Characterization of the mechanical behaviour of Tempcore 500C rebar steel during tensile test necking: experimentation and simulation

1. INTRODUCTION

Understanding the true stress-strain behaviour of ductile materials is extremely important when evaluating mechanical behaviour during welding processes or extreme structural failure loads, as the safety factors of concrete-reinforced bars of a structure submitted to extreme conditions is indispensable to comply with standards. Enhancement of structures ductility, both in terms of design and mechanical behaviour of materials, remains a priority for researchers, manufactureres and governments. According to their engineering strain values at the onset of necking, ribbed reinforcement bars are classified by EN 1992:1-1 (2004)EN 1992-1-1 (2004). Design of concrete structures. European Committee for Standardization, Brussels..

Ascertaining true stress and strain values on the minimum cross-section is a complex issue due to the state of triaxial stress in the neck. A first approximation based on neck geometry was proposed by Bridgman (1944)Bridgman, P.W. (1944). The stress distribution at the neck of a tension specimen. Trans. Am. Soc. Met. 32, 553-574. , from the assumption of axial-symmetrical behaviour, which leads to a symmetrical axial profile, close to an arc of a circle. True stress and strain values are calculated through the equationsequations 1, 2:

Equivalent stress, defined as that necessary to achieve plasticizing of materials and which is constant throughout the cross sections, is calculated, according to Von Misses yield criterion, through Eq. (3) from true stress values obtained through Eq. (2), depending on the radius of the minimum cross section and the neck-profile curvature radius.

In order to obviate the need for instantaneous curvature radius measure through the necking process, Bridgman (1944)Bridgman, P.W. (1944). The stress distribution at the neck of a tension specimen. Trans. Am. Soc. Met. 32, 553-574. proposed the following relationship:

