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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Shape of wall's deformation wave caused by baseball's impact
Clicking through this year's top sports pictures, I stumbled upon this one. I was wondering about the shape the baseball is leaving on the wall.
What phenomenon causes this peculiar shape? Why is ...
8
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2answers
274 views
Symmetry of the stress tensor
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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7answers
933 views
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 ...
7
votes
1answer
761 views
water flow in a sink
When one turns on the tap in the kitchen, a circle is observable in the water flowing in the sink. The circle is the boundary between laminar and turbulent flow of the water (maybe this is the wrong ...
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3answers
437 views
Why are Navier-Stokes equations needed?
Can't we picture air or water molecules individually? Then, why are Navier-Stokes equations needed, after all? Can't we just aggregate individual ones? Or is it computationally difficult, or ...
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8answers
849 views
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 ...
4
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1answer
107 views
Normal modes of a flexible rod clamped at only one point
I am interested in the vibrations of a thin, flexible rod that would only be clamped at one point, properly I'd like to calculate its eigenvalue. But the way I learned it in wave mechanics doesn't ...
4
votes
2answers
755 views
Stress tensor in a cube with shear forces
I want to calculate stress matrix in a cube with two faces parallel to x axis and perpendicular to z axis (sorry I don't know how can I put a picture in this post).
There are two force uniform ...
4
votes
1answer
402 views
A conceptual problem with Euler-Bernoulli beam theory and Euler buckling
Euler-Bernoulli beam theory states that in static conditions the deflection $w(x)$ of a beam relative to its axis $x$ satisfies
$$EI\frac{\partial^4}{\partial x^4}w(x)=q(x)\ \ \ \ (1)$$
where $E$ is ...
4
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0answers
356 views
Interpretation of Stiffness Matrix and Mass Matrix in Finite Element Method
I would like to have a general interpretation of the coefficients of the stiffness matrix that appears in FEM. For instance if we are solving a linear elasticity problem and we modelize the relation ...
3
votes
2answers
638 views
Good books on elasticity
Can someone suggest good books/textbooks/treatises/etc on elasticity? Thanks.
3
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2answers
308 views
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 ...
3
votes
1answer
46 views
References on wave solutions in continuum mechanics [closed]
I am interested in literature on known wave solutions in continnum mechanics, precisely the following mechanical equation:
$$\rho\partial_t^2u_i = C_{ijkl}\nabla_j\nabla_ku_{l}$$
My interest is spread ...
3
votes
2answers
92 views
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 ...
3
votes
2answers
69 views
Extension to continuous in proofs of rigid body mechanics
I'm studying rigid body mechanics and I've seen several proofs of properties related to total angular momentum, kinetic energy, etc. that all regard discrete set of points. For example, to show that ...
3
votes
1answer
265 views
Explain $\rho_{0}\dot{e} - \bf{P}^{T} : \bf{\dot{F}}+\nabla_{0} \cdot \bf{q} -\rho_{0}S = 0$
I am trying to understand the balance of energy -law from continuum mechanics, fourth law here. Could someone break this a bit to help me understand it? From chemistry, I can recall $$dU = \partial Q ...
3
votes
3answers
103 views
Why is the (nonrelativistic) 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 linear ...
3
votes
1answer
119 views
Decomposition of deformation into bend, stretch and twist?
I'm wondering is there any way to decompose the deformation of an object into different components? For example, into stretching, bending and twisting part respectively? The decomposition could be ...
3
votes
1answer
128 views
(Botanical) branch bending under gravity
I'm a PhD student in maths, and attended my last physics class some 15 years ago, so you can imagine my competences in the field.
My supervisor (also not a mechanist) cant tell me how to proceed ...
3
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0answers
153 views
Does a thermally expanding torus experience internal stress?
I'm trying to learn continuum mechanics and thermo-mechanics.
