Questions tagged [continuum-mechanics]
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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Are there any books/portions of books/particular topics of study that go into great detail to describe the physics of a rope (or any ropelike body)?
I don't know why, but lately, I can't get the physics of a rope of my mind, specifically what happens to a coiled, or otherwise not taught/pulled straight rope when it is pulled taught. I don't really ...
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Pressure in a fluid at rest
I hope this is the right place to post such a question. I'm studying Continuum Mechanics from Gurtin's book, and so far in my class we've seen Cauchy's the about the existence of stress, and nothing ...
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Real and imaginary parts of derivative of complex velocity in fluids
Just by chance, I noticed that for an ideal fluid which has a complex velocity say $w=u-iv$, that its derivative with respect to $z=x+iy,$ when written out in real and imaginary parts, looks like
$$\...
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Taylor Series Expansion of unknown, fraction function
I am learning about deformation, and the deformed state between two points can be defined as
$$E(x) = \frac{(f(x+dx) - f(x))^2 - (dx)^2}{2(dx)^2}$$
My textbook says
When $dx \to 0$ we can use a ...
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Why does a short pulse sent on a rope speed up as it climbs height?
Following from the hanging rope model here, there was a question I was doing in which the rope is jerked from the bottom , and from the relation between velocity, tension and mass density, we get ...
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How is there tension difference in a rope if we assume external force is zero?
Following from the hanging rope model here, I understood that the tension only varies with length if the rope is accelerated by external forces. Now, I was reading the proof of wave equation in this ...
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Can we say normal stress and pressure are the same? [duplicate]
Now if your Ans is yes then I have one more question to ask.
.ie don't u agree that we should not say stress as a tensor of 2nd order rather only tangential stress as tensor of 2nd order and normal ...
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Sheet deformation under gravity
When explaining space-time curvature in physics outreach, one often uses the analogy between space-time curvature and the deformation of a sheet (thus a 2d surface) which is topped with a mass $M$ (...
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Determining local area change from surface displacement field
In continuum mechanics, deformed area elements are related to initial area elements by Nanson's formula:
$$\vec{n}\ da = (\det{\pmb{F}})(\pmb{F}^{-T}\cdot \vec{N})dA$$
Assuming one can experimentally ...
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Ogden Hyperelastic Model - units of pressure
I have come across the following equation in a material science paper for the pressure $p$ and is given below.
I am trying to figure out
1)what the units are for pressure, ie $Pa$ or $MPa$?
2) where ...
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Model to understand tension varying in an hanging rope with mass
From page-90 of Kleppner and Kolenkow,
An idealized model of a string is a single long chain of molecules
bound together by intermolecular forces. Suppose that force F is applied
to molecule 1 at the ...
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Doubt on Cauchy Stress tensor: a partial derivative of metric tensor?
In the reference $[1]$ the author presented a definition of Stress tensor:
$$ \sigma = 2 \rho \frac{\partial e}{\partial g} \tag{1}$$
In a local chart we have:
$$ \sigma_{ab} = 2 \rho \frac{\...
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Sum of Stresses in a Control Volume along $x$-axis
We were looking in class at the sum of forces, acting on surface $x$, of a control volume $dV$.
I see from the equation, the first term is the normal force, and then the rate of change of the force ...
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Volumetric Dilatation Rate, Material derivatives, and Divergence
in class we derived the following relationship:
$$\frac{1}{V}\frac{dV}{dt}= \nabla \cdot \vec{v}$$
This was derived though the analysis of linear deformation for a fluid-volume, where:
$$dV = dV_x +...
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Reference request for piezoelectricity
I'm looking for a mathematically rigorous treatment of piezoelectricity (preferably both the linear theory as well as nonlinear electroelasticity). I've already seen Yang's (2018) An Introduction to ...
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Continuity of Cauchy stresses
The question about the reason of the Cauchy stress tensor has been brought up a couple of times here. I've read through the explanations, and they all seem to rely on the fact that the six independent ...
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Does āā Ļ = Ī¼Īv in the Cauchy Momentum Equation?
