Characteristics of selected measures of stress triaxiality near the crack tip for 145Cr6 steel - 3D issues for stationary cracks

1 Intoduction (based on [3, 4, 5, 6])

The 145Cr6 steel is the basic material used for the production of cold working tools. In the previous nomenclature, 145Cr6 steel was designated as NC6. This material is considerably resistant to abrasion, is characterized by high hardness, shows medium hardenability. The 145Cr5 steel characterized by as well as low tendency to deformation and significant regularity of dimensional changes during hardening. In addition, 145Cr6 steel is steel with good machin-ability and blade durability, but only under operating conditions that do not cause excessive heating of the working tool. This material also tends to form a carbide mesh at the boundaries of austenite grains, which remains after soft annealing and hardening, reducing tool ductility. The 145Cr6 steel is used to produce tools and workpieces up to 15 mm thick. It is used to make cutting tools, pressing and measuring tools that should not change dimensions after hardening, such as gauges, taps, dies, reamers, file cutters, paper and tobacco knives, circular saws for cold metal cutting, deep drawing cold stamping dies, cutting dies. In some cases, this steel is sometimes used for the production of bolts, dowels, small beams that carry quite significant loads.

Professional literature and industry portals provide some characteristics of the mechanical properties of this steel, however, information about the parameters characterizing the fracture of 145Cr6 steel is quite poor, which results from the fact that this material does not appear in Eurocodes [6] as the basic construction material used in the construction of machinery and equipment or in the construction of buildings (civil engineering, road engineering, etc.). A rather interesting fact about 145Cr6 steel is shown in [5],when after quenching and next after tempering, this material characterized by the decrease in hardness as the tempering temperature increases and will become a material more and more plastic [5], with quite a high yield point and a high ultimate strength.

2 About experimental research and critical values of the J - integral – J_C (based on [3, 4, 5, 6])

In papers [3, 4], the topic of assessment of parameters characterizing the fracture of 145Cr6 steel was discussed, using experimental tests and numerical calculations. In the first paper [3], the experimental research program was discussed, while in the second paper the details of numerical calculations were presented, which were carried out for the case of three-dimensional models and the case of plane strain state dominance, after which the results of preliminary numerical calculations were presented. Among the experimental results presented in [3], one can find information on the determined J-R curves for three groups of SEN(B) specimens made of 145Cr6 steel, characterized by different thickness. For each thickness, experimental program was included the use of specimens with different relative crack length a/W. In addition, the critical value of the J-integral, determined by J_C, was estimated for each specimen, which, according to the requirements of ASTM [7], can be considered to be fracture toughness for plane strain state domination. Among the results of numerical calculations, which were given in [4] for critical moments (characterized by critical value of J-integral denoted as J_C), values of measures of geometrical constraints denoted as T_m and Q^*_m [8, 9, 10] were determined:

• T_m – the average value of the T_z parameter, which is definition of the level of stress triaxiality according to the T_z formula T_z = σ_xx /(σ_yy + σ_zz) [11, 12, 13];

• Q^*_m – the thickness-averaged and normalized by the yield stress difference between the actual stress value – estimated by numerical calculations, and the solution proposed by Guo Wanlin [11, 12, 13].

Both parameter can be used to more accurately describe the stress field near crack tip in elastic-plastic materials in the case of analysis of three-dimensional problems [8], or to propose new fracture criteria [9, 10], allowing the estimation of real fracture toughness.

The concept of critical moment in fracture mechanics requires clarification here. The critical moment is the point on the J-R curve, which according to ASTM assumptions [7] corresponds to the increase in the crack length – crack growth – da = 0.2mm. It is, according to ASTM recommendations [7], the conventional moment of initiation of the crack growth, for which the critical value of fracture toughness is determined - in this case, the critical fracture toughness is critical value of the J-integral, denoted as J_C.

The paper [4] also presents in graphic form the changes in these measures of geometric constraints along with the increasing external load, expressed by J-integral, as well as the impact of the relative crack length and specimen thickness on the changes in the T_m = f (J) and Q^*_m = f (J) curves [4].

