Crack Problems Research Articles

Evolutionary Optimization Under Uncertainty: The Strategies to Handle Varied Constraints for Fluid Catalytic Cracking Operation.

This article studies an operational optimization problem of the fluid catalytic cracking (FCC) unit under uncertainty. The objective of this problem is to quickly reoptimize the operating variables that control the operational condition of the FCC unit when fossil fuel yield constraints or prices change. To solve this problem, based on the challenges caused by the varied constraints, we establish a mathematical model and propose a fast adaptive differential evolution algorithm with an adaptive mutation strategy, a parameter adaptation strategy, a repaired strategy, and an enhanced strategy. In the proposed algorithm, we integrate the status information of each solution into the mutation strategy and parameter adaptation scheme to search for the best solution in the irregular feasible region of the operating variables. In addition, a repaired strategy is proposed to repair the infeasible operating variables with unknown bounds, and an enhanced strategy is presented to further improve the objective function value of the best solution. The experimental results on ten test scenarios with different fossil fuel yield constraints and prices demonstrate the robustness of the proposed algorithm for optimizing the operating variables of the FCC unit under uncertainty.

The Erdogan fundamental solution-based hybrid boundary node method for fracture problems

Based on the Erdogan fundamental solutions for infinite cracked plates, a new hybrid boundary node method for linear fracture problems is proposed in this paper. The origin singular intensity factor for the Erdogan fundamental solutions is developed to overcome the singular when the field points are coincided with the source points, and no virtual source points are needed, a new scheme for calculating the origin singular intensity factor for the Erdogan fundamental solutions is developed. Based on the Erdogan fundamental solutions, the zero traction boundary condition on crack surfaces is naturally and strictly satisfied in this method, and no nodes are arranged on the crack surface in the entire calculation process. Based on the Erdogan fundamental solution of stress intensity factor for the mixed mode crack, the stress intensity factor of the present method can be easily interpolated by the Erdogan fundamental solutions. As a result, no complex scheme for calculating stress intensity factor is needed. Based on those theories and methods, the proposed method is further applied to analyze some linear crack problems, and the computational accuracy, convergence rate and the versatility of the present method are demonstrated in details.

Experimental study on electro-thermal and compaction properties of electrically conductive roller-compacted concrete overwintering layer in high RCC dams

To solve the thermal crack problem of overwintering layer of roller-compacted concrete (RCC) dams. A new type of intelligent hydraulic concrete, namely electrically conductive roller-compacted concrete (ERCC), was designed. ERCC layer could provide a new way to control the temperature distribution and reduce the cracking likelihood of RCC dams by its electro-thermal properties, closely related to the variation of resistivity of ERCC. Thus, the relationships between design factors and the resistivity were investigated to obtain the real resistivity of ERCC layer of RCC dams. Then the temperature increase of ERCC layer with the size of 3000 × 700 × 200 mm was measured under 30 V direct current voltage. The temperature rises 14 ~ 16 ℃ within 235 min. It means that the electro-thermal properties of ERCC are enough to resist cold waves in winter. On the other hand, the mechanical and compaction properties of ERCC were investigated through experiments to evaluate whether the roller compaction method can overcome the poor fluidity and improve the construction quality of ERCC. So the density and strength of roller compacted device formed ERCC were compared with those of vibration formed ERCC, which shows the superiority of rolling construction method. According to unit compaction energy and parameters of the roller machines, the onsite compaction passes NS are calculated, which provides guidance for the construction of overwintering layer. In summary, selecting ERCC as the overwintering material instead of traditional thermal insulation materials could provide a more flexible and intelligent temperature control way for RCC dams in the severe cold areas, signifying ERCC is a promising material in hydraulic engineering.

Modeling the entire progressive failure process of rock slopes using a strength-based criterion

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.

