# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...