X-ray diffraction studies on fatigue crack plastic zones developed under plane strain state conditions

An overview of the X-ray fractography technique, as performed on fatigue crack surfaces of several steels and Al-alloys under different loading conditions, is presented. The plastic zone sizes of fatigue cracks, for plane strain conditions, are measured from the in-depth distribution of residual stresses and X-ray diffraction peak broadening. In addition to the usual monotonic plastic zone size determination methodology, a model for the estimation of the reverse plastic zone size was established in the case of fatigue softening materials. Monotonic and cyclic plastic zone sizes are related to the stress intensity by, respectively, r_{pm = α (Kmax} /σ_ys )² and r_{pc = α} (ΔK/2σ′ys )2. The α-value, in the monotonic plastic zone size equation, increases as the yield strength of the material increases, following the relationship α = 0.196 [σ_ys /(129 + 0.928σ_ys )]2. The α-value versus σ_ys evolution has been understood through the influence of the hardening rate of materials on the plastic zone size. X-ray fractography has been applied to actual failure analyses to predict some aspects of the actual loadings.     

Monotonic and cyclic crack tip plasticity

Monotonic and cyclic plastic zone sizes were measured in a medium strength ferrite-pearlite steel (BM 45) tested in fatigue at 25 Hz at room temperature. Two methods were applied: microhardness and the recently developed ‘fatigue in compression’ technique. The results obtained are discussed in terms of accuracy and reliability.The retardation effect due to overloads was also studied in the same material and is illustrated experimentally as a $\text{d}\text{a}\text{d}\text{N}$ vs ΔK curve. This effect emphasizes the importance of an accurate evaluation of both the size and shape of the overall plastic zone. The shape and dimensions of the cyclic plastic zones seem to indicate that in ductile metals the steady state of fatigue crack growth occurs under plane strain conditions.     

The prediction of KIc from tensile data

A theory of fracture toughness is developed which is based on the Dugdale crack model. Work hardening is considered. By using the stress-strain curves as input data, $\text{K}{}_{\mathrm{Ic}}$ values are predicted for a wide range of materials including uranium alloys, beryllium, aluminum, a nickel-cobalt alloy, and a titanium alloy. The calculated values agree exceptionally well with the experimental values. Plastic-zone size (including a qualitative estimate of shape) and stress and strain in the plastic zone are also predicted.     

The plastic zone and growth of fatigue cracks in magnesium MA12 alloy at room and low temperatures

The plastic zone formed at the fatigue crack tip and the fracture topography in MA12 magnesium alloy samples, tested at 293 and 140 K in air and in vacuum, were analysed. It was found that the plastic zone formed in vacuum is characterized by a greater size ($\text{h}$) and degree of plastic strain that in air, and the crack growth rate ($\text{d}\text{l/}\text{d}\text{N}$) is lower. Temperature reduction leads to a decrease in $\text{h}$, while $\text{d}\text{l/}\text{d}\text{N}$ and the fracture mechanism are affected by temperature ambiguously, depending on the alloy microstructure and the $\text{K}{}_{\mathrm{<mtext>max</mtext>}}$ value. It was established that the size of the plastic zone can be described by the equation:

$\text{h=A}\text{(}\text{K}{}_{\max }\text{σ}{}_{\mathrm{0.2ps}}\text{)}{}^2$

where $\text{A}$ is a coefficient dependent on the alloy structural state, environment and test temperature. Evaluation of the cyclic plastic zone size at $\text{K}{}_{\mathrm{<mtext>max</mtext>}}$, corresponding to the transition from a low temperature region to a ‘Paris’ region, showed that this transition occurred when the cyclic plastic zon reached the structural parameter of the material.     

