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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Constitutive relations and strain energy in finite strain viscoelastic solid mechanics
I'm an applied math graduate student, and my research is straying into hyperviscoelastic models of materials. I've had trouble finding an answer to this question I have about the mathematical theory ...
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Do real numbers exist in reality? [duplicate]
In mathematics, one way we derive the real numbers using (Dedekind) cuts of rational numbers so that we can get an uncountably infinite set of numbers such as $\sqrt 2$. This represents the "...
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Which side of the surface does surface tension act on?
I just realized there's something extremely basic about surface tension that I don't understand. Surface tension is a property of the interface between two different materials, such as water and air, ...
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Speed of longitudinal waves depend upon the quality of the medium. Isn’t that un-intuitive?
The formula for speed of longitudinal wave is $v = \sqrt {\frac E\rho}$ where $\rho$ is density. So basically the speed of longitudinal wave in a given medium is fixed, isn’t that unintuitive . Like ...
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Rate of deformation and spin tensors
I am studying a set of notes by Ellen Kuhl of Stanford university on continuum mechanics, where I encountered the rate of deformation and spin tensors, as discussed in this set of notes. This set of ...
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Are there fractal solutions in continuum mechanics?
Continuous media belongs to the continuum and could have an elaborate structure. So I guess there is a fractal solution. But at present, I do not know a specific example, so I would like to ask my ...
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Describing the deformation of a medium as a diffeomorphism
In this paper online, the author models the deformation of a medium as a diffeomorphism $ \mathbb{R}^3 \rightarrow \mathbb{R}^3$ as given by:
$$ y^i \mapsto x^i(y)=y^i + u^i(x) $$
as given by equation ...
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Euler's Equation applied to Perfect Fluid
Consider this form of Euler's Equation:
$$\rho \vec{a}=\nabla \cdot T+\rho \vec{f}$$
Where: $\rho$ is the density, $\vec{a}$ is the acceleration, $T$ is Cauchy's Stress Tensor and $\vec{f}$ is the ...
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A Problem Regarding Cauchy's Fundamental Lemma
In Continuum Mechanic Cauchy's Fundamental Lemma states that:
The stress vectors acting on opposite sides of the same surface are equal in magnitude and opposite in direction. Cauchy's fundamental ...
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If an object is pulled at one end, how fast would an impulse travel to the other end in a realistic scenario?
In SR, there are a couple of paradoxes such as the bug-rivet paradox that require us to take into account a finite speed of impulse.
Suppose I have a rod at rest, and I yank one end to some speed. I ...
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Advection-diffusion with periodic boundary conditions
Context: Consider the advection-diffusion equation with periodic boundary conditions (PBC) over a flat square domain $L \times L$.
The scalar density $\rho $ is transported by a prescribed field $\...
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Does adding waves to a plane affect how energy in a twist is dissipated?
I am learning about a model of the human spine I saw and was doing a thought experiment with a ruler to try to understand the physics.
Suppose I have a plastic ruler that's not rigid but also not ...
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How does shear-stress act on a spring hanging vertically with a mass attached to its end?
I have come to known that both longitudinal and shear strain act on a spring when it is hanging.
I can understand how longitudinal strain acts but I do not understand why shear stress/strain can act ...
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Proof of the Piola Transform
As I understand it, the relationship between the second order tensor $\bf T$ over a reference configuration and the same tensor in a deformed configuration $\bf T^\prime$ is given defined as follows:
$...
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1answer
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Self-deformation due to own weight [closed]
The goal is to calculate the total deformation due to self-weight.
I understand the calculation that is shown in the picture. I forgot to include that $x$ is the length of the object's portion under ...
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Is this expression for bulk modulus of a fluid incorrect?
Equation (6) of Lagrangian and Eulerian Representations of Fluid Flow:Kinematics and the Equations of Motion [PDF], by James F. Price is this:
A convenient measure of the stiffness or inverse ...
