Quasi-static Crack Propagation Research Articles

Quasi-static straight and oscillatory crack propagation in ferroelectric ceramics due to moving electric field: experiments and theory

A novel experimental setup was developed which allows controlled quasi-static crack propagation in initially unpoled ferroelectric ceramics as a result of a moving electric field. Applying an electric field on unpoled specimens partially covered with electrodes induces mechanical stresses that drive crack propagation. Different crack morphologies from single straight and oscillating crack propagation to straight double crack propagation were observed. The load for the transition from straight to oscillating crack propagation as well as the wave length for the oscillating crack are determined and compared with bifurcation theory. The experimental results are in good agreement with theoretical predictions. Comparing static with quasi-static experiments, the observed crack lengths were significantly smaller in the latter case. This behavior is attributed to the time dependency of domain switching.

Modeling quasi-static crack growth with the extended finite element method Part I: Computer implementation

The extended finite element method (X-FEM) is a numerical method for modeling strong (displacement) as well as weak (strain) discontinuities within a standard finite element framework. In the X-FEM, special functions are added to the finite element approximation using the framework of partition of unity. For crack modeling in isotropic linear elasticity, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are used to account for the crack. This enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces, and hence quasi-static crack propagation simulations can be carried out without remeshing. In this paper, we discuss some of the key issues in the X-FEM and describe its implementation within a general-purpose finite element code. The finite element program Dynaflow™ is considered in this study and the implementation for modeling 2-d cracks in isotropic and bimaterial media is described. In particular, the array-allocation for enriched degrees of freedom, use of geometric-based queries for carrying out nodal enrichment and mesh partitioning, and the assembly procedure for the discrete equations are presented. We place particular emphasis on the design of a computer code to enable the modeling of discontinuous phenomena within a finite element framework.

Dynamical instabilities of quasistatic crack propagation under thermal stress

We address the theory of quasistatic crack propagation in a strip of glass that is pulled from a hot oven towards a cold bath. This problem had been carefully studied in a number of experiments that offer a wealth of data to challenge the theory. We improve upon previous theoretical treatments in a number of ways. First, we offer a technical improvement of the discussion of the instability towards the creation of a straight crack. This improvement consists in employing Padé approximants to solve the relevant Wiener-Hopf factorization problem that is associated with this transition. Next we improve the discussion of the onset of oscillatory instability towards an undulating crack. We offer a way of considering the problem as a sum of solutions of a finite strip without a crack and an infinite medium with a crack. This allows us to present a closed form solution of the stress intensity factors in the vicinity of the oscillatory instability. Most importantly we develop a dynamical description of the actual trajectory in the regime of oscillatory crack. This theory is based on the dynamical law for crack propagation proposed by Hodgdon and Sethna. We show that this dynamical law results in a solution of the actual crack trajectory in post-critical conditions; we can compute from first principles the critical value of the control parameters, and the characteristics of the solution such as the wavelength of the oscillations. We present detailed comparison with experimental measurements without any free parameters. The comparison appears quite excellent. Finally we show that the dynamical law can be translated to an equation for the amplitude of the oscillatory crack; this equation predicts correctly the scaling exponents observed in experiments.

Extended finite element method for quasi‐brittle fracture

AbstractA methodology for the simulation of quasi‐static cohesive crack propagation in quasi‐brittle materials is presented. In the framework of the recently proposed extended finite element method, the partition of unity property of nodal shape functions has been exploited to introduce a higher‐order displacement discontinuity in a standard finite element model. In this way, a cubic displacement discontinuity, able to reproduce the typical cusp‐like shape of the process zone at the tip of a cohesive crack, is allowed to propagate without any need to modify the background finite element mesh. The effectiveness of the proposed method has been assessed by simulating mode‐I and mixed‐mode experimental tests. Copyright © 2003 John Wiley & Sons, Ltd.

Brittle fracture in polycrystalline microstructures with the extended finite element method

AbstractA two‐dimensional numerical model of microstructural effects in brittle fracture is presented, with an aim towards the understanding of toughening mechanisms in polycrystalline materials such as ceramics. Quasi‐static crack propagation is modelled using the extended finite element method (X‐FEM) and microstructures are simulated within the framework of the Potts model for grain growth. In the X‐FEM, a discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Hence, crack propagation can be simulated without any user‐intervention or the need to remesh as the crack advances. The microstructural calculations are carried out on a regular lattice using a kinetic Monte Carlo algorithm for grain growth. We present a novel constrained Delaunay triangulation algorithm with grain boundary smoothing to create a finite element mesh of the microstructure. The fracture properties of the microstructure are characterized by assuming that the critical fracture energy of the grain boundary (Gcgb) is different from that of the grain interior (Gci). Numerical crack propagation simulations for varying toughness ratios Gcgb/Gci are presented, to study the transition from the intergranular to the transgranular mode of crack growth. This study has demonstrated the capability of modelling crack propagation through a material microstructure within a finite element framework, which opens‐up exciting possibilities for the fracture analysis of functionally graded material systems. Copyright © 2003 John Wiley & Sons, Ltd.

