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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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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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Differential in Lagrangian description, missing term?

First, let's consider some function $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ as $$\mathbf{f}=\mathbf{f}(x_1,\,...,\,x_n)$$ Clearly, its differential is given by $$d\mathbf{f}=\frac{\partial \mathbf{f}}...
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How to get the height of a constrained, infinite elastic surface?

Suppose I have an infinite elastic sheet, constrained by several poles of height 1. The height of the surface would be 1 at the coordinates of each of the poles, and converge to 0 as the distance from ...
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What is the error in the potential energy calculation of an elongated rod (if any)?

When we calculate the energy stored in the rod after elongation by an external force, we only consider longitudinal expansion. My question is that according to Poisson, there is also a sideways/...
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Can I combine two shear-stresses acting on one plane?

For example, I have an axial force and a torsion force applied on a rock socket installed in bedrock. So, the socket surface will have two shear stress generated, one in axial direction, and the other ...
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Why does my baby's feeding bottle get smashed in fewer pieces when fuller?

After and after again my baby has tossed in the air his glass made feeding bottle and get it smashed on the floor, I realized that the more the milk the bottle has the fewer fragments I had to ...
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8answers
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Why do we bend a book to keep it straight?

I noticed that I have been bending my book all along, when I was reading it with one hand. This also works for plane flexible sheets of any material. Illustration using an A4 sheet Without bending ...
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Why standing wave conditions don't include a phase difference of π

So in order to set up a standing wave on string I can set a travelling wave on it first then let this travelling wave get reflected from a fixed end and this reflected wave interferes with the ...
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Reflection of transverse wave from free end?

I have been using David Morin' drafts on waves along with French's wave book and Fox Smith's book for my undergrad wave course and one thing I don't understand is the physical intuition behind ...
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Prove from first principles that a guitar string will vibrate at a constant frequency

From experience I am aware that a taught string will generally vibrate with a constant frequency. I wanted to prove this by considering the relation of distance from the resting position, and its ...
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1answer
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Doubt in the derivation of the relation between Young's modulus and Modulus of rigidity

Please refer to the image attached. My doubt is marked in red. What i don't understand in this part of the derivation is that how and why is the extension and compression is equivalent to a shear ...
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Relation between electric charge and electric current distributions via velocity

Assume we have a charge $\rho(\vec{r},t)$ and current $\vec{\jmath}(\vec{r},t)$ distributions given in some region of space. From Maxwell's equations we know that the only relation between these two ...
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1answer
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FEM: Distributed loads over adjacent quadratic bar elements

I am an electrical engineering student trying to teach myself Finite Element Methods (FEM) through a couple of textbooks and independent study. While I believe that I understand the basic ideas ...
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1answer
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Unclear assumption in deriving fluid energy conservation laws

I am currently working through Alexandre Chorin's Mathematical Introduction to Fluid Mechanics. In the first chapter, he treats the change in Kinetic energy of a fluid region $W\subset D$ subject to ...
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Question regarding expression for speed of a transverse wave in a string?

How to prove the expression for speed of transverse wave in a string that is :- $$ v= \sqrt{\frac Tμ }$$ Without loss of mathematics rigor, that is without using infinitesimals or differentials in ...
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1answer
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Stress tensor of a rope in equilibrium [closed]

First of all, let me say that I'm not a student of physics but of mathematics, so please be comprehensive. I'm currently studying continuum mechanics. As an example of applications of Cauchy stress ...
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1answer
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Why is the partial derivative of strain energy function with respect to strain equal to stress

In Elasticity, we have a strain energy function , $W$, that is a function of strain tensor, $E$. Then the cauchy stress tensor, $T$ can be determined by: $$T_{ij}=\frac{\partial W}{\partial E_{ij}} \...
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1answer
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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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1answer
57 views

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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2D Linear Elastostatics with Displacement Boundary Conditions

I am new to this type of problem and feel as if I am going in circle with regards to boundary conditions. I am interested in finding an analytic solution for: $\mu\nabla^{2}\underline{u}+(\lambda+\mu)...
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How are fluid membranes stretchable?

The Poisson's ratio of fluid lipid membranes is exactly $\nu=0.5$, because fluids can flow. As answered in this question, under the assumption of a finite bulk modulus $K$, this is the value of $\nu$ ...
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Aggregation phenomena : How to get from a discrete to a continuous point of view

I'm studying a diffusion limited aggregation phenomenon. The $N$ particles diffuse in a box and when there is a contact they stick with a probability $p$, and let's say to simplify $p=1$. Meaning that ...
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1answer
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Divergence of a displacement vector field multiplied by delta function

I'm trying to work out why $$ \boldsymbol{\nabla\cdot[u}\,\delta^3(\mathbf{r})]=0, $$ where $\boldsymbol{u}$ is the displacement field of a source of stress, $\boldsymbol{\nabla\cdot u}\ne 0$, and $\...
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1answer
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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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1answer
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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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1answer
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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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1answer
178 views

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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1answer
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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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1answer
673 views

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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1answer
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Relation between strain and velocity

The strain tensor writes $\epsilon_{ij}=\frac{1}{2}\Big(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_i}{\partial x_j}\Big)$ with $u_i$ the displacement in the $i$ direction. Then $\frac{\...
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1answer
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Integral formula for inertia tensor

Writing down the balance of angular momentum, we introduce the inertia tensor by the formula \begin{equation} J(t)a \cdot b = \int_{S(t)} \rho (t,x)\left( a \times \left( x - X(t) \right)\right)\cdot ...
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1answer
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Continuity equation in the Lagrangian flow picture approach

In deriving continuity equations using Lagrangian. We consider the element of fluid which occupied a rectangular parallelopiped having its centre at the point $(a,b,c)$ and its edges $\delta a$ , $\...
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1answer
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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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1answer
330 views

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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2answers
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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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0answers
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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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1answer
139 views

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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2answers
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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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2answers
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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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2answers
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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 ...
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1answer
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Spring Constant of Two-Sided Fixed Beam [closed]

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{...
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1answer
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eigenvectors of two deformation measures described in the same basis system

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 $\textbf{F1}$ ...