Why does a ductile rod in torsion fail at an angle perpendicular to its axis? I know the reason for brittle material failing in pure shear but I don't understand the reason for ductile material.
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$\begingroup$ materials are rarely 100% ductile $\endgroup$– ratchet freakAug 16, 2017 at 15:26
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$\begingroup$ ok but assuming it is.Can you give me the theoritical answer? $\endgroup$– gateprepAug 16, 2017 at 15:59
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$\begingroup$ Do a Mohr's circle and note the direction of the principal stresses. They are 45° from the axis of the rod (90° on Mohr's circle). $\endgroup$– John AlexiouAug 16, 2017 at 19:43
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2 Answers
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As a rule of thumb: When brittle materials are subjected to torsion they fail in the plane, where tension is at its highest, i.e. at a 45° angle. Ductile materials on the other hand fail in the plane of maximum shear stress.
Take a look at Mohr's Circle for pure shear.
Maximum-Shear-Stress Theory states:
[...] yielding of the [ductile] material begins when the absolute maximum shear stress in the material reaches the shear stress that causes the same material to yield when it is subjected only to axial tension.
– R.C. Hibbeler, Mechanics of Materials, p.525
for ductile materials: $$ \tau_{max}=\frac{\sigma_y}{2} $$ As you can see in the graphic, $\tau_{max}$ is reached before $\sigma_y$, thus the plane of failure is perpendicular to the primary axis.
For brittle materials on the other hand, Maximum-Normal-Stress Theory says:
[...] a brittle material will fail when the maximum tensile stress, $\sigma_1$, in the material reaches a value that is equal to the ultimate normal stress the material can sustain […]
– R.C. Hibbeler, Mechanics of Materials, p.528
for brittle materials: $$ \tau_f=\sigma_f $$ So, under pure shear it fails in tension at a 45° angle.
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The stresses of a rod in shear and in torsion are of the same kind. So they fail of the same reason.
There is a conversion factor for materials. It is sqrt(3). So with the same stress induced to a rod under torsion compared to one under tensile load is more likely to break by this factor.