The form of crack tip plastic zones

The “radius” of the plastic zone at a crack tip is a parameter that has numerous applications in fracture mechanics. However, attention is drawn here to the confusion that is apparent, even in text-books, concerning the calculation of the plastic zone “radius” under plane strain conditions. The aim of this work has been to resolve this point, to determine the actual shape and size of the zone and to investigate the influence of stress state and other factors.The plastic zone dimensions have been simply calculated, over a range of values of Poisson's ratio, for isotropic materials subjected to loading under plane stress and plane strain conditions; the analysis has been further extended to cover some effects of anisotropy. It has been demonstrated that, for isotropic materials, the maximum extent of the plastic zone directly ahead of, and in the plane of, a crack is ($\text{K}{}_{\mathrm{I}}\text{/Y)}{}^2\text{18π}$ under plane stress loading and is ($\text{K}{}_{\mathrm{I}}\text{/Y)}{}^2\text{18π}$ under plane strain loading. This latter result is smaller, by a factor of $\text{1}\text{3}$ than the plastic zone “radius” under plane strain conditions that is widely quoted in fracture mechanics texts. That “radius”, ($\text{K}{}_{\mathrm{I}}\text{/Y)}{}^2\text{6π}$ is, in fact, the maximum size of the zone parallel to, but not in, the plane of the crack, if Poisson's ratio is taken to be $\text{1}\text{3}$.A lower value of Poisson's ratio or an increased material anisotropy can lead to an enlarged plastic zone; this latter conclusion suggests that test-pieces for valid fracture toughness measurements on anisotropic materials could be required to be larger than defined in the relevant British Standard.     

CRACK TIP PLASTIC ZONE SIZES IN CYLINDRICAL BARS SUBJECTED TO TORSION

Abstract— Plastic zone sizes in circumferentially cracked round bars in torsion are estimated using an adaptation of the strip yield model. Results are compared with those from elastic-plastic finite element solutions for elastic-perfectly plastic and strain hardening material conditions, as well as with previously published data. The strip yield model gives results which are in close agreement with the finite element solutions over a wide range of torques.     

Organizational scheme for corrosion-fatigue crack propagation data

A model, proposed earlier, is modified in an attempt to explain a number of curious behaviors of corrosion-fatigue crack propagation (CFCP). The behaviors include effects of load ratio R in air and salt water vs vacuum, and effects of loading frequency at fixed R in these environments. Assumptions of the modeling are reviewed in detail in view of earlier objections to them. The ingredients of CFCP per this model: Poisson contraction, strain hardening, ligament surface attack/annihilation, and stress relaxation are developed and related to conditions of the crack tip locale. In the modeling, a parameter G, for growth rate factor, is developed solely as a function of the form of the ordinary or of the cyclic stress-strain curve. Previous work had developed a G₁ for the ordinary curve, to be associated with the surface attack effect as in stress-corrosion cracking, and one G₂ for the cyclic curve, to be associated with the stress relaxation effect as in fatigue crack propagation (FCP). A hybrid $\text{G}{}_{\mathrm{<mtext>2</mtext><mtext>1</mtext>}}$ is developed, combining attributes of both, which seems to successfully describe the corrosion induced augmentation of G₂. Parametric curves of $\text{G}{}_{\mathrm{<mtext>2</mtext><mtext>1</mtext>}}\text{(+G}{}_2\text{)}$ correspond well with stage II frequency-dependent growth in CFCP. However, alone they do not explain the frequency-wise stage II threshold shift nor the frequency-independent air-environment FCP rate. It is found that these trends can be represented by loci of constant plastic strain rate, due to crack loading and propagation, relative to the surface annihilation rate. Such loci are determined by comparing growth rate factor maps with strain rate maps, using parametric curves of equal geometric-series spacing. Maps of this sort are used to analyze about a dozen cases of CFCP including two titanium alloys and three steels, with one of the steels of four different tempers. Stress-strain curves of the low strength steels are processed to remove the Lüder band effect to facilitate the modeling. The scheme for data organization involves a representation of indexes of the two kinds of parametric curves fitting the data, and the process zone size implied by the fitting. Model predication of load-ratio effects on the fatigue crack growth threshold is in good correspondence with literature data. Comparison of estimated process zone sizes with literature data of microstructural and fractographic size measurements is encouraging.     

循环荷载下奥氏体型和双相型不锈钢材料本构关系研究

为研究循环荷载下不锈钢材料的本构关系,对奥氏体型S30408不锈钢和双相型S220503不锈钢材料进行了单调拉伸和大应变超低周循环加载试验。采用三种常用的单调拉伸本构模型对所得应力-应变曲线进行拟合,得到相应单调荷载下材料本构参数;采用Ramberg-Osgood本构模型对循环骨架曲线进行拟合,得到材料循环强化参数;利用Chaboche塑性本构模型,标定了两种材料的循环本构参数。结果表明:在单调拉伸荷载下,G-R-O本构模型更适用于拟合不锈钢材料的单调拉伸本构;在循环荷载下,不锈钢材料滞回曲线饱满,且随着应变增大,两种材料在加载后期均表现出了明显的循环强化现象;Ramberg-Osgood本构模型对骨架曲线拟合较好,有限元计算结果和试验滞回曲线吻合度高;表明该文标定出的强化参数、循环本构参数可用于结构体系地震响应分析之中,为准确分析不锈钢结构在地震作用下的受力性能提供参考。     

