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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Degree of anisotropy of crystal tensors

Does there exist a scalar that can describe how anisotropic the elasticity of a crystal is? What about other tensors such as the permittivity or susceptibility? I found a Wikipedia article that was ...
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1answer
193 views

What equation predicts at what point a stretched object comes apart?

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
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1answer
108 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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1answer
349 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 ...
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922 views

How many atoms exist within a continuum body?

Materials, such as solids, liquids and gases, are composed of molecules separated by "empty" space. On a microscopic scale, materials have cracks and discontinuities. However, certain physical ...
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3answers
513 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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3k 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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1answer
173 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
577 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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2answers
288 views

Jaumann deviatoric stress rate

Background about terms in this question: Hookes law and objective stress rates From my understading, the Jaumann rate of deviatoric stress is written as: $$dS/dt = \overset{\bigtriangleup}{{S}} = ...
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3answers
147 views

Why is it said that standing waves do not transfer energy?

The author of my physics textbook writes that standing waves, unlike travelling waves, do not transfer energy. He says that this is because a standing wave is composed of two travelling waves carrying ...
2
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2answers
158 views

pure compression or pure traction?

I know that if we are given a stress tensor that is diagonal, the sign on the diagonal entries tell us whether we have traction or compression. Now, imagine that we are given a non diagonal stress ...
2
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1answer
406 views

Viscosity coefficients

I'm using the 2nd edition of "Transport Phenomena" by Bird and Stewart. I am having trouble with one of the equations: $$\tau_{ij} = \sum_k \sum_l \mu_{ijkl} \frac{\partial v_k}{\partial x_l} $$ ...
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1answer
50 views

physical meaning of major symmetry of the stiffness tensor

What happens if a stiffness tensor does not have the "major symmetry" $C_{ijkl}=C_{klij}$? Background: In linear elasticity (generalising Hooke's law from a spring to a continuous medium), the ...
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1answer
107 views

Derive the Boltzmann factor in classical statistical mechanics

In both quantum and classical statistical mechanics, the probability of an NVT system having an energy $E$ is proportional to $$ p(E)\propto e^{-E/T} $$ However, all of the derivations (that I can ...
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1answer
138 views

What does it mean for shear modulus to be less than bulk modulus?

It is known that Shear Modulus is generally less than Young's modulus for most materials. What does this mean? Does this mean that it is easier to change shape of a fixed body by applying force than ...
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2answers
146 views

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 ...
2
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1answer
751 views

How to determine plastic strain rate

Equivalent plastic strain rate is defined as $$ \dot{\bar{\epsilon}}=\sqrt{\frac{2}{3}\dot{\epsilon_{ij}}^{p}\dot{\epsilon_{ij}}^{p} } $$ Where, $ \dot{\bar{\epsilon}}$ is equivalent plastic strain ...
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2answers
299 views

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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2answers
278 views

Understanding incompressibility (of rubber or viscoelastic material)

Literature gives a lot of explanation why rubber is incompressible. However, I still need some thinking to understand physical behavior of rubber or any such material. Often, incompressibility is ...
2
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1answer
82 views

Superior attachment of Möbius strip

In this DIY project, the Möbius strip is used to make a spill-proof coffee cup carrier. The author uses a Möbius strip as the handle of this carrier and says If you attach a Möbius strip to an ...
2
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1answer
259 views

Physics of Wrinkling: Understanding inextensibility condition

I'm reading this very cool paper on the formation of wrinkles in elastic materials. The key result of the paper is a set of scaling laws for the amplitude and wavelength of wrinkles based on the ...
2
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1answer
138 views

Why does shape of elements matter in finite elements analysis? [closed]

I have used FEA for a couple of years now, but using it and using it correctly are two different things, safety factor is not the solution to everything. I have the feeling I won't be using it right ...
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1answer
131 views

Current density in phase space

$\newcommand{\dd}{{\rm d}}$ I have a question which arises from looking at the impact free Boltzmann equation. Let $(\vec{x},\vec{v})$ be a vector in our phase space $\Gamma^N = \mathbb{R}^{6N}$. The ...
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1answer
581 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
589 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 ...
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2answers
367 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 ...
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1answer
74 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 ...
2
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1answer
89 views

Why did we make equations dimensionless? [closed]

I study a paper on propagation of plane wave, in which equations are made dimensionless. Equation of motion is \begin{equation*} c_{ijmn}u_{m,nj} = \ddot{u_i} \end{equation*} where $c_{ijmn}$ are ...
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1answer
498 views

Hookes law and objective stress rates

Often, in papers presenting updated Lagrangian simulation methods for solid dynamics, the following procedure for updating the (Cauchy) stress tensor is presented: First, the Cauchy stress tensor is ...
2
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1answer
148 views

buckling of tube - shell thickness vs. momentum of inertia optimum

is there any simple formula (perhabs semi emperical, or aproximatively derived model) for buckling of tube under axial compression load given its crossection and wall thickness? ( and naturraly ...
2
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1answer
252 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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0answers
81 views

What is “Accumulated plastic strain rate” in Current yield Norton law?

I'm doing FEA of steel under high strain rates and using Elasto-ViscoPlastic material model, with Von-mises yield criterion along with Isotropic hardening. The strain rate sensitivity is addressed by ...
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80 views

Golf ball impact

A golf ball is said to be "compressed" when hit by a golf club and makes a characteristic "thwack-hiss" sound coming off of the club when impacted by professional golfers (whose impact conditions have ...
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74 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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3answers
211 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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2answers
198 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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2answers
207 views

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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1answer
175 views

Variation in spring constant with respect to the length and no. of coils

Do the spring constant depend upon the length of the spring? No. of coils? Like what happens to the spring constant if you cut it in the half?
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1answer
360 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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1answer
202 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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2answers
59 views

Why an impact exerts so much force? [closed]

If an object of velocity $v$ and mass $m$ moves towards a resting object of mass $M$, then if the object which is hit might break. Why? What is the reason that a collision has more power than a ...
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1answer
343 views

Why liquids and solids are mostly regarded as incompressible?

In many continuum-mechanical Problems it is assumed that liquid and solid substances cannot Change the total value of volume where it holds $\rho = const, \vec{\nabla}\cdot \vec{v} = 0$. In the ...
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2answers
58 views

Can $U_{ij}$ or $v_{ij}$ in continuum mechanics be negative?

In continuum mechanics, we have the deformation gradient $\mathbf F$ to be: $$d\mathbf x = \mathbf F d \mathbf X$$ And then, we do a polar decomposition (A good reference here would be ...
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2answers
52 views

On the isotropy of materials

I am working on honeycomb structures and first of all I would like to understand whether it is isotropic or not, and, if the latter holds, which kind of anisotropy does it have? How to do it? I don't ...
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2answers
278 views

Is there a way to calculate strain energy based on stress and deformation gradient?

We know that we can obtain stress from strain energy density and deformation gradient, for example: $$\mathbf P=\frac{\partial W}{\partial \mathbf F}$$ However, is there a way to calculate $W$ from ...
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1answer
565 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
139 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
11 views

Is deviatoric strain associated with thermal effects?

Does temperature have any effects on deviatoric strain for a linearly elastic isotropic material?
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1answer
47 views

Metric Tensor and Strain Rate Tensor- Comparison of Units

Is there any way the metric tensor can have a dimension in general relativity? I ask because there is an equation where the strain rate tensor of continuum mechanics is expressed as a difference of ...