Approximation of the crack-tip field in fatigue cracks in bridge steel specimens: DIC analysis of different constraint levels

A study on the accuracy of the approximation of the displacement field around of crack tip in a sample made from bridge steel (S355) is main objective of contribution. Linear elastic fracture mechanics (LEFM) theory in framework of multi-parameter formulation, i.e. postulated by Williams is used to determine of coefficients of Williams power series terms. Over deterministic method was used to calculate the terms based on the least squares regression technique, applied on data from numerical simulation and experiment on S355 steel grades. Comparison between the stress fields (by principal stress s1 and von Mises stress sHMH) obtain from experimental measurement DIC, Hybrid method and obtain from reconstruction by using various number of Williams power terms are quantified in order to get key information around the crack tip region on bridge steel specimens.


INTRODUCTION
on-destructive inspections of fatigue loaded components or civil engineering structures are usually performed to predict the size and the locations of cracks that are created from singular stress concentrator/defects during the manufacturing process or service life. The existence of cracks changes predictable load-bearing capacity and service interval [1][2][3][4] of the civil engineering components especially during the fatigue loading. Characterisation of stress fields near crack tip has been studied for many decades; see for example a short state-of-art [5] where an Over Deterministic Method (ODM) is presented to evaluate the combination of mode I and mode II by using photo-elasticity for drawing isoline and given number of WE terms [6]. The ODM has also been employed successfully to analyse bulk strain data obtained with synchrotron tests [7,8]. Understanding the evolution of higher order terms for different geometries is key to predict the constraint of elastic-plastic crack tip fields [9]. Consequently, a large effort has been devoted in recent times to develop algorithms for fast determination of higher order terms and to improve the accuracy in the estimation of higher order terms [10][11][12][13] both quasi-statically [14] and in dynamic situations [15]. Digital Image Correlation technique (DIC) was applied successfully on NiTi pseudoelastic alloy for evaluation of fracture toughness and another fracture properties [16]. In the papers [17,18,[23][24][25][26][27], the displacement field in the surrounding of a crack tip of single edge geometry specimens was employed in conjunction with Williams power series to estimate the stress intensity factor (SIF). The higher-order terms effects were analysed by using Williams expansion (focusing of T-stress) for various sizes of the crack tip distance of applied region for fitting data. A comparison between the SIF results and the analytical solution was established. The aim of this contribution is to analyse the various stress fields in the vicinity of the crack tip that were constructed from the data obtained from measurement of the displacement field by DIC by comparing the crack tip field approximations obtained with the ODM [28] and hybrid element method [29,30]. The displacement field in the vicinity of the crack tip, which is necessary for the following analysis, was measured in a standard compact tension (CT- [31]) specimen made of S355 J2 under the constant value of stress intensity factor K I =15 MPa√m.(according ASTM E647, [32]). The linear elastic fracture mechanics (LEFM, [38]) theory was applied on calculation of displacement vector components to be more precisely the multi-parameter formulation given by Williams [39] is applied. Calculation of terms coefficients of Williams power series was applied the least squares-based regression technique marked ODM, [28] for which displacements data obtained experimentally from optical measurements are taken as inputs. The influence of the number of terms considered for the power series reconstruction of two kinds of definition for the stress fields for the calculation of the values of coefficients of those terms are investigated in the current paper.

