Int J Mater Form (2008) Suppl 1:213 –216 DOI 10.1007/s12289-008-0366-8 # Springer/ESAFORM 2008
New Constitutive Equation for Plasticity in High Speed Torsion Tests of Metals J.D. Bressan 1 , K. Lopez 1 1
Department of Mechanical Engineering - Centre for Technological Sciences - Santa Catarina State University - Campus Universitário - 89223-100 Joinville, Brazil. URL: www.joinville.udesc.br e-mail:
[email protected] ABSTRACT: Present work investigates a phenomenological plasticity equation for the plastic behavior of metals deformed in high speed torsion tests. The tests were carried out at room temperature in a laboratory torsion test equipment and also in an universal tensile test machine on annealed commercial pure copper and aluminum specimens. The tensile tests were performed at room temperature by an universal testing machine at low strain rate of 0.017/s. The experimental torsion tests were carried out at constant angular speed that imposed a constant shear strain rate to the specimen. In the tests, the rotation speed were set to 61 rpm and 200 rpm which imposed high strain rates of about 2/s and 6.5/s respectively in tubular specimens. The torsion test lasted between 0.5 s and 1 s. The experimental hardening curves of equivalent stress versus strain in torsion were sigmoidal type and were fitted by different constitutive equations for plasticity that takes into account the strain hardening and thermal softening effects due to local temperature increase as the JohnsonCook model, the modified Voce equation and others. The specimen local temperature increase was calculated assuming adiabatic deformation process. It is proposed a new constitutive equation or modified Voce equation for the equivalent flow stress which considers the effects of strain hardening, strain rate hardening and thermal softening for the best fit to the present experimental data of annealed aluminum and copper. Key words: Torsion Test, Flow stress, Work hardening, Strain rate, Aluminum, Copper.
1 INTRODUCTION Laboratory torsion testing of metals are commonly used to determine the mechanical properties related to shear behaviour of metals such as the shear elasticity modulus (G), the shear yield stress, the shear rupture stress, the shear rupture strain or the total strain to shear and work hardening. However, torsion testing of metal alloy can be also used for investigating the plastic behavior of metals under high temperatures and high strain rates for obtaining the work hardening laws in different straining conditions such as high strain velocities and low, medium or high temperatures. These operation conditions arise in metal forming processes as sheet metal rolling and metal upsetting. In these processes, strain rates values can attain 10/s. This level of strain velocity can be obtained in the torsion testing of solid or tubular specimens as reported in the previous works by the author [1,2,3]. Furthermore, torsion test can produce larger work hardening curve than tensile test of metals, allowing a better curve fit.
Both strain and strain rate hardening effects are very important in the plastic behaviour of metals at low and high deformation velocities. However, material thermal softening due to temperature increase generated inside the metal with the development of plastic deformations is also important. The aims of the present work were to investigate a new constitutive equation for plasticity for metals under high straining speed and compare it with the experimental results from torsion testing. Also, the goal was to investigate phenomenological equations from literature for the best fit of work hardening curves or the equivalent stress versus equivalent strain of metals under high strain rates. 2 STRESS, STRAIN AND TEMPERATURE IN TORSION TESTING During torsion testing the hydrostatic stress is zero; the shear stresses and strains can be determined from the instantaneous measures of torque and rotation angle of the solid or tubular specimen. The shear
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stress Wa and the shear strain Ja at the external surface of the tubular specimen were calculated using the following equations [2],
Wa Ja
M 3 n m 2S a 23 a13
(1)
T § a1 a 2 ·
¨ Lu ©
2
(2)
¸ ¹
where M = torque, a1= internal radius, a2 = external radius, n = work hardening coefficient, m = strain rate sensitivity coefficient, T = angle of torsion in radians and Lu = useful length of specimen. The von Mises equivalent stress and strain used in the present work for isotropic materials and neglecting the hydrostatic stress component were, V
