Steady-state Crack Growth Research Articles

A comparison of fatigue crack growth in human enamel and hydroxyapatite

Cracks and craze lines are often observed in the enamel of human teeth, but they rarely cause tooth fracture. The present study evaluates fatigue crack growth in human enamel, and compares that to the fatigue response of sintered hydroxyapatite (HAp) with similar crystallinity, chemistry and density. Miniature inset compact tension (CT) specimens were prepared that embodied a small piece of enamel ( N = 8) or HAp ( N = 6). The specimens were subjected to mode I cyclic loads and the steady state crack growth responses were modeled using the Paris Law. Results showed that the fatigue crack growth exponent ( m) for enamel ( m = 7.7 ± 1.0) was similar to that for HAp ( m = 7.9 ± 1.4), whereas the crack growth coefficient ( C) for enamel ( C = 8.7E−04 (mm/cycle) × (MPa m 0.5) − m ) was significantly lower ( p < 0.0001) than that for HAp ( C = 2.0E+00 (mm/cycle) × (MPa m 0.5) − m ). Micrographs of the fracture surfaces showed that crack growth in the enamel occurred primarily along the prism boundaries. In regions of decussation, the microstructure promoted microcracking, crack bridging, crack deflection and crack bifurcation. Working in concert, these mechanisms increased the crack growth resistance and resulted in a sensitivity to crack growth ( m) similar to bone and lower than that of human dentin. These mechanisms of toughening were not observed in the crack growth response of the sintered HAp. While enamel is the most highly mineralized tissue of the human body, the microstructural arrangement of the prisms promotes exceptional resistance to crack growth.

C∗-controlled creep crack growth by grain boundary cavitation

Quasi steady-state creep crack growth is widely associated with the nucleation and growth of voids on grain boundaries ahead of the crack tip. In this paper, a micromechanics-based constitutive law is used to study the velocity-dependent fracture toughness of porous solids under extensive creep conditions. Void growth and coalescence in the fracture process zone is modeled by a nonlinear viscous microporous strip of cell elements. Under steady-state crack growth, two dissipative processes contribute to the macroscopic fracture toughness: the work of separation in the fracture process zone, and creep dissipation in the background material. Under extensive creep conditions, the competition between these two processes produces an inverted U-shaped C ∗–velocity curve. The effects of rate sensitivity, initial porosity as well as hydrogen attack on fracture toughness are studied. The numerically simulated fracture toughness vs. crack velocity curves show good agreement with existing experimental results.

J-integral and crack driving force in elastic–plastic materials

This paper discusses the crack driving force in elastic–plastic materials, with particular emphasis on incremental plasticity. Using the configurational forces approach we identify a “plasticity influence term” that describes crack tip shielding or anti-shielding due to plastic deformation in the body. Standard constitutive models for finite strain as well as small strain incremental plasticity are used to obtain explicit expressions for the plasticity influence term in a two-dimensional setting. The total dissipation in the body is related to the near-tip and far-field J-integrals and the plasticity influence term. In the special case of deformation plasticity the plasticity influence term vanishes identically whereas for rigid plasticity and elastic-ideal plasticity the crack driving force vanishes. For steady state crack growth in incremental elastic–plastic materials, the plasticity influence term is equal to the negative of the plastic work per unit crack extension and the total dissipation in the body due to crack propagation and plastic deformation is determined by the far-field J-integral. For non-steady state crack growth, the plasticity influence term can be evaluated by post-processing after a conventional finite element stress analysis. Theory and computations are applied to a stationary crack in a C(T)-specimen to examine the effects of contained, uncontained and general yielding. A novel method is proposed for evaluating J-integrals under incremental plasticity conditions through the configurational body force. The incremental plasticity near-tip and far-field J-integrals are compared to conventional deformational plasticity and experimental J-integrals.

