Crack Bifurcation Research Articles

Study of rubber‐modified brittle epoxy systems. Part II: Toughening mechanisms under mode‐I fracture

AbstractThe toughening mechanisms in grafted‐rubber concentrate (GRC), dispersed acrylic rubber (DAR), and Proteus rubber‐modified brittle epoxy (i.e., highly crosslinked) systems are examined using scanning electron microscopy, optical microscopy and transmission electron microscopy techniques. The toughening of the GRC‐modified brittle epoxy system is found to be due to cavitation of the GRC rubber particles, followed by formation of limited shear yielding when the crack propagates. Crack bifurcation and crack deflection are also observed in this system. Only crack bifurcation, crack deflection, and possibly crack/particle bridging mechanisms are operative in the DAR‐modified brittle epoxy system. In the case of the Proteus rubber‐modified system, the rubber appears to be rigid (Tg ≈ 28°C). As a result, the crack/particle bridging mechanism is not observed. Only crack deflection and crack pinning mechanisms are found. These observations are in agreement with the toughness measurement results (see Part I), which indicate that the GRC rubber provides the most effective toughening, followed by the DAR rubber, and then by the Proteus rubber. An approach for toughening brittle epoxies is also discussed.

Crack propagation modes in fiber and particulate composites in the microstructure

AbstractThe influence of the hard or soft inclusions and the mesophase layers in either a soft‐hard‐soft or hard‐soft‐hard combination of biphase plates submitted to dynamic tensile loads on the fracture mode and bifurcation process in both phases was investigated in this paper. It was assumed that the soft or hard matrix is infolding the hard or soft inclusion of the plate, so that the plate constitutes a meridional section of the representative volume element of a unidirectional fiber composite, or a principal section of a particulate. The influence of the mechanical properties of either phase on the crack propagation velocity and the initiation of crack bifurcation was studied by using high‐speed photography and dynamic caustics. The results showed that the propagating crack tended to bifurcate either in the brittle or in the mesophase layer under certain conditions of propagation velocity. It was shown that bifurcation of a propagating crack depends on the elastic moduli and Poisson's ratios of the phases, as well as on the extent of the mesophase layer, which depended on the adhesion quality of phases.

Cryogenic failure mechanisms of fiber‐epoxy composites for energy applications

AbstractFiber‐reinforced composites used in the production, storage, and transport of energy are often exposed to extreme environments such as cryogenic temperatures and/or nuclear radiation. The fraction resistance of composites subjected to these conditions are of fundamental interest. A study was undertaken to examine the effects of Gammaradiation on the structure and properties of fibers, matrix and their interfacial bonding in terms of their influence on composite failure mechanisms. Also thermomechanical analysis was conducted to estimate the residual thermal stresses due to differential thermal expansion coefficients both between fiber and matrix and between two laminae. Results indicate that the impact energy of a composite laminate could be increased by 100 percent by immersing the specimen in liquid nitrogen for only five min. Samples impact‐tested at cryogenic temperatures were found to possess a great degree of delamination and crack bifurcation. Thermomechanical analysis and SEM investigation both reveal that microcracking and small‐scale delamination are promoted by differential thermal contraction. Such effects could be responsible for the observed crack branching and delamination phenomena during impact loading. Although under certain circumstances the Gammaradiation may yield a small increase in fracture resistance it generally degrades the cohesive strength of the matrix and reduces the interfacial bonding between fiber and matrix. Results of a mechanical and microscopic analysis are presented and discussed.

Preliminary measurements of terminal crack velocity in ice

Direct measurements of ice crack velocity obtained from double cantilever beam fracture toughness specimens gave a terminal velocity of approximately 20 m/s or 0.01C d (0.005C s) where C d is the dilatational wave velocity and C s the shear wave velocity. This value is much lower than both theoretical predictions, and experimental values obtained in other brittle materials, of approximately 0.3 C d. It is shown that this low velocity is due to the low hardness of ice compared to glasses. Crack bifurcation was not observed in any of the samples and this is explained by the low crack velocity — in other materials bifurcation only occurs at high velocities. It is concluded that fragment size in ice is, therefore, not the result of bifurcation, but rather a consequence of the various stress fields developed.