T-stress, mixed-mode stress intensity factors, and crack initiation angles in functionally graded materials: a unified approach using the interaction integral method

For linear elastic functionally graded materials (FGMs), the fracture parameters describing the crack tip fields include not only stress intensity factors (SIFs) but also T-stress (nonsingular stress). These two fracture parameters are important for determining the crack initiation angle under mixed-mode loading conditions in brittle FGMs (e.g. ceramic/ceramic such as TiC/SiC). In this paper, the mixed-mode SIFs and T-stress are evaluated by means of the interaction integral, in the form of an equivalent domain integral, in combination with the finite element method. In order to predict the crack initiation angle in brittle FGMs, this paper makes use of a fracture criterion which incorporates the T-stress effect. This type of criterion involves the mixed-mode SIFs, the T-stress, and a physical length scale r c (representative of the fracture process zone size). Various types of material gradations are considered such as continuum models (e.g. exponentially graded material) and micromechanics models (e.g. self-consistent model). Several examples are given to show the accuracy and efficiency of the interaction integral scheme for evaluating mixed-mode SIFs, T-stress, and crack initiation angle. The techniques developed provide a basic framework for quasi-static crack propagation in FGMs.

Pseudodomain Fracture Analysis of Instrumented Analog Rocket Motors

An advanced stress analysis technique has been developed for evaluating propellant fracture in solid-rocket motors subjected to thermal cooling. When a strain-to-pseudostrain transformation is performed, it is shown that the time- and temperature-dependent relaxation of the solid propellant can be taken into account to yield a nonlinear elastic material model. This material model can be used in a finite element analysis of a solid-rocket motor grain to quantify the driving force for quasi-static crack propagation under a variety of thermal loads by using elastic-plastic fracture mechanics. The analysis technique was assessed by subjecting analog rocket motors instrumented with miniature bond-stress sensors to a sequence of quasi-isothermal temperatures. The data measured by the sensors allowed the stress predictions to be verified and the crack length and fracture resistance of the propellant to be deduced. It is hypothesized that compatibility must be maintained between the crack tip constraint conditions in the fracture specimen used to characterize a propellant and the crack tip constraint conditions in the structure being analyzed to use this analysis technique in a predictive manner.

Three-dimensional numerical investigation of brittle bond fracture

A fracture mechanics based analysis of interface bond failure is presented. The bond edge is regarded as an interface crack front loaded under combined mode 1, 2 and 3 loading, and results are obtained for the critical stress for initiation of bond failure and the location along the bond edge where failure is initiated. A numerical procedure is formulated to study the propagation of the interface crack following initiation. Assuming that the crack propagates at the interface, a criterion for propagation is formulated, and it is shown that the crack front shape predicted is consistent with the basic interface fracture mechanics assuming quasi-static crack propagation. Results for the bond strength are presented for different fracture criteria and different bond shapes.

Scaling of fracture length and distributed damage

The linear theory of elasticity formulated in terms of dimensionless strain components does not allow the introduction of any space scaling except linear relations between fracture length and displacements and thus the determination theoretically of the strength of a body or structure directly. Self-similarity of a fracture process means the existence of a universal faulting mechanism. However, the general applicability of universal scaling to field observations and rock mechanics measurements remains the subject of some debate. Complete self-similarity of a fracture process is hardly ever found experimentally, except in some aluminium alloys. At early stages of the loading, material degrades due to increasing microcrack concentrations. Later, these microcracks where distributed in the process zone localize into a subcritically growing macrocrack, and finally the fracture process accelerates and rupture runs away, producing dynamic fracture. The macroscopic effects of distributed cracking and other types of damage require treatment by constitutive models that include non-linear stress–strain relations together with material degradation and recovery. The present model treats two physical aspects of the brittle rock behaviour: (1) a mechanical aspect, that is, the sensitivity of the macroscopic elastic moduli to distributed cracks and to the type of loading, and (2) a kinetic aspect, that is, damage evolution (degradation/recovery of elasticity) in response to ongoing deformation. To analyse the scaling of a fracture process and the onset of the dynamic events, we present here the results of numerical modelling of mode I crack growth. It is shown that the distributed damage and the process zone created eliminate the stress–strain crack-tip singularities, providing a finite rate of quasi-static crack growth. The growth rate of these cracks fits well the experimentally observed power law, with the subcritical crack index depending on the ratio between the driving force and the confining pressure. The geometry of the process zone around a quasi-static crack has a self-similar shape identical to that predicted by universal scaling of the linear fracture mechanics. At a certain stage, controlled by dynamic weakening and approximated by the reduction of the critical damage level proportional to the rate of a damage increase, the self-similarity breaks down and crack velocity significantly deviates from that predicted by the quasi-static regime. The subcritical crack growth index increases steeply, crack growth accelerates, the size of the process zone decreases, and the rate of crack growth ceases to be controlled by the rate of damage increase. Furthermore, the crack speed approaches that predicted by the elastodynamic equation. The model presented describes transition from quasi-static crack propagation to the dynamic regime and gives proper time and length scales for the onset of the catastrophic dynamic process.