Edge crack forming mechanism and the countermeasures in as‐sintered 2024 aluminium alloy during cold rolling

AbstractIn order to solve the problem of edge cracking of the as‐sintered 2024 aluminium alloy during cold rolling, the initiation mechanisms of edge cracking were studied. The results indicate that the “herringbone” crack parallel to the rolling direction and the “zigzag” crack perpendicular to the rolling direction easily formed during cold rolling. The former mainly occurred during the early stage of cold rolling and was related to the poor plasticity of the as‐sintered alloy and the stress state at the end of the rolled plate, whereas the latter appeared in a sample with a rolling reduction of more than 60%, mainly because the grains at the edge of the sample were subjected to three‐direction shear stress, which results in cracking along the grain boundaries. However, both herringbone and zigzag cracks propagated along the grain boundary. Finally, the crack‐free products were obtained by optimizing the cold rolling process.

A phase field modeling approach of cyclic fatigue crack growth

Phase field modeling of fracture has been in the focus of research for over a decade now. The field has gained attention properly due to its benefiting features for the numerical simulations even for complex crack problems. The framework was so far applied to quasi static and dynamic fracture for brittle as well as for ductile materials with isotropic and also with anisotropic fracture resistance. However, fracture due to cyclic mechanical fatigue, which is a very important phenomenon regarding a safe, durable and also economical design of structures, is considered only recently in terms of phase field modeling. While in first phase field models the material’s fracture toughness becomes degraded to simulate fatigue crack growth, we present an alternative method within this work, where the driving force for the fatigue mechanism increases due to cyclic loading. This new contribution is governed by the evolution of fatigue damage, which can be approximated by a linear law, namely the Miner’s rule, for damage accumulation. The proposed model is able to predict nucleation as well as growth of a fatigue crack. Furthermore, by an assessment of crack growth rates obtained from several numerical simulations by a conventional approach for the description of fatigue crack growth, it is shown that the presented model is able to predict realistic behavior.

Stress intensity factors for bonded two half planes weakened by thermally insulated cracks

The thermally insulated inclined or circular arc cracks problems subjected to axial stress in the upper part of bonded two half planes are considered. The modified complex potential function method with the continuity conditions of the resultant force, displacement, and heat conduction functions is used to develop the new system of hypersingular integral equations (HSIEs) for the problems. The new system of HSIEs is solved numerically for the unknown crack opening displacement function and the known traction along the crack as the right-hand term using the appropriate quadrature formulas. Numerical results for the nondimensional stress intensity factors (SIFs) at all cracks tips are presented. A comparison between the nondimensional SIFs for cracks with and without thermal influence is also given

Three-dimensional closed-form solution to elliptical crack problem in magneto-electro-elasticity: Electrically and magnetically induced Maxwell stress boundary condition

An elliptical mode-I crack problem in an infinite multi-ferroic composite material with transverse isotropy under remote uniform mechanical, electric and magnetic loadings is analyzed. The electro-magnetically semi-permeable crack boundary condition and Maxwell stress on the crack faces caused by the electric and magnetic fields are considered, which makes the problem nonlinear. On the basis of the general solution for multi-ferroic composite materials and modern potential theory method, the nonlinear integro-differential boundary equations are established and solved analytically. The field variables in the three-dimensional space and some important fracture mechanics quantities, are obtained in explicit forms. From numerical examples, it is found that the combination of the applied electric and magnetic loadings should be in a elliptical region in the electric displacement-magnetic induction coordinate plane to keep the crack open. Furthermore, in the vicinity of the center of this region, the electro-magnetically semi-permeable crack boundary condition without the Maxwell stress could provide relatively accurate results. Otherwise, the Maxwell stress would have a significant influence, and thus should not be neglected.

Certain cases of elastic equilibrium of a composed wedge with an interface crack

Problems of interface cracks starting from the common corner points of pairs of perfectly glued wedges of different isotropic elastic materials are addressed. It is demonstrated that for a few particular configurations and a restrictive condition imposed on values of elastic constants (corresponding to vanishing of the second Dundurs parameter), the problem of elastic equilibrium may be solved by Khrapkov’s method. These configurations are: (i) the wedges forming a half-plane; (ii) the wedges forming a plane; (iii) one of the wedges being a half-plane. In all cases, the external boundaries are supposed to be free of stresses. By applying Mellin’s transform for all three configurations the problem has been reduced to vector Riemann’s problem, and the matrix coefficient has been factorized for the case of the mentioned restrictive condition. The first configuration, i.e. the problem of an inclined edge crack located along the boundary separating two wedges of different elastic isotropic materials forming a half-plane is considered in more detail. The solution has been obtained for both uniform (corresponding to remote loading) and non-uniform (loading applied at the crack faces) problems. Numerical results are presented and compared with the available results obtained by other authors for particular cases. The obtained solutions appear especially valuable for analysing extreme cases of parameters.

