Optical method of caustics applied in viscoelastic fracture analysis

The optical method of caustics is developed here to study the fracture of viscoelastic materials. By adopting a distribution of viscoelastic stress fields near the crack tip, the method of caustics is used to determine the viscoelastic fracture parameters from the caustic patterns near the crack tip. Two viscoelastic materials are studied. These are PMMA and ternary composites of HDPE/POE-g-MA/CaCO₃. The transmitted and reflective methods of caustics are performed separately to investigate viscoelastic fracture behaviors. The stress intensity factors (SIFs) versus time is determined by a series of shadow spot patterns combined with viscoelastic parameters evaluated by creep tests. In order to understand the viscoelastic fracture mechanisms of HDPE/POE-g-MA/CaCO₃ composites, their fracture surfaces are observed by a Scanning Electron Microscope (SEM). The results indicate that the method of caustics can be used to characterize the fracture behaviors of viscoelastic materials and further to optimize the design of polymer composites.     

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.     

The influence of singular field dominance on the caustic methods

The caustic method as applied to fracture mechanics for measuring the stress intensity factor, contains several approximations which limit its applicability and accuracy. The accuracy of the caustic method for the determination of the stress intensity factor is based on the restriction that the K-dominant singular field is fully established at which the experimental data are taken. In this paper, we attempt to identify the regions in which experimental measurements by caustic methods can be performed with confidence. The reliability of the method of caustics applied to the static problem and the case of a stationary crack subjected to dynamic loading are investigated in detail. The results demonstrate that significant deviation can occur in the determination of the stress intensity factor from shadow spot measurements. Conclusions regarding the method of caustics in the determination of the stress intensity factor are extracted from these results.     

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.     

Structural fracture scales in shock-loaded epoxy composites

Shock fracture mechanisms of different scales were investigated on epoxy composite materials reinforced with silicon carbide microparticles of different concentrations. It is shown that the high heterogeneity of the epoxy composites at different structural scales is one of the factors responsible for their physical and mechanical properties. Under dynamic loading, the material reveals a developed structural scale hierarchy which provides self-consistent deformation and fracture of the material bulk with the lead of rotational deformation modes. As a result, microcracks develop due to low shear strain limited in addition by reinforcing particles. At the start of a main crack, microscale mechanisms dominate, whereas the propagation of its front is governed by macroscale fracture mechanisms.     

Fracture Analysis in Orthotropic Thermoelasticity Using Extended Finite Element Method

In this paper, a method for extracting stress intensity factors (SIFs) in orthotropic thermoelasticity fracture by the extended finite element method (XFEM) and interaction integral method is presented. The proposed method is utilized in linear elastic crack problems. The numerical results of the SIFs are presented and compared with those obtained using boundary element method (BEM). The good accordance among these two methods proves the applicability of the proposed approach and conforms its capability of efficiently extracting thermoelasticity fracture parameters in orthotropic material.     

Photoelastic investigation of interfacial fracture between orthotropic and isotropic materials

Static and dynamic fracture of interfaces between orthotropic and isotropic materials were studied using photoelasticity. In this study, a bi-material specimen made of PSM-1^® and Scotchply^® 1002, a unidirectional glass fiber reinforced epoxy composite, was used. Two fiber orientations, fibers parallel to the interface (α=0°) and fibers perpendicular to the interface (α=90°) were considered. Center crack bi-material specimens having different crack lengths were loaded quasi-statically and the full-field isochromatics were recorded using a digital camera. The complex stress intensity factor corresponding to each crack length was calculated from the isochromatics and the values were compared to that obtained from boundary collocation method. Dynamic interfacial fracture was studied with an edge crack bi-material geometry for the two different fiber orientations. The isochromatics around the propagating crack were recorded using a digital high-speed camera. The fracture parameters such as crack speed, complex stress intensity factor and energy release rate were extracted from the isochromatics using the asymptotic stress field equations. The complex stress intensity factor obtained from the static experiments was in close agreement with that calculated using the boundary collocation method. The results also indicated that the fiber orientation with respect to the interface influences the fracture parameters for stationary and propagating cracks.     

