Caustics method in dynamic fracture problem of orthotropic materials

Caustics method is a powerful optical technique in fracture mechanics because of its high sensitivity to stress gradients. In this paper, it is applied to resolve dynamic fracture problems in orthotropic composites. Considering most orthotropic materials are opaque, reflective caustics method is derived here by combining the fundamental principle of caustics method with the mechanical properties of orthotropic materials. Meanwhile, corresponding experiments are carried out for typical glass fiber-reinforced composites, where mode I and mixed-mode fracture states are taken into account. By recording and analyzing shadow spot patterns during the crack propagation process carefully, crack onset time, dynamic fracture toughness and crack growth velocity of orthotropic composite are determined. These results will be useful to evaluate the dynamic fracture properties of composites and further to optimize their designs.     

The optical method of caustics

An overview of the basic principles of the optical method of caustics for the determination of stress intensity factors in crack problems is presented. The method is based on the assumption that the state of stress in the neighborhood of the crack tip is plane stress. However, the state of stress changes from plane strain very close to the tip to plane stress at a critical distance from the tip through an intermediate region where the stress field is three-dimensional. The caustic is the image of the so-called initial curve on the specimen and, therefore, depends on the state of stress along the initial curve. For the determination of stress intensity factors the values of the stress-optical constants are needed. These values depend strongly on the state of stress being plane stress, plane strain or three-dimensional. This complicated the experimental determination of stress intensity factors. For the characterization of the state of affairs near the crack tip a phenomenological triaxiality factor is introduced. A methodology based on the use of optically birefringent materials is developed for the determination of stress intensity factors without paying attention to the location of the initial curve in the plane stress, plane strain or three-dimensional region. Finally, a comparison of the methods of photoelasticity and caustics takes place, and the potentialities and limitations of both methods for the solution of crack problems are explored.     

Toughening of Propylene‐ethylene Copolymer/Calcium Carbonate Composites with High‐Density Polyethylene

Propylene‐ethylene copolymer/calcium carbonate (CaCO₃) composites (weight ratio=50/50) toughened with high density polyethylene (HDPE) were prepared using a twin‐screw extruder; the HDPE content in composites was in the range of 0–4 wt.%. The notched impact strength of propylene‐ethylene copolymer/CaCO₃ composites with 1.5 wt.% HDPE was 46% higher than that of propylene‐ethylene copolymer/CaCO₃ composites. Differential scanning calorimetry (DSC) experiments showed that good miscibility between propylene‐ethylene copolymer and HDPE enhanced the interpenetration of the macromolecules located in the interface. It was shown that debonding of the small HDPE particles within the propylene‐ethylene copolymer matrix resulted in the formation of small voids; the subsequent plastic deformation of the propylene‐ethylene copolymer matrix next to the voids thinned the ligaments and led to large energy consumption.     

Elliptically polarized white light photoviscoelastic technique and its application to viscoelastic fracture

The present paper demonstrates the successful application of the photoviscoelastic technique using elliptically polarized white light to the stress field evaluation of the crack growth in a viscoelastic strip. Using the proposed technique, which can determine both isochromatic and isoclinic parameters simultaneously from a single color image, the time-dependent stress state around a slowly propagating crack tip in a viscoelastic strip plate is successfully analyzed. Then, the time-dependent stress intensity factor extended for linearly viscoelastic materials is evaluated from the experimental results using a method based on the least squares. The results show that the proposed critical stress intensity factor for fast crack growth may be considered as a characteristic property of the material under monotonically increasing load.     