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Continuity equation derivation from trajectories of point particles
The continuity equation, where $\rho$ is a conserved density advected by the velocity field $\mathbf{v}$,
$$
\partial_t \rho(\mathbf{x},t) +\nabla \cdot [ \mathbf{v}(\mathbf{x},t) \rho(\mathbf{x},t)]=...
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Is possible to derive electromagnetic terms in energy balance from a molecular prespective?
In 'Physical foundations of continuum mechanics', A. Ian Murdoch shows that,
starting from the Newton's law of a single molecule, modeled as a material point inserted in a system of material points, ...
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How long would it take for something really far away, say $300,000,000 \ \text{meters}$, to feel the force from a pull of a rope attached to it?
My friend phrased the original question like this:
If I tied a rope to myself and wrapped it all the way around the earth and then pulled (assuming the rope was already taut), which direction would I ...
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Why is the continuity equation hardly used in solid mechanics when it is essential in fluid mechanics?
For any continuum, fluid or solid, we can express mass conservation through the continuity equation
$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 ,$$
where $\rho$ is density ...
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Resources for continuum mechanic [duplicate]
I'm looking for textbooks/resources that deal with the general aspects behind continuum mechanics. I'm currently studying fluid mechanics and in the future will move on to study the theory of ...
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Recasting integrals from Lagrangian to Eulerian frame
Working on a research problem in the continuum mechanics of fluids. For clarity, uppercase will be used for tensors in the reference configuration, and corresponding spatial items will be in lowercase....
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Physical limits or empirical data on ratios of bulk modulus, shear modulus and density
The elastic dynamics of an isotropic continuum (solid or fluid) can be described in terms of the bulk modulus ($\kappa$), shear modulus ($\mu$, zero in a fluid) and density. (The two elastic moduli ...
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Equation of Motion for a Structure Vibrating near the Speed of Light
For a standard $N$ DOF system we can find the eigenfrequencies and eigenmodes of the system by setting up an eigenvalue problem in the form of
$$
([M]-\omega^{2}[K])\phi=0.
$$
For a continuous ...
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Design an Experiment based on Castigliano’s Theorem to Determine Young’s modulus of the Unknown Member
The question states:
"You need to design an experiment based on Castigliano’s theorem to determine Young’s modulus of the vertical member (unknown material) without destroying the structure."
Is ...
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Hyperelasticity: relating two PK2 stress tensors in terms of two nonlinear displacements
I am working on numerical analysis for a nonlinear hyperelasticity problem. Given that the second Piola Kirchhoff stress tensor $S$ depends on the Green strain tensor $E$, which in turn depends on the ...
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1answer
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How are body deformations modeled in Lagrangian mechanics?
With rigid-body systems, we choose a finite number of generalized coordinates to model a system, i.e. a pendulum. However, I've read that deformable bodies like elastomers have "infinite" degrees of ...
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Independent Elements of Elastic Stiffness and Compliance Tensor for ALL Space Groups
In short: Does anybody know if there exists a compendium, a document, a book or a stone tablet listing the independent elements of the elastic stiffness and compliance tensors ( that is, naming the ...
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3answers
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Why do two 'transverse waves' travelling in same medium have the same wave speed?
Why is it that two transverse waves travelling in same medium have same wave speed? What does the medium has to do with it? I have searched through the web, but I couldn't understand. I have found ...
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1answer
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Interpretation of vector identity in the context of fluid dynamics
I know that
$(v \cdot\nabla)v = \nabla(\frac{1}2v^2)+\omega \times v $
is just a general vector identity with $v$ substituted for some general vector field and $\omega$ for $\boldsymbol \nabla \times ...
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3answers
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What makes a transverse wave maintain its shape?
Consider a simple transverse wave propagating along a rope. We understand it's propagation stating that each point is pulled by its neighbors, making it act along with them. This will create a similar ...
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1answer
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Why is tension needed to create a wave in a string? [closed]
Suppose, a long straight string is present in vacuum. I oscillate one end of the string with a certain frequency. Shouldn't a wave be formed? If it is formed, what will be the velocity of the wave?
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Why does a fluid flows?
