裂尖超钝化:实验、理论与数值模拟

本文报道了在改性聚丙烯裂尖平面应变段发生超钝化的实验发现。利用环向冷拉模型,可说明裂尖超钝化的形成取决于冷拉传播速率与加载率之比。裂尖超钝化可使材料韧性,尤其是冲击韧性的改善,达到一个新的水平。本文在超弹性-粘塑性各向异性损伤的有限变形本构模型下进行了详尽的数值计算。模拟结果成功地再现了裂尖超钝化的基本特征。     

On the higher order asymptotic analysis of a non-uniformly propagating dynamic crack along an arbitrary path

Transient mixed-mode elastodynamic crack growth along arbitrary smoothly varying paths is considered. Asymptotically, the crack tip stress field is square root singular with the angular variation of the singular term depending weakly on the instantaneous values of the crack tip speed and on the mode-I and mode-II stress intensity factors. However, for a material particle at a small distance away from the moving crack tip, the local stress field will depend not only on the instantaneous values of the crack tip speed and stress intensity factors, but also on the past history of these time dependent quantities. In addition, for cracks propagating along curved paths the stress field is also expected to depend on the nature of the curved crack path. Here, a representation of the crack tip fields in the form of an expansion about the crack tip is obtained in powers of radial distance from the tip. The higher order coefficients of this expansion are found to depend on the time derivative of crack tip speed, the time derivatives of the two stress intensity factors as well as on the instantaneous value of the local curvature of the crack path. It is also demonstrated that even if cracks follow a curved path dictated by the criterion K ₁₁ ^d =0, the stress field may still retain higher order asymmetric components related to non-zero local curvature of the crack path.     

Theoretical development and experimental validation of a thermally dissipative cohesive zone model for dynamic fracture of amorphous polymers

A thermally dissipative cohesive zone model is developed for predicting the temperature increase at the tip of a crack propagating dynamically in a nominally brittle material exhibiting a cohesive-type failure such as crazing. The model assumes that fracture energy supplied to the crack tip region that is in excess of that needed for the creation of new free surfaces during crack advance is converted to heat within the cohesive zone. Bulk dissipation mechanisms, such as plasticity, are not accounted for. Several cohesive traction laws are examined, and the model is then used to make predictions of crack tip heating at various crack propagation speeds in the nominally brittle amorphous polymer PMMA, observed to fail by a crazing-type mechanism. The heating predictions are compared to experimental data where the temperature field surrounding a high speed crack in PMMA was measured. Measurements are made in real time using a multi-point high speed HgCdTe infrared radiation detector array. At the same time as temperature, simultaneous measurement of fracture energy is made by a strain gauge technique, and crack tip speed is monitored through a resistance ladder method. Material strength can be estimated through uniaxial tension tests, thus minimizing the need for parameter fitting in the stress-opening traction law. Excellent agreement between experiments and theory is found for two of the cohesive traction law temperature predictions, but only for the case where a single craze is active during the dynamic fracture of PMMA, i.e. crack tip speed up to approximately 0.2c_R. For higher speed fracture where subsurface damage becomes prominent, the line dissipation model of a cohesive zone is inadequate, and a distributed damage model is needed.     

Analysis of dynamic growth of a tensile crack in an elastic-plastic material

The influence of inertia on the stress and deformation fields near the tip of a crack growing in an elastic-plastic material is studied. The material is characterized by the von Mises yield criterion and J₂ flow theory of plasticity. The crack grows steadily under plane strain conditions in the tensile opening mode. Features of the stress and deformation state at points near the moving crack tip are described for elastic-perfectly plastic response and for several crack propagation speeds. It is found that inertia has a significant effect on the elastic-plastic response of material particles near the crack tip, and that elastic unloading may occur behind the crack tip for higher speeds. The relationship between the applied crack driving force, represented by a remote stress intensity factor, and the crack tip speed is examined on the basis of a critical crack tip opening angle growth criterion. The calculated result is compared with dynamic fracture toughness versus crack speed data for a 4340 steel.     

