226

Understanding why this works turns out to be quite deep. This answer is kind of a long story, but there's no maths. At the end ('A more formal approach') there is an outline of how the maths works: skip to that if you don't want the story. Insect geometry Consider a little insect or something who lives on the surface of the paper. This insect can't see ...


64

The breaking of dry spaghetti was discussed in a 2005 Phys. Rev. Lett. by French physicists Audoly and Neukirch. Bottom line is that elastic (flexural) waves propagating along the spaghetti cause local increases in curvature leading to multiple breaking points: abstract to article. In essence, your assumption "that vibrations from a first break could ...


64

There are two separate concepts here: the Young's modulus, which determines the force needed to stretch the material the elastic limit, aka yield strain, which determines how far the material can be stretched As you say, the term elastic tends to be used in a vague way that conflates these two properties. Generally a high Young's modulus means the material ...


50

Both the OP and John Rennie have well illustrated the imperfections in the usage of the word "elastic" in physics and how the word can create confusion between "stiffness" and a material's ability to brook strain. But an important point to be made is that the one important field where one hears the vague statement that "steel is more elastic than rubber" is ...


26

You have essentially discovered principles behind bending moments and structural engineering. As another poster stated, physically the structure you made is stronger, because to bend something (for example, a beam loaded at the top) layers at the top are compressed whereas layers at the bottom are stretched. This is simply due to geometry and the physical ...


22

Uniform bodies are idealizations like frictionless surfaces or no air resistance. They make the work easier. In reality there will be slight deviations in material properties (such as density, tensile strength, etc.) along various parts of the body. In your example these deviations become more and more important and extreme as the body is stretched (which ...


21

Pressure is defined as force per unit area applied to an object in a direction perpendicular to the surface. And naturally pressure can cause stress inside an object. Whereas stress is the property of the body under load and is related to the internal forces. It is defined as a reaction produced by the molecules of the body under some action which may ...


21

For each surface on a unit cube (see below), the stress on that surface can point in each of the three directions. (source) Since it is not necessarily the case that $\sigma_{11}=\sigma_{31}=\sigma_{21}$ (all pointing the in the same $\mathbf{e}_1$ direction)--or any of the other $\sigma_{ij}$ combinations, we need to have 9 components describing it, hence ...


20

When you bend a piece of material, the resistance is provided by stretching the material on the outside part of the bend, and compressing the material on the inside of the bend. A thin flat sheet bends easily because, physically, not of lot of stretching or compressing occurs when it bends. When you give your book a fold, like a trough, that shape can not ...


15

You could consider it as one more demonstration of the underlying quantum mechanical frame keeping atoms and molecules bonded together. Quantum mechanics is a probabilistic theory, and which bond will "break" depends on the square of the wavefunction describing the rod, with a probability which manifests in this one instance of breakage. To get the ...


12

Applying a force in the $x$-direction might change the shape of the material in the $y$-direction. The only way to capture such an effect is through a tensor. If you have a general force acting on your body $$ \vec F = (F_x, F_y, F_z)^T$$ and you are interested in the reaction of the body by looking at its deformation $$ \vec \epsilon = (\epsilon_x, \...


12

I'll take the question to be referring to solid rock. In reality, I think small asteroids are loose jumbles of rubble with a lot of vacuum between the rocks, and larger bodies like Ceres may have been liquid when they formed. Googling turned up [Scheuer 1981], which can be found online for free by googling. S/he estimates the maximum height of a mountain to ...


12

The tensor itself is not the model, but the the tensor is used to model (one could also say describe or quantify) the anisotropy. One example is an anisotropic electric conductor. The conductivity $\sigma$ describes which current occurs in response to an electric field $\vec j = \sigma \vec E$. In isotropic materials (e.g. glass, microcrystalline metals ...


11

You are asking two questions really 1) How is PE actually stored in a steel spring at the atomic level? The explanation for this lies in quantum mechanics 2) Could you explain in detail how/where potential energy is actually stored in a steel spring and why the material never surrenders to the bending forces taking a new shape? Replying to 1) one ...


11

The other answers so far are technically correct, but none of them really seem to give a commonsense/intuitive and simple answer. So I'll have a go at one. Imagine very slightly bending some kind of object downwards at one end, while holding the other end firmly horizontal. (It could be almost any object, could be paper, a branch off a tree, some plastic ...


