Regularities of plastic deformation development in the crack tip vicinity

. When determining the service life of cyclically loaded metal structures at the stage of fatigue crack development, an important role is played by the accuracy of assessing the plastic deformation zone size at the crack tip formed by the tensile overload. Within this zone, after reducing the load to the initial level, the fatigue crack development slows down. Despite the large number of studies in this area, none of the proposed dependencies is currently fully confirmed by experimental results in determining the plasticity zone size. The aim of the work is to assess the size of the plastic deformation zone formed by a single tensile overload. In the work of the finite element method for central-notch specimens and compact tension specimens, an analysis of the stress-strain state in the crack top vicinity is performed. As a calculation, a flat and three-dimensional model was used. Calculations are performed in the Ansys 14.0. Data on the mechanical properties of steels were used: St20, VSt3sp, 09G2S, 15G2SF. With increasing load, the plastic zone size in front of the crack tip was estimated, in the process of reduction, the distribution and length of the fields of residual compressive stresses and the size of the cyclic plastic zone were recorded, the length of which was also determined experimentally using strain gauges with a base of 0.5 mm. To assess the plastic zone size formed by a single tensile overload, a dependence is proposed that takes into account the patterns of plastic deformation development in the vicinity of the fatigue crack tip, and the results of the calculation agree well with the experimental data given in the literature. The obtained expression makes it possible to predict with greater reliability the development of fatigue cracks at variable load amplitudes.


Introduction
Almost all industries operate welded metal structures that perceive variable (cyclic) effects.Operation in alternating stresses conditions leads to the initiation and fatigue crack development in them.
When predicting the crack development taking into account overloads, an important role is played by the accuracy of estimating the monotonous and cyclic plastic zone size in front of the crack top when exposed to tensile overload.After reducing the load to the initial the fatigue crack growth, it slows down sharply.The return to the original speed occurs after the crack passes the plastic zone formed by the overload.
The fatigue crack development in metal structures during cyclic loading with a constant amplitude is accompanied by a formation in front of its apex of a monotonic (the area in front of the crack front, for which the σ i σ y ) and cyclic plastically deformed regions (the area at the front of the crack tip, for which in the half-cycle of unloading Δσ i S T (S T , where S T is the cyclic yield strength) [1,2], and behind its top there is a trace of plastically deformed material (Fig. 1,a).Fig. 1.Plastic deformations at the apex: (a) fatigue crack; fixed fissure at (b) monotonic stationary and (c) at cyclic loads Several different approaches have been proposed to calculate the size of the monotonic plastic zone in front of its tip.For bodies with fixed cracks subjected to static loading (Fig. 1b), with the development of relatively small monotonic zones of plasticity in their vertices, Irivin [3] analytically obtained a dependence for determining the plastic zone in a plate of finite width with a central crack: where α = 1 in the case of a plane-stressed state and α = 3 -a flat deformation.
Guo [4] on the basis of a three-dimensional strip yield model, an integral equation for estimating the size of a monotonous plastic zone is proposed: where β = (for plane strain); μ -poisson's ratio.Voorwald et al. [5] proposed a parametric function for β, taking into account all the constraints at the top of the crack.This function depends on the maximum applied stress, the yield strength of the material and the thickness of the specimen: In the work [6], an elastic-plastic calculation was performed by the finite element method in order to verify the expressions proposed in the works [4,5].Using the specimens model with a central crack l/W = 0.4÷1.0;t/W = 0,05÷1,0; t = 2÷50 mm, σ y = 200÷800 MPa made of ideally plastic material E = 206 GPa, μ = 0.3 it was established that the expressions (2) and (3) are valid only in a narrow range of variable parameters.To calculate the coefficient, a dependence β proposed: β = 0,35 − 0,29 This expression is obtained from the calculation of FEM without taking into account the deformation hardening of the material in the vicinity of the crack top.Therefore, the size values of the plastic deformation zone , calculated by the Eqs.(4), basically exceed (up to 50%) the experimentally obtained values.
The authors of the work [7] investigated the dependence of the β parameter on the maximum SIF, the material yield strength, the specimen thickness, the hardening of the material in the crack tip and proposed the dependence: The paper notes the coincidence of experimental and calculated values of the sizes of the plastic deformation zones formed during static loading, and it is also assumed that the plastic deformation zone before the crack top will decrease if a compression load follows after a tensile load.However, the works results [2,8] do not support this assumption.In addition, the sizes of the plastic deformation zones calculated by the Eqs.( 5) are significantly less than the values obtained experimentally in the works [14,15,16,17,18,19,20].
Nicoletto [9] experimental measurements of the monotonous plastic zone sizes in front of the fatigue crack front under cyclic loading with a constant amplitude on the specimen surface with a one-sided lateral notch 5 mm and 10 mm thick from NiCrMoV and Fe-3Si steels, respectively, found that the size of the plastic zone in the fracture plane coincides well with the calculation results according to Eqs. (1) at α = 2.
The purpose of the work is to assess the monotonic and cyclic zone size plastic deformations before the fatigue crack top formed by the tensile overload and, accordingly, to assess the zone of delayed crack development during cyclic loading of metal structures elements.