3 Wa
(3)
H
1 Ja 3
(4)
The plastic torsion tests were performed with constant rotation speed, thus, the equivalent strain rate was calculate by, H
1 J a 3
(5)
2.1 Adiabatic temperature calculations The plastic deformation process of specimen during the torsion testing was considered adiabatic due to the very short time interval of maximum 0.5 s for rupture. The temperature increase in the deformed useful portion of the specimen was calculated by equating the rate of heat generated by the shear plastic deformations to the total experimental power delivered by the torsion shaft to the specimen, T.2.S.N , discounting the initial elastic energy portion. Thus, the equation for the temperature increase from the initial yield shear stress was, T
To
2N (a 2 2 a12 ). U . c p . Lu
t
³ M dt
(6)
to
where To = room temperature, M = applied torque, N = rotation speed in rps, t = time interval in seconds, a1 = internal radius, a2 = external radius, cp = specific heat, U = density and Lu = useful length of specimen. This temperature increase in the
specimen produce material thermal softening, lowering its hardening curve with plastic strain. 3 NEW PLASTICITY CONSTITUTIVE EQUATION FOR TORSION BEHAVIOR The Johnson-Cook phenomenological equation [4] is the constitutive equation widely used by researchers for high speed metal forming processes. In this approach, the material behavior is modeled as thermo-visco-plastic, i.e., it is considered dependent upon the plastic strain, the plastic strain rate and temperature variations. The temperature increase is due to plastic deformation that cause material softening. The equivalent flow stress V for the current plastic yielding is,
A B H n
V
ª § H ·º «1 m "n ¨¨ ¸¸» © Ho ¹¼ ¬
elasto-plastic
p ª § TT · room ¸¸ «1 ¨¨ «¬ © Tmelt Troom ¹
viscosity
º » »¼
softening (7)
where A = initial yield stress (MPa), B = coefficient of strength (MPa), n = work hardening exponent, m = coefficient of strain rate sensitivity, H = strain rate, H o = reference strain rate, T = temperature of material (oC), Troom = room temperature (oC), Tmelt = material melting temperature (oC) and p = thermal softening coefficient. Alternatively, another constitutive model for flow stress that considers both the effects of work hardening and softening due to local increase in material temperature is [5], V
wV K ( H o H ) n H m ( T To ) o wT
(8)
where To = room or reference temperature (oC), T = current temperature of material (oC), Ho = prestrain , K = coefficient of material strength (MPa) and w V o / w T is the rate of material softening with temperature increase. Thus, the hardening behaviour depend on the imposed strain, strain rate and the material temperature increase (T – To). Also, for torsion testing, Gavrus et al. [6] have proposed the following constitutive equations based on the Norton-Hoff law, V
3 K. H n . 3 H
m. exp(E / T)
(9)
where the work hardening coefficient n may vary
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with strain through the linear relation n n o n 1 H and the strain rate sensitivity m can vary with temperature T as m m o m1 T . The parameter E is related to the material thermal softening. However, considering that some experimental curve is sigmoidal for metals exhibiting strain hardening which can be fitted by the Voce equation and the curves obtained for copper and aluminum, it is proposed the following modified phenomenological equation for flow stress which take into account strain, strain rate and thermal softening effects,
In table 2, the relevant physical properties and the mechanical properties obtained from tensile tests at strain rate of 0.17x10-2 /s and 3.34 x10-2/s are shown. The hardening equations from tensile test are also presented. The correspondent equivalent flow stress for aluminum in torsion test at 61 rpm was 0.295 V 120 0.035 H H 0.27 MPa and for copper in torsion at 61 rpm was V 450 (0.025 H )0.40 H 0.015 MPa.
m § H · wV V [Vf (Vf V0 ) exp n H ]¨¨ ¸¸ (T T0 ) 0 Ho wT © ¹ (10)
In Fig.1, the time interval for rupture and the calculated temperature increase during torsion testing of aluminum and copper specimens at 200 rpm can be observed. The specimen initial or room temperature was 25 oC. The maximum temperature
where V maximum limit stress, V0 = initial yield stress. Other parameters are defined as above.