Crack bifurcation modes in composite plates under impact

Cracked plates with bonded elastic circular inclusions inside a matrix were subjected to an impact tensile loading. The influence of the inclusion on the propagation of an initial edge-crack was discussed in this paper. Two types of specimens were tested, that is with hard or soft inclusions surrounded by soft or hard matrices respectively. It has been shown that bi-, or tri-furcations of the propagating crack have persistently occured at the region of mesophase between matrix and circular inclusions. The phenomenon of splitting of the initial crack was common in both types of specimens. For the definition of the prevailing stress field, the dynamic stress intensity factors were evaluated at different positions of the crack, as well as the crack-velocities along the main crack or its branches during their propagation. It was observed that the crack initiation in soft plates with hard inclusions was followed by the emission of Rayleigh waves, which contributed to the mode of fracture of the composite. Isochromatic patterns were also used to simulate the stress-field of the propagating cracks against the inclusions at distinct time-instants where the propagating crack was frozen at this instant and static photoelastic analysis was applied. These approximations gave satisfactory results with the cracks advancing inside the composite plates.

Crack Propagation and Bifurcation in Fiber-Composite Models-II: Hard-Soft-Hard Sequence of Phases

An experimental study of the influence of a soft-fiber inclusion and the mesophase layer surrounding it in a hard-soft-hard bi-phase representative volume element (RVE), on the fracture and bifurcation processes in both phases was undertaken. The influence of the mechanical properties of each phase and the effect of the external loading-rate on the crack propagation velocity and the initiation of the crack bifurcation was studied by using high speed photography and dynamic caustics. It was shown that the propagating crack tended to bifurcate either in the hard or in the mesophase layer, and the bifurcation of the crack depended strongly on the strain rate, the elastic modulus and the Poisson ratio of the phases, as well as on the extent of the mesophase layer, which relies upon the degree of adhesion between phases.

Dynamic effects in mode III crack bifurcation

Transient elastodynamic nonplanar self-similar Mode III crack growth in brittle materials is examined. The dynamic similarity and Chaplygin's transformation reduce the class of problems considered to the solution of Laplace's equation in a semi-infinite strip. The Schwarz-Christoffel transformation is subsequently employed to map the semi-infinite strip on a half-plane. The theory of analytic functions can then be used. Elastodynamic influences in the vicinity of a rapidly moving tip after branching are examined in a rather general fashion.To reveal the sensitivity of one branch's crack tip elastodynamic stress intensity factor to the relative velocity and orientation of the other, the problem of asymmetric crack bifurcation under stress wave loading is chosen for study. The development of specific analytical solutions requires the solutions to the symmetric bifurcation problem subjected to loading that induces either antisymmetrical or symmetrical deformations, respectively, about the original crack plane. For asymmetric geometries the solution required the numerical evaluation of several integrals in the final stages. It is ultimately shown that the stress intensity factor of one branch is significantly altered by changes in the velocity and orientation of the other and by the angle of stress wave incidence.

The Rapid Tearing of a Half Plane

The surface of an elastic half space is subjected to sudden antiplane mechanical disturbances. Crack initiation and subsequent crack instability are examined via two idealized problems; the first is concerned with instantaneous crack bifurcation and the second with instantaneous skew crack propagation. In either problem, crack propagation occurs at a constant subsonic velocity under an angle κπ with the normal to the surface. For the externally applied disturbances that are considered here, and for contstant crack-tip velocities, the particle velocity and τθz are functions of r/t and θ only, which allows Chaplygin’s transformation and conformal mapping to be used. The problems can then be solved using analytic function theory. For various values of the angle of crack propagation, the dependence of the elastodynamic stress intensity factors on the crack propagation velocity is investigated. For certain specific geometries, fully analytical solutions are derived to provide check cases.

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.

How to use damage mechanics

The background of continuum damage mechanics is first presented in the framework of thermodynamics with some examples of constitutive equations for ductile damage, creep damage and fatigue damage. After the general scheme of structural calculations for macro-crack initiation, through non-coupled or coupled strain damage equations, some examples of “simple applications” are given: fracture limits of metal forming, surface initial damage in fatigue, creep fatigue interaction, and bifurcation of cracks.

Influence of temperature rise at the crack tip on crack bifurcation

Influence of temperature rise at the crack tip on crack bifurcation

Two Analytical Solutions for Dynamic Crack Bifurcation in Antiplane Strain

A semi-infinite crack is subjected to constant magnitude, dynamic antiplane loading at time t = 0. At the same instant the crack is assumed to bifurcate and propagate normal to its original plane or to propagate without branching. For constant crack-tip velocities the stresses and particle velocity are functions of r/t and θ only, which allows Chaplygin’s transformaton and conformal mapping to be used to obtain two Riemann-Hilbert problems which can be solved analytically. Expressions for the elastodynamic Mode III stress-intensity factors are then computed as functions of the crack-tip velocity and some conclusions concerning crack initiation are drawn.

Computer simulation of fast crack propagation in brittle material

A method is presented to simulate the behavior of a rapidly running crack by employing the finite element method with a singular element. The crack-tip is assumed to move so as to satisfy a dynamic fracture criterion that the dynamic energy release rate be equal to a specific fracture energy during failure. Simulation is carried out for a crack in a rectangular plate subjected to a suddenly applied load, and it has indicated that the dynamic behavior of a crack, such as onset of crack propagation, bifurcation, or stopping, is deeply influenced by the magnitude and the time duration of the applied load.

