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Analysis of standard fracture toughness test based on digital image correlation data
Engineering Fracture Mechanics xxx (2017) xxx–xxx
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Analysis of standard fracture toughness test based on digital image correlation data I. Jandejsek ⇑, L. Gajdoš, M. Šperl, D. Vavrˇík The Institute of Theoretical and Applied Mechanics AS CR, v. v. i., Prosecká 809/76, Prague 9 19000, Czech Republic
a r t i c l e
i n f o
Article history: Received 30 March 2016 Received in revised form 26 May 2017 Accepted 29 May 2017 Available online xxxx Keywords: DIC Full-field measurement J-integral CTOD ASTM standard
a b s t r a c t This paper deals with an experimental methodology based on the 2D Digital Image Correlation method for precise full-field in-plane stress-strain analysis of a region containing a crack. The presented methodology is employed as a supporting and comparative method for the standard J-integral testing defined by the ASTM. The methodology utilizing the DIC-based displacements for a subsequent evaluation of strain, stress and final extraction of the J-integral from its definition. This J-integral is compared with the value obtained by the ASTM standard and with the value obtained by the DIC-based CTOD measurement. Several techniques improving the conventional DIC and data post-processing are proposed to achieve reliable full-field strain/stress results. A direct comparison between the experimental and FEM full-field results is carried out. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Stress-strain analysis of materials and structures subjected to various loadings is an essential task in experimental solid mechanics. Various well-known image-based, non-contact and full-field methods like Photoelasticity [10], Moiré [18] and Speckle interferometry [29] have been developed and applied for this purpose in addition to the widely-used strain gauge technique, in which strains are measured locally. Among others, 2D Digital Image Correlation (DIC) [17] has become very popular nowadays in the experimental solid mechanics community for its accessibility and easy-to-use implementation. An important discipline in which DIC-based full-field measurements find usage is experimental fracture mechanics (FM) [24,31]. The main benefit of full-field measurements in the region surrounding a crack is the possibility to extract fracture parameters such as the stress intensity factor or the J-integral. Among the first works, Chiang and Asundi [5] attempted to evaluate the stress intensity factor (SIF) from the displacement field measured by the speckle method. Huntley and Field [11] employed laser speckle photography for full-field measurement, and then optimized numerical results on the basis thereof evaluated the SIF and J integral. The direct extraction of the J-integral and J-R curve from strain/stress analysis obtained by MI was carried out by Dadkhah [6] and Kang [12]. One of the first works utilizing DIC for the extraction of fracture characteristics is by McNeil [15] and Han [8]. Noteworthy works from recent times are by Abanto-Bueno and Lambros [1] and Réthoré [20], in which they deal with the determination of SIF. Becker [4] attempted to measure the J integral using DIC, utilizing the equivalent expression of the J-integral as an area integral. Most of the works mentioned, which have tried to extract the fracture parameters from measured fields, have faced the problem of the insufficient accuracy of DIC results. For that reason, the proposed techniques often combine experimental ⇑ Corresponding author. E-mail address: [email protected] (I. Jandejsek). http://dx.doi.org/10.1016/j.engfracmech.2017.05.045 0013-7944/Ó 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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Nomenclature ASTM CT DIC FM FEM ITP NCC SSD SSY a Apl b B CTOD (d) De Dep E I J, JC KI m P p n r t T u U w W W
C
e eH gpl v
r ts ry r
American Society for Testing and Materials compact tension specimen digital image correlation fracture mechanics finite element method incremental theory of plasticity normalized cross-correlation sum-squared difference small scale yielding crack length plastic area under load-displacement curve remaining ligament specimen thickness crack tip opening displacement elastic stiffness matrix elastic-plastic stiffness matrix modulus of elasticity image J-integral, critical J-integral stress intensity factor for mode I plastic constraint factor load, remote force affine transformation coefficients normal vector to C correlation coefficient traction vector image template displacement vector (field) 2 right stretch tensor strain energy density specimen width affine transformation path around crack tip strain tensor (field) Hencky finite strain tensor (field) geometric factor Poisson’s ratio tensile strength yield stress stress tensor (field)
results with numerical simulations based on FEM, trying to optimize the simulations so that the results match the measured fields. Subsequently, on the basis of the best fit, the parameter is extracted from the numerical solution. The low degree of accuracy of DIC can arise from multiple factors: the low resolution of image data, poor quality of the ‘‘speckled” pattern, errors caused by out-of-plane motion, or insufficient smoothing procedures. All these factors can be significantly improved with the use of suitable techniques and corrections, improving DIC accuracy so that there will be no need for FEM. Here is a list of difficulties that one is encountered when attempting to measure displacements and to evaluate strain and stress fields in the region surrounding a crack: Selection of an appropriate surface coating to create speckled pattern with sufficient speckle size and density at a relatively high optical magnification. The pattern has to enable measurement of a wide range of strain (from elastic to plastic deformation). Moreover, due to large strains, such coat has to ensure good adhesion. The presence of high displacement/strain gradients in the vicinity of the crack. The set of points in which the displacements are measured using DIC should reflect this feature. In DIC, the reliability of the evaluated strains depends strongly on smoothing of the measured displacements which are inevitably noisy (locally non-monotonic). The presence of a crack or a notch causes a discontinuity within the displacement field, making smoothing difficult in such a region. The problem of discontinuity has been solved in [19] introducing the X-DIC method for example.
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Evaluating the stress in the plastic regime. This mainly concerns ductile materials, in which large-scale yielding accompanies the fracture process. In addition, the accuracy of DIC itself is susceptible to numerous factors [26,22,28]. It is therefore important to be able to compare DIC-based results with other methods and with FEM simulations. This work addresses all these issues. An experimental methodology based on DIC intended for precise in-plane analyses of a stress-strain field with a crack was proposed and developed. The methodology was employed as a supporting and comparative method for the elastic-plastic standard fracture testing defined by ASTM [3]. The test was carried out with a ductile low-C steel commonly used in the pipeline industry. The principle of the methodology allows a direct comparison between experimental results and the FEM simulation. On the basis of measured fields, the J-integral is evaluated directly as the line integral. This direct evaluation is compared with values obtained by the ASTM standard, and with a simultaneous CTOD measurement. CTOD is also measured by DIC using two templates located above and below the crack tip. In contrast to COD measurement performed in [9,23], templates are located coaxially with the crack tip. 2. Methodology The experimental-numerical methodology for a precise analysis of the stress-strain field with a crack that is proposed and developed in this work encompasses the following constituent parts:
An in-house developed customizable DIC algorithm. Surface coating of the specimen using an airbrush gun. The use of an adaptive mesh for full-field displacement measurements in the vicinity of the stress concentrator. A novel solution for smoothing of measured displacements. An evaluation of finite strain and stress (including plasticity). A direct evaluation of the J-integral from its definition. Measurement of CTOD using DIC.
