenience and accuracy.
The numerical solution of the hydro-mechanical-chemical (HMC) fully coupled equations in porous media faces significant challenges due to spurious oscillation in pore pressure and concentration ...caused by locking and convection dominance. This study proposes a combination of two different discretization schemes: (1) the Galerkin discretization on the primal mesh for the soil skeleton deformation, and (2) the finite volume method (FVM) on the dual mesh for solute transport and fluid flow, named G-FVM, where the approximations of skeleton displacement (u), pore pressure (p), and concentration(c) are established by NMM, to reflect compressible and incompressible deformation. Typical examples of chemo-osmotic and chemo-mechanical consolidation are simulated to verify the accuracy of the G-FVM. Through the numerical tests of 1D and 2D chemo-mechanical consolidation problems, it is evident that when convection-dominated solute transport is associated with Biot's consolidation law, two different numerical oscillations are observed in both pore pressure and concentration if only the Galerkin method is applied. Nevertheless, the G-FVM did not produce oscillations in either pore pressure or concentration and is free of locking and convection dominance, accurately predicting the response of low-permeability porous media.
•The numerical manifold method (NMM) is further developed to solve 2D rock hydraulic fracturing problems.•A “cubic law” is incorporated into the NMM for modeling fluid flow through fractures.•The ...asymptotic fracture-tip functions are used to enrich the global displacement function space of NMM.•The present results agree well with the existing experimental and analytical results.•The advantages of the NMM in discretization and accuracy are demonstrated.
Recent attempts to solve rock mechanics problems using the numerical manifold method (NMM) have been regarded as fruitful. In this paper, a coupled hydro-mechanical (HM) model is incorporated into the enriched NMM to simulate fluid driven fracturing in rocks. In this HM model, a “cubic law” is employed to model fluid flow through fractures. Several benchmark problems are investigated to verify the coupled HM model. The simulation results agree well with the analytical and experimental results, indicating that the coupled HM model is able to simulate the hydraulic fracturing process reliably and correctly.
To simulate the entire process of the progressive failure of rock slopes, a series of techniques are incorporated into the original NMM (numerical manifold method). To reflect the stress ...concentration near the crack tip, the Williams' displacement series is adopted to enrich the global displacement function of the NMM. Furthermore, the most recently proposed strength-based LT criterion, which can account for tensile cracks, tensile-shear cracks and compressive-shear cracks, is adopted to determine the crack propagation direction and length. Three typical numerical examples, including a Mode-I crack problem, a Mode-II crack problem and a Brazilian disc problem are adopted to verify the numerical model. The numerical results indicate that the numerical model is capable of accurately simulating the Mode-I crack propagation problem, the mode-II crack propagation problem and the failure process of Brazilian disc. Furthermore, the numerical model is adopted to investigate the entire progressive failure process of two rock slopes. The corresponding results indicate that the numerical model can not only simulate the propagation and coalescence of multiple cracks in rock masses but also the opening/sliding of rock blocks along discontinuities. The proposed numerical model warrants further investigation.
An improved grain-based numerical manifold method (NMM) is developed to investigate deformation and damage of intact rocks at the meso‑scale. The grain boundaries are embedded into the numerical ...manifold method using a random Voronoi tessellation technique to approximate the microstructure of rocks at the meso‑scale. To enhance efficiency, an improved contact loop updating algorithm is proposed, which only preserves the corners of polygonal blocks and deletes the rest of the loop boundary nodes, thus greatly reducing the number of loop nodes involved in contact retrieval. An interface contact model considering cohesion and tensile strength between rock grains is incorporated into the numerical manifold method to simulate fracturing. With the newly developed grain-based numerical manifold method, Brazilian tests and uniaxial compression tests are numerically simulated to validate failure pattern and macroscopic response against laboratory tests. Sensitivity analysis is conducted using the proposed model to further investigate the influence of different number of grains and different stiffness ratio on the macroscopic response of rocks. The results indicate that the improved grain-based numerical manifold method can be effectively used to study deformation, damage and fracturing of rocks at the meso‑scale.
A micro-mechanical based numerical manifold method (NMM) is proposed in this study to investigate the micro-mechanisms underlying rock macroscopic response and fracture processes. The Voronoi ...tessellation technique is adopted to create randomly-sized polygonal rock micro-grains. A rock micro-grain based broken criterion is proposed and a corresponding grain breaking technique is developed. To better represent the contact behavior of rock grain bonds, a cohesive fracture model that considers tensile, shear and compressive behaviors together, is adopted to interpret the failure of rock grain bonds. The developed program is first validated by reproducing biaxial tests of Transjurane sandstone. Finally, the influences of micro-parameters on the rock macroscopic response and failure modes are investigated. The results show that the developed micro-based model can mimic the deformation and failure characteristics of the test closely. A parameter study shows that the grain contact cohesion has significant effects on the model uniaxial compressive strength. The fracture process and failure mode of rock are dependent on the ratio of grain contact shear stiffness to normal stiffness. With the increase of the contact stiffness ratio, the failure mode of rock under uniaxial compression changes from a diffuse pattern to a concentrated shear band.
•A micro-mechanical based NMM is proposed.•Voronoi tessellation technique is used to create the polygonal rock grains.•A rock micro-grain based broken criterion is proposed.•A cohesive fracture model is adopted to model the failure of rock grain bonds.•Influences of micro-parameters on rock macroscopic response are studied.
•An monolithic two-scale homogenization numerical manifold model is proposed for transient nonlinear hydro-mechanics.•Two-level simulations are computed in the same newton loop, avoiding microscale ...iterations.•Microscale problems are decoupled from each other, reducing size of the resultant system of equations greatly.•Around 40% of computational costs are reduced in comparison with the conventional staggered homogenization model.
This paper presents an efficient monolithic computational homogenization model for transient nonlinear hydro-mechanical analysis within the framework of Numerical Manifold Method (NMM). The proposed model is on the same theoretical basis as the FE2 method. The scale transitions are achieved through the extended Hill-Mandel theorem so that the microscopic fluid and solid dynamic effects are fully incorporated. The two-scale simulations are solved in a monolithic manner and the microscopic problems of all macroscopic integration points are decoupled from each other to prevent size of the system of equations from soaring to exceedingly large. By conveying microscale unbalanced forces and tangent operators to the macroscale level, the micro- and macroscale problems are solved in the same Newton loop such that unnecessary microscopic iterations based on estimated macroscopic variables in the conventional nested homogenization m are avoided. By solving benchmark numerical examples, the proposed model proves to be capable of capturing transient hydro-mechanical responses accurately. Moreover, in contrast to the conventional nested homogenization model, the proposed model saves around 40% of computational costs for nonlinear hydro-mechanical analysis. Using the framework of numerical manifold, the presented model can be easily extended to multiscale analyses involving complex boundaries, interfaces and fractures.
Summary
Aiming to accurately simulate seismic dynamic response of rock masses using the numerical manifold method (NMM), boundary settings must be treated carefully. In this paper, 4 issues in ...boundary settings are investigated to improve the performance of NMM: (1) Nonreflecting boundaries including the viscous boundary and viscoelastic boundary are considered; (2) A free‐field boundary is incorporated into NMM to accurately simulate external source wave motion; (3) A seismic input boundary is considered, and the force input method is introduced; and (4) A static‐dynamic unified boundary is incorporated for the convenience of transforming displacement boundary into other types of boundaries, such as nonreflecting boundaries and seismic input boundary. Several benchmark problems are solved to validate the improved NMM. Simulation results agree well with analytical ones, indicating that the improved NMM is able to simulate seismic dynamic response of rock masses reliably and correctly.
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