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If you cut something by pushing a blade directly into it, here's what happens: On first contact of blade with material, only the very thin edge of the blade is touching the material, the force per unit area is very high, and the blade cleaves the material very easily. That's why it's almost trivially easy to make score marks in things like aluminum using a ...

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Here is an intuitive / qualitative answer. Maybe someone else will add some math. I wonder if it's instructive to look at diamond cleaving. As you know, diamond is extremely hard, and conventional machining is very difficult. But if you can find the right fracture plane ((111) and its symmetrical cousins), it's possible to cleave the diamond along that ...

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Strictly speaking, Young's modulus is not always greater than the shear modulus, but it does tend to work out that way. You can see the reason why if you look at the relation between the two quantities (and Poisson's ratio). $$G = \frac{E}{2(1+\nu)}$$ Combined with the knowledge that $\nu$ can be anywhere in the range $(-1, \frac{1}{2})$, one can see ...

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When people study continuum mechanics they usually do so at first in $\mathbb{R}^3$ where we have usually implied the usual metric tensor $(g_{ij}) = \operatorname{diag}(1,1,1)$ and the Levi-Civita connection associated with it. In that case vectors and covectors are equivalent: the metric tensor induces the musical isomorphism and allows one to convert ...

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What you observe as mechanical deformation of a steel spring is an actual displacement (motion) of the atoms constituting the spring. In places, atoms will be slightly closer to their neighbors (compression) and in some other places actually futher apart (tension). The combination of compression on one side and compression on the other side of a beam or a ...

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There are two forces in presence and, because the system is in equilibrium (=0), the change in one will be compensated by a change in the other. A change in this context is decoded to 'the rate of change in relation to space: $\frac{d}{dx_i}$ ' $\frac{d\sigma}{dx_i}$ is a force per unit length and $\frac{d\psi}{dx_i}$ is an acceleration per unit length, ...

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whenever external force is applied on the object automatically a restoring force is developed inside the object to restrict the deformation of the object.The ratio of restoring force perpendicular to the surface to the area is known as stress.The ratio of external force perpendicular to the surface to the area is known as pressure. for example if you press ...

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