Creep fracture toughness using conventional and cell element approaches

The mechanical response of a polymeric material is loading rate sensitive especially near the glass transition temperature. In this work, the polymeric material is modeled as an elastic-nonlinear viscous solid. A computational scheme based on the finite element method is used to simulate steady-state crack growth in the polymeric material under plane strain mode I, small-scale yielding conditions. The scheme is validated by reproducing the Hui–Riedel singularity field around a steadily growing crack in the elastic-nonlinear viscous solid. It is observed that the Hui–Riedel singularity has a limited range of validity. Thereafter, two fracture approaches are employed to study viscoelastic creep crack growth. One is the conventional critical strain over critical distance ahead of the crack. The other is the cell element approach for crack growth caused by void growth and coalescence, in which the rate-dependent fracture process zone is represented by a nonlinear viscous microporous strip of cell elements. These cell elements are described by a Gurson-like micromechanics model recently proposed for void growth in nonlinear viscous matrix. The computed toughness-velocity curves using strain criterion exhibit mesh dependence while those obtained by cell element approach appear to be robust.

Mode mixity and nonlinear viscous effects on toughness of interfaces

This paper examines steady-state crack growth at interfaces between polymeric materials and hard substrates under quasi-static conditions. The polymeric material is taken to be an elastic nonlinear viscous solid while the substrate is treated as a rigid material. Void growth and coalescence in the rate-dependent fracture process zone is modeled by a nonlinear viscous porous strip of cell elements. In the first part of this paper, the polymeric background material surrounding the process zone is assumed to be purely elastic. Under fixed mode mixity, the computed interface toughness is found to be a monotonically increasing function of crack velocity; toughness also increases rapidly with higher rate sensitivity. This behavior can be explained in terms of voids growing in a strain-rate strengthened process zone. In the second part of the paper, the background material is also treated as an elastic nonlinear viscous solid. The competition between work of separation in the process zone and energy dissipation in the background material leads to a U-shaped toughness–crack velocity curve. Effects of mode mixity, initial porosity, rate sensitivity, as well as the initial yield strain on toughness are studied. The simulations produce trends that agree with interface toughness vs. crack velocity data reported in experimental studies for rubber toughened epoxy-paste adhesive and urethane acrylate adhesive.

The dependency of the crack growth rate on the loading pattern and temperature in stress corrosion cracking of strain-hardened 316L stainless steels in a simulated BWR environment

The stress corrosion cracking (SCC) growth rate of a warm-rolled (WR) 316L stainless steel contoured double cantilever (CDCB) specimen was measured in high purity water at various temperatures and under various loading patterns. An alternating current potential drop (ACPD) technique was used to monitor the crack growth kinetics throughout the tests. The fracture surface exhibited typical intergranular SCC characteristics. Depending on the test conditions, three kinds of crack growth kinetics, i.e., increasing with time then becoming steady, being constant during the whole period, decreasing with time then becoming steady, were identified and are described. The steady state crack growth rate (CGR) values are used to quantify the effects of the loading pattern and the environmental temperature. A moderate increase in the crack growth rate was encountered by employing periods of unloading and reloading to form a trapezoidal loading pattern and the enhancement factor was found to depend on the holding time and the times for unloading and reloading. It was found that the crack growth is thermally activated; however, the apparent activation energy is not constant but seems to be greater at higher temperatures. Several types of temperature-dependent crack growth kinetics are proposed based on the rate-determining step for the crack growth. The present experimental results can be rationalized by considering multiple element processes such as aqueous mass transport and solid-state mass transport in the crack growth. The cracking mode, the temperature dependence of the crack growth rate, and the transient crack growth behavior for WR 316L SS after changing the environmental temperature are quite similar to those for a cold-worked(CW) 316L SS tested in the same environment, despite their different absolute crack growth rate values. The effect of yield strength on CGR is more significant at lower temperatures and the apparent activation energy for the crack growth rate seems to be lower in the material with a higher yield strength. Time-lag crack growth behavior was found at points during several test steps on WR 316L SS, for example, just after in situ pre-cracking and after increasing or decreasing the temperature, which is quite consistent with the results obtained with CW 316L SS. The importance of in situ monitoring of the crack growth for obtaining steady state crack growth rates is emphasized, especially for those steps for which a nonlinear crack growth period occurs after changing the test condition.