Dynamic caustics and its applications

For the study of elastodynamic problems of propagating cracks it is necessary to evaluate the dynamic stress intensity factor K^d_I which depends on the form of expressions for the stress components existing at the running crack tip at any instant of the propagation of the crack and the corresponding dynamic mechanical and optical properties of the material of the specimen under identical loading conditions. In this paper the distortion of the form of the corresponding reflected caustic from the lateral faces of a dynamically loaded transparent and optically inert specimen containing a transverse crack running under constant velocity was studied on the basis of complex potential elasticity theory and the influence of this form on the value of the dynamic stress intensity factor was given. The method was applied to the study of a propagating Mode I crack in a PMMA specimen under various propagation velocities and the corresponding dynamic stress intensity factor K^d_I evaluated. Also, crack propagation behaviour of notched composites in dynamic loading modes are reviewed and evaluated. A relatively large data base using metal-epoxy particulates, rubber-toughened poly(methyl methacrylate), and Sandwich plates are given. In all cases, a combination of high-speed photography and the optical method of dynamic caustics has been used. Results on the dynamic crack propagation mode, fracture toughness and crack propagation velocities of several rubber-modified composite models are presented. The composite models studied include specimens with one and/or two ‘complex’ two-stage inclusions, i.e. PMMA round inclusions surrounded by concentric rubber rings, one and/or press-fifting inclusions without rubber interface, all under dynamic loading. In all cases both qualitative and quantitative results were obtained. Also, results on crack propagation mode, crack propagation velocity, stress intensity factors and on the influence of the sandwich phases on crack propagation mode are presented.     

Crack propagation in amorphous polymers and their composites

The fracture energy of a polymer depends strongly on the viscoelastic responses of the material, and therefore is a function of temperature and crack velocity. The toughness of a composite is determined by the way in which the reinforcing filler modifies the energy dissipating mechanisms of the polymeric matrix.

The fracture toughness of a variety of polymeric glasses and their composites with glass beads, glass fibers, and rubber particles was measured. The velocity of rapidly moving cracks and the crack propagation rates under controlled loading conditions were also measured.

It was found that the crack propagation velocities in unfilled and glass bead filled materials were controlled by the longitudinal stress waves in the matrix and that the only effects of the glass beads were to blunt the crack tip and limit the viscous deformation. The effect on fracture toughness was relatively small and either positive or negative, depending on which of the above two factors dominated.

The presence of rubber particles as a second phase lowered terminal crack propagation velocities and greatly increased the fracture toughness, indicating a crack retarding effect of the rubber particles. This is related to the induction of crazes in the matrix by the rubber phase.

Glass fibers had a tendency to bridge the tip of a propagating crack, thereby greatly increasing the fracture toughness. In this case the work of fracture comes from a combination of the elastic strain energy stored in the fibers, the energy dissipated in debonding the fibers from the matrix, and the fracture energy of the matrix itself.     

Dynamic optical visualization on the interaction between propagating crack and stationary crack

Dynamic interactions between the propagating crack and the static crack in PMMA material are studied by combining high-speed Schardin camera with optical caustic method. A series of dynamic optical bifocal patterns (the specimen-focused image and the off-focused image) around the propagating crack tip and the static crack tip are recorded for PMMA thin strip which contains two collinear-edge-cracks subjected to tensile loading, the variations of the caustic diameter and the distortion of the caustic shape are revealed due to the influence of local stress singularity at the crack tip. Interactions between the moving crack and the static crack are analyzed by means of the evolution of dynamic fracture parameters. The influence of crack interaction on fracture parameters is discussed based on both a K-dominance assumption and a higher order transient crack-tip expansion. These results will be useful to the evaluation of dynamic properties and the design of structures in the cracked polymer material.     