I was reading about viscosity and momentum diffusion in Physical Chemistry and I came up with the title. Suppose we have some amount of liquid inside a container with a piston (initially fixed) in the ...
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Problems regarding Searle's Experiment
Recently I came across a problem on Searle's Experiment where we had to find the maximum possible error in Young's modulus calculation due to an error in the calculation of extension of string. ...
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Euler beam equation sign confusion
Euler beam equation is defined as follows:
$$M=-EI{\frac {d^{2}w}{dx^{2}}}$$
There is a negative sign, so if the second derivative of the deflection is negative, the moment is positive. When the ...
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Lagrangian of a piezoelectric material
I am confused about the Lagrangian of a piezoelectric material. I have seen multiple sources that seem to contradict each other.
The way I personally approach the problem is the following:
$$L = K - ...
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What is single mode approximation in fluid solid interactions?
Before redirecting me to the other questions, I am asking this with respect to fluid solid interactions. In order to obtain an understanding between the fluid - solid interactions, governing equations ...
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Measure-theoretic force
I understand classical mechanics as a science of moving masses. So I decided to work out it formulation based on measure there just for fun. In this framework the classical mechanical system would be ...
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Plastic section modulus and yield strength relationship
Wikipedia and many material science books claim that:
The plastic section modulus is used to calculate the plastic moment, $M_p$, or full capacity of a cross-section. The two terms are related by ...
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Entropy as a state function for irreversible paths
Searching Physics Stackexchange for entropy I have found several posts regarding entropy, lately most of the questions why entropy is a state variable.
This got me thinking. I have understood so far ...
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1answer
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Doubt on calculation of a “general expression” on the energy of a travelling wave in a column filled with gas
I) SOME FACTS
When we study mechanical waves, we essentially have to deal with $dm$ mass increments; then we can "expand" they as:
$$ dm = \rho(x,t)dV.\tag{1}$$
Also, I think that it's pretty ...
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1answer
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Is there any deference between the directional derivatives of isotropic and anisotropic tensors?
I need to calculate an elasticity tensor which can be expressed by the directional derivate of a tensor with respect the other. Is the anisotropy of the tensor affects this derivative and how?
Many ...
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Continuum mechanics - deformation gradient confusion
I have seen several different approaches to describing continuum mechanics that are all very similar, yet some differences (that I see, not sure if they are true) keep confusing me.
The picture that ...
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1answer
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Continuum assumption: validity & motivation
I still can't grasp why the so-called continuum assumption can be taken as reasonable under the proper conditions.
Let's consider the space-time microscopic distribution of a generic tensor field. If ...
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1answer
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Why stresses on a flywheel are similar to a pressure vessel?
A spinning wheel or an engine car flywheel has the same maths, regarding the stresses developed when spinning, with a pressure vessel
Maybe someone knows the underlying mechanism for this similarity?
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1answer
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Wave Equation for Flexural Waves in Thin Rods from Navier's Equations of Elasticity?
The equation for flexural waves in thin rods is $$\frac{\partial^4 v}{\partial x^4} + \frac{\rho A}{EI}\frac{\partial^2 v}{\partial t^2} = 0$$
where $I$ is the $2nd$ moment of area.
Now, as Navier's ...
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How to know theoretically if a certain liquid will flow through a vertical tube of certain diameter with or without applying force to the liquid? [duplicate]
Does a certain liquid having any value of viscosity, surface tension, temperature and other fluid properties flow through a vertical tube of any given diameter from the top end to the bottom end of ...
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1answer
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In mean-field theory, why are the collisions of particles in the mean-field neglected?
Mean field theory is a tractable framework for analyzing parameters of a continuum or an infinite number of identical micro-particles or agents. The former has been treated extensively in statistical ...
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What is the speed of Pushing/Pulling? [duplicate]
If there was a huge rod the length of a light-year in space and someone pulled one end of it, when would the other end move?
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The force on loose end of a membrane [closed]
If I have a membrane that has one loose end, does it mean that the tension on that end is equal to zero and because of that the the force on that end is also equal to zero?