Transient and steady state crack growth kinetics for stress corrosion cracking of a cold worked 316L stainless steel in oxygenated pure water at different temperatures

The stress corrosion cracking (SCC) growth kinetics for a cold worked 316L stainless steel was continuously monitored in high purity water at different temperatures and dissolved oxygen (DO) levels under a K (or K max) of 30 MPa m 0.5. The total SCC test time was more than 8000 h to make sure the steady state crack growth rate under each test condition could be reached. Crack growth rate (CGR) increases with increasing temperature in the range 110–288 °C. A typical intergranular-cracking mode is identified. Depending on the previous test condition, especially the temperature, three kinds of crack growth kinetics, i.e., increasing with testing time then becoming steady, being constant during the whole period, or decreasing with test time then becoming steady, are identified and discussed. Time-dependent and testing history-dependent crack growth modes were confirmed in two series of tests in 2 ppm DO and 7.5 ppm DO pure water. The apparent activation energies are calculated and compared with other data in different environments under different applied loading levels for understanding the cracking mechanism.

Rate effects on toughness in elastic nonlinear viscous solids

A micromechanics-based constitutive relation for void growth in a nonlinear viscous solid is proposed to study rate effects on fracture toughness. This relation is incorporated into a microporous strip of cell elements embedded in a computational model for crack growth. The microporous strip is surrounded by an elastic nonlinear viscous solid referred to as the background material. Under steady-state crack growth, two dissipative processes contribute to the macroscopic fracture toughness—the work of separation in the strip of cell elements and energy dissipation by inelastic deformation in the background material. As the crack velocity increases, voids grow in the strain-rate strengthened microporous strip, thereby elevating the work of separation. In contrast, the energy dissipation in the background material decreases as the crack velocity increases. In the regime where the work of separation dominates energy dissipation, toughness increases with crack velocity. In the regime where energy dissipation is dominant, toughness decreases with crack velocity. Computational simulations show that the two regimes can exist in certain range of crack velocities for a given material. The existence of these regimes is greatly influenced by the rate dependence of the void growth mechanism (and the initial void size) as well as that of the bulk material. This competition between the two dissipative processes produces a U-shaped toughness–crack velocity curve. Our computational simulations predict trends that agree with fracture toughness vs. crack velocity data reported in several experimental studies for glassy polymers and rubber-modified epoxies.

Mechanics of Matrix Cracking in Bonded Composites

AbstractIn this paper, a general matrix cracking model including the effect of fiber/matrix debonding in the crack wake is developed for a unidirectional fiber reinforced composite. The debonding mechanics is incorporated into matrix cracking model by treating the crack-wake debonding as a particular crack propagation problem along the interface. Then, the closed-form analytical solution of the critical stress for the onset of widespread matrix cracking is derived, based on the analysis of steady state crack growth in the matrix. The fracture mechanics approach adopted in the present analysis is compared with the analysis in which the crack-wake debonding mechanics was modeled by energy balance approach. The conditions for attaining no-debonding and debonding as onset of widespread matrix cracking are discussed in terms of the interfacial properties of debond toughness and frictional shear stress. The theoretical results are compared with experimental data that are available in the literature.

Analysis of in-plane transonically propagating interface crack with a finite contact zone

The report of Lambros and Rosakis [(1995) J Mech Phys Solids 43(2): 169–188] has focused attention on steady-state transonic interfacial crack growth accounting for the phenomenon of crack face contact in elastic/rigid bimaterial but could not handle issues relating to energy transmission across the interface. The present paper attempts to provide a complete explicit expression of the asymptotic fields induced by transonically propagating interfacial crack in elastic/elastic bimaterial for in-plane case. The energy distribution on the contact area, crack tip and two singular characteristic lines is analysed thoroughly and compared with the dynamic separated J-integrals. The length of the contact zone is also discussed briefly by establishing energy fracture criterion that satisfies contact condition. The two-dimensional in-plane asymptotic deformation field surrounding the contact area of a crack propagating transonically along an elastic/elastic bimaterial interface is observed and discussed thoroughly.