The results presented in [3, 4] may prove useful in solving engineering problems in the field of assessing the actual fracture toughness, determined according to one of many fracture criteria, as well as in estimating the stress distribution near the crack front in structural elements made of 145Cr6 steel. However, for all real specimens analyzed in [3, 4, 5], for which bidirectional numerical analysis was performed, the stress state prior to the crack point was not assessed (both for calculations in the case of assumption of plane deformation and calculations for 3D cases), as well as changes in the values of stress triaxiality parameters for three-dimensional cases were not discussed. For this reason, in recent years the area of analysis of collected experimental and numerical data has been expanded and it was decided to supplement the conducted research with new elements. This paper presents selected results of numerical analysis conducted for three-dimensional cases, focusing on stress distribution near the crack tip and the distribution of selected measures of stress triaxiality parameters.

Presented in [3] laboratory tests data, show, that each specimen used in the experimental research program ensures domination of the plane strain state. That means, that determined for all specimens critical value of the J-integral, denoted as J_c, may be considered as fracture toughness for plane strain state domination, determined in accordance with standard as J_IC. For the purposes of this study, in order to analyze the level of stress triaxiality, it was decided to use the experimental results for three SEN(B) specimens characterized by different thicknesses B = {5, 10, 15} mm, characterized by relative crack lengths a/W ≈ 0.30 – for subsequent thicknesses the relative crack length was a/W = {0.32, 0.35, 0.29} respectively. The width of the specimens W was equal to W = 25mm and the support spacing S = 100mm. The total length of the specimen was L = 120mm. Figure 1a [3, 5] shows the geometry of the SEN(B) specimen, while Figure 1b shows the stress-strain curves registered in the laboratory, on the basis of which the material constants required to perform the numerical analysis were determined [3, 5]. For the purpose of numerical calculations, the stress-strain curves were described by the power law (1):

Figure 1

a) The geometry of the SEN(B) specimen used to assess the fracture toughness of 145Cr6 steel; b) Engineering tensile diagrams of 145Cr6 steel, together with the determined values of material constants assumed in further FEM analysis and the stress-strain curve which was used in FEM analysis; (based on [3, 4, 5])

it was assumed that the constant α = 1.

For the purposes of this paper, the 145Cr6 steel used in the tests can be characterized by the following values of material constants: Young’s modulus E = 200GPa, yield point σ₀ = 927MPa, strain corresponding to the yield point ϵ₀ = 0.00464, tensile strength σ_m = 1040MPa, corresponding strain tensile strength ϵ_m = 0.09775 Poisson’s ratio ν = 0.3 and the strain hardening exponent in the Ramberg-Osgood law n = 28. The assumed values of material constants guarantee obtaining a model of stress-strain curve which is the lower boundary of the experimentally determined charts σ = f(ϵ), shown in Figure 1b [3, 4, 5], which is consistent with the desire to obtain conservative solutions, referred to in [1, 2, 11, 12].

Figure 2 presents examples of the results of experimental tests, which are presented in [3, 5] – all the curves presented relate to specimens with a relative crack length a/W close to a/W ≈ 0.3. Figure 2a shows the change in the potential drop signal φ as a function of load line displacement v_LL – the potential drop signal in accordance with [7, 13] is used to determine the increase in crack length denoted as da, which is required to plot J-R curves, which are necessary to determine the critical value J-integral - J_C. Figure 2b presents a comparison of forces P acting on the specimen, as a function of load line displacement v_LL – the thicker the specimen, the greater the force value. In contrast, Figure 2c is a summary of the determined J-R curves for the three mentioned SEN(B) specimens. There are slight differences between J-R curves for specimens with thickness B = 10mm and B = 15mm, for crack length increments da > 5mm. These curves significantly differ in terms of the increase in crack length da = 〈0; 3〉mm [3, 5].

Figure 2

The comparison of the signals (quantities) of: a) potential drop φ in the function of the load line displacement v_LL; b) force P in the function of the load line displacement v_LL; c) J-integral (determined based on laboratory tests) as a function of the increase of the crack length da; for three SEN(B) specimens made of 145Cr6 steel; (based on [3, 4, 5])

Figure 3 is a summary of the signals registered in the laboratory and the J-R curves determined on their basis, separately for each of the SEN(B) specimen analyzed in the numerical program, of different thickness B. These quantities can be found on the comparative graphs shown in Figure 2, however in Figure 3, the change of some quantities can be analyzed much more clearly. Table 1 presents a summary of the results of experimental tests and selected results of numerical calculations in the scope of assessment of geometrical constraints measures – values averaged over thickness (T_m – measure of constraints in the thickness direction (the out-of plane constraints), Q^*_m – measure of in-plane constraints, Q – the Q-stress defined by O’Dowd and Shih [14, 15] - also a measure of in-plane constraints determined at the dominance plane strain state), for SEN(B) specimens (used further in the numerical program), for the moment of crack initiation, which corresponds to the moment for which critical value of the J-integral, denoted by J_C [3, 4, 5]. The impact of thickness on the change in the critical value of the J-integral and the measures of geometrical constraints determined in [4] are presented in Figure 4 [3, 4, 5].