As we know, heating an object increases the mean atomic distance $a_0$ of the atoms in a rigid body. Let's assume it is a linear elastic ...
3
votes
1answer
67 views
Can convection cells evolve in stably stratified fluid?
Assume stably stratified fluid but not in equilibrium, e.g. with non-constant temperature gradient for example. Can convection cells be present? Typical example of convection cells is Rayleigh–Bénard ...
3
votes
1answer
219 views
Can we have non continuous models of reality? Why don't we have them?
This question is about Godel's theorem, continuity of reality and the Luvenheim-Skolem theorem.
I know that all leading physical theories assume reality is continuous. These are my questions:
1) Is ...
2
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1answer
166 views
What is the two dimensional equivalent of a spring?
I'm trying to model isotropic linear elastic deformation in two dimensions. In one dimension, I know that a linear elastic material can be thought of as a spring which obeys Hooke's law $F=-k\Delta ...
2
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1answer
117 views
What is Relativistic Navier-Stokes Equation Through Einstein Notation?
Navier-Stokes equation is non-relativistic, what is relativistic Navier-Stokes equation through Einstein notation?
2
votes
1answer
96 views
Dispersion relation in continuum mechanics
I'm looking at the vibration of a solid having a lattice structure, they obey the following equation:
$$\rho\partial_t^2u_i = C_{ijkl}\nabla_j\nabla_ku_{l}$$
with $u(\vec{x},t)$ the displacement to ...
2
votes
1answer
174 views
A differential equation of Buckling Rod
I tried to solve a differential equation, but unfortunately got stuck at some point.
The problem is to solve the diff. eq. of hard clamped on both ends rod.
And the force compresses the rod at both ...
2
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0answers
52 views
stress work of uniformly deforming continuum
I have a volume which is deforming (using explicit time-integration scheme) uniformly with velocity gradient $L$ and stress tensor $\sigma$. I would like to determine work done by the volume ...
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1answer
51 views
Strain and stress tensor
I have problem by definition of strain and stress.
From Gockenbach's book that our reference for FEM, we have
$$\epsilon=\frac{\nabla u+ \nabla u^T}{2},$$
that $u$ is vector displacement, and ...
1
vote
2answers
95 views
How local is the stress tensor?
I am confused by the definition of the stress tensor in a crystal (let's say a semi-conductor), I don't see how it could be "more local" than over an unit cell. I know that in field theory the stress ...
1
vote
1answer
58 views
Forms of the first law of thermodynamics
The first law of thermodynamics states that
$$\frac{D}{Dt}(K+U)=W+H,$$
where K is the kinetic energy, U is the internal energy, W is the power of the external forces and H is the heat flux. I have ...
1
vote
1answer
99 views
How wide does a wall of ice need to be to stay in place?
Let us say that we have unlimited manpower to construct a huge wall of water ice e.g. 200 m tall (700 feet). -and that the wall is placed in a climate, where the temperature never (for your purpose) ...
1
vote
1answer
68 views
Difference between using displacement and current configuration as unknown?
We could use either the current configuration $x$ or the displacement $u$ as unknown while solving for the deformation, for example, of a solid object. I want to know what's the difference between ...
1
vote
2answers
590 views
Calculate the weight a simple plank can support
I'd like to build a simple desk; just a single plank of wood (or a few side-by-side) with solid supports on each end of the desk. What I'm trying to figure out is how thick a plank I want to use for ...
1
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1answer
203 views
Tensors: relations between physics and linear algebra
In continuum mechanics we use finite deformation tensors to exprime deformations in a point. The 9 components of the tensor (in reality 6 because of its symmetry) are defined as
$$
...
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1answer
396 views
Continuity equation for compressible fluid
A question is given as
Consider a fluid of density $ \rho(x, y, z, t) $ which moves with velocity $v(x, y, z, t) $ without sources or sink. Show that $ \nabla \cdot \vec J + \frac{\partial \rho ...