Field structure at mode III dynamically propagating crack tip in elastic-viscoplastic materials

An elastic-viscoplastic mechanics model is used to investigate asymptotically the mode Ⅲ dynamically propagating crack tip field in elastic-viscoplastic materials. The stress and strain fields at the crack tip possess the same power-law singularity under a linear-hardening condition. The singularity exponent is uniquely determined by the viscosity coefficient of the material. Numerical results indicate that the motion parameter of the crack propagating speed has little effect on the zone structure at the crack tip. The hardening coefficient dominates the structure of the crack-tip field. However, the secondary plastic zone has little influence on the field. The viscosity of the material dominates the strength of stress and strain fields at the crack tip while it does have certain influence on the crack-tip field structure. The dynamic crack-tip field degenerates into the relevant quasi-static solution when the crack moving speed is zero. The corresponding perfectly-plastic solution is recovered from the linear-hardening solution when the hardening coefficient becomes zero.     

Kinetics of microcrack blunting ahead of macrocrack approaching shear wave speed

The fracturing of glass and tearing of rubber both involve the separation of material but their crack growth behavior can be quite different, particularly with reference to the distance of separation of the adjacent planes of material and the speed at which they separate. Relatively speaking, the former and the latter are recognized, respectively, to be fast and slow under normal conditions. Moreover, the crack tip radius of curvature in glass can be very sharp while that in the rubber can be very blunt. These changes in the geometric features of the crack or defect, however, have not been incorporated into the modeling of running cracks because the mathematical treatment makes use of the Galilean transformation where the crack opening distance or the change in the radius of curvature of the crack does not enter into the solution. Change in crack speed is accounted for only via the modulus of elasticity and mass density. For this simple reason, many of the dynamic features of the running crack have remained unexplained although speculations are not lacking. To begin with, the process of energy dissipation due to separation is affected by the microstructure of the material that distinguishes polycrystalline from amorphous form. Energy extracted from macroscopic reaches of a solid will travel to the atomic or smaller regions at different speeds at a given instance. It is not clear how many of the succeeding size scales should be included within a given time interval for an accurate prediction of the macroscopic dynamic crack characteristics. The minimum requirement would therefore necessitate the simultaneous treatment of two scales at the same time. This means that the analysis should capture the change in the macroscopic and microscopic features of a defect as it propagates. The discussion for a dual scale model has been invoked only very recently for a stationary crack. The objective of this work is to extend this effort to a crack running at constant speed beyond that of Rayleigh wave. Developed is a dual scale moving crack model containing microscopic damage ahead of a macroscopic crack with a gradual transition. This transitory region is referred to as the mesoscopic zone where the tractions prevail on the damaged portion of the material ahead of the original crack known as the restraining stresses, the magnitude of which depends on the geometry, material and loading. This damaged or restraining zone is not assumed arbitrarily nor assumed to be intrinsically a constant in the cohesive stress approach; it is determined for each step of crack advancement. For the range of micronotch bluntness with 0 < β < 30° and 0.2 σ_∞/σ₀ 0.5, there prevails a nearly constant restraining zone size as the crack approaches the shear wave speed. Note that β is the half micronotch angle and the applied stress ratio is σ_∞/σ₀ with σ₀ being the maximum of the restraining stress. For σ_∞/σ₀ equal to or less than 0.5, the macrocrack opening displacement COD is nearly constant and starts to decrease more quickly as the crack approaches the shear wave speed. For the present dual scale model where the normalized crack speed v/c_s increases with decreasing with the one-half microcrack tip angle β. There prevails a limit of crack tip bluntness that corresponds to β 36° and v/c_s 0.15. That is a crack cannot be maintained at a constant speed if the bluntness is increased beyond this limiting value. Such a feature is manifestation of the dependency of the restraining stress on crack velocity and the applied stress or the energy pumped into the system to maintain the crack at a constant velocity. More specifically, the transitory character from macro to micro is being determined as part of the unknown solution. Using the energy density function dW/dV as the indicator, plots are made in terms of the macrodistance ahead of the original crack while the microdefect bluntness can vary depending on the tip geometry. Such a generality has not been considered previously. The macro-dW/dV behavior with distance remains as the inverse r relation yielding a perfect hyperbola for the homogeneous material. This behavior is the same as the stationary crack. The micro-dW/dV relations are expressed in terms of a single undetermined parameter. Its evaluation is beyond the scope of this investigation although the qualitative behavior is expected to be similar to that for the stationary crack. To reiterate, what has been achieved as an objective is a model that accounts for the thickness of a running crack since the surface of separation representing damage at the macroscopic and microscopic scale is different. The transitory behavior from micro to macro is described by the state of affairs in the mesoscopic zone.     