Figure 3

List of signals (quantities) registered during experimental tests (φ = φ(v_LL), P = P(v_LL)) and determined J-R curves (J = J(da)) for three SEN(B) specimens made of 145Cr6 steel: a) B = 5mm, a/W = 0.32; b) B = 10mm, a/W = 0.35; c) B = 15mm, a/W = 0.29; (based on [3, 4, 5])

Figure 4

The influence of thickness on the change of the critical value of the J - integral denoted as J_C and the measures of geometrical constraints determined in [4] (based on [3, 4, 5])

As can be seen (Figure 4), the increase in specimen thickness is usually accompanied by a decrease in the value of Q-stress determined for the dominance of the plane strain state – the lower the value of the Q-stress, the lower the level of geometric constraints. As the thickness of the specimen increases, a slight increase in the T_m parameter is observed,which is almost constant for the thickness B=10mm and B = 15mm. The increase in thickness is also accompanied by a decrease in the value of the Q^*_m parameter, which is a measure of in-plane constraints determined for the case of three-dimensional structural elements [8] – this change may seem insignificant, but it is worth noting here.

The parameters of geometrical constraints, mentioned above, can be considered as some measures of stress tri-axiality, however, currently in the literature various researchers are increasingly focusing on stress triaxiality parameters determined directly from the estimated FEM of stress distributions to be independent in this undertaking from the adopted material model, which generally determines quantities that can be further analyzed. Below in the next section the characteristics of the quantities considered to be a stress triaxiality parameters will be clarified, which will be analyzed in the following sections of this paper.

3 Some aspects of numerical calculation (based on [4])

In the 3D Finite Element Method (FEM) numerical analysis, the geometry of the SEN(B) specimens used in the experimental program was accurately reproduced. All calculations were carried out assuming small deformations and small displacement. Calculations for the 3D case were also made with the help of the ADINA SYSTEM 8.4 program [32, 33], solving the contact problem. A quarter of the specimen was modelled [4]. The standard twenty-node three-dimensional finite elements (FEs) with 27 numerical integration points, assuming “MIXED" interpolation formulation in FE. The region close to crack tip, with a radius equal to (0.5÷1) mm, was divided into (36÷50) FEs, the smallest of which at the crack tip was (20÷100) times smaller than the last. This meant that this element was (1/2000÷1/10000) of the specimen width W, and the largest one modelling the crack tip region was (1/100÷1/1000) of the specimen width. The crack tip was modelled as a quarter of an arc with a radius of r_w = (1 ÷ 2) · 10⁻⁶m. It turned out to be 40000 times smaller than the specimen width. This arch is divided into 7 equal parts [4].

The thickness quadrant of the SEN(B) specimen was divided into nine layers, located in the following coordinates: x₃/B = { 0; 0.119; 0.222; 0.309; 0.379; 0.434; 0.472; 0.483; 0.494; 0.5}, whereas the thickness of the layers decreased in the direction from the specimen axis (x₃/B = 0.0) to the edge of the specimen (x₃/B = 0.5). It should be noted that the layer located in the specimen axis was (20÷50) times larger than the layer on the edge of the specimen, which is associated with a large stress gradient in this area. In total, the specimen was modelled using 2488 FEs, which consisted of 12142 nodes [4].

All parameters and details of the numerical model (such as the radius of rounding at the crack tip, size of the FE, type of FE, division of the structural element into FEs, division in the direction of thickness) are the result of a number of tests carried out during calculations (attempts were made to determine such parameters that the results obtained coincide), as well as the guidelines contained in [8]. The calculation used a homogeneous, isotropic material model with the Huber-Misses-Hencky plasticity condition. The constitutive relationship describing the material analyzed is described by the relationship (1). The J-integral during calculations was determined using the virtual increase in crack length. It is the J-integral calculated in accordance with Guo’s recommendations [25], i.e. it is the far field J-integral (J^far) determined using the integration contour passing through the area dominated by plane stress state. Figure 5a presents an example of the numerical model of the SEN(B) specimen used in the 3D FEM calculations [4].