5 RESULTS AND DISCUSSIONS
4 MATERIALS AND EXPERIMENTAL PROCEDURES In table 1, the investigated materials, annealed aluminium and copper, and the tubular specimen dimensions are presented. All specimens were annealed at 450 oC, the same condition as presented in previous work by the author [3]. The torsion testing were carried out at two different rotation speed of 61 and 200 rpm which correspond to shear strain rate of 2/s and 6.5/s respectively. Table 1. Specimen dimensions and materials used in the present torsion testing [3]. Tubular Specimen Dimensions / Materials Aluminum - 5 Aluminum - 6 Copper - 6 Copper - 7
Ltotal 2 a1 Lu (mm) (mm) (mm) 16 50 8.0 16 50 8.0 16 50 7.96 16 50 8.0
2 a2 Thickness rpm (mm) (mm) 12.15 2.07 61 12.04 2.02 200 12.07 2.05 200 12.07 2.035 200
Fig. 1. Specimens temperature increase during torsion testing calculated by equation (6).
calculated for copper is 312 oC and for aluminum is 151 oC. These temperature increase curves were calculated using the properties from table 2 in equation (6) and the time integral term was evaluated from the torque curves presented in Fig. 2. The time interval for specimen rupture was 0.53 s for aluminum and 0.90 s for copper. In Fig.3, comparisons of aluminum experimental
Table 2. Physical and mechanical properties obtained from tensile tests at strain rates of 0.17x10-2/s and 3.34 x10-2/s . Material
Elasticity Density modulus E
Specific heat cp
U 3
o
Melting temperature o
( K)
Yield stress Vesc
Ultimate Elongastrength tion Vt % (MPa)
Hardening equation
V K( Ho H )n H m (MPa)
(GPa)
(kg/m )
Copper
82
8900
380
1356
182
257
21
V
Aluminum
74
2690
850
933
73
138
26
V 235 (0.01 H)0.21 H 0.027
(J/Kg C )
(MPa)
450 (0.025 H)0.20
216
curve with the present model and the Johnson-cook model for torsion test at 61 and 200 rpm are shown. The present model has a better agreement with the experimental curve than the Johnson-Cook which
do not capture the stress softening from the point of maximum stress for 200 rpm. The equations are: - Present model or modified Voce equation (MPa) : 0.28
V [ 92 77 exp 0.79 H ] 380
(T 25)(2.65)
- Johnson-Cook model (MPa) : V
ª
25 · 8 90 H >1 0.2 "n 380 @ «1 §¨ T635 ¸ 0.35
«¬
©
¹
0.7
º » »¼
The comparisons of the experimental curves for copper and the Johnson-Cook and the present models are seen in Fig.4: both models are similar to 0.15 V [ 212 140 exp 0.85 H ] 380 (T 25)(0.5) . 6 CONCLUSIONS Fig. 2. Experimental torque versus time curves from torsion tests at 200 rpm of annealed copper and aluminum.
From the comparisons of torsion experimental curves with the phenomenological models for flow stress the following conclusions can be drawn: - present model has better agreement with the aluminum and copper experimental curves than the Johnson-Cook model for high strain rate of 6.5/s. ACKNOWLEDGEMENTS The authors would like to gratefully acknowledge the financial suport received from CNPq of Brazil and the University of Santa Catarina State/Brazil. REFERENCES 1.
Fig.3. Comparisons of aluminium experimental curve of equivalent stress versus strain with the present model and the Johnson-cook model for torsion test at 200 rpm.
2. 3.
4.
5. 6. Fig. 4. Comparisons of copper experimental curve of equivalent stress versus strain with the present model and the Johnson-cook model for torsion test at 200 rpm.
J.D. Bressan and R.K. Unfer, “Plastic Instabilities and Fracture of Metals in Torsion and Tensile Tests”, In: Proc. ESAFORM 2006, edited by Neal Juster & Andrzej Rosochowski, Glasgow, Scotland, 2006, pp.327-330. J.D. Bressan and R.K. Unfer, Construction and validation tests of a torsion test machine. J. Mater. Proc. Technol., v.179, 2006, pp.23-29. J.D. Bressan, Plastic Behavior and Fracture of Aluminum and Copper in Torsion Tests, In: Proc. ESAFORM 2007, edited by E. Cueto, Zaragoza, Spain, 2007, pp.. G.R. Jonhson and W.H. Cook, Fracture characteristic of three metals subjected to various strain, strain rates, temperatures and pressures. Engineering Fracture Mechanics, v.21/1, 1985, pp.31-48. J.D. Bressan and M. Vaz Jr., A computational approach to blanking process. J. Mater. Proc. Technol., v.125-126, 2002, pp.206-212. A. Gavrus and E. Massoni, Improvement of material behaviour analysis using a general parameter identification model based on the inverse method. In: Proc. ESAFORM 1999, edited by J.A. Covas, Guimarães, Portugal, 1999, pp.607-610.