Effect of orthotropy on crack propagation

The tendency of moving cracks to spread along the preferred directions of material anisotropy is treated. Depending on the velocity of crack propagation, the change of material properties in orthogonal planes is shown to affect the bifurcation characteristics. The problem is reduced to a system of dual integral equations that can be solved in a standard fashion. Of particular interest is the dynamic stress field near the tip of a moving crack in an orthotropic material. Although the 1√r stress singularity is preserved with r being the radial distance measured from the crack tip, the angular variations of the stresses are dependent on crack speed and material anisotropy. The possibility of crack bifurcation is examined by application of the strain energy density criterion for several composite systems. Crack branching is found to be enhanced by material anisotropy, a phenomenon that is not uncommon in composite materials.     

Dynamic self-similar debonding of interface at very high velocity: Anti-plane case

This paper analyzes the anti-plane problem of dynamic self-similar debonding of interface at very high velocity. The debonding is modeled as an interface crack propagating self-similarly from zero-length. The extending speed is assumed to be transonic or supersonic. We first consider the dynamic debonding under moving concentrated loads. The moving dislocation model of self-similar propagation of an interface crack is used to formulate the problem to a singular integral equation which is solved analytically. The singularity of stresses near the crack tip is discussed and the dynamic stress intensity factors are presented. Finally the solution of dynamic debonding underx ²-type loads is obtained by using the superposition method.     

Field structure at mode Ⅲ dynamically propagating crack tip in elastic-viscoplastic materials

An elastic-viscoplastic mechanics model is used to investigate asymptotically the mode Ⅲ dynamically propagating crack tip field in elastic-viscoplastic materials. The stress and strain fields at the crack tip possess the same power-law singularity under a linear-hardening condition. The singularity exponent is uniquely determined by the viscosity coefficient of the material. Numerical results indicate that the motion parameter of the crack propagating speed has little effect on the zone structure at the crack tip. The hardening coefficient dominates the structure of the crack-tip field. However, the secondary plastic zone has little influence on the field. The viscosity of the material dominates the strength of stress and strain fields at the crack tip while it does have certain influence on the crack-tip field structure. The dynamic crack-tip field degenerates into the relevant quasi-static solution when the crack moving speed is zero. The corresponding perfectly-plastic solution is recovered from the linear-hardening solution when the hardening coefficient becomes zero.     

Special approach for the determination of fracture mechanical parameters at an interface crack tip

An interface crack of finite length is considered between two semi-infinite planes with an artificial contact zone at one of the two crack tips. A transcendental equation and certain simple asymptotic formulas are established for the real contact zone (in the Comninou-Dundurs sense) in terms of the stress intensity factors (SIFs) of the considered model. In these terms analytical expressions are also provided for the energy release rate and for the SIF of the classical interface crack model with an oscillating singularity at the crack tip. The appropriate length of the artifical contact zone is shown to be attainable on the basis of the analysis of the stresses at the crack tip. The use of the proposed model is suggested for integrity assessment of inhomogeneous structural elements of composites containing interface cracks. Received 26 March 1997; accepted for publication 12 September 1997     