Figure 5

Exemplary numerical model of the SEN(B) specimen used in the FEM 3D calculations; b) A fragment of the mesh around the crack tip for the FEM 3D analysis case; (based on [4])

4 Numerical results

The analysis of the obtained numerical results was carried out for such a load level that corresponds to the moment of reaching the critical point on the J-R graph,which is identified with the conventional moment of crack initiation. As part of the analysis, the distribution of the main components of the stress tensor (which determine the value of normal stresses, values of stress triaxiality parameters, effective stress values) in a limited range of the normalized distance from the crack tip ψ = 〈0; 10〉 (where ψ = r · σ₀/J, where r is the physical distance of the measuring point from the crack tip) for each model layer. In addition to assessing the normal component distributions of the stress tensor, the analysis of the distribution of the mentioned effective stresses, mean stresses and selected measures of the level of stress triaxiality, listed in section 1.2 (among them the ratio of mean stress and effective stress, parameter T_z and parameter LODE) were done. The analysis of all these quantities was illustrated in graphical form using charts, which are shown below.

As can be seen, as the distance from the crack tip increases, the level of normal components of the stress tensor decreases (Figure 6) - as the crack tip approaches, the singularity in the stress distribution over the crack tip is observed. As expected, the highest normal components of the stress tensor reach in the specimen axis (x₃/B = 0.000) - Figure 6,where theoretically we are dealing with the dominance of the plane strain state. With the distance from the specimen axis, going towards the edge of the specimen (x₃/B = 0.000), the value of stress decreases, hypothetically striving to achieve the values characteristic of a plane stress state. Analyzing Figures 6c, 6f and 6i, it can be seen that the value of the component σ_zz on the edge of the specimen is almost 0, which is in line with expectations and confirms the correctness of the numerical calculations.

Figure 6

The normal stress distributions near the crack tip for three SEN(B) specimens made of 145Cr6 steel: a-c) B = 5mm, a/W = 0.32; d-f) B = 10mm, a/W = 0.35; g-i) B = 15 mm, a/W = 0.29 (x₃/B = 0.500 – the edge of the specimen; x₃/B = 0.000 – the middle plane of the specimen thickness)

Figure 7 shows a summary of selected stress triaxiality measures. For the 145Cr6 steel considered in the paper, the level of effective stress (Figures 7a, 7d, 7g) at a normalized distance from the crack tip ψ = 〈1; 3〉 (in which the maximum stress opening the crack surfaces generally occurs when analyzing large deformations and large displacements), is at the level of σ_eff /σ₀ ≈ 1.0, in the layer thickness range of the specimen x₃/B ≤ 0.472.As the specimen edge approaches, a local decrease in effective stress is observed, whose value in the considered range is about σ_eff /σ₀ ≈ 0.7.

Figure 7

The distributions of the effective stress (σ_eff /σ₀), mean stress (σ_m/σ₀) and the ratio of the mean and effective stress (σ_m/σ_eff) – (which is also called as triaxiality parameter), near the crack tip for three SEN(B) specimens made of 145Cr6 steel: a-c) B = 5mm, a/W = 0.32; d-f) B = 10mm, a/W = 0.35; g-i) B = 15mm, a/W = 0.29 (x₃/B = 0.500 – the edge of the specimen; x₃/B = 0.000 – the middle plane of the specimen thickness)

As can be seen in Figures 7a, 7d, 7g, in the presented range of normalized distances from the crack tip, the level of effective stress is generally almost σ_eff /σ₀ ≈ 1.0, for layers characterized by the location x₃/B ≤ 0.472. As the crack tip rises farther away, in each subsequent layer of the specimen thickness, the level of mean stress (also called hydrostatic stress) decreases, and as it approaches the edge of the specimen (x₃/B = 0.500), it tends to the value of σ_m/σ₀ = 0.5 (see Figures 7b, 7e, 7h). Similar conclusions can be drawn from the analysis of Figures 7c, 7f, 7i, presenting changes in the ratio of average stresses and effective stresses σ_m/σ_eff , which results from the fact that the value of effective stresses generally for layers characterized by the location x₃/B ≤ 0.472 almost is equal to the yield point. The average stress decreases as the specimen edge approaches (x₃/B = 0.500) – their largest value is observed in the specimen axis (x₃/B = 0.000), which results from the previously described normal components of the stress tensor.