I find two versions of the Cauchy momentum equation (1, 2):
$$
\rho \frac{D\vec{v}}{Dt}=\rho\vec{g} - \nabla{p} + \mu\nabla^2\vec{v}
$$
$$
\rho \frac{D\vec{v}}{Dt}=\rho\vec{g} - \nabla{p} + \nabla \...
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How do I account for damping on structural vibrations?
I want to know what terms to add to the differential equation for structural vibrations to account for structural damping. Not external damping, but internal damping. I consider the transfer of energy ...
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The narrowing of a column of water from a faucet $r=r_0\left(1+\frac{2gs}{u^2}\right)^{-\frac{1}{4}}$ [closed]
A tap (faucet) pushes out a stream of water of circular cross section of radius r0
directly downwards at an exit speed u.
Assuming a constant downward acceleration, g, of water, the
radius of the ...
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Geometrical linearization in continuum mechanics
In continuum mechanics, we often make use of "physical and geometrical linearization", e. g. during derivation the Navier-Cauchy equations (c. f. https://en.wikipedia.org/wiki/...
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What is the mathematical justification for introducing the surface traction/stress tensor in the derivation of the momentum balance?
In fluid dynamics and continuum mechanics it is common to derive the the momentum balance by the following argument:
For some open domain $V$, use the Reynolds Transport Theorem to show:
$$\frac{d\...
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How is flow velocity defined in Navier-Stokes equations?
I know Navier-Stokes equations rely on the continuum assumption. In this context, how is the flow velocity mathematically defined? Is it merely a spatial average of the micrscopic particles velicities ...
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Finding the radius of curvature and length of catenary [closed]
A uniform massive string of mass $m$ and length $l$ is fixed between two rigid supports. Angles made by the tangents on the string at points of suspension are shown in the figure. The point C is the ...
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Meaning of the notation $\sigma_{ji,j}$
In page 28 of the book Introduction to Linear Elasticity, 4ed by Phillip L. Gould Ā· Yuan Feng, it says
$$
\int_V{\left( f_i+\sigma _{ji,j} \right) \text{d}V=0}
$$
What does it mean by writing $\sigma ...
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Fundamentals of body deformations (recommend some books? soft robots?)
I'm researching about soft robots but most of the papers I've found have a very complex theory, which is making my progress quite slow. I was wondering if any of you who have some background on ...
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Using Persistence length to establish straightness of position data
In the following picture I am showing 3D position data for 4 different tracks just to illustrate how the tracks looks like.
As we can see that we have all kind of behaviors occurring from helical ...
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2answers
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How derive a spring constant of a helically turned helix based on elasticity of the material and the geometry? [closed]
The structure has helicities on two levels and looks like the tungsten wire in an incandescent light bulb:
How, if at all, can the spring constant be derived from elastic properties of the material, ...
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Rubber's resistance to compression along one axis
I'm concerned with application of rubber sealings. I need to find out what Force is needed to compress this piece of rubber (the rubber cannot move in the z direction, consider the width of rubber ...
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2answers
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Deriving mechanical energy balance law from assumed pairwise potential
I'm currently working through a book on continuum mechanics that derives mechanical balance laws by considering the particles that compose the continuum. One of the balance laws, pertaining to the ...
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Friction in continuum mechanics
How does one include friction forces in the context of continuum mechanics? I imagine that one could rewrite the relation
$$
F\leq N
$$
from the classical mechanics in terms the strain on one surface ...
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Maxwell equations: Lagrangian or Eulerian description
I am just wondering the 4 Maxwell equations (i.e Fadaray Law, Maxwell-Ampere) are Lagrangian or Eulerian description? Does it really matter?
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Why is String Stiffness Proportional to the Fourth Spatial Derivative of String Displacement?
I recently watched a lecture on dispersive medium. I understand that if you disregard the idea that a string is ideal, you must add a stiffness factor into the wave equation. From that you can derive ...
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Beam stiffening when twisted
For a particular cylindrical beam that is bent and twisted, its bending stiffness is found to increase with twist. I have a limited knowledge of continuum mechanics. Can the theory explain this, ...
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Why are shear-stress and momentum-flux the same in the GR?