Thermally induced quasi-static wavy crack propagation in a brittle solid

Morphological aspects of thermally induced brittle cracks are revisited. The formation of a growth pattern of cracks, which are subjected to the thermal loading due to the penetration of surface cooling, has formerly been studied for a system of parallel interacting straight cracks. In the present paper, attention is focused on the transition from straight to wavy crack propagation. The incremental extension of a curved crack is predicted by using the local symmetry criterion, where the step-by-step stress analyses are carried out by the finite-element method with the aid of an automatic mesh generation which is based on the paving method. Varieties of numerical results are obtained so that the influencing factor which may control the wavy crack paths is identified. Discussions are also made by comparing the numerical results with experimental observation.

Quasi-Static Propagation of Subinterfacial Cracks

The problem of quasi-static crack propagation in a three-point bend specimen containing an initial crack that is parallel to and offset from a bimaterial interface is considered. An approximate dislocation-based fracture model is used to identify conditions under which such cracks are attracted to or repelled by the interface. Possible configurations (material/geometry) where such subinterfacial cracks experience pure mode I conditions are determined. Experimental results are presented showing quasi-static crack propagation of subinterfacial cracks for three regimes: (attractive) into the interface; (repulsive) away from the interface; as well as (equilibrium) parallel to the interface. [S0021-8936(00)01903-6]

Instabilities of quasi-static crack patterns in quenched glass plates

We have produced controlled quasi-static crack propagation in thin glass plates to study crack instabilities. Quasi-static fractures can exhibit drastic changes in morphology when the control parameter is varied. The thermal stress and the velocity of the crack tip depend on the temperature difference between two heat reservoirs (Δ T) and the speed of descent of the sample ( V). For large V and Δ T, cracks went from straight to oscillating and then to branched in a phase diagram for these three patterns. The transition from straight to oscillating is a super-critical Hopf bifurcation. Approaching the transition line from below in the phase diagram, the relaxation time of the damped oscillating crack scaled with Δ T and V. Above the transition line, amplitude of oscillations scaled with Δ T and V. The wavelength λ of the crack oscillation was measured at the onset of oscillation. Introducing two non-dimensional parameters R and μ, we successfully compared the conditions for the onset of straight crack propagation and oscillation with theory and numerical simulation. Further above the onset of oscillation we observed period doubling.

Quasistatic Crack Propagation in Heterogeneous Media

The dynamics of a single crack moving through a heterogeneous medium is studied in the quasi-static approximation. Equations of motion for the crack front are formulated and the resulting scaling behaviour analyzed. In a model scalar system and for mode III (tearing) cracks, the crack surface is found to be self affine with a roughness exponent of $\zeta=1/2$. But in the usual experimental case of mode I (tensile) cracks, local mode preference causes the crack surface to be only logarithmically rough, quite unlike those seen in experiments. The effects of residual stresses are considered and found, potentially, to lead to increased crack surface roughness. But it appears likely that elastic wave propagation effects may be needed to explain the very rough crack surfaces observed experimentally.

Theory of brittleness in stress softening models of fracture mechanics

A differential equation for quasi static crack propagation is presented and discussed analytically. The result of the discussion is used to develop a first order theory of brittleness in stress softening models of fracture mechanics. The theory is valid for materials, such as concrete, with strength in compression much higher than strength in tension. The problem is whether a given crack propagation process is ductile with gradual loss of bearing capacity or a sudden rupture takes place at ultimate load without any warning in advance in the form of great displacements and open visible cracks. The idea is to separate the effect of plastic deformation from the effect of crack propagation on the elastic compliance of the structure. When the length of the initial crack is small compared to other length in the plane of the structure, the theory gives almost exact solutions. In the limiting case when linear elastic fracture mechanics is applied as the model of fracture mechanics, the theory gives exact solutions. At the other limit when the process zone is so large that the open crack does not propagate the theory of brittleness does not apply.