Fatigue crack growth retardation inconel 600

The effect of single-cycle overloads on the subsequent fatigue crack growth behavior of Inconel 600 is studied. Overloads ranging from 10 to 50% are applied to a sample undergoing baseline fatigue crack growth at constant Δ$\text{K}$. In all cases, the crack growth rate increases slightly immediately after the overload and then decreases rapidly to a minimum value before later returning to the pre-overload value. The plastic zone size, affected crack length and the crack growth increment at minimum crack growth rate, $\text{a}\text{?}$, are measured for each overload.The affected crack length is considerably larger than the overload plastic zone size for overloads greater than 20%. Consequently, although the minimum crack growth rate occurs within the plane stress overload plastic zone, the effect of the overload extends well beyond the overload region.Within the overload plastic zone, contact occurs between the crack faces due to the excessive deformation produced during the overload cycle. The size of the contact region agrees very well with the overload plastic zone size. Beyond the overload region, Δ$\text{K}{}_{\mathrm{eff}}$ remains less than the applied Δ$\text{K}$ for some time due to the wedge action of the plastically deformed overload region, delaying recovery of the pre-overload crack growth rate. The crack growth rate recovers only after the crack grows out of the region of influence of the wedge.     

On the cyclic behavior of cast and extruded aluminum alloys. Part B: Fractography

In a prior study [1], the fatigue crack propagation (FCP) response of a cast and an extruded aluminum alloy was examined as a function of mean stress and specimen orientation while crack closure data were collected. In this work, extensive electron fractographic studies were conducted on the previously generated fatigue fracture surfaces using both scanning and transmission electron microscopy. The threshold micromorphology revealed crisp, cleavage-like facets. Striation spacing measurements at intermediate and high $\text{Δ}\text{K}$ levels were obtained to determine microscopic growth rates; these measurements were seen to vary with $\text{R}$ ratio and were best correlated with $\text{ΔK}{}_{\mathrm{EFF}}$ rather than $\text{Δ}\text{K}{}_{\mathrm{APP}}$. Slope changes in the $\text{da/da-}\text{Δ}\text{K}$ plots were identified and attempts made to establish correlations between the associated plastic zone sizes and microstructural dimensions. Of particular note, a stage IIa to IIb transition in the extruded material was found to correspond to a micromechanism change from faceted growth to striated growth when the reversed plastic zone size was similar to the subgrain dimension.     

Finite deformation cohesive polygonal finite elements for modeling pervasive fracture

In the present study, mode I crack subjected to cyclic loading has been investigated for plastically compressible hardening and hardening–softening–hardening solids using the crack tip blunting model where we assume that the crack tip blunts during the maximum load and re-sharpening of the crack tip takes place under minimum load. Plane strain and small scale yielding conditions have been assumed for analysis. The influence of cyclic stress intensity factor range (\(\Delta \hbox {K})\), load ratio (R), number of cycles (N), plastic compressibility (\({\upalpha })\) and material softening on near tip deformation, stress–strain fields were studied. The present numerical calculations show that the crack tip opening displacement (CTOD), convergence of the cyclic trajectories of CTOD to stable self-similar loops, plastic crack growth, plastic zone shape and size, contours of accumulated plastic strain and hydrostatic stress distribution near the crack tip depend significantly on \(\Delta \hbox {K}\), R, N, \({\upalpha }\) and material softening. For both hardening and hardening–softening–hardening materials, yielding occurs during both loading and unloading phases, and resharpening of the crack tip during the unloading phase of the loading cycle is very significant. The similarities are revealed between computed near tip stress–strain variables and the experimental trends of the fatigue crack growth rate. There was no crack closure during unloading for any of the load cycles considered in the present study.     

An experimental study of crack tip plastic flow in mild steel

The paper deals with an experimental study on the nature of plastic flow at the root of a crack in mild steel beams, for the non-valid $\text{K}{}_{\mathrm{IC}}$ test regime, under three point hending loads. Photoelastic coating technique has been used to measure the plasticity spread ahead of the tip in relation to the load-COD record. It is observed that in all cases there is a sudden increase in specimen compliance near the maximum linear load due to an abrupt increase in plastic zone size on some preferential planes ahead of the crack tip. This abrupt increase in plastic flow was seen to occur along the 45° planes (with respect to the plane of the crack) for thicker beams and/or with longer cracks. In contrast, the plastic zone extended more on the plane of the crack for thin section beams with relatively shorter cracks. The stress intensity factor required to cause this sudden loss of resistance to localized deformation is found to be remaining constant beyond a certain crack length for a given specimen thickness. These observations suggest that a critical stress intensity factor () concept can be introduced to describe the abrupt flow localization ahead of the crack tip. This () can be taken as a new parameter in addition to those commonly used in characterising the overall “fracture” behaviour of large scale yielding materials like mild steel, especially in the non-valid $\text{K}{}_{\mathrm{IC}}$ test regime.     

