# Tag Info

45

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 ...

7

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 ...

5

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 ...

5

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 ...

5

First we need to understand the force between individual atoms. At relatively large separations (e.g., a few atomic diameters) atoms attract each other with a force that does, as you suggest, get weaker with distance due to polarization and ionic effects that we needn't go into here. If that was all there was to the story, however, collections of atoms ...

4

http://www.physicsforums.com/archive/index.php/t-37701.html says "Most of the strength of a cylinder comes from the outer portions. I think the contribution goes like the cube of the radial position. So, if you took a solid rod and drilled out a half the volume from the center, you do not lose half the strength. Strength to weight ratio is ...

4

Dave, to answer your question, we need to know the wall thickness and weight of each cube. BUT I can tell you that if you give any reasonable answer (like wall thickness = 1/8 of an inch = about 3.2mm), and choose steel bolts with right-sized steel washers, that it will be pretty hard to fail. Also, assuming your bolts are 60ksi tensile strength steel ...

4

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 ...

4

It provides a convenient graphical means of finding the maximum and minimum shear stress, which are important for determining material failure. You don't absolutely need it, but the graphical interpretation of the circular relationship between normal and shear stress is somewhat convenient. I've read good solid mechanics books that give little if any ...

4

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 ...

3

Examples: stress is zero but strain is present= when component is loaded beyond the elastic limit it shows plastic deformation which can not be regained. after unloading the specimen in plastic deformation zone material will follow slope similar to the elastic slope and will come back to zero stress (as load is removed now). but during this process it has ...

3

Zero strain does not always imply zero stress and visa versa. There are matterials that display stress-strain, $\sigma-\epsilon,$ hysteresis behaviour. In matterials like this, when you start loading them, they behave normally, i.e increasing the stress increases the strain. However, when you start to unload them (remove the load), instead of the stress ...

3

Yes, the bulk modulus $B$ is the inverse of the isothermal compressibility $c$, $$B = \frac {1}{c}.$$ See e.g. Wikipedia. The "bulk modulus" is more typical terminology in mechanics where we don't care about heat much and where the typical assumption is that the temperature is kept fixed (because mechanical engines start to malfunction if their ...

3

Start with differential form of Poisson's ratio: $$\frac{\text{d} x}{x}=- \nu \frac{\text{d} l}{l}$$ $$\int_{x_0}^{x_0+\Delta x} \frac{\text{d} x}{x}=- \nu \int_{l_0}^{l_0+\Delta l} \frac{\text{d} l}{l}$$ $$\ln \frac{x_0+\Delta x}{x_0}=- \nu \ln \frac{l_0+\Delta l}{l_0}$$ $$1 + \frac{\Delta x}{x_0}=\left(1+\dfrac{\Delta l}{l_0}\right)^{-\nu}$$

3

Your assumptions are correct (but $r$ is often defined as the distance from the pipe centerline). However, this is a very specific case: laminar pipe flow. In general, the stress will be a tensiorial quantity, defined as $$\tau_{ij}= \eta \frac{\partial u_i}{\partial x_j}$$ which is true for turbulent flow, in arbitrary geometries. Where $i,j$ are in the ...

3

The Cauchy stress matrix $\Sigma$ is a $3 \times 3$ real symmetric matrix. It is interesting that we may without problems generalize $\Sigma$ to a $3 \times 3$ Hermitian matrix. It has three mutually orthogonal principal stress directions with principal stresses (eigenvalues) $\lambda_1\geq\lambda_2\geq \lambda_3$. Consider an arbitrary unit vector ...

2

I know this question is a little old, but so far there is no correct answer posted. It happens that I used to argue this question with Civil Engineering students years ago. They had trouble with it because they learned about "bending moment", which is basically the same thing as curvature only no one told them that. The curvature, or the bending moment if ...

2

You need to consider the elasticity of materials in order to avoid reduntant constraints. If the forces acting on body are completely known (as well as moments), then you might be able to distribute the applied forces onto internal stresses on the basis of hertz contact pressures and subsurface stresses on the elastic half spaces. To get bulk stresses you ...