碳纤维布补强梁裂尖奇异性的光学焦散线实验研究

通过光学焦散线方法对碳纤维布加固含裂纹梁进行三点弯荷载下的静荷载和冲击实验,研究了其断裂特性。在静态实验研究中,记录了不同载荷状态下裂纹的焦散斑光学图像,计算了与之对应的裂纹尖端的应力强度因子,分析了试件在粘贴加固不同长度的碳纤维布情况下焦散斑和应力强度因子演化规律。在动态实验中,记录了裂纹扩展的过程和在此过程中不同时刻的焦散斑和裂纹扩展的路线。通过各瞬间的焦散斑图像,分析裂纹的扩展长度及速度,并得到各时刻的动态应力强度因子。     

A model of a belt chamber cracking based on crack profiles calculations

Abstract

The problem of a Belt chamber matrix cracking is presented. The influence of crack surface quality on the effective values of near crack tip stress is discussed. It is shown that under working conditions of the vessel, the existing shear friction between upper and lower crack surfaces caused by crack surface roughness can prevent the crack surface sliding displacement. Therefore, the control variable for matrix cracking is the value of stress intensity factor K_I corresponding to normal node of loading only. The calculations are performed by finite element method within the range of linear elastic fracture mechanics.     

Linear least squares approach for evaluating crack tip fracture parameters using isochromatic and isoclinic data from digital photoelasticity

In this work, digital photoelasticity technique is used to estimate the crack tip fracture parameters for different crack configurations. Conventionally, only isochromatic data surrounding the crack tip is used for SIF estimation, but with the advent of digital photoelasticity, pixel-wise availability of both isoclinic and isochromatic data could be exploited for SIF estimation in a novel way. A linear least square approach is proposed to estimate the mixed-mode crack tip fracture parameters by solving the multi-parameter stress field equation. The stress intensity factor (SIF) is extracted from those estimated fracture parameters. The isochromatic and isoclinic data around the crack tip is estimated using the ten‐step phase shifting technique. To get the unwrapped data, the adaptive quality guided phase unwrapping algorithm (AQGPU) has been used. The mixed mode fracture parameters, especially SIF are estimated for specimen configurations like single edge notch (SEN), center crack and straight crack ahead of inclusion using the proposed algorithm. The experimental SIF values estimated using the proposed method are compared with analytical/finite element analysis (FEA) results, and are found to be in good agreement.     

Finite strain stress fields near the tip of an interface crack between a soft incompressible elastic material and a rigid substrate

We present a numerical study of finite strain stress fields near the tip of an interface crack between a rigid substrate and an incompressible hyperelastic solid using the finite element method (FEM). The finite element (FE) simulations make use of a remeshing scheme to overcome mesh distortion. Analyses are carried out by assuming that the crack tip is either pinned, i.e., the elastic material is perfectly bonded (no slip) to the rigid substrate, or the crack lies on a frictionless interface. We focus on a material which hardens exponentially. To explore the effect of geometric constraint on the near tip stress fields, simulations are carried out under plane stress and plane strain conditions. For both the frictionless interface and the pinned crack under plane stress deformation, we found that the true stress field directly ahead of the crack tip is dominated by the normal opening stress and the crack face opens up smoothly. This is also true for an interface crack along a frictionless boundary in plane strain deformation. However, for a pinned interface crack under plane strain deformation, the true opening normal stress is found to be lower than the shear stress and the transverse normal stress. Also, the crack opening profile for a pinned crack under plane strain deformation is completely different from those seen in plane stress and in plane strain (frictionless interface). The crack face flips over and the tip angle is almost tangential to the interface. Our results suggest that interface friction can play a very important role in interfacial fracture of soft materials on hard substrates.     

STICK-SLIP TYPE CRACK GROWTH DURING INSTRUMENTED HIGH-SPEED IMPACT OF HDPE AND HDPE/SELAR® DISCONTINUOUS LAMINAR MICROLAYER COMPOSITES

Stick-slip type crack growth was triggered in high-density polyethylene (HDPE) and HDPE/Selar® discontinuous laminar microlayer composites by instrumented Charpy impact under suitable test conditions (temperature ?40°C, hammer speed 3.7 m/s). Fractographic analysis showed that crazing is responsible for this peculiar fracture. The onset of this stick-slip phenomenon was favored by a mixed plane strain/plane stress condition prevailing in the specimens. The relative orientation of the Selar microlayers in respect to the crack growth direction affected the stick-slip type crack growth and the related failure considerably. No stick-slip type crack propagation was observed when gasoline-plasticized specimens were impacted, which failed by ductile tearing instead of crazing.     