10

Given a stress tensor $\mathbf{\sigma}$, which has 9 components in general, the pressure (in continuum mechanics at least) is defined as $P = 1/3 tr(\mathbf{\sigma})$. So the pressure at a point in the continuum is the average of the three normal stresses at the point. The off-diagonal terms manifest as shear stress. It's hard to say "stress" without ...


9

If you experience such a uniform force, e.g. when an astronaut on a space walk near the ISS (just earth's gravity), you don't experience any forces at all. That's freefall. Even with 10G, you'd experience a rapid freefall, but that is still harmless. It's the hitting the ground which kills you - that's not a uniform force.


8

Suppose you bend a perfect, i.e. unscratched, piece of glass, the forces on it look like: The top of the glass is in tension and the bottom in compression, but the stress is spread over a large area of glass so the local stress at any point isn't enough to break the glass. Now put a scratch in the top surface and bend it again: This time the stress is ...


8

I would expand upon Sebastian's nice answer to point out that any orientation-sensitive quantity $f(\hat v)$ may be expanded by spherical harmonics and the symmetric rank-2 tensor can be often used to represent the first nonzero term. To understand this, start by noting that all of these spherical harmonics come with a polynomial structure factor. You may ...


8

You should consider the implicances of your question, which is mostly a philosophical issue regarding our human subjectivity. You are implying an ideal uniformity of states which wouldn't allow quantum behavior. That would mean that atoms should have all the exact same state before the breaking, that is, considering them as particles, they should be all at ...


7

Waves on strings combine linearly. This means that you can split up a string's motion into two (or more) superimposed waves. The two superimposed waves behave independently, as if the other one was not there. So if you have a standing wave set up on a string, and then you also introduce a travelling pulse, you get something like the following. (The arrows ...


7

Paper is made from wood, and wood is made from long fibers. Typically the manufacturing process leaves the fibers are more or less parallel. So it is easier to tear in the direction that separates fibers from neighboring fibers than in the direction that breaks fibers. Wood is the same. It is easier to split a log than chop it. Creasing paper breaks and/...


7

It is a quite famous theorem due to Cauchy. Consider an internal portion $S$ of a continuous body $C$. There are two kinds of forces acting on it: Forces proportional to the mass, of the form $$\int_V \mu(x) \vec{f}(x) d^3x\tag{0}$$ where $\vec{f}(x)$ is the density of force acting on $x \in V$. And forces acting through the surface $\partial V$, the ...


7

Referring to your graph which is for a ductile material I suggest the following. A is the limit of proportionality up to which the stress and strain are proportional to one another and when unloaded the material goes back to its original length. B is the elastic limit. With stresses below this the material behaves elastically ie when unloaded returns ...


6

Hooke's Law is frequently used to model multi-dimensional materials because the stress tensor is simple (linear). The full expression can be found on Wikipedia. The simplification for 2D is straight forward (drop any terms with a 3 in the subscript). Note that whether deformation in one dimension affects the others is a property of the material and shows up ...


6

Indeed, both the strain tensor $$\epsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right) \tag{1}$$ and the stress tensor $$\sigma_{ij}=2\mu\epsilon_{ij}+\lambda\epsilon_{kk}\delta_{ij} \tag{2}$$ are symmetric by definition. However, bear in mind that these definitions are not always valid; $(1)$ assumes ...


6

I am rather surprised that neither link posted above gives a simple discussion of the effect, so here it goes. Let us consider many asteroids of cubic shape, of constant density $\rho$, and of varying side $l$. We ask when, roughly, self-gravity will be able to perturb this shape into a spherical one. A cube of side $l$ has the same volume as a sphere of ...


6

The link you provided already had enough information. Well, unlike hardness, which denotes only resistance to scratching, diamond's toughness or tenacity is only fair to good. That is, it is easily breakable by a hammer. The toughness of diamond is about 2.0 MPa which is good compared to other gemstones, but poor compared to most engineering materials. So ...


6

Isotropy and homogeneity are different. The former is a consequence of invariance under rotations while the latter comes from invariance under translations. The stress tensor of an isotropic fluid then must be invariant under any orthogonal transformation, and this implies that it is a multiple of the "identity" tensor. More precisely, assume matrix notation ...


Only top voted, non community-wiki answers of a minimum length are eligible