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Mr. Rajkumar D. Patil^{1}, Prof. P. N. Gore^{2}

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The shear stressstrain curve of materials in torsion is very important in the design, analysis and manufacture of a wide variety of products and components. When the shear loadings are applied, the shear modulus of elasticity and shear yield strength must be known in order to use the equations commonly used in engineering design and analysis. These are measured according to an ASTM torsion test. In this test a material specimen of solid or hollow round cross section is used in a torsion testing machine and the applied torque and angle of twist are recorded simultaneously. The torquetwist diagram is drawn from these data, and elementary mechanics theory is then used to construct the shear stressstrain diagram. Precise determination of shear modulus depends on specimen dimensions namely gauge length and root radius. This paper presents torquetwist curve and true stress strain curve for 1020 steel and aluminium using different specimen geometry. Also torsion tests are simulated by using ABAQUS and experimentally validated
Keywords 
torsion test, ASTM, Nadai method, specimen geometry, torque, twist angle, stress–strain curve, FEA 
INTRODUCTION 
Mechanical testing plays an important role in finding fundamental properties of engineering materials. These can be used in development of new materials. For a material to be used in engineering structure subjected to a load, it is important to know the strength and rigidity of the material to withstand the loads. As a result, numbers of experimental techniques have developed by engineers for mechanical testing of engineering materials. These materials may be subjected to tension, compression, bending or torsion loading. To test the material in torsion the proper test procedure is needed. It involves mounting a specimen on the testing machine. The torque is applied incrementally and both the applied torque and the corresponding angle of twist are measured. Using the appropriate formulae, relationships and the measured dimensions, shear stress and shear strain can be determined. Then, plot the torque vs. angle of twist, and shear stress vs. shear strain curve. From which material shear properties can be determined. [1] 
Adequate construction of the failure criterion involves an experimental study of fracture in different types of stress state, while the experiments in tension and compression traditionally used for this purpose are not enough. For the construction of the fracture surface in this case, besides compression and tension experiments, additional shear tests as well as combined biaxial tests with thinwalled tubular specimens of different geometry should be conducted.[2] Zarroug et al. [3] examines the results obtained from combined tension–torsion loading tests carried out on Mild steel (En8) specimen. The results from the finite element analysis (FEA) are validated against experimental results. 
Graber et al. [4] applied new approximate method for calculating stressstain curves from torsion test results. For obtaining best accuracy the use of thin walled tubular specimens is recommended especially when the flow curve deviates strongly from the exponential LudwigHolloman law. The calculation also makes use of the concept of the “effective” length during deformation for short specimens. 
Pavel et al. [5] conducted experiments for the failure criterion construction in a torsion test on solid or thickwalled cylindrical specimens. 
Berto et al. [6] conducted 70 new fracture tests on notched specimens made of polymethylmethacrylate. All torsion tests are carried out at room temperature under torsion loading conditions. Semicircular notches as well as U and Vnotches (with an opening angle equal to 120°) are used, with a root radius ranging from 0.1 to 7.0 mm. Plots of torque loads versus twist angles are recorded varying the notch root radius and the notch depth. 
HanChin Wu et al. [7] conducted series of experiments on cast and extruded high purity aluminium material under monotonic large strain torsion condition. Both freeend and fixedend torsions studied using tubular specimens of different gage lengths (long, medium and short). The torque versus angle of twist curves were recorded and converted into true shear stress–strain curves by use of the modified Nadai method. The axial extension for freeend torsion and the axial stress developed during fixedend torsion are recorded. The effect of specimen geometry is studied. It is found that the long, thickwalled tubular specimen suitable for torsion test in the large strain range. 
Bressan et al. [8] worked for developing, testing and operating a computerized prototype machine. The equipment consists of a horizontal torsion machine composed of an electric motor, a wheel, a horizontal shaft and a control and acquisition data system. An encoder obtains the torsion angle and the torque is measured by a load cell. Experimental tests are carried out at a constant angular speed that imposed a constant shear strain rate to the test specimen. The torsion tests have been performed on annealed materials such as 1020 steel, brass, pure copper and pure aluminium. The strain rate sensitivity parameter m has been evaluated from the equivalent stress versus strain curve from tensile and torsion tests. In torsion tests the mvalues were 0.072 and 0.045 for steel and brass. 
Utpal Borah et al. [9] developed multiaxial state of deformation kinematics and stresses in bulk metal forming processes, evaluation of workability calls for materials testing using combined axial and torsion tests. A high temperature combined axial and torsion testing system (ATTS) is developed for assessing workability of various materials under multiaxial state of stress and deformation conditions up to 1300 °C. In this paper, results from finite element modelling of axial torsion tests and seamless tube extrusion are presented. 