Effects of MgO Expansive Agent and Steel Fiber on Crack Resistance of a Bridge Deck.

To prevent cracks caused by shrinkage of the deck of the Xiaoqing River Bridge, MgO concrete (MC) and steel fiber reinforced MgO concrete (SMC) were used. The deformation and strength of the deck were measured in the field, the resistance to chloride penetration of the concrete was measured in the laboratory, and the pore structure of the concrete was analyzed by a mercury intrusion porosimeter (MIP). The results showed that the expansion caused by the hydration of MgO could suppress the shrinkage of the bridge deck, and the deformation of the deck changed from −88.3 × 10−6 to 24.9 × 10−6, effectively preventing shrinkage cracks. At the same time, due to the restriction of the expansion of MgO by the steel bars, the expansion of the bridge deck in the later stage gradually stabilized, and no harmful expansion was produced. When steel fiber and MgO were used at the same time, the three-dimensional distribution of steel fiber further limited the expansion of MgO. The hydration expansion of MgO in confined space reduced the porosity of concrete, optimized the pore structure, and improved the strength and durability of concrete. The research on the performance of concrete in the in-situ test section showed that MgO and steel fiber were safe for the bridge deck, which not only solved the problem of shrinkage cracking of the bridge deck but also further improved the mechanical properties of the bridge deck.

An improved singular curved boundary integral evaluation method and its application in dual BEM analysis of two- and three-dimensional crack problems

In this paper, an improvement is made to an efficient direct method for numerical evaluation of high order singular curved boundary integrals. Then this improved singular integral evaluation method is employed to solve the strongly and hypersingular integrals involved in the dual boundary element method (Dual BEM), which combine the use of displacement and traction boundary integral equations to solve crack problems in a single domain formulation. The singular integral evaluation method is carried out based on a parameter plane expansion and radial integral approach, this paper proposed a new strategy for treating the singular radial integral, which plays a vital role in this method. In isoparametric coordinate system, the singular curved boundary integral is mapped into a singular square plane integral in intrinsic coordinates, then the radial integration method (RIM) is employed to transform the singular square plane integral into a regular line integral over the contour of intrinsic square plane and a singular radial integral over the path from source point to the contour of intrinsic square plane. A singularity isolation technique is utilized to divide the singular radial integral into two parts, the regular radial integral can be evaluated normally using Gauss quadrature and the singular radial that can be evaluated analytically by expanding the non-singular part of the integrand function into a power series. Compared with conventional local interpolation approach to deal with the singular radial integral, the newly proposed method has a more rigorous mathematical derivation, and can achieve more stable and precise results. Based on the successful implementation of direct evaluation of singular boundary integrals, Dual BEM is successfully applied to solve two- and three-dimensional elastic crack problems including straight and curved crack paths with continuous or discontinuous elements. Two different approaches, geometrical extrapolation method and J-integral method are used in the evaluation of stress intensity factors. Several numerical examples are given to validate effectiveness of the presented method.

Solving stress tensor fields around multiple pressure-loaded fractures using a linear superposition method (LSM)

The present paper presents the key steps to model internally pressurized fractures in a homogeneous elastic medium. Internal pressure in the cracks transforms the displacement field, which alters the associated stress concentration. The displacements are solved analytically using a Linear Superposition Method (LSM), and stresses are solved under the assumption of linear elasticity. The method allows for the fractures to have any location, geometry, and orientation. Additionally, each crack may be pressurized by either equal or individual pressure loads. Solution methodology are explained, and results are generated for several cases. Selected LSM model results show excellent matches against other independent methods (photo-elastics for multiple crack problems, and prior analytical solutions for single crack problems). The grid-less, closed-form LSM solution is able to achieve fast computation times by side-stepping adaptive grid-refinement, while achieving high target resolution.