Methods and materials
The stress state elastoplastic analysis of near the crack tip was carried out by the finite element method (FEM) for a standard specimens (central-notch specimen M(T) and compact tension (CT) specimen), for which expressions for calculating the SIF are known [10].As a design, a flat and three-dimensional model with a fixed crack was used.The final element minimum size at the crack tip was 0.05÷0,1 mm.Stepwise loading made it possible to separate the elastoplastic task into a series of consistent elastic tasks with varying material characteristics, determined by the elastoplastic deformation diagram [11].Stress calculation is based on the work of V.V. Moskvitin [12] with a cyclic change in the load in the n-th half-cycle of loading.
Data on the mechanical, elastic and plastic properties of steels St20, VSt3sp, 09G2S, 15G2SF (table 1) and their deformation diagrams were used in solving this task.Deformation diagrams were obtained by a uniform technique using low-base strain gauges.This is presented in work [13].All selected steels are cyclically stable, and their mechanical properties cover the mechanical property entire range of construction steels.The load ratio R (from -5,5 to 0,8) and the stress intensity factor range (SIF) ΔK (in accordance with the deformation diagrams in the range 9,639-24,098 MPa m 1/2 for steel VSt3sp, 14,459-28,918 MPa m 1/2 for steel St20, 24,098-38,557 for steel 09G2S and 33,738 -48,165 MPa m 1/2 for steel 15G2SF) varied in the calculations.The monotonic plastic zone size (the place ahead of the crack front, for which σ i σ y ) in the crack propagation direction was estimated at the maximum load in the entire range of changing load and the load ratio.We fixed the distribution and dimension of compressive residual stress field in the process of reducing the load, and also we fixed the cyclic plastic zone size (the place ahead of the crack front, for which, in a half-cycle of unloading, Δσ i  S y ; S y is the cyclic yield strength).

Results
Determination of the plastic deformation zone size in front of the static crack top was carried out by FEM for the plane-stressed state.The results of the calculation in determining the plasticity zones for specimens with a central crack and eccentrically stretched for all the studied steels are presented in Fig. 2 in the coordinates: the plastic zone size r p -relative stresses σ net ⁄σ y .For each type of sample, curves independent of the steel grade are obtained in the given coordinates.
Using the Irwin Eqs.(1) and the obtained values of the monotonous plastic zone sizes rp (Fig. 2) for the studied steels and specimens, the values of the α coefficients were calculated.
Fig. 3 shows the change in the α parameter depending on the load level σ net ⁄σ y .It follows from the graph that at the loading levels of (CT) specimens, σ net ⁄σ y = 0.3÷1.0α varies in the range from 1.0 to 1.2, and the average value of the coefficient is α=1.10.Under cyclic loading, the monotonous plastic zone size in front of the vertex of the fatigue crack, as shown in the experimental study [9], coincides well with the values calculated by formula (1) at α≈2.
The cyclic plastic zone size in front of the crack apex for the studied steels in (CT) and M(T) specimens at different load ratio was determined with a decrease in load.
In Fig. 4 for steels VSt3sp and 15G2SF shows the dependences of the size of the zone of cyclic plastic deformations in the direction of crack advancement, obtained by FEM during elastoplastic deformation of the material, on ΔK at different values of R. In the same figures, the dashed line shows the curve of the value of the zone of cyclic plastic deformations, obtained by calculation by the equation: where S y is the cyclic yield strength of steel; ∆K is the stress-intensity factor (SIF) range, determined by the equations of linear fracture mechanics.The results of the r c calculation are confirmed by experimental studies of the stressstrain state in front of the crack apex by the strain gauge method.
In Fig. 5 shows diagrams of the distribution of strain range and normal stresses in front of the crack front in its plane in the M(T) specimen, obtained by the strain gauge method using chains of two component resistors with a base of 0.5 mm.The specimen was made of VSt3sp steel.The transition from measured deformations to stresses was carried out using a cyclic steel strain diagram.The size of the cyclic plastic zone obtained using resistance strain gauge r exp = 1 mm (Fig. 5,b) practically coincides with the size calculated by the Eqs.( 6) r cal = 0.985 mm.
A good match between the results of the FEM calculations and Eqs.(6) indicates that the cyclic plastic deformation intensity at the crack tip is fully controlled by the SIF range.
The results obtained coincide with the results of Rice [1] and Nicoletto's experimental measurements [9] of the cyclic plastic zone size in the fatigue crack tip on the surface of steel specimens.