2

As you mention, the concept of rigid bodies and stresses don't fit together. The stress distribution is dependent on the elastic properties of the material. For a relatively simple 2D shape (say, less than 700 nodes, 1000 triangles), doing FE-simulations in real time can be feasible (~1/100 seconds). Also, since the stiffness matrix wouldn't change, this ...

2

I'd swap the meaning of x and y to make the sparsity structure more conspicuous (block diagonal). The I'd try to identify the paramaters as in the isotropic case, where they exist, and itnroduce new ones for the others by analogy (note that each column has the same denominator). Then consider special stress vectors that affect only few strain components, and ...

2

Application: A strain gauge is a device used to measured the strain (change in length as a proportion of the original length) in an object as a result of an applied load. Most strain gauges are designed to measure strains in only one direction. How it works: A common type of strain gauge consists of thin metallic foil cut into a pattern such that most of ...

2

So, buckling is the bifurcation of static equilibrium. And thus: More technically, consider the continuous dynamical system described by the ODE $\dot x=f(x,\lambda)\quad > f:\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}^n.$ A local bifurcation occurs at $(x0,λ0)$ if the Jacobian matrix $\textrm{d}f_{x_0,\lambda_0}$ has an ...

2

The key idea here is the concept of "power-conjugate" stress and strain-rate measures. For the Cauchy stress $\sigma$, the stress power is given by: $$\dot W/V = \sigma:D$$ where $D$ is the rate of deformation tensor defined as the symmetric part of the velocity gradient. $$D = sym(L) = \frac{1}{2}(L + L^T)$$ The quantity $\sigma:D$ gives the stress ...

2

Though I generally agree with whoplisp's answer, it is worth to note that obtaining the (lower) limit of the thickness is rather tricky as long as it is defined by stability under 'strong' deformations. Where 'strong' is compared with the tube wall thickness. Which is obvious from common sense point of view: thin rod is easier to deform than the hollow ...

2

Starting with the basic concept of Gaussian curvature, we look to the metric given in the question, and try to identify some concept of strain from that. This is what hyperbolic curvature looks like for 2D in 3D, and I believe it should extend to the present problem of 3D to 4D without any material change. From this picture we could imagine calculating ...

2

Any cross section of your wall is supporting the weight of all the wall above it. In a first approximation, every cross section will be in a state of pure axial compression. The most heavily solicited cross section will be the one at the very bottom, which will be supporting a compressive pressure of $\rho h g$, where $\rho$ is the density of the ice, $h$ ...

2

From the above definition for tensile strain, $L$ is $L=\epsilon L_{0}+L_{0}$. Thus $dL=L_{0}d\epsilon$ replacing in the above $L_{0}$ with $L_{0}=\frac{L}{1+\epsilon}$ you get $dL=\frac{L}{1+\epsilon}d\epsilon$ Now, if the deformation of the material is very small you can expand $\frac{1}{1+\epsilon}$ as $\frac{1}{1+\epsilon}=1-O(\epsilon)$ Hence you ...

2

I think you are looking for something like this: We measured the elastic properties and intrinsic breaking strength of free-standing monolayer graphene membranes by nanoindentation in an atomic force microscope. The force-displacement behavior is interpreted within a framework of nonlinear elastic stress-strain response, and yields second- and ...

2

Yes, when two objects collide, there's some excess pressure in the impact zone but the pressure propagates by the wave equation through the material – it's nothing else than the mechanism by which sound propagates in a medium. Also, waves may be transverse or longitudinal (the variation of the position is going in parallel with the direction of the wave). ...

2

I think it's safe to say yes for all elastic (non-rigid) material. I believe it depends on the elasticity. Deformation (Strain) occurs in pretty much every body because of the force you apply and it could be temporary at a macroscopic level due to the elasticity. So an impulse results in a compression wave that propogate through the material. Deformation ...

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