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I was reading the book Mechanics of Materials by Beer and Johnston.The author points out in torsion chapter that ductile materials generally fail in shear.And brittle materials materials are weaker in tension than in shear.From this they concluded that when subjected to torsion a circular shaft made of ductile material breaks along a plane perpendicular to it's longitudinal axis and the brittle material break along surfaces forming angle 45° with the longitudinal axis.

I couldn't get how they come to the fact that ductile material fail in shear and brittle material fail in tension.Any ideas?Thanks.enter image description here

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  • $\begingroup$ So, how do you think they should fail? $\endgroup$ – Solar Mike Jan 12 '18 at 13:07
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    $\begingroup$ Consider the stress-strain to failure curves. Relatively speaking, 'so-called' ductile materials will have a large zone of plasticity, whereas 'so-called' brittle materials will have a relatively small, perhaps non-existant zone of plasticity - but a large elastic zone. Therefore one would surmise that ductile materials would experience failure within the plastic zone, whereas brittle materials would experience failure at the limit of elasticity. $\endgroup$ – AsymLabs Jan 12 '18 at 15:05
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    $\begingroup$ On a micro scale, plasticity (generally) is the result of a slip along the grain boundaries, the larger the grains the less the resistance to slip and the greater the ductility, whereas finely grained materials would resist slippage (be more brittle). The grains would split apart in the latter case - this leads to a tension failure. This is a simplification of course. $\endgroup$ – AsymLabs Jan 12 '18 at 15:15
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    $\begingroup$ You may wish to supplement Beer & Johnston with an Intro to Materials book such as Callister's, which will explain the atomic-scale origins of failure. Briefly, ductile materials fail by dislocation slip, which is a shear phenomenon. Brittle materials fail by crack propagation generally triggered by tensile loads. $\endgroup$ – Chemomechanics Jan 14 '18 at 1:01
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    $\begingroup$ @AsymLabs Grain boundaries aren't required; single crystals also fail in shear via dislocation glide. The Hall-Petch effect that you describe is a result of dislocations moving through grains, not along grain boundaries. $\endgroup$ – Chemomechanics Jan 14 '18 at 1:02
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Backing off of micro-crack fracture theory, what your engineering introduction book is trying to hint at is the implications of Mohr's circle. To illustrate, from Wikipedia:

enter image description here

As you can see, when you have a stress in x-direction, a stress in y-direction, and a shear stress, you can convert the stress to a new arbitrary axis (x' and y') with a different normal and shear stress. Basic statics can resolve these forces:

$$\sigma_n = \frac{1}{2} ( \sigma_x + \sigma_y ) + \frac{1}{2} ( \sigma_x - \sigma_y )\cos 2\theta + \tau_{xy} \sin 2\theta$$

$$\tau_n = -\frac{1}{2}(\sigma_x - \sigma_y )\sin 2\theta + \tau_{xy}\cos 2\theta$$

In the case of a shaft under pure torsion, the equations simplify:

$$\sigma_n = \tau_{xy} \sin 2\theta$$ $$\tau_n = \tau_{xy}\cos 2\theta$$

If the primary failure method is shear failure, such as the case for most ductile materials, then this occurs when we look at the axis where $\theta = 0$, and the stress is entirely in shear. However, if we look at the axis where $\theta = 45$°, then the stress is all in tension normal stress. In this case, when the material fails easily in tension, it will fail in this direction.

The real world is actually more complicated with micro-crack fracture theory, stoichastic methods, and fatigue stress. However, when running designs, this is practice to keep in mind - just because the shear stress is low doesn't mean that the brittle material can handle the tensile forces. As the introduction to machine design book, this is simply to make you aware of this phenomenon and know there should be a difference. However, to handle this in the real world, design criteria methods such as Mohr-Coulomb Theory would be utilized for brittle materials, while von Mises yield criteria are used on ductile materials. Creep is used in plastics. A nice summary of other methods is listed at Colorado State.

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