Fracture mechanics analysis of cracked 2-D anisotropic media with a new formulation of the boundary element method

A new formulation of the boundary element method (BEM) is proposed in this paper to calculate stress intensity factors for cracked 2-D anisotropic materials. The most outstanding feature of this new approach is that the displacement and traction integral equations are collocated on the outside boundary of the problem (no-crack boundary) only and on one side of the crack surfaces only, respectively. Since the new BEM formulation uses displacements or tractions as unknowns on the outside boundary and displacement differences as unknowns on the crack surfaces, the formulation combines the best attributes of the traditional displacement BEM as well as the displacement discontinuity method (DDM). Compared with the recently proposed dual BEM, the present approach doesn't require dua elements and nodes on the crack surfaces, and further, it can be used for anisotropic media with cracks of any geometric shapes. Numerical examples of calculation of stress intensity factors were conducted, and excellent agreement with previously published results was obtained. The authors believe that the new BEM formulation presented in this paper will provide an alternative and yet efficient numerical technique for the study of cracked 2-D anisotropic media, and for the simulation of quasi-static crack propagation.

Crack instabilities of a heated glass strip.

Recently, Yuse and Sano [Nature (London) 362, 329 (1993)] have observed that a crack traveling in a glass strip submitted to a nonuniform thermal diffusion field undergoes numerous instabilities. We study two cases of quasistatic crack propagation. The crack extension condition in straight propagation is determined. An asymptotic analysis of the elastic free energy is introduced and scaling laws are derived. A linear stability analysis of the straight propagation is performed, based on the assumption that the crack tip propagation deviates from the centered straight one as soon as it is submitted to a ``physical'' singular shear stress. It is shown that a straight propagation can become unstable after which a wavy instability appears. The condition for instability as well as the selected wavelength is calculated quantitatively. The results are compared with experiments and the agreement is favorable.

Fatigue behaviour of glass bead filled epoxy

There is a relatively abundant literature on the mechanical properties of particle filled thermosets. Detailed experimental data are available on the effect of variables, such as the filler volume fraction, its surface treatment or shape factor, on the usual properties. In the case of epoxy matrix composites, data have been published on elastic properties. Kinetic studies on thermoplastics, as well as microscopic investigations clearly show that each particle acts as a crack initiation site. The present study deals with thermoset epoxy-glass bead composites. A noticeable advantage of the sphericity of the glass beads over the previously studied mineral fillers is that theoretical calculations, for instance of interparticle average distance, are easier. Some results are reported concerning the eventual role of the geometrical characteristics, including particle diameter, number of particles per volume unit, particle-matrix contact area, interparticle distance, on the fatigue characteristics of the composite as assessed from Paris or Wohler plots. In addition, quasi-static crack propagation characteristics will be compared with dynamic ones. It is clearly shown that glass beads improved the fatigue crack propagation. Despite this fact, it is also shown that even a small amount of mineral filler, acting as crack initiator, can considerably reduce the fatigue life of epoxy composites.

Shear induced transformation toughening in ceramics

There is experimental evidence that the tetragonal to monoclinic transformation in zirconia (ZrO 2) can be triggered not only by hydrostatic pressure but also by shear stresses. In this paper we consider the implications of a transformation criterion which depends on both mean stress and shear stress on transformation toughening in ceramics reinforced with zirconia inclusions. We treat zirconia toughened ceramics (ZTC) as dilute composites of randomly oriented inclusions in an elastic matrix and develop the constitutive model for the composite from the behaviour of single inclusions by orientation averaging. We thenstudy quasistatic crack propagation and predict the transformation zone, distribution of transformed inclusions and toughness enhancement.

Energy-release rates and path-independent integrals in electroelastic crack propagation

Two energy-release rates due to quasistatic crack propagation in electroelastic materials are obtained by using a Griffith-type energy approach. Their difference represents the free electric field contribution to the propagation of the crack. Appropriate conditions for mechanical and electrical fields allow us to express the energy-release as path independent integrals, already obtained by Pak and Herrmann, and by Maugin and Epstein. The assumptions are verified on the solution of an antiplane crack problem and the integrals are evaluated in terms of electrical and mechanical stress intensity factors.

A self-consistent description of ruptures in an elastic medium: an application to earthquakes

We present a general self-consistent procedure using lattice Green's functions to find the stress redistribution due to multiple ruptures in a discrete elastic medium. We use this method to study a dipole version of the quasi-static crack-propagation model for earthquakes proposed by Chen et al. (1991) Unlike the earlier treatment we do not assume that ruptures occur independently. We obtain the stresses self-consistently and this changes the scaling behaviours of the frequency-size (seismic moment, energy) distribution. We make a careful comparison of our results with recent analysis of seismological data.