Material S355 J2 and method of measurement
he experiments were conducted on a Compact Tension (CT) geometry that were machined in the T-L direction that is the rolling direction was parallel to the crack growing direction. The specimen was machined according to ASTM E647 [32]. The material was S355 J2 structural steel that is commonly used offshore, marine and structural applications. The chemical composition of the steel grade is specified in EN 10025-2:2004 standard and is presented in [33]. Chemical composition of the experimental material was verified by producer, and it is in agreement with the standard presented in this paper and measured mechanical properties of the S355 J2 are summarized in Tab. 2, the fatigue properties were published e.g. [34][35][36][37].  Specimens were prepared from a metal sheet in two different orientations to the sheet rolling direction. Specimens were marked with letters "A-crack propagating along rolling direction" and "B-crack propagating across to rolling direction" to see the influence of the material structure orientation on the crack propagation rate. In case of the specimens marked with letter "A", the initial crack (notch) is oriented parallel to rolling direction, whereas with letter "B" the initial crack (notch) is oriented perpendicular to the rolling direction (rotated by 90°), the microstructure of S355 J2 for different rolling direction is shown in Fig. 2   The fatigue propagation test was conducted on a 100 kN Instron sevo-hydraulic rig. The precracking was implemented for 120,000 cycles, a frequency of 20Hz and a load ratio, R, of 0.1. Attention was paid throughout the test to fulfil small scale yielding conditions (see the paragraph entitled Justification of using LEFM). The value of stress intensity has changed each 1 mm of crack growing to be at given point 15 MPam 1/2 . The selected values of forces for relative crack length, calculated for given range of K according ASTM E647 [32], are mentioned in Tab. 3.

Justification of using LEFM
Condition of small scale yielding is a very useful concept [38], which has been used to describe that the linear extension of the plastic region is small compared to significant dimensions of the body, particularly the crack length Due to the high gradient that takes place at the crack tip because of the crack tip singularity, a certain plastic deformation will occur. The extension of this plastic deformation was estimated with Irwin's model where the plastic radius can be inferred along the x-axis, by equating the stresses acting on the crack opening direction to the yield stress of the material: In Eqns. 2 and 3, KI is stress intensity factor and y is yield stress. Therefore, valid points from used DIC measurement were taken in higher distance than 0.25 mm.

Full-field experimental measurement of displacements
Key experimental information of the displacement field around the crack tip was obtained with Digital Image Correlation (DIC). The study was conducted for various crack lengths and one load level (see Tab. 3). DIC is a powerful full-field technique that enables the in-plane displacement field between two different strain states to be measured. DIC technique works by splitting the image of the analysed surface into a small squared images also called interrogation windows. It is based on the cross-correlation function [40] that evaluates the similarity between the different interrogation windows during the different strain states: where c is the cross-correlation function that depends on the horizontal, u, and vertical, v, displacement vectors that join the centre of all interrogation windows in the two different strain states. IA and IB are distributions of the intensity across the two images captured in the two different strain states. N is the total number of interrogation windows included in each image. The maximum of the cross-correlation function indicates the most likely displacement vector for all the interrogation windows. The experimental setup is shown in Fig. 3. Fig. 3 shows the loading fixture employed to apply mode I loading on the CT geometry and the DIC experimental system. This includes the digital camera, the macro-lens and fibre-optics co-axial illumination. The camera was positioned so that the crack propagation direction was parallel to the positive x coordinate. The crack opening direction was parallel to the y coordinate [17]. Such pre-arranging simplifies the post-processing of the results [18], in particular for non pure mode I cracks (i.e. for mixed mode cracks, e.g. [19,20]. Unlike most previous works where speckle pattern was obtained by spraying the surface with black and white paint [21,22], here we obtain a random pattern in the studied surface by abrading the surface with different grades SiC papers [23,24]. Such procedure yielded excellent results in the past with similar types of alloys [25]. Moreover, the recommendations generated in previous studies were incorporated in the tests [26]. The digital images were acquired with a 2452×2052 pixels CCD camera with 8-bit dynamic range. The frame rate of the camera was set to 12. Overall the camera/sample/illumination/loading rig setup was analogous to that employed previously [10,27]. In order to not introduce additional sources of error, the raw information measured by DIC, was fed into the elastic analytical model to conduct the fracture mechanics analysis (see following section). That is, horizontal, u, and vertical, v, displacement information was used, rather than strain information that requires an additional step.