I am investigeting the meaning of the components of the Stress-Energy tensor:
My source also states, that this matrix is always symmetric in the General Relativity. That looks obvious on the image - ...
asked Nov 13 '20 at 13:17
peterh - Reinstate Monica
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Tower of building blocks
Why does a tower of building blocks (cubes) fall?
Theoretically, as long as the center-of-mass is above the blocks' bottom faces, and as long as one does not shake the tower too much, it should not ...
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Continuum Mechanics: Why would dashpot kinematics require 5 equations to specify deformation for a 2D axisymmetric problem?
I work with PEG hydrogels and use the material to recapitulate cartilage biology and I am really interested in modeling the soft tissue mechanics in COMSOL. I have been studying the work of Caccavo et ...
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Problematic step in proof that Cauchy stress tensor can be expressed as a function of the left Cauchy-Green tensor
I'm studying this proof that Cauchy stress tensor $\sigma$ can be expressed as a function of the left Cauchy-Green tensor, i.e.
$$\sigma({\bf F}) = \sigma({\bf B}) \;\;\;\;\;\;\;\text{where} \;\;\;\;\;...
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Do you use the material derivative in the Eulerian or Lagrangian description of continuum mechanics?
When do you use the material derivative, in the Eulerian point of view when looking at a constant position in space, or in the Lagrangian point of view, following a particle?
Since I saw that another ...
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Why is there a density instead of mass in the Navier-Stokes Equation, if it's analogue to Newton's Second Law?
I read in Ian Stewart's 17 Equations that Changed the World book that Navier-Stokes equation (I know it's not exactly a scientific book, but still, I'd like clarification on what is wrong if it's the ...
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Calculation of elastic constants - when do we need to account for Poisson?
I try to calculate the elastic constant tensor for some organic molecular crystals.
There are plenty of accounts in the literature where people do that, using atomic
resolution models and DFT, EAM, or ...
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Thickness of the boundary layer generated on the flat plate
The shear stress produced in a thin sheet Ļwall, that is, when y = 0, is
given by the integral of the Von-Karman momentum:
$$
\tau=-\frac{d}{d x}\left(\int_{0}^{\delta} \rho\left(u^{2}-U u\right) d y\...
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Shape change of buckled sheet under change of load
I need to solve the following problem: a sheet of flexible but inextensible material (can be modelled as cable or chain in 2D) is fixed in endpoints and buckles up. Then a variable force is applied in ...
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On the continuity of the normal component of the diffusion flux
Is the normal component of diffusion flux is always continuous? I know the continuity at any surface would mean the amount of fluid that is entering through the surface is the same amount that is ...
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Origins of Tension
First of all, I have a confusion about the definition and idea of Tension.
For example, in my Physics Textbook, the idea of tension is written like this:
"Let's say there is a wire with a cross-...
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Proof of RivlināEricksen representation theorem relies on arbitrary tensors
I am having philosophical difficulties with the use of arbitrary orthogonal tensors in the proof of the RivlināEricksen representation theorem on page 6 of the these lecture notes (author unknown; ...
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Tension in a string at an atomic level [closed]
Please explain, at atomic level, What is happening inside a string when a tension force is created?
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Does the rotating space skyhook allow for a periodic motion?
Take a look at the relevant wikipedia-page to get accustomed to the concept of the skyhook. My question is whether it is possible to distribute the mass along the tether in such a way and kickstart ...
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Understanding Newton's second law of motion for 'massive' bodies
I find Newton's second law of motion for point particles quite easy to grasp. However, I run into a lot of confusion when I deal a discrete particle/ continuous body system.
In these notes by Jaan ...
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Equation for stationary string
I have some doubts on the following derivation of the EOM of a stationary string.
Let $F_x, F_y$ be horizontal and vertical tension of the string
$\mu$ be the mass per unit length of the string [kg/m]
...
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What is a “point” in continuum mechanics? [closed]
In continuum mechanics, a continuum is defined as a set of points filling a space (or a part of it). This seems a bit confusing to me since in a mathematical sense, points are zero-dimensional but ...