II. EXPERIMENTATION 
The problems associated with torsion tests are : the homogeneity of the material, the strain measurement, the specimen geometry, and the determination of the shear stressstrain curve from the experimental torque versus angle of twist curve. These problems are explained as follows: 
1. The material homogeneity: If the material is not homogeneous, nonuniform deformation will occur, leading to regions that remain almost undeformed while plastic deformation occurring in the remaining part of the specimen. The strain concentration will lead to an early development of shear band localization in the specimen. This is a very serious problem, especially when long gagelength specimens are used. 
2. The strain measurement: An important problem associated with the torsion test and the combined axialtorsion test is the strain measurement. The method does not lead to accurate strain measurements due to the geometry of the specimen which has enlarged ends. 
3. The specimen geometry: Torsion tests have been conducted by use of tubular specimen, solid shaft. Tubular specimens are used in most of research. The advantages for this type of specimen are: (a) accurate shear strain measurement may be obtained because of the long gage section that this type of specimen has; (b) accurate strain measurements may be obtained in axial and hoop strains; (c) the specimen may be used for combined axialtorsional testing; (d) the specimen is suitable for test involving unloading/reloading/cyclic loading; and (e) the specimen may be used for investigation of shear band localization. The axial strain mentioned in (b) is greatly influenced by the specimen geometry Solid shafts can achieve very large strain without buckling. 
4. The determination of shear stressstrain curve: It uses a theory to convert the torque versus angle of twist curve into the shear stressstrain curve. Most of the theories such as Nadai [1950] or Canova et al. [1982] do not account for length change of the specimen subjected to torsion and, therefore, do not lead to accurate results for freeend torsion. Although the method can be used to obtain accurate shear stressstrain curve for the fixedend torsion test [Wu et al., 1997]. Therefore, the freeend torsion of tubular specimens is still a practical test for the determination of true shear stressstrain curve. For this purpose, experimental data for axial and hoop strains are also needed. Based on the Nadai method, Wu et al. [1992, 1997] developed a method which, by accounting for the axial and hoop strains, provides a true shear stressstrain curve from the freeend torsion test. This true shear stressstrain curve is consistent with that determined from the fixedend torsion test. 
The loading of the specimen is carried out in different modes: (i) maintaining tensile force or axial displacement constant and increasing torque or angle of twist (ii) maintaining torque or angle of twist constant and increasing load or axial displacement.[3] Experimentation is carried out for different specimen geometry of different materials like mild steel, aluminium. The torque and angle of twist is recorded by varying load. The stressstrain curve plotted from recorded data by using Nadai method and value of modulus of rigidity is determined. Also the strain rate parameter is determined at room temperature. To study the effect of specimen geometry on torsion test results. The dimensions of 
specimens such as gauge length, fillet radius and diameters are varied and the effect on deformation curves is studied. A new effective radius is proposed for calculation of strains and shear stresses from toquetwist data for solid specimens. 
III. EXPERIMENTAL RESULTS 
In Fig. 1, the torsion test specimens for materials like aluminium, steel, brass and copper are shown. The specimens are hollow and specimens are designed according to stiffness criteria. 
In table 1, different dimensions for hollow specimens are shown. 
In Figs.2 and 4, the torsion curve results of the present torque versus the twist angle for the plastic torsion tests for commercial pure aluminium and 1020 steel are shown. 
The graphs in Figs. 3 and 5 represent the shear stress versus shear strain curves for these materials. 
In Fig. 3, the difference between the solid and tubular specimens plastic behaviour can be observed. The solid specimen yields a lower curve and a larger elongation or a shear strain to rupture. Thus, the hardening curves are different, the solid specimen showing a lower workhardening coefficient. In the tubular specimen, the shear stress gradient across its thickness is negligible but in the solid specimens are not. The shear stress rupture point shown in the plotted curve is very clear. Thus, the tubular specimens are more reliable. 
In Fig. 4, the plastic behaviour for solid and tubular aluminium specimens are quite diverse. The solid specimen has presented larger elongation and a hardening curve with three maximum load points. Therefore, some softening has been present during the deformation process in the torsion test. This softening is possibly due to the onset of adiabatic shear bands. 
In table 2, different dimensions for solid specimens of 1020 steel and aluminium are shown. 
The solid specimen has an increased ductility or elongation, although it has the same useful length Lu as the tubular specimen. This may be due to the specimen compression in its longitudinal axis as the specimen is constrained at its ends and, generally, there is some axial strain. 
For the tubular specimens, fracture occurred at the point of maximum load or at the instability point, as seen in Fig. 5. This phenomenon repeated and has occurred in all tested materials for tubular specimens. The torsion test curve is superior to the tensile test due to its higher strain rate and the strain rate hardening mechanism or a positive mvalue. On the other hand, the rupture strain for the tensile test is greater than the torsion test. Therefore, higher strain rate has decreased the steel ductility or the rupture strain. Besides, the workhardening coefficient nvalue is different. However, the situation is quite different for aluminium. The equivalent fracture strain for the torsion test is greater than the tensile test: about three times the fracture strain in tensile test. The parameter m from tensile tests for brass and copper are zero, however, for 1020 steel and aluminium are 0.013 and 0.027, respectively. In torsion tests themvalues were 0.072 for steel, 0.045 for brass and 0 for aluminium and copper. 