Caustics,pseudocaustics and the related illuminated and dark regions with the computational method of quantifier elimination

The method of caustics is a powerful experimental method in elasticity and particularly in fracture mechanics for crack problems. The related method of pseudocaustics is also of interest. Here we apply the computational method of quantifier elimination implemented in the computer algebra system Mathematica in order to determine (i) the non-parametric equation and two properties of the caustic at a crack tip and especially (ii) the illuminated and the dark regions related to caustics and pseudocaustics in plane elasticity and plate problems. The present computations concern: (i) The derivation of the non-parametric equation of the classical caustic about a crack tip through the elimination of the parameter involved (here the polar angle) as well as two geometrical properties of this caustic. (ii) The derivation of the inequalities defining the illuminated region on the screen in the problem of an elastic half-plane loaded normally by a concentrated load with the boundary of this illuminated region related to some extent to the caustic formed. (iii) Similarly for the problem of a clamped circular plate under a uniform loading with respect to the caustic and the pseudocaustic formed. (iv) Analogously for the problem of an equilateral triangular plate loaded by uniformly distributed moments along its whole boundary, which defines the related pseudocaustic. (v) The determination of quantities of interest in mechanics from the obtained caustics or pseudocaustics. The kind of computations in the applications (ii) to (iv), i.e. the derivation of inequalities defining the illuminated region on the screen, seems to be completely new independently of the use here of the method of quantifier elimination. Additional applications are also possible, but some of them require the expansion of the present somewhat limited power of the quantifier elimination algorithms in Mathematica. This is expected to take place in the future.     

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.     

Viscoelasticity of a polymer matrix and fracture of heat-resistant fiber composites

This paper reports on the results of investigations into the general regularities of deformation and fracture of fiber composite materials based on new heat-resistant polymer binders. Fiber composites based on these binders can find wide application in various fields of engineering. It is established that an increase in the loss modulus of the polymer matrix decreases the probability of formation of a brittle crack in the matrix at the fiber break and increases the time interval between breakages of adjacent fibers. This leads to retardation of the correlated breakage of the fibers in fiber composite materials under loading, i.e., to an increase in their strength and fracture toughness. The inference is made that the matrix of high-strength heat-resistant fiber composites with a high fracture toughness should possess not only a high elasticity (this has long been known) but also good dissipative properties over the entire temperature range of operation.     

The caustic approach to Rayleigh-waves and their interaction with surface irregularities

During the past 25 years the method of caustics has matured to the stage where it offers a highly potential tool for qualitative as well as quantitative solutions of engineering problems in many fields. In conjunction with high-speed recording techniques such as high-speed cinematography, the method of caustics serves as a means for visualizing the highly complex interaction processes that occur when elastic stress waves interact with material inhomogeneities or discontinuities such as inclusions, surface or embedded cracks and contacting surfaces during impact.In this contribution, the method of caustics is applied to a number of dynamic problems all of which are associated with the propagation and interaction of Rayleigh-waves:

• the pseudo-caustic of a Rayleigh-wave;

• Rayleigh-wave generation during dynamic impact of two elastic bodies;

• Rayleigh-wave interaction with a shallow embedded crack;

• Rayleigh-wave interaction with various configurations of sets of surface cracks.

Sequences of high-speed cinematographic recordings of the evolution of a dynamic process provide the raw data for data and image analysis for the determination of stress intensity factors, contact forces, load-time traces or other quantities of physical interest.The paper also discusses some of the advantages and disadvantages of the method of caustics over other methods currently used in photomechanics such as dynamic photoelasticity. Data reduction and analysis aspects will also be highlighted.     

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.     

New types of reinforced composite materials

The physical properties of solids determine their usefulness as structural materials. Metals have some disadvantageous characteristics which reduce their effectiveness in critical engineering applications. These limitations can be overcome by the use of certain types of fibrous reinforced composites which have become available over the last few years. However, these materials in turn have their own inherent limitations, particularly in their mode of fracture under overload conditions. In this review the basic properties of conventional fibrous composites are discussed, particular emphasis being given to physics of these failure processes.In an attempt to overcome some of these limitations a new type of fibre reinforced composite has been designed and preliminary research data on the physical properties of these systems has now been obtained. The primary reinforcing elements extend throughout the whole length of the composite but differ from others in that they do not fracture when the composite is subjected to a wide ranged of loading and deformation conditions.These characteristics are achieved because the interface between the primary reinforcing members and the rest of the composite structure is responsive to the local stress carried by the reinforcing members. This stress controlled decoupling/recoupling process is very broadly analogous to the transition between elastic and plastic deformation in metals and the physical principles underlying the design of the reinforcing members are outlined.Because the reinforcing members do not fracture any failure process is confined to the rest of the composite structure. Various interactions occur between the non-fracturing and fracturable parts of the system and these suppress crack growth in the latter, thus producing a structure possessing very considerable damage tolerance. A preliminary analysis of the basic physics of these interactions is given.Possible future developments of these materials are outlined. Also ways are discussed in which the simple analytical models, developed in the study of the fracture processes occuring these materials, amy be applied to more conventional fibrous composites.     