Equation (3) has been subsequently adjusted by Davidenkov and Spiridnova (1946)Davidenkov, N.N., Spiridnova, N.I. (1946). Analisis of the state of stress in the neck of a tension specimen. Proceedings_American Soc. Test. Mater. 46, 1147-1158. and Eisenberg and Yen (1983)Eisenberg, M.A., Yen, C.F. (1983). An anisotropic generalization of the bridgman analysis of tensile necking. J. Eng. Mater. Technol. 105 (4), 264-267. https://doi.org/10.1115/1.3225656.. However, according to Valiente (2000)Valiente, A. (2000). On Bridgman’s Stress Solution for a Tensile Neck Applied to Axisymmetrical Blunt Notched Tension Bars. J. Appl. Mech. 68 (3), 412-419. https://doi.org/10.1115/1.1360689., the best approximation is still given by Bridgman´s equation. On the other hand, Eq. (4) has been questioned by several authors (Hang and Rosenfield, 1975Hahn, G.T., Rosenfield, A.R. (1975). Metallurgical factors affecting fracture toughness of aluminum alloys. Metall. Mater. Trans. A 6, 653-668. http://doi.org/10.1007/BF02672285.; Norris et al., 1978Norris, D.M., Moran, B., Scudder, J.K., Quiñones, D.F. (1978). A computer simulation of the tension test. J. Mech. Phys. Solids 26 (1), 1-19. https://doi.org/10.1016/0022-5096(78)90010-8.; Le Roy et al., 1981Le Roy, G., Embury, J.D., Edwards, G., Ashby, M.F. (1981). A Model of Ductile Fracture Based on the Nucleation and Growth of Voids. Acta Metall. 29 (8), 1509-1522. https://doi.org/10.1016/0001-6160(81)90185-1.; La Rosa et al., 2003La Rosa, G., Risitano, A., Mirone, G. (2003). Postnecking elastoplastic characterization: Degree of approximation in the Bridgman method and properties of the flow-stress/true-stress ratio. Metall. Mater. Trans. A 34 (3), 615-624. https://doi.org/10.1007/s11661-003-0096-y.; Celentano et al., 2004Celentano, D.J., Cabezas, E.E., García, C.M., Monsalve, A.E. (2004). Characterization of the mechanical behaviour of materials in the tensile test: Experiments and simulation. Model. Simul. Mater. Sci. Eng. 12 (4), 425-444. https://doi.org/10.1088/0965-0393/12/4/S09.; Celentano et al., 2005Celentano, D.J., Cabezas, E.E., García, C.M (2005). Analysis of the Bridgman procedure to characterize the mechanical behavior of materials in the tensile test: Experiments and simulation. J. Appl. Mech. Trans. 72 (1), 149-152. https://doi.org/10.1115/1.1827243.; Bueno and Villegas, 2011Bueno, R., Villegas, D. (2011). Ductility in reinforcing steel: New parameter and applications. Conference Proceedings for the 81st Annual Convention of the Wire Association International, 46, pp. 70-76.). Norris et al. (1978)Norris, D.M., Moran, B., Scudder, J.K., Quiñones, D.F. (1978). A computer simulation of the tension test. J. Mech. Phys. Solids 26 (1), 1-19. https://doi.org/10.1016/0022-5096(78)90010-8., La Rosa et al. (2003)La Rosa, G., Risitano, A., Mirone, G. (2003). Postnecking elastoplastic characterization: Degree of approximation in the Bridgman method and properties of the flow-stress/true-stress ratio. Metall. Mater. Trans. A 34 (3), 615-624. https://doi.org/10.1007/s11661-003-0096-y., Celentano et al. (2004)Celentano, D.J., Cabezas, E.E., García, C.M., Monsalve, A.E. (2004). Characterization of the mechanical behaviour of materials in the tensile test: Experiments and simulation. Model. Simul. Mater. Sci. Eng. 12 (4), 425-444. https://doi.org/10.1088/0965-0393/12/4/S09., and Celentano et al. (2005)Celentano, D.J., Cabezas, E.E., García, C.M (2005). Analysis of the Bridgman procedure to characterize the mechanical behavior of materials in the tensile test: Experiments and simulation. J. Appl. Mech. Trans. 72 (1), 149-152. https://doi.org/10.1115/1.1827243. found divergences of between 3 and 10% for the equivalent stress value on fracture. Hanh and Rosenfield (1975)Hahn, G.T., Rosenfield, A.R. (1975). Metallurgical factors affecting fracture toughness of aluminum alloys. Metall. Mater. Trans. A 6, 653-668. http://doi.org/10.1007/BF02672285., Le Roy et al. (1981)Le Roy, G., Embury, J.D., Edwards, G., Ashby, M.F. (1981). A Model of Ductile Fracture Based on the Nucleation and Growth of Voids. Acta Metall. 29 (8), 1509-1522. https://doi.org/10.1016/0001-6160(81)90185-1. and Bueno and Villegas (2011)Bueno, R., Villegas, D. (2011). Ductility in reinforcing steel: New parameter and applications. Conference Proceedings for the 81st Annual Convention of the Wire Association International, 46, pp. 70-76. propose alternative relationships between a and R. Finally, concerning the neck profile, the proposal of Chen (1978)Chen, J.L. (1978). Study on metal fracture. Metallurgical Industry Press, Beijing. and Dong et al. (2019)Dong, S., Xian, A., Mohamed, H.S., Chen, Y., Yao, J. (2019). A Simplified Analytical Solution for the Necking Semi-Empirical Stresses based on Aramis System. KSCE J. Civ. Eng. 23 (1), 268-279. https://doi.org/10.1007/s12205-018-0307-0. are close to a hyperbola.

Other authors (Mirone, 2004Mirone, G. (2004). Approximate model of the necking behaviour and application to the void growth prediction. Int. J. Damage Mech. 13 (3), 241-261. https://doi.org/10.1177/1056789504042592.; Ganharul et al., 2012Ganharul, G.K., De Braganza Azevedo, N., Donato, G.H. (2012). Methods for the experimental evaluation of true stress-strain curves after necking of conventional tensile specimens: Exploratory investigation and proposals. ASME Pressure Vessels and Piping Conference 6, 163-172. https://doi.org/10.1115/PVP2012-78856.; Donato and Ganharul, 2013Donato, G.H., Ganharul, G.K. (2013). Methodology for the experimental assessment for true stress-strain curves after necking employing cylindrical tensile specimens: Experiments and parameters calibration. ASME 2013 Pressure Vessels and Piping Conference, Vol 6A, Paris. https://doi.org/10.1115/PVP2013-97993.) analyse the stress distribution in the neck independently of its profile, and propose an empirical relationship between true and equivalent stress depending solely on instantaneous true strain values. Finally, additional relationships are proposed by Ling (1996)Ling, Y. (1996). Uniaxial True Stress-Strain after Necking. AMP J. Technol. 5, 37-48. and Mirone et al. (2019)Mirone, G., Verleysen, P., Barbagallo, R. (2019). Tensile testing of metals: Relationship between macroscopic engineering data and hardening variables at the semi-local scale. Int. J. Mech. Sci. 150, 154-167. https://doi.org/10.1016/j.ijmecsci.2018.09.054. in order to obtain true stress values from several parameters such as the strain and stress ones on the onset of necking and the increase in specimen gauge length.