Plastic J-integral calculations using the load separation method for the center cracked tension specimen

The load separation method was used to determine the plastic work factor (η_pl) for the center crack tension geometry for power law hardening materials with Ramberg-Osgood hardening exponents n ranging from 2 to 20 and crack sizes a/W ranging from 0.4 to 0.8. The resulting expression for η_pl compares favorably to the analytical work of Rice, Paris, and Merkle and to Sharobeam and Landes. The results were also compared to η_pl calculated from the plastic J results in EPRI Handbook NP-1931. Unlike the EPRI results, η_pl from the load separation method are not a function of crack size.     

A fatigue crack propagation model for strain hardening materials

In this paper a crack propagation model based on Tomkins concept (dl/dN ∝ Δε_p · ω) has been developed using the theoretically developed cyclic plastic zone sizes. The crack propagation rates are found to be functions of stress intensity factor, Elber's effective stress range ratio, cyclic yield strength of material, crack length, specimen width and cyclic strain hardening exponent. Suitably grouped to give the crack growth rate in terms of five constants termed as Loading Constant, Material constant, Crack size constant, specimen Width Constant and Stress Intensity Exponent. The crack growth rates found by theory are compared with the experimental results available in literature and a good agreement is found.     

Effects of grain size on cyclic plasticity and fatigue crack initiation in nickel

Fatigue experiments were conducted on polycrystalline nickel of two grain sizes, 24 and 290 μm, to evaluate the effects of grain size on cyclic plasticity and fatigue crack initiation. Specimens were cycled at room temperature at plastic strain amplitudes ranging from 2.5×10⁻⁵ to 2.5×10⁻³. Analyses of the cyclic stress–strain response and evolution of hysteresis loop shape indicate that the back stress component of the cyclic stress is significantly affected by grain size and plastic strain amplitude, whereas these parameters have little effect on friction stress. A nonlinear kinematic hardening framework was used to study the evolution of back stress parameters with cumulative plastic strain. These are related to substructural evolution features. In particular, long range back stress components are related to persistent slip bands. The difference in cyclic plasticity behavior between the two grain sizes is related to the effect of grain size on persistent slip band (PSB) morphology, and the effect this has on long range back stress. Fine grain specimens had a much longer fatigue life, especially at low plastic strain amplitude, as a result of the influence of grain size on fatigue crack initiation characteristics. At low plastic strain amplitude (2.5×10⁻⁴), coarse grain specimens initiated cracks where PSBs impinged on grain boundaries. Fine grain specimens formed cracks along PSBs. At high plastic strain amplitude (2.5×10⁻³), both grain sizes initiated cracks at grain boundaries.     

Strain rate dependence of HPFRCC cylinders in monotonic tension

High-Performance Fiber-Reinforced Cementitious Composite (HPFRCC) materials exhibit strain hardening in uniaxial, monotonic tension accompanied by multiple cracking. The durability of HPFRCC materials under repeated loading makes them potentially suitable for seismic design applications. In this paper, the strain rate dependence of tensile properties of two HPFRCC materials in cylindrical specimens is reported from a larger study on strain rate effects in tension, compression and cyclic tension–compression loading. The cylindrical specimens were loaded in monotonic tension at strain rates ranging from quasi-static to 0.2 s⁻¹. To evaluate the impact of specimen geometry on tensile response, coupon specimens loaded in monotonic tension under a quasi-static strain rate were compared to corresponding cylindrical specimens made from the same batch of material. Tensile strength and ductility of the HPFRCC materials were significantly reduced with increasing strain rate. Multiple cracking, strain hardening, strain capacity, and the shape of the stress–strain response were found to be dependent on specimen geometry. SEM images taken of the fracture plane of several specimens indicated that pullout and fracture of the fibers occurred for both HPFRCC materials studied here.     