Two-dimensional scaling properties of experimental fracture surfaces

The self-affine properties of postmortem fracture surfaces in silica glass and aluminum alloy were investigated through the 2D height-height correlation function. They are observed to exhibit anisotropy. The roughness, dynamic, and growth exponents are determined and shown to be the same for the two materials, irrespective of the crack velocity. These exponents are conjectured to be universal.     

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.     

Ultrasonic polar scans: numerical simulation on generally anisotropic media

Ultrasonic polar scans are based on the recording of the reflected or transmitted amplitude of sound, impinging a fiber reinforced composite from every possible angle of incidence. The mechanical anisotropy of such materials makes the reflection coefficient direction dependent, whence an ultrasonic polar scan forms a fingerprint of the investigated material. Such scans have already proved to be very valuable in the characterization of composites. Simulations have been performed for single layered and multi-layered systems, for pulsed and harmonic waves. Fiber reinforced composites are mostly orthotropic. The current report presents simulations not only on orthotropic materials but on materials of any kind of anisotropy. These extended numerical simulations are not only valuable in the characterization of highly sophisticated composites, but may also be used to characterize thin slices of crystals and even layered crystals.     

Stiffness matrix invariants to validate the characterization of composite materials with ultrasonic methods

This paper presents a method of testing the ultrasonic measurements of the stiffness matrix, and the identification of the anisotropic behaviour, of composite materials. Some linear combinations of elastic constants are invariants for a rotation around an axis of symmetry. If the stacking sequence is the only parameter which changes in a set of long-fibre composites made of the superimposition of plies, the composites must own these invariants. So, PEEK-carbon fibre composite samples were constructed in this way to measure their elastic properties by immersion and contact ultrasonic methods, and to compare the results with predicted invariants. By changing the stacking sequence of plies three anisotropic models are tested: orthotropic, hexagonal and quadratic. Measurements of ultrasonic velocities in various planes of propagation permit the identification of the elastic constant and invariants. From the invariance of these linear combinations, the precision of the three-dimensional effective moduli can be estimated.     

Nanoscale periodic morphologies on the fracture surface of brittle metallic glasses

Out-of-plane, nanoscale periodic corrugations are observed in the dynamic fracture surface of brittle bulk metallic glasses with fracture toughness approaching that of silica glasses. A model based on the meniscus instability and plastic zone theory is used to explain such dynamic crack instability. The results indicate that the local softening mechanism in the fracture is an essential ingredient for controlling the formation of the unique corrugations, and might provide a new insight into the origin of fracture surface roughening in brittle materials.     

Molecular dynamics simulation of brittle fracture in silicon

Brittle fracture in silicon is simulated with molecular dynamics utilizing a modified embedded atom method potential. The simulations produce propagating crack speeds that are in agreement with previous experimental results over a large range of fracture energy. The dynamic fracture toughness is found to be equal to the energy consumed by creating surfaces and lattice defects in agreement with theoretical predictions. The dynamic fracture toughness is approximately 1/3 of the static strain energy release rate, which results in a limiting crack speed of 2/3 of the Rayleigh wave speed.     

Predicting Interfacial Strengthening Behaviour of Particulate-Reinforced MMC — A Micro-mechanistic Approach

The fracture properties of particulate-reinforced metal matrix composites (MMCs) are influenced by several factors, such as particle size, inter-particle spacing and volume fraction of the reinforcement. In addition, complex microstructural mechanisms, such as precipitation hardening induced by heat treatment processing, affect the fracture toughness of MMCs. Precipitates that are formed at the particle/matrix interface region, lead to improvement of the interfacial strength, and hence enhancement of the macroscopic strength properties of the composite material. In this paper, a micro-mechanics model, based on thermodynamics principles, is proposed to determine the fracture strength of the interface at a segregated state in MMCs. This model uses energy considerations to express the fracture toughness of the interface in terms of interfacial critical strain energy release rate and elastic modulus. The interfacial fracture toughness is further expressed as a function of the macroscopic fracture toughness and mechanical properties of the composite, using a toughening mechanism model based on crack deflection and interface cracking. Mechanical testing is also performed to obtain macroscopic data, such as the fracture strength, elastic modulus and fracture toughness of the composite, which are used as input to the model. Based on the experimental data and the analysis, the interfacial strength is determined for SiC particle-reinforced aluminium matrix composites subjected to different heat treatment processing conditions.