Toughening due to domain switching in single crystal ferroelectric materials

In this paper Mode I steady state crack growth in single crystal ferroelectric materials is investigated. Specifically, the fracture toughness enhancement due to domain switching near a steadily growing crack tip is analyzed. For this purpose, an incremental phenomenological constitutive law for single crystal ferroelectric materials is implemented within a finite element model to calculate the stress and remanent strain fields around the crack tip. Also, the ratio of the far field applied energy release rate to the crack tip energy release rate, i.e. the toughening, is calculated. The numerical computations are carried out for single crystal ferroelectric materials of tetragonal or rhombohedral structure with different switching hardening and irreversible remanent strain levels. Toughening levels for crack growth along different crystallographic directions and planes are obtained and compared. Results from numerical computations for the toughening anisotropy for both tetragonal and rhombohedral crystals are presented and discussed.

Mechanism of electric fatigue crack growth in lead zirconate titanate

A series of experiments was performed with through-thickness cracks in ferroelectric double cantilever beam (DCB) specimens. Cyclic electric fields of different amplitudes were applied which resulted in cyclic crack propagation perpendicular to the electric field direction. Crack propagation was observed optically and three regimes were identified: a pop-in from a notch, steady-state crack growth and a decrease of the crack growth rate with increasing cycle number. Crack growth only occurred if the applied field exceeded the coercive field strength of the material. Furthermore, the crack extended during each field reversal and the crack growth rate increased with increasing field. Based on the experimental observations, a mechanistic understanding was developed and contrasted with a nonlinear finite element analysis which quantified the stress intensity in the DCB specimens. The driving forces for crack formation at the notch and subsequent fatigue crack growth were computed based on the distribution of residual stresses due to ferroelectric switching. The finite element results are in good agreement with the experimental observations and support the proposed mechanism.

Effects of in-plane electric fields on the toughening behavior of ferroelectric ceramics

Mode-I steady state crack growth in poled ferroelectric ceramics subjected to simultaneous electrical and mechanical loading is analyzed to investigate the effect of in-plane electric fields and polarization on the toughening behavior. A multiaxial, incremental constitutive law for domain switching is implemented within the finite element method to obtain the electromechanical crack tip fields. Simulation results are presented for the cases of initial remanent polarization states and applied electric fields perpendicular to the crack plane and parallel to the crack growth direction. Specific results from the calculations include the shapes and sizes of switching zones, and the toughening effects due to domain switching near the crack tip. Ferroelectric ceramics have been widely used in smart structure applications due to their large electromechanical coupling effects. Since ferroelectricdevices oftenoperate understrong mechanicaland electrical loading conditions, the brittle ferroelectric ceramic material is susceptible to fracture. Therefore, an understanding of ferroelectric fracture is a key issue for the efficient and reliable design of these devices. This paper is concerned with the study of the Mode-I fracture behavior of ferroelectric ceramics under combined in-plane electrical and mechanical loading. The scenario investigated here is illustrated in Figure 1. A ferroelectric material is initially poled by an electric field either perpendicular or parallel to the direction of crack growth. After the initial poling, an electric field is applied along (positive electric field) or opposite (negative electric field) to the initial poling direction. Finally, mechanical loading is applied and crack growth occurs. Experimental observations of the fracture properties of ferroelectrics under such conditions have been obtained on several materials from indentation tests and compact tension specimens. Using indentation tests on a PZT-8 material composition, Tobin and Pak [1993] showed that for cracks perpendicular to the poling direction, the apparent fracture toughness decreases with positive electric field and increases with a negative field. For cracks parallel to the poling direction, their results indicated that both positive and negative electric fields have little influence on the toughening. Tobin and Pak [1993] also observed that with no applied electric field the fracture toughness was greater for cracks parallel to the poling direction than for cracks perpendicular to the poling direction. Park and Sun [1995] investigated electric field effects on crack growth in a PZT-4 ceramic by using conventional compact tension fracture tests. Their results agree with those of Tobin and Pak [1993] for the case of electric field applied perpendicular to the cracksurface. However, theVickersindentationtestsofWangandSingh[1997]showedthatiftheapplied electricfields are perpendicular to the crack in a PZT EC-65 ceramic, then a positive electricfield impedes