1
vote
1answer
118 views
In continuum mechanics, what is work potential in the context of total potential energy?
I'm reading a book on the finite element method. Specifically I'm looking at the background material where they are discussing potential energy, equilibrium, and the Rayleigh-Ritz method.
The book ...
1
vote
0answers
36 views
Explain the Föppl–von Kármán equations
I am a newbe to elasticity.
Could someone please explain to me briefly how the Föppl–von Kármán equations work?
What are we trying to solve for?
Is there some kind of intuition to the way they look?
...
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0answers
98 views
2-D Turbulence - how does it look like?
Consider parallel flow in the X direction over a 2D semi infinite flat plate. If turbulence is 2-D, in which axes should we expect the vortices to form.
Also, are there any experimental/visualization ...
1
vote
0answers
74 views
Physics for taffy pulling?
I am creating a simulation and am interested in pulling stretchy things and when they break, like taffy. I imagine this is a bit tougher then a simple equation like gravity, but I have no idea.
Is ...
0
votes
1answer
52 views
Equivalence of turbulence in solid materials
The governing equations for a fluid and a solid are effectively the same and many times analysis can be done for a solid using the Navier-Stokes equations with the equation of state and/or the stress ...
0
votes
3answers
81 views
Hooke's law limitation question
Let's consider a spring. I am a strong man(well, lets assume) and I am pulling the spring. the work I do is being stored in the spring in the form of its elastic potential energy. Then suddenly, ...
0
votes
1answer
445 views
Calculation of a bending moment
I'd like to calculate the bending moment of a cantilever, fixed at its base, and submitted to a certain stress on a specific spot, but I can't find the proper definition of this bending moment (first ...
0
votes
1answer
225 views
Boundary conditions of Navier-Cauchy equation
I'm having difficulties with Neumann boundary conditions in Navier-Cauchy equations (a.k.a. the elastostatic equations). The trouble is that if I rotate a body then Neumann boundary condition should ...
0
votes
2answers
811 views
Calculation of the maximum load to the bar
Looking for a way of calculating the maximum weight (W) to the rod with the given length (L) where the rod did not break and that only bend for (b) mm.
Need only approximative solution (read: ...
0
votes
3answers
287 views
2d soft body physics mathematics [duplicate]
Possible Duplicates:
Modern references for continuum mechanics
Good books on elasticity
The definition of rigid body in Box2d is
A chunk of matter that is so strong
that the distance ...
0
votes
0answers
158 views
How to solve fixed-fixed beam with finite difference method?
What equations to use on this system to form a matrix $A$ with dimensions $[n,n]$ and load vector $q$ with dimension $[n]$ ? I am trying to get vertical displacement $w$.
$$w = A^{-1}\times q$$
...
0
votes
0answers
56 views
Continuum mechanics and effects of stress
Going to word this question a bit more straightforward than I may have before. Also, I'm trying to use baby formulas so I can grasp exactly what's going on.
Object A has an elasticity of ...
0
votes
0answers
63 views
Can a wave propagate in an elastic fluid in the absence of volume forces?
A motion (wave) $\mathbf{x}: \mathcal{B}_0 \times [t_0,t_1] \to \mathcal{E}:$ such that
$q-o = \mathbf{x}(p,t)=(p-o)+\mathbf{a}_0 cos(\mathbf{k}_0\cdot(p-o) - \omega_0 t)$
can propagate in an elastic ...
-1
votes
1answer
89 views
Why is this thought experiment flawed: A vast lever rotating faster than the speed of light [duplicate]
If there were a vast lever floating in free space, a rigid body with length greater than the width of a galaxy, made of a hypothetical material that could endure unlimited internal stress, and this ...
-2
votes
1answer
55 views
what is a difference in the width of the spinning bar?
The bar with length l, density r, diametr d, Young's modulus E, Poisson's ratio mu, is spinning around the cross-section, what is the change in the width of this bar?