Principles of the well-known DIC method have already been described in many papers, see [25] for instance. The essential principle is the tracking (displacement) of small templates through an image sequence representing the deformation of the surface of a specimen. The in-house DIC algorithm developed in this work is based on two basic approaches: Normalized cross-correlation (NCC) [13], and Sum-squared difference (SSD) [14]. The code of our DIC is written in MATLABÒ. The algorithm is described in detail in the Appendix. A telecentric lens is used to reduce errors caused by out-of-plane motion [26]. These errors are much more significant when using a standard lens due to the parallax. 2.1. Surface coating The surface of the specimen has to be treated to exhibit a random ‘‘speckled” pattern. For the magnification required for the experiment carried out in this work, a conventional spray was found to be insufficient for the speckle size. Moreover, tests showed that a coat of this kind tends to peel off the surface during higher deformation. Finally, a fine airbrush with matt black and white acrylic paint (RevellÒ – Aqua Color) was tested, and was found to yield an excellent coat in terms of fineness of the speckled pattern and good adhesion. The usability of the created pattern is estimated using NCC of a typical template with itself. The estimation is associated with the peak width of NCC at a pre-defined value, typically at 0.5. The standard says [25] that the appropriate size of this peak width is in the range of <3, 6> pixels. Fig. 2.1 shows the typical template prepared using the airbrush and NCC thereof. It can be seen that the peak width of NCC at a value of 0.5 is about 5 pixels, thus optimal. 2.2. Use of adaptive mesh An advantage of the DIC method is that it can be used for measuring the displacement almost anywhere within the speckled pattern. This makes it possible to form a set of measurement points as an adaptive mesh. This is particularly beneficial in cases when large displacement/strain gradients are expected, e.g. in the vicinity of a stress concentrator. In such regions, the mesh is refined. For this purpose, the mesh was created using a linear quadrilateral element (so-called Q4). An example is shown in Fig. 2.2; the mesh is added to a raw image of the vicinity of a crack. Each node of the mesh defines the centre of a square template, the displacement of which is measured applying the DIC algorithm. In this work, the mesh was prepared using the ANSYSÒ meshing, but any other meshing software can be used for this purpose. The experimental results can therefore be compared directly with an FEM simulation of the same problem. Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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1.1 mm
1.1 mm
0.5
≈ 5 px
Fig. 2.1. The image template of the airbrush-prepared spackle pattern of size 41 41 pixels (1.1 1.1 mm2 physical area) (left); auto-NCC function of the template with itself (middle); Cross-section of NCC and determination of the peak width at value of 0.5 (right).
Displacement of the node using DIC
Strain and stress of the element
u fague crack p template
Q4 element
Fig. 2.2. A raw image of the crack vicinity taken by a camera (left); Image data with a mesh; in each node, the displacement is measured employing the DIC algorithm (middle). The strain and the stress are evaluated for each element (right).
2.3. Displacement smoothing When the displacement vector at each node is known, the strain tensor at each element can be calculated. However, the measured displacements are intrinsically always noisy. For a more reliable evaluation of the strain, a smoothing procedure based on a spline function approximation [7] is employed. This procedure is performed in the polar coordinate system (PCS), with the centre at the tip of the crack to avoid discontinuity of the displacement field within the notch. Fig. 2.3 shows an example of the vertical displacement field around a crack (loading in mode I). There is a step discontinuity of the displacements within the crack that complicates the smoothing procedure. When smoothing in conventional Cartesian coordinate system (CCS), the region above and below the crack are corrupted by the discontinuity, see Fig. 2.3. The problem disappears when the PCS is used. Moreover, the field smoothed in PCS reflects the raw field more accurately, see Fig. 2.3.
Raw displacement field
Smoothing in CCS
Smoothing in PCS
Cross secon
Corrupted
Disconnuity Due to crack
Corrupted field
Raw displacement Smoothing in CCS Smoothing in PCS
Fig. 2.3. The raw measured vertical displacement field (left); Smoothing of the field in CCS and PCS, the field is corrupted above and below the crack in the case of CCS (middle); Cross section over the crack (right).
Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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2.4. Strain and stress After the displacement smoothing procedure, the strain tensor at each element is evaluated. In this work, due to the presence of large displacement gradients, the finite Hencky (true) finite strain tensor is used:
eH ¼ lnðUÞ
ð2:1Þ
where U is the right stretch tensor. The strain tensor is associated with the centre of mass of the element in the deformed configuration (Fig. 2.2). When the strains are known, the stress can be evaluated using material constitutive relations. The high ductility of the material tested in this work in conjunction with the plane stress loading condition (specimen surface) predetermined that there would be a significant plastic region in the vicinity of the crack tip. The Von Mises yield criterion is used as the condition for material yielding:
f ðr; ry Þ ¼ rm ry ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2xx rxx ryy þ r2yy þ 3s2xy ry ¼ 0
ð2:2Þ
where rm is the Von Misses stress and ry is the yield strength of the material. Plasticity occurs when the Von Misses stress meets the yield strength of the material. Where the Von Misses stress does not reach the yield value, the stress tensor is computed using well-known Hooke’s law (plane stress) as:
r ¼ De e
ð2:3Þ
where De is the elastic stiffness matrix. When the stress exceeds the yield strength, the incremental theory of plasticity (ITP) is employed:
dr ¼ Dep de
ð2:4Þ
where Dep is the elastic-plastic stiffness matrix. The uniaxial stress-strain curve of the material serves for evaluating Dep in each step. The resulting stress tensor, and also the strain, are associated with the centre of mass of the element in the deformed configuration (Fig. 2.2). 2.5. Evaluating the J-integral If we know the displacement, strain and stress fields, the J-integral can be evaluated directly from the well-known definition [21]:
J¼
Z @u ds wdy t @x C
ð2:5Þ
Re where w ¼ 0 rde is the strain energy density, t is the traction vector defined according to the outward normal n along path C as: t ¼ r n. Finally, u is the displacement. The integral is evaluated in the counter clockwise direction, starting from the lower crack boundary and continuing along path C to the upper boundary of the crack, see Fig. 2.4. For the purposes of evaluating the J-integral from the experimental data, the nodal values are interpolated onto a regular orthogonal grid. In this case, the J-integral can be evaluated simply along a rectangular-shaped path, as is shown in Fig. 2.4. Due to the discrete nature of the data, the original J-integral relation (2.5) is replaced by the summation of the discrete increments of length Ds, given by the pitch of the orthogonal grid onto which the nodal values are interpolated, see Fig. 2.4.
Interpolaon of mesh values to regular orthogonal grid
y
crack
ds
.
n
x
Γ
Path for the J-integral Fig. 2.4. Line J-integral around a crack (left); Interpolation of nodal data onto an orthogonal grid and the rectangular path for evaluating the J-integral (right).
Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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The interpolation is performed using MATLAB function TriScatteredInterp. The pitch of the grid is selected so as to have approximately the same size as the smallest element of the mesh. 2.6. Measurement of CTOD CTOD [30] is the opening displacement of the original crack tip due to blunting, see Fig. 2.5. In this work, CTOD is measured directly from the acquired images. For this purpose, displacements of just two points are measured, employing the DIC method. These points are located coaxially with the initial crack tip, as illustrated in Fig. 2.5. Fig. 2.5 shows that there has to be some initial non-zero pitch between two points, otherwise square templates would interfere with one another. This pitch is given by the smallest possible size of the template that makes the DIC measurement possible. However, the proper definition of CTOD has zero initial pitch. It is assumed that the initial pitch is so small that it is a sufficient approximation of the proper CTOD. Naturally, the initial pitch is subtracted from the measured displacement values. 2.7. Error estimation The experimental system used in this work was tested for uncertainties according to Standards Project for Optical Techniques of Strain Measurement (SPOTS) [16]. For the template of size 41 41 pixels, the uncertainty of strain was determined as 55 lStrain. 3. Experimental work 3.1. Material and specimen A fracture experiment was carried out with a compact tension (CT) specimen made of low-C steel (CˇSN 411373 which corresponds to Fe 360 B according to ISO 630:1980), which is widely used in the pipeline industry. The experiment met the ASTM standards for J-integral testing [3]. The essential mechanical properties of the material are summarized in Table 3.1. The uniaxial true stress–strain curve of the material (as supplied by the producer) is shown in Fig. 3.1. The geometry of the specimen, and the dimensions and the rolling direction of the sheet from which the specimen was cut are depicted in Fig. 3.2. Using ASTM standard labelling of the CT specimen, the three important characteristic dimensions were the width W = 50 mm, the crack length a = 25 mm, and the thickness B = 5 mm. Other dimensions depend on these values. The initial crack 1 mm in length was created by the standard fatigue method [2]. A speckled pattern was prepared on the optically observed side of the specimen, using an airbrush gun. 3.2. Fracture test The specimen was subjected to monotonic tension loading (opening mode I) under conditions of a constant grip displacement velocity of 1 mm/min until the load point displacement reached 3.8 mm, and then it was unloaded. The load was measured using a load cell with a read-out frequency of 1 Hz. The experimental setup with the necessary equipment for the DIC method is shown in Fig. 3.3left. The speckled side of the specimen was optically observed during the test. Images were acquired with the frame rate of 1 fps, using a 5 MPixel monochromatic camera (MANTAÒ) with a mounted telecentric lens (Opto engineeringÒ). The surface of the specimen was illuminated using a circular diffusion light to obtain sufficient light intensity in the images, and also to avoid reflection artefacts. The images were stored in monochrome RAW format. The resolution of the images was 2056 2452 pixels, and the resulting image scale was 1 pixel = 28 lm. The resulting loading curve of the test is plotted in Fig. 3.3. The maximum load reached in the test was 9.1 kN. The force measurement error given by the accuracy of the
Fig. 2.5. Definition of CTOD (left). CTOD measurement by DIC. Displacement measurement of two points located coaxially with the crack tip (right).
Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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I. Jandejsek et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx Table 3.1 Mechanical properties of low-C steel (CˇSN 411373, Fe 360 B according to ISO 630:1980). Modulus of elasticity E [MPa]
Poisson’s ratio m [–]
Yield strength ry [MPa]
Tensile strength rts [MPa]
Ultimate elongation [%]
206,000
0.3
260
380
13
ˇ SN 411373, Fe 360 B according to ISO 630:1980). Fig. 3.1. The uniaxial true stress-strain curve of steel (C
Fig. 3.2. CT specimen geometry with dimensions (left); orientation of the specimen to the rolling direction of the sheet.
Lucas 10 kN load cell is specified as to be <0.2%. Similarly, the displacement measurement error given by the accuracy of the Instron clip-on gauge is specified as to be <0.5%.
4. Results The image data provides input for measurements of full-field displacements and for the subsequent evaluation of strains and stresses and the final extraction of the J-integral. The data processing methodology is described in Section 2, so only the results of the particular steps will be shown and commented on here.
4.1. Mesh A model of the CT specimen was created in ANSYSÒ. The model was meshed using a linear quadrilateral element (Q4), see Fig. 4.1. A part of the mesh covering the optically observed region was exported from ANSYSÒ to MATLABÒ, where the DIC measurement was performed. The nodal points are added ‘‘virtually” to the reference image (see Fig. 4.1). The resulting mesh for the DIC measurement consisted of 696 nodes and 641 elements. Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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Load cell
10
light
8 P
Load [kN]
telecentric lens specimen
camera
max
= 9.1 kN
6 4 2 0 0
Posioning device
1
2
3
4
Load point displacement [mm]
Fig. 3.3. Experimental setup (left); Loading curve of the fracture test (right).
Fig. 4.1. A model of the specimen created and meshed in ANSYSÒ (left); a raw image of the observed area taken by the camera (middle); image data, together with a part of the mesh (green crosses are nodes of the mesh) (right). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
4.2. Displacement A displacement vector was measured in each node of the mesh, employing the DIC algorithm. Each node defined the centre of a square template 41 41 pixels in size used for the correlation process. The number of evaluated loading states was equal to the number of images acquired during the test (126). A smoothing procedure was employed for a more reliable evaluation of the strain (Chapter 2.3). The resulting full-field displacements at the loading state of the maximum load (9.1 kN) reached in the test are shown in Fig. 4.2. 4.3. Strain The Hencky strain tensor (2.1) was evaluated for each element and for each loading state. The resulting full-field strain tensor components at the maximum load reached in the test (9.1 kN) are shown in Fig. 4.3. Component eyy revealed that the region very close to the tip of the crack reached 1314% deformation at the maximum load. Another part with a high degree of deformation was the centre of the vertical boundary, where the compressive deformation reached 4%. In comparison with the experimental results, Fig. 4.4 shows the strain results within the same region obtained by an FEM simulation performed in ANSYSÒ (using the model from Fig. 4.1 loaded by the same force of 9.1 kN). The problem was solved in ANSYSÒ as 2D plane stress with thickness. The material was modelled as multi-linear, using the stress-strain curve from Fig. 3.1. Calculated component ezz prevalently driven by the component eyy was not plotted here as it was not measured experimentally, therefore direct comparison was not possible. Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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u
v
[mm]
[mm]
1
20
0.6
15
[mm]
0.4
0.8
10
0.2
5
0.6
0 -5
0 -0.2
0.4
-10
-0.4
0.2
-15
-0.6
-20 0
10
[mm]
20
0
10
20
[mm]
Fig. 4.2. Resulting full-field displacements at the maximum load of 9.1 kN; the displacement vector (red arrow) of each nodal point (left), interpolated horizontal (middle) and vertical (right) components of displacement within the whole ligament. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4.3. Experimental results: full-field strain tensor components at the maximum load of 9.1 kN.
It should be emphasized that the results in Fig. 4.3 and in Fig. 4.4 arose from the same mesh. The graphs are scaled in the same ranges, so it is clear to see that the results are in good qualitative and quantitative agreement. There is some discrepancy between the results only in the exx component in the vicinity of the crack tip. Further analysis revealed that the compressive part of exx in the vicinity of the crack tip (the blue V-region in Fig. 4.3) is also present in the FEM simulation, but to a much smaller extent. This discrepancy can be explained in the following way: the simulation is solved as a 2D problem, so it cannot involve an actual triaxial stress at the crack tip, and the simulation cannot include the effect of crack tunneling which was detected analysing the fracture surface in SEM. An error caused by an out-of-plane motion due to the localized contraction at the crack tip is negligible, because it would have a lower order of magnitude. Nevertheless, this inconsistency will be a subject for further investigation. 4.4. Stress The high ductility of the material in conjunction with the plane stress loading condition (specimen surface) predetermined that there would be a significant plastic region in the vicinity of the crack tip. The Von Mises yield criterion (2.2) was used as the condition for material yielding. The expansion of the plastic region during loading, which was revealed by the full-field evaluation of the Von Mises stress, is shown in Fig. 4.5. Global loading curve with marked loading states analysed is plotted left: 1 – linear behaviour of the specimen; 2 – significant plastic region; 3 – maximal force loading. Crack Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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Fig. 4.4. FEM results: full-field strain tensor components at a load of 9.1 kN, obtained by ANSYSÒ.
plasc region
1.
2.
3.
[MPa]
Loading curve
3.
2. 1.
Fig. 4.5. The evaluated Von Mises stress field at three loading states; the plastic region, where the value exceeds the yield strength (260 MPa), is highlighted. Global loading curve is plotted left.
advancing at such loading states was estimated as to be 0.0, 0.1 and 0.7 mm respectively (related experimental work was done with equivalent specimens without DIC measurement). Fig. 4.5 shows clearly that almost the entire ligament in the transverse direction was in the plastic regime at the maximum load reached in the test. This case cannot be considered to be SSY. Another notable area is in the central part of the right vertical boundary line, where the Von Mises stress exceeded the yield strength of the material, but in the compressive manner. Simple elasticity (2.3) was applied in order to evaluate the stress outside the plastic region. Where the Von Mises stress exceeded the yield strength (plastic region), ITP (2.4) was applied. The resulting full-field stress tensor components within the entire ligament at the maximum load of 9.1 kN, evaluated using ITP, are shown in Fig. 4.6. For a comparison with the experimental results, Fig. 4.7 shows the stress results obtained by an FEM simulation performed in ANSYSÒ. In this case, the experimental stress results are far noisier than the strain results. This is mainly because the experimental stress was actually evaluated incrementally (ITP) between particular loading states. Therefore, the measurement error in each loading state is accumulated. In contrast to this, the FEM stress results are obtained considering only the reference state and the final state, and the stress is evaluated using virtual increments along the material uniaxial stress-strain curve. Nevertheless, the qualitative character and the quantitative character of the fields are in agreement.
Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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Fig. 4.6. Experimental results: stress tensor components at the maximum load of 9.1 kN.
Fig. 4.7. FEM results: stress tensor components at the maximum load of 9.1 kN, computed by ANSYSÒ.
4.5. J-integral The J-integral in each loading state was evaluated by (2.5). The size of the integration step was given by the pitch of the orthogonal grid onto which the nodal values were mapped (0.5 mm). Rectangular integration paths for evaluating the Jintegral were led through the elastic region where possible. However, at the state of the maximum load, the right vertical part of the path had to be led unavoidably through the plastic region (state 3. in Fig. 4.5). For this reason, three integration paths (C1, C2, C3) were evaluated, differing only in the position of the right vertical part (see Fig. 4.8), to show whether the plastic region will affect the value of the J-integral. The evolution of the resulting J-integral values during loading for all three paths is plotted in Fig. 4.8. Despite the large plasticity, the J-integral results showed good independence of the integration path. This finding is consistent with an observation made in related work [27]. It has to be emphasized that specimen was loaded monotonically (without unloading), therefore basic condition to be able describe material as non-linear elastic was satisfied (supposed by the J-integral theory). The J-integral value of 310 kJ/m2, when the maximum load was reached for the first time, was taken as the critical J-integral, see Fig. 4.8. 4.6. CTOD CTOD was measured using the technique described in Section 2.6. The smallest achievable initial pitch of the measurement points was 0.6 mm, with templates 17 17 pixels in size. Fig. 4.9 captures the process of crack blunting in three Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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Fig. 4.8. Three rectangular integral paths around a crack (left); Evolution of the resulting J-integral values during loading for all three integration paths, and the determination of the critical value at the maximum load (right) reached in the test.
Fig. 4.9. Blunting of the crack tip and measurement of CTOD employing DIC in two points.
loading states, global loading curve with related loading states is plotted left. The initial pitch was subtracted from the measured values to obtain a correct CTOD. For the purposes of comparison, the J-integral was computed from the measured CTOD, using the well-known relation:
J ¼ mry CTOD
ð4:1Þ
where ry is the yield strength of the material. The dimensionless plastic constraint factor m is known for the CT specimen type, and is defined as [32]:
2 3 rts rts rts þ 4:33 m ¼ 3:62 4:21 2
ry
ry
ry
ð4:2Þ
The resulting evolution of such a CTOD-based J-integral during loading, in comparison with a direct evaluation, is plotted in Fig. 4.10. The results showed good agreement, although the determined critical value of 330 kJ m2 is 6% higher than the value obtained by a direct evaluation. 4.7. The J-integral according to the ASTM standard The J-integral obtained by the proposed experimental methodology was compared with the J-integral obtained by the ASTM standard [3]. In this standard, the evaluation of the J-integral is based on decomposing the J-integral into elastic and plastic components, as:
J ¼ J el þ J pl ¼
K 2I ð1 m2 Þ gpl Apl þ E Bb
ð4:3Þ
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Fig. 4.10. Evolution of the J-integral during loading obtained by three different approaches.
Table 4.1 Results for the critical J-integral evaluated employing three approaches. Low-Carbon Steel (CˇSN 411373) (Fe 360 B: ISO 630:1980)
Direct evaluation as a line integral
Computed from measured CTOD
ASTM standard
JC [kJ m2]
310
330
320
where KI is the stress intensity factor, gpl is the geometry factor, and Apl is the plastic area under the load–load point displacement curve (Fig. 3.3). For a standard CT specimen, the relations for gpl and KI are:
gpl ¼ 2 þ 0:522
b W
ð4:4Þ
P K I ¼ pffiffiffiffiffiffi f ða=WÞ B W
ð4:5Þ
where f(a/W) is a dimensionless function of a/W, defined for the CT specimen as:
f ða=WÞ ¼
½ð2 þ a=WÞð0:886 þ 4:64ða=WÞ 13:32ða=WÞ2 þ 14:72ða=WÞ3 5:6ða=WÞ4 Þ ð1 a=WÞ3=2
ð4:6Þ
The results obtained in this way are shown in Fig. 4.10 in a comparison with the direct evaluation of the J-integral. The results showed very good agreement, although the determined critical value of 320 kJ m2 is 3% higher than the value obtained by the direct evaluation. One aspect of the standard evaluation that should be mentioned is the big difference in the values of the particular components of the J-integral (relation 4.3). For example, the critical value of 320 kJ m2 consists of the elastic part Jel, which is only 20 kJ m2, while the plastic part Jpl is 300 kJ m2. Thus, the plastic component of the J-integral plays a dominant role in the fracture behaviour of a ductile material of this kind. An alarming consideration from the comparison of the three different approaches for evaluating the J-integral (Fig. 4.10) is that after the peak of the maximum load, the values begin to differ significantly. This would result in different J-R curves. The obtained critical values of the J-integral represent the elastic-plastic fracture toughness of the tested material. Table 4.1 summarizes these values evaluated employing the three approaches described above. 5. Conclusions An experimental methodology based on optical measurements and the DIC method for precise stress-strain analysis of a field with a crack and successive extraction of the fracture parameters without the need of FEM has been presented. Several techniques for improving conventional 2D DIC have been developed to achieve a reliable full-field results. Displacement measurements using an adaptive mesh that can reflect displacement/strain gradients in the vicinity of a crack have been introduced. This feature also makes it possible to make a direct comparison with FEM simulations. On the basis of the measured nodal displacements, an extension has been developed for evaluating finite strains and stresses, including plasticity, which is necessary when dealing with ductile metals. The novel displacement smoothing technique, based on a transformation from the Cartesian to the Polar coordinate system, has been introduced. It has been shown that the J-integral can be Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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evaluated as a line integral directly from the fields that are obtained. The optical nature of the method allows simultaneous measurements of the CTOD parameter. The methodology has been employed successfully in an experimental study of the standard fracture test. The test was performed with a CT specimen type made of ductile low-C steel. The measurement mesh for DIC was extracted from an FEM model of the specimen. This enabled a direct comparison of the experimentally evaluated strain and stress fields with the results of an FEM simulation. Both a qualitative comparison and a quantitative comparison showed good agreement. The evaluated stress field revealed a large degree of plasticity that developed during loading, so the material undergoes largescale yielding. The evaluated J-integral exhibited good path independency, even though one segment had to cross the plastic region which developed throughout the ligament during loading. The critical value of the J-integral was determined at the maximum load as 310 kJ m2. The direct evaluation of the J-integral was found to be consistent with the ASTM approach and the CTOD-based measurement. The critical values obtained by the ASTM standard are only 3% higher, while the critical values obtained by the CTOD-based approach are 6% higher. The obtained values correspond to the known values for this class of steels. Finally, it can be concluded that the proposed experimental methodology can be taken as a strong tool in the area of fullfield measurements of crack containing regions. It can be used as a verification tool for engineering approaches, or as the only option for measuring fracture parameters where conventional methods cannot be employed. Acknowledgement This work has been supported by the Technological Grant Agency of the Czech Republic (Grant No.TE02000162) and by the Grant Agency of the Czech Republic (Grant No. GA15-07210S). 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[22] Schreier HW, Braasch JR, Sutton MA. Systematic errors in digital image correlation caused by intensity interpolation. Opt Eng 2000;39(11):2915–21. [23] Sutton MA, Helm JD, Boone ML. Experimental study of crack growth in thin sheet material under tension-torsion loading. Int J Fract 2001;109:285–301. [24] Sutton MA, McNeill SR, Helm JD, Boone ML. Measurement of crack tip opening displacement and full-field deformations during fracture of aerospace materials using 2D and 3D image correlation methods. IUTAM Symp Adv Opt Methods Appl Solid Mech 2000;82:571–80. [25] Sutton MA, Orteu JJ, Schreier HW. Image correlation for shape, motion and deformation measurements. NY: Springer; 2009. [26] Sutton MA et al. The effect of out-of-plane motion on 2D and 3D digital image correlation measurements. Opt Lasers Engng 2008;46(10):746–57. [27] Vavrˇík D, Jandejsek I. Experimental evaluation of contour J integral and energy dissipated in the fracture process zone. Engng Fract Mech 2014;129:14–25. 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Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Analysis of standard fracture toughness test based on digital image correlation data I. Jandejsek ⇑, L. Gajdoš, M. Šperl, D. Vavrˇík The Institute of Theoretical and Applied Mechanics AS CR, v. v. i., Prosecká 809/76, Prague 9 19000, Czech Republic