All the experimental techniques used by these authors are mainly based on 2D image analysis and a validation of the results through Finite Element Simulation (FEM). Nowadays, 3D Digital Image Correlation (DIC) is employed to analyse smooth-specimen necking (Kamaya and Kawakubo, 2011Kamaya, M., Kawakubo, M. (2011). A procedure for determining the true stress-strain curve a large range of strains using digital image correlation and finite element analysis. Mech. Mater. 43 (5), 243-253. https://doi.org/10.1016/j.mechmat.2011.02.007.; Li et al., 2013Li, Z., Shi, J., Tang, A. (2013). Experimental verification and analysis for Bridgman formula. Chinese J. Appl. Mech. 30 (4), 488-492.; Zhu et al., 2014Zhu, F., Bai, P., Zhang, J., Lei, D., He, X. (2014). Measurement of true stress-strain curves and evolution of plastic zone of low carbon steel under uniaxial tension using digital image correlation. Opt. Lasers Eng. 65, 81-88. https://doi.org/10.1016/j.optlaseng.2014.06.013.; Genovese et al., 2016Genovese, K., Cortese, L., Rossi, M., Amodio, D. (2016). A 360-deg Digital Image Correlation system for materials testing. Opt. Lasers Eng. 82, 127-134. https://doi.org/10.1016/j.optlaseng.2016.02.015.; Rossi et al., 2018Rossi, M., Cortese, L., Genovese, K., Lattanzi, A., Nalli, F., Pierron, F. (2018). Evaluation of Volume Deformation from Surface DIC Measurement. Exp. Mech. 58 (7), 1181-1194. https://doi.org/10.1007/s11340-018-0409-0.). These researchers also found axial- symmetrical behaviour during the necking process.

Nevertheless, the application of all these theories on ribbed bars remains questionable due to their irregular geometry. Paul et al. (2014a)Paul, S.K., Majumdar, S., Kundu, S. (2014a). Low cycle fatigue behavior of thermo-mechanically treated rebar. Mater. Design 58, 402-411. https://doi.org/10.1016/j.matdes.2014.01.079. and Paul et al. (2014b)Paul, S.K., Rana, P.K., Das, D., Chandra, S., Kundu, S. (2014b). High and low cycle fatigue performance comparison between micro-alloyed and TMT rebar. Constr. Build. Mater. 54, 170-179. https://doi.org/10.1016/j.conbuildmat.2013.12.061. analyses the high-and-low-cycles fatigue performance of quenched and self-tempered rebar steel manufactured by the Tempcore process using 2D Finite Element Simulation, founding stress accumulation at the root of the transverse ribs. Concerning the 3D simulation of ribbed bars, only Rocha et al. (2016)Rocha, M., Brühwiler, E., Nussbaumer, A. (2016). Geometrical and material characterization of quenched and self-tempered steel reinforcement bars. J. Mater. Civ. Eng. 28 (6), 04016012. https://doi.org/10.1061/(ASCE)MT.1943-5533.0001. research is available, which concludes that residual stress concentrations are present on the ribbed profile of Tempcore steel bars.

In this work, the mechanical behaviour up to failure of ribbed bars of Tempcore carbon steel is studied with the help of 3D Finite Element Simulation of the necking process. It is shown that a non-symmetrical neck profile develops in correlation with the influence of the ribs on stresses and strains throughout this phenomenon. The results obtained are compared with those of smooth bars of similar steel.