A Peridynamic Model for Analyzing Fracture Behavior of Functionally Graded Materials Used as an Interlayer

Functionally graded materials (FGMs) can be used as an interlayer. The interlayer connects two homogeneous materials and slows mutations in material properties, owing to the variability of parameters and microstructure. In this numerical simulation, a bond-based peridynamic (PD) model was used to perform dynamic fracture analysis of a continuous beam and a sandwiched structure with an FGM interlayer. Some simulation results of two convergence studies, which include grid refinement (m-convergence) and the reducing radius of PD horizon ( $$\delta $$ -convergence), were discussed. Under a four-point bending load, the propagation behaviors of single pre-crack in the continuous beam with the FGM interlayer at different locations are simulated to compare with the experimental results. For the sandwiched structure, the problems of a single crack and double cracks were considered. The numerical results show that the crack can deflect toward areas where the energy release rate is low. Additionally, the crack on the weak side for symmetric cracks only has fracture behavior, and if the double cracks are asymmetrical, all these cracks have fracture behaviors and propagation of the initial crack in the stiff region has a higher sensitivity than in the compliant region.

Concrete Cracking Control of Chamber wall in Jiepai Ship-lock Project

Based on analysis of crack problem of chamber wall ship lock expansion project in Jiepai province, the related research of countermeasures have been made, to solve the cracks occurs in the structure. The countermeasure of cracks are analysed by monitoring internal-external temperature and shrinkage stress of chamber wall. It is provided that the rational measures could make less cracks occur, including fiber lattice with curing of insulation and moisture materials and optimization of proportion mix. A series of investigations are carried out insitu and prove countermeasures of cracking control effectively. This paper states the technical details and developing of inner temperature and shrinkage in chamber wall by adopting above measures.

Low-cement concrete dams: construction, structures and innovations

Introduction. Low-cement concrete dams are water-retaining structures frequently used in modern hydraulic engineering. The number of facilities of this type goes up every year due to their simple design, speedy construction and high economic efficiency. However, a number of problems may arise in the construction and operation of such facilities. In particular, reduced strength and water permeability of interlayer joints may constitute a problem. Temperature effects arising in the course of construction and operation may cause additional deformation of their structure, changes in their stress-strain state, opening of existing cracks and further cracking. The study of the design experience, building and operation of dams made of low-cement concrete will lay the groundwork for the development of similar structures and defect elimination methods. It will also provide an opportunity to learn more about thebehaviour of structures exposed to various conditions, including climatic ones.
 Materials and methods. Data on existing dams made of low-cement concrete, research articles, technical reports and conference proceedings, including those issued by the International Commission on Large Dams (ICOLD), have been collected and analyzed.
 Results. The comparative analysis of designs of dams in operation and future-oriented solutions is performed. The co-authors have demonstrated the need for an integrated approach to solving temperature cracking problems by using filtration through the dam body.
 Conclusions. The results can be used as the basis for further detailed studies. The comparative analysis of traditional and innovative waterproofing liners will help to effectively choose the protection solution for an upstream face of a facility. New forecasting methods and recommendations aimed at the reduction of negative temperature effects on the operation of facilities can solve the temperature cracking problem.

Mode-I stress intensity factors for Periodic Cracked Stiffened Panels under tension

Stress intensity factor is the nuclear physical parameters in fracture mechanics. Stiffened panels structure is heavily used in actual engineering field, such as aviation, marine and other bearing structures. The SIF for periotic cracked stiffened panels under tension is a representative three-dimensional crack problem. In the present article, the solution of SIFs for stiffened panels periodic cracks are derived on the basis of materials mechanics and the conservation law J 2- integral.

