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.
5
votes
0answers
64 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 ...
1
vote
0answers
26 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?
...
-1
votes
1answer
79 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 ...
3
votes
3answers
86 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 ...
2
votes
1answer
94 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?
1
vote
0answers
96 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 ...
0
votes
3answers
75 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
49 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 ...
3
votes
1answer
42 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 ...
2
votes
1answer
91 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 ...
4
votes
1answer
102 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 ...
0
votes
1answer
430 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 ...
1
vote
2answers
90 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 ...
0
votes
0answers
151 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
51 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 ...
8
votes
2answers
181 views
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 ...
3
votes
1answer
124 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 ...
5
votes
3answers
419 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 ...
1
vote
1answer
98 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) ...
2
votes
1answer
171 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
votes
1answer
160 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 ...
1
vote
1answer
191 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
$$
...
1
vote
1answer
64 views
Difference between using displacement and current configuration as unknown?
We could use either the current configuration $x$ or the displacement $u$ as unknown while solving for the deformation, for example, of a solid object. I want to know what's the difference between ...
3
votes
1answer
112 views
Decomposition of deformation into bend, stretch and twist?
I'm wondering is there any way to decompose the deformation of an object into different components? For example, into stretching, bending and twisting part respectively? The decomposition could be ...
3
votes
2answers
91 views
A problem of approximation [duplicate]
Possible Duplicate:
Why are continuum fluid mechanics accurate when constituents are discrete objects of finite size?
When we apply differentiation on charge being conducted with respect to ...
1
vote
1answer
365 views
Continuity equation for compressible fluid
A question is given as
Consider a fluid of density $ \rho(x, y, z, t) $ which moves with velocity $v(x, y, z, t) $ without sources or sink. Show that $ \nabla \cdot \vec J + \frac{\partial \rho ...
4
votes
1answer
399 views
A conceptual problem with Euler-Bernoulli beam theory and Euler buckling
Euler-Bernoulli beam theory states that in static conditions the deflection $w(x)$ of a beam relative to its axis $x$ satisfies
$$EI\frac{\partial^4}{\partial x^4}w(x)=q(x)\ \ \ \ (1)$$
where $E$ is ...
0
votes
1answer
215 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 ...
4
votes
0answers
345 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 ...
0
votes
0answers
61 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
vote
0answers
70 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 ...
3
votes
2answers
301 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 ...
0
votes
2answers
736 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: ...
-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?
3
votes
0answers
142 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
260 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 ...
1
vote
1answer
116 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 ...
0
votes
3answers
278 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 ...
2
votes
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 ...
1
vote
2answers
559 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 ...
3
votes
1answer
218 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 ...
3
votes
2answers
622 views
Good books on elasticity
Can someone suggest good books/textbooks/treatises/etc on elasticity? Thanks.
7
votes
1answer
741 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 ...
4
votes
2answers
743 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 ...
7
votes
7answers
923 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 ...
4
votes
8answers
828 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 ...
