用能量方法研究混凝土断裂过程区的力学性能

准脆性混凝土自由裂缝前缘断裂过程区的发展与其非线性断裂特征及尺寸效应现象密切相关。它的物理力学行为的量化分析对理解混凝土断裂破坏机理和建立适用于混凝土结构裂缝稳定分析和安全评估断裂准则尤为重要,一直是混凝土断裂力学研究的核心问题。该文依据Hillerborg给出的断裂能定义,给出了计算单位长度断裂过程区发展能量耗散的通用表达式。以三点弯曲梁为例,采用非线性软化本构关系,进一步给出了计算此平均能量耗散的具体步骤及对应的公式。在根据实测的三点弯曲梁的断裂能回归拟合了特征裂缝张开位移w0后,计算了每个试件整个断裂全过程中不同荷载时刻断裂过程区耗能的平均值。结果表明:随着裂缝扩展,断裂过程区能量耗散的变化和试件尺寸无关,可描述断裂过程区混凝土材料的力学性能。     

西气东输管道裂纹的韧性减速机理研究

将动态断裂力学的基本原理、基本方法和止裂判据与壳体动力学的有限元方法相结合,形成了求解输气管道裂纹扩展问题的数值分析方法。对于高韧性钢,塑性功卸载引起的热耗散不可忽略。为此根据瞬态裂纹扩展条件下的整体能量平衡方程,在有限元中构造了求解瞬时速度的迭代算法。其中以材料的热耗散比率为主要构成的动态断裂韧性由实验测得,在计算中作为裂纹扩展速度的函数代入,形成韧性减速机理。为了给出计算需要的参数,推导了双试件DWTT法测定热耗散比率的公式。     

Softening and snap-back instability in cohesive solids

The nature of the crack and the structure behaviour can range from ductile to brittle, depending on material properties, structure geometry, loading condition and external constraints. The influence of variation in fracture toughness, tensile strength and geometrical size scale is investigated on the basis of the π-theorem of dimensional analysis. Strength and toughness present in fact different physical dimensions and any consistent fracture criterion must describe energy dissipation per unit of volume and per unit of crack area respectively. A cohesive crack model is proposed aiming at describing the size effects of fracture mechanics, i.e. the transition from ductile to brittle structure behaviour by increasing the size scale and keeping the geometrical shape unchanged. For extremely brittle cases (e.g. initially uncracked specimens, large and/or slender structures, low fracture toughness, high tensile strength, etc.) a snap-back instability in the equilibrium path occurs and the load–deflection softening branch assumes a positive slope. Both load and deflection must decrease to obtain a slow and controlled crack propagation (whereas in normal softening only the load must decrease). If the loading process is deflection-controlled, the loading capacity presents a discontinuity with a negative jump. It is proved that such a catastrophic event tends to reproduce the classical LEFM-instability (K_I = K_IC) for small fracture toughnesses and/or for large structure sizes. In these cases, neither the plastic zone develops nor slow crack growth occurs before unstable crack propagation.     

Insight into the fracture evolution behavior of pre-flawed hollow-cylinder granite under multi-stage increasing-amplitude cyclic loads: A lab-scale testing

The structural deterioration and associated fracture evolution behavior of pre-flawed hollow-cylinder granite subjected to multi-stage increasing-amplitude (MSIA) cyclic loads are studied herein. The influences of rock structure on volumetric deformation, damage accumulation, energy dissipation, and failure pattern were investigated. It is shown that the volumetric deformation is relatively large for rock having high flaw angle, and it is the minimum and maximum for rock having a 10° and 70° flaw angle. A damage evolution model that can describe a first fast and then steady damage propagation was proposed based on the irreversible axial strain. Much energy needs to be consumed to drive crack propagation and hole collapse for rock having high angle flaws. A series of 2D computed tomography (CT) images reveal the different crack network pattern and how it is affected by the rock structure. A more complicated crack network is found for rock having a high flaw angle.     

Numerical analysis of dynamic crack propagation in rubber

In the present paper, dynamic crack propagation in rubber is analyzed numerically using the finite element method. The problem of a suddenly initiated crack at the center of stretched sheet is studied under plane stress conditions. A nonlinear finite element analysis using implicit time integration scheme is used. The bulk material behavior is described by finite-viscoelasticity theory and the fracture separation process is characterized using a cohesive zone model with a bilinear traction-separation law. Hence, the numerical model is able to model and predict the different contributions to the fracture toughness, i.e. the surface energy, viscoelastic dissipation, and inertia effects. The separation work per unit area and the strength of the cohesive zone have been parameterized, and their influence on the separation process has been investigated. A steadily propagating crack is obtained and the corresponding crack tip position and velocity history as well as the steady crack propagation velocity are evaluated and compared with experimental data. A minimum threshold stretch of 3.0 is required for crack propagation. The numerical model is able to predict the dynamic crack growth. It appears that the strength and the surface energy vary with the crack speed. Finally, the maximum principal stretch and stress distribution around steadily propagation crack tip suggest that crystallization and cavity formation may take place.     