Influence of High-Density Polyethylene and Nano-Calcium Carbonate of Various Content Ratios on the Crystallization of Polypropylene

The influence of high-density polyethylene (HDPE) and nano-CaCO₃ of various content ratios on the crystallization of polypropylene (PP) was investigated by differential scanning calorimetry, dynamic rheology, wide angle X-ray diffraction (WAXD), and Izod impact strength measurements. The results showed that HDPE and PP were phase separated in their blends and the additive CaCO₃ filler mainly dispersed in the PP phase, acting as a nucleation agent to promote the crystallization of PP. For the samples HDPE/ nano-CaCO₃ 30/0 and 25/5, the β crystals content was much higher than the other samples. The reason is that the viscosity difference between HDPE and PP led to a velocity difference, which could induce shear stress at the interfaces of HDPE and PP during injection molding. The intensive shear stress at their phase interfaces is advantageous for orientation of the chains, inducing the formation of β crystals. However, with the increment of CaCO₃ content, there were dual effects of CaCO₃ on the crystallization of PP: at low CaCO₃ content, it would hamper the orientation of PP chains, thus leading to a decrease of β crystals; at high CaCO₃ content, it would induce β crystals by itself.     

Scaling effect on the mixed-mode fracture path of rock materials

In this paper, a new approach is presented to predict the crack growth path in the rock materials by taking into account the size effect. The proposed approach is an incremental method in which the crack initiation angle for each step is determined from the modified forms of the maximum tangential stress criterion. These modified maximum tangential stress criteria take into account the influence of the higher order terms of the stress series at the crack tip in addition to the singular terms. As an important parameter in the proposed method, the critical distance r _c is also assumed to be size dependent. Finally the incremental method is evaluated by experimental results obtained from Guiting limestone and CJhorveh marble specimens reported in the previous studies. It is shown that the proposed approach can predict the fracture trajectory of cracked specimens with different sizes in good agreement with the experimental results when three terms of Williams series expansion are considered for characterizing the stress field around the crack tip.     

Modeling of thermomechanical fracture of functionally graded materials with respect to multiple crack interaction

Different aspects of thermomechanical fracture of functionally graded materials (FGMs) are considered. Among them are the crack interaction problems in a functionally graded coating on a homogeneous substrate (FGM/H). The interaction between systems of edge cracks is investigated, as well as, how this mutual interaction influences the fracture process and the formation of crack patterns. The problem is formulated with respect to singular integral equations which are referred to the boundary equation methods. The FGM properties are modeled by exponential functions. The main fracture characteristics are calculated, namely, the stress intensity factors, the angles of deviation of the cracks from their initial propagation direction and the critical stresses when the crack starts to propagate. The last two characteristics are calculated using an appropriate fracture criterion. The problem contains different parameters, such as the geometry (location and orientation of cracks, their lengths, and the width of the FGM layer) and material parameters, i.e. the inhomogeneity parameters of elastic and thermal coefficients of the functionally graded material. The influence of these parameters on the thermo-mechanical fracture of FGM/H is investigated. As examples the following real material combinations are discussed: TiC/SiC, Al₂O₃/MoSi₂, MoSi₂/SiC, ZrO₂/nickel and ZrO₂/steel.     

Unstable Fracture Behavior of Rubber Toughened Poly(Styrene-co-Acrylonitrile)

A linear elastic fracture mechanics (LEFM) approach was used to study fracture characteristics of ABS materials. The effects of crack (ligament) length and rubber content on the microscopic deformations taking place at the front of crack tip and in the bulk of the specimens were investigated. The results of fractography studies showed that, in addition to rubber content, the microscopic deformations are influenced by crack length. For some materials this manifests itself as a change in macroscopic response. The ligament length dependent behavior was increased for the samples with higher rubber contents. The results also showed that, although the elastic behavior with unstable crack growth is the dominant micromechanism of deformation, stable crack propagation still occurred in some compositions. All the fracture parameters, including fracture toughness, fracture energy, plastic zone size, and crack tip opening, increased with rubber content. The changes in microscopic and, as a consequence, in the macroscopic deformation behavior of a given specimen with ligament length were attributed to changes in yield stress of the sample and maximum stress on the ligament.     