2. MATERIALS AND METHODS

The quality (grade) of Tempcore bars tested in this work, provided by the Spanish company Siderurgica Sevillana S.A., is similar to that of rebar steel 500C, according to European standard EN 1992:1-1 (2004)EN 1992-1-1 (2004). Design of concrete structures. European Committee for Standardization, Brussels.. Both smooth and ribbed specimens, 14 mm in nominal diameter, 8 for each type, have been tested. Chemical composition, as well as the maximum reference values set out in EN 10080 (2005)EN 10080 (2005).Steel for the reinforcement of concrete. Weldable reinforcing steel. General. International Organization for Standardization, Geneva. standard, are shown in Table 1. The geometrical properties of the specimens, as well as the gauge length, can be observed in Fig. 1.

Tensile tests have been carried out accordance with to standards ISO 15630-1 (2010)ISO 15630-1 (2010). Steel for the reinforcement and prestressing of concrete. Test methods. Part 1: Reinforcing bars, wire rod and wire. International Organization for Standardization, Geneva. and ISO 6892-1 (2016)ISO 6892-1 (2016). Metallic materials. Tensile testing. Part 1: Method of test at room temperature. International Organization for Standardization, Geneva.. A test speed of 0.167 mm s^-1 was selected for the whole process. Young`s modulus was obtained by measuring 3 specimens of each type of steel with a Class 1 extensometer. Axial and transversal displacements have been measured by a high resolution camera synchronized with the test machine.

The evolution of the neck profile has been obtained from 4 specimens of each typology. Twelve images per specimen were captured and processed by means of high-precision image analysis software, resulting in a total of 60 images per each material. As an experimental measure of necking development, the outer diameter profile was chosen to be recorded on ribbed specimens, as shown in Fig. 2a.

It should be noted that, according to the results obtained from the recordings, there is no axial symmetry on rebar profiles. Therefore, axial-symmetrical behaviour must be rejected and, consequently, a 2D analysis for these bars should also be rejected. Figure 2b shows one instantaneous neck profile shape for one of the tested ribbed bars. Concerning Bridgman´s hypothesis (Bridgman, 1944Bridgman, P.W. (1944). The stress distribution at the neck of a tension specimen. Trans. Am. Soc. Met. 32, 553-574. ), both right-hand and left-hand profiles present a suitable fit (over 90%) to an arc of a circle. Nevertheless, curvature radius values (R₁ =84.15 mm, R₂ =72.60 mm) and minimum cross section fail to coincide with these two profiles. Due to these results, an experimental methodology by means of a 3D analysis has been carried out in order to obtain more accurate data.

Four tested specimens were joined by drilling a hole in both parts of the fractured specimens that were coincident with the longitudinal axis of the bars (Fig. 3a). Subsequently, a bolt was inserted in order to join these two parts, which had previously been reinforced with adhesive (Figs. 3b and 3c). Finally, the neck area of the joined specimens was scanned using 3D equipment. The cloud of point obtained was processed by means of Catia v5-6, in order to obtain the cross-sections area and the outer diameter data on 19 planes at right angles to the longitudinal axis. (Fig. 3d).

3. RESULTS

3.1. Strain-hardening behavior

 ⌅

The engineering stress-strain relationship of smooth and ribbed bars is shown in Fig. 4. Average engineering mechanical values are shown in Table 2. Nomenclature follows the recommendations as per ISO 6892-1 (2016)ISO 6892-1 (2016). Metallic materials. Tensile testing. Part 1: Method of test at room temperature. International Organization for Standardization, Geneva..

Figure 4.  Engineering stress-strain relationship for smooth and ribbed bars.

Table 2.  Average engineering mechanical values

	Rp,0.2 (MPa)	Rm (MPa)	Rm/ Rp,0.2	Agt	At	E (GPa)
Smooth	515.45±11.42	627.65±3.06	1.22±0.03	0.105±0.17	0.174±0.005	195±1.86
Ribbed	521.86±11.13	647.19±1,37	1.24±0.03	0.156±0.008	0.221±0.012	200±2.21

True stress and strain values up to the onset of necking have been obtained from Eqs. (5) and (6), proposed by Nadai (1950)Nadai, A. (1950). Theory of flow and fracture of solids. Vol. 1, Mc Graw-Hill, New York., which are based on the material incompressibility and, therefore, cross section reduction is taken into account.