Moving line crack accompanied with damage zone subject to remote tensile loading

In the 1920s, a closed-form solution of the moving Griffith crack was first obtained by Yoffe. Based on Yoffe's solution, the Dugdale model for the moving crack case gives a good result. However, the Dugdale model fails when the crack speed is closed to the Rayleigh wave speed because of the discontinuity occurred in the crack opening displacement (COD). The problem is solved in this paper by introducing a restraining stress zone ahead of the crack tip and two velocity functions. The restraining stresses are linearly distributed and related to the velocity of the moving crack. An analytical solution of the problem is obtained by use of the superposition principle and a complex function method. The final result of the COD is continuous while the crack moves at a Rayleigh wave speed. The characteristics of the strain energy density (SED) and numerical results are discussed, and conclusions are given.     

A simple model of the double cantilever beam crack propagation specimen

A very simple model of the double cantilever beam (dcb) dynamic crack propagation specimen is studied. The main assumption on which the model is based is that the arms of the dcb specimen deform as shear beams. The particular problem studied is the dynamic growth of a sharp crack from a blunt pre-crack with the ends of the specimen arms held at a fixed separation distance. It is demonstrated that this simple model predicts crack motion which is qualitatively consistent with the results of more detailed numerical analyses of the problem and with experimental results. The analysis employs an energy balance crack propagation criterion and both constant specific fracture energy and a class of crack speed dependent fracture energies are considered. Among the features exhibited by the model is that, for constant specific fracture energy, the crack tip speed is constant from initiation up to arrest. On the other hand, for the same geometry and loading conditions, but a strongly crack speed dependent specific fracture energy, the crack speed decreases gradually between fracture initiation and crack arrest.     

Oscillatory fracture paths in thin elastic sheets

We report a novel mode of quasi-static oscillatory crack propagation when a cutting tip of moderately large width is driven through a thin brittle polymer film. Experiments show that the amplitude and wavelength of the oscillatory crack paths scale linearly with the width of the cutting tip over a wide range of length scales but are independent of the width of the sheet and of the cutting speed. We propose a mechanism for this instability, based on the coupling between crack propagation and out-of-plane deformations of the film. To cite this article: B. Roman et al., C. R. Mecanique 331 (2003).     

Dynamic stress intensity factor K III and dynamic crack propagation characteristics of anisotropic materials

Based on the mechanics of anisotropic materials, the dynamic propagation problem of a mode Ⅲ crack in an infinite anisotropic body is investigated. Stress, strain and displacement around the crack tip are expressed as an analytical complex function, which can be represented in power series. Constant coefficients of series are determined by boundary conditions. Expressions of dynamic stress intensity factors for a mode Ⅲ crack are obtained. Components of dynamic stress, dynamic strain and dynamic displacement around the crack tip are derived. Crack propagation characteristics are represented by the mechanical properties of the anisotropic materials, i.e., crack propagation velocity M and the parameter ~. The faster the crack velocity is, the greater the maximums of stress components and dynamic displacement components around the crack tip are. In particular, the parameter α affects stress and dynamic displacement around the crack tip.     

An improved experimental method for the determination of the critical stress intensity factor KIc of brittle materials

The critical stress intensity factor K_Ic is determined by a simple and accurate method, using small test specimens and a simple procedure in this paper.Single edge V-notched tension specimens made of PMMA are subjected to a load which is slowly increased until the crack begins to move from the notch tip. During the crack propagation event shadow patterns at the tip of the crack are recorded in a video recorder. Under these loading conditions, the creating real crack propagate slowly until the crack propagation velocity take an abrupt increase and the entire fracture of the specimen takes place. The stress intensity factor which correspond to the transition from the slow to fast crack speed, is the critical stress intensity factor K_Ic and it can be the fracture toughness of the material.The results are accurate and in good agreement with those values of K_Ic which are calculated by approximate theoretical expressions.The purpose of this paper is to introduce an improved, simple and accurate experimental method for the determination of fracture toughness of brittle materials.     