a r t i c l e
i n f o
Article history: Received 30 March 2016 Received in revised form 26 May 2017 Accepted 29 May 2017 Available online xxxx Keywords: DIC Full-field measurement J-integral CTOD ASTM standard
a b s t r a c t This paper deals with an experimental methodology based on the 2D Digital Image Correlation method for precise full-field in-plane stress-strain analysis of a region containing a crack. The presented methodology is employed as a supporting and comparative method for the standard J-integral testing defined by the ASTM. The methodology utilizing the DIC-based displacements for a subsequent evaluation of strain, stress and final extraction of the J-integral from its definition. This J-integral is compared with the value obtained by the ASTM standard and with the value obtained by the DIC-based CTOD measurement. Several techniques improving the conventional DIC and data post-processing are proposed to achieve reliable full-field strain/stress results. A direct comparison between the experimental and FEM full-field results is carried out. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Stress-strain analysis of materials and structures subjected to various loadings is an essential task in experimental solid mechanics. Various well-known image-based, non-contact and full-field methods like Photoelasticity [10], Moiré [18] and Speckle interferometry [29] have been developed and applied for this purpose in addition to the widely-used strain gauge technique, in which strains are measured locally. Among others, 2D Digital Image Correlation (DIC) [17] has become very popular nowadays in the experimental solid mechanics community for its accessibility and easy-to-use implementation. An important discipline in which DIC-based full-field measurements find usage is experimental fracture mechanics (FM) [24,31]. The main benefit of full-field measurements in the region surrounding a crack is the possibility to extract fracture parameters such as the stress intensity factor or the J-integral. Among the first works, Chiang and Asundi [5] attempted to evaluate the stress intensity factor (SIF) from the displacement field measured by the speckle method. Huntley and Field [11] employed laser speckle photography for full-field measurement, and then optimized numerical results on the basis thereof evaluated the SIF and J integral. The direct extraction of the J-integral and J-R curve from strain/stress analysis obtained by MI was carried out by Dadkhah [6] and Kang [12]. One of the first works utilizing DIC for the extraction of fracture characteristics is by McNeil [15] and Han [8]. Noteworthy works from recent times are by Abanto-Bueno and Lambros [1] and Réthoré [20], in which they deal with the determination of SIF. Becker [4] attempted to measure the J integral using DIC, utilizing the equivalent expression of the J-integral as an area integral. Most of the works mentioned, which have tried to extract the fracture parameters from measured fields, have faced the problem of the insufficient accuracy of DIC results. For that reason, the proposed techniques often combine experimental ⇑ Corresponding author. E-mail address: [email protected] (I. Jandejsek). http://dx.doi.org/10.1016/j.engfracmech.2017.05.045 0013-7944/Ó 2017 Elsevier Ltd. All rights reserved.
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Nomenclature ASTM CT DIC FM FEM ITP NCC SSD SSY a Apl b B CTOD (d) De Dep E I J, JC KI m P p n r t T u U w W W
C
e eH gpl v
r ts ry r
American Society for Testing and Materials compact tension specimen digital image correlation fracture mechanics finite element method incremental theory of plasticity normalized cross-correlation sum-squared difference small scale yielding crack length plastic area under load-displacement curve remaining ligament specimen thickness crack tip opening displacement elastic stiffness matrix elastic-plastic stiffness matrix modulus of elasticity image J-integral, critical J-integral stress intensity factor for mode I plastic constraint factor load, remote force affine transformation coefficients normal vector to C correlation coefficient traction vector image template displacement vector (field) 2 right stretch tensor strain energy density specimen width affine transformation path around crack tip strain tensor (field) Hencky finite strain tensor (field) geometric factor Poisson’s ratio tensile strength yield stress stress tensor (field)
results with numerical simulations based on FEM, trying to optimize the simulations so that the results match the measured fields. Subsequently, on the basis of the best fit, the parameter is extracted from the numerical solution. The low degree of accuracy of DIC can arise from multiple factors: the low resolution of image data, poor quality of the ‘‘speckled” pattern, errors caused by out-of-plane motion, or insufficient smoothing procedures. All these factors can be significantly improved with the use of suitable techniques and corrections, improving DIC accuracy so that there will be no need for FEM. Here is a list of difficulties that one is encountered when attempting to measure displacements and to evaluate strain and stress fields in the region surrounding a crack: Selection of an appropriate surface coating to create speckled pattern with sufficient speckle size and density at a relatively high optical magnification. The pattern has to enable measurement of a wide range of strain (from elastic to plastic deformation). Moreover, due to large strains, such coat has to ensure good adhesion. The presence of high displacement/strain gradients in the vicinity of the crack. The set of points in which the displacements are measured using DIC should reflect this feature. In DIC, the reliability of the evaluated strains depends strongly on smoothing of the measured displacements which are inevitably noisy (locally non-monotonic). The presence of a crack or a notch causes a discontinuity within the displacement field, making smoothing difficult in such a region. The problem of discontinuity has been solved in [19] introducing the X-DIC method for example.
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Evaluating the stress in the plastic regime. This mainly concerns ductile materials, in which large-scale yielding accompanies the fracture process. In addition, the accuracy of DIC itself is susceptible to numerous factors [26,22,28]. It is therefore important to be able to compare DIC-based results with other methods and with FEM simulations. This work addresses all these issues. An experimental methodology based on DIC intended for precise in-plane analyses of a stress-strain field with a crack was proposed and developed. The methodology was employed as a supporting and comparative method for the elastic-plastic standard fracture testing defined by ASTM [3]. The test was carried out with a ductile low-C steel commonly used in the pipeline industry. The principle of the methodology allows a direct comparison between experimental results and the FEM simulation. On the basis of measured fields, the J-integral is evaluated directly as the line integral. This direct evaluation is compared with values obtained by the ASTM standard, and with a simultaneous CTOD measurement. CTOD is also measured by DIC using two templates located above and below the crack tip. In contrast to COD measurement performed in [9,23], templates are located coaxially with the crack tip. 2. Methodology The experimental-numerical methodology for a precise analysis of the stress-strain field with a crack that is proposed and developed in this work encompasses the following constituent parts:
An in-house developed customizable DIC algorithm. Surface coating of the specimen using an airbrush gun. The use of an adaptive mesh for full-field displacement measurements in the vicinity of the stress concentrator. A novel solution for smoothing of measured displacements. An evaluation of finite strain and stress (including plasticity). A direct evaluation of the J-integral from its definition. Measurement of CTOD using DIC.