${\epsilon }_{true}=ln\left(1+A\right)$  (5)

${\sigma }_{true}=R\left(1+A\right)$  (6)

The average values achieved for true strain on the onset of necking are 0.101(±0.003) on smooth bars and 0.145 (±0.007) on ribbed bars. Nevertheless, similar mechanical behaviour is observed for both materials through strain hardening phase. According to Hollomon and Jaffe (1945)Hollomon, J.H., Jaffe, L.D. (1945). Time-temperature Relations in Tempering Steel. Trans. Am. Inst. Min. Eng. 162, 223-249., in the equation , shown in Fig. 5, similar values for strain hardening exponent (n) are obtained for each of the two materials (Eqs. (7) and (8)), around 0.18, whereby the yield and creep phases are rejected on this fit. Additional information on this issue can be found in Hortigon et al. (2017)Hortigón, B., Gallardo, J.M., Nieto-García, E.J., López, J.A. (2017). Elasto-plastic hardening models adjustment to ferritic, austenitic and austenoferritic Rebar. Rev. Metal. 53 (2), e094. https://doi.org/10.3989/revmetalm.94. and Hortigon et al. (2019)Hortigón, B., Gallardo, J.M., Nieto-García, E.J., López, J.A. (2019). Strain hardening exponent and strain at maximum stress: Steel rebar case. Constr. Build. Mater. 196, 175-184. https://doi.org/10.1016/j.conbuildmat.2018.11.082..

$\mathrm{Smooth: }{\sigma }_{true}=1060.5{\epsilon }_{true}^{0.1802}$  (7)

$\mathrm{Ribbed: }{\sigma }_{true}=1065.8{\epsilon }_{true}^{0.1806}$  (8)

Figure 5.  True stress-strain relationship for smooth and ribbed bars.

3.2. Necking

 ⌅

In order to obtain equivalent stress values from true strain ones, Eq. (9) , as proposed by Mirone (2004)Mirone, G. (2004). Approximate model of the necking behaviour and application to the void growth prediction. Int. J. Damage Mech. 13 (3), 241-261. https://doi.org/10.1177/1056789504042592., has been used, which is independent of the shape of the neck profile.

$\raisebox{1ex}{${\sigma }_{equ}$}\!\left/ \!\raisebox{-1ex}{${\sigma }_{true}$}\right.=0.9969-0.6058{\left({\epsilon }_{true}-{\epsilon }_N\right)}^2+0.6317{\left({\epsilon }_{true}-{\epsilon }_N\right)}^3-0.2107{\left({\epsilon }_{true}-{\epsilon }_N\right)}^4$  (9)

Given the symmetrical development of necking in round, smooth bars, minimum values of cross section area have been achieved from minimum diameter measured in the images recorded during the test. Subsequently, true stress and strain values during necking have been calculated through Eqs. (1) and (2).

Regarding ribbed bars, relationship between cross section area and outer diameter (Fig. 6), as obtained from planes at right angles to the longitudinal axis of 3D-scanned neck of bars, shown in Table 3, results in a fit R²=0.9:

$S=0.4974d_{out}^2+28.1334$  (10)

Figure 6.  Neck cross section area vs. measured outer diameter.

TABLE 3.  Outer diameter and cross section area data obtained by means of 3D-scanned images of ribbed bars neck