IV. RESULTS OF FINITE ELEMENT ANALYSIS 
AxialTorsion Testing 
Various authors have proposed multiple plasticity models for torsional deformation [9]. However, literature on numerical modeling of thermomechanical axial torsion deformation in steels is scarce. The geometry of the axial torsion specimen is axisymmetric; the process is not amenable to conventional axisymmetric FEM modeling due to the torsional loading. The generalized axisymmetric element formulation in ABAQUS allows for an additional degree of freedom to apply torsional loading to the axisymmetric elements about the axis. These elements in fully integrated bilinear form, designated as CGAX4, were used in the FE models of the following tests: 
(i) Pure torsion at 1rpm speed: gauge diameter = 15 mm and gauge length =78.2 mm, fillet radius = 25 mm, one revolution. (ii) Pure torsion at 1rpm speed: gauge diameter = 10 mm and gauge length =10 mm, fillet radius = 5 mm, one revolution. (iii) Torsion at 1rpm with compression at 0.083 mm/s: gauge diameter = 10 mm and gauge length =10 mm, fillet radius = 5 mm, one revolution, 5 mm stroke. (iv) Torsion at 1rpm with tension at 0.083 mm/s: gauge diameter = 10 mm and gauge length =10 mm, fillet radius = 5 mm, one revolution, 5 mm stroke. Elastoplastic FEM modeling are carried out for room temperature torsion testing of 316L Stainless Steel conforming to test case (i). The model is validated with experimental torsion test data generated by the ATTS. Subsequently, elastoviscoplastic FEM modeling of axialtorsion tests are carried out using this validated FEM model for tests cases (ii) to (iv). 
V. CONCLUSIONS 
The true shear stressstrain curve can be drawn using modified Nadai method. This method accounts for length change in freeend torsion and it is a method to determine the true shear stressstrain curve without measurement of radial and hoop strains. 
When the axial displacement is maintained constant and torque is increased, the axial load carrying capacity of the specimen decreases. A rapid drop in the axial load carrying capacity is observed once yield torque is reached. When the angle of twist is maintained constant and the axial load was increased, the torque carrying capacity of the specimen drops rapidly once the yield load is reached. 
When the initially applied torque is closer to the yield torque, the maximum axial load that can be applied to the specimen decreases significantly. After the specimen becomes fully plastic, even without any increase in load the axial stress and shear stress increases rapidly as necking commences and fails rapidly. 
Tubular specimens are more reliable for torsion tests than solid specimens. The gauge length and wall thickness may have a significant influence on the axial stress in fixedend torsion. 
Accurate determination of shear modulus depends on specimen dimensions and errors involved in torsion test so we have a scope to study the effect of specimen dimensions namely root radius and gauge length on torsion test results. 
ACKNOWLEDGEMENT 
The experiments presented in this paper are performed by authors listed below. So we would like to express our sincere gratitude to these authors. 
References 
[1] S. P.Timonshenko, D.H.Young, “Elements of strength of materials”, Fifth Edition,citton educational publishing, INC., pp 70 94,1988 [2] G.E. Dieter, "The Torsion Test in Mechanical Metallurgy”, Chapter 10, Third Edition, McGrawHill, Inc., New York, 1986. [3] N.M Zarroug, R Padmanabhan, B.J MacDonald, P Young, M.S.J Hashmi, “Mild steel (En8) rod tests under combined tension–torsion loading”, Journal of Materials Processing Technology, vol. 143–144, pp 807813,2013. [4] A. Graber, K. Pohlandt, K. Lange, “A New Approach to the Torsion Test for Determining Flow Curves”, CIRP Annals  Manufacturing Technology, vol. 38, pp 223226,1989. [5] Pavel A.,Mossakovsky Fedor K., Antonov Lilia A., Kostyreva, “ Investigation of Failure Criterion in Dynamic Torsion Tests with Solid Cylindrical Specimens”, 8th European LSDYNA Users Conference, pp 14231430,2011. [6] F. Berto, M. Elices, P. Lazzarin, M. Zappalorto, “Fracture behaviour of notched round bars made of PMMA subjected to torsion at room temperature”, Engineering Fracture Mechanics, vol. 90, pp 143160,2012 [7] HanChin Wu, Zhiyou Xu, Paul T Wang, “Torsion test of aluminium in the large strain range”, International Journal of Plasticity, vol.13, pp 873 892,1997. [8] Jose Divo Bressan,Ricardo Kirchhof Unfer, “Construction and validation tests of a torsion test machine”, Journal of Materials Processing Technology, vol.179, pp 2329, 2006. [9] Utpal Borah, Diptimayee Samantaray, Shaju K. Albert, A. K. Bhaduri ,T. Jayakumar, “ThermoMechanical AxialTorsion Testing to Assess Workability Modelling Using Finite Element Method and Experimental Validation”,International Journal of Metallurgical Engineering , vol.2, pp 214220,2013. 