# Tagged Questions

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles.

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### Derivation of elastic energy per unit volume

So I basically asked this question a little while back and didn't get much help, but I really need help, so I'm coming back and asking again. Looking at the section on Continuum Systems on the ...
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### How to do continuum approximation?

Assume you have $N$ matrix fields $T_{j}$ on a 1d lattice with lattice constant unity. Now consider a sum like the following (you can think of the traces as supertraces), and subject it to a continuum ...
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### Why is temperature a function of $y$ and $t$ only?

Say you have an incompressible thermal conducting fluid contained between two infinite horizontal plates separated by a distance $H$. Initially both the plates and the fluid are at rest at ...
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### Show that the boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$ [closed]

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by transformation techniques. Any help would be ...
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### Derivative of deformation gradient with respect to Green-Lagrangian strain?

For hyperelastic material, the elastic energy $\Psi$ is related to the deformation gradient $F$ and other internal variables (e.g. temperature $\theta$) In many literatures (including Malvern's and ...
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### In continuum mechanics, why is the stress vector $T=\sigma\cdot n$ not a covector?

In continuum mechanics, the stress vector (see Cauchy stress tensor) $T=\sigma\cdot n$ is the surface density of a force. Forces are covectors, since they map a displacement vector to a scalar energy. ...
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### How to derive this equation? [closed]

How to derive this equation? It comes from a book named Seismic Wave Propagation in Stratified Media.
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### What does the continuum hypothesis of fluid mechanics mean?

I'm a bit confused by the continuum hypothesis stating that fluid are continuous objects rather than made out of discrete objects. Say for $\rho (x,t)$ (density) is there more than one fluid ...
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### Continuity equation in fluid mechanics

The continuity equation in fluid mechanics states that $$\frac{\partial\rho}{\partial t} + \nabla\cdot(ρ\mathbf u)=0$$ Can you explain to me what is the physical meaning of each term of the ...
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### Why are stress forces considered as acting on a cross-sectional area through a solid?

I'm trying to understand the Cauchy-Stress tensor, in which the stress acting on a body at a point is analyzed by considering the cross-sectional area through which a force passes. And my question is -...
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### Derive Equation For a Cantilever in SHM

I am currently investigating how a hacksaw blade's time period of oscillation changes when I add mass to the end of it or when I change the length it is clamped at. I found the following equation ...
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### Can surface waves exist near a fixed surface

(I'll phrase this question in terms of waves in an elastic medium, but this is a more general question.) Surface waves are waves near the surface of a medium whose amplitude decreases as you go away ...
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### Complex potentials in plane polar coordinates - stream function

Determine the stream function and the potential in plane polar coordinates and sketching streamlines. We need to take the value of m=1. I know how to find the stream function and velocity potential, ...
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### Where does this formula for sagging of a beam come from?

In one of my physics textbooks there is a chapter on the elasticity of materials which contains pretty basic outline about Young's modulus, stress-strain, elastic potential energy and related stuff. ...
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### Variational calculus, bending a stick and stationary states

We have a horizontal stick, one of its ends is on the wall, and we can apply a force to the other end. We assume that anything that we can do will leave this in the same plane. Our question is to ...
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### Superior attachment of Möbius strip

In this DIY project, the Möbius strip is used to make a spill-proof coffee cup carrier. The author uses a Möbius strip as the handle of this carrier and says If you attach a Möbius strip to an ...
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### Prove Poisson's Ratio is 0.5 [closed]

Poisson's ratio is the negative ratio of the transverse strain (_T) to the axial strain (_A). For an incompressible (density doesn't change), homogeneous (everything is the same molecule), isotropic (...
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### Degree of anisotropy of crystal tensors

Does there exist a scalar that can describe how anisotropic the elasticity of a crystal is? What about other tensors such as the permittivity or susceptibility? I found a Wikipedia article that was ...
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### How to define the heat current in an isotropic continuum material

I'm doing a FDTD (finite difference time domain) simulation of an isotropic continuum material. And I have several questions. How do you define the energy transferred through an isotropic continuum ...
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### Divergence of Cauchy Stress Tensor

On the wikipedia page for the Cauchy Momementum Equation, it's stated that the equation can be written as $$\rho \frac{D\,\textbf{v}}{D\,t} = \nabla \cdot \sigma + \textbf{f}$$ Where $\sigma$ is ...
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### Confusion in Euler-Bernoulli beam theory

Euler-Bernoulli beam equation is given by $$EI \frac{\mathrm d^2 u}{\mathrm d x^2} = M'(x) \\ EI \frac{\mathrm d u}{\mathrm d x} = xM'(x) + C_1$$ Where, $E$ is modulus, $I$ is second moment of area,...
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### Stress Force - Understanding Cauchy Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation (I'...