	dout (mm)	S (mm2)		dout (mm)	S (mm2)		dout (mm)	S (mm2)		dout (mm)	S (mm2)
Sp1	11.70	99.44	Sp2	11.74	97.07	Sp3	11.60	95.22	Sp4	11.91	97.46
	11.53	97.01		11.54	94.14		11.49	92.82		11.65	94.58
	11.22	94.45		11.24	91.40		11.31	90.46		11.46	91.97
	11.05	92.07		11.03	88.92		11.16	88.60		11.43	89.81
	10.69	90.16		10.85	86.96		11.03	86.85		11.21	87.79
	10.55	88.10		10.79	85.14		10.87	85.43		10.95	86.81
	10.45	85.98		10.63	83.71		10.78	84.47		10.89	86.14
	10.40	84.31		10.59	82.87		10.76	83.93		10.81	85.14
	10.36	82.98		10.53	82.21		10.73	83.56		10.75	84.71
	10.30(*)	82.41(*)		10.50(*)	81.88(*)		10.49(*)	83.77(*)		10.66(*)	84.18(*)
	10.34	81.86		10.58	82.00		10.57	84.48		10.77	83.92
	10.39	81.56		10.76	82.85		10.63	85.96		10.85	84.26
	10.48	81.67		10.84	84.19		10.69	87.56		11.05	85.17
	10.50	82.57		10.98	85.75		10.88	89.07		11.19	86.28
	10.73	83.80		11.07	87.66		11.05	91.14		11.45	87.45
	10.87	85.26		11.28	89.83		11.22	93.78		11.60	89.80
	11.01	87.01		11.43	92.24		11.45	95.93		11.78	92.25
	11.23	88.97		11.67	94.67		11.68	98.52		12.00	94.89
	11.34	91.27		11.71	97.22		11.90	101.08		12.16	97.30

(*)Minimum cross section

Despite the fact of a good fit is achieved for Eq. (10), Fig. 7 shows the geometry of some of the cross sections obtained from a specimen, where obvious differences can be observed in the shape of certain sections due to dissimilarities in the ribs strain.

Figure 7.  Sample of cross section of ribbed bar neck obtained from 3D scanned images.

Finally, in order to achieve true stress and strain values during the necking, F-d_out data obtained from the recording of the evolution of neck profile during the tests, determined by outer diameter (see Fig. 2), were extrapolated to F-S data through Eq. (10). Subsequently, true stress and strain values were obtained, from this last relationship, using Eqs. (1) y (2).

The true stress-strain relationship for each type of specimens, including necking data, is compared below. Figure 8 shows results obtained for smooth bars. It should be noted that, taking into account the values of strain hardening phase, fitting to Hollomon´s equation (Hollomon and Jaffe, 1945Hollomon, J.H., Jaffe, L.D. (1945). Time-temperature Relations in Tempering Steel. Trans. Am. Inst. Min. Eng. 162, 223-249.) (Fig. 8a), as used by several authors (La Rosa et al., 2003La Rosa, G., Risitano, A., Mirone, G. (2003). Postnecking elastoplastic characterization: Degree of approximation in the Bridgman method and properties of the flow-stress/true-stress ratio. Metall. Mater. Trans. A 34 (3), 615-624. https://doi.org/10.1007/s11661-003-0096-y.; Celentano et al., 2004Celentano, D.J., Cabezas, E.E., García, C.M., Monsalve, A.E. (2004). Characterization of the mechanical behaviour of materials in the tensile test: Experiments and simulation. Model. Simul. Mater. Sci. Eng. 12 (4), 425-444. https://doi.org/10.1088/0965-0393/12/4/S09.; Celentano et al., 2005Celentano, D.J., Cabezas, E.E., García, C.M (2005). Analysis of the Bridgman procedure to characterize the mechanical behavior of materials in the tensile test: Experiments and simulation. J. Appl. Mech. Trans. 72 (1), 149-152. https://doi.org/10.1115/1.1827243.; Bueno and Villegas, 2011Bueno, R., Villegas, D. (2011). Ductility in reinforcing steel: New parameter and applications. Conference Proceedings for the 81st Annual Convention of the Wire Association International, 46, pp. 70-76.; Mirone et al., 2019Mirone, G., Verleysen, P., Barbagallo, R. (2019). Tensile testing of metals: Relationship between macroscopic engineering data and hardening variables at the semi-local scale. Int. J. Mech. Sci. 150, 154-167. https://doi.org/10.1016/j.ijmecsci.2018.09.054.) to characterize metallic materials up to failure, remains inappropriate in this case. However, a greater fit (R²=0.99) is obtained by means of the following polynomial equation taking into account only necking values:

${\sigma }_{true}=-115.82{\epsilon }_{true}^2+631.89{\epsilon }_{true}^{}+634.59$  (11)

Figure 8.  True stress-strain relationship up to failure for round smooth bars. Adjustment to: a) the Hollomon equation, b) polynomial equation (+ equivalent stress).