Orientation dependence of cyclic deformation behavior and dislocation structure in Ti–5at.% Al single crystals

Cyclic deformation behavior and substructure in single crystal Ti–5at.% Al alloy with different orientations fatigued at Δε_t/2=0.4% were examined. The selected loading orientation includes: single prism slip, A; double prism slips [$\text{2}\text{1}\text{̄}\text{1}\text{̄}\text{0}$], B; pyramidal slip, C; and twinning [0001], D. The testing results show that the crystallographic orientation has a strong effect on the cyclic stress response and the plastic deformation mode of Ti–5at.% Al single crystals. The crystals displayed an initial cyclic hardening followed by a striking softening period, and then a saturation stage was reached in specimens A and B. In contrast, an obvious cyclic saturation stage was obtained after first cyclic hardening until to fracture in specimens C and D. Trace analysis on the surface of specimens with an optical microscope shows that the ($\text{1}\text{1}\text{̄}\text{00}$) single prism slip was operated in specimen A during cycling. The ($\text{10}\text{1}\text{̄}\text{0}$) and ($\text{1}\text{1}\text{̄}\text{00}$) double prism slips can be distinguished from the traces on the (0001) surface in specimen B. The ($\text{1}\text{1}\text{̄}\text{01}$) pyramidal slip and the ($\text{1}\text{1}\text{̄}\text{00}$) prism slip were activated simultaneously in specimen C. Twinning is the primary plastic deformation mode in specimen D. The twinning type includes: {$\text{11}\text{2}\text{̄}\text{1}$}, {$\text{10}\text{1}\text{̄}\text{1}$}, {$\text{11}\text{2}\text{̄}\text{2}$} and {$\text{10}\text{1}\text{̄}\text{2}$}. The substructure in the fatigued specimens was examined using TEM. Typical dislocation configuration is well developed saturation bundle structure (SBS) in specimen A, while it is the planar edge dislocations which are tangled on the primary ($\text{10}\text{1}\text{̄}\text{0}$) plane and arranged parallel to the [0001] direction in specimen B. Fully developed loop patches were formed in specimen C. Typical deformed structure was the twin bundles and dislocations among twins in specimen D. The effects of plastic deformation mode on the cyclic stress response and the corresponding dislocation configuration of Ti–5at.% Al single crystals are then discussed.     

Plane strain crack closure and cyclic hardening

This study is focused on the understanding of the mechanical effects of cyclic hardening on crack tip plasticity and on plasticity-induced crack closure. Various finite element analyses were conducted using abaqus. Cyclic hardening is found to affect both crack closure and the shape of the plastic zone at the crack tip. Crack growth modelling in plane strain conditions in a cyclically hardening material is discussed. An empirical formula is provided which allows the calculation of the crack tip plastic zone size under plane strain conditions in a cyclically hardening material. The effects of overloads are also examined.     

Analytic and numerical studies on mode I and mode II fracture in elastic-plastic materials with strain gradient effects

This paper presents a study on fracture of materials at microscale (∼1 μm) by the strain gradient theory (Fleck and Hutchinson, 1993; Fleck et al., 1994). For remotely imposed classical K fields, the full-field solutions are obtained analytically or numerically for elastic and elastic-plastic materials with strain gradient effects. The analytical elastic full-field solution shows that stresses ahead of a crack tip are significantly higher than their counterparts in the classical K fields. The sizes of dominance zones for mode I and mode II near-tip asymptotic fields are 0.3l and 0.5l,while strain gradient effects are observed within land 2l to the crack tip, respectively, where l is the intrinsic material length in strain gradient theory and is on the order of microns in strain gradient plasticity (Fleck et al., 1994; Nix and Gao, 1998; Stolken and Evans, 1997). The Dugdale–Barenblatt type plasticity model is obtained to provide an estimation of plastic zone size for mode II fracture in materials with strain grain effects. The finite element method is used to investigate the small-scale-yielding solution for an elastic-power law hardening solid. It is found that the size of the dominance zone for the near-tip asymptotic field is the intrinsic material lengthl. For mode II fracture under the small-scale-yielding condition, transition from the remote classical K _IIfield to the near-tip asymptotic field in strain gradient plasticity goes through the HRR field only when K _IIis relatively large such that the plastic zone size is much larger than the intrinsic material length l. For mode I fracture under small-scale-yielding condition, however, transition from the remote classical K _I field to the near-tip asymptotic field in strain gradient plasticity does not go through the HRR field, but via a plastic zone. This revised version was published online in July 2006 with corrections to the Cover Date.