Properties in Shear. Figure 7c. Figure 7b. Figure 7a
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1 Properties in Shear Shear stress plays important role in failure of ductile materials as they resist to normal stress by undergoing large plastic deformations, but actually fail by rupturing under shear stress. Shear stresses are categorized into two main types: 1. Direct (transverse) shear in rivets (Fig. 7a) and beams (Fig. 7c).. Pure (torsional) shear in shafts subjected to pure torsion (Fig. 7b). Figure 7a Figure 7b Figure 7c Direct or transverse shear tests are employed to obtain approximate measure of shear strength for specific applications while torsion test is usually employed to evaluate shear behaviour and properties of materials. 1
2 Direct Shear Tests In direct shear tests, the specimen is clamped in a fixture and shearing force is applied through a shear tool so that the maximum load is determined. Tests are done in three ways: Single shear test (Fig. 8a) Double shear test (Fig. 8b) Punch shear test (Fig. 8c) Figure 8 single shear : double shear : F r punch shear : F dh F r (a) (b) (c) These tests do not provide reliable information on properties of materials in shear (i.e. yield, stiffness, resilience). They only provide the approximate ultimatestrengthofmaterialinshear. Even so, it is not always reliable due to certain factors (e.g. hardness, sharpness and correct setting of shearing tools as well as bending stresses and friction between the parts).
3 Torsion Test Torsion tests are carried out by applying a twisting moment (torque T) to one end of a specimen while measuring angular deformation (angle ) at the other end. A torsion test machine (Fig. 9) consists of units for loading (A) and indicating (B). Unit A is stationary (mounted on bed) while unit B is movable (to permit adjustment of varying specimen lengths and automatic compensation of changes in length). Cylindrical specimens having square/hexagon shaped ends are usually used. Figure 9 3
4 Torsion Test Torsion test is useful in determining the material properties such as shear modulus of elasticity, torsional yield strength and shear modulus of rupture. Such tests can also be carried out on full-sized engineering components to determine their behaviour under service conditions. Torsion test offerscertainadvantagesovertensiletest: During torsion test, no necking occurs. Therefore, the torque increases up to the moment of failure. Plastic deformation is almost uniform over entire length of specimen, which enables the determination of deformations and stresses reliably for highly ductile materials (especially pure metals). Brittle or low ductility materials, that are often difficult to test in tension, can undergo quite measurable deformation in torsion test, which enables the determination of their mechanical properties. In addition, torsion tests can easily be conducted at high strain rates. 4
5 Torsion Test During torsion test, torque (T) and angle of twist () are measured. Based on this data, torque-twist diagram is constructed. Figure 10 Fig. 10 gives torque-twist diagrams for ductile (normalised mild steel) and brittle (cast iron) material. Normalised mild steel (D = 6.35 mm, L = 15.4 mm) Using torque-twist diagram, shear stresses and corresponding shear strains are calculated(byequations given in the next slide). Cast iron (D = 6.35 mm, L = 76. mm) 5
6 Shear Stress & Shear Strain Take a cylindrical bar (Fig. 11a) that is fixed at one end and subjected to a torsional moment (torque, T) at the other end, and hence will be twisted through an angle (). Such torsional deformation (S) induces shear stress () resulting in shear strain () along the circumference (Fig. 11b). Figure 11 (a) (b) T O C B A T R C L B S T R J J R 4 (for solid bar) tan S L R L : shear stress (kg/mm ) : shear strain (mm/mm) T : applied torque (kg*mm) R : radius of the bar (mm) J : polar moment of inertia (mm 4 ) : angle of twist (radians) L : length of the bar (mm) 6
7 Solid vs. Tubular (Hollow) Specimens Torsion test is not used in material specifications to the same extent as tension test since no uniform shear stress can be generated. Magnitude of shear stress () varies from zero at symmetry axis to the maximum at surface (Fig. 1a). When the surface reaches to elastic limit, the interior will still be in elastic range. Thus, start of yielding at the surface cannot be detected until large amount of plastic deformation occurs. In order to overcome this problem, thin-walled tubular specimens are used so that shear stress is Figure 1 (a) Solid (b) Tubular (thin-walled) assumed to be uniform along cross section (Fig. 1b). However, there is a danger of buckling when using tubular (hollow) specimens if ratios of length-to-diameter and diameterto-thickness are not kept within min Tmax min Tmax elastic limits. max >> min = 0 max min 7
8 Stiffness in Torsion Stiffness in torsion is measured by shear modulus or modulus of rigidity denoted by G: G T L J It can also be obtained from Young s Modulus (E) provided that Poisson s Ratio () of that material is known: G E 1 Poisson s ratio and modulus of rigidity for various materials are given in table (isabout0.3formostmaterials). 8
9 Elastic Shear Strength Elastic shear strength is measured by the maximum stress in specimen corresponding to torque during the transition from elastic to plastic range. In torsion test, the first onset of yielding is usually not apparent for most materials, and hence offset yield point is employed (point P in Fig. 13). Offset T T sy Figure 13 P angle of twist is usually taken as 4 * 10-5 rad/mm of gauge length. Thus, elastic shear strength (S sy ) is defined by: // S T R J sy sy T sy : torque at proportional limit (i.e. at offset angle of twist) O offset Tubular specimens are also used to precisely determine torsional elastic limit or yield strength. Using a specimen having a thin-walled circular crosssection, shear stress along the wall is uniform for all practical purposes as such specimens do not benefit from strengthening effect of inner fibers. 9
10 Elastic Shear Strength Strengthening effect of inner fibers is illustrated in Fig. 14 for Al alloy 6061-T4. In theory, thinner is the wall thickness, more reliable is the measurement of elastic shear strength as all fibers are at about the same stress. However, if a thin-walled specimen is subjected to torsion, it would first fail by buckling before the shear strength of material is reached. Thus, a tubular specimen with ratio of length-to-diameter of at least 10 and ratio of diameter-towall thickness of about 10 is recommended for torsion test without buckling. For a tubular specimen, the elastic shear strength is defined by: Figure 14 S sy T sy Rt t d d o R d d o i i 10
11 Resilience, Ultimate Strength, Toughness The area under torque-twist diagram represents the total work in stressing specimen to the proportional limit. Thereby, modulus of resilience (U s ) is defined as: U s Tsy sy Ssy AL 4G T sy : torque at proportional limit sy : corresponding angle of twist S sy : elastic shear strength A : cross-sectional area L : gauge length G : modulus of rigidity Plastic shear strength ( u ) is the approximate definition of modulus of rupture or ultimate shear strength (S su ): solid specimen : S T R J tubular specimen : S T su su u u Rt For accurate definition of ultimate shear strength, tubular specimens with short reduced sections are recommended (i.e. L/d = 0.5 and d/t = 10). Definition for toughness based on torque-twist diagram is also approximate. Toughness index number in torsion (T o ) can be defined as: T o T u f AL f : fracture angle of twist 11
12 Failure Types in Torsion The torsional fracture is quite distinct from either tension or compression fracture. There is almost no localized reduction or area (i.e. no necking). Possible fractures are as follows: Ductile materials (Fig. 15a) fracture at 90 to the torsional axis in maximum shear plane. Brittle materials (Fig. 15b) fracture at 45 to the torsional axis in maximum tension plane. Buckling (Fig. 15c) occurs in hollow specimen if L/d and d/t ratios are not kept withing limits. (a) (b) (c) Figure 15 Torsional failure of a brittle material Torsional failure of a ductile material 1
13 Failure Examples in Torsion 13
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