Both true stress-strain (obtained through Eq. (11)) and equivalent stress-true strain relationships, where equivalent stress are achieved from true stress values through Eq. (9), are shown in Fig. 8b.

Nevertheless, on ribbed bars, fitting true stress-strain data up to failure to Hollomon´s equation (Hollomon and Jaffe, 1945Hollomon, J.H., Jaffe, L.D. (1945). Time-temperature Relations in Tempering Steel. Trans. Am. Inst. Min. Eng. 162, 223-249.) results in a R² value of 0.96 (Fig. 9). In this equation, the value of strain hardening exponent (0.1569) is lower than that obtained on fitting the same equation up to the onset of necking (0.1806). Equivalent stress values, according to Eq. (9) are also shown in Fig. 9.

${\sigma }_{true}=1022.5{\epsilon }_{true}^{0.1569}$  (12)

Figure 9.  True and equivalent stress vs. true strain for ribbed bars.

A comparative analysis of the behaviour of both type of bars, leads to the conclusion regarding the influence of ribs during the necking phase, which causes an early failure. Although the onset of necking occurs earlier for round smooth bars, the neck development range is broader, and reaches an average true strain value at failure equal to 1.028 (±0.027), while the value for ribbed bars is equal to 0.522 (±0.022). True and equivalent stress values associated to these average strain ones, obtained from Eqs. (11) , (12) y (9), are shown in Table 4.

Table 4.  Average strain and stress values at failure for smooth and ribbed bars

	εtrue	σtrue, (MPa)	σequ, (MPa)
Smooth	1.028±0.027	1175.37	969.08
Ribbed	0.522±0.022	923.69	867.95

On the other hand, clear differences can be observed on the fracture shape (Fig. 10). Smooth bars show a cope & cone failure, characteristic of ductile metals. However, the failure of ribbed bars occurs on the root of one transverse rib, due to the concentration of stresses generated in this area. This phenomenon on ribbed bars has already been observed by several authors (Paul et al., 2014aPaul, S.K., Majumdar, S., Kundu, S. (2014a). Low cycle fatigue behavior of thermo-mechanically treated rebar. Mater. Design 58, 402-411. https://doi.org/10.1016/j.matdes.2014.01.079.; Paul et al., 2014bPaul, S.K., Rana, P.K., Das, D., Chandra, S., Kundu, S. (2014b). High and low cycle fatigue performance comparison between micro-alloyed and TMT rebar. Constr. Build. Mater. 54, 170-179. https://doi.org/10.1016/j.conbuildmat.2013.12.061.; Rocha et al., 2016Rocha, M., Brühwiler, E., Nussbaumer, A. (2016). Geometrical and material characterization of quenched and self-tempered steel reinforcement bars. J. Mater. Civ. Eng. 28 (6), 04016012. https://doi.org/10.1061/(ASCE)MT.1943-5533.0001.). A very similar failure shape is observed by Paul et al. (2014a)Paul, S.K., Majumdar, S., Kundu, S. (2014a). Low cycle fatigue behavior of thermo-mechanically treated rebar. Mater. Design 58, 402-411. https://doi.org/10.1016/j.matdes.2014.01.079., and Paul et al. (2014b)Paul, S.K., Rana, P.K., Das, D., Chandra, S., Kundu, S. (2014b). High and low cycle fatigue performance comparison between micro-alloyed and TMT rebar. Constr. Build. Mater. 54, 170-179. https://doi.org/10.1016/j.conbuildmat.2013.12.061. on high- and low-fatigue cycles. On the other hand, residual stress concentration is found by Rocha et al. (2016)Rocha, M., Brühwiler, E., Nussbaumer, A. (2016). Geometrical and material characterization of quenched and self-tempered steel reinforcement bars. J. Mater. Civ. Eng. 28 (6), 04016012. https://doi.org/10.1061/(ASCE)MT.1943-5533.0001. in the same area.

Figure 10.  Type of failure for smooth bars (top) and ribbed bars (bottom).

