基于断裂力学的裂隙岩体强度分析初探

基于断裂力学讨论裂隙岩体的强度一直是断裂力学与岩石力学等学科的重要课题。通过分析单个线性裂纹在压缩载荷作用下的剪切断裂条件,明确了压缩状态下张破裂剪切断裂韧性的物理意义及求解办法。使用复变函数和边界配置法求解了裂尖应力强度因子无量纲系数,进而对张破裂压剪准则进行了改进,使之更加便捷地进行有限裂纹体断裂的预测。基于修正的压剪判据推导了裂隙岩体试件的抗压强度求解公式,算例表明该文方法所建立的强度公式是准确、可靠的。最后,详细讨论了摩擦系数、裂隙倾角、裂隙长度、黏聚力以及围压对裂隙岩体试件抗压强度的影响机理。     

Relationship between the Stress Intensity Factors and Bond σ in Graphene Sheet

Relationship between the stress intensity factors and the bond σ in the plane of interatomic hexagonal structures within graphite is given numerically. Under the consideration in this plane, the bond π and van der Waals force are ignored, and the plane strain of macroscopical elasticity is considered. The molecular mechanics is used to describe the displacements of atoms in the area near the tip of a crack, and the linear elastic fracture mechanics is used outside this area. Connection between these two theories is based on an assumption that the displacements of atoms along the boundary described by the molecular mechanics are equal to those described by the fracture mechanics. The nonlinear equations of molecular mechanics derived by the modified Morse potential function are solved by using Matlab software. When choosing the maximum stretching force of atomic bond as a failure criterion, the fracture toughness is obtained for Mode I and Mode II problems.     

Energy-based criterion of long-term static strength of materials with nonlinear physical properties

We generalize the existing approaches of physical materials science and the mechanics of deformable bodies and suggest a mathematical model of the long-term strength of solid materials with nonlinear physical properties and new formulas for energy-based criteria of bulk brittle and ductile failure. The experimental data on the fracture of numerous polymeric materials and nonferrous metals presented in the work are in good agreement with the theoretical data of the criteria computed for the exponential type of deformation.     

Fracture mechanics as a tool in failure analysis — Prospects and limitations

Although fatigue crack propagation and fracture cause a large part of failure events in industrial practice, fracture mechanics in failure analysis seems to be still a side issue. Starting from an introduction into important basic questions of failure analysis and fracture mechanics, the authors specify what kind of questions in failure analysis can be effectively solved by fracture mechanics (and which can't). They illustrate their discussion with a number of 13 case studies from the literature. Much more pronounced than in the design stage the benefit of fracture mechanics in failure analysis depends on its accuracy. This is limited by both, intrinsic factors of the method and the availability and quality of the input information. The authors discuss the various aspects and provide the reader with some background information which, as they believe, will be helpful for better understanding the prospects and limitations of fracture mechanics in failure analysis and the conditions of its application.     

Failure of amorphous polystyrene

The failure behaviour of amorphous polystyrene was studied in methanol and ambient air under constant load and strain rate conditions. The good correlation found between fracture mechanics theory and the test results of both crazing and fracture indicates that fracture mechanics theory can be used in predicting failure of amorphous polystyrene. From the fracture mechanics analysis of the results it is inferred that the kinetics associated with craze initiation and crack propagation are similar and that the inherent flaw responsible for failure first initiates the craze in which a crack is then formed. Both the distribution of inherent flaws and the kinetics of crazing and fracture are dependent on the test environment.     

Three-dimensional numerical investigation of brittle bond fracture

A fracture mechanics based analysis of interface bond failure is presented. The bond edge is regarded as an interface crack front loaded under combined mode 1, 2 and 3 loading, and results are obtained for the critical stress for initiation of bond failure and the location along the bond edge where failure is initiated. A numerical procedure is formulated to study the propagation of the interface crack following initiation. Assuming that the crack propagates at the interface, a criterion for propagation is formulated, and it is shown that the crack front shape predicted is consistent with the basic interface fracture mechanics assuming quasi-static crack propagation. Results for the bond strength are presented for different fracture criteria and different bond shapes.     

