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.

411 questions
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Derivation of dynamic equation for continuous media

I'm trying to understand Newton's law $\frac{\partial{\mathbf{j}}}{\partial{t}}=\mathcal{F}$ for continuous media. Here $\mathcal{F}$ is a force density, and the momentum density $\mathbf{j}$ is given ...
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If atoms are mostly vacuum, why are things so rigid around us? [duplicate]

I can't say confidently that an atom is mostly vacuum, but I am somewhat sure of it because electrons and nucleons cover little space, and everything other than these elementary particles in an atom ...
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How to measure deformation in tissue?

I read some paper and they measured the deformation from 4D data in tissue (live imaging experiment). I was wondering is there any way to measure deformation in 3D data itself that means I have only ...
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Can dispersion relations be derived using Euler-Lagrange equations?

The question is long because of the demonstrations I give, but the problem is simple, so bear with me for a minute. I am trying to derive the dispersion relation of a semi-infinite system using Euler-...
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Relation between the dimensions of a pressurized cylinder and stress on its walls

I don't have much of a background in physics so apologies if i use the terms incorrectly. Given a pressurized cylinder, if the cylinder increases in length the stress on its walls doesn't change but ...
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What are the assumptions for porous media to be considered as continuum?

The poroelastic models like - Biot theory- are built on the continuum approach. But I am not able to understand, how this heterogeneous media can be considered as a continuum. However, If the porous ...
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Differential tetrahedron, area of slanted face?

The above diagram shows a cuboid containing a tetrahedron OABC. When the cuboid represents a differential volume element all sides are small in size. With the length of line $OA$ denoted by $|OA|$ ...
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Are acoustic phonons always the lowest energy vibrational modes in solids?

In solids with unit cells containing more than one atom, the normal modes show acoustic and optical branches. The number of optical branches is proportional to the number of atoms in the unit cell, ...
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What type of stresses occur during bending of steel rod? [closed]

When an external weight (and/or sometime by weight of its own) is tied to a horizontal metal rod, whose ends are supported by two opposite walls, the rod sags, bends due to the weight. My question is ...
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Overdamped Approximation

I am reading the paper Information transfer and behavioural inertia in starling flocks (Attanasi et al, 2014) in the field of active matter for an undergraduate course I am taking. The paper describes ...
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What causes the wave velocity of longitudinal waves in a spring to changes as its length changes?

This video shows the velocity of longitudinal waves changing in a slinky spring as its length changes: https://youtu.be/y7qS6SyyrFU?t=17 Something else responsible for the change in velocity ...
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General question about making differential equations dimensionless

Suppose you have a set of differential equations that you wish to normalize/make dimensionless. From what I've seen, you can usually use dimensional analysis to figure out a good choice of constants ...
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Lorentz Reciprocal Theorem: Why No Pressure

So in my reading on the Lorentz reciprocal identity, I got lost in a step. Why can he remove the pressure term with the continuity equation? Any advice or guidance would be great! Pozrikidis
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Guitar strings struck out of phase

Do guitar strings struck out of phase with one another force each other to begin vibrating in phase with one another? I ask because wouldn’t chords sound more dissonant from time to time if this did ...
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Force analysis of falling spring [duplicate]

As picture below, I want to explain why the bottom of spring will be motionless a little while after release. But in my view, the bottom of spring will be falling in a low speed after release, since ...
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How to calculate the max load a metal bar can hold?

How would one calculate the amount of weight a steel bar could hold before breaking? Apologies for the terrible diagram. So if I had a steel bar of the length 18in, and the cross-section with a ...
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determine the shear stress and bending moment of the next axis

I have problems to raise the equations. They could recommend me a text to be able to solve it.
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Young's modulus and shear modulus relation - temperature effect

Young's modulus and shear modulus are related by $E=2G(1+\nu)$ (for isotropic and homogeneous materials), $E$ is Young's modulus, $G$ is shear modulus and $\nu$ is Poisson's ratio. I can do experiment ...
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Terminology for how bendable an object is and what affects the bendable-ness of an object

I was wondering what the term is for how bendable an object is. Also, does this feature vary depending on the thickness of the object? Say, for example, I want to know how bendable a ruler is. Does ...
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Fixing the Poisson equation to match the deformation of elastic sheet with experimental observation

I am working on the calculation of the deformation of a circular elastic sheet with radius $R=1.2~m$ when a plate with mass $M$ and radius $r_0 = 4~cm$ is sitting in the center of the sheet. I used ...
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Helfrich energy derivation

The Helfrich elastic energy of membranes is given by $F = \int dS (\kappa H^2 + \kappa_G K)$ where $H$ is the mean curvature and $\kappa_G$ is the Gaussian curvature. The derivation in the original ...
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Is $\sin\left[2\alpha\right]\cos\left[2\alpha\right]\ge0$ a valid restriction on the angles of the principal stresses in 2D elasticity?

This question pertains to Elasticity: Tensor, Dyadic, and Engineering Approaches By: Pei Chi Chou, Nicholas J. Pagano, Section 1.4. The objective under discussion is to find the directions of ...
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Boundary condition: displacement

I have a controversial case. I have a rod, which is fixed from one end (constraint). From another end, I apply a compressive force, by pressing the rod down. So in a way I have a constraint, but at ...
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Why stress always flows through the shorter/stiffer path?

Why load has a preference at which way to act? What is the reason behind this preference? Any link or comment would be welcome
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Is continuum mechanics a generalization or an approximation to point particle mechanics?

Newtonian Mechanics is usually presented as a theory of point particles (and forces). My impression of the status of continuum mechanics is that it is mostly taken as an approximate description for ...
Let's suppose I have a two-sided fixed beam: ...and I want to find the equivalent spring constant... can I do the following: I know that the maximum deflection (at the center is): \delta =\frac{...
I have two deformation gradient tensors, corresponding to the simple shear of a cube; one at time t (say $\textbf{F0}$) and another at time $t+\Delta t$ (say $\textbf{F1}$). The basis of $\textbf{F1}$ ...