Discussion
The stress-strain state of the material in the vicinity of the fatigue crack tip during cyclic loading is stabilized in the same way as it occurs near conventional stress concentrates in repeated loading.
With an increase and decrease in the load at the fatigue crack tip inside the monotonic plastically deformed region, a region of the alternating in sign (reversible) flow is formeda cyclic plastic zone (Fig. 1a).The cyclic plastic deformation size occurring at the crack tip in the half-cycles of increasing and decreasing load, regardless of the stress ratio loading, is determined by the range ΔK.
In half-cycles of load reduction, the local monotonic plastically deformed zone at the fatigue crack tip is affected by the elastic working material surrounding it (the development of fatigue cracks occurs against the background of elastic deformations) and a field of residual compressive stresses is formed in it each time.In half-cycles of increasing load, the field of residual compressive stresses makes it difficult for plastic deformations to occur in the monotonic plastic zone in front of its top.Experimental measurements of the monotonous plastic zone dimensions on the surface of steel specimens in the work [9] established that they coincide well with the theoretical ones, calculated by equation ( 1) at α = 2.
When exposed to a single tensile overload in the range of SIF change from K max to K ol , an increase in the monotonic plastic zone occurs.Its size in the specified range of change SIF according to the analysis in the kinetics of plastic deformations at crack tip coincides well with the theoretical values calculated according to Irwin's equation at α = 1 for a static crack.
Experimental studies [14,15,16,17,18,19,20] of crack development after exposure to overload show that a return to the original crack growth rate occurs when the top of the crack under cyclic loading reaches the boundary of the plastic zone from overload (Fig. 6).Comparison of the calculated values of the zones of delayed crack growth according to Eqs. (7) with the experimental results obtained in [14,15,16,17,18,19,20] is presented in Fig. 7.The difference between the experimental values of the zones of delayed development of cracks in the area of plastic deformations formed by overload and the calculated values does not exceed 20%.2. With a cyclic change in the load, the size of the permanently formed cyclic plastic zone inside the monotonous plastically deformed region at the top of the fatigue crack, regardless of the asymmetry of the loading cycle, is determined by the range ΔK.
3. When predicting the crack development, an important role is played by the accuracy of estimating the plastic zone size in front of the crack top after exposure to tensile overload, within which the development of the fatigue crack slows down sharply.To estimate the plastic zone size formed by a single tensile overload, a dependence is proposed that takes into account the regularities of the plastic deformation flow at the top of the fatigue crack under the influence of tensile overload.The obtained expression makes it possible to predict the development of fatigue cracks with greater reliability at variable load amplitudes.

Fig. 2 . 3 .
Fig. 2. Changing the size of the plastic zone Fig. 3. Changing the α parameter from from the relative stress level σ net ⁄σ y the relative stress leve σ net ⁄σ y For (CT) specimens in the range σ net ⁄σ y = 0.3÷0.8α varies from 1.0 to 0.85 and α = 0.95.The values α obtained taking into account the hardening of the material in the vicinity of the vertex for a static crack.Under cyclic loading, the monotonous plastic zone size in front of the vertex of the fatigue crack, as shown in the experimental study[9], coincides well with the values calculated by formula (1) at α≈2.

Fig. 4 .
Fig. 4. Dependence of the size of the zone of cyclic plastic deformations in the direction of the crack movement on the value ∆K in (CT) specimen

Fig. 6 .
Fig. 6.Scheme of changing the size of the plastic zone in the crack tip when exposed to tensile overload Thus, the zone length of delayed crack development is determined by the difference in the monotonous plastic zone size from overload and cyclic loading.

Fig. 7 . 5 Conclusions 1 .
Fig. 7. Comparison of experimental dimensions (r ol exp. ) and theoretical (r p cal ) sizes of zones of plastic deformation after exposure to overload
Ψ − reduction at fracture, S s − rupture strength in the neck of the sample