Hydraulic fracturing of a saturated porous medium-I: General theory

This investigation deals with the problem of steady state hydraulic fracture in an infinite isotropic fluid-saturated elastic porous medium induced by a uniform pressure applied to the crack surfaces. A quasi-static approach is employed in the study. A boundary value problem is formulated and then analyzed by means of the Fourier transform associated with the Wiener-Hopf technique. Stress intensity factor and potential energy release rate are found by asymptotic analysis and the superposition principle as functions of the speed of crack propagation. The material breakdown process at the crack tip is discussed based on Dugdale's model. Finally, a brief discussion of the effect of pressure drop on the hydraulic fracture process and the decrease in crack speed during crack extension is included.     

Multiscale modeling of crack initiation and propagation at the nanoscale

Fracture occurs on multiple interacting length scales; atoms separate on the atomic scale while plasticity develops on the microscale. A dynamic multiscale approach (CADD: coupled atomistics and discrete dislocations) is employed to investigate an edge-cracked specimen of single-crystal nickel, Ni, (brittle failure) and aluminum, Al, (ductile failure) subjected to mode-I loading. The dynamic model couples continuum finite elements to a fully atomistic region, with key advantages such as the ability to accommodate discrete dislocations in the continuum region and an algorithm for automatically detecting dislocations as they move from the atomistic region to the continuum region and then correctly “converting” the atomistic dislocations into discrete dislocations, or vice-versa. An ad hoc computational technique is also applied to dissipate localized waves formed during crack advance in the atomistic zone, whereby an embedded damping zone at the atomistic/continuum interface effectively eliminates the spurious reflection of high-frequency phonons, while allowing low-frequency phonons to pass into the continuum region.The simulations accurately capture the essential physics of the crack propagation in a Ni specimen at different temperatures, including the formation of nano-voids and the sudden acceleration of the crack tip to a velocity close to the material Rayleigh wave speed. The nanoscale brittle fracture happens through the crack growth in the form of nano-void nucleation, growth and coalescence ahead of the crack tip, and as such resembles fracture at the microscale. When the crack tip behaves in a ductile manner, the crack does not advance rapidly after the pre-opening process but is blunted by dislocation generation from its tip. The effect of temperature on crack speed is found to be perceptible in both ductile and brittle specimens.     

The interaction between crack and electric dipole of piezoelectricity

Discrete dipoles located near the crack tip play an important role in nonlinear electric field induced fracture of piezoelectric ceramics. A physico-mathematical model of dipole is constructed of two generalized concentrated piezoelectric forces with equal density and opposite sign. The interaction between crack and electric dipole in piezoelectricity is analyzed. The closed form solutions, including those for stress and electric displacement, crack opening displacement and electric potential, are obtained. The function of piezoelectric anisotropic direction,p _a (θ)=cosθ+p _a sinθ, can be used to express the influence of a dipole's direction. In the case that a dipole locates near crack tip, the piezoelectric stress intensity factor is a power function with −3/2 index of the distance between dipole and crack tip. Supported by National Natural Science Foundation of China(No. 10072033)     

Loss of adhesion at the tip of an interface crack

A model is constructed to analyze adhesive bond failure at the tip of an interface crack. The model is based on the assumption that there are zones of bounded cohesive tensile and shear stresses near a crack tip. Within the context of certain broad a-priori assumptions on the distributions of certain stress and displacement components in the cohesive zones, the requirement thatall stresses in the two materials remain bounded provides a method to compute the specific details for these zones. It is assumed that bond failure occurs when the extension of the bond fiber at the crack tip exceeds a critical value. For an interface crack in a uniform tension field computations for two alternate formulations suggest that this failure criterion is independent of the precise distribution of the cohesive stresses, but rather depends only upon their averaged values. Combined loading with a dominant tensile component has also been analyzed. If the critical extension of bond fibers and the maximum value of the cohesive tensile stress are known, the model provides the maximum allowable interface stresses for given crack dimension and material parameters.