Dual approach in non-linear fracture mechanics

A dual approach in fracture mechanics based on complementary energy is proposed. The analysis of the dissipation shows that the thermodynamical force associated with the evolution of a crack is an energy release rate, form of which depends on the presence or not of mechanical discontinuities. This energy release rate is given as a integral based on free or complementary energy. The invariance of sintegrals is analysed and the obtained results in elastoplasticity are discussed. The energy release rate is determined in terms of potential energy or complementary energy, first in elasticity and secondly in elastoplasticity. Associated to these definitions, the law of propagation of the crack is chosen as a Griffith law and the propagation is governed by a normality rule. In this framework we formulate the evolution problem concerning crack propagation in an elastoplastic material. Variational formulations are obtained in terms of rate of displacement, of stresses and of crack length.     

A model of brittle fracture propagation part I—continuum aspects

The micromechanism of crack propagation in steel is described and analyzed in continuum terms and related to the macroscopic fracture behavior. It is proposed that propagation of cleavage microcracks through favorably oriented grains ahead of the main crack tip is the principal weakening mode in brittle fracture. This easy cleavage process proceeds in the Griffith manner and follows a continuous, multiply connected, nearly planar path with a very irregular front which spreads both forward and laterally and leaves behind disconnected links which span the prospective fracture surface. A discrete crack zone which extends over many grains thus exists at the tip of a running brittle crack. Final separation of the links is preceeded by plastic straining within the crack zone and occurs gradually with the increasing crack opening displacement. It is suggested that in low stress fracture, straining of the links is the only deformation mode. However, it is recognized that under certain conditions plastic enclaves may adjoin the crack zone. This deformation mode is associated with high stress fracture, energy transition and eventually with crack arrest.

Energy dissipation resulting from the two deformation mechanisms is related to crack velocity, applied load and temperature and the crack velocity in a given material is expressed as a function of the external conditions. Fracture initiation and crack arrest are then discussed in terms of the conditions which are necessary to maintain the propagation process. Finally, the dimensions of a small scale crack tip zone for a steady state, plane strain crack are evaluated as functions of material properties and the elastic stress intensity factor.

The microstructural aspects of brittle fracture will be discussed in a separate Part 2 [1].     

Coupled hygro-thermo-viscoelastic fracture theory

A coupled hygro-thermo-viscoelastic fracture theory is developed for quasi-static and dynamic crack propagation in viscoelastic materials subject to combined mechanical loading and hygrothermal environmental exposure based on fundamental principles of thermodynamics. The Helmholtz free energy is taken to be a functional of the histories of strain, temperature and fluid concentration with the crack parameter being introduced as an internal state variable. A thermodynamically consistent time-dependent fracture criterion for crack propagation in the presence of thermally and mechanically assisted fluid transport is obtained from the global energy balance equation and the requirement of non-negativity of the global energy dissipation rate, which is generally applicable to both quasi-static and dynamic loading and both isothermal/isohumidity and non-isothermal/non-isohumidity conditions with classic fracture criteria as special cases. On the basis of the developed theory, the generalized energy release rate method, the generalized contour integral method and the extended essential work of fracture method are proposed for fracture characterization of load-carrying viscoelastic materials in hygrothermal environments, and the interrelation of these methods and their correlation with conventional methods and existing models, simulations and experiments are discussed.     

Crack propagation analysis for orthotropic non-homogeneous materials

The fracture mechanics of orthotropic non-homogenous materials like welds involves the use of a fracture criterion with the stress analysis of the specimen. A maximum energy release rate criterion is proposed in this analysis. A finite element model with special crack tip elements has been developed for this study. Two techniques have been developed to model the crack propagation. The model was then applied to a compact tension specimen with a hypothetical weldment and to a weldment with an edge crack.The maximum energy release rate criterion seems to be the most suitable for fracture prediction of non-homogeneous materials, since it gives the crack propagation direction as well as the maximum value for the energy released. The virtual displacement technique for modeling the crack propagation saves computer time and gives accurate results. The analysis indicates the effect of orthotropy on the crack propagation direction and maximum energy release rate.     