The value of the T_z triaxiality stress parameter, which was defined by Guo Wanlin [23, 24, 25] - formula (3) decreases with distance from the crack tip, with the highest values being achieved in the layer located in the specimen axis (x₃/B = 0.000), and the lowest on the edge of the specimen, tending with the distance from the crack tip to the value of T_z = 0, which is characteristic for the dominance of the plane stress state (Figure 8a-8c). Quite interesting are the changes in the LODE parameter near the crack tip for the moment of crack initiation (Figure 8d-8f). The LODE parameter in close proximity to the crack tip ψ < 2.0, takes negative values, then increases, striving for a set value of LODE ≈ 0.20. It should be noted that the value of the LODE parameter changes with the distance from the specimen axis (x₃/B = 0.000) – the closer the specimen edge (x₃/B = 0.500), the higher the value of the LODE parameter (Figure 8d-8f).

Figure 8

The distributions of the T_z parameter defined by Guo Wanlin [23, 24, 25] and the LODE parameter, near the crack tip for three SEN(B) specimens made of 145Cr6 steel: a-b) B = 5mm, a/W = 0.32; c-d) B = 10mm, a/W = 0.35; e-f) B = 15mm, a/W = 0.29 (x₃/B = 0.500 – the edge of the specimen; x₃/B = 0.000 – the middle plane of the specimen thickness)

Figure 9 presents changes along the front of the crack, the quantities previously analyzed (normal components of the stress tensor, effective stresses, mean stresses and selected measures of stress triaxiality) for the moment of crack initiation, equated when the J-integral reaches the critical value denoted as J_C. These charts were prepared for two normalized distances from the crack tip ψ = 1.0 and ψ = 2.0, where ψ = r · σ₀/J_C.

Figure 9

Changes in normal components of the stress tensor (σ_yy /σ₀, σ_zz /σ₀, σ_xx /σ₀) and selected measures of stress triaxiality parameters (σ_eff /σ₀, σ_m/σ₀, σ_m/σ_eff , T_z, LODE) along the crack front, for two selected distances from the crack tip (r = 1.0 · J/σ₀ and (r = 2.0 · J/σ₀) for three SEN(B) specimens made of 145Cr6 steel: a-c) B = 5mm, a/W = 0.32; d-f) B = 10mm, a/W = 0.35; g-i) B = 15mm, a/W = 0.29 (x₃/B = 0.500 – the edge of the specimen; x₃/B = 0.000 – the middle plane of the specimen thickness)

The analysis of Figures 9a, 9d and 9g leads to the following conclusions: the highest values near the crack tip are achieved by the stresses opening the crack surfaces, marked by σ_zz, and the smallest stresses by σ_xx. The further from the crack tip, the lower the stress value. Along the crack front, slight changes in effective stress are observed (Figures 9b, 9e, 9h), whose value is usually equal to the yield stress of the material and practical in the considered range ψ = 1.0 and ψ = 2.0 is independent of the distance from the crack tip. The nature of changes in mean stress normalized by the yield stress or effective stress, presented in Figures 9b, 9e and 9h, coincides with the nature of changes in the normal components of the stress tensor – faster changes in the values of the parameters σ_m/σ₀ and σ_m/σ_eff are observed for specimens with smaller thickness B = 5mm. In the case of specimens with a thickness of B = 15mm, in the range of distance x₃/B = 〈0.0, 0.35) from the specimen axis, one can speak of an almost constant value of the parameters σ_m/σ₀ and σ_m/σ_eff (Figure 9h). Figures 9c, 9f and 9i show changes along the crack front of the T_z and LODE parameters. Faster changes in both parameters are observed for a specimen with thickness B = 5mm (Figure 9c). Both parameters maintain a nearly constant value for the distance x₃/B < 0.4 from the specimen axis in the case of a specimen thickness B = 15mm (Figure 9i). The further from the crack tip, the smaller the value of the T_z parameter and the greater the value of the LODE parameter (Figure 9c, 9f, 9i). A detailed analysis of mutual dependencies is left to the reader.

The analysis of the obtained numerical results indicates that in the considered thickness range of SEN(B) specimens, for the analyzed 145Cr6 steel, the effective stresses practically do not depend on the specimen thickness (Figure 10a). In close proximity to the specimen axis (x₃/B < 0.1), mean stresses also do not depend on the specimen thickness (Figure 10a). Moving along the front of the crack from the specimen axis (x₃/B = 0.000) to its edge (x₃/B = 0.500) is accompanied by the fact that the mean stress value depends on the specimen thickness (Figure 10a-c) - the thicker the specimen, the greater the mean stress value. For the material considered in the paper, it can be stated also, that the T_z parameter along the front of the crack starting from the specimen axis is initially almost independent of thickness, however, after exceeding the position x₃/B > 0.300, the value of the T_z parameter begins to decrease and tends to zero, wherein with the thicker the specimen, the greater the value of the T_z parameter (Figure 10b). The thicker the specimen, the lower the LODE parameter value (Figure 10c). For specimens of greater thickness, we should note about a constant value of the LODE parameter along the front of the crack, for layers distant from the specimen axis that meet the condition x₃/B < 0.450.