4. FINITE ELEMENTS ANALYSIS

The aim of this analysis involves determining the mechanical behavior of each of the two models by means of numerical methods in order to validate experimental results shown previously. In this analysis, software Ansys Workbench v16 has been used. A mechanic no-linear analysis has been introduced working, therefore, with a stress-strain relationship up to failure. In addition, bars have been modeled as an isotropic material up to yield strength, set in 500 MPa, where Young´s modulus value is equal to 210 GPa and Poisson´s ratio equal to 0.3. Finally, a multi-lineal isotropic hardening has been used from the yield strength, based on the experimental results (Fig. 11). Regarding smooth bars, values of stress up to the onset of necking have been determined through Eq. (7). Throughout necking phase, equivalent stress values have been used, and hence Eq (9) has been applied to true stress values obtained through Eq. (11). On the other hand, Eq. (12) has been used for the behaviour of ribbed bars throughout the whole process. Similar to smooth bars, equivalent stress values from the onset of necking have been introduced through Eq. (9).

Due to its axial-symmetrical behaviour, a 2D analysis has been carried out for smooth round bars. The high order element Plane 183, with 8 nodes and 2 degrees of freedom at each one, was adopted. On the other hand, a higher order 3D element, Solid 186, with 20 nodes and initial hexaedrical shape, has been adopted for ribbed bars, which model has been obtained by means of 3D-scanned images of pre-test specimens. Although this shape is predominant in the core of the bar, this last element can change into a tetrahedron with 10 nodes, into a pyramid with 13 nodes or into a prism with 15 nodes, in order to adapt to the irregular ribbed geometry. Shape functions are adapted automatically when necessary. Concerning the properties of these elements, both have plasticity, stress stiffening and large strain capabilities.

On the other hand, a parametric study of tested ribbed bars has been carried out for their modeling. A higher mesh scale of critical zones, the transverse ribs and their roots have been proposed, as observed in Fig. 12.

Axial displacement has been used as a control parameter during the simulation. Concerning round smooth bars, axial force has been applied upon the top of cross-section nodes, with a restricted axial displacement on the bottom cross-section ones. On the other hand, concerning ribbed bars, axial displacement has been allowed on both the top and bottom cross-section nodes, thereby applying axial forces upon both sections in order to guarantee the development of the minimum cross section of the neck in the half-plane bar. Furthermore, in order to block the condition of free solid and, therefore, allow the computation, a zone of 1 mm in diameter has been placed in the middle of the bar, with forbidden displacements at right angles to the longitudinal axis. Under these conditions, a very good fit between experimental and simulated stress-strain relationship is achieved for both models (Fig. 13), despite the earlier failure of ribbed one.

Regarding the stress distribution, Figs. 14 and 15 shows the results obtained for longitudinal paths in the center of smooth model and in the core boundary of ribbed one. By comparing both figures, it is evident that the existence of ribs clearly affects the behaviour of the bar through the necking. A progressive decrease in stress values can be observed in the ribbed model from the root to the middle of the transverse ribs affected by the neck, and hence, minimum values are reached at the top of this one (Fig. 15). This stress distribution has already been proved by Rocha et al., (2016)Rocha, M., Brühwiler, E., Nussbaumer, A. (2016). Geometrical and material characterization of quenched and self-tempered steel reinforcement bars. J. Mater. Civ. Eng. 28 (6), 04016012. https://doi.org/10.1061/(ASCE)MT.1943-5533.0001. with reference to residual stresses due to manufacturing, and it suggests a large concentration of stresses at the intersection between the core and the ribs affected by necking, despite the fact of stress values are no constant along the length of this ribs. On the other hand, an irregular stress distribution is obtained in the neck cross sections (Fig. 16), reaching the higher values in the core of the bar, in a similar way to those obtained at the root of the transverse rib coincident with the minimum cross section.

It is worth to note that the differences between the experimental cross section shapes and necking profile of ribbed bars and those achieved by means of FEM simulation. Nevertheless, true stress-strain relationship up to failure is very similar in both cases, as far as the stress distribution provided by FEM model can be supposed as a reliable approximation to the mechanical behaviour of these ribbed bars throughout the necking.

5. CONCLUSIONS

In this research, the necking behavior of ribbed bars of Tempcore 500C carbon steel is ascertained using room-temperature tensile tests. The following conclusions can be enunciated: