Study of the Evolution of Elastoplastic Zone by Volumetric Method

International Journal of Mechanics and Applications 2015, 5(1): 1-9 DOI: 10.5923/j.mechanics.20150501.01 Study of the Evolution of Elastoplastic Zone by Volumetric Method M. Moussaoui 1,*, S. Meziani 2 1 Mechanical Engineering Department, University Ziane Achour, Djelfa, Algeria 2 Laboratory of Mechanic, University Constantine 1, Campus Chaab Erssas, Constantine, Algeria Abstract In this work a study was carried out to analyze the effect of changes in parameters characterizing the notch, such that the radius and angle, on the evolution of plastic zone near of notch root. A steel CT specimen under uni-axial loading is taken as a finite element model. Two theoretical approaches are applied to calculate the extent of the plastic zone for different values of angle and notch radius using the volumetric and Irwin approaches. The elastoplastic analysis was used to determine the effective distance and the relative stress gradient by applying the finite element method (FEM). The size of the effective plastic zone deduced from elastic calculation reaches its maximum when the radius is close to zero and angle equal to zero. The variation of the angle modifies the notch extended plastic zone. Keywords Plastic zone, Effective distance, Stress gradient, Irwin method, Volumetric method 1. Introduction The discontinuities, filets, notches and cracks in particular, generate always high stress concentrations. Near the root of defect, the singularity of stresses enables to activate strong plastic deformation. From the equations of stress distribution, derived from the linear elastic fracture mechanics (LEFM), the stress singularity at the crack tip leads to an area for which the conditions of plasticity will be achieved. This area driven by the crack tip from the high stress plays an important role in fracture of materials. However, analytical determined stresses the crack tip exceeds the elastic limit of the real materials and induce a small volume that undergoes plastic deformation. If the plastic zone is large, a high quantity of energy is dissipated during crack growth, whereas if the plastic zone is small, the propagation of cracks requires less energy, therefore the size of the plastic zone is directly related to the material s hardness [8]. The size of the plastic zone depends strongly on the stress mode and it is much more important under condition of plane stresses that in plane strain. The later will be the most critical for resistance to fracture propagation. The mechanical properties of the material and the stress state, affect together the size and shape of the plastic zone. Many researchers have contributed their work to calculate the size of the plastic zone. Irwin and Dugdale [8], proposed respectively, different models to estimate the size * Corresponding author: moussaoui_must@yahoo.fr (M. Moussaoui) Published online at http://journal.sapub.org/mechanics Copyright 2015 Scientific & Academic Publishing. All Rights Reserved of the plastic zone near the crack tip. Huang Yi et al [9] introduced a new method to calculate the size of the plastic zone. They used the maximum crack opening displacement (MCOD) to determine the size of plastic zone. And others leaned on the effect of notch radius, by focusing on effects of notch radius and the size of specimen (thickness and ligament) on the apparent strength of fracture of low-alloy steels. They concluded that there is a critical radius of notch below the value of the apparent fracture resistance independently on notch radius, in addition that there is a dependency between the apparent fracture resistance and the dimensions of the specimen [6]. The works of T. Fett [7] treats the influence of notch radius on fracture resistance. He shows that there is a characteristic radius notch, below which an apparent constant tenacity occurs and increases with increasing initial length of the crack. This is calculated by assuming that the notch has a small crack in notch root and acting as a long crack in the same total size. A study was conducted by S. Banebjee [8] in order to evaluate the effect of thickness, width and geometry of the fracture resistance and the resultant displacement due to the growth of the plastic zone and the crack. The result of analysis shows that if the plastic zone size decreases, the stress decreases as the width of the specimen increases. The geometry of the notch is then characterized by the length, the notch angle ψ and the notch radius, [4]. The classification of notches is made by two parameters ψ and as follows (Figure 1): Crack : = 0 and ψ = 0 Sharp notch V: = 0 and ψ # 0 Blunt notch U: # 0 and ψ = 0 Simple notch : # 0 and ψ # 0

2 M. Moussaoui et al.: Study of the Evolution of Elastoplastic Zone by Volumetric Method Figure 1. Parameters defining the simple notch In this study, we considered a finite plate specimen CT (Figure 2a), with 20x20 [mm] dimensions, containing a notch with a length equal to 7.25 [mm], a radius which ranges from 0 to 3.5 [mm] and an angle ψ equal respectively to 0, 5, 10, 30 and 45, subject to tensional stresses σ=125mpa. Under a mode of plane stress, the finite element model used in the elastoplastic analysis has the following mechanical properties: Young's modulus E = 230E03MPa, yield strength 670MPa, Poisson s coefficient ν = 0.293. Figure 2a. Finite element model A discretization was made with triangular elements with six nodes and an appropriate refinement was applied around the end of the notch using the castem software program. In Figure 2c a sharp notch has an angle ψ 0 and radius =0 and in Figure 2b a blunt notch contains ψ 0; 0. When ψ = 0; =0 notch is similar to a crack. Figure 2c. Sharp notch: ψ 0; =0 2. The Field of Stress in Notch Root According to the extension of Griffith theory to the ductile materials in particular the metal alloys, a part of energy is dissipated in the formation of the plastic zone due to a high stress concentration near the tip of the notch. In the case of the existence of the plasticized zone, near the bottom of notch, the stress distribution has an elastoplastic character, and the use of the finite element method enables to describe the stress distribution at the notch root. The results obtained by finite element and analytical approaches show that the maximum elastoplastic stress is not at the notch root but a certain distance from it (Figure 3b), unlike the elastic distribution where we notice that is characterized by a maximum stress at the notch root (Figure 3a). Various functions representing the stress distribution can be found in literature and are presented as following: Table 1. Various Functions of Elastic Stress Distribution Authors [13] Chen and Pen, 1978 [14] Kujawski, 1991 σ Elastic stress distribution yy = σ max + 8x 1/2 3/2 2x 2x σ yy = fσ max 1+ + 1+ x if 0.2 f = 1 ( / 2k ) x tan π t x if 0.2,f = 1+ 0.2 2.8 [12] Neuber and Weiss σ = σ + 4 yy max 1962 x Figure 2b. Blunt notch: ψ 0; 0 [15] Usami 1985 2 4 σ max 1 x 3 x σ yy = 1+ 1 1 3 2 + + + 2

International Journal of Mechanics and Applications 2015, 5(1): 1-9 3 Figure 3. Stress distribution, a) Elastic behaviour, b) Elastic-Plastic behaviour 3. Modeling of the Plastic Zone by Analytical Approach In the volumetric approach, the fundamental parameter that represents the region of elaboration fracture process is characterized by the effective distance X eff, representing the diameter of a volume assumed to have a cylindrical shape by analogy with the notch plastic zone which has a similar shape. The effective distance X eff is determined using the distribution of the normal stress and relative stress gradient reported in a bi-logarithmic diagram (Figure 4) [1, 2] where the stress normal to the notch plane is plotted against the distance ahead of notch. In Figure 4, the opening stress distribution versus distance is plotted in bi-logarithmic axes; the relative stress gradient is also plotted on the same graph relating to an opening notch angle equal to 45 and notch radius equal to 0.6 [mm]. As it may be seen, the stress distribution at the notch tip decreases with the distance from the notch tip, and is characterised by three zones: the first one very near the notch tip, where the stress is constant and/or increasing to its maximum value; the second zone between the first and third zones; and the third, which can be simulated as a pseudo-stress singularity [11].

4 M. Moussaoui et al.: Study of the Evolution of Elastoplastic Zone by Volumetric Method Figure 4. Bi-logarithmic diagram of the elastic-plastic stress distribution and relative stress gradient distribution and relative stress gradient For determination of X eff, it has been observed that the effective distance is related to the minimum value of the relative stress gradient χ. Hence, the position of the minimum relative stress gradient allows obtaining an effective distance precise value. Studies have shown that this distance is not related to the geometry of the notch, but to the stress distribution. This distance is in relation to a zone of pseudo-stress singularity appearing at effective distance X eff, of notch bottom. To determine this distance [5], we use the relative stress gradient, defined as follows: d σ (x) 1 yy χ(x) = (1) σ (x) dx yy An analytical formulation is given by interpolating the discrete points of the stress distribution under the polynomial form: σ ( ) = n i x ax (2) yy i= 0 The stress gradient is expressed by: 1 dσyy ( x) χ ( x) = σ yy ( x) dx i n i 1 ( ia i= 0 ix ) 1 = n i ax i= 0 i The function of the stress gradient represents a minimum that corresponds to the effective distance, X eff : (3) d χ( x) = 0 dx Irwin considered that the presence of a plastic zone at the bottom of crack, fact that the length of the crack behaves as if it was longer than its physical size and the stress distribution is equivalent to a crack elastic length (a + r), [8, 10], so its effective length, a eff is a a + = With R p = 2rE eff r E For simple estimation of the size of plastic zone along θ equal to zero degree, considering a first approximation that plastic zone is circular with diameter Rp, for a perfectly elastic plastic material, according to: (4) 2 1 Κ R = 2 = I p r E (5) π σe 4. Analysis of Notch Effect on the Maximum Elastic-Plastic Stress Figure 5 shows the evaluation of influence of notch radius and angle of the maximum elasto-plastic stress at the point of notch. The y-axis shows the maximum elasto-plastic stress and the x-axis presents the notch radius for a well-defined notch angle. This stress takes large values if the notch radius tends to zero. We can see that when the notch radius increases, the look of curves of the maximum elastoplastic stress decreases from a maximum, relative to the lower values.

International Journal of Mechanics and Applications 2015, 5(1): 1-9 5 Figure 5. Evolution of the maximum stress elasto-plastic under the variation of notch angle and radius Figure 5b shows for an angle of notch equals to zero, the maximum stress equals to 312.5 MPa and the minimum value is 168.8 MPa and from a radius 1.5mm, elastoplastic stress re-establishes moderate growth. Figure 5c shows for ψ angle of notch equals to 45, the maximum and minimum elastoplastic stresses are 252.1MPa, 164.8MPa respectively, accordingly an increase of notch angle decreases the elastoplastic stress. The superposition of curves shows the effect of variation of angle and notch radius on evolution of elastoplastic maximum stress, consequently for small radii where the notch approaches to acute form, the value of the elastoplastic stress is important and increasing of notch angle decreases the elastoplastic maximum stress, Figure 5 shows, for angles 30 and 45, the stresses are lower relative to angles 0, 5 and 10. Beyond the radius 1.5 [mm] stress resumes its growth. Given that the angle of notch is a factor that intervenes in variation of amplitude of maximum elastoplastic stress, its increase reduces the stress in a field of notch radius of low value (notch sharp), and furthermore in the field of large values of notch radius, the maximum elastoplastic stress tends to important values. 5. Effect of Changing the Notch Parameters on Plastic Zone We admit that the approximation given by Irwin and volumetric approach to the shape of plastic zone are similar. The first considers the diameter R p of circular form and the

6 M. Moussaoui et al.: Study of the Evolution of Elastoplastic Zone by Volumetric Method second admits the cylindrical shape of diameter X eff effective distance. The curves given in Figure 6 show a comparison between the evolution of effective distance and the distance of the plastic zone Rp of Irwin, versus notch radius. The calculation by finite elements method leads to assessment of effective distance, and the dimension of diameter Rp of the plastic zone. For small radii where the notch tends to a more acute form, the size of the plastic zone tends to large values for the two approaches, but when the notch approximates the shape of a crack, i.e., when the radius tends to zero and the angle becomes zero, the estimate of effective distance is approximately twice the value of the dimension of Irwin plastic zone (R p ). Increasing the notch radius increases the extent of the plastic zone. The maximum is given by the volumetric approach, and is equal to 3.504[mm], and the minimum is given by the approach of Irwin, is equal to 1.247[mm], with an angle of notch equal to 0, (Figure 6). Furthermore increasing of the angle of notch tends to attenuate the extent of plastic zone at an angle of 45. The maximum of effective distance reaches a value of 3.344[mm] and the minimum is obtained by approach of Irwin, and is equal to 0.8467[mm], (Figure 6). Figure 6. Effective distance Xeff & plastic zone Rp versus radius and angle of notch

International Journal of Mechanics and Applications 2015, 5(1): 1-9 7 Figure 7. Development of plastic zone.a) Dimension Rp calculated by Irwin. b) Dimension of X eff calculated by volumetric method Figure 7 and Figure 8 show the effective distance and the dimension plastic zone of Irwin, for different angles of notch: 0, 5, 10, 30 and 45. Figure 8a shows the comparison between the different effective distances obtained as a function of notch radius. The acute angles develop a larger effective dimensions compared to less acute notches (blunt notches) and the high stress concentration will take place under the effect of morphology of the notch The Maximum of effective distance is equal to 3.758[mm], obtained for an angle of ψ=5, unlike the approach of Irwin where the maximum is obtained for an angle equal to zero and it is equal to 2.343[mm]. The change in parameters of notch, radius and angle, changes the morphology of plastic zone near the bottom of notch. Smaller the radius decreases, the plastic zone increases, and when the notch shall be infinitely acute (lower to 1.5[mm] radius) plus the notch angle increases, the plastic zone decreases. This rule is valid for angles greater than 15. Calculating the dimension of the plastic zone depends essentially on the type of the test specimen and the values of radius and angle of notch.