THEORETICAL BACKGROUND: MULTI-PARAMETER FRACTURE MECHANICS
Similarly, the displacement vector can be written as:     In Eqns. 5 and 6, r and θ are polar coordinates with the start of coordinate system in the crack tip),  is shear modulus defined as  = E/2(1+), where n denotes the index of the term in the series development and  is the Kolosov constant defined for plane stress ((3- or plane strain (3-4 conditions. Coefficients of WE marked A n depend on relative crack length  = a/W. The start of polar coordinate system was positioned at the crack tip [40] and curvature of crack was not considered [41]. It is good to mention, that bulk effect was not taken in account like in [42]. The used shortened form of the WPS used N terms is applied on description of approximation of stress/displacement fields. When the crack is loaded by mixed-mode [44], Eqns. 5 and 6 have to include a shear mode terms, see e.g. [45,46].

Over-deterministic method (ODM)
In the scientific literature [29,30,47,48], various numerical procedures for calculation of the coefficients of the Williams power series can be found. Some of them are based on the hybrid crack elements, boundary collocations or other mathematically more difficult definitions and techniques. Contrary to that, there exist a relatively simple method that can utilize the basic results of data postprocessing included in each commercial finite element software. The method is called T over-deterministic (ODM) [28], because the coefficients are obtained from an over-determined system of equation. These equations are formed based on Eqn. 6. The principle consists in choosing a set of particular nodes around the crack tip. The number of nodes, k, corresponds to the number of Williams expansion terms, N, that should be determined (it has to hold that 2k  N + 1 for mode I configurations). The displacements of these nodes obtained experimentally from the DIC measurements are used as inputs for Eqn. 6 together with their polar coordinates (the total number of nodes between the radius from 0.25 up 1 mm around the crack tip was approximately 120 for each configuration). Thus, the coefficients An could be calculated, which was programmed in Wolfram Mathematica code [49] and php software [50]. More details on advantages, restrictions and accuracy of the ODM can be found in [10][11][12][13].

Methods of Reconstruction of stress fields in vicinity of a crack tip
The knowledge of higher-order terms of the WE from Eqn. 5 allow us to plot the stress isolines for given values of stress.
In the contribution, two methods of approximation are used: the principal stress  1 and von Mises stress  HMH. The following equations show how to calculate these stress values: where x, y are normal stress components in x and y direction and xy is shear stress component. For comparison of stress distribution around the crack tip, the coefficients of the WE obtain via the hybrid crack elements for a normalized CT specimen were taken from the literature [29].

Stress intensity factor (SIF)
IF values obtain from various methods were compared, see Tab [29].

Comparison of stress fields in vicinity of a crack tip
For comparison of experimentally obtain data and numerically calculated, the relative crack length 0.86 were selected due to maximal value of constraint (T-stress), see Tab. 3 and Fig. 4. In Fig. 5, σ1 and σHMH the approximation of the crack tip stresses taking into account one and two terms of the Williams power series on specimen are shown: a) stress values calculated from the coefficients determined theoretically [29]; b) stress values computed from the coefficients that were calculated through the ODM from the experimental displacement information.  : σ 1 and σ HMH crack tip stress approximation considering one (thick lines) and two (dashed lines) initial terms of the Williams expansion on specimen with the relative crack length 0.86: a) isoline of stresses calculated from the coefficients determined theoretically [29]; b) isoline of stresses calculated from the coefficients calculated by ODM from the DIC measurements.

CONCLUSION
his work presents an investigation regarding the stress distribution around a fatigue crack tip. The study has been conducted on a S355 steel often used in offshore, marine and structural applications. The experimental program is based on CT specimens analysed by means of DIC full-field technique. Different cases with constant stress intensity factor, K = 15 MPam 1/2 , were analysed. One of the main outputs has been inferring the Fourier coefficients that are required to approximate the Williams expansion describing the field. This has allowed the crack tip stress field distribution for be reconstructed for a real fatigue crack. The results have been validated by comparison between the Williams based analysis and theoretical solutions. The current research is part of a broader investigation program aimed at multi-parameter characterisation of failure in metals.