Numerical study of geometric constraint and cohesive parameters in steady-state viscoelastic crack growth

This paper presents a finite element study of cohesive crack growth in a thin infinite viscoelastic strip to investigate the effects of viscoelastic properties, strip height, and cohesive model parameters on the crack growth resistance. The results of the study show that the dependence of the fracture energy on the viscoelastic properties for the strip problem is similar to that obtained for the infinite body problem even when the cohesive zone length is large compared to the height of the strip. The fracture energy also depends on the crack speed v through the dimensionless parameter v τ/L ∞ where L ∞ is the characteristic length of the cohesive zone and τ is the characteristic relaxation time of the bulk material. This relationship confirms that at least two properties of the fracture process must be prescribed accurately to model viscoelastic crack growth. In contrast, the fracture energy and crack speed are insensitive to the strip height even in situations where the growth of the dissipation zone is severely constrained by the strip boundaries. We observe that at high speeds, where the fracture energy asymptotically approaches the maximum value, the material surrounding the cohesive zone is in the rubbery (equilibrium) state and not the glassy state.

Theoretical and numerical modelling of creep crack growth in a carbon–manganese steel

This paper presents an analytical and numerical study of time dependent crack growth at elevated temperatures. A triaxiality dependent damage model is used to represent the multiaxial creep ductility of the material and an analytical model to predict steady state crack growth in terms of the fracture parameter C ∗, designated the NSW-MOD model, is presented. This model is an enhancement of the earlier NSW model for creep crack growth as it accounts for the dependence of stress and strain on angular position around the crack tip. Elastic–creep and elastic–plastic–creep finite element analyses are performed for a cracked compact tension specimen and the crack propagation rate in the specimen is predicted. It is found that in general the NSW-MOD model gives an accurate estimate of the crack growth rate when compared to the finite element predictions and experimental data for a carbon–manganese steel. However, crack growth rates predicted from the finite element analysis at low values of C ∗ may be higher than those predicted by either the NSW or NSW-MOD model. This enhanced level of crack growth may be associated with the non-steady state conditions experienced at the crack tip.

Fatigue damage in bulk metallic glass II: Experiments

Cyclic fatigue was investigated in metallic glasses using fracture mechanics-based crack-growth and stress-life experiments. Steady-state crack growth in a Zr-based bulk metallic glass occurred with a Paris law exponent of ∼1.5. Growing planar fatigue cracks were often noted to destabilize into a wavy crack front geometry, and the conditions determining the stability of these cracks are analyzed. Transient crack-growth rates were measured during variable-amplitude fatigue. Stress-life experiments showed low endurance limits. Fatigue damage initiated at an angle to the applied load, but changed direction at a critical length and grew as mode I cracks. However, the initiation stage of fatigue damage was very short, and almost the entire fatigue life was spent in the crack-growth stage.