A cumulative damage theory for fatigue crack initiation and propagation

A method for evaluating the cumulative damage resulting from the application of cyclic stress (or strain) sequences of varying amplitude is presented. Both the crack initiation and propagation stages of the fatigue failure process are included. The development is based on the concept of plastic strain energy dissipation as a function of cyclic life. The damage accumulated at any stage is evaluated from a knowledge of the fatigue limit in the initiation phase and an ‘apparent’ limit obtained through fracture mechanics for the propagation phase. The proposed damage theory is compared with two-level strain cycle test data of thin-walled specimens, and is found to be in fairly good agreement.     

A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization

A variational formulation of quasi-static brittle fracture in elastic solids at small strains is proposed and an associated finite element implementation is presented. On the theoretical side, a consistent thermodynamic framework for brittle crack propagation is outlined. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip. On the numerical side, we first consider a standard finite element discretization in the two-dimensional space which yields a discrete formulation of the global dissipation in terms of configurational nodal forces. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity for two-dimensional problems is performed by the doubling of critical nodes and interface segments of the mesh. A crucial step for the success of this procedure is its embedding into a r-adaptive crack-segment re-orientation algorithm governed by configurational-force-based directional indicators. Here, successive crack propagation is performed by a staggered loading-release algorithm of energy minimization at frozen crack state followed by nodal releases at frozen deformation. We compare results obtained by the proposed formulation with other crack propagation criteria. The computational method proposed is extremely robust and shows an excellent performance for representative numerical simulations.     

A non-equilibrium thermodynamic model for the crack propagation rate

A non-equilibrium thermodynamics-based evolution model describes the nonsteady, crack propagation rate for both brittle fracture and for viscoplastic behavior at the crack tip. This model for dynamic crack propagation under dynamic or quasi-static loading is developed from an energy functions viewpoint and extends a non-equilibrium thermodynamics construction based on a instantaneous maximum dissipation criterion and a thermodynamic relaxation modulus that permits multi-scale modeling. The evolution equations describing the non-equilibrium fracture process are generated from a generalized energy function whose zero gradient manifold gives the assumed quasi-static crack propagation equations. The class of models produced includes the classical Freund model and a modification that is consistent with the experimental maximum crack velocity. In unstable propagation, the non-equilibrium process is repelled from the quasi-static manifold. If the initial state is stable, then the crack growth process approaches the quasi-static manifold and eventually the crack is arrested. An application of the construction gives the craze growth in PMMA. A simple viscoplastic model for metals predicts the change in temperature at the crack tip as the crack grows.     

On the use of embedded discontinuity elements with crack path continuity for mode-I and mixed-mode fracture

In this paper, strong discontinuities embedded in finite elements are used to model discrete cracking in quasi-brittle materials. Special attention is paid to (i) the constitutive models used to describe the localized behaviour of the discontinuities, (ii) the enforcement of the continuity of the crack path and (iii) mixed-mode crack propagation. Different constitutive relations are adopted to describe the localized behaviour of the discontinuities, namely two damage laws and one plasticity law. A numerical algorithm is introduced to enforce the continuity of the crack path. In the examples studied, an objective dissipation of energy with respect to the mesh is found. Examples of mode-I and mixed-mode crack propagation are presented, namely a double notch tensile test and a single-edge notched beam subjected to shear. In the former case different crack patterns are obtained depending on the notch offset; in the latter case special emphasis is given to the effect of shear on the global structural response. In particular, both the peak load and the softening response of the structure are related to the amount of shear tractions allowed to develop between crack faces. The results obtained are compared to experimental results. As a general conclusion, it is found that crack path continuity allows for the development of crack patterns similar to those found in experiments, even when reasonably coarse meshes are used.     

Further studies on fracture process zone for mode I concrete fracture

The development of a fracture process zone associated with stable crack growth in single edge-notched (SEN), three-point bend concrete specimens of different tensile strengths was monitored with moire interferometry. The experimental data was then used to drive a finite element model of the fracture specimen in its generation/propagation mode and to determine the crack closure stress due to aggregate bridging, the energy release and dissipation rates and the resistance curve.     

Initiation, propagation and arrest of an interface crack subjected to controlled stress wave loading

An experimental study has been conducted to investigate the initiation, propagation, and arrest of bimaterial interface cracks subjected to controlled stress wave loading in the form of a tensile dilatational stress wave pulse. The tensile pulse is generated by detonating lead azide explosive in a specially designed specimen. Dynamic loading of the bimaterial interface results in crack initiation, propagation, and arrest, all in the same experiment. This failure event is observed using photoelasticity in conjunction with high speed photography. Full field data from the experimentally obtained isochromatic fringe patterns is analyzed to determine time histories of various fracture parameters such as the crack tip speed, the dynamic complex stress intensity factor, the energy release rate, and the mixity. The experimental data is also used to quantify the values of the dynamic initiation and arrest toughness and to evaluate a recently proposed dynamic interface fracture criterion. This revised version was published online in August 2006 with corrections to the Cover Date.     