Crack bifurcation under hydrostatic pressure

Glass tubes were fractured by internal pressure under atmospheric or superimposed hydrostatic pressure. It is found that the crack length at bifurcation is significantly increased by superimposed hydrostatic pressures. The dynamic energy release rate for a crack propagating with superimposed hydrostatic pressure is calculated. It is shown that the pressure effect on crack length at bifurcation may be explained by combining the energy balance condition and the Yoffe's concept for crack bifurcation.

Size effect on the strength of glassy carbon

Three-point bend tests were conducted on three sets of commercial glassy carbon specimens having volumes of V, 2V, and 4V, and mean strengths of these specimens decreased with increasing volume (from 37 to 32 to 28×103 psi). These results are in good agreement with Weibull predictions of σ(V)=1.12σ(2V)=1.26σ(4V), which are based on a uniform distribution of flaws throughout a volume of material that is characterized by a Weibull modulus of m=6.0. Moreover, the resulting strength formulation for any volume V(in.3), σ=6 V−1/6×103 psi, correlates well with widerspread data from other sources. In common with other brittle materials, glassy carbon satisfies the crack bifurcation relationship of σr1/2=constant, and this was used to provide additional support for the validity of the volume-dependent Weibull theory. Failures in this material usually originated at interior spherical pores, and the mean size of these flaws for different sets of specimens increased with test volume.

Strain rate effects on viscoelastic crack bifurcation

The effects of applied strain rate on the viscoelastic crack bifurcation phenomenon in Polymethyl Methacrylate (PMMA) were investigated. It was still verified that the product σ fC b 1 2 was constant, as was already observed by Congleton and Petch, and Anthony, Chubb and Congleton, for brittle elastic materials, for any strain rate, where σ f = the gross fracture stress and C b = the main crack length until the bifurcation starts. However, it was found that the higher strain rate increases the main crack length C b resulting in the decrease in the gross fracture stress σ f and vice versa. This might be interpreted that the higher stress concentration at the initiation crack tip, which is realized by becoming more brittle due to the higher strain rate owing to the predominance of the elastic element in the viscoelastic material, decreases the gross fracture stress leading to the longer main crack length.

Observations on microstructurally sensitive fatigue crack growth in a widmanstätten Ti-6Al-4V alloy

Fatigue crack growth rates in commercial purity Ti-6A1-4V can be substantially reduced with a beta anneal from levels associated with the mill anneal, owing primarily to crystallographic crack bifurcation in the Widmanstatten packets. This microstructurally sensitive fatigue crack growth occurs when the reversed plastic zone is less than the packet size and results in a fracture surface with a faceted morphology. It appears from replica and scanning electron microscopic examination that the facets are comprised of three superposed features,viz cleavage-like river-line patterns, very fine striations and traces of slip lines (and slip-band cracks). Limited X-ray evidence suggests a facet orientation some 8 to 10 degrees off the basal plane.

Stress pulses emitted during fracture in tension

This paper is concerned with the stress pulses which are emitted during the tensile fracture of glass rods of rectangular cross-section. Both a longitudinal and a flexural pulse were observed. The experimentally recorded pulse shapes are found to be in close agreement with those predicted by using a theoretical model proposed by Phillips and Kolsky. It is shown that the deviations noted in the earlier work with circular glass rods are due, primarily, to the effect of crack bifurcation and not to the geometry of the cross-section. Further, the phenomenon of crack bifunction is found to depend upon the magnitude of the applied stress required to initiate fracture, and to be essentially independent of the section geometry.

Bifurcation of a running crack in antiplane strain

An elastodynamic explanation of running crack bifurcation is explored. The geometry is a semi-infinite body in a state of antiplane strain, which contains a two-dimensional edge crack. It is assumed that a quasi-static increase of the external loads gives rise to rapid crack propagation at time t = 0, with an arbitrary and time-varying speed, but in the plane of the crack. A short time later the crack is assumed to bifurcate at angles −κπ and +κπ, and with velocities v. The elastodynamic intensity factors are computed, and the balance of rates of energies is employed to discuss the conditions for bifurcation.

Crack bifurcation in brittle solids

Classic thermodynamic considerations are employed to derive the condition for bifurcation of a crack situated within a brittle solid deforming in an anti-plane strain mode. Criteria are determined for bifurcation along circular segments at right angles to the initial crack, and also along planar segments inclined at an arbitrary angle with respect to the original crack, the results being used in a brief discussion of physical situations where bifurcation occurs.