Principles of the well-known DIC method have already been described in many papers, see [25] for instance. The essential principle is the tracking (displacement) of small templates through an image sequence representing the deformation of the surface of a specimen. The in-house DIC algorithm developed in this work is based on two basic approaches: Normalized cross-correlation (NCC) [13], and Sum-squared difference (SSD) [14]. The code of our DIC is written in MATLABÒ. The algorithm is described in detail in the Appendix. A telecentric lens is used to reduce errors caused by out-of-plane motion [26]. These errors are much more significant when using a standard lens due to the parallax. 2.1. Surface coating The surface of the specimen has to be treated to exhibit a random ‘‘speckled” pattern. For the magnification required for the experiment carried out in this work, a conventional spray was found to be insufficient for the speckle size. Moreover, tests showed that a coat of this kind tends to peel off the surface during higher deformation. Finally, a fine airbrush with matt black and white acrylic paint (RevellÒ – Aqua Color) was tested, and was found to yield an excellent coat in terms of fineness of the speckled pattern and good adhesion. The usability of the created pattern is estimated using NCC of a typical template with itself. The estimation is associated with the peak width of NCC at a pre-defined value, typically at 0.5. The standard says [25] that the appropriate size of this peak width is in the range of <3, 6> pixels. Fig. 2.1 shows the typical template prepared using the airbrush and NCC thereof. It can be seen that the peak width of NCC at a value of 0.5 is about 5 pixels, thus optimal. 2.2. Use of adaptive mesh An advantage of the DIC method is that it can be used for measuring the displacement almost anywhere within the speckled pattern. This makes it possible to form a set of measurement points as an adaptive mesh. This is particularly beneficial in cases when large displacement/strain gradients are expected, e.g. in the vicinity of a stress concentrator. In such regions, the mesh is refined. For this purpose, the mesh was created using a linear quadrilateral element (so-called Q4). An example is shown in Fig. 2.2; the mesh is added to a raw image of the vicinity of a crack. Each node of the mesh defines the centre of a square template, the displacement of which is measured applying the DIC algorithm. In this work, the mesh was prepared using the ANSYSÒ meshing, but any other meshing software can be used for this purpose. The experimental results can therefore be compared directly with an FEM simulation of the same problem. Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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1.1 mm
1.1 mm
0.5
≈ 5 px
Fig. 2.1. The image template of the airbrush-prepared spackle pattern of size 41 41 pixels (1.1 1.1 mm2 physical area) (left); auto-NCC function of the template with itself (middle); Cross-section of NCC and determination of the peak width at value of 0.5 (right).
Displacement of the node using DIC
Strain and stress of the element
u fague crack p template
Q4 element
Fig. 2.2. A raw image of the crack vicinity taken by a camera (left); Image data with a mesh; in each node, the displacement is measured employing the DIC algorithm (middle). The strain and the stress are evaluated for each element (right).
2.3. Displacement smoothing When the displacement vector at each node is known, the strain tensor at each element can be calculated. However, the measured displacements are intrinsically always noisy. For a more reliable evaluation of the strain, a smoothing procedure based on a spline function approximation [7] is employed. This procedure is performed in the polar coordinate system (PCS), with the centre at the tip of the crack to avoid discontinuity of the displacement field within the notch. Fig. 2.3 shows an example of the vertical displacement field around a crack (loading in mode I). There is a step discontinuity of the displacements within the crack that complicates the smoothing procedure. When smoothing in conventional Cartesian coordinate system (CCS), the region above and below the crack are corrupted by the discontinuity, see Fig. 2.3. The problem disappears when the PCS is used. Moreover, the field smoothed in PCS reflects the raw field more accurately, see Fig. 2.3.
Raw displacement field
Smoothing in CCS
Smoothing in PCS
Cross secon
Corrupted
Disconnuity Due to crack
Corrupted field
Raw displacement Smoothing in CCS Smoothing in PCS
Fig. 2.3. The raw measured vertical displacement field (left); Smoothing of the field in CCS and PCS, the field is corrupted above and below the crack in the case of CCS (middle); Cross section over the crack (right).
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2.4. Strain and stress After the displacement smoothing procedure, the strain tensor at each element is evaluated. In this work, due to the presence of large displacement gradients, the finite Hencky (true) finite strain tensor is used:
eH ¼ lnðUÞ
ð2:1Þ
where U is the right stretch tensor. The strain tensor is associated with the centre of mass of the element in the deformed configuration (Fig. 2.2). When the strains are known, the stress can be evaluated using material constitutive relations. The high ductility of the material tested in this work in conjunction with the plane stress loading condition (specimen surface) predetermined that there would be a significant plastic region in the vicinity of the crack tip. The Von Mises yield criterion is used as the condition for material yielding:
f ðr; ry Þ ¼ rm ry ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2xx rxx ryy þ r2yy þ 3s2xy ry ¼ 0
ð2:2Þ
where rm is the Von Misses stress and ry is the yield strength of the material. Plasticity occurs when the Von Misses stress meets the yield strength of the material. Where the Von Misses stress does not reach the yield value, the stress tensor is computed using well-known Hooke’s law (plane stress) as:
r ¼ De e
ð2:3Þ
where De is the elastic stiffness matrix. When the stress exceeds the yield strength, the incremental theory of plasticity (ITP) is employed:
dr ¼ Dep de
ð2:4Þ
where Dep is the elastic-plastic stiffness matrix. The uniaxial stress-strain curve of the material serves for evaluating Dep in each step. The resulting stress tensor, and also the strain, are associated with the centre of mass of the element in the deformed configuration (Fig. 2.2). 2.5. Evaluating the J-integral If we know the displacement, strain and stress fields, the J-integral can be evaluated directly from the well-known definition [21]:
J¼
Z @u ds wdy t @x C
ð2:5Þ
Re where w ¼ 0 rde is the strain energy density, t is the traction vector defined according to the outward normal n along path C as: t ¼ r n. Finally, u is the displacement. The integral is evaluated in the counter clockwise direction, starting from the lower crack boundary and continuing along path C to the upper boundary of the crack, see Fig. 2.4. For the purposes of evaluating the J-integral from the experimental data, the nodal values are interpolated onto a regular orthogonal grid. In this case, the J-integral can be evaluated simply along a rectangular-shaped path, as is shown in Fig. 2.4. Due to the discrete nature of the data, the original J-integral relation (2.5) is replaced by the summation of the discrete increments of length Ds, given by the pitch of the orthogonal grid onto which the nodal values are interpolated, see Fig. 2.4.
Interpolaon of mesh values to regular orthogonal grid
y
crack
ds
.
n
x
Γ
Path for the J-integral Fig. 2.4. Line J-integral around a crack (left); Interpolation of nodal data onto an orthogonal grid and the rectangular path for evaluating the J-integral (right).
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The interpolation is performed using MATLAB function TriScatteredInterp. The pitch of the grid is selected so as to have approximately the same size as the smallest element of the mesh. 2.6. Measurement of CTOD CTOD [30] is the opening displacement of the original crack tip due to blunting, see Fig. 2.5. In this work, CTOD is measured directly from the acquired images. For this purpose, displacements of just two points are measured, employing the DIC method. These points are located coaxially with the initial crack tip, as illustrated in Fig. 2.5. Fig. 2.5 shows that there has to be some initial non-zero pitch between two points, otherwise square templates would interfere with one another. This pitch is given by the smallest possible size of the template that makes the DIC measurement possible. However, the proper definition of CTOD has zero initial pitch. It is assumed that the initial pitch is so small that it is a sufficient approximation of the proper CTOD. Naturally, the initial pitch is subtracted from the measured displacement values. 2.7. Error estimation The experimental system used in this work was tested for uncertainties according to Standards Project for Optical Techniques of Strain Measurement (SPOTS) [16]. For the template of size 41 41 pixels, the uncertainty of strain was determined as 55 lStrain. 3. Experimental work 3.1. Material and specimen A fracture experiment was carried out with a compact tension (CT) specimen made of low-C steel (CˇSN 411373 which corresponds to Fe 360 B according to ISO 630:1980), which is widely used in the pipeline industry. The experiment met the ASTM standards for J-integral testing [3]. The essential mechanical properties of the material are summarized in Table 3.1. The uniaxial true stress–strain curve of the material (as supplied by the producer) is shown in Fig. 3.1. The geometry of the specimen, and the dimensions and the rolling direction of the sheet from which the specimen was cut are depicted in Fig. 3.2. Using ASTM standard labelling of the CT specimen, the three important characteristic dimensions were the width W = 50 mm, the crack length a = 25 mm, and the thickness B = 5 mm. Other dimensions depend on these values. The initial crack 1 mm in length was created by the standard fatigue method [2]. A speckled pattern was prepared on the optically observed side of the specimen, using an airbrush gun. 3.2. Fracture test The specimen was subjected to monotonic tension loading (opening mode I) under conditions of a constant grip displacement velocity of 1 mm/min until the load point displacement reached 3.8 mm, and then it was unloaded. The load was measured using a load cell with a read-out frequency of 1 Hz. The experimental setup with the necessary equipment for the DIC method is shown in Fig. 3.3left. The speckled side of the specimen was optically observed during the test. Images were acquired with the frame rate of 1 fps, using a 5 MPixel monochromatic camera (MANTAÒ) with a mounted telecentric lens (Opto engineeringÒ). The surface of the specimen was illuminated using a circular diffusion light to obtain sufficient light intensity in the images, and also to avoid reflection artefacts. The images were stored in monochrome RAW format. The resolution of the images was 2056 2452 pixels, and the resulting image scale was 1 pixel = 28 lm. The resulting loading curve of the test is plotted in Fig. 3.3. The maximum load reached in the test was 9.1 kN. The force measurement error given by the accuracy of the
Fig. 2.5. Definition of CTOD (left). CTOD measurement by DIC. Displacement measurement of two points located coaxially with the crack tip (right).
Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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I. Jandejsek et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx Table 3.1 Mechanical properties of low-C steel (CˇSN 411373, Fe 360 B according to ISO 630:1980). Modulus of elasticity E [MPa]
Poisson’s ratio m [–]
Yield strength ry [MPa]
Tensile strength rts [MPa]
Ultimate elongation [%]
206,000
0.3
260
380
13
ˇ SN 411373, Fe 360 B according to ISO 630:1980). Fig. 3.1. The uniaxial true stress-strain curve of steel (C
Fig. 3.2. CT specimen geometry with dimensions (left); orientation of the specimen to the rolling direction of the sheet.
Lucas 10 kN load cell is specified as to be <0.2%. Similarly, the displacement measurement error given by the accuracy of the Instron clip-on gauge is specified as to be <0.5%.
4. Results The image data provides input for measurements of full-field displacements and for the subsequent evaluation of strains and stresses and the final extraction of the J-integral. The data processing methodology is described in Section 2, so only the results of the particular steps will be shown and commented on here.
4.1. Mesh A model of the CT specimen was created in ANSYSÒ. The model was meshed using a linear quadrilateral element (Q4), see Fig. 4.1. A part of the mesh covering the optically observed region was exported from ANSYSÒ to MATLABÒ, where the DIC measurement was performed. The nodal points are added ‘‘virtually” to the reference image (see Fig. 4.1). The resulting mesh for the DIC measurement consisted of 696 nodes and 641 elements. Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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Load cell
10
light
8 P
Load [kN]
telecentric lens specimen
camera
max
= 9.1 kN
6 4 2 0 0
Posioning device
1
2
3
4
Load point displacement [mm]
Fig. 3.3. Experimental setup (left); Loading curve of the fracture test (right).
Fig. 4.1. A model of the specimen created and meshed in ANSYSÒ (left); a raw image of the observed area taken by the camera (middle); image data, together with a part of the mesh (green crosses are nodes of the mesh) (right). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
4.2. Displacement A displacement vector was measured in each node of the mesh, employing the DIC algorithm. Each node defined the centre of a square template 41 41 pixels in size used for the correlation process. The number of evaluated loading states was equal to the number of images acquired during the test (126). A smoothing procedure was employed for a more reliable evaluation of the strain (Chapter 2.3). The resulting full-field displacements at the loading state of the maximum load (9.1 kN) reached in the test are shown in Fig. 4.2. 4.3. Strain The Hencky strain tensor (2.1) was evaluated for each element and for each loading state. The resulting full-field strain tensor components at the maximum load reached in the test (9.1 kN) are shown in Fig. 4.3. Component eyy revealed that the region very close to the tip of the crack reached 1314% deformation at the maximum load. Another part with a high degree of deformation was the centre of the vertical boundary, where the compressive deformation reached 4%. In comparison with the experimental results, Fig. 4.4 shows the strain results within the same region obtained by an FEM simulation performed in ANSYSÒ (using the model from Fig. 4.1 loaded by the same force of 9.1 kN). The problem was solved in ANSYSÒ as 2D plane stress with thickness. The material was modelled as multi-linear, using the stress-strain curve from Fig. 3.1. Calculated component ezz prevalently driven by the component eyy was not plotted here as it was not measured experimentally, therefore direct comparison was not possible. Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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u
v
[mm]
[mm]
1
20
0.6
15
[mm]
0.4
0.8
10
0.2
5
0.6
0 -5
0 -0.2
0.4
-10
-0.4
0.2
-15
-0.6
-20 0
10
[mm]
20
0
10
20
[mm]
Fig. 4.2. Resulting full-field displacements at the maximum load of 9.1 kN; the displacement vector (red arrow) of each nodal point (left), interpolated horizontal (middle) and vertical (right) components of displacement within the whole ligament. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4.3. Experimental results: full-field strain tensor components at the maximum load of 9.1 kN.
It should be emphasized that the results in Fig. 4.3 and in Fig. 4.4 arose from the same mesh. The graphs are scaled in the same ranges, so it is clear to see that the results are in good qualitative and quantitative agreement. There is some discrepancy between the results only in the exx component in the vicinity of the crack tip. Further analysis revealed that the compressive part of exx in the vicinity of the crack tip (the blue V-region in Fig. 4.3) is also present in the FEM simulation, but to a much smaller extent. This discrepancy can be explained in the following way: the simulation is solved as a 2D problem, so it cannot involve an actual triaxial stress at the crack tip, and the simulation cannot include the effect of crack tunneling which was detected analysing the fracture surface in SEM. An error caused by an out-of-plane motion due to the localized contraction at the crack tip is negligible, because it would have a lower order of magnitude. Nevertheless, this inconsistency will be a subject for further investigation. 4.4. Stress The high ductility of the material in conjunction with the plane stress loading condition (specimen surface) predetermined that there would be a significant plastic region in the vicinity of the crack tip. The Von Mises yield criterion (2.2) was used as the condition for material yielding. The expansion of the plastic region during loading, which was revealed by the full-field evaluation of the Von Mises stress, is shown in Fig. 4.5. Global loading curve with marked loading states analysed is plotted left: 1 – linear behaviour of the specimen; 2 – significant plastic region; 3 – maximal force loading. Crack Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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Fig. 4.4. FEM results: full-field strain tensor components at a load of 9.1 kN, obtained by ANSYSÒ.
plasc region
1.
2.
3.
[MPa]
Loading curve
3.
2. 1.
Fig. 4.5. The evaluated Von Mises stress field at three loading states; the plastic region, where the value exceeds the yield strength (260 MPa), is highlighted. Global loading curve is plotted left.
advancing at such loading states was estimated as to be 0.0, 0.1 and 0.7 mm respectively (related experimental work was done with equivalent specimens without DIC measurement). Fig. 4.5 shows clearly that almost the entire ligament in the transverse direction was in the plastic regime at the maximum load reached in the test. This case cannot be considered to be SSY. Another notable area is in the central part of the right vertical boundary line, where the Von Mises stress exceeded the yield strength of the material, but in the compressive manner. Simple elasticity (2.3) was applied in order to evaluate the stress outside the plastic region. Where the Von Mises stress exceeded the yield strength (plastic region), ITP (2.4) was applied. The resulting full-field stress tensor components within the entire ligament at the maximum load of 9.1 kN, evaluated using ITP, are shown in Fig. 4.6. For a comparison with the experimental results, Fig. 4.7 shows the stress results obtained by an FEM simulation performed in ANSYSÒ. In this case, the experimental stress results are far noisier than the strain results. This is mainly because the experimental stress was actually evaluated incrementally (ITP) between particular loading states. Therefore, the measurement error in each loading state is accumulated. In contrast to this, the FEM stress results are obtained considering only the reference state and the final state, and the stress is evaluated using virtual increments along the material uniaxial stress-strain curve. Nevertheless, the qualitative character and the quantitative character of the fields are in agreement.
Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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Fig. 4.6. Experimental results: stress tensor components at the maximum load of 9.1 kN.
Fig. 4.7. FEM results: stress tensor components at the maximum load of 9.1 kN, computed by ANSYSÒ.
4.5. J-integral The J-integral in each loading state was evaluated by (2.5). The size of the integration step was given by the pitch of the orthogonal grid onto which the nodal values were mapped (0.5 mm). Rectangular integration paths for evaluating the Jintegral were led through the elastic region where possible. However, at the state of the maximum load, the right vertical part of the path had to be led unavoidably through the plastic region (state 3. in Fig. 4.5). For this reason, three integration paths (C1, C2, C3) were evaluated, differing only in the position of the right vertical part (see Fig. 4.8), to show whether the plastic region will affect the value of the J-integral. The evolution of the resulting J-integral values during loading for all three paths is plotted in Fig. 4.8. Despite the large plasticity, the J-integral results showed good independence of the integration path. This finding is consistent with an observation made in related work [27]. It has to be emphasized that specimen was loaded monotonically (without unloading), therefore basic condition to be able describe material as non-linear elastic was satisfied (supposed by the J-integral theory). The J-integral value of 310 kJ/m2, when the maximum load was reached for the first time, was taken as the critical J-integral, see Fig. 4.8. 4.6. CTOD CTOD was measured using the technique described in Section 2.6. The smallest achievable initial pitch of the measurement points was 0.6 mm, with templates 17 17 pixels in size. Fig. 4.9 captures the process of crack blunting in three Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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Fig. 4.8. Three rectangular integral paths around a crack (left); Evolution of the resulting J-integral values during loading for all three integration paths, and the determination of the critical value at the maximum load (right) reached in the test.
Fig. 4.9. Blunting of the crack tip and measurement of CTOD employing DIC in two points.
loading states, global loading curve with related loading states is plotted left. The initial pitch was subtracted from the measured values to obtain a correct CTOD. For the purposes of comparison, the J-integral was computed from the measured CTOD, using the well-known relation:
J ¼ mry CTOD
ð4:1Þ
where ry is the yield strength of the material. The dimensionless plastic constraint factor m is known for the CT specimen type, and is defined as [32]:
2 3 rts rts rts þ 4:33 m ¼ 3:62 4:21 2
ry
ry
ry
ð4:2Þ
The resulting evolution of such a CTOD-based J-integral during loading, in comparison with a direct evaluation, is plotted in Fig. 4.10. The results showed good agreement, although the determined critical value of 330 kJ m2 is 6% higher than the value obtained by a direct evaluation. 4.7. The J-integral according to the ASTM standard The J-integral obtained by the proposed experimental methodology was compared with the J-integral obtained by the ASTM standard [3]. In this standard, the evaluation of the J-integral is based on decomposing the J-integral into elastic and plastic components, as:
J ¼ J el þ J pl ¼
K 2I ð1 m2 Þ gpl Apl þ E Bb
ð4:3Þ
Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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Fig. 4.10. Evolution of the J-integral during loading obtained by three different approaches.
Table 4.1 Results for the critical J-integral evaluated employing three approaches. Low-Carbon Steel (CˇSN 411373) (Fe 360 B: ISO 630:1980)
Direct evaluation as a line integral
Computed from measured CTOD
ASTM standard
JC [kJ m2]
310
330
320
where KI is the stress intensity factor, gpl is the geometry factor, and Apl is the plastic area under the load–load point displacement curve (Fig. 3.3). For a standard CT specimen, the relations for gpl and KI are:
gpl ¼ 2 þ 0:522
b W
ð4:4Þ
P K I ¼ pffiffiffiffiffiffi f ða=WÞ B W
ð4:5Þ
where f(a/W) is a dimensionless function of a/W, defined for the CT specimen as:
f ða=WÞ ¼
½ð2 þ a=WÞð0:886 þ 4:64ða=WÞ 13:32ða=WÞ2 þ 14:72ða=WÞ3 5:6ða=WÞ4 Þ ð1 a=WÞ3=2
ð4:6Þ
The results obtained in this way are shown in Fig. 4.10 in a comparison with the direct evaluation of the J-integral. The results showed very good agreement, although the determined critical value of 320 kJ m2 is 3% higher than the value obtained by the direct evaluation. One aspect of the standard evaluation that should be mentioned is the big difference in the values of the particular components of the J-integral (relation 4.3). For example, the critical value of 320 kJ m2 consists of the elastic part Jel, which is only 20 kJ m2, while the plastic part Jpl is 300 kJ m2. Thus, the plastic component of the J-integral plays a dominant role in the fracture behaviour of a ductile material of this kind. An alarming consideration from the comparison of the three different approaches for evaluating the J-integral (Fig. 4.10) is that after the peak of the maximum load, the values begin to differ significantly. This would result in different J-R curves. The obtained critical values of the J-integral represent the elastic-plastic fracture toughness of the tested material. Table 4.1 summarizes these values evaluated employing the three approaches described above. 5. Conclusions An experimental methodology based on optical measurements and the DIC method for precise stress-strain analysis of a field with a crack and successive extraction of the fracture parameters without the need of FEM has been presented. Several techniques for improving conventional 2D DIC have been developed to achieve a reliable full-field results. Displacement measurements using an adaptive mesh that can reflect displacement/strain gradients in the vicinity of a crack have been introduced. This feature also makes it possible to make a direct comparison with FEM simulations. On the basis of the measured nodal displacements, an extension has been developed for evaluating finite strains and stresses, including plasticity, which is necessary when dealing with ductile metals. The novel displacement smoothing technique, based on a transformation from the Cartesian to the Polar coordinate system, has been introduced. It has been shown that the J-integral can be Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045
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evaluated as a line integral directly from the fields that are obtained. The optical nature of the method allows simultaneous measurements of the CTOD parameter. The methodology has been employed successfully in an experimental study of the standard fracture test. The test was performed with a CT specimen type made of ductile low-C steel. The measurement mesh for DIC was extracted from an FEM model of the specimen. This enabled a direct comparison of the experimentally evaluated strain and stress fields with the results of an FEM simulation. Both a qualitative comparison and a quantitative comparison showed good agreement. The evaluated stress field revealed a large degree of plasticity that developed during loading, so the material undergoes largescale yielding. The evaluated J-integral exhibited good path independency, even though one segment had to cross the plastic region which developed throughout the ligament during loading. The critical value of the J-integral was determined at the maximum load as 310 kJ m2. The direct evaluation of the J-integral was found to be consistent with the ASTM approach and the CTOD-based measurement. The critical values obtained by the ASTM standard are only 3% higher, while the critical values obtained by the CTOD-based approach are 6% higher. The obtained values correspond to the known values for this class of steels. Finally, it can be concluded that the proposed experimental methodology can be taken as a strong tool in the area of fullfield measurements of crack containing regions. It can be used as a verification tool for engineering approaches, or as the only option for measuring fracture parameters where conventional methods cannot be employed. Acknowledgement This work has been supported by the Technological Grant Agency of the Czech Republic (Grant No.TE02000162) and by the Grant Agency of the Czech Republic (Grant No. GA15-07210S). 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Please cite this article in press as: Jandejsek I et al. Analysis of standard fracture toughness test based on digital image correlation data. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.045