Resistance Against the Intrinsic Rate of Fracture Mechanics Parameters for Polymeric Materials Under Moderate Impact Loading

This study contributes towards understanding crack toughness as resistance against the intrinsic rate of fracture mechanics parameters. Up to now only few investigations have been done under moderate impact loading conditions. Based on experimental investigations using the crack resistance (R) concept, it has been shown that the stop block method combined with the multiple-specimen technique is a unique method for polymers under impact loading conditions in comparison with different R-curve methods. Other methods for the determination of R curve such as the low-blow technique are normally not applicable for polymers due to their time-dependent mechanical properties. The crack-tip opening displacement (CTOD) rate is a measurement of the rate sensibility of stable fracture process depending on the type of deformation, which can provide deep insights into the micromechanics and activation mechanisms during the fracture processes. In the polymeric materials mostly investigated, one can understand the stable crack propagation with three-stage processes; crack-tip blunting/crack initiation, non-stationary stable crack growth and steady-state stable crack growth (an equilibrium state). In this stable crack propagation, the values of normalized CTOD rate converge rapidly to a ‘matrix’-specific threshold. The stop block method in the multiple-specimen technique assures the criteria of the time-independent strain field around the crack tip and constant crack speed therewith and the J-integral is a valid toughness parameter.

Simplified analysis of steady-state crack growth of piezoelectric ceramics based on the continuum damage mechanics

A simplified analysis of crack growth of piezoelectric ceramics was performed by using the double cantilever beam (DCB) model. The analysis was intended for the crack growth in a steady state by using some assumptions to simplify the analysis. In the present paper, damage within piezoelectric ceramics was represented by a damage variable based on the continuum damage mechanics, and the effect of the damage on material properties was taken into account in a constitutive equation of the piezoelectric ceramics. The constitutive equation and an evolution equation of the damage variable were applied to the DCB model. Then, the crack growth in the DCB model was dealt with within the framework of the continuum damage mechanics. As a result of the analysis, a constant crack growth rate and distribution of stress, strain and damage variable in front of the crack tip were elucidated, and the effects of the applied electric field on the crack growth were discussed in particular.

Experimental determination of cohesive failure properties of a photodegradable copolymer

In this paper we present a methodology to measure the material traction-separation relation for a poly(ethylene carbon monoxide) copolymer, ECO. This material exhibits a ductile-to-brittle transition when subjected to ultraviolet irradiation and undergoes a change of failure mechanism, from shear yielding to crazing, in the process. Single edge notched tension specimens of ECO irradiated for 50 h were used to generate slow-speed stable crack growth, predominantly from material crazing. Full-field measurements of the in-plane deformation around the growing crack tip were performed using the optical technique of digital image correlation. A multicamera setup was used in which simultaneous measurement of both the far-field displacement and that directly surrounding the craze was performed. The far-field results were used to obtain a value of the energy release rate supplied to the crack tip region. The near-tip results were used to extract a material traction-separation law in the regime of steady-state crack growth. A softening traction-separation relation was measured. The area under the traction-separation curve was then compared with the simultaneous far-field measurements and the agreement was very good (within 6.5%) validating the experimental methodology used.

Simulation of hydride-induced steady-state crack growth in metals ? Part II: general near tip field

Hydride-induced steady-state crack propagation in metals is investigated under conditions of constant temperature, plane strain, small-scale yielding and small-scale hydride precipitation, by taking into account the coupling of the operating physical processes. It is shown that the near-tip field depends on a normalized stress intensity factor, which incorporates both effects of the applied stress intensity factor and the crack velocity. According to Part I of the present study, when the normalized stress intensity factor tends to zero, the crack-tip field near the threshold stress intensity factor is produced, which is characterized by a constant hydrostatic stress in the hydride precipitation zone. As the value of the normalized stress intensity factor increases, the evolution of the near-tip field for crack propagation from stage-I to stage-II regime is produced: the actual size of the hydride precipitation zone decreases, the hydrostatic stress increases, deviating from the level of the plateau, and the near-tip field tends to that of a hydrogen-free metal. The near-tip field depends strongly on hydrogen concentration, far from the crack tip. The stage-II crack growth velocity is predicted and the experimentally observed effect of metal yield stress and temperature on crack velocity is confirmed.