Mode-I pressurized axisymmetric penny-shaped crack in graded interfacial zone with variable modulus and Poisson’s ratio

This paper presents an analytical treatment to the fracture problem of a Mode-I pressurized penny-shaped crack in a graded interfacial zone or FGMs interlayer with variable shear modulus and Poisson’s ratio. An efficient multilayered model is proposed to address the general form of material inhomogeneity. The mixed boundary value problem is reduced to a Fredhom integral equation of the second kind. The solution procedure is also extended to address the Leonov-Panasyuk-Dugdale plastic crack. Analytical solutions in explicit form are derived for the full stress fields at the crack plane, Mode-I stress intensity factor (SIF), crack opening displacement (COD), T-Stress and plastic zone in terms of the solution of the integral equations. Numerical results of SIF, elastic stress fields, COD, T-Stress and length of plastic zone are presented to quantitively explore the effects of material inhomogeneity on the fracture response. It is demonstrated that the graded Poisson’s ratio has non-negligible influences on COD and T-Stress under certain condition. The present solutions can be used to advanced fracture analysis of pressurized crack in engineering composite material.

Modeling method for periodic cracks in functionally graded strips with arbitrary properties

A piecewise exponential model is developed to investigate the problems of periodic cracks in functionally graded strips with arbitrary material properties. The problem of a row of embedded cracks vertical to strip surfaces under a mode I load in a functionally graded strip with arbitrary nonhomogeneous properties is considered in a plane elasticity state. The true properties of functionally graded materials (FGMs) are approached by a series of sub-layers with material properties varying exponentially and without losing continuity. The thicknesses of different sub-layers can be set to improve calculation accuracy and efficiency. The formulation for doing so is derived using Fourier integral transformation and Fourier series, and the problem is finally reduced to an integral equation with Cauchy-type singularity. Crack-surface displacement derivative is selected as an auxiliary function and is numerically solved to calculate the stress intensity factors (SIFs) at crack tips. The results of previous research are compared with those of the current study to verify the present model. Four representative functions of material properties are considered and the SIFs are presented. The effects of crack spacing, nonhomogeneous properties and crack length on SIFs are discussed. The conclusions are beneficial to the knowledge of fracture behaviors of FGMs.

The interface crack problem under steady heat flux for a functionally graded coating-substrate structure with general coating properties

In this manuscript, the interface crack problem is investigated for a functionally graded coating-substrate structure under steady heat flux. The general thermal-elastic properties of the functionally graded material (FGM) coating are considered. A piecewise-exponential model (PE model) is developed for the thermal fracture problem. In the PE model, the piecewise-exponential function is used to approach the generally continuous variation of coating’s material properties, including shear modulus, thermal conductivity and thermal expansion coefficient, and then the problem is turned into solving three singular integral equations after a series of mathematical manipulation. The thermal insulation coefficient of the crack is considered, which can indicate the influence of the crack on the heat flux. With the present PE model, the mixed-mode thermal stress intensity factors (TSIFs) are obtained for the crack problem of FGMs with general thermal-elastic property distributions. After analyzing the calculation results, the influences of the thermal-elastic property distributions of the FGM coating and the geometric parameters of the structure on the thermal stresses and the mixed-mode TSIFs are discussed comprehensively. The present study can provide a method for the fracture behavior evaluation and the design of FGM coating-substrate structures.

A new crack-tip singular element for cracks in three-dimensional elastic bodies

Symplectic analytical singular element (SASE) is a special crack-tip element in the framework of finite element method (FEM) for modelling cracks. A group of SASEs, which have been developed by the authors, for various crack problems almost in two-dimensional (2D), but not touched yet more complex ones, i.e., in three-dimensional (3D) domain. The underlying reason lies in the fact that analytical symplectic eigen solution for 3D elastic crack problem is not yet available, a 3D SASE thus cannot be constructed through a simple generalization from the 2D SASEs. This study aims to fill out this 3D gap, we thus use a trail displacement field to construct a 3D SASE. The derived strain and stress fields still possess singularity in the vicinity of crack-tip. Finite element formulation is derived based on the minimum total potential energy principle. It is found that some of key features of a 2D SASE still exist in the developed 3D one, e.g., the SIFs can be calculated accurately without any post-processing. Numerical examples are considered to show the accuracy and performance of the proposed element.