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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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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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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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Plastic flow rule for large deformation [closed]

Is this equivalent plastic strain formula valid for Finite strain (Large deformation)? I mean can i use the green strain tensor for the same formula?
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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
34 views

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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40 views

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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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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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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55 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
55 views

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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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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SHM time period with spring mass [duplicate]

Prove that time period of SHM for spring block system for small displacement is $T=2\pi\sqrt{m_{eq}/k}$ where $$m_{eq}=m_{block}+m_{spring}/3.$$ I am able to prove for massless spring (by both ...
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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
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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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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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1answer
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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
183 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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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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124 views

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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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
194 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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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 ...
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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 $\textbf{F1}$ ...
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What is wrong with my deformation gradient calculation?

I created two ellipses,$\hspace{150px}$,where the red ellipsis is as the blue one, except translated to the right and rotated by ${30}^{\circ} .$ Using rotation matrix, $$ \left[ \begin{array}{cc} ...
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3answers
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Stress Energy Tensor in language of differential forms

The motivation for this is that quantities like the electric current $J$ in maxwell's equations of motion can be expressed as a differential 3-form, so that the continuity equation can be written just ...
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1answer
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Strain compatibility

From this link, strain compatibility for 2D problem with strains as $$ \varepsilon_{11} = \cfrac{\partial u_1}{\partial x_1} ~;~~ \varepsilon_{12} = \cfrac{1}{2}\left[\cfrac{\partial u_{1}}{\...
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1answer
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Landau and Lifshitz argument for symmetry of stress tensor

In Landau and Lifshitz's book on the theory of elasticity (vol 7, theoretical physics series), specifically section 2 of the first chapter, the authors present an argument for justifying the symmetry ...
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1answer
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Continuum Physics

How distinct is fluid dynamics from continuum physics? I've heard it is a subset and by the definition of the subject's name, it seems likely to be the case. Can anyone please clarify?
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Are there any proved consistent quantum continuous cellular automata/game of life model?

There are quantum cellular automata (continuous game of life, which is a type of cellular automata): https://hal.archives-ouvertes.fr/hal-00542373/document There are continuous cellular automata (...
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How to map a (virtual) truss deformation to a surrounding object

I am trying to craft a way of simulating deformable objects without using finite element method (despite it is the goldstar method for such purpose). The goal is to rapidly get an estimate of how a ...
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3answers
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Why are diagonal elements of stress tensor equal to pressure for fluids?

In fluid mechanics it is assumed, that the normal components of the stress tensor are all the same and identical to the pressure p: $\sigma_{xx}= \sigma_{yy}=\sigma_{zz} = p$ Where does this come ...
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Statistical mechanics arguments for the Newtonian stress-strain model in continuum mechanics

In traditional continuum mechanics, it can be shown using frame-invariance and the Cayley-Hamilton theorem that an isotropic, inelastic* fluid embedded in Galilean space-time must possess the ...
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Making sense of the stress tensor for elastic deformations

I've seen this kind of formula a number of times, in the context of elastic deformations. $$ -\nabla \sigma = f $$ whete $\sigma$ is "the stress tensor" and $f$ is force. I never understood it even ...
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Simple Harmonic Motion - spring with mass [duplicate]

I'm reading about the following set-up: We have a body of mass $m$ attached to a spring. The restoring force is proportional to the displacement of the body from equilibrium. Using Newton's 2nd law ...