Figure 10

Comparison of: a) the effective stress (σ_eff /σ₀) and mean stress (σ_m/σ₀); along the crack front; b) stress triaxiality parameter (σ_m/σ_eff ) and Guo Wanlin stress triaxility parameter (T_z); c) stress triaxiality parameter (σ_m/σ_eff ) and LODE parameter (LODE); for two selected distances from the crack tip (r = 1.0·J/σ₀ and (r = 2.0·J/σ₀) for three SEN(B) specimens made of 145Cr6 steel (x₃/B = 0.500 – the edge of the specimen; x₃/B = 0.000 – the middle plane of the specimen thickness)

5 Conclusions and summary

The paper presents a summary of experimental tests that allow assessing the fracture toughness of 145Cr6 steel (based on [3, 4, 5]), as well as analysis of the selected measures of stress triaxiality parameters. Based on the numerical calculations scheme presented in [4], the most important aspects of modeling three-dimensional problems in the field of fracture mechanics in the finite element method were indicated, details of the numerical model were given, and then the values were selected that will be analyzed in quantitative and qualitative assessment [4]. Based on the recommendations in the assessment of the measures of stress triaxiality parameters given in [16, 17, 26, 27], for three three-dimensional SEN(B) specimens with different thicknesses (B = {5,10,15}mm) and relative crack length a a/W ≈ 0.30, diagrams of normal components of the stress tensor, mean stresses, effective stresses and selected measures of stress traxiality (T_z and LODE parameters, quotient of mean stress and effective stress) were prepared. The moment when the value of the J-integral, considered as the crack driving force, reaches the critical value J_C, determined by the ASTM standard [7], was evaluated. Changes in these quantities were analyzed along with the distance from the crack tip, as well as changes in these sizes along the crack front for two normalized distances from the crack tip – ψ = 1.0 and ψ = 2.0 – for these distances O’Dowd and Shih [14, 15] recommended determination of Q-stress, while in [8] a catalogue of numerical solutions is given in the scope of changes in the parameters of geometrical constraints – T_z and Q^*_m parameters, which were discussed in section 1.1.

The numerical analysis and evaluation of the obtained results, lead to the following conclusions:

• the increase in distance from the crack tip is accompanied by a decrease in the values of stress tensor components, mean stresses, effective stresses and the T_z parameter – while the value of the LODE parameter increases in the considered range of distances from the crack tip;

• among the normal components of the stress tensor, the highest value is observed for the component opening the crack surfaces, and the lowest for the stresses acting along the thickness of the specimen;

• the increase in specimen thickness is usually accompanied by an increase in the value of normal stresses, mean stresses or the T_z parameter;

• effective stresses are not sensitive to changes in specimen thickness;

• an increase in specimen thickness reduces the value of the LODE parameter in the same normalized position relative to the crack tip and relative to the specimen axis;

• smaller thickness of the specimen, means that the determined along the cracks front quantities (stress components, stress triaxiality parameters) change faster – in the case of large thicknesses (B = 15m), starting from the specimen axis, initially the stability of the determined values is observed, after which the change (increase / decrease) occurs in the layer characterized by relative position x₃/B > 0.400.

In summary, carrying out a comprehensive numerical analysis of the mechanical fields near crack tip, should include not only an analysis of stress tensor component distribution, but also an assessment of stress triaxiality parameters. The stress triaxiality parameters can be considered as the as measures of geometric constraints [34, 35, 36, 37, 38, 39, 40, 41, 42, 43], affecting the distribution of mechanical fields near the crack tip, and but real fracture toughness. Such an analysis should be carried out in a quantitative and qualitative way, and all the conclusions drawn can be used in the correct description of mechanical fields, or in predicting actual fracture toughness using proper fracture criteria [1, 2, 9, 10, 12, 25] or assessing the strength of a structure containing certain defects, using FAD (Failure Assessment Diagrams) or CDF (Crack Driving Force) diagrams [1, 2, 11, 12].