Study of mode I crack dynamic propagation behaviour and rock dynamic fracture toughness by using SCT specimens

To study crack dynamic propagation behaviour and rock dynamic fracture toughness, a single cleavage triangle (SCT) specimen was proposed in this paper. By using these specimens and a drop‐weight test system, impact experiments were conducted, and the crack propagation velocity and the fracture time were measured by using crack propagation gauges. To examine the effectiveness of the SCT specimen and to predict the test results, finite difference numerical models were established by using AUTODYN code, and the simulation results showed that the crack propagation path agrees with the test results, and crack arrest phenomena could happen. Meanwhile, by using these numerical models, the crack dynamic propagation mechanism was investigated. Finite element code ABAQUS was applied in the calculation of crack dynamic stress intensity factors (SIFs) based on specimen dimension and the loading curves measured, and the curves of crack dynamic SIFs versus time were obtained. The fracture toughness (including initiation toughness and propagation toughness) was determined according to the fracture time and crack speeds measured by crack propagation gauges. The results show that the SCT specimen is applicable to the study of crack dynamic propagation behaviour and fracture toughness, and in the process of crack propagation, the propagation toughness decreases with crack propagation velocity, and the crack arrest phenomena could happen. The critical SIF of an arrest crack (or arrest toughness) was higher than the crack propagation toughness but was lower than the initiation toughness.     

半圆盘构件冲击断裂特性的动态焦散线实验研究

采用动态焦散线实验系统,对有机玻璃（PMMA）在冲击载荷下的I型和I-II混合型裂纹在起裂和扩展时的动态断裂特性进行了研究。结果表明：随着PMMA由I型断裂转变为I-II混合型断裂,从落锤作用在试件上到裂纹起裂所需时间不断增加,说明裂纹起裂需要的能量有所增加,同时从裂纹起裂到最终贯通所需时间不断减少,说明裂纹平均扩展速度也不断增大;在I型断裂中,PMMA的断裂韧度KIC为2.04 MN/m3/2,而在I-II混合型断裂中,PMMA的断裂韧度KIC低于I型断裂时的断裂韧度KIC,但是KIIC有所增大;对于I-II混合型断裂,PMMA极限扩展速度约为366m/s,当达到极限扩展速度后,裂纹尖端出现微裂纹增韧现象,使裂纹的表面能迅速增大,随后裂纹的扩展速度迅速减小。     

Integral estimates to the fast brittle fracture mechanics

In the terms of a general approach the problem of the fast crack propagation in elastic bodies is considered. The time dependence of, so called domain of possible fractures due to the fast crack propagation, is obtained. The dynamic domain of possible fractures is the upper estimate of the zone of real fractures and thereby it takes into account such important features of the fast brittle fracture as the instability of the fast crack propagation, branching, waving, the influence of boundary conditions and so on. The problem of dynamic crack initiation is considered in some details. Within the linear fracture mechanics the dynamic criterion of crack initiation is offered.     

Computational analysis of dynamically propagating cracks in axisymmetric solids

Dynamic crack propagation behavior in axisymmetric solids is investigated using an effective computational procedure. First, an accurate method to extract energy release rate of a dynamically propagating crack from finite element solutions is formulated for axisymmetric geometries. The method is applied to an analysis of a radially growing circular crack under remote tension in an infinite medium. The computed dynamic energy release rate shows an excellent agreement with the exact solution. Next, we have developed an iterative technique to propagate a crack whose velocity history is initially unknown. With the iterative procedure, the crack propagates according to a condition specified by a dynamic fracture criterion. At each increment of crack growth, an optimum velocity at which the crack growth condition satisfies is obtained by the iterative scheme. This procedure requires no artificial input or no preset crack tip speed in a simulation study. The iterative scheme is employed in a dynamic fracture analysis of ceramics. The computational analysis is carried out for simulation of fracture experiments using circumferentially notched round bars. In the test, a ceramic specimen is subjected to tensile stress wave loading and after crack growth initiation, the external crack propagates to tear the specimen. The computational simulation is carried out for the entire fracture process including the crack growth initiation and the dynamic propagation. The iterative procedure enables us to predict the crack tip velocity which is unmeasurable in the experiment. Suitabilities of proposed fracture toughness criteria for the ceramics are investigated by comparing measured and computed transmitted pulses through the uncracked ligament. This study proves the usefulness of the computational procedures for dynamic fracture analysis. It is most effective in characterizing dynamic fracture toughness where determination of every relevant parameter is difficult in the experiments.