# Torsion Tests. Subjects of interest

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1 Chapter 10 Torsion Tests Subjects of interest Introduction/Objectives Mechanical properties in torsion Torsional stresses for large plastic strains Type of torsion failures Torsion test vs.tension test Hot torsion testing

2 Objectives This chapter provides fundamental knowledge of torsion test and significant parameters such as torque, modulus of rupture in torsion and angle of rotation will be highlighted. Different types of torsion failures will be evaluated and differentiated from tension failures.

3 Introduction Torsion test is not widely accepted as much as tensile test. Torsion tests are made on materials to determine such properties as the modulus of elasticity in shear, the torsion yield strength and the modulus of rupture. Often used for testing brittle materials and can be tested in full-sized parts, i.e., shafts, axles and twist drills which are subjected to torsional loading in service.

4 Introduction Examples: Driveshaft is subjected to torsion in service. GKN driveline Bar shaft

5 Torsion-testing equipment Torsion-testing equipment consists of 1) A twisting head, with a chuck for gripping the specimen and for applying the twisting moment to the specimen. 2) A weight head, which grips the other end of the specimen and measures the twisting moment of torque. Specimen used A circular cross section specimen is normally used since in the elastic range, shear stress varies linearly from a value zero at the centre of the bar to a maximum value at the surface. Note: Thin-walled tubular specimen is frequently used.

6 Determination of torsion test Applying twisting moment to the specimen and measure the torque. Determination is made of the angular displacement (or degree of rotation) of a point near one end of the test section of the specimen with respect to a point on the same longitudinal element at the opposite end (using Troptometer). The shear strain γ is given by Where γ =tanφ = rθ L Eq.1 θ is the angle of twist or degree of rotation, radian L is the test length of the specimen. Torque-degree of rotation diagram

7 Mechanical properties in torsion Consider a cylindrical bar subjected to a torsional moment at one end. The twisting moment is resisted by shear stresses set up in the cross section of the bar. (zero at centre, max at surface) M T r a τ τrda = r 0 r = = = o a r 2 da Eq.2 r 2 da But is the polar moment of inertia of the area with respect to the axis of the bar. Thus, Where τ = shear stress, Pa M T = torsional moment, Nm r = radial distance measured from centre of bar, m J = polar moment of inertia, m 4 τ J MT =, or τ = r M J T r Eq.3 Torsion of a solid bar

8 Shear stress The maximum shear stress at the surface of the bar is MTD/ 2 τ max = = 4 πd / 32 16M πd T 3 Eq.4 For a tubular specimen, the shear stress on the outer surface is 16M 1 τ = T π 4 ( D D D ) Eq.5 Where D 1 = Outside diameter of tube D 2 = Inside diameter of tube Note: Eq.4 and Eq.5 is applied only for a linear relationship.

9 Elastic properties and yield strength The elastic properties in torsion can be obtained by using the torque at the proportional limit (~ 0.04 rad.m -1 ) where the shear stress is calculated corresponding to the twisting moment from Eq.4 or Eq.5. The torsional elastic limit or yield strength can be obtained from testing a tubular specimen since the stress gradient are practically eliminated. Note: To measure the shearing yield strength and modulus of elasticity Length of reduced section Outside diameter ~ 10 Diameter Thickness ~ 8-10 To prevent specimen to fail in bulking rather than in torsion: Torque-twist diagram

10 Modulus of rupture Modulus of rupture or the ultimate torsional shearing stress is determined by substituting shearing measured torque in to Eq.4 or Eq.5. slightly overestimate. The modulus of elasticity in shear G or the modulus of rigidity is addressed as follows: G = τ γ Eq.6 Or G M Jθ L T = Eq.7 Torque-twist diagram

11 Torsional stresses for large plastic strains Beyond the torsional yield strength, the torque-twist angle relationship is no longer linear and in the plastic range the shear stress in the bar at the outer fiber can be calculated from ' T τ θ + 3M Eq.8 Or τ ( BC CD) a 1 = 3 ' T 2πa dm dθ Where θ = θ /L 1 = 2 π a a dm T =0 ' At maximum point, dθ. The ultimate torsional shear strength or modulus of rupture in the bar at the outer fibre is 3Mmax τ u = 3 Eq.9 2πa Method of calculating shear stress from torque-twist diagram.

12 Types of torsion failures State of stress in torsion on the surface of a bar occurs on two mutually perpendicular planes (longitudinal yy and transverse xx). The principal stresses σ 1,(longitudinal) σ 3 (compressive) make an angle of 45 o and σ 3 = -σ 1. (intermediate stress σ 2 = 0). Torsion failures are different from tensile failures in that there is little localised reduction of area or elongation. State of stress in torsion (a) Shear (ductile) failure (b) Tensile (brittle) failure a) Shear (ductile) failure is along the maximum shear plane. b) Tensile (brittle) failure is perpendicular to the maximum tensile stress (at 45 o ), resulting in a helical fracture.

13 Types of torsion failures GKN driveline Drive shaft failure Brittle failure

14 Torsion test vs. tension test There is better chance for ductile behaviour in torsion than in tension. Critical τ max Critical σ max Plastic flow Fracture Effect of the ratio τ max /σ max in determining ductility.

15 Hot torsion testing Under hot-working condition (T > 0.6T m, ε up to 10 3 s -1 ), torsion test is often used to obtain data on the flow properties and fracture. Cold-working condition Warm-working condition Hot-working condition Torque-twist curves at different temperatures for a nickel-based alloy in torsion at ε =2.5 s Hot torsion testing machine In hot working condition (m>>n), the maximum shear stress is expressed as follows; M T a + = (3 + m n) 3 2 π a τ Eq.10 Where m is strain rate sensitivity n is the strain hardening exponent.

16 References Dieter, G.E., Mechanical metallurgy, 1988, SI metric edition, McGraw-Hill, ISBN

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