Grundlagen der Bruchmechanik

Basis principles of fracture mechanics. Based upon the linearelastic equations for the stress field in a tension stressed body containing a crack and upon the energy condition for the unstable crack propagation the quantities stress intensity factor and crack extension force are derived. Taking into account the plastic zone combined with the crack propagation in real bodies the linear-elastic relations hold furthermore with a sufficient accuracy if the length of the plastic zone is small compared with the crack length (small scale yielding). Then the fracture toughness is a materials constant. The nonlinear extension of these equations for the partly plastic range exhibit that the critical crack opening stretch is likewise a materials constant up to a length of the plastic zone comparable with the crack length. In all these cases the fracture stress decreases with increasing crack length and has a value below about 0.7 times the yield stress under uniaxial tension (low stress fracture). Approaching to the fully plastic yielding these statements become invalid, the fracture stress is greater than the uniaxial yield stress and is independent of the crack length. To obtain a perceptible picture of these statements the numerical values of the quantities are presented for a pressure vessel steel and for a heat-treatable steel. —The principles of the fracture mechanics and of the COS-concept are described. In this connection the plastic instability of pressure gas pipelines are mentioned and furthermore it is discussed if it is necessary – at all – to examine the structural steels with the methods of the fracture mechanics.     

Development of the Local Approach to Fracture over the Past 25 years: Theory and Applications

This review paper is devoted to the local approach to fracture (LAF) for the prediction of the fracture toughness of structural steels. The LAF has been considerably developed over the past two decades, not only to provide a better understanding of the fracture behaviour of materials, in particular the failure micromechanisms, but also to deal with loading conditions which cannot easily be handled with the conventional linear elastic fracture mechanics and elastic–plastic fracture mechanics global approaches. The bases of this relatively newly developed methodology are first presented. Both ductile rupture and brittle cleavage fracture micromechanisms are considered. The ductile-to-brittle transition observed in ferritic steels is also briefly reviewed. Two types of LAF methods are presented: (i) those assuming that the material behaviour is not affected by damage (e.g. cleavage fracture), (ii) those using a coupling effect between damage and constitutive equations (e.g. ductile fracture). The micromechanisms of brittle and ductile fracture investigated in elementary volume elements are briefly presented. The emphasis is laid on cleavage fracture in ferritic steels. The role of second phase particles (carbides or inclusions) and grain boundaries is more thoroughly discussed. The distinction between nucleation and growth controlled fracture is made. Recent developments in the theory of cleavage fracture incorporating both the effect of stress state and that of plastic strain are presented. These theoretical results are applied to the crack tip situation to predict the fracture toughness. It is shown that the ductile-to-brittle transition curve can reasonably be well predicted using the LAF approach. Additional applications of the LAF approach methods are also shown, including: (i) the effect of loading rate and prestressing; (ii) the influence of residual stresses in welds; (iii) the mismatch effects in welds; (iv) the warm-prestressing effect. An attempt is also made to delineate research areas where large improvements should be made for a better understanding of the failure behaviour of structural materials.     

Prediction of Type IV creep failure of a seam-welded mod. 9Cr-1Mo elbow based on microscopic damage simulation

Utilising the random-fracture-resistance model of grain boundaries, micro-macro combined creep damage simulation was applied to the prediction of the distribution of small defects in the FGHAZ (fine-grained heat-affected zone) of longitudinal welds in an actual-size elbow of modified 9Cr-1Mo (9Cr-1MoVNb) steel subject to internal pressure at 923 K. Based on the simulation results, a prediction scheme for the final rupture life of welds was considered using the damage mechanics concept together with effective stress. The applicability of nonlinear fracture mechanics was also discussed, assuming the initial crack length determined from the microscopic simulation results. The results thus obtained are summarized as follows: As the simulation results showed, the peaks of small defect density in the subsurface could be predicted, corresponding well with the observed results. Final failure life prediction based on the damage mechanics concept was found to be applicable, by considering both the final failure surface connecting the weakest grain boundaries and the effective stress against this surface. The fracture mechanics approach was also found applicable when assuming the initial crack length from the high peaks of the simulated small defects in the last stage of creep life.     

A thermodynamic framework of fracture mechanics

A rigorous theory of non-linear fracture mechanics consistent with the basic laws of thermodynamics is proposed. It is shown that the local balance equation(s) can be derived from the postulated global balance equation using a generalized Reynolds transport equation appropriate to a body containing a growing crack. The necessary conditions to be fulfilled on the newly created fracture surfaces are derived. Within the framework of local theory the governing equations are applicable to crackproblems in materials with internal variables or fading memory and to geometrically and/or physically non-linear problems. As a special case, the governing equations for infinitesimal thermoviscoelastic fracture mechanics are given.     

Multiscale modeling of intergranular fracture in aluminum: constitutive relation for interface debonding

Intergranular fracture is a dominant mode of failure in ultrafine grained materials. In the present study, the atomistic mechanisms of grain-boundary debonding during intergranular fracture in aluminum are modeled using a coupled molecular dynamics—finite element simulation. Using a statistical mechanics approach, a cohesive-zone law in the form of a traction–displacement constitutive relationship, characterizing the load transfer across the plane of a growing edge crack, is extracted from atomistic simulations and then recast in a form suitable for inclusion within a continuum finite element model. The cohesive-zone law derived by the presented technique is free of finite size effects and is statistically representative for describing the interfacial debonding of a grain boundary (GB) interface examined at atomic length scales. By incorporating the cohesive-zone law in cohesive-zone finite elements, the debonding of a GB interface can be simulated in a coupled continuum–atomistic model, in which a crack starts in the continuum environment, smoothly penetrates the continuum–atomistic interface, and continues its propagation in the atomistic environment. This study is a step toward relating atomistically derived decohesion laws to macroscopic predictions of fracture and constructing multiscale models for nanocrystalline and ultrafine grained materials.     

Aspects of ductile fracture and adaptive mesh refinement in damaged elasto‐plastic materials

Numerical simulation of elasto‐plastic problems involving multi‐fracturing materials requires a reliable failure prediction technique and a robust solution algorithm. This work approaches ductile fracture by means of continuum damage mechanics, from which two new failure criteria based on coupled and uncoupled damage analysis are derived. A two‐parameter stress update algorithm for damaged materials based on a Newton–Raphson iterative procedure is presented. A posteriori error estimators using ductile failure concepts are also discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Energy-based criterion of long-term static strength of materials with nonlinear physical properties

We generalize the existing approaches of physical materials science and the mechanics of deformable bodies and suggest a mathematical model of the long-term strength of solid materials with nonlinear physical properties and new formulas for energy-based criteria of bulk brittle and ductile failure. The experimental data on the fracture of numerous polymeric materials and nonferrous metals presented in the work are in good agreement with the theoretical data of the criteria computed for the exponential type of deformation. Deceased. Translated from Problemy Prochnosti, No. 5, pp. 23–29, September–October, 1997.     

THE EFFECT OF MISALIGNMENT ON THE FATIGUE STRENGTH OF WELDED CRUCIFORM JOINTS

Abstract— The effect of axial misalignment on the fatigue strength of load-carrying transverse cruciform welded joints was investigated using experimental and fracture mechanics methods. Where failure occurred by cracking from the weld toe, misalignment significantly reduced the fatigue strength. The reduction could be predicted using a nominal stress concentration factor (SCF). Misalignment had less effect where failure was due to cracking through the weld metal; an expression was deduced for the SCF in this case. For fracture mechanics assessments, an expression for an effective stress intensity factor using the SCF and stress intensity factors for aligned welds was shown to agree with the finite element (FE) results. Predictions of the effect of misalignment using the FE results agreed with experimental data. Misaligned transverse load-carrying cruciform joints should be assessed for fatigue failure from the toe using the same SCF as for a butt weld with the same misalignment. For failure through the throat, an alternative expression for the SCF is recommended. Fracture mechanics assessments of misaligned joints should be carried out using an effective stress intensity factor derived from the SCF and stress intensity factors for aligned joints. These recommendations are now incorporated in British Standard PD 6493:1991.     

The residual strength characteristics of stiffened panels containing fatigue cracks

In heavy structural members, where plane strain conditions prevail, linear fracture mechanics can be used for predicting residual strength. Aircraft structures consist largely of sheet structures with plane stress conditions where linear fracture mechanics do not seem to apply. Yet it is in the aircraft main structure that large fatigue cracks can develop and that has to be designed fail-safe. The present paper describes a method to predict the residual strength of a cracked sheet structure.Contrary to an unstiffened sheet, the sheet structure contains stiffening elements that can act as crack stoppers. This crack arresting action and its consequences for the residual strength are considered in the analysis.The paper proposes a method that relates the crack resistance of a stiffened panel to that of an unstiffened sheet. It takes full account of sheet-stringer interaction in the cracked region. A criterion for crack arrest is put forward. Ultimate panel failure after crack arrest is initiated either by subsequent unstable crack growth or by stiffener failure. Critical load conditions for both failure modes are presented. In case crack arrest does not occur, the residual strength of the unstiffened panel constitutes a safe lower bound.Computational results of the interacting rivet forces by both analytical and numerical (finite element) methods are presented. From these the load concentration in the stiffener and the reduction of the stress intensity at the crack tip can be determined. This enables the complete residual strength characteristics to be predicted.The results of residual strength tests on bonded and riveted panels with symmetric strip stiffeners or eccentric Z-stringers fully substantiate the method proposed for residual strength calculations.     

