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.

learn more… | top users | synonyms

1
vote
1answer
135 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 ...
3
votes
1answer
96 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 ...
10
votes
2answers
756 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
203 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
233 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 ...
4
votes
3answers
922 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 ...
7
votes
1answer
635 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
1answer
167 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
4answers
533 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, ...
2
votes
1answer
134 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
92 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
176 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
209 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
624 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
168 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
318 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
109 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
257 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 ...
4
votes
1answer
203 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 ...
6
votes
3answers
739 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 ...
7
votes
3answers
241 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) ...
3
votes
4answers
171 views

Particles scattering on fluids: breakdown of the effective continuum description

When does the macroscopic continuum description of a medium like a fluid break down? Say I'm interested in a scattering process of some particles with momentum $p$ and energy $E$ off a fluid of ...
2
votes
1answer
374 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
321 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
290 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
106 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
193 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
117 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
1k 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
661 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 ...
1
vote
1answer
375 views
+50

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 ...
5
votes
0answers
519 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
85 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
103 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
646 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
3k 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
56 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
236 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
352 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 ...
2
votes
2answers
207 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
419 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
1answer
66 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 ...
0
votes
2answers
1k views

Calculate the weight a simple plank can support [closed]

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
281 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 ...
4
votes
2answers
1k views

Good books on elasticity

Can someone suggest good books/textbooks/treatises/etc on elasticity?
8
votes
1answer
1k 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
938 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 ...
17
votes
3answers
4k views

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. ...
10
votes
6answers
1k 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 ...
6
votes
3answers
1k 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 ...