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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Classical point particles to classical fields

I often hear that in the continuum limit we can study large numbers of particles as fields. I always imagined that by removing all bounds on the number of particles (while keeping total energy, ...
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165 views

Derivation of normal shear stress

I am self-studying this note and I am stuck in the derivation of the normal shear stress. Specifically I can't see how the relations (23) and (24) come about. Specifically, what I don't understand is ...
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1k views

Physical description of momentum flux tensor

In the field of fluid mechanics, what is the momentum flux tensor? Is there an easy explanation for how it "works"?
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436 views

Conservation of energy and continuity equation

When physicists say energy is conserved, do they mean that energy satisfies the continuity equation: $$\triangledown \cdot j+\dot{\rho}=0$$ On the internet there is plenty of talk of how the ...
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64 views

Pressure derivative of bulk modulus

Hi all what is the definition of pressure derivative of bulk modulus if it is a pressure derivative of bulk modulus at zero pressure. if the pressure is zero how it is derivative by pressure?
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1answer
1k views

Problem with Velocity of efflux [closed]

I am stuck in this problem- I need to find the velocity of efflux at the hole of the container. [We can assume that the area of the hole is negligible in comparison with the base area of the ...
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1answer
99 views

Metric of following spacetime and refractive index

Let's have metrics $$ ds^{2} = f(\mathbf r)dt^{2} - h(\mathbf r )\delta_{ij}dx^{i}dx^{j}. $$ Hot to show that motion of light in spacetime with this metrics is equal to motion in continuous media with ...
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1answer
85 views

Show that detF(X,t) is positive in continuum mechanics?

I want to show that the determinant of the field $detF$ at the point $X \in B$ is positive, when the following motion. I think that time derivative of Jacobin is positive for $t > 0$. However, I ...
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59 views

If I roll an elastic plate into a cylinder, does it shrink?

Suppose I start with a rectangular elastic (to keep things simple, zero Poisson's ratio) sheet of length $2\pi R$, thickness $h$, and (immaterial) width $W$. I roll it up into a cylinder of radius ...
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1answer
93 views

Why is general relativity only formulated in continuum terms?

So, when we are discussing Newtonian mechanics, we treat particles as point particles. In continuum mechanics, which I understand to be a version in which mass is continuously distributed, we have ...
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3answers
194 views

Configuration space of particles in the box

The notion of entropy says that we can count microstates that correspond to macrostate. But, I do not understand how this can be done. Does it imply that the state space is discrete (finite or ...
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1answer
77 views

Free energy variations

In a paper, I found this: $\mathbf{h}=\mathbf{h}(\mathbf{r})$ is called molecular field and is defined as the variation field of the Frank free energy functional $F_{d}$ with respect to the ...
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2answers
637 views

Forces acting on a body in equilibrium

The resultant force acting on a body in equilibrium is 0: $$\iiint_R \rho {\bf b}\ dV + \iint_S {\bf t}^{(n)} ds = 0,$$ in which $R$ is a region inside the body, $\rho {\bf b}$ the body force per ...
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2answers
370 views

What is the motivation for Mohr's circle?

I am very puzzled by the motivation for Mohr's circle in Wikipedia here. Please, explain why we need something called "Mohr's circle". Use as little words as possible and be precise. Helper questions ...
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1answer
105 views

What is the $n$ in the formula in Solid Mechanics? [closed]

The formula is about the critical force for the elastic beam that is supported by its joints: $$ P_{cr} ~=~ n^2 \pi^2 \frac{EI}{ L^2} $$ It should be based on the book Parnes - Solid ...
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1answer
179 views

Can Smoothed-Particle Hydrodynamics (SPH) be used to simulate porous media flow and deformation?

I am trying to use Smoothed-Particle Hydrodynamics (SPH) to study fluid flow in and around porous media. The aim is to observe how it causes erosion and failure. For this, from my understanding, there ...
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1answer
216 views

Equations of motion of displacement field

We have an action: $$S[\boldsymbol{u}] = \frac{1}{2} \int dt \int d^3x \left\{ \mu (\frac{\partial u_{i}}{\partial t})^{2} - \nu (u_{ii})^{2} - \rho(u_{ij})^{2}\right\} $$ Where $u_{ij} = ...
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1answer
8k views

Neutral axis of T shaped beam? [closed]

I am not a mechanics or physics student but a computer science student. I came across a question related to find neutral axis of figure but I do not have slightest idea of what it is and how to find ...
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2answers
5k views

Conservation Vs Non-conservation Forms of conservation Equations

I understand mathematically how one can obtain the conservation equations in both the conservative $${\partial\rho\over\partial t}+\nabla\cdot(\rho \textbf{u})=0$$ ...
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1answer
198 views

How practical is fracture mechanics?

I have been reading fracture mechanics recently and have encountered many beautifully elegant theories. However, one thing keeps bothering me: How practical is fracture mechanics in the real world? ...
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1answer
181 views

Einstein Notation for Strain Energy Function

I encounter the following formula (for strain energy function) a lot in physics literature: $$ W(\epsilon_{kl}) = \int_0^{\epsilon_{kl}} \sigma_{ij} \textrm{d}\epsilon_{ij} $$ where all indices ...
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1answer
250 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 ...
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247 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 ...
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1answer
146 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 ...
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1answer
98 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 ...
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Symmetry of the $3\times 3$ Cauchy 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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0answers
241 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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1answer
329 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 ...
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3answers
1k 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 ...
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1answer
769 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?
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1answer
174 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 ...
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4answers
898 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, ...
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1answer
140 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 ...
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1answer
101 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 ...
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1answer
199 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 ...
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1answer
230 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 ...
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1answer
676 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 ...
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180 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 ...
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369 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 $w$. $$w = A^{-1}\times q$$ ...
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123 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 ...
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2answers
263 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 ...
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1answer
218 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 ...
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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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3answers
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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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4answers
185 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 ...
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1answer
432 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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1answer
367 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 ...
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1answer
316 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
111 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 ...
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1answer
221 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 ...