Cohesive stresses and size effects in elasto-plastic and quasi-brittle materials

A novel approach to the derivation of Baant's size effect law is presented. Contrarily to the original Lagrangian derivation which hinged on energetic consideration, a Newtonian approach based on local stress intensity factors is presented. Through this approach, it is shown that Baant's size effect law is the first (and dominant) term in a series expansion for the nominal stress. Furthermore, analytical expressions for B are derived for selected specimen geometries. It is also shown that size effect is exhibited not only by quasi-brittle materials (such as concrete), but by elasto-plastic materials too. Finally, for three point bend concrete specimens, it is shown that a nonlinear fracture mechanics analysis (with non-zero tensile strength) is practically identical to a linear elastic fracture mechanics analysis with zero fracture toughness.     

Probabilistic model for fatigue crack growth and fracture of welded joints in civil engineering structures

This paper presents a probabilistic assessment model for linear elastic fracture mechanics (LEFM). The model allows the determination of the failure probability of a structure subjected to fatigue loading. The distributions of the random variables for civil engineering structures are provided, and the relative importance of these random variables is determined. An example of a bridge detail is provided in order to show the application of the model. Partial factors are derived for the case of fatigue of welded joints in civil engineering structures. The failure probability appears to be relatively insensitive to the failure criterion (attainment of a through-thickness crack or fracture) when considering the total fatigue life.     

Level-set topology optimization for maximizing fracture resistance of brittle materials using phase-field fracture model

Fracture is one of the most common failure modes in brittle materials. It can drastically decrease material integrity and structural strength. To address this issue, we propose a level-set (LS) based topology optimization procedure to optimize the distribution of reinforced inclusions within matrix materials subject to the volume constraint for maximizing structural resistance to fracture. A phase-field fracture model is formulated herein to simulate crack initiation and propagation, in which a staggered algorithm is developed to solve such time-dependent crack propagation problems. In line with diffusive damage of the phase-field approach for fracture; topological derivatives, which provide gradient information for the topology optimization in a LS framework, are derived for fracture mechanics problems. A reaction-diffusion equation is adopted to update the LS function within a finite element framework. This avoids the reinitialization by overcoming the limitation to time step with the Courant-Friedrichs-Lewy condition. In this article, three numerical examples, namely, a L-shaped section, a rectangular slab with predefined cracks, and an all-ceramic onlay dental bridge (namely, fixed partial denture), are presented to demonstrate the effectiveness of the proposed LS based topology optimization for enhancing fracture resistance of multimaterial composite structures in a phase-field fracture context.     

Nonlinear fracture mechanics and plasticity of the split cylinder test

The split cylinder test is subjected to an analysis combining nonlinear fracture mechanics and plasticity. The fictitious crack model is applied for the analysis of splitting tensile fracture, and the Mohr-Coulomb yield criterion is adopted for modelling the compressive crushing/sliding failure. Two models are presented, a simple semi-analytical model based on analytical solutions for the crack propagation in a rectangular prismatic body, and a finite element model including plasticity in bulk material as well as crack propagation in interface elements. A numerical study applying these models demonstrates the influence of varying geometry or constitutive properties. For a split cylinder test in load control it is shown how the ultimate load is either plasticity dominated or fracture mechanics dominated. The transition between the two modes is related to changes in geometry or constitutive properties. This implies that the linear elastic interpretation of the ultimate splitting force in term of the uniaxial tensile strength of the material is only valid for special situations, e.g. for very large cylinders. Furthermore, the numerical analysis suggests that the split cylinder test is not well suited for determining the tensile strength of early age or fibre reinforced concrete.     

Nonlinear failure analysis of carbon nanotubes by using molecular-mechanics based models

Equivalent nonlinear fracture models for pristine and reconstructed one- and two-atom vacancy defected single wall carbon nanotubes are developed by using the molecular mechanics based models where the initial reconstructed nanotube models are obtained by using molecular dynamic simulations and nonlinear characteristic of the covalent bonds are obtained by using the modified Morse potential. As a result of analyses, it is concluded that fractures of all types of nanotubes are brittle, armchair nanotubes are stiffer than zigzag nanotubes and vacancy defects significantly affect the mechanical behavior of nanotubes. In brief, fracture stress and strain values of pristine armchair nanotubes are respectively 30% and 32% larger than those of pristine zigzag nanotubes, and predicted failure stress and strain values of vacancy defected nanotubes are respectively 27% and 52% smaller than those of pristine ones. It is shown that large deformation and nonlinear geometric effects are important on fracture behavior of nanotubes. Comparisons are made with the failure stress and strain results reported in literature that show good agreement with our results.