8 M. Moussaoui et al.: Study of the Evolution of Elastoplastic Zone by Volumetric Method Figure 8. Evolution of plastic zone, as a function of notch parameters (radius, angle): a) Effective distance X eff, b) Dimension Rp Irwin 6. Conclusions The present study was based on the use of two theoretical approaches, volumetric and Irwin, in order to model and analyze the evolution of plastic zone around the tip of notch. The variation of radius and angle of notch affects directly on the distribution of maximum elastic-plastic stress and on extent of plastic zone. The change in parameters of notch, radius and angle, changes the morphology of plastic zone, near the notch root. More radius decreases more the plastic zone increases and when the notch is infinitely acute (less than 1.5mm radius) more the notch angle increases and the size of the plastic zone decreases. This rule is valid for angles greater than 15. The calculation of the size of the plastic zone depends mainly on the type of specimen, the radius and angle of notch. Given that the angle of notch is a factor that intervenes in variation of amplitude of maximum elastic-plastic stress, its increase reduces the stress in a field of notch radius of low value (notch sharp), and furthermore in the field of large values of notch radius, the maximum elastic-plastic stress tends to important values.

International Journal of Mechanics and Applications 2015, 5(1): 1-9 9 Nomenclature ψ Notch angle E Young s module Notch radius a i Polynomial coefficients χ Relative stress gradient ϕ(x,χ) Weight function K I Notch stress intensity factor under mode I σ yy (x) Stress in the direction y σ Tension applied σe Elastic stress σ eff Effective stress Rp Plastic zone size according to Irwin X eff Effective distance σ max Maximum elastic stress υ Poisson s ratio kt Stress concentration factor a Notch length REFERENCES [1] H. Adib, G. Pluvinage. 2003, Theoretical and numerical aspects of the volumetric approach for fatigue life prediction in notched components. International Journal of fatigue 25 67-76. [2] Guy Pluvinage, Joseph Gilgert, 2003, Fracture emanating from stress concentrators in materials: links with classical fracture mechanics. Materiali in Tehnologue 37 3-4, MTAEC9, 37(3-4)117(2003). [3] G. Qylafiku, Z. Azari, N. Kadi, M. Gjonaj, G. Pluvinage, 1999, Application of a new model proposal for fatigue life prediction on notches and key-seats. International journal of fatigue 21 753-760. [4] G. Pluvinage, 1998, Fatigue and frature emanating from notch; the use of the notch stress intensity factor. Nuclear Engineering and Design 185 173-184. [5] KADI Nawar, 2001, La fiabilité des arbres entaillés dans les machines agricoles: applications sur les arbres clavetés. Thèse de Doctorat, Université de Metz, France. [6] Abdel-Hamid I Mourad, Aly El-Domiaty, 2011, Notch radius and specimen size effects on fracture toughness of low alloy steel. Procedia Engineering 10 1348-1353. [7] T. Fett, 2005, Influence of a finite notch root radius on fracture toughness. Journal of the European Ceramic Society 25 543 547 T. [8] S. Banejee, 1981, Influence of specimen size and configuration on the plastic zone size: toughness and crack growth. Eng. Fract. Meek. 15:3-4-390. [9] Huang Yi, Chen Jingjie, Liu Gang, 2010, A new method of plastic zone size determined based on maximum crack opening displacement. Engineering Fracture Mechanics 77 2912 2918. [10] E.E. Gdoutos, 2005, Fracture Mechanics an Introduction Solid Mechanics and its Applications, Second Edition, Published by Springer, Springer, 101 Philip Drive, Norwell, MA 02061, U.S.A. [11] Vratnica M, Pluvinage G, Jodin P, Cvijovic Z, Rakin M, Burzic Z, 2010, Influence of notch radius and microstructure on the fracture behavior of Al Zn Mg Cu alloys of different purity. Materials and Design 31 1790 1798. [12] Neuber N, Weiss V, 1962. Trans. ASME paper No 62-WA-270. [13] Chen, S.I, C.C, Pan, HI, 1978. Collection of paper on Fracture of metals (in Chinese). Metallurgy Industry Press, Bejing, pp.197-239. [14] Kujawski, D, 1991, Estimation of stress intensity factors for small cracks at notches. Fatigues Fract. Eng. Mater. Struct.14, 953-965. [15] Usami, Tanaka, M. Jono, Komai, K (Eds.), 1985, Current Research on Fatigue Cracks. The Society of Material Science, Kyoto, Japan, p.119.