A unified theory for brittle and ductile shear mode fracture

A unified theory captures both brittle and ductile fracture. The fracture toughness is proportional to the applied stress squared and the length of the crack. For purely brittle solids, this criterion is equivalent to Griffith's theory. In other cases, it provides a theoretical basis for the Irwin-Orowan formula. For purely ductile solids, the theory makes direct contact with the Bilby-Cottrell-Swinden model. The toughness is highest in ductile materials because the shielding dislocations in the plastic zone provide additional resistance to crack growth. This resistance is the force opposing dislocation motion, and the Peach-Koehler force overcomes it. A dislocation-free zone separates the plastic zone from and the tip of the crack. The dislocation-free zone is finite because molecular forces responsible for the cohesion of the surfaces near the crack tip are not negligible. At the point of crack growth, the length of the dislocation-free zone is constant and the shielding dislocations advance in concert. As in Griffith's theory, the crack is in unstable equilibrium. The theory shows that a dimensionless variable controls the elastoplastic behaviour. A relationship for the size of the dislocation-free zone is derived in terms of the macroscopic and microscopic parameters that govern the fracture.     

On the application of the optical method of caustics to the investigation of transient elastodynamic crack problems: Limitations of the classical interpretation

Several possible sources of inaccuracy that occur in the classical interpretation of caustics patterns generated during transient crack growth in elastic materials are examined using a ‘Bifocal Caustics’ set-up and a new full field optical technique called ‘Coherent Gradient Sensing’. During unsteady dynamic crack growth, strict K^d_I-dominance is generally absent, especially at times close to crack initiation and arrest, even in regions outside the crack-tip 3-D zone where plane stress conditions persist. In such cases a truly transient higher order expansion is found to be essential for correctly describing stress fields outside the 3-D zone.     

Shadow optical analysis of dynamic mixed mode loading conditions in impacted izod-type specimens

The crack tip loading conditions in impacted Izod-type specimens are investigated by means of the shadow optical method of caustics. The analysis for evaluating mode-II and mixed mode-I mode-II double caustics for optically anisotropic birefringent materials is developed to correct for erroneous solutions reported in the literature. The long time response of the specimen results in an almost undisturbed mode-I loading, as is expected from quasi-static considerations. In the early time range, however, superimposed mode-II loading conditions are observed; these result from the influences of dynamic wave propagation phenomena. The larger the crack and the nearer the impact point to the crack the larger these effects. Even changes in sign of the loading conditions are observed. Master curves and criteria are developed for establishing specific test conditions with Izod-type specimens leading either to practically pure mode-I conditions of loading or to well-defined controllable mixed mode-I mode-II loading conditions. The results are presented in the form of normalized parameters to allow for a transferability of the established data to any practical problems.     

相干梯度敏感干涉测量技术及在静态断裂力学实验中的应用

介绍了相干梯度敏感(CGS)干涉测量技术的基本原理及其在静态Ⅰ型断裂实验中的应用,验证了该方法对裂纹尖端局部变形场和断裂特性进行定量研究的可行性。给出了代表静态Ⅰ型裂纹尖端奇异场光力学信息的CGS控制方程,模拟并分析了Ⅰ型裂纹尖端的CGS条纹模式,对静态Ⅰ型裂纹尖端变形场和断裂特性进行了三点弯曲的CGS试验,并提取了应力强度因子KⅠ。结果表明,试验结果与理论分析结果相吻合。