Questions tagged [continuum-mechanics]
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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Rotate a long bar in space and get close to (or even beyond) the speed of light $c$
Imagine a bar
spinning like a helicopter propeller,
At $\omega$ rad/s because the extremes of the bar goes at speed
$$V = \omega * r$$
then we can reach near $c$ (speed of light)
applying some ...
Hernan Eche
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Why is the (non-relativistic) stress tensor linear and symmetric?
From Wikipedia:
"[...] the stress vector $T$ across a surface will always be a linear function of the surface's normal vector $n$, the unit-length vector that is perpendicular to it. [...] The ...
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Navier-Stokes Derivation
Someone knows a physical derivation of the Navier-Stokes equation? Mainly the stress tensor. A lot of authors simply "jumps" the stress tensor and it's the more important of physical motion and ...
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How does hot air rise?
If a balloon is filled with hot air, it is rising due to buoyancy: the mass of the hot air inside the balloon is lower than the mass of the same volume of the cold air outside the balloon cavity.
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How can transverse waves on a string carry longitudinal momentum?
In general, if a wave carries energy density $u$ with velocity $v$, it also carries momentum density $u/v$. I've seen this explicitly shown for electromagnetic waves and (longitudinal) sound waves.
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Classical Field Theory - Continuum limit in forming the Lagrangian density and the elasticity modulus
I have been looking at taking the continuum limit for a linear elastic rod of length $l$ modeled by a series of masses each of mass $m$ connected via massless springs of spring constant $k$. The ...
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Why are continuum fluid mechanics accurate when constituents are discrete objects of finite size?
Suppose we view fluids classically, i.e., as a collection of molecules (with some finite size) interacting via e&m and gravitational forces. Presumably we model fluids as continuous objects that ...
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Derivation of continuum expression of the first law of thermodynamics
Continuum expression of first law of thermodynamics:
$$\frac{D E_t}{D t}=\nabla\cdot({\bf \sigma\cdot v}) - \nabla\cdot{\bf q}$$
(I've seen it in my physics book)
How this equation is derived?
...
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Forms of the first law of thermodynamics
The first law of thermodynamics states that
$$\frac{D}{Dt}(K+U)=W+H,$$
where K is the kinetic energy, U is the internal energy, W is the power of the external forces and H is the heat flux. I have ...
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Symmetry of the $3\times 3$ Cauchy Stress Tensor [duplicate]
When presenting the stress tensor (say in a non-relativistic context), it is shown to be a tensor in the sense that it is a linear vector transformation: it operates on a vector $n$ (the normal to a ...
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Modern references for continuum mechanics
I'm wondering what some standard, modern references might be for continuum mechanics. I imagine most references are probably more used by mechanical engineers than physicists but it's still a ...
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Fluid Mechanics from a variational principle
It is possible to define a good variational principle to describe Fluid Mechanics? If so, what is the correct treatment of the issue. I guess something like:
$$I=\int d^4x \left(\frac{1}{2}\rho v^2-P-...
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Continuity equation in fluid mechanics
The continuity equation in fluid mechanics states that
$$
\frac{\partial\rho}{\partial t} + \nabla\cdot(ρ\mathbf u)=0
$$
Can you explain to me what is the physical meaning of each term of the ...
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Conservation Vs Non-conservation Forms of conservation Equations
I understand mathematically how one can obtain the conservation equations in both the conservative
$${\partial\rho\over\partial t}+\nabla\cdot(\rho \textbf{u})=0$$
$${\partial\rho{\textbf{u}}\over\...
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In wave motion of a string both kinetic energy and potential energy are minimum at $y=y_\text{max}$ then why does the string come down again?
In wave motion of a string both kinetic energy and potential energy are minimum at $y=y_\text{max}$ then why does the string come down again?
As everything in nature tries to attain the lowest energy ...
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Kinetic energy of a massive spring
Suppose we had a spring-mass system where the spring isn't assumed to be massless (has mass $M$) and is of length $L$. One end of the spring is held fixed and the other end I guess is left to freely ...
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How wide does a wall of ice need to be to stay in place?
Let us say that we have unlimited manpower to construct a huge wall of water ice e.g. 200 m tall (700 feet). -and that the wall is placed in a climate, where the temperature never (for your purpose) ...
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Good books on elasticity
Can someone suggest good books/textbooks/treatises/etc on elasticity?
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Rigorously proving equal tension on both ends
There is always one point in introductory mechanics that has been continuously bothering me ever since I had taken my freshman physics course - Why does a massless rope have its tension equal on both ...
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Origins of Tension
First of all, I have a confusion about the definition and idea of Tension.
For example, in my Physics Textbook, the idea of tension is written like this:
"Let's say there is a wire with a cross-...
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Breaking the sound barrier underwater
Theoretically, if I were to launch something faster than the speed of sound in water (around 5 times that of air), what would happen?
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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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Wave speed of a hanging rope
Let us consider a homogeneous rope hanging from the ceiling. I will call the vertical direction $x$ and the horizontal displacement $y$. When we apply the second Newton's Law to a portion of mass $\...
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Why does destructive interference not stop a wave?
In this picture, the two waves keep moving even after undergoing destructive interference.
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Resource(s) for developing a good understanding of surface tension?
I have read through several junior undergraduate level explanations of surface tension. Here is a typical presentation at that level:
Molecules at the surface of a fluid experience approximately ...
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Understanding Newton's second law of motion for 'massive' bodies
I find Newton's second law of motion for point particles quite easy to grasp. However, I run into a lot of confusion when I deal a discrete particle/ continuous body system.
In these notes by Jaan ...
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Why is there an emergent quasi Lorentz symmetry in classical string lagrangian?
For a chain with spacing $a$, the action is
$$S=\int dt\sum_i \left(\frac{M}{2}\dot\phi^2_i-\frac{k}{2}(\phi_{i+1}-\phi_i)^2\right)$$
which only has a translational symmetry. When you take the ...
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Potential energy of an infinitesimal length of elastic rod
I am having an embarrassingly hard time with the derivation for the potential energy of an infinitesimal element of an elastic rod of area $A$. The picture shown below is an element of the rod that ...
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Closure conditions in the form of equation of state
While reading the book "Riemann solvers and numerical methods for fluid dynamics" By E. Toro, the very first paragraph is:
"In this chapter, we present the governing equations for the ...
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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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Liquid flow in a vertical tube
A cylindrical vertical tube has uniform cross section $A_1$, and length $l$. It is open at both ends. Water enters from the top with a constant velocity $v_1$, and allowed to flow out from the bottom. ...
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Why do we bend a book to keep it straight?
I noticed that I have been bending my book all along, when I was reading it with one hand.
This also works for plane flexible sheets of any material.
Illustration using an A4 sheet
Without bending ...
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Why aren't the lengths of the bars on a toy glockenspiel proportional to the wavelengths?
As you might already know, frequency of musical notes is arranged in a such a way that if, for example, an A note has frequency of $x$, another A note which is placed one octave higher would produce ...
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Why can't a piece of paper (of non-zero thickness) be folded more than $N$ times?
Updated:
In order to fold anything in half, it must be $\pi$ times longer than its thickness, and that depending on how something is folded, the amount its length decreases with each fold differs. –...
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Shape of a rotating rope with one free-end [closed]
One end of a uniform rope (with total mass $M$) is fixed on the edge of a cylinder. The cylinder has a radius $R$ and rotates with angular velocity $\omega$. The axis is vertical in a gravitational ...
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Similarity between Schrodinger and Euler-Bernoulli equations - any possible physical meaning?
I noticed a long time ago the similarity between Schrodinger equation and Euler-Bernoulli beam equation. Namely, Euler-Bernoulli equation is equivalent to the system of Schrodinger equation for a free ...
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What is Relativistic Navier-Stokes Equation Through Einstein Notation?
Navier-Stokes equation is non-relativistic, what is relativistic Navier-Stokes equation through Einstein notation?
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What is the equation for a string fixed at both ends without simplifying assumptions?
The book Mathematical Physics by Eugene Butkov has, on Chapter 8, the equation for a held string (by held I mean with endpoints fixed and both at the same height) as being
$$T\frac{\partial^2 u}{\...
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Advection-diffusion with periodic boundary conditions and tilt
Context: Consider the advection-diffusion equation with periodic boundary conditions (PBC) over a flat square domain $L \times L$.
The scalar density $\rho $ is transported by a prescribed field $\...
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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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Model to understand tension varying in an hanging rope with mass
From page-90 of Kleppner and Kolenkow,
An idealized model of a string is a single long chain of molecules
bound together by intermolecular forces. Suppose that force F is applied
to molecule 1 at the ...
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A problem of approximation [duplicate]
Possible Duplicate:
Why are continuum fluid mechanics accurate when constituents are discrete objects of finite size?
When we apply differentiation on charge being conducted with respect to time,...
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Two formulations of Reynolds transport theorem
I am super confused about two different formulations of RTT (Reynolds transport theorem) that yield two different results when used in the same class of problem. The first is found on wiki and ...
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Viscoelastic Kelvin-Voigt model and inelastic strain
In a viscoelastic medium, the total strain can be assumed as the sum of elastic strain and inelastic strain (ref1,2):
\begin{align} \label{eq1}
(1): \mathcal{E}^t_{ij}= \mathcal{E}^e_{ij}+\mathcal{E}^...
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Continuity equation derivation from trajectories of point particles
The continuity equation, where $\rho$ is a conserved density advected by the velocity field $\mathbf{v}$,
$$
\partial_t \rho(\mathbf{x},t) +\nabla \cdot [ \mathbf{v}(\mathbf{x},t) \rho(\mathbf{x},t)]=...
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heat generation due to viscosity in a 3D fluid flow
Consider an arbitrary 3D fluid flow:
$$\vec{\nu}=\vec{V}\left( \vec{x} ,t \right) \tag{1}$$
where velocity at each point $\vec{\nu}$ is a function $\vec{V}$ of position $\vec{x}$ and time $t$ (non-...
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Conservation of heat equation: what represent heat, enthalpy or internal energy?
I'm trying to write the heat transfer equation in an arbitrary fluid (compressible and viscous). Consider an adiabatic system where the only heat generated is due to the internal friction/viscosity. ...
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What are the partial differential equations for Solid Stress Analysis?
When using Finite Element Analysis for Fluids we solve the Navier Stokes Equation and continuity equation, when solving for temperature we solve the heat equation and fouriers law, when dealing with ...
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What is the physical meaning of Navier-Stokes equations?
What is the physical meaning of Navier-stokes equations?
I am trying to understand the physical meaning of Navier-stokes equations. But I did not get any reasonable answer so far.
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Strain rate tensor derivation
The strain (deformation) tensor $\varepsilon_{ij}$ is given by
\begin{align}
\varepsilon_{ij}&=\left[\begin{array}{ccc}\varepsilon_{xx} & \varepsilon_{xy} &\varepsilon